Generalized Musical Intervals and Transformations
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Generalized Musical Intervals ...

Author:
David Lewin

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Generalized Musical Intervals and Transformations

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Generalized Musical Intervals and Transformations

David Lewin

OXFORD UNIVERSITY PRESS

2007

OXFORD UNIVERSITY PRESS

Oxford University Press, Inc., publishes works that further Oxford University's objective of excellence in research, scholarship, and education. Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam

Copyright © 2007 by Oxford University Press, Inc. Originally published 1987 by Yale University Press Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Lewin, David, 1933-2003. Generalized musical intervals and transformations / David Lewin. p. cm. Originally published: New Haven: Yale University Press, c1987. Includes bibliographical references and index. ISBN 978-0-19-531713-8 1. Music intervals and scales. 2. Music theory. 3. Title. ML3809.L39 2007 781.2'37—dc22 2006051121 1 3 5 7 9 8 6 4 2 Printed in the United States of America on acid-free paper

For June and Alex

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Contents

Foreword by Edward Gollin Preface

xiii

Acknowledgments Introduction 1. 2.

ix

xxvii

xxix

Mathematical Preliminaries

1

Generalized Interval Systems (1): Preliminary Examples and Definition

16

3.

Generalized Interval Systems (2): Formal Features

31

4.

Generalized Interval Systems (3): A Non-Commutative GIS; Some Timbral GIS models

60

5. Generalized Set Theory (1): Interval Functions; Canonical Groups and Canonical Equivalence; Embedding Functions

88

6. 7. Transformation Graphs and Networks (1):

Generalized Set Theory (2): The Injection Function

Intervals and Transpositions

123

157

8. Transformation Graphs and Networks (2): Non-Intervallic Transformations

175

193 9. Transformation Graphs and Networks (3): Formalities 10. Transformation Graphs and Networks (4): Some Further Analyses 11. Appendix A: Melodic and Harmonic GIS Structures; Some Notes on the History of Tonal Theory

245

12. Appendix B: Non-Commutative Octatonic GIS Structures; More on Simply Transitive Groups Index

255

251

220

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Foreword to the Oxford Edition Edward Gollin

It has been nearly twenty years since the initial publication of David Lewin's Generalized Musical Intervals and Transformations (GMIT), and the work has aged well. This is due in part to the foundational nature of the book's subject matter. The work, a methodical examination of the concept of a musical interval, explores how the familiar notion of interval as "a distance extended between pitches in a Cartesian space" is merely one specific case of a more general idea, one that can embrace different kinds of musical objects (durations, meters, Klangs, timbres, and so on), different (i.e. non-Euclidean) geometries, and different orientational perspectives (interval as action or gesture rather than as simply measurement of distance between things). Along the way, the work recasts set theory, the concepts of transposition and inversion, and notions of musical time in this generalized image. But the work has maintained its relevance and importance as well because of the brilliance and musicality of its author. David had a gift for finding musically significant examples for his sometimes abstract concepts, and a gifted musical imagination that delighted in finding new ways to hear and understand familiar musical passages. While GMIT does not offer the extended musical analyses of his later books, Musical Form and TransformationorStudies in Music with Text, the work is nonetheless rich with smaller analytical gems. To be sure, transformational theory has evolved in the years sinceGMITfirst appeared—the analytical use of Klumpenhouwer networks, the development of neo-Riemannian theory, and the resurgence of spatial methodologies and metaphors in analysis all postdate David's seminal study. But each of these subsequent developments can find its basis in the framework David sets forth in GMIT: Klumpenhouwer networks apply the Generalized Interval System (GIS) concept recursively to create networks of networks; neo-Riemannian theory, which emerged from explorations begun in chapter 8 of GMIT, takes families of contextual transforma-

ix

Foreword to the Oxford Edition

tions to be the formal intervals between the familiar set of harmonic triads or seventh chords; spatial methodologies simply extend the idea of transformational networks to create graphs that embrace all members of a family of objects (pitches, pitch sets, rhythmic durations, and so on) related by certain contextually significant intervals.1 One notable new feature of this edition is an author's addendum (the preface), drawn from a previously unpublished typescript titled "Updating GMIT," which presents, in a sometimes synoptic form, concepts or musical examples David had planned for a future edition of GMIT. The document was likely written in the summer of 1987 and was used as the handout for a talk given at the Eastman School of Music in the fall of that same year. It should not be surprising to those who knew David's incredible industry and the speed with which he could read and suggest revisions to others' work that David would have been drafting plans for a new edition of GMIT so soon after its publication—for David, it was often difficult to stop thinking about a project, or tinkering with its ideas, once begun, and the document clearly represents David's residual energy following the writing of GMIT. The examples explored in the addendum are diverse, although certain themes recur. For one, David seems to have been particularly concerned with examples that involve non-commutative groups of operations, no doubt because such groups often defy our accustomed and familiar intuitions about the way intervals work. For another, David seems to have been interested in finding examples that do not simply involve individual pitch classes (transformations of melodies, of Lagen in triple counterpoint, of ordered hexachords), again because these are less familiar, and often reveal less intuitive aspects of interval. Although the document is perfectly intelligible, some sections of "Updating GMIT" deserve additional comment. 1. The error in figure 8.2 (g minor instead of g# minor) that prompted David's commentary in section I has been corrected in this edition. The first section of David's notes was expanded to become his article "Some Notes on Analyzing Wagner: The Ring and Parsifal" (19th-century Music 16.1, 1992, reprinted in David Lewin, Studies in Music with Text [Oxford University Press, 2006]). 2. David developed and expanded section IV into a pair of unpublished exercises for his math and music course at Harvard University. Exercise 5 (2 pages) directs the student to discover the elements of the Q-X group acting on the augmented triads of sc (014589) and then find transformations of the "rapture of the

x

1. David has written articles on each of these topics subsequent to the publication of GMIT. Klumpenhouwer networks are the topic of two articles: "Klumpenhouwer Networks and Some Isographies that Involve Them," Music Theory Spectrum 12.1 (1990): 83-120, and "A Tutorial on Klumpenhouwer Networks, Using the Chorale in Schoenberg's op. 11, no. 2," Journal of Music Theory 38.1 (1994): 79-101. David's most significant post-GMIT contribution to neo-Riemannian theory is the article "Cohn Functions," Journal of Music Theory 40.2 (1996): 181-216. Two of David's contributions to graphical methods of analysis are "The D-major Fugue Subject from WTCII: Spatial Saturation?" Music Theory Online 4.4 (1998), and "Notes on the Opening of the F# Minor Fugue from WTC I," Journal of Music Theory 42.2 (1998): 235-239.

Foreword to the Oxford Edition

strife" figure under Q4, Q8, and X5 in Schoenberg's Ode to Napoleon (as in David's example 2 from the addendum). An optional part of that exercise encourages students to explore transformations of characteristic tetrachords in Schoenberg's Ode using the members of the same Q-X group. Exercise 8 (3 pages) explores the simple transitivity of the Q-X group and has the student find the (interval-preserving) elements of the commuting group, {T0, T4, T8, I,, I5, I9). David's article "Generalized Interval Systems for Babbitt's Lists, and for Schoenberg's String Trio" (Music Theory Spectrum 17.1 [1995]: 81-118), in particular "Part 5: Background on Non-Commutative GISs," explores the relationship between non-commutative GISs and their commuting groups. 3. David similarly developed and extended section V into an exercise for his math and music course (exercise 9,4 pages). The Daniel Harrison article to which David refers was published as "Some Group Properties of Triple Counterpoint and Their Influence on Compositions by J. S. Bach" (Journal of Music Theory 32.1 [1988]: 23-49). David inserted a manuscript page into the "Updating GMIT" typescript that presents a TPERM and VPERM analysis of Bach's c-minor fugue from the Well-Tempered Clavier, Book I. The manuscript notes that the diagram is modeled after Schenker's "Table of Voices" from "Das Organische der Fuge" in Das Meisterwerk in derMusik, Band II, p. 59, and further observes that the Lagen symbol "'A' can mean 'Subject,' 'B' can mean 'Countersubject' and 'C' can mean 'any third part of roughly characteristic rhythm'" (emphasis Lewin's), suggesting that the methodology is not bound to works in strict triple counterpoint. David's diagram, however, has not been incorporated into the author's addendum of this volume because David wrote no accompanying text for it—creating new text would have adversely disrupted David's prose in the rest of the section. David, however, did use the c-minor fugue analysis as part of exercise 9 in his math and music course, which I present below for interested readers to explore if they wish (terminology has been adapted to conform to the text of "Updating GMIT"): PART I OF EXERCISE 9: (a) Complete the partially-filled diagram below, which pertains to the c-minor fugue in Book I: Meas.

Stufe

1

i

11 15 20 26.5

m V

i i

Lage

TPERM interval

VPERM interval

(b) Discuss features of the construction which you find revealed by the double intervallic analysis. For instance, does the use of 3-cycles bring out any aspect of the structure? Do the TPERM and VPERM

xi

Foreword to the Oxford Edition

analyses coincide as they did [in the A-major Prelude]? What aspects of the piece are bound together by repetition of TPERM intervals? By repetition of VPERM intervals? 4. Section VI considers the GIS structure of a family of 12-tone-row transformations that David first explored in his article "On Certain Techniques of ReOrdering in Serial Music" (Journal of Music Theory 10.2 [1966]: 276-287). David refers in the section to "an excellent work, as yet unpublished" by Andrew Mead. That work was published in two parts as "Some Implications of the PitchClass/Order-Number Isomorphism Inherent in the Twelve-Tone System: Part One" (Perspectives of New Music 26.2 [1988]: 96-163) and, more pertinent to Lewin's addendum, "Some Implications of the Pitch-Class/Order-Number Isomorphism Inherent in the Twelve-Tone System Part Two: The Mallalieu Complex: Its Extensions and Related Rows" (Perspectives of New Music 27.1[1989]: 180-233). David, of course, never created a second edition ofGMIT, an undertaking that, he wrote, would have involved "[fixing] a lot of errata & corrigenda; some major rewrites here and there; a reasonable amount of bibliographic updating."2 This edition ofGMIT, while retaining the text of the original, does incorporate the corrections indicated by David's errata list. Moreover, while it does not attempt to identify or alter passages that David felt needed rewriting, the articles cited in this foreword give a picture of David's evolving ideas about transformational theory. And while David may have wanted a new edition of GMIT, rather than a second printing, he was also eager to make GMIT available to students and scholars. In these respects, this Oxford edition fulfills David's wishes—that his ideas be available to all who seek them, so that they may grow, evolve and multiply. 2. 1995 e-mail correspondence, recipient unknown.

xii

Preface

I. The following figures redo those of figure 8.2 on p. 179. Music examples la and b present scores of the relevant passages.

L = LEITTONWECHSEL; +- = MAJOR-MINOR; S = "BECOMES SUBDOMINANTOF".

EXAMPLE la

xiii

Preface b) Modulating section of Valhalla, Rheingold II,5ff.

EXAMPLE Ib

The analysis is better than that in the book. It brings out a clear isography between the passages. Figure 8.2a in the book is not a well-formed "graph" by the later definition. (SUBM is not = LT SUED on major as well as minor Klangs: (C,+) SUBM = (e,-) but (C,+)LT SUED = (e,-)SUBD = (b,-).) The symbol "(G,-)" on figure 8.2a is a misprint for (Gt,-).1 The discussion of section 8.1.2, pages 179-180, still applies: a group that contains L, S, and +— operations on Klangs will not be simply transitive in equal temperament. (For instance, (C,+)SSSS =(E,+), but (C,+)L +- also = (E,+).) xiv

1. See item 1 in the foreword, p. x.

Preface

Later in the Ring, Wagner develops the relationship of Valhalla and Tarnhelm themes very ambitiously. Figures c) through f) below analyze a transformation that occurs at the climax of Walkure 11,2: Woton, coming to realize the full implications of Valhallagate, ironically gives his blessing to Hagen ("So nimm meinen Segen, Niblungen Sohn!"). Music examples Ic through le are coordinated with the figures.

EXAMPLE Ic-e Figure Ic) shows the Valhalla Kopf put into At major and 4/4 meter, with the original harmonization. Figure d) is the +- transform of c). Figure e) transforms d) so that the subdominant inflection of c)—d) is applied not to the tonic but to the Leittonwechsel of the tonic; also the inflected Klangs change mode as they go, via + — . Music example le is essentially the upper part of the accompaniment for Wotan's pronouncement (there is more beneath!). The Tarnhelm network infects the diatonic aspect of Valhalla here. Figure f) brings that out by rewriting e) in a format that suggests a). In the Waltraute scene of Gotterdammerung, the idea gets even more overloaded ... rather like the picture of Dorian Grey.

JCV

Preface

II. An interesting transformation network is used by Lora L. Gingerich, ". . . Melodic Motivic Analysis in ... Charles Ives," MTS 8 (1986), 75-93. The network appears as her example 24, page 90.

III. Let s be the twelve-tone row of Schonberg's Fourth Quartet. Let S be the family comprising the 48 forms of s. Let TTO be the group of forty-eight twelve-tone operations. TTO is simply transitive on S (given forms s and t, there exists a unique member OP of TTO such that OP(s) = t.) It follows that we can develop a CIS structure for S in such wise that the members of TTO are exactly the formal transposition operations for the GIS (GMIT1A. 1, pp. 157-58). The standard practice, in which forms of the row are labeled by their TTO-intervals from a "tonic" referential row-form—as "RI3," "17," etc.—instances the LABELing practice discussed in chapter 3 of GMIT. If s is any one of the 48 forms, then there exists a unique inverted form of s (in this case) which shares the same three tetrachordal segments with s. Define a transformation TETRA on S: given a sample s, TETRA transforms s into this inverted tetrachordal associate. For instance: TETRA (Ob78 312a 6549) = 780b 4659 123a; TETRA (5012 6a9b 4378) = 6ba9 5120 7843. The transformation TETRA is a formal interval-preserving operation of the GIS under discussion (GMIT 3.4.6, p. 48). Similar operations for this particular row, like TRI and HEXA, are also interval-preserving operations. In his dissertation on Moses undAron (Yale, 1983), Michael Cherlin argues that transformations of this sort, engaging the forms of the Moses row, are highly constructive features of Schonberg's compositional method in the opera.

xvi

IV. Appendix B in GMIT outlines two possible non-commutative GIS structures for the octatonic set. It develops two simply transitive groups of operations on that set; either may be taken as the group of formal transpositions for a GIS; the other then becomes the group of formal interval-preserving operations. A similar situation obtains for set-class 6-20. Taking S as [modeled by] the six numbers 0,1,4,5,8, and 9 mod 12, two simply transitive groups of operations may be defined on S as follows. The group Gl comprises the operations R0= identity, R4 = pc transposition by 4, R8 = pc transposition by 8, Jl = pc inversion with index number 1, J5 = pc inversion with index number 5, and J9 = pc inversion with index number 9. (In the GIS determined by this simply transitive group, all the six operations are formal "transpositions" for that GIS.) The group G2 comprises the six operations RO, Q4, Q8, XI, X5, and X9, defined as follows:

Preface

RO = identity operation. Q4 takes pcs 0,4, and 8 to pcs 4, 8, and 0 resp.; takes pcs 1,5, and 9 topes 9,1, and 5 resp. Q8 takes pcs 0,4, and 8 to pcs 8,0, and 4 resp.; takes pcs 1,5, and 9 to pcs 5, 9, and 1 resp. The Qs are "queer" operations, as opposed to the "rotations" R. XI exchanges each pc of 6-20 with that pc which lies ic 7 away. Thus XI maps 0 to 1,1 to 0,4 to 5, 5 to 4, 8 to 9, and 9 to 8. X5 exchanges each pc with the pc that lies ic 5 away. X9 exchanges each pc with the pc that lies ic 3 away. Both the groups Gl and G2 are simply transitive on S. Either group may be taken as the group of formal transpositions for a formal GIS involving S; the other group thereupon becomes the group of interval-preserving transformations. The pertinence of G2 is manifest in Schonberg's Ode to Napoleon. Music example 2 shows some prominent thematic motives of the piece, all interrelated by operations of G2. Example 2a projects a six-note series that is mapped into example 2b by Q8. 2b' retrogrades 2b; 2c shows the series of 2b' in action. Example 2d is the Q4-transform of series 2a; 2e shows series 2d in action. Example 2f is the

EXAMPLE 2

xvii

Preface

X5-transform of series 2a; 2g is a combinatorial inversion of series 2f, and 2h shows series 2g in action. The motives of 2a, 2c, 2e, and 2h appear frequently in the work, at a variety of pitch levels, retrograded (= inverted), etc.; the various sixnote series generate characteristic tetrachordal segments that are ubiquitous motivic germs in the music. (These tetrachords are all G2-forms of one another).

V. Daniel Harrison, in a recent study of triple counterpoint, has made interesting analytic use of GIS structures.2 I adapt his procedures to my terminology here. Let us suppose three tunes, A, B, and C, that work in triple counterpoint. Let us suppose three voices, 1, 2, and 3, in which the tunes can appear. We can consider the six various possible dispositions of the three tunes in the three voices; let us call each such disposition a "Lage." We can model each Lage by a three-element series: thus the series models "tune B in voice 1, tune C in voice 2, and tune A in voice 3." Let LAGEN be the family of the six possible Lagen. Given two members of LAGEN, there are two "natural" ways to conceptualize a transformation taking the first Lage into the second. For instance, suppose s and t are the Lagen and respectively. We can imagine the tunes as being permuted, to get from s to t: tune B (in voice 1) becomes tune C; tune C (in voice 2) becomes tune A; and tune A (in voice 3) becomes tune B. Thus, in getting from s to t, we permute tune B to tune C, tune C to tune A, and tune A to tune B. We can symbolize this permutation of tunes by the symbol (ABC): A becomes B, B becomes C, and C becomes A. But there is also another "natural" way of conceptualizing getting from s to t: we can imagine the voices as being permuted. Thus, in passing from s = to t = , we can note that tune B, in voice 1 for s, goes into voice 3 for t; tune C, in voice 2 for 5, goes into voice 1 for t; tune A, in voice 3 for 5, goes into voice 2 for t. In sum, voice 1 of s becomes voice 3 of t; voice 3 of s becomes voice 2 of t; and voice 2 of s becomes voice 1 of t. We can symbolize this permutation of voices by the symbol (132): 1 becomes 3, 3 becomes 2, and 2 becomes 1. There are six possible permutations on the symbols {A,B,C}; the six permutations can be used to label six transformations on LAGEN; those six transformations form a group of operations on Lagen which we shall call TPERMS, for "tune-permutations." There are six possible permutations on the symbols {1,2,3}; those six permutations can be used to label six transformations on LAGEN; and those six transformations form a group of operations on Lagen which we shall call VPERMS, for "voice-permutations." Both the groups TPERMS and VPERMS are simply transitive on LAGEN. Either group can be taken as the group of formal transpositions for a GIS whose

xviii

2. See item 3 in the foreword, p. xi.

Preface

family is LAGEN; the other group thereupon becomes the group of formal intervalpreserving operations for the GIS. This situation is as in the last paragraph of appendix B, GMIT. Harrison analyzes the D-major 3-part invention, observing most of the following structure. "A" is the lead-off theme in the rh; "B" is the counterpoint that runs along in sixteenths; "C" is the counterpoint which steps down in leisurely suspensions. Meas.

Stufe

Lage

TPERM interval

VPERM interval

(ACB)

(123)

(ACB) (BIG MIDDLE SECTION) (ACB) (ACB)

(123)

3.5

V

6

I

10

vi

19

IV

21.5

I

23.5

I

(132) (132)

The columns headed "TPERM interval" and "VPERM interval" are read as follows: from Lage (m.3.5) to Lage (m.6) the formal interval of transposition in the TPERM GIS is (ACB), while the formal interval of transposition in the VPERM GIS is (123). From Lage (m.19) to Lage (m.21.5) the formal interval of transposition in the TPERM GIS is (ACB), while the formal interval of transposition in the VPERM GIS is (132). Harrison points out that all six Lagen appear. He notes that the articulation into the two subfamilies of 3 Lagen each, before and after the middle of the piece, is "natural." He points out that in the first half of the piece, the tunes "sweep down" through the voices, while in the second half of the piece, the tunes "sweep up" through the voices. (He does not use the VPERM GIS to discuss this, but expresses it by investigating specific properties of the group TPERMS.) He makes a number of other cogent observations about the TPERM structure of the piece. Among those, he notes that the second half of the piece is TPERM-isographic to the first half, even though the tunes "sweep down" the voices in the first half and "sweep up" the voices in the second half. In GMIT terminology, this can be expressed by noting that in the VPERM GIS, the second half of the piece is ann'-isographic to the first half: (132) is the inverse of (123) in VPERMS. Harrison analyzes other works, including the f-minor invention. Here is my analysis of Lagen in the A-major Prelude from Book I:

xix

Preface Meas.

Stufe

Lage

1

I

4

V

8.5

I

12

vi

16.5

I

19

I

TPERM interval

VPERM interval

(ABC)

(132)

(ABC)

(132)

(ABC)

(132)

(ABC)

(132)

(AB)

(13)

This analysis is useful to contrast to the D-major invention. Here only four Lagen are used. The idea seems to be that the final tonic Lage has a special function here: it breaks the otherwise incessant chain of (ABC) or (132) intervals. Harrison's analysis of the f-minor invention provides still a different idea, for laying out various Lagen. The whole enterprise smells of Marpurg; perhaps the way in which he formulated "Rameau's" (i.e. his) theories of chord inversion might bear similar updating, perhaps even in a somewhat isomorphic vein. VI. Let us consider the family SPECIAL of 12-tone rows whose order-rotation beginning on order-number 4 is the same as their T4-transpose. An example of a SPECIAL row is Ob56439a8712: starting the row at order-number 4 and proceeding therefrom, we derive 439a8712 [and around the end to] Ob56; this order-rotation is the same as T4 of the original row. To fix a notation, we consider each SPECIAL row as a function s mapping the order number ord [mod 12] into the pc number s(ord) [mod 12]. The SPECIAL row of the above paragraph is thus conceived as a function s: s(0) = 0, s(/) = b, s(2) = 5, ...,s(a) = l,s(fc) = 2. Using this notation, we can write out the algebraic property that characterizes SPECIAL rows: SPECIAL PROPERTY: for all ord, s(ord + 4 = s(ord) + 4. [all addition mod 12]

xx

Of interest to us here is the fact that the family of SPECIAL rows admits a simply transitive group-of-operations G in a natural way. Therefore, according to the discussion of GMIT, the family of SPECIAL rows has a natural GIS structure, a structure in which the operations of G play the role of formal transpositions. What follows is a semi-formal development of the group G, and a semi-formal indication that G is simply transitive on SPECIAL.

Preface

To begin with we consider certain operations ADD{j,m}, where j is some multiple of 4 mod 12 and m is some multiple of 3 mod 12; i.e. j = 0, 4, or 8, and m = 0,3,6, or 9. The operation ADD{j,m}, when applied to the SPECIAL row s, adds the pc interval j to the mm, the (m+4)th, and the (m+S)th notes of s. For example, let us take the SPECIAL row of the first paragraph above and apply the operation A{8,3} to it, adding the pc interval 8 to its 3rd, 7th, and b\h notes: o r dnumbers: 0 1 2 3 4 5 6 7 8 9 a b pc numbers of s: O b 5 6 4 3 9 a 8 7 1 2 We add 8 to 3rd, 7th, and Mi, +8 +8 +8 obtaining pc numbers of ADD{8,3}(s): O b 5 2 4 3 9 6 8 7 1 a In the example, we note that ADD {8,3}(s) is still a row. That is because s is SPECIAL: since s(ord + 4) = s(ord) + 4, it follows that the pc numbers s(3), s(7), and s(&)—that is the 3rd, 7th, and 6th notes of s—form an augmented triad. In the example above the augmented triad comprises the pc numbers 6, a, and 2. When we add the interval j = 8 to each of these pc numbers, we simply permute the members of that augmented triad among themselves, without disturbing the other pcs in the other order-positions of the row. Thus, in the above example,

order-positions contain pcs of the row s; when 8 is added to each of those pc numbers, the same order-positions then contain pcs of the row ADD{ 8,3}(s), while the other pcs of s "carry on down" to ADD{8,3}(s), unchanged in their order-positions.

3, 6,

7, a,

b 2

2,

6,

a

This observation can be made rigorous and general, to show that each operation ADD{j,m}, when applied to any SPECIAL row s, yields a row. Furthermore, it can be proved what is intuitively obvious: the new row ADD{j,m}(s) will itself be SPECIAL. The following formulas are easily verified, for j and k any multiples of 4 mod 12, and for m and n any multiples of 3 mod 12: FORMULA 1: ADD{j,m} ADD{k,m} = ADD{j+k,m} FORMULA 2: ADD{j,m} ADD{k,n} = ADD{k,n}ADD{j,m} ADD{0,m} is the identity operation, for each m: it leaves [the pcs of] any sample SPECIAL row unchanged. It follows, via formulas 1 and 2, that the collection of all operations that can be written in form ADD{jO,0} ADD{j3,3} ADD{j6,<5} ADD{j9,9} is a group of operations. We will call this group "ADDINGS." The group is commutative. It has 3-times-3-times-3-times-3 members, ie 81 members.

xxi

Preface

Now we shall develop another group of operations on SPECIAL rows, a group we shall call "PERM." A PERM operation X{p} is defined by any permutation p that acts upon the four symbols 0,3, (5, and 9. Here the permutation p is to be considered as any 1-to-l function that maps the family of four symbols onto itself. The PERM operation X{p} is determined by the PERM DEFINITION: X{p}(s)(m +/) = s(p(m) + j ) where m symbolizes a multiple of 3 mod 12 and; symbolizes a multiple of 4 mod 12. Fix m = 0 in the formula of the definition, and let; run through the values 0, 4, and & The formula tells us that the Oth, 4th, and 8th notes of the X{p}(s) will be respectively the p(0)th, (p(0)+4)th, and (p(0)+5)th notes of s. Similarly [for m = 3] the 3rd, 7th, and £th notes of X{p}(s) will be respectively the p(3)rd, (p(3)+4)th, and (p(3)+S)th notes of s. And so forth [for m = 6 and m = 9]. For an example, fix p to be the permutation p(0) = 3, p(3) = 0, p(6) = 6, p(9) = 9. Then, according to the work we have just gone through, X{p}(s) will have in its Oth, 4th, and 5th order-positions the 3rd, 7th, and bth notes of s respectively, while X{p}(s) will have in its 3rd, 7th, and bth order-positions the Oth, 4th, and 5th notes of s respectively; otherwise X{p}(s) will maintain the [other] notes of s in their respective order positions. The diagram below shows this X{p} applied to the specimen special row used before. o r dnumbers: 0 1 2 3 4 5 6 7 8 9 a b pc numbers of s: 0 4 8 b 5 39 71 6 a 2 pc numbers of X{p}(s): 6 a 2 b 5 39 71 0 4 8 For any SPECIAL row s, and any permutation p, X{p}(s) is a SPECIAL row. If p and q are permutations, then we have FORMULA 3: X{p} X{q) = X{qp}. The PERM operations on SPECIAL rows form a group (anti)-isomorphic to the group of permutations on the four symbols 0,3,6,9. PERM therefore has 4! = 24 members. The group is not commutative. The following formula can be proved: FORMULA 4: ADD{j,m} X{p) = X{p} ADD{j,p(m)}]. In general, therefore, members of PERM do not commute with members of ADDINGS. However, formula 4 tells us that the collection of all operations which can be expressed as some-ADDING-following-some-PERM is a closed family of xxii

Preface

EXAMPLE 3

xxiii

Preface

operations: following one such by another such will generate a third such. It follows that this collection of operations is a group of operations. It is our desired group G. A specimen member of G can be written in CANONICAL FORM: ADD{jO,0} ADD{j3,3} ADD{j6,<5} ADD{J9,P} X{p}. The chromatic scale is one SPECIAL row. It is straightforward, if tedious, to show that given any SPECIAL row s, there is a unique member of our group G which transforms s into the chromatic scale. (Set kO = s(0), k3 = s(3), k6 = s(6), k9 = s(9); since s is SPECIAL, each of the k's must lie within a different augmented triad; apply four appropriate ADDs to obtain a new s' in which the set of k'-values is 0,3,6, and 9; permute the row s' into the chromatic scale. Etc. etc.) It follows that the group G is simply transitive on SPECIAL rows: given any two SPECIAL rows s and t, there is a unique member of G, in the canonical form above, which transformes s to t. (Transform s into the chromatic scale; then transform the chromatic scale into t.) Thus the family of SPECIAL rows has a natural GIS structure, as discussed above. The group G has cardinality 81-times-24 = 1944; that then is also the number of SPECIAL rows. SPECIAL rows become more interesting when one notes their relation to "semi-Mallalieu" rows. Andrew Mead, in excellent work as yet unpublished, has investigated semi-Mallalieu rows exhaustively; some interesting insight can be shed on his work by placing it in a GIS setting.3 Pertaining to our SPECIAL rows are those semi-Mallalieu rows whose every-third-note transform is identical with their T4-transposition. Every-ninth-note of such a row will then be its T8-transposition. (Every-ninth-note = retrograde-of-every-fourth-note.) Such a row, for example, is a premise of my piano piece Just a Minute, Roger [PNM 16.2 (SS 1978), 143-45]: Ob45732681a9. Right at the opening of the piece, one hears quite clearly that every-third-note of this row is its T4-transpose: in meas. 1-4 (music example 3a), the total texture is governed by the row, while the right hand picks out every-third-note, thereby projecting T4-of-the-row. Later on, in meas. 30-35 (example 3b), the total texture is governed by the T4-form of the row, while the right hand picks out every-fourth-note-of-T4, thereby projecting the retrograde of every-ninth-note-of-T4 = the retrograde of T8(T4) = the retrograde of the original row. We shall focus in on these rows for the nonce, calling them SEMI-MALLALIEU. The point is, that the family of SEMI-MALLALIEU rows and the family of SPECIAL rows, along with their characteristic properties, are mathematically equivalent in structure under a transformation that makes each row in one family correspond

xxiv

3. See item 4 in the foreword, p. xii.

Preface

uniquely to a row in the other. I discuss that transformation in ITO 2.7 (October 1976), 8. The upshot of this is that the SEMI-MALLALIEU rows also form a natural GIS, whose formal transposition operations are the members of a natural simply transitive group G' that corresponds to the group G we have just explored. Mead's procedures enable one to find the equivalent of our G or G' for any operation OP on order numbers that can be written as four permutation-cycles of order 3 on the order numbers. OP generalizes "rotation-by-4" in the case of SPECIAL rows, or "every-third-note" in the case of SEMI-MALLALIEU rows.

VII. The injection function, discussed in chapter 6 of GMIT, can be generalized even farther. Let S and S' be any two families of objects—we allow here for the possibility that S' may be a different family from S. Let X be a (finite sub)set of S; let Y' be a (finite sub)set of S'; let f be any function from S into S'. Then the injection number into Y' for f, denoted INJ(X,Y')(f), is the number of elements s in X such that f(s) is a member of Y'. We can also develop a twin concept, not developed in GMIT: the surjection number of X into Y' for f, denoted SURJ (X,Y')(f), is the number of elements s' in Y' such that s' = f(s) for some member s of X. If f is not oneto-one or onto, SURJ(X,Y')(f) may be a very different number from INJ(X,Y')(f). EXAMPLE: Take S to be the family of numbers {0,1,2,4,6,8,10,14,16,18,19,20,22,23,24,26,27,28}. Take S' to be the family of pitches {A3,CH,D4,E4,F4,G4,A4,BI4,D5}. Take the function f to map S into S' according to the following table: s= 0 1 2 4 6 8 10 14 16 18 19 20 22 23 24 26 27 28 f(s)= D4 E4 F4 E4 D4 A4 D5 A4 BW G4 E4 A4 F4 D4 G4 E4 C#4 A3

The function f is onto but not one-to-one, f models certain aspects of the theme from Bach's d-minor Concerto. Take X to be the set of all numbers in S divisible by 4; thus X = {0,4,8,16,20,24,28}. Take Y' to be the triad {D4,F4,A4} within S'.

XXV

Preface

We have f(0) = D4, f(8) = A4, and f(20) = A4. Otherwise f(s) is not a member of the set Y' when s is a member of the set X. The three members 0,8, and 20 of X are mapped by f into members of Y'. Hence INJ(X,Y')(f) = 3. The two members D4 and A4 of Y' are attacked, during this passage, at time-points that are multiples of 4. Hence SURJ(X,Y')(f) = 2.

xxvi

A cknowledgmen ts

The time and leisure I needed to do this work were afforded by a Senior Faculty Fellowship from Yale University and by a Guggenheim Fellowship. The Guggenheim Foundation also provided a subvention toward publication. The excerpt from the score of Elliott Carter's String Quartet No, 1 is used by permission of Associated Music Publishers, New York. Material from Arnold Schoenberg's songs, Opus 15, from his Six Little Piano Pieces, Opus 19, and from Pierrot Lunaire, Opus 21, is used by permission of Belmont Music Publishers, Los Angeles, California 90049. Arnold Schoenberg's Phantasiefor Violin with Piano Accompaniment, Opus 47, is copyright 1952 by Henmar Press Inc. It has been used by permission of C. F. Peters Corporation. The excerpt from the score of Anton Webern's Four Pieces for Violin and Piano, Opus 7, and material from the third movement of his Variations for Piano, Opus 27, are used with these permissions: "Anton Webern—Four Piecesfor Violin and Piano, Op. 7. Copyright 1922 by Universal Edition. Copyright renewed 1950. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. agent for Universal Edition" and "Anton Webern—Variations for Piano, Op. 27. Copyright 1937 by Universal Edition. Copyright Renewed 1965. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. agent for Universal Edition." Analytic sketches for works by Bartok and Prokofieff appear with the following permissions: "Syncopation # 133 from Mikrokosomos (volumes 1-6) Bela Bartok © copyright 1940 by Hawkes & Son (London) Ltd.; Renewed 1967. Reprinted by permission of Boosey and Hawkes, Inc." and "Prokofieff: Melody # 1 from Melodies Op. 35. Reprinted by permission of Boosey & Hawkes, Inc. Copyright Owner."

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Introduction

The following overview of the book will provide a good point of departure. Chapter 1 is purely mathematical; it presents terminology and notation that will be needed later, along with a few important theorems. I am not happy to begin a book about music with a mathematical essay. On the other hand, I do feel that it is helpful for the reader to have this material collated and isolated from the rest of the book. Chapter 1 can be used for quick reference where it stands, and the material obtrudes only minimally into musical discussions later on. Readers who find themselves put off or fatigued in the middle of this chapter are urged to move on into the rest of the book; they can return to chapter 1 later, when later applications of the material make the reference back seem natural or desirable. Chapter 2 takes as its point of departure the general situation portrayed schematically by figure 0.1.

FIGURE 0.1 The figure shows two points s and t in a symbolic musical space. The arrow marked i symbolizes a characteristic directed measurement, distance, or motion from s to t. We intuit such situations in many musical spaces, and we are used to calling i "the interval from s to t" when the symbolic points are

xxix

Introduction

xxx

pitches or pitch classes. Chapter 2 begins by running through twelve examples of musical spaces for which we have the intuition of figure 0.1. Six involve pitches or pitch classes in melodic or harmonic relations; six involve aspects of measured rhythm. The general intuition at hand is then made formal by a mathematical model which I call a Generalized Interval System, GIS for short. A few basic formal properties of the model are explored. Then the twelve examples are reviewed to see how each (with one exception) instances the generalized structure. Chapter 3 concerns itself with further formal properties of the GIS model. In that model, the points of the space may be labeled by their intervals from one referential point; this has advantages and disadvantages. New GIS structures may be constructed from old in various ways. A passage from Webern is examined in connection with a combined pitch-and-rhythm GIS constructed in one such way. Generalized analogs of transposition and inversion operations are explored. So are "interval-preserving operations"; these coincide with transpositions in some GIS models but not in others, specifically not in GISs that are "non-commutative." The bulk of chapter 4 explores one non-commutative GIS of musical interest. The elements of the system are formal time-spans. Extended discussion of a passage from Carter's First Quartet demonstrates the pertinence of this GIS to exploring music in which there are functional measured relations among time spans, but no one overriding time span that acts as a unit to measure all others. After that, chapter 4 presents two examples of timbral GISs, and ends with a methodological note on the relations of music theory, perception, and the intuitions of a listener. Some motivic work by Chopin is considered in this connection. Chapter 5 begins a study of generalized set theory, that is, the interrelationships among finite sets of objects in musical spaces. The first construction studied is the Interval Function between sets X and Y; this function assigns to each interval i in a GIS the number of ways i can be spanned between a member of X and a member of Y. Then the Embedding Number of X in Y is studied; this is the number of distinct forms of X that are subsets of Y. To study that number, we have to establish what we mean by a "form" of the set X, a notion that involves stipulating a Canonical Group of operations. Both the Interval Function and the Embedding Number generalize Forte's Interval Vector. Passages from Webern, Chopin, and Brahms illustrate applications of the constructs. Chapter 6 continues the study of set theory, generalizing the work of chapter 5 even farther. The basic construction is now the Injection Function: Given a space S, finite subsets X and Y of S, and a transformation f mapping S into itself, INJ(X, Y) (f) counts how many members of X are mapped by f into members of Y. This number is meaningful even when S does not have a GIS structure, and even when the transformation f js not so well behaved as are

Introduction

transpositions, inversions, and the like. Passages from Schoenberg and from Babbitt are studied by way of illustration. Instead of starting with a GIS and deriving certain characteristic transformations therefrom, it is possible to start with a family of characteristic transformations on a musical space and derive a GIS structure therefrom. That is, instead of regarding the i-arrow on figure 0.1 as a measurement of extension between points s and t observed passively "out there" in a Cartesian res extensa, one can regard the situation actively, like a singer, player, or composer, thinking: "I am at s; what characteristic transformation do I perform in order to arrive at t?" Chapter 7 explores this conceptual interrelation between interval-as-extension and transposition-as-characteristicmotion-through-space. After developing the mathematics that shows a logical equivalence between GIS structures and certain structures of transformations on spaces, the work proceeds by example. Passages from Schoenberg, Wagner, Brahms, and Beethoven indicate how suggestive it can be to consider networks of "intervals" and networks of "transpositions" (modulations, and so forth) as various aspects of the same basic phenomenon. The morphology of such networks can be carried over to that of networks involving other sorts of transformations. Chapter 8 studies networks involving transformations of Klangs in the sense of Riemann, networks involving serial transformations of various sorts, and networks involving inversional transformations. The Beethoven example from chapter 7 is reconsidered, and there are further examples from Wagner, Webern, and Bach. Chapter 9 develops the formalities of transformation networks in a rigorous way. The structure of a network allows us to assign a formal "input" function to some things and a formal "output" function to other things; these functions seem of considerable musical interest in some cases. The networks have intrinsic rhythmic properties which can also be studied formally. Network structure can accommodate hierarchic levels in a quasi-Schenkerian setting, as an example shows. Chapter 10 applies the network concept in a variety of ways to passages from Mozart, Bartok, Prokofieff, and Debussy.

Note on Musical Terminology All references to specific pitches in this book will be made according to the notation suggested by the Acoustical Society of America: The pitch class is symbolized by an upper-case letter and its specific octave placement by a number following the letter. An octave number refers to pitches from a given C through the B a major seventh above it. Cello C is C2, viola C is C3, middle C is C4, and so on. Any B# gets the same octave number as the B just below it; thus B#3 is enharmonically C4. Likewise, any Cb gets the same octave number as the C just above it; thus Q?4 is enharmonically B3.

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1

Mathematical

Preliminaries

A mathematician would begin by saying, "Let S be a set." Unfortunately, music theory today has expropriated the word "set" to denote special musictheoretical things in a few special contexts. So I shall avoid the word here. Instead I shall speak of a "family" or a "collection" of objects or members. When I do so, I mean just what mathematicians mean by a "set." For present purposes, it will be safe to leave the sense of that concept to the reader's intuition. 1.1 DEFINITION: Let S and S' be families of objects. The Cartesian product S x S' is the family of all ordered pairs (s, s') such that s is a member of S and s' is a member of S'. 1.2.1 DEFINITION: A function or mapping from S into S' is a subfamily f of S x S' which has this property: Given any s in S, there is exactly one pair (s, s') within the family f which has the given s as the first entry of the pair. We say that s', in this situation, is the value of the function f for the argument s; we shall write f (s) = s'. If we think of fas a table, listing members of S (arguments) in a column on the left and corresponding members of S' (values) in a column on the right, then the defining property for functionhood stipulates that each member of S appear once and only once in the left-hand column. (Some members of S' may appear more than once in the right-hand column. Some members of S' may not appear at all in the right-hand column.)

1

1.2.2

Mathematical Preliminaries

1.2.2 DEFINITION: Given families S and S', we shall say that the functions f and g from S into S' are the same, writing f = g, if f and g are the same subsets of S x S', that is if they produce the same table. This special definition of functional equality is worth stressing. We shall soon see why. 1.2.3 DEFINITION: Let f be a function from S into S', and let f' be a function from S' into S". Then the composition function f'f is defined from S into S" as follows: Given an argument s in S, the value (f'f)(s) is f'(f(s)). 1.2.4 Let me draw special attention to the orthographic convention whereby f' appears to the left of fin the notation for the composition function f'f. That convention follows logically from another orthographic convention, the convention of writing the function name to the left of the argument in the expression "f (s)." The reader is no doubt used to this convention. One can read "f (s)" as "the resulting value, when function f is applied to argument s." Then "f'f(s)" is "the result when f is applied to the result of applying f to s." These conventions will be called left (functional) orthography. Right functional orthography is preferred by some mathematicians for all contexts and by most mathematicians for some contexts. In right orthography, one writes "sf" or "(s)f" for "the operand s, transformed by the function f." This value is what was written "f(s)" in left orthography. The composition function which we called "f'f" in left orthography is called "ff " in right orthography, so as to be consistent: "(s)ff'" in right orthography is "s-transformed-by-f, all transformed by f'." This is what was notated "f f (s)" in left orthography. In the following work we shall use left orthography almost exclusively. We shall use right orthography only once, when its intuitive pertinence seems overwhelming. At that point in the text, the reader will be reminded of this discussion. Right orthography would abstractly be more suitable for our eventual purposes, but the reader's presumed familiarity with left orthography seemed decisive to me in making my choice.

2

1.2.5 Suppose that f t and f 2 are functions from S to S'; suppose that f{ and f'2 are functions from S' to S"; suppose that f" is a function from S to S". We can consider the truth or falsity of functional equations like f^ = f", fif t = f 2 f 2 , and so on. Our discussion of "functional equality" in 1.2.2 tells us how to understand these equations, in evaluating their truth or falsity. The first equation above asserts, "for any sample s, the result of applying f[ to ^(s) is the same as the result of applying f" to the given s." The second equation above asserts, "for any sample s, applying f{ to f^ (s) yields the same result as applying f 2 to f2(s)."

Mathematical Preliminaries

1.3.1

For an example, let us take S, S', and S" all to be the family of positive integers. Let f^s) = s + 3, f{(s) = 2s, f2(s) = 2s, and f^s) = s + 6. The four specified functions satisfy the functional equation f'1f1= f^. That is, given any integer s, if we compute fjfi(s), multiplying by two the result of adding 3 to s, we obtain the same net result as we do when we compute f^Cs), adding 6 to the result of multiplying s by two. For another example, let us take S, S', and S" all to be the family of the twelve pitch-classes. Let f(s) = s transposed by 2, f(s) = s inverted with respect to the pitch class C, and f"(s) = s inverted with respect to the pitch class B. The three specified functions satisfy the functional equation f'f = f". That is, given any pitch class s, if we compute f T(s), inverting about C the result of transposing s by 2, we obtain the same net result as we do when we compute f"(s), inverting the given s about B. 1.2.6.1 DEFINITION: The function f from S into S' is onto S' if every member of S' is the value of some argument. (Every member of S' appears at least once in the right-hand column of the function table.) 1.2.6.2 DEFINITION: The function f from S into S' is 1-to-l if no two distinct arguments share the same value. (No member of S' appears more than once in the right-hand column of the function table.) 1.2.6.3 DEFINITION: Let f be a 1-to-l function from S onto S'. Then f"1, the inverse function off, is defined as the family of pairs (s', s) within S' x S such that (s, s') is a member of f. 1.2.6.4 THEOREMS: Given the situation as in 1.2.6.3 above, then f ~* is indeed a function in the sense of 1.2.1. f - 1 is in fact a 1-to-l function from S' onto S. The inverse function off" 1 is, of course, f. The theorems are stated without proof. 1.2.6.5 THEOREM: Let f and f' be functions from S into S' and from S' into S respectively. Suppose that the functions satisfy the two conditions (A) and (B) following. (A): for every s in S, f'f(s) = s. (B): for every s' in S', ff'(s') = s'. Then f and f' are both 1 -to-1; they are respectively onto S' and onto S; and they are inverse functions, each of the other. The theorem is given without proof. 1.3.1 DEFINITION: A function from a family S into S itself will be called a transformation on S. If the function is 1-to-l and onto, it will be called an operation on S.

3

1.3.2

Mathematical Preliminaries

1.3.2 DEFINITION: Given a family S, a collection F of transformations on S is called closed if, given any members f and g of F, the composition fg is a member of F. A closed collection of transformations on S will also be called a semigroup of transformations on S. 1.3.3.1 DEFINITION: The identity operation on a family S is that operation 1 on S which assigns the value 1 (s) = s to any argument s. 1.3.3.2 THEOREM: For any transformation f on S, the functional equations If = f and fl = fare true (in the sense of 1.2.5 above). 1.3.3.3 THEOREM: A transformation f on S is an operation (i.e., 1-to-l and onto) if and only if there exists a transformation f on S satisfying the functional equations f T = 1; ff' = 1. If this be the case then f' is the inverse operation of f. The theorem follows from the various matters studied over section 1.2.6. 1.3.4 DEFINITION: By a group of operations on S we shall mean a family (i.e. collection) G of transformations on S which satisfies conditions (A) and (B) following. (A): G is a closed family, a semigroup of transformations in the sense of 1.3.2. (B): Given any member f of G, there exists a member f of G satisfying f'f = ff = 1. Condition (B) guarantees that the members of G are indeed operations, via 1.3.3.3. (B) also guarantees that G contains the inverse operation for each of its member operations. (A) and (B) together imply that G contains the identity operation 1, provided that G contains any members. Whether we call G a "collection" or a "family" is immaterial; for us the terms are synonymous with each other as they also are with the terms "ensemble" and "set-in-themathematical-sense."

4

1.3.5 The work of section 1.3 so far has explored certain algebraic behavior characteristic of transformations on S. The transformations compose one with another, f with g to form the transformation fg. There is an identity transformation 1, which composes left or right with any f to yield f itself: If = fl = f. Certain transformations, the operations, have inverses; if f is such then f"1 is characterized by the algebraic relations f -1 f = ff" 1 = 1. These algebraic features of the situation are abstracted and generalized by the study of "abstract" semigroups and groups, a study we shall shortly commence. Before we do so, we should note one more aspect of transformation algebra which the abstract study will generalize. This is the associativity

Mathematical Preliminaries

1.5.1

of transformational composition. That is, the composition of transformations obeys the Associative Law f(gh) = (fg)h: Given any sample s, the result of applying f to the (gh)-transform of s is the same as the result of applying (fg) to the h-transform of the given s. 1.4 Now we begin the abstract study. We fix a family (i.e. collection) X of abstract objects x, y, z,..., and develop abstract algebraic systems that model the behavior of transformational algebra. First we must specify how the objects of X are to "compose" one with another. 1.4.1 DEFINITION: A binary composition on X is a function BIN that maps X x X into X. We write BIN(x, y) for the value of BIN on the pair (x, y). 1.4.2 DEFINITION: A binary composition on X is associative if BIN(x, BIN(y, z)) = BIN(BIN(x, y), z) for all x, y, and z. A familiar non-associative binary composition on the natural numbers is exponentiation: BIN(x, y) = x-to-the-y-power. For example BIN(3, BIN (2,3)) = 3-to-the-(2-cubed)-power, or 3-to-the-eighth-power, while BIN(BIN(3,2), 3) = (3-squared)-to-the-third-power, or 9-cubed. Nine-cubed is 3-to-the-sixth, not 3-to-the-eighth. 1.4.3 DEFINITION: A semigroup is an ordered pair (X, BIN) comprising a family X and an associative binary composition BIN on X. It is traditional to write the binary composition for a semigroup using multiplicative notation when there is no reason to use some specific other notation. Thus we shall generally write "xy" to signify BIN(x, y) in a semigroup, failing some reason to write "x + y" or "x * y" and the like. The Associative Law for BIN then reads "x(yz) = (xy)z." This notational convention simplifies the look of the page. It is important, though, not to carry over into our general study intuitions about numerical multiplication which may not be valid within a specific semigroup at hand. It is also important to remember that in order to define a particular semigroup, we must specify not only the family X of elements but also the composition BIN under which the elements combine. Despite this, it is customary to refer (improperly) to "the semigroup X" when the binary composition is clearly understood in a given context. 1.5.1 DEFINITION: A left identity for a semigroup is an element 1 such that for every x, Ix = x. A right identity is defined dually: For every x, xr = x. An identity is an element e which is both a left identity and a right identity.

5

7.5.2

Mathematical Preliminaries

1.5.2 THEOREM: If a semigroup has both a left identity 1 and a right identity r, then 1 and r must be equal. Hence there can be at most one identity for a semigroup. If a semigroup has one, we can therefore speak of "the" identity element. Proof:lr must equal r since 1 is a left identity. Ir must also equal 1 since r is a right identity. There are, incidentally, semigroups that have an infinite number of left identities. (By the theorem above, a semigroup that has more than one left identity cannot have any right identities.) There are, in fact, both finite and infinite semigroups in which every element is a left identity. To illustrate this, take any family X and define on X the composition BIN(x, y) = y for all x and all y; (X, BIN) is such a semigroup. 1.6.1 DEFINITION: Given a semigroup with identity e; given an element x, a left inverse for x is an element 1 satisfying Ix = e. A right inverse for x is an element r satisfying xr = e. An inverse for x is an x' which is both a left inverse and a right inverse. 1.6.2 THEOREM: If an element x of a semigroup with identity has both a left inverse 1 and a right inverse r, then 1 = r. Hence x can have at most one inverse. If x has one, we can therefore call it "the" inverse of x. Proof:1 = le = l(xr) = (lx)r = er = r. 1.6.3 In multiplicative notation for a semigroup with identity, the inverse of an element x that has one is denoted x"1. 1.7 DEFINITION: A group is a semigroup with identity in which every element has an, inverse. The abstract definitions of "semigroup" and "group" (1.4.3; 1.7) are consistent with the earlier use of those terms in connection with families of transformations (1.3.2; 1.3.4). 1.8.1 DEFINITION: Given a binary composition BIN on a family X, elements x and y commute if BIN(y, x) = BIN(x, y), that is, if yx = xy in multiplicative notation. The composition BIN is commutative if every pair of elements commutes. A semigroup or group is commutative if its binary composition is commutative.

6

The group of transposition and inversion operations on the twelve pitchclasses is non-commutative. To illustrate this, let T2 be the operation of

Mathematical Preliminaries

1.9.2

transposing-by-2; let I, J, and K be the respective operations of invertingabout-C, inverting-about-B, and inverting-about-Cft. Then, as we observed earlier, IT2 = J (1.2.5). On the other hand, T2I = K. Thus the operations T2 and I do not commute. (Remember that we are using left orthography. "IT2 = J" means: "Given any sample pitch-class s, if you invert-about-C the 2-transpose of s, you will obtain the inversion-about-B of the given s." "T2I = K" means: "Given any sample pitch-class s, if you transpose-by-2 the inversion-about-C of s, you will obtain the inversion-about-C# of the given s.") 1.8.2 DEFINITION: Given a binary composition BIN on a family X, an element c of X is central if c commutes with every x in X. The family of all central c is the center of the system (X, BIN). 1.9 In this section we shall develop the conceptual structure and terminology for equivalence relations on a family S (not necessarily a semigroup). We shall see in particular how the notion of an equivalence relation is intimately connected with the idea of mapping S onto another family S' by some function f. 1.9.1 DEFINITION: Given a family S, an equivalence relation on S is a subfamily EQUIV of S x S that satisfies conditions (A), (B), and (C) following. (A): For every s in S, (s, s) is in EQUIV. (B): If (s, t) is in EQUIV, then so is (t, s). (C): If (r, s) and (s, t) are in EQUIV, then so is (r, t). The three conditions are called the "reflexive," "symmetric," and "transitive" properties of the relation. The conditions express formally some of our intuitions about things that are "equivalent." (A) matches our intuition that any object s should be equivalent to itself. (B) matches our intuition that if s is equivalent to t, then t should be equivalent to s. (C) matches our intuition that if r is equivalent to s and s is equivalent to t, then r should be equivalent to t. 1.9.2 THEOREM: Let f be a function from S onto S'. Define a relation EQUIV on S by putting the pair (s, t) in the relation if and only if f (s) = f (t). Then EQUIV is an equivalence relation. Proof: (A) f (s) = f (s), so (s, s) is in the defined relation. (B) If (s, t) is in the defined relation, f (s) = f (t). Then f (t) = f (s), so that (t, s) is in the defined relation. (C) If f(r) = f(s) and f(s) = f(t) then f(r) = f(t). We shall see soon that every equivalence relation on S can be regarded as being generated in precisely the above fashion, for some suitable choice of S' andf.

7

1.9.3

Mathematical Preliminaries

1.9.3 THEOREM: Let EQUIV be an equivalence relation on a family S. For each s in S let E(s) be the subfamily of S comprising exactly those members of S which are in the EQUIV relation to s, i.e. those t such that (s, t) is in the EQUIV relation. Then, giveu any s and any t in S, either (A) or (B) below will be true. (A): s and t are equivalent; E(s) and E(t) are the same collection. (B): s and t are not equivalent; E(s) and E(t) are disjoint (have no common member). Proof: Suppose first that s and t are equivalent. Then, by the symmetric and transitive laws, r is equivalent to s if and only if r is equivalent to t. In other words, r is a member of E(s) if and only if r is a member of E(t). Thus E(s) and E(t) are the same collection. (A) of the theorem obtains. Now suppose that s and t are not equivalent. Then there can be no r which is both a member of E(s) and a member of E(t). For if there were such an r, then r would be equivalent to both s and t; by the symmetric and transitive laws, we could infer that s was equivalent to t, which we have supposed is not the case. Thus E(s) and E(t) are disjoint. (B) of the theorem obtains, q.e.d. 1.9.4 By virtue of Theorem 1.9.3, an equivalence relation partitions S into the set-theoretic union of mutually disjoint subfamilies E^,.. .,En,... These subfamilies are called the equivalence classes of the relation. For each s in S, there is precisely one equivalence class En to which s belongs, s is a member of the class En if and only if En = E(s), where E(s) is the family defined in 1.9.3, the family of objects equivalent to s. Indeed, it would be possible to define any equivalence relation by partitioning S into mutually disjoint subfamilies S l f ..., S n , ... One could then define the pair (s, t) to be in a relation REL if both s and t lie in the same subfamily of the partition. One could show that REL is an equivalence relation, and that the members S j , . . . , S n ,... of the given partition are exactly the equivalence classes for that equivalence relation. 1.9.5 DEFINITION: Given an equivalence relation EQUIV on a family S, the family of equivalence classes is called the quotient family of S modulo EQUIV. We shall denote it symbolically by S/EQUIV. The function E of 1.9.3 maps S onto S/EQUIV, mapping each argument s to the value E(s), the member of the quotient family that contains s. The function E is called the natural map of S onto S/EQUIV.

8

We may now regard every equivalence relation as potentially generated in the manner of 1.9.2. Given EQUIV on S, take S' = S/EQUIV and take f = E, the natural map of S onto S'. s and t are then equivalent under the given relation if and only if f (s) = f (t).

Mathematical Preliminaries

1.9.7

1.9.6.1 EXAMPLE: Let S be the family of all pitches under twelve-tone equal temperament. Define EQUIV by putting (s, t) in EQUIV if s and t have the same letter name, give or take enharmonic equivalence. The quotient family S/EQUIV comprises the twelve pitch classes. The natural map E takes each pitch s into its pitch class E(s). 1.9.6.2 EXAMPLE: Let S be the family of all beats in a certain waltz. Define a function f from S into the numbers 1,2, and 3: f (s) = 1 if s is the first beat of its measure; f(s) = 2 if s is the second beat of its measure; f(s) = 3 if s is the third beat of its measure. A dancing master might construct this function by calling "one-two-three," over and over again as the beats go by. The function f induces an equivalence relation on S by the method of 1.9.2: s and t are EQUIValent if they share the same f-value. The three equivalence classes can be called the "beat classes" of the relation; they comprise the first beats, the second beats, and the third beats of the waltz. 1.9.6.3 EXAMPLE: Let S be the family of all collections of pitch classes. Put the pair (s, t) into the relation SAMETYPE if the collection t is a transposed or inverted form of the collection s. (Transposition-by-zero is considered a formal transposition here.) SAMETYPE is an equivalence relation. The equivalence classes are Forte's set-types.1 The class 3-11, in Forte's nomenclature, contains the twenty-four major and minor triads. The class 3-12 contains the four augmented triads. And so on. 1.9.7 OPTIONAL: This section of the work is for those who are curious to explore the material in a bit more depth. Given a function f from S onto S', define EQUIV as in 1.9.2; i.e. put (s, t) into the EQUIV relation if and only if f(s) = f(t). Given any member s' of S'; since f is onto, there is some s in S satisfying f(s) = s'. The family of all s satisfying f(s) = s' is an equivalence class En; En contains just those arguments for f having the given s' as their f-value. We write En = ARGS(s'). ARCS is a function from S' into S/EQUIV. ARGS maps S' onto S/EQUIV: Given any equivalence class En, let s be a member of En and let s' = f(s); then ARGS(s') = En; the given En is an ARGS-value. The function ARGS is also 1-to-l: If s' and t' are distinct members of S', then the equivalence class ARGS(s'), comprising those s such that f(s) = s', is obviously different from the equivalence class ARGS(t'), comprising those t such that f (t) = t'. By the method of its construction above, the function ARGS satisfies 1. Allen Forte, The Structure of Atonal Music (New Haven and London: Yale University Press, 1973).

9

1.10

Mathematical Preliminaries

formula (A) below. (A): ARGS(f (s)) = E(s)

for all s in S.

We have observed that the function ARGS is 1-to-l and onto. Hence it has an inverse function, which we shall call f/EQUIV. ARGS maps S' 1-to-l onto S/EQUIV as in formula (A) above. f/EQUIV maps S/EQUIV 1-to-l onto S'. Applying f/EQUIV to both sides of formula (A), we obtain formula (B). (B): f (s) = (f/EQUIV) (E(s)) for all s in S. f/EQUIV is called the induced map on S/EQUIV. While f may map many of its arguments onto the single value f (s) in S', f/EQUIV maps exactly one of its arguments onto that value. Via formulas (A) and (B), the mutually inverse functions ARGS and f/EQUIV set up a 1-to-l correspondence between the members f (s) of the image family S' and the members E(s) of the quotient family S/EQUIV, the family of equivalence classes. 1.10 When we shift our attention from an arbitrary family S to a semigroup (X, BIN), certain sorts of equivalence relations on X are of special interest because of the ways they interact with the algebraic structure of the semigroup. We shall study here some special equivalence relations called congruences. They interrelate with special sorts of functions on semigroups, functions called homomorphisms. Homomorphisms map semigroups one into another in a special way that engages algebraic structure. 1.10.1 DEFINITION: An equivalence relation on a semigroup is a congruence if it has this property: Given xt equivalent to y x and x2 equivalent to y 2 , then \1x2 is equivalent to y^a1.10.2 THEOREM: Given a congruence on a semigroup, let C^ and C2 be any congruence classes (equivalence classes for the congruence). Then there is a unique congruence class C3 such that whenever X A and x2 are members of C t and C2 respectively, the composition x t x 2 is a member of C3. Proof: Take any specimen y: in Q and any specimen y 2 in C 2 . Let C3 be the congruence class containing y^. C3 is the class whose existence the theorem asserts. To see this, suppose that x t and x 2 are any members of C x and C 2 respectively. Since Xj is congruent to y : and x 2 is congruent to y 2 , x t x 2 will be equivalent to y^ (1.10.1). Hence XjX 2 will lie within the same congruence class as y x y 2 . That is, x x x 2 will lie within the constructed C3. q.e.d.

w

1.10.3 THEOREM: Let CONG be a congruence on the semigroup (X, BIN). Then the quotient family X/CONG (i.e. the family of congruence classes) becomes a semigroup itself under the binary composition BIN/CONG defined as follows. Given congruence classes Cj and C2 (members of X/CONG),

Mathematical Preliminaries

J.JO.4.2

the composition (BIN/CONG) (C 1 ,C 2 ) is the congruence class C3 of Theorem 1.10.2, that is the unique congruence class which contains BIN(x 1 ,x 2 ) whenever x1 belongs to C t and x2 belongs to C 2 . Sketch of proof: The heart of the theorem is that the binary composition BIN/CONG is well defined. Given Cj and C 2 , the value of C3 does not depend at all on the specimen \v and x2 we may select to represent C t and C2. C3 depends only upon the classes C1 and C2 themselves. Having noted that, it is not hard to prove that BIN/CONG is associative. 1.10.4.1 EXAMPLE: Let (X, BIN) be the group of all integers, positive, negative, or zero, under addition. Define a relation CONG: the pair of integers (x, y) is in this relation if the difference y — x is an integral multiple of 12. CONG is reflexive: (x, x) is in the relation since x — x = 0 = 0-times-12 is an integral multiple of 12. CONG is symmetric: If (x,y) is in the relation, then there is some integer n such that y — x = n-times-12; then there is some integer m such that x — y = m-times-12 (take m = — n); then (y, x) is in the relation. CONG is transitive: If y — x = m-times-12 and z — y = n-times-12, then z — x = (z — y) + (y — x) = (n + m)-times-12. So CONG is an equivalence relation. It is in fact a congruence, for it satisfies the requirement of 1.10.1: If y t — x t is a multiple of 12 and y2 — x2 is a multiple of 12, then (yt + y 2 ) — (xj + x 2 ) is a multiple of 12. We write C(x) for the congruence class containing x. Since C(x) = C(xplus-or-minus-any-multiple-of-12), every congruence class is one of the twelve classes C(0), C(l), ..., C(ll). The quotient semigroup, then, contains just those twelve members. For each i between 0 and 11 inclusive, the class C(i) contains exactly those integers that can be written as i-plus-some-multiple-of12. Composition of the twelve congruence classes within the quotient semigroup follows the rule of addition modulo 12. That is, C(i) + C(j) = C(i + j) if i + j is less than 12; otherwise C(i) + C(j) = C(i + j - 12). Thus C(5) + C(8) = C(l). According to 1.10.2, we can read this as stating correctly: "Any number divisible by 12 with a remainder of 5, added to any number divisible by 12 with a remainder of 8, produces some number divisible by 12 with a remainder of 1." The equation "C(5) + C(8) = C(l)" in the above context is customarily abbreviated: "5 + 8 = 1 (mod 12)." The quotient semigroup is called "the integers modulo 12." It is in fact a group. We shall see later that the quotient semigroup of any group must itself be a group. If we replace the modulus 12 in the above construction by an arbitrary integer N greater than 1, we obtain "the integers modulo N" as a quotient group. 1.10.4.2 EXAMPLE: Let (X, BIN) be the group of all rational numbers that can be expressed as x = 2a3b5c, where a, b, and c are integers (positive, negative, or zero); BIN is multiplication. We can consider these numbers to model all possible ratios of pitches in just intonation.

11

1.11

Mathematical Preliminaries

Define a relation CONG: The pair (x, y) is in this relation if the number y is some power of 2 (positive, negative, or zero) times the number x. In our intervallic model, this will be the case when the intervals x and y differ by some number of octaves. For example, any one of the numbers 12/5, 6/5, 3/5, and 3/10 is in this relation to itself or to any other one. The four numbers model the four intervals of a minor tenth up, a minor third up, a major sixth down, and a major thirteenth down. As an exercise, using the procedure of 1.10.4.1 as a guide, the reader may verify that CONG is a congruence. (Remember to verify first that it is an equivalence relation!) The quotient group models all pitch-class intervals in just intonation. That is, each congruence class consists of one interval, give or take any number of octaves. Mathematically, C(x) = C(2x) = C(4x) = • • • = C(x/2) = C(x/4) = . . From this, it can be proved: Given any x, there is a unique member x' of C(x) which lies between 1 (inclusive) and 2 (exclusive). In this way, the members x' of X that lie between 1 and 2 provide a plausible system of labels for the congruence classes C(x'). (The various pitch-intervals between the unison and the rising octave can be used to label the various intervals-modulo-theoctave.) It can also be proved: Given any x, there is a unique member x" of C(x) which can be expressed as x" = 3b5c. So the numbers x" that have factors of 3 and 5 only in their rational expressions provide another plausible system of labels for the congruence classes C(x"). (x" = 3b5c labels the pitch-class interval of "b dominants and c mediants, modulo the octave.") 1.11 When we studied an equivalence relation on a family S, we made a number of observations about the natural map E, the function that maps each element s of S into the equivalence class E(s) of which s is a member. When the family S is a semigroup X and the equivalence relation is a congruence, we shall replace the name "E" of this natural map by the name "C": C maps each element x of the semigroup X into the congruence class C(x) of which x is a member. We have already used this nomenclature in examples 1.10.4.1 and 1.10.4.2 above. Everything that we observed earlier about the natural map E (1.9.3,1.9.4, 1.9.5,1.9.7) is true for the natural map C, which is only a special notation for E in the particular event that S is a semigroup and the equivalence relation is a congruence. Beyond that, C has special properties that engage the algebraic structure of the semigroup X and the quotient semigroup X/CONG. Specifically, the natural map C of X onto X/CONG satisfies law (A) below. (A): C(x!)C(x2) = C(XiX 2 ) for all x x and x 2 . 72

Indeed we defined the binary composition "C(x1)C(x2)" in X/CONG

Mathematical Preliminaries

1.11.3

precisely so as to satisfy this law. That was the work of 1.10.2 and 1.10.3. Mathematicians express property (A) above by saying, "C is a homomorphism of X onto X/CONG." The crucial term "homomorphism" is defined in 1.11.1 below. 1.11.1 DEFINITION: A function f from a semigroup (X, BIN) into a semigroup (X', BIN') is a homomorphism if it satisfies the law: BIN'(f( Xl ),f(x 2 )) = f(BIN( Xl ,x 2 )) for all Xi and all x2 in X. One can express this law colloquially by saying, "The combination of the images is the image of the combination." Using multiplicative notation for both semigroups, the law looks simpler: f( Xl )f(x 2 ) = f( Xl x 2 ). Certain sorts of homomorphisms are of special interest. 1.11.2 DEFINITION: A homomorphism is an isomorphism (into) if it is 1 -to-1 If f is an isomorphism of (X, BIN) onto (X', BIN'), we say the two semigroups are isomorphic (via f). In that case the inverse map f -1 is an isomorphism of (X', BIN') onto (X, BIN). 1.11.3 OPTIONAL: Let f be a homomorphism of a semigroup (X, BIN) onto a semigroup (X', BIN'). We have already seen (in 1.9.2) that an equivalence relation is defined if we select as equivalent just those pairs (x, y) satisfying f (x) = f (y). We can show that the relation in this case is in fact a congruence CONG. From earlier work (1.9.7) we know that the mapping ARGS of X' into X/CONG is 1-to-l and onto. (ARGS(x') is the congruence class comprising exactly those x such that f (x) = x'.) In this case, f being a homomorphism, we can show that ARGS is a homomorphism of the semigroup (X', BIN') into the quotient semigroup (X, BIN)/CONG. Here is a sketch for the proof of that. We want to show that for all \l and for allx 2 ,ARGS(f(x 1 ))ARGS(f(x 2 )) = ARGS(f(x1)f(x2)). In this equation, the symbolic product of the two ARGS-values on the left means the binary composition of those values in the quotient semigroup; the symbolic product f(Xi)f(x 2 ) within the equation means the binary composition of those two f-values in the semigroup (X',BIN'). Now f(x t )f(x 2 ) = f(x 1 x 2 ), since f is a homomorphism, and ARGS (f (any thing)) = C(that thing), as per 1.9.7(A). Hence the equation we have to show reduces to the equation C(x1)C(x2) = C(\1x2). And the latter equation is indeed true, since CONG is a congruence (1.11 (A)). Since ARGS is 1-to-l, onto, and a homomorphism, it is an isomorphism of the two semigroups (X', BIN') and (X, BIN)/CONG. Colloquially speak-

13

1.11.4

Mathematical Preliminaries

ing, we can say that the image semigroup is isomorphic with the quotient semigroup in this context. f/CONG, the inverse map of ARGS, the induced map of the quotient semigroup onto the image semigroup, is therefore also an isomorphism. This is very significant. It means that any homomorphic image (X', BIN') of a semigroup (X, BIN) "is essentially" some quotient semigroup of (X, BIN), and the generic homomorphism f of (X, BIN) onto that image "is essentially" the natural map of (X, BIN) onto that quotient. The words "is essentially" here must be interpreted with some care. They express the idea of identification up to within isomorphism of the image semigroups. With that understanding, we can say that it suffices to study the possible congruence relations on (X, BIN), in order to know all possible homomorphisms which can map (X, BIN) onto other semigroups, and all possible other semigroups which are homomorphic images of (X, BIN). 1.11.4 One more term should be introduced here. An anti-homomorphism of one semigroup into another is a function f satisfying f(x!)f(x 2 ) = f^x^. Given a semigroup (X, BIN) we can define another binary composition ANTIBIN on the family X: ANTIBIN (x l 5 x 2 ) = BIN(x 2 ,x 1 ). ANTIBIN is associative, so (X, ANTIBIN) is a semigroup. (X, ANTIBIN) is antiisomorphic to (X, BIN) under the map f (x) = x. In the obvious sense, every anti-homomorphism of (X, BIN) is a homomorphism of (X, ANTIBIN) and vice-versa. Thus we will not normally have to concern ourselves with antihomomorphisms. We will only have to do so when we have to deal with both homomorphisms and anti-homomorphisms of the same semigroup at the same time. Such a situation will in fact arise later on. We shall be studying a certain group whose elements are i, j, k ...; we shall also be studying various families of transformations on a certain family of objects. One such family will be called "transpositions"; for each i there will be a corresponding transpositionoperation TJ. Another such family will be called "interval-preserving operations"; for each i there will be a corresponding interval-preserving operation PJ. The P-operations will combine according to the rule PjPj = P^; the Toperations will combine according to the rule T;Tj = T^. The map of i to P; will be an isomorphism, while the map of i to T, will be an anti-isomorphism of the same group. In such a situation we must perforce deal with the concept of antihomomorphism. We could change BIN to ANTIBIN in the index group i, j, k . . . so as to make the mapping of i to Tj an isomorphism, but then the mapping of i to PJ would become an anti-isomorphism. Using right orthography for the operations T; and P; would have the same effect. 14

1.12.1 THEOREM: Let f be a homomorphism of the semigroup (X, BIN) onto the semigroup (X', BIN'). If e is an identity for (X, BIN) then f (e) is an identity

Mathematical Preliminaries

1.13

for (X', BIN'). In that case, if x has an inverse x * in (X, BIN) then f(x-1) is an inverse for f(x) in (X', BIN'). Proof: f(e)f(x) = f(ex) = f(x); f(x)f(e) = f(xe) = f(x). Thus f(e) is an identity for the family of all f(x). Since f is onto, every member of X' can be written as the value f (x) of some argument x. Hence f (e) is an identity for all of X'. If x has an inverse then f(x -1 )f(x) = f(x-1x) = f(e) = the identity in X'; likewise f(x)f(x-1) is the identity in X'. So f(x-1) is the inverse in X' for f(x). Thatis, f(x~ 1 ) = (f(x)r1. 1.12.2 THEOREM: A homomorphic image of a group is a group. The theorem follows at once from 1.12.1. 1.12.3 THEOREM: Any quotient semigroup of a group is a group. The theorem follows from 1.12.2, since the natural map of the given group onto its quotient semigroup is a homomorphism (1.11 (A)). The quotient construction is one common way to derive new semigroups or groups from old. Another way is to form "direct products" as sketched below. 1.13 Let SGPi = (X t , BINO and SGP2 = (X2, BIN2) be semigroups. The direct product of SGPj and SGP2 is a semigroup SGP3 = (X3,BIN3) constructed as follows. X3 is the Cartesian product Xt x X 2 . Given (x ls x 2 ) and (y1}y2) in X3, BIN3((xl5x2), (y l5 y 2 )) is defined as the element (BIN^x^yj), BIN 2 (x 2 ,y 2 )) of X3. In multiplicative notation, (Xi,x 2 )(y 1 ,y 2 ) is defined = (x 1 y 1 ,x 2 y 2 ). BIN3 as defined is associative, so that SGP3 is indeed a semigroup. To symbolize that SGP3 is the direct product of SCPj and SGP2 we write SGP3 = SGP 1 ®SGP 2 . If ej and e2 are identities for SGPj and SGP2, then e3 = (e ls e 2 ) is an identity for SGP3. If xl in Xt and x2 in X2 have inverses in their respective semigroups, then (x^SxJ 1 ) is an inverse for the element (Xj,x 2 ) of SGP3. It follows: If SGPj and SGP2 are both groups, then so is their direct product SGP3.

75

2

Generalized Interval Systems (1): Preliminary Examples and Definition

In conceptualizing a particular musical space, it often happens that we conceptualize along with it, as one of its characteristic textural features, a family of directed measurements, distances, or motions of some sort. Contemplating elements s and t of such a musical space, we are characteristically aware of the particular directed measurement, distance, or motion that proceeds "from s to t." Figure 0.1 on page xi earlier symbolized this awareness, using an arrow marked i extending or moving from a point marked s to a point marked t to help us render visible our intuition. When s and t are pitches or pitch classes we are comfortable with the word "interval" as a term to use in connection with the i-arrow. That is why I have decided to keep using the word "interval" when generalizing our intuitions about the i-arrow to musical spaces whose objects s, t, and the like are not necessarily pitches or pitch classes. It will be helpful to explore some specific musical spaces informally in this connection, before proceeding more formally later on. "int(s, t)" will provisionally denote our intuition of a directed measurement or motion behaving like an "interval from s to t." Later on we shall attach a more formal significance to the expression "int(s, t)."

16

2.1.1 EXAMPLE: The musical space is a diatonic gamut of pitches arranged in scalar order. Given pitches s and t, int(s, t) is the number of scale steps one must move in an upwards-oriented sense to get from s to t. Thus int(C4, C4) = 0, int(C4,D4) = 1, int(C4,E4) = 2, and int(C4,C5) = 7. Int(C4,A3) = — 2, since moving " — 2 steps up" amounts to moving 2 steps down. Using these measurements, if we take 2 steps up (e.g. from C4 to E4) and then take 2 more steps up (in this case, from E4 to G4), we have taken 4 steps up in all (in this case, from C4 to G4). Symbolically, int(C4, E4) = 2,

Generalized Interval Systems (1)

2.1.5

int(E4, G4) = 2, int(C4, G4) = 4, and 2 + 2 = 4. The intervallic measurements of the model thus interact effectively with ordinary arithmetic. This obviates a defect in the traditional measurements which tell us, for example, that a "3rd" and another "3rd" compose to form a "5th." (3 + 3 = 5 ???) 2.1.2 EXAMPLE: The musical space is a gamut of chromatic pitches under twelve-tone equal temperament. Given pitches s and t, int(s, t) is the number of semitones one must move in an upwards-oriented sense to get from s to t, not counting s itself. Thus int(C4, D4) = 2, int(C4, G4) = 7, int(C4, C5) = 12, int(C4, F3) = -7, and int(C4, F2) = -19. 2.1.3 EXAMPLE: The musical space comprises the twelve pitch-classes under equal temperament. If we arrange the pitch classes around the face of a clock following the order of a chromatic scale, then int(s, t) is the number of hours that we traverse in proceeding clockwise from s to t. For instance, if s is at 8 o'clock and t is at 1 o'clock, int(s, t) = 5. Note that the number int(s, t) does not depend on which pitch class is positioned at 12 o'clock. In any case, int(E, E) = 0, int(E, F) = 1, and int(F, E) = 11. 2.1.4 EXAMPLE: The musical space comprises seven pitch-classes, corresponding to the seven mode degrees of system 2.1.1. If we wrap the scale around the face of a seven-hour clock, then int(s, t) is the number of hours that we traverse on that clock, in proceeding clockwise from s to t. Thus int(D, D) = 0, int(D, E) = 1, and int(D, C) = 6. We could produce analogs for the linear spaces of examples 2.1.1 and 2.1.2, using other sorts of scales. And, for systems in which octave equivalence is functional, we could derive analogs for the modular spaces of examples 2.1.3 and 2.1.4. For example, we could investigate octatonic-scale space in the manner of 2.1.1 and 2.1.2; we could derive there from a modular space of eight pitch-classes, wrapping the octatonic scale around an eight-hour clock and measuring intervals modulo 8. 2.1.5 EXAMPLE: This musical space, harmonic rather than melodic, comprises pitches available from a given pitch using just intonation. If we write FQ(s) to denote the fundamental frequency of the pitch s, then int(s, t) is the quotient FQ(t)/FQ(s). That quotient will be some number of the form 2a3b5c, where a, b, and c are integers, positive, negative, or zero. It is not immediately clear what intuitions of "distance" or "motion" we are measuring by these intervals. Personally, I am convinced that our intuitions are highly conditioned by cultural factors. In particular, I do not think that the acoustics of harmonically vibrating bodies provide in themselves an adequate basis for grounding those intuitions. For instance, when we

17

2.7.5

Generalized Interval Systems (1)

write int(C4, F#4) = 45/32 (= 2~ 5 32 5), I do not believe that we are intuiting a common partial frequency F#9 for both C4 and F#4, a partial which is intuited forthwith in some harmonic space as both the 32nd partial of F#4 and the 45th partial of C4. Nor do I believe that we intuit a path in harmonic space which corresponds directly to a compound series of individual multiplications and divisions by 2,3, and 5. That is, if we take the 5th partial of the 3rd partial of the 3rd partial of C4, and then find the frequency of which that is the 2nd partial, and then find the frequency of which that is the 2nd partial, continuing on in this way and so arriving eventually at F#4,1 do not believe that the way of getting from C4 to F#4 which we have intellectually reconstructed in harmonic space is in any sense an intuition of distance or gesture being measured by the composite ratio 45/32 = 5 times 3 times 3 divided by 2 divided by 2 and so on. In order to describe an actual harmonic intuition, I would rather proceed as follows. When we hear or imagine the succession C4-F#4 in its own context and try to intuit a harmonic sensation, we intuit a tonic followed by the leading tone of its dominant. And we intuit the secondary leading tone harmonically as the third of a harmony whose root is the dominant of the dominant. Constructing a fundamental bass representative for that root, i.e. some D below the F$4, we will locate that D in register as D3, to keep it completely beneath the "soprano line" C4-F#4. For the same reason, in constructing a fundamental bass for the note G4 that we imagine following F#4 in the soprano, a fundamental bass representing the implicit role of dominant harmony in the context, we will locate the bass G as G3. In this way we intuit the enlarged harmonic context of figure 2.1 (a) from the given stimulus C4-F#4.

FIGURE 2.1

18

The arrows on figure 2.1(b) show the path in harmonic space which I believe we actually intuit in this enlarged context. Starting at C4 as a local tonic, the first arrow takes us to G3, a fundamental bass for the dominant harmony where we hear the implicit enlarged context closing. The second arrow shows G3 inflected by its own dominant immediately preceding; the arrow points to the fundamental bass D3 for that event. The third arrow points to F#3, the major third of the harmony over D3. The fourth arrow points to F#4, the octave above F#3. Collating the entire path, we can retrace it and express it in prose: F#4 lies an octave above the major third of that

Generalized In terval Systems (1)

2.1.5

dominant which lies a fourth below that dominant which lies a fourth below C4. Now we can finally see in what sense the number 45/32 is a valid measurement for some intuition of a characteristic way from C4 to F#4 in harmonic space. We do have clear intuitions for a number of basic harmonic moves; we can measure those moves, and we have intuited (not just constructed) a chain of them. Our belief in the validity of mathematics carries us the rest of the way. Specifically, we intuit clearly the relation "t lies an octave above s," and we accept empirically the measurement FQ(t) = 2FQ(s) as a valid reflection of that intuition. We also intuit clearly the relation "t is the major third of the s harmony," and we accept the measurement FQ(t) = (5/4)FQ(s) as valid in connection with that intuition. Finally, we intuit clearly the relation "t is that dominant which lies a fourth below s," and we accept the measurement FQ(t) = (3/4)FQ(s) as valid in that connection. Applying those basic measurements to the arrows of figure 2.1(b), we get FQ(F#4) = 2FQ(F#3), FQ(F#3) = (5/4)FQ(D3), FQ(D3) = (3/4)FQ(G3), and FQ(G3) = (3/4)FQ(C4). Applying mathematics to this chain of measurements, we infer that the equation FQ(F#4) = 2FQ(F#3) = 2(5/4)FQ(D3) = 2(5/4)(3/4)FQ(G3) = 2(5/4)(3/4)(3/4)FQ(C4) is valid as measuring an intuited chain of intuitions, that is, not simply as an empirical fact. Observe that the number int(C4, F#4) = FQ(F#4)/FQ(C4) = 45/32 arises here not as the product of 2~ 5 and 32 and 5, which is its most "natural" mathematical factorization. Rather, 45/32 arises as the product of the four factors 2, (5/4), (3/4), and (3/4), reflecting its "natural" way of measuring an intuited chain of intuitions in the given situation. I should stress again not only the sophistication and complexity of this system (compared, for example, to the melodic system of example 2.1.1) but also its heavy reliance on cultural conditioning. A brief review of just how the noteheads got onto figure 2.1 will emphasize the point: Cultural conditioning is obviously important in our construction of the extended mental/aural context, given only the acoustical stimulus C4-F#4 and the intent to think/hear "harmonically." That intent, in turn, is itself a cultural phenomenon. Imagine a culture whose members, when they hear the notes of figure 2.1, are able to override the melodic relation of F#4 to the G4 which follows, as a primary structural determinant for a system of music theory that addresses such passages! The amount of time it takes the reader to discover that the last sentence is ironic indicates the greater or lesser extent to which we are all still within the grip of that culture. Then too, our interest in the harmonic system under study depends to a considerable extent on a cultural predilection for harmonically resonating instruments (with one degree of freedom) producing sustained (steady-state) sounds. In fact, much of the traditional theory pertinent to the system was developed under the supposition that the domain of investigation could be

19

2.1.6

Generalized In terval Systems (1)

adequately represented by aspects of one or more stretched strings. Despite its problems, that representation had the great advantage of enabling theorists to connect their intuitions of musical intervals with the measurement of visible, tangible material distances along the strings. This Cartesian modelling of harmonic intervals as res extensae, in a space outside the minds and bodies of the musicians, enabled harmonic theorists of earlier times to avoid having to confront such complex and problematic gestural intuitions as those of figure 2.1.

20

2.1.6 EXAMPLE: The musical space comprises the pitch classes generated by the space of example 2.1.5 above. Given pitch classes s and t, int(s, t) is the ordered pair of integers (b, c) such that t lies b dominants and c mediants from s. Here, if p and q are pitches belonging to the classes s and t respectively, then there is some integral power of 2, say 2a, such that the interval from p to q in the system of 2.1.5 above is 2a3b5°. We have int(C, G) = (1,0), int(G, D) = (1,0), int(D, F#) = (0,1), int(C, F#) = (2,l), int(C,F) = (-l,0), int(C,Ab) = (0,-l), and int(C,Db) = (—1, —1). (Since we are in just intonation, there are many distinct pitch classes that have any given letter name C, G, D, F#, F, A(7, D|?, and so on. I am supposing above that we are considering the simplest possible harmonic relationship in each case. Later discussion will clarify the exact issues involved.) This system modularizes the system of 2.1.5 by reducing out all octave relationships among both pitches and intervals. Its utility is clear in connection with figure 2.1. The most salient aspect of our harmonic intuition there was that the pitch class F# represented the mediant of the dominant of that dominant of the pitch class C. Our having to manipulate precise registers in the figure, particularly registers for a fundamental bass, made the working out of that intuition more complicated and problematical than necessary in this case. Now that we have available the modular harmonic system of the present example, we could get from C4 to F#4 more simply by a new chain of intuitions. Contemplating the succession of pitches C4-F#4 as before, we arrive as before at the intuition that F#4 is a leading tone to a dominant of C, and therefore functions harmonically as a mediant for a dominant of the dominant. We have now intuited int(C, F#) = (2,1) in modular harmonic space. We know that the pitch interval between two pitches whose pitch classes are in a tonic/dominant relation is validly measured by some-powerof-2 times 3/2. We also know that the pitch interval between two pitches whose pitch classes are in a root/mediant relation is validly measured by somepower-of-2 times 5/4. Hence we infer from our intuition of the pitch-clas interval int(C, F#) = (2,1) the validity of measuring the pitch interval int(C4, F#4) as = some-power-of-2 times (3/2)2 times (5/4). We know that a rising interval between two pitches lying closer than an octave is validly

Generalized Interval Systems (1)

2.1.6

measured by a number between 1 and 2. So we infer that int(C4, F|4) is validly measured by the number 2a times (3/2)2 times (5/4), where the integer a makes the product greater than 1 and less than 2. We can visualize modular harmonic space as a two-dimensional map or game board in the format of figure 2.2.

FIGURE 2.2 On this map, if int(s, t) = (b, c) then the pitch class t lies b places to the right of s and c places above s. (—b places to the right is b places to the left.) The subscripts help us keep track of the mediant dimension: If s has subscript m and t has subscript n then int(s, t) will be of form (something, n - m). The subscripts and the visual format more generally clarify the functional distinctions in this space between pitch classes with the same letter-name but different subscripts; they are at different places on the map. Since we are assuming just intonation, the frequencies involved will differ by factors involving the syntonic comma and the pitch classes C_,, C0, C,, ... will be distinct acoustically. But even in equal temperament, the visual format of figure 2.2 portrays a conceptually infinite harmonic space, whose distinct places correspond to subscripted pitch-classes. (Subscripted pitch-classes can be expressed formally as ordered pairs; for example A\>3 can be expressed as the pair (A|>, 3).) We can conceptualize the intervals of modular harmonic space as characteristic "moves" on the game board of figure 2.2. For example, in going from CQ to Ffj, we move 2 squares to the right and 1 square up. The number-pair (2, 1), which is int(C0, F|j), can thereby be conceptualized as a label for this particular form of the "knight's move" on the game board, i.e. the knight's move east by northeast. The same move (interval) takes us from Al to D#2, or from DL, to G0.

27

2.2.7

Generalized Interval Systems (1)

We could reduce the system of 2.1.6 farther if we considered pitch classes to be equivalent when they shared the same letter name, differing only by subscript. Then C_ l 5 C0, Cj, C 2 ,... C n ,... would all mean the same thing; so would £_!, E0, E l 5 E 2 , . . . , E n . In this case, moving one square north on the game board of figure 2.2 would be functionally equivalent to moving four squares east. The north/south dimension of the board would functionally disappear, and we could reduce our map to a one-dimensional east/west succession of dominant-related pitch classes... E|?0, 6(70, F0, C0, G0, D0, A 0 , E0, ... Because of the equivalence relation that led to this series, we may as well consider the reduced pitch classes to represent pitches in quarter-comma mean-tone temperament: Four new "fifths" (that were steps east on figure 2.2) are pitch-class equivalent to a "major third" (that was a step north on figure 2.2). There is no reason to keep the zero subscripts in the series of mean-tone pitch classes, so we can just write... E[?, 8(7, F, C, G, D, A, E,... The intervals of this reduced system are integers measuring steps "east" on that chain; since we have lost the north/south dimension of our earlier figure, we may as well say "to the right" rather than "east." We could reduce the mean-tone system even farther by declaring the enharmonic equivalence of Gl? with F#, of Dfr with C#, and so on. The infinite series of mean-tone pitch classes thereby gets wrapped around the face of a clock, and we find ourselves back at the system of 2.1.3, only now measuring intervals-modulo-the-octave by (equally tempered) fifths rather than by semitones.1 We have explored six examples, and suggested some further examples, of musical spaces in connection with which we traditionally use the word "interval" to denote a directed measurement, distance, or motion. All six of these musical spaces, melodic or harmonic, had pitches or pitch classes for their elements. Now we transfer our attention to some musical spaces whose elements are measured rhythmic entities of various sorts. We presuppose a context that makes us sensitive to time in segments that can be measured by some temporal unit, whether this unit is some local pulse within a piece or some conceptual span, like the minute that underlies metronome markings. 2.2.1 EXAMPLE: The musical space is a succession of time points pulsing at regular temporal distances one time unit apart. Given time points s and t,

22

1. Maps like figure 2.2 have been especially common in German theories of tonality since the eighteenth century, generally in connection with key relationships rather than root relationships (though some theories do not dwell on such a distinction). The closest precedent I can find for the actual configuration of figure 2.2 itself appears in Hugo Riemann, Grosse Kompositionslehre, vol. 1, Der homophone Satz (Melodielehre und Harmonielehre) (Berlin and Stuttgart: W. Spemann, 1902). Riemann's map is on page 479. He illustrates intervals as moves on the board, on page 480.

Generalized Interval Systems (1)

2.2.3

int(s, t) is the number of temporal units by which t is later than s. (—x units later is x units earlier.) 2.2.2EXAMPLE: The musical space is the preceding one, wrapped around the face of an N-hour clock. We can imagine this as modeling the imposition of an N-unit meter on the earlier space, so that barlines appear regularly every N pulses. The present space has N members, which we shall call "beat classes," labeling them by numbers from O through N — 1. Beat-class 0 comprises all the pulses of 2.2.1 that occur at some bar-line; beat-class 1 comprises all the pulses of 2.2.1 that^ occur one unit after some barline;...; beat-class (N — 1) comprises all the pulses of 2.2.1 that occur one unit before some barline. If s and t are beat classes, int(s, t) is the number of hours clockwise that t lies from s on the N-hour clock. Thus, in twelve-eighths meter (N = 12) the interval from beat-class 10 to beat-class 5 is 7. We discussed the notion of beat classes earlier (1.9.6.2), as exemplifying the concept of equivalence classes. We observed there that a dancing master often calls out beat classes over and over as the pulses go by, using numbers 1 through N rather than 0 through N — 1, e.g. "ONE-two-three, ONE-twothree,..." Conductors and conducting students will also be familiar with the notion kinetically. For them, beat classes are associated with definite spatial positions of the hand, positions which are numbered on pedagogical diagrams. Intervals of 1 between beat classes correspond to minimal unbroken hand gestures for the conductor, gestures that proceed from each beat class to the next along smooth arrows on the diagrams, tracing a characteristic gestural path through this modular space over and over again. This is the path along which we ride "from s to t," making int(s, t) gestural articulations along the way. Milton Babbitt has worked with a system of 12 beat classes that behaves formally exactly like the traditional 12-tone system for pitch classes.2 2.2.3 EXAMPLE: The musical space is a family of durations, each duration measuring a temporal span in time units. And int(s, t) is the quotient of the t and s measurements, t/s. If s spans 4 time units and t spans 3 time units, then int(s, t) = 3/4. t is "3/4 the length of" s. We may, if we wish, identify each duration with the beat for a certain tempo. The numerical quotients of our durations then measure the inverse quotients of the corresponding tempi in tempo-space.3 2. He describes the system in "Twelve-Tone Rhythmic Structure and the Electronic Medium," Perspectives of New Music vol. 1, no. 1 (Fall 1962), 49-79. 3. Influential compositions whose rhythmic textures involve such proportions include Elliott Carter's String Quartet no. 1 (1950-51), Conlon Nancarrow's Studies for Player Piano (1951-), and Gyorgy Ligeti's Poeme symphonique for 100 metronomes (1962).

23

2.2.4

Generalized Interval Systems (I)

2.2A EXAMPLE: We reduce the system of 2.2.3 by a durational modulus M greater than 1. Two durations are conceived as equivalent if one is some integral power of M times the other. This leads us to a modular musical space whose elements are duration-classes (i.e. equivalence classes of durations under the defined equivalence relation). The intervals of 2.2.3 are reduced in the same manner: Two numerical quotients or proportions are conceived as equivalent if one is some power of M times the other. The ratio-classes can be used as formal intervals in the reduced system. Mathematically, the reduction from 2.2.3 to 2.2.4 is exactly the same as the reduction from a system of pitches and pitch-ratios, to a system of pitch classes and ratio-classes modulo powers of M = 2, that is pitch classes and intervals-modulo-the-octave. To illustrate, let us take M = 2 for a rhythmic modulus: Two durations, or two numerical proportions, are conceived as equivalent if one is twice the other, or four times the other, or eight times the other, or half the other, or one-quarter of the other, and so on. One equivalence class of durations is then the family r = (...,5/32,5/16,5/8,5/4,5/2,5,10,20,...). Another equivalence class is the family s = (..., 1/96,1/48,1/24,1/12,1/6,1/3,2/3,4/3, 8/3,...). Yet another equivalence class is the family t = (..., 7/80,7/40,7/20, 7/10,7/5,14/5,28/5,...). With reference to these particular classes r, s, and t, int(r, s) is the ratio-class that contains the number 16/15. The members of this ratio-class are exactly the numbers of form (some-power-of-2)-times-(16/15). If p is any member of class r and q is any member of class s, then the ratio q/p is (some-power-of-2)-times-(16/15). In similar wise, int(s, t) is the ratio-class that contains 21/20; the members of this class are exactly the numbers of form (some-power-of-2)-times-(21 /20). If we allow irrational durations (or tempi with respect to the time-unit), we can consider the equivalence class u = (..., n/4, n/2, n, 2n, 4rc,...). Int(s, u) is the ratio-class that contains the number 3?r/8; the members of this class are exactly the numbers of form (some-power-of-2)-times-(37c/8).4 2.2.5 EXAMPLE: The musical space is a family of durations. Int(s, t) is the difference (NB not the quotient) of time units between s and t: Int(s, t) = (t — s) units. So if r, s, and t are respectively 3, 4, and 8 units long, then int(r, s) = (4 — 3)units = 1 unit, int(s, t) = (8 — 4)units = 4 units, and int(t,r) = (3 — 8)units = — 5 units. In the earlier system of quotients (2.2.3), the corresponding intervals would have been 4/3, 2, and 3/8.

24

4. Karlheinz Stockhausen argues the plausibility of system 2.2.4, with M = 2, in "... how time passes ...," trans. C. Cardew, Die Reihe (English version) vol. 3, pp. 10-40. He also argues there the interrelatedness of his rhythmic system with traditional pitch systems. A clear view of how Stockhausen uses these ideas for the tempi of Gruppen (1955-57) is provided by Jonathan Harvey, The Music of Stockhausen (Berkeley and Los Angeles: University of California Press, 1975), 55-76.

Generalized Interval Systems (1)

2.2.6

The additive system now under study measures intervals in units of time; the earlier system of quotients measured intervals as pure numbers (ratios). The difference between the systems here becomes striking if we set the time unit as "one sixteenth note." Then the durations r = 3, s = 4, and t = 8 can be symbolized respectively by a dotted eighth, a quarter note, and a half note. The additive intervals int(r, s), int(s, t), and int(t, r), computed above as 1 unit 4 units, and — 5 units, can be expressed as "plus a sixteenth," "plus a quarter," and "minus a-quarter-tied-to-a-sixteenth." The corresponding multiplicative intervals are simply the numbers 4/3, 2, and 3/8, numbers that express ratios involving the durations. 2.2.6 EXAMPLE: To simplify matters, we restrict our attention to the durations of 2.2.5 that are exactly the positive integral multiples of some basic small duration, which we take as the temporal unit. We wrap these durations around an M-hour clock, accordingly reducing the system to a modular system. The modular space comprises M duration-classes: Two durations belong to the same duration-class if their lengths differ by some integral multiple of M. The interval between duration classes s-units-mod-M and tunits-mod-M is (t — s)-units-mod-M. t — s is the number of hours clockwise from s to t on the M-hour clock. The duration t is int(s, t) units longer than s, give or take any number of M-unit "measures". For example take M = 16; take s = 8 and t = 4 units mod 16. Then int(s,t) = 4 — 8= — 4 = 12 units mod 16. If we represent the unit as a sixteenth note, then the M-unit "measure" lasts a whole note. The duration-class s = 8 is represented by a half note, give or take any number of whole notes tied on. The duration-class t = 4 is represented by a quarter note, give or take the same. The interval int(s, t) = 12 is represented by "plus a dotted half," give or take the same. Our arithmetic mod 16 above reflects this observation: A quarter note, tied to an extra whole note for free, is a dotted half longer than a half note. We have now explored six rhythmic spaces as well as six tonal ones. To the extent we intuit these spaces, we intuit "intervals" in connection with them. Later on we shall explore yet other spaces, including some more rhythmic ones and some timbral ones. At this point, though, it will be helpful to stop and develop some formal generalities. All of the examples in this chapter so far have certain structural features in common. Foremost among these is our intuition in each case of a group, explicitly or implicitly defined, within which the intervals lie. If i and j are intervals (characteristic measurements, distances, motions, or the like) we intuit being able to compose them in some characteristic way (e.g. by addition, addition mod 12, multiplication, multiplication mod powers of 2, concatena-

25

2.3.1

Generalized Interval Systems (1)

tion of moves on a game board, and so on). And we intuit the composition ij of the intervals i and j to be itself an interval of the system (characteristic measurement, distance, motion, and the like). Indeed, we intuit that for any elements r, s, and t of the musical space, the interval-from-r-to-s composes with the interval-from-s-to-t to yield the interval-from-r-to-t. Symbolically: int(r, s)int(s, t) = int(r, t). We intuit the composition of intervals to be associative: i(jk) = (ij)k. We intuit an identity interval e, that composes with any interval j to yield j: ej = je = j. Indeed, we intuit that each object s of the space lies the identity interval from itself: int(s, s) = e. We intuit that each interval has an "inverse interval" in the sense of measurement, distance or motion: i""1 measures, extends, or moves in the reverse sense from i. We intuit that this intuitive inverse is also a group inverse: 'i-1i = ii"1 = e. Indeed, we intuit that if i is the interval from s to t, then i"1 will be the interval from t to s. Symbolically: int(t, s) = int(s, t)"1. We can collate all these intuitions to construct a formal generalized system. As we shall see, all our examples so far with one exception suggest specific instances of the generalized system. 2.3.1 DEFINITION: A Generalized Interval System (GIS) is an ordered triple (S, IVLS, int), where S, the space of the GIS, is a family of elements, IVLS, the group of intervals for the GIS, is a mathematical group, and int is a function mapping S x S into IVLS, all subject to the two conditions (A) and (B) following. (A): For all r, s, and t in S, int(r, s)int(s, t) = int(r, t). (B): For every s in S and every i in IVLS, there is a unique t in S which lies the interval i from s, that is a unique t which satisfies the equation int(s, t) = i. Condition (B) of the definition is a new idea. We shall discuss it shortly. Condition (A) has already been discussed. But what about the other equations involving the function int, equations we also discussed above? Should we not also stipulate these other equations in defining a GIS? It turns out that we do not have to, because they are logically implied by the group structure and Condition (A). We demonstrate that in the form of a theorem. 2.3.2 THEOREM: In any GIS, int(s, s) = e and int(t, s) = int(s, t)"1 for every s and t in S. Proof: int(s, s)int(s, s) = int(s, s), via Condition (A). Multiply both sides of that equation by int(s, s)"1; we obtain int(s, s) = e as asserted.

26

int(s, t)int(t, s) = int(s, s) via Condition (A). We have just proved that int(s, s) = e; hence int(s, t)int(t, s) = e. Multiply both sides of that equation on the left by int(s, t)"1; we obtain int(t, s) = int(s, t)"1 as asserted.

Generalized Interval Systems (1)

2.3.2

Now let us turn our attention to Condition (B) of the definition. Essentially, it guarantees that the space S is large enough to contain all the elements we could conceive of in theory. The idea is: If we can conceive of an element s and if we can conceive of a characteristic measurement, distance, or motion i, then we can conceive of an element t which lies the interval i from s. In certain specific cases, application of this idea may require enlarging practical families of musical elements, to become larger formal spaces that are theoretically conceivable while musically impractical. For instance, we shall need to conceive supersonic and subsonic "pitches" in order to accommodate the idea of being able to go up or down one scale degree from any note, in connection with example 2.1.1. Figure 2.2 affords another good example: Obviously no finite musical context can explore the entire extent of this map, which accommodates the idea of being able to conceive the dominant, mediant, subdominant, and submediant of any pitch class. This is the methodological point: We must conceive the formal space of a GIS as a space of theoretical potentialities, rather than as a compendium of musical practicalities. In a specific compositional or theoretical context, the space S of a GIS might be perfectly accessible in practice. Such is the case, for example, with the "twelve-tone" GIS pertaining to example 2.1.3: Every one of its twelve pitch-classes is easily referenced by any pertinent music. On the other hand, in other compositional or theoretical contexts, the space S of a GIS might be pertinent as an entirety only to the extent it is suggested or implied by the actually stated musical material, plus the characteristic relationships actually employed. In just this way a painting or statue might suggest or imply the entire extension of Euclidean two-or-three-dimensional space, or some other geometrical space. (I am thinking in particular of the parabolic space in some of Van Gogh's late work.) Let us consider figure 2.2 yet again in this connection. In order to conceive the extension of the entire map, we need only three things: one (tonic) place on the map, the characteristic idea of a "just dominant" relation involving pitch classes, and the characteristic idea of an independent "just mediant" relation. In other words, we need only the pitch classes and the intervals we can infer from one tonic triad (!) in order to generate the entire conceptual group of intervals, and thereby to infer the conceptual extension of the entire map, as a terrain within which a particular composition or theory may occupy some particular region. Another feature of Condition (B) also requires discussion. Given s and i, the condition demands not just some t that satisfies int(s, t) = i, but a unique such t. We might consider weakening the condition, replacing the words "a unique," where they appear in 2.3.1(B), with the word "some." Let us call the weakened condition "(weak B)". Could we gain even greater generality by using (weak B) instead of (B)? Not really. Under condition (weak B), the space S would be partitioned into equivalence classes: s and s' would be equivalent if and only if int(s, s') = e. Given s' equivalent to s and t' equivalent to t, it would

27

2.4

Generalized Interval Systems (1)

be true that int(s', t') = int(s, t). We could thus think of the intervals as being from one equivalence class to another. We could replace S by the quotient family S/EQUIV, the family of equivalence classes, and obtain a GIS thereby. (That is, Condition (B) would apply to the function int on equivalence classes.) It is hard to see what we could possibly want to do with S that we could not do as well or better with the reduced space S/EQUIV of equivalence classes.

28

2.4 At this point, let us briefly review all our examples, noting the various GIS structures which they suggest. 2.1.1 suggests a GIS in which S is the indicated gamut, extended indefinitely both up and down. In this GIS, IVLS is the group of integers under addition and int(s, t) is the number of steps up from s to t. — i steps up is i steps down. The reader may verify that Conditions (A) and (B) of Definition 2.3.1 are satisfied. As we observed informally during the earlier discussion of this example, the non-traditional numbering of the scalar intervals is necessary so that the algebra of Condition (A) can obtain. The GIS suggested by example 2.1.2 consists of the indicated space S, a chromatic scale extended indefinitely up and down, the group IVLS = integers under addition, and the function int(s, t) = number of semitones up from s to t, not counting s. Again, — i up is i down. The reader may verify that Conditions (A) and (B) of 2.3.1 are satisfied. The GIS for example 2.1.3 consists of the space S = the twelve pitch classes as indicated, the group IVLS = the integers under addition modulo 12, and the function int(s, t) = number of hours clockwise from s to t on a 12-hour clock. The GIS for example 2.1.4 consists of the space S = the seven mode degrees as indicated, the group IVLS = the integers under addition modulo 7, and the function int(s, t) = number of hours clockwise from s to t on a 7-hour clock. The GIS for example 2.1.5 has for its space S the extended family of all "pitches" conceptually available from a given pitch using just intonation. The group IVLS here is the multiplicative group comprising all rational numbers that can be expressed in the form 2a3b5c, where a, b, and c are integers. The function int here is the quotient int(s, t) = FQ(t)/FQ(s). To conceive the GIS for example 2.1.6, we take as our space S the "game board" of figure 2.2. IVLS here is the group of ordered pairs of integers (b,c) under componentwise addition: (b^c^ + (b 2 ,c 2 ) = (bt + b 2 ,C! + c2). (IVLS is thus the direct product of the-integers-under-addition with itself.) The function int is as discussed: int(s, t) = (b, c), where t lies b squares east and c squares north of s on the game board. The GIS for example 2.2.1 has as its space S the indicated succession of time points, conceptually extending indefinitely both backwards and forwards in time. IVLS here is the integers under addition, and int(s, t) is the number of time units by which t is later than s, — i later meaning i earlier. For example 2.2.2, the GIS consists of S = the N beat-classes, IVLS = the integers

Generalized Interval Systems (1)

2.4

under addition modulo N, and int(s, t) = the number of hours clockwise that t lies from s on an N-hour clock. The GIS for example 2.2.3 depends upon just what proportions among durations we wish to allow. IVLS in any case will be some multiplicative group of positive numbers. If we allow only the basic proportions of 2 and 3, then IVLS will comprise those numbers of form 2a3b, where a and b are integers; S will then comprise exactly such durations as are in those proportions to any given duration. S is conceptually extended to allow indefinitely short and indefinitely long "durations." If we allow basic rhythmic proportions of 5 and 7, as well as 2 and 3, then IVLS will comprise all numbers of form 2a3b5c7d, and S will contain the corresponding extra members. Or we might take, as basic rhythmic proportions, the square root of 2 (2(1/2)) and the square root of 3 (3(1/2)); then IVLS will contain exactly those numbers of form 2(a/2)3(b/2), where a and b are integers, and S will contain the corresponding durations in proportion to any one given duration. And so on. In each case, the function int(s, t) is given by the quotient t/s = (t units)/(s units). When we reduce the space of 2.2.3 to the space of 2.2.4 by the modulus M, we also reduce the group of intervals for 2.2.3 to a quotient group, the new group of intervals for 2.2.4. Specifically, intervals i' and i for 2.2.3 are declared congruent if one is the other multiplied by some integral power of M: i' = iMa. This relation is indeed a congruence in the group-theoretic sense (1.10.1): It is an equivalence relation, and if j' is congruent to j (f =jM b ), then i'j' is congruent to ij (i'j' = (ij)M(a+b)). The congruence generates a quotient group, imposing a group structure on the family of congruence-classes. Those congruence-classes are the "ratio-classes" of example 2.2.4. The function int for 2.2.4 can be derived mathematically from the function int for 2.2.3, along with the declared equivalence and congruence relations on the space and group of 2.2.3. We shall explore this derivation formally in chapter 3, where we shall study the general construction of a "quotient GIS" from a given GIS and a given congruence on its group of intervals. Example 2.2.5, exceptionally, does not lead at once to a GIS structure. If we try to find a pertinent GIS, we will take IVLS to be some additive group of numbers i, j, ..., for as durations s and t vary in the family S of durations, t may be ±i, ±j,... units longer or shorter than s. And int(s, t) = (t — s)units But S, IVLS, and int here cannot satisfy Condition (B) of Definition 2.3.1. For instance, try s = 3 units and i = — 8 units; then there is no duration t in S satisfying int(s, t) = i. If there were such a t, then (t — s)units = (t — 3)units would have to equal i units = — 8 units, t — 3 would equal — 8, and t would be — 5 units in duration. But S does not contain "negative durations," and failing some convention not yet specified, it is not clear what intuition we could possibly be modeling, when we stipulate a duration t that lasts not only less than no time at all, but also measurably less than no time at all. The modeling problem here is different in kind from those involved in other spatial exten-

29

2.4

Generalized In terval Systems (1)

sions we have made conceptually. E.g., while we cannot hear a "pitch" of .001 Hertz fundamental, or of five trillion Hz fundamental, we can conceive such pitches. That is, we are at ease with the notion of periodic vibration at those rates, and we can imagine that other creatures might be sensibly aware of them. In the same sense, we can conceive indefinitely short and indefinitely long "durations," and we can conceive "time points" that lie indefinitely far back in the past or ahead in the future. But we can not conceive, in such a sense, a duration lasting precisely 5 units less than no time at all, which is thereby precisely 2 units longer than a duration lasting precisely 7 units less than no time at all. So example 2.2.5 does not lead to a GIS. Example 2.2.6 can be regarded as one means of salvaging example 2.2.5 in this connection, by providing a convention that attaches meaning to the concept of a negative duration-class. E.g. we can think of duration-class " — 5" as that class containing all durations lasting just 5 units less than some multiple of the modulus duration. " — 5" thus means the same as "M — 5," or the same as " — 5, modulo M." Example 2.2.6 has a GIS structure. The space S comprises the M duration-classes. The group IVLS is the additive group of integers modulo M. int(s, t) is the number of hours t lies clockwise from s when the duration-classes are wrapped around an M-hour clock. The reader can verify that (S, IVLS, int) is indeed a GIS. In particular, Condition (B) of the definition, which failed for example 2.2.5, obtains for example 2.2.6.

30

3

Generalized Interval Systems (2): Formal Features

There is a familiar convention whereby the twelve pitch classes in equal temperament are labeled by their intervals from a referential pitch class C. C, C#, D, ..., Bb, B are thereby labeled 0, 1, 2, ..., 10, 11 (mod 12). This convention can be generalized. That is, in any GIS we can always use the intervals of the group IVLS to label the members of the space S by their respective intervals from an assumed referential object in S. We can make the notion formal by introducing some new terminology. 3.1.1 DEFINITION: Given a GIS (S, IVLS, int) and a fixed referential member "ref" of S, the function LABEL, mapping S into IVLS, is defined by the equation LABEL(s) = int(ref, s). 3.1.2THEOREM: Whatever the element ref, the function LABEL maps S 1-to-l onto IVLS, and it satisfies the formula int(s,t) = LABEL(s)~1LABEL(t). Proof: Given the element ref and any interval i, there is one and only one s in S satisfying int(ref,s) = i, by Condition (B) of Definition 2.3.1 for a GIS. Since int(ref, s) is LABEL(s), we have observed the following: Given any i, there is some s satisfying LABEL (s) = i; furthermore, there is only one such s. Thus the function LABEL is onto; furthermore it is 1-to-l. Now we prove the formula of the theorem. By definition 3.1.1, LABEL(s)"1LABEL(t) = int(ref,s)~1int(ref,t). In that equation we can substitute int(ref, s)"1 = int(s, ref), via 2.3.2. The equation then states: LABEL(s)~1LABEL(t) = int(s, ref)int(ref, t). And the expression on the

31

3.1.2

Generalized Interval Systems (2)

right side of that equation equals int(s, t) as desired, via Condition (A) of Definition 2.3.1. q.e.d. The LABEL function can be very useful, particularly for computations involving members of S. On the other hand, its use in some contexts can be problematic, both conceptually and computationally. Conceptually, there may not always be adequate musical reasons for assigning a special referential status to ref. Why, for example, should I assign this status to the pitch class C a priori? Perhaps, in a certain context, I hear no referential pitch class at all, but rather notes related only to each other, and not to any one given note. Or perhaps I hear E as referential in this piece; then why should I call E "4" in a CLABELing system, rather than "0" in an E-LABELing system? But the violins and string basses have actually decided how E will sound by tuning their Es a certain interval from A; then would it not be methodologically most accurate to use an A-LABELing system? And so on. Some of the conceptual issues will be familiar from similar issues in fixed-do and movable-do systems of solfege. Conceptual difficulties aside, computations themselves can be muddled, when we use a LABELing system, by the algebraic influence of irrelevant intervals, intervals arising from irrelevant relations of ref to the objects s, t , . . . , whose interrelations we actually wish to compute.1 Now we shall explore some ways in which we can formally construct new GIS structures from old. We have observed one such construction used informally again and again during our survey of examples 2.1 and 2.2 earlier; that is the construction of a "quotient GIS" from a given GIS and a stipulated congruence relation on IVLS. Whenever we spoke informally about "modularizing" one system to obtain another, we invoked such a notion. Soon we shall make the notion formal. By way of preparation before doing so, we shall examine a bit more carefully just how it applies in one specific and familiar case. We consider for this purpose how the "chromatic scale" GIS of 2.1.2 gives rise to the "twelve-tone" GIS of 2.1.3. Let us call the two systems respectively GISj = (S^IVLS^intO and GIS2 = (S 2 ,IVLS 2 ,int 2 ). S t is a chromatic scale extended conceptually up and down indefinitely. IVLSi is the integers under addition. int: (s, t) is the number of semitones up from s to t. When "modularizing" GISX to obtain GIS2, we use a certain congruence on the group IVLSi. Intervals i and i' are congruent if i' = i plus or minus some integral multiple of 12. This relation is indeed a congruence in the grouptheoretic sense of 1.10.1. That is, it is an equivalence relation; furthermore, i' + j' is congruent to i + j whenever i' is congruent to i and j' is congruent to

32

1. I discuss these matters further in "A Label-Free Development for 12-Pitch-Class Systems," Journal of Music Theory vol. 21, no. 1 (Spring 1977), 29-48.

Generalized Interval Systems (2)

3.1.2

j. As we saw in chapter 1, the congruence gives rise to a quotient group IVLSJCONG. The elements of this group are the congruence classes, and the binary combination of these elements in the quotient group is well defined (!) by the formula (class-containing-i) 4- (class-containing-j) = (classcontaining-(i + j)). In this case the quotient group is the integers modulo 12. And that is IVLS2, the group of intervals for GIS2. In modularizing GlSi to GIS2 we also invoked a certain equivalence relation on S^ Pitches s and s' were declared equivalent if they differed by some integral number of octaves. The condition for equivalence of s and s' can be expressed by using the congruence relation on IVLSj (and that is important): s and s' are equivalent as defined if and only if int^s, s') is divisible by 12, which is the case if and only if intl (s, s') is congruent to the identity interval oflVLSj. The equivalence relation reduces the family S! of pitches into the family S2 of pitch classes. S2 = S1/EQUIV: The 12 pitch classes are precisely the 12 equivalence classes of pitches, under the constructed equivalence relation. Finally, in modularizing GlSt to GIS2, we implicitly invoked a significant interrelation between the congruence on IVLS^ and the equivalence relation on Sv: If pitches s and s' are equivalent (different by some number of octaves), and if pitches t and t' are also equivalent, then the intervals int^s.t) and int^s'jt') are congruent (different as integers by some multiple of 12). We can put this another way, significant for our purposes: The congruence class to which intj(s, t) belongs depends only on the equivalence class to which s belongs and the equivalence class to which t belongs, not on any specific s' and t' chosen to represent those equivalence classes. This feature of the situation enables us to see how the function int2 works for our present example: Given pitch classes (equivalence classes) p and q, int2(p, q) in IVLS2 is the congruence class (interval mod 12) to which int1(s, t) belongs, whenever s and t are any pitches belonging to the pitch classes (equivalence classes) p and q respectively. For example, let p and q be the pitch classes C and F. If s and t are middle C and Queen-of-the-Night F, then int^s.t) = 29 (semitones). If s' and t' are high C and contra F, then intjCs'.t') = -55 (semitones). The integers 29 = 24 + 5 and -55 = — 60 + 5 belong to the same congruence class mod 12, "congruence-class 5." If s" and t" are any other pitches belonging to p and q (i.e. named C and F), then intj (s", t") also belongs to congruence-class 5. That is what gives rigorous meaning to our saying "int2(C, F) = 5." Otherwise, our clock-face model for GIS2 would remain only arbitrarily or vaguely related to GISj. Indeed, we could use the relationships under present discussion to define int2'. int 2 (p,q) is that unique congruence class which contains any and all values of inti(s, t), s being any member of p and t being any member of q. Everything noted about OK^ and GIS2 in the example just studied can be generalized so as to define a "quotient GIS," given any GIS and any con-

33

3.2.1

Generalized Interval Systems (2)

gruence relation on its group of intervals. The work of section 3.2 following will be devoted to this generalization. 3.2.1 THEOREM AND DEFINITION: Let (S, IVLS, int) be a CIS; let CONG be a congruence on the group IVLS. Then an equivalence relation EQUIV is induced on S by declaring s and s' to be equivalent whenever int(s, s') is congruent to the identity e in IVLS. EQUIV will be called the induced equivalence on S. Proof: int(s, s) = e, so EQUIV as defined is reflexive. If int(s, s') is congruent to e, then int(s', s) = int(s, s')"1 is congruent to e; thus EQUIV as defined is symmetric.2 Finally, if int(s, s') and int(s', s") are both congruent to e, then int(s, s") = int(s, s')int(s', s") is congruent to e-times-e = e. Thus EQUIV is transitive. 3.2.2 LEMMA: Let (S, IVLS, int) be a GIS; let CONG be a congruence on IVLS. Then the following is true. Suppose s and s' are equivalent in S under the equivalence induced by CONG; suppose t and t' are also equivalent under that equivalence relation; then int(s, t) and int(s', t') are congruent members of IVLS. Proof: int(s,t) = int(s,s')int(s',t')int(t',t). Now both int(s,s') and int(t', t), by supposition, are congruent to e. Hence int(s, t) is congruent to e • int(s', t') • e. That is, int(s, t) is congruent to int(s', t'). q.e.d. 3.2.3 THEOREM: Let (S^IVLS^inti) be a GIS. Let CONG be any congruence on IVLSj; let EQUIV be the induced equivalence relation on S^ Let S2 be the quotient space Sj/EQUIV, the family of equivalence classes within §! under the induced equivalence. Let IVLS2 be the quotient group IVLS ! /CONG, whose members are the congruence classes within IVLS^ Then a function int2 from S2 x S2 into IVLS2 is well defined by the following method: Given equivalence classes p and q (members of S2), the value int 2 (p, q) is that congruence class (member of IVLS2) to which int1(s, t) belongs, whenever s and t are members of p and q respectively. Furthermore, (S 2 ,IVLS 2 ,int 2 ) is itself a GIS. Proof (optional): Lemma 3.2.2 assures us that int2 is well defined by the indicated procedure: Given p and q, if s and s' are any members of p, and if t and t' are any members of q, then int^s', t') belongs to the same congruence class as int^s, t). That congruence class thus depends only on the equivalence classes p and q, and not on the particular s-and-t, or s'-and-t', which we choose

34

2. The reader may prove as an exercise the necessary lemma: If x is congruent to e in a group, then x"1 is also congruent to e. After that, prove this: If x is congruent to y, then x"1 is congruent to y-1. (Hint: x"1 y is congruent to x"1 x; apply the preceding lemma. Alternatively, but less elegantly, one can bludgeon out all the desired results as corollaries of 1.11, 1.12.1, and related results already established.)

Generalized Interval Systems (2)

3.2.4

to represent p and q. And it is that congruence class which is thus well defined, asint 2 (p,q). We now prove that (S2, IVLS 2 ,int 2 ) is a GIS. To do so, we must show that Conditions (A) and (B) of Definition 2.3.1 are satisfied. We show first that (A) is satisfied. Given equivalence classes o, p, and q in S 2 , we want to show that int2(o, p)int2(p, q) = int2(o, q). Let r, s, and t be elements of Sj that are members of classes o, p, and q respectively. The congruence class containing inti(r, s) combines in the quotient group with the congruence class containing int^s, t), to yield the congruence class containing int^r, s)int1(s, t) (1.11). And, since (S^IVLS^intJ is a GIS, int^r, 8)111^(8,1)1 = int^M). Hence the congruence class containing int^r, s) combines in the quotient group with the congruence class containing int^s, t), to yield the congruence class containing intj(r, t). Or, by the definition of int 2 in the theorem, int 2 (o,p)int 2 (p,q) = int 2 (o,q), as desired. So the system (S 2 ,IVLS 2 ,int 2 ) satisfies Condition (A) of 2.3.1. It remains to show that the system also satisfies Condition (B) of that definition. Given any equivalence class p (member of S 2 ) and any congruence class J (member of IVLS2), we must show that there is a unique equivalence class q that satisfies the equation int2(p, q) = J. Let s be a member of the given p; let j be a member of the given J. Since (S:, IVLSi, intl) is a GIS, we can find a (unique) t in Sj which satisfies the equation int t (s, t) = j. Let q be the equivalence class containing this t. q is that class (member of S 2 ) for which we are searching. First of all, q satisfies the equation int2(p, q) = J. That is so because int^s, t) = j, s belongs to p, t belongs to q, and j belongs to J; int2 was defined precisely so as to make this happen. Furthermore, q is a unique solution for the equation int2(p, q) = J. To see this, let us suppose that int 2 (p,q') = J; we shall show that q' must equal q. Let t' be a member of q'; then by the nature of int 2 , in^ (s, t') lies in the congruence-class J. So intj (s, t') is congruent to int^s, t). Thence, applying the second lemma of footnote 2, we infer that int^s, t')"1 is congruent to int^s, t)"1. Or: int^t', s) is congruent to int t (t,s). Then int^t', s^nt^s,t) is congruent to int^t,s)int 1 (s,t). Or: inti(t', t) is congruent to int^t, t) = e. But then t' is equivalent to t under the induced equivalence relation: The interval between them is congruent to e. Hence t' and t lie in the same equivalence class; that is, q' = q as asserted, q.e.d. 3.2.4 DEFINITION: Given the situation as in 3.2.3, the GIS(S 2 ,IVLS 2 ,int 2 ) will be called the quotient GIS of (S^IVLS^intJ modulo CONG. We write GIS2 = GIS1/CONG. Thus, to review yet once more the specific example discussed just preceding 3.2.1, let (S l5 IVLS15 int x ) be the GIS such that S t is an infinite chromatic scale, IVLSj is the additive group of integers, and int x (s, t) is the number of

35

3.2.4

36

Generalized Interval Systems (2)

semitones up from s to t. Let CONG be the relation on IVLSj that makes i congruent to i' when the intervals differ by any integral multiple of 12 semitones. Then the quotient GIS(S2, IVLS2, intj), constructed by the method of section 3.2, has these components: S2 is the family of twelve pitch-classes, IVLS2 is the integers modulo 12, and int^p, q) is the reduction modulo 12 of the integer intj(s, t), where s and t are any pitches belonging to the pitch classes p and q respectively. Here is another specific example. S, is the family of all pitches that can be conceptually generated from a given pitch using just intonation. IVLSj is the group under multiplication of all rational numbers that can be written in the form 2a3b5c, where a, b, and c are integers. int,(s, t) is the fundamental frequency of t divided by the fundamental frequency of s. (Sp FVLSj, intj) is a GIS. It was studied earlier in example 2.1.5. When we informally "modularized" that GIS to obtain the "modular harmonic space" of example 2.1.6, we were actually constructing a quotient GIS, using a certain congruence relation on IVLSj. Specifically, we declared intervals i and i' within IVLSj to be congruent if i' was some-power-of-2 times i. As we noted earlier (in example 1.10.4.2), this relation is indeed a congruence on IVLSj: It is an equivalence relation; further, if I' is some-power-of-2 times I and j' is some-power-of-2 times j, then i'j' is some-power-of-2 times ij. Using this congruence, we can note how the formal constructions of section 3.2 go through for the GIS at hand, producing the GIS for example 2.1.6 as a quotient GIS. By Definition 3.2.1, the induced equivalence on Sj makes s equivalent to s' if and only if intj(s,s') is some-power-of-2. Thus s and s' enjoy the induced equivalence relation here if and only if the pitches lie some number of octaves apart. The equivalence classes (members of S2) are thus precisely the pitch classes determined by the pitches of S,. And, given pitch classes p and q, int^p^) in the quotient group is well defined as the congruence class to which intj(s, t) belongs, whenever s and t are members of p and q. If the intervals, i, i', i", . . . all belong to this congruence class, then we can write i = 2a3b5c, i' = 2a'3b5c, i" = 2a"3b5c, ... and so on. So we may identify the congruence class with the number-pair (b, c). When s and t are members of p and q, then t, give or take some number of octaves, lies b twelfths and c majorseventeenths from s. Taking some octaves, we can say that t, give or take some number of octaves, lies b fifths and c major-thirds from s. Returning our attention to the pitch classes p and q, of which s and t are members, we can say that q lies b dominants and c mediants from p. That is, on the game board of figure 2.2, q lies b steps east and c steps north from p. The quotient GIS here is thus the GIS associated with that game board in example 2.1.6. As a further specific example, the reader can work out how the formal constructions of section 3.2 apply to the "diatonic scale" GIS associated with example 2.1.1, together with the congruence which collects octave-related intervals (those here which differ by multiples of 7 scale steps), to give rise to a

Generalized Interval Systems (2)

3.3.1

quotient GIS which is in fact the GIS of example 2.1.4, a GIS of seven scaledegrees. In similar fashion the GIS associated with the time-point space of example 2.2.1, together with the congruence which collects time intervals differing by multiples of "N beats," gives rise to a quotient GIS which is in fact the GIS of example 2.2.2, the GIS associated with the space of N beat-classes. Likewise, the GIS of example 2.2.3, together with a suitable congruence, gives rise to the GIS of example 2.2.4 as a quotient GIS; the discussion of example 2.2.4 may be reviewed in this connection. The reader may also review further the discussion of example 2.1.6, to see how the GIS of just pitch-classes, corresponding to the game board of figure 2.2, can give rise to a quotient GIS whose space is an infinite one-dimensional chain of dominant-related pitch classes in quarter-comma mean-tone temperament. The congruence at issue declares two "moves" (b, c) and (b', c') on the game board of figure 2.2 to be congruent if there is some integer N such that b' = b + 4N while c' = c — N, that is, if b' steps east is the same as b steps east plus 4N steps east, while c' steps north is the same as c steps north and then N steps south. ( — N east, north, etc. means N west, south, etc.) If we take b = 0, c = 0, and N = 1 in the above arithmetic, then b' = 4 and c' = — 1. Thus going four squares east and one square south on the game board is a move congruent to "staying still." The reader may verify that the defined relation is a congruence, and that the induced equivalence relation on the game board renders equivalent just those pitch classes that share the same letter-name (differing, if at all, only by subscript). Each of the specific examples above instances, in one setting or another, the general and abstract relation of a given GIS, modulo a given congruence, to a quotient GIS. We can see thereby how ubiquitous the quotient construction really is, as a method of generating new GIS structures from old. Another useful method is the construction of a "direct-product GIS." Before proceeding to an abstract discussion of that idea, let us study some examples of it. 3.3.1 EXAMPLE: Let GISj be the GIS of example 2.1.3: Sj is the twelve equally-tempered pitch classes; IVLSi is the integers modulo 12; int^p, q) is the number of semitones up, mod 12, from any pitch within class p to any pitch within class q. Let GIS2 be the GIS of example 2.2.1: S2 is an indefinite series of equally spaced time-points; IVLS2 is the integers under addition; int2(s, t) is the number of beats from time-point s to time-point t. We construct a new GIS, GIS3, as follows. S3 is the Cartesian product of S x and S2, that is, S3 = Si x S2. S3 is thus the family of ordered pairs (p, s), where p is a pitch class in Sj and s is a time point in S2. IVLS3 is the direct-product group of the groups IVLS! and IVLS2; that is, IVLS3 = IVLSj (g) IVLS2. A sample member of IVLS3 thus has the form (ij, i 2 ), where ij is an integer mod 12 (member of IVLSj) and i 2 is an integer (member of IVLS2). The members of IVLS3 combine under the law of 1.13 for direct-

37

3.3.1

Generalized Interval Systems (2)

product groups: (ilt\2) + (JiJ 2 ) = Oi + Ji,i 2 + J2>; here the sum h + Ji is computed mod 12 in IVLS l9 while the sum i 2 + J 2 is the ordinary integer sum in IVLS2. The function int3 works as follows. Given two members (p, s) and (q, t) of S3, each a pitch-class-cum-time-point pair, the interval int3((p, s),(q, t)) is taken as the pair (int 1 (p,q),int 2 (s,t)) within the group IVLS 1 (g)IVLS 2 = IVLS3. (S3, IVLS3, int 3 ) as constructed is in fact a GIS. The reader may take this on faith or verify it as an exercise. Later on we shall call this GIS the direct product GIS of GlSi and GIS2, writing GIS3 = GISj ® GIS2. Right now, let us explore what GIS3 models. One sample member of S3 is the pair (C#, 35), which models a reference to pitch class C# at time 35. Another member of S3 is the pair (F,46), which models a reference to pitch class F at time 46. The interval between the two cited objects is (int^Cft, F),int 2 (35,46)) = (4,11). The compound interval (4,11) models the spanning of pitch-class interval 4 between events happening at time-points 11 beats apart.

FIGURE 3.1

38

Figure 3.1 will be used to suggest the relevance of GIS3 for musical analysis and theory. It transcribes the opening of the third movement from Webern's Piano Variations op. 27. The brackets on the figure display some GIS3-intervals of interest, using the written quarter note as the temporal unit. Temporal intervals between time-points are calculated between attacks of the notes. Thus the bracket from E|? to D on figure 3.1 is labeled by the GIS3interval (11,5): A pitch-class interval of 11 is spanned between the attack of the Eb and the attack of the D, 5 beats later. Similarly, the bracket on the figure between D and C# is labeled by the GIS3-interval (11, 5): A pitch-class interval of 11 is again spanned between two time-points 5 beats apart, here between the attack of the D and the attack of the C#. The GIS3-interval (11,1) labels the brackets extending from B to B[? attacks, from C# to C attacks, and from A to G# attacks; in each case, the pitch-class interval 11 is spanned between events happening 1 beat apart. The recurrences of GIS3-intervals on figure 3.1 are of analytic interest,

Generalized Interval Systems (2)

3.3.1

for not many GIS3-intervals occur more than once in this passage. The recurrent GIS3-interval (11, 1) is associated with a quiet slurred figure in the left hand which we shall call "the accompaniment figure"; this figure occurs in the music at B-B|> and again A-G#. However the musical presentation of C#-C, also associated with the GIS3 interval (11, 1), does not project the accompaniment figure; rather, it is loud and staccato. The recurrence of the GIS3 interval (11, 1) on figure 3.1 imbues the pitch-class interval 11 with a special function: As the passage unfolds in time, pitch-class interval 11 becomes bound up with defining the beat. That is, pitch-class interval 11 recurs significantly in connection with beat-interval 1, forming the GIS3-interval (11, 1). And no other pitch-class interval recurs in conjunction with beat-interval 1. This special beat-defining function for pitch-class interval 11 gives special meaning in turn to the recurrence of GIS3-interval (11, 5) on figure 3.1, a GIS3-interval which also involves pitch-class interval 11. Via the recurrent GIS3-interval (11, 5), the temporal interval of 5-beats-later becomes associated with the pitch-class interval 11, a pitch-class interval of special mensural status. "5 beats later" thereby acquires a special mensural status itself, linking it with "1 beat later" via the pitch-class interval 11 that is shared by the recurrent GIS3-intervals (11, 5) and (11, 1). This special mensural status for "5 beats later" is not the only reason many analysts hear the music "in * meter," but it does endow the possibility of such a hearing with a special meaning and thematic richness.3 I myself believe that pertinent statements involving GIS3-intervals provide a more exact and less problematic account of the mensural structures at issue here, then does the notion of "^ meter." Figure 3.1 also shows the recurrence of the GIS3-interval (3, 2), spanning B-attack to D-attack and also E-attack to G-attack. The recurrent (3, 2) seems to have a cadential function in the music, to my ear, "2 beats later," the "2" of the GIS3-interval (3, 2), engages the notated meter of the music, which "5 beats later" does not. In these connections, the pitch-class interval 3, as it recurs within the GIS3-interval (3, 2), has its own special function, to my ear a cadential function. Hearing a cadence on the D and another on the G is consistent with the maximal amount of time we wait, after the attack of the D and again after the attack of the G, before hearing the next attack in each case. It is curious that the next attack after this maximal wait comes precisely "5 beats later" in each case. Indeed if we start measuring spans of 5 beats starting 3. A \ "cross-meter" is heard as one of several contending temporal patterns by Edward T Cone, "Analysis Today," Problems of Modem Music, ed. Paul Henry Lang (New York: W. W. Norton, 1962), p. 44. Elsewhere, we can read that the first three notes we hear form a rhythmic group which "is clearly *; the motion of the two quarter notes into the next downbeat defines the meter precisely." This is the hearing of James Rives Jones, "Some Aspects of Rhythm and Meter in Webem's Opus 27," Perspectives of New Music vol. 7, no. 1 (Fall-Winter 1968), p. 103. Later, still on page 103, he refers to "the \ meter" as "already ... established" by the cited criterion.

39

3.3.1

Generalized Interval System (2)

at the attack of the opening El., then the D and the G on figure 3.1, with the rests that follow them, each fill one such span. I do not know what to make of this. The idea of quintuple "perfections" seems a better metaphor for my hearing this aspect of the music than does the idea of quintuple "meter." Figure 3.1 also shows the recurrent GIS3-interval (2, 7), interlocking the mensural function of "7 beats later" with the pitch-class interval 2. The first (2, 7) recurrence links the two notes of the B-B[> accompaniment figure to the corresponding two notes of the C|-C figure, 7 beats later. The second recurrence (third occurrence) of (2, 7) links the attack of the cadential G to the beginning of the accompaniment figure A-G| 7 beats later. In sum, the recurrent GIS3-interval (2, 7) links aspects of the recurrent accompaniment figure with other events of the music.

FIGURE 3.2

40

Figure 3.2 summarizes the discussion of figure 3.1 so far, in the form of a table. It shows how the recurrence of GIS3-intervals gives special meanings to the pitch-class intervals 11, 3, and 2, as those interrelate with the temporal

Generalized Interval Systems (2)

3.3.1

intervals 1,5,2, and 7, all in connection with various compositional features of the music. 83-elements

GIS3-interval vectors of S3 sets

FIGURE3.3 Figure 3.3 applies to the opening of the passage a theoretical construction suggested by the temporal aspect of GIS3. The left-hand column of the figure lists the first six members of S3 in their order of appearance during the passage. First comes pitch-class Eb at time-point 0, instancing element (Eb,0) of S3. Next comes pitch-class B at time-point 3, instancing element (B, 3) of S3. At this time (i.e. just after time-point 3), we become aware of a 2-element S3-set, that is, the set ((Eb, 0), (B, 3)). The elements of this set form one GIS3-interval with a positive time-component, that is, the interval (8,3) from (Eb,0) to (B, 3). The GIS3-interval (8,3) is listed in the second row of the second column in figure 3.3; that interval is the sole constituent within the interval vector of the 2-element S3-set. Now the pitch class 8(7 occurs at time-point 4, providing the new S3 element (Bb,4) for the third row of column 1 on figure 3.3. At this time (i.e. just after time-point 4), we become aware of a 3-element S3-set, ((Eb,0),(B,3),(Bb,4)). Besides the GIS3-interval of (8,3) already listed in row 2, the elements of the 3-member set produce new GIS3-intervals of (7,4) (= int 3 ((Eb,0),(Bb,4))) and (11, !)(= int 3 ((B,3),(Bb,4))). These new constituents for the interval vector of the expanded 3-element S3-set are listed in the third row of column 2 on figure 3.3; the old interval vector expands to include the occurrences of the two new intervals. When pitch class D enters at time-point 5, the 3-element S3-set expands to a 4-element S3-set ((Eb,0),(B,3),(Bb,4),(D,5)), and the interval vector of the 3-element set expands to adjoin new occurrences of the GIS3-intervals (11,5)(= int 3 ((Eb,0),(D,5))), (3,2)(= int 3 ((B,3),(D,5))), and (4,1) (= int 3 ((Bb,4),(D, 5))). The new intervals are listed in the fourth row of column 2, on figure 3.3. At this time (i.e. just after time-point 5), we become

41

3.3.1

Generalized Interval Systems (2)

aware for the first time that some pitch-class intervals are predominating over others, and that some temporal intervals are predominating over others, as we note the various intervals going by. In earlier work I have suggested that our becoming aware of such predominances is associated with our marking such a time-point as an "ictus."4 The present analysis supports that theoretical idea, since time-point 5 is both a notated barline, indeed the first notated barline, and also audible to some extent as marking the attack of an intuitively "strong" quarter. These considerations lie behind my bracketing the first four entries of column 1 on figure 3.3, as belonging together in a special way. According to the theory just sketched, it is only when the 4-element S3-set has been completely exposed, i.e. it is at and only at a moment just after time-point 5, as we listen along, that we first become sensitive to any regular mensural structuring in the passage. Just after time-point 5 we become aware that temporal interval 1 is predominating over other temporal intervals; we can then (and only then) hear temporal interval 1 as a beat with which to measure other temporal intervals. In that sense the GIS3-structure of figures 3.1-3.3 really "begins" for a listener at (and only at) time-point 5, the first written barline and the first perceived ictus; any GIS3-structure preceding time-point 5, according to this theory, is reconstructed^ a listener at (or following) time-point 5. Not only is the beat established at time-point 5, the pitch-class interval 11 is also established at the same time, as a predominating pitch-class interval. Pitch-class interval 11 is thus bound psychologically to the establishment of mensural structure in the piece, as part of one and the same Gestaltist experience that a listener will be having just after time-point 5.

FIGURE 3.4

Figure 3.4 symbolically collates the ideas discussed just above. The figure also suggests how the temporal interval of "5 beats later" has already acquired a special significance at the moment the listener hears time-point 5. "5 beats later," namely, spans the temporal distance from the opening attack to the first ictus. Figure 3.4 shows how that temporal distance is already clearly associated with the structural function of the recurrent pitch-class interval 11,

42

4. David Lewin, "Some Investigations into Foreground Rhythmic and Metric Patterning," Music Theory, Special Topics, ed. Richmond Browne (New York: Academic Press, 1981), 101-37.

Generalized Interval Systems (2)

3.3.1

the pitch-class interval between the opening Eb and the D of the first ictus, 5 beats later. These considerations enable us to analyze with greater precision how the idea of "being in 4" might arise, and how it would become associated with the compositionally thematic E(?-D gesture.5 Let us now inspect the fifth row of figure 3.3, investigating how our impressions develop when C# is attacked at time-point 10, introducing new manifestations of the GIS3-intervals (10,10), (2,7), (3,6), and (11,5). We now hear a second pitch-class interval of 3, but simultaneously we also hear a third pitch-class interval of 11. The latter interval, by recurring yet again, continues to predominate over other pitch-class intervals. Indeed, its predomination itself recurs. Furthermore, we now (i.e. just after time-point 10) hear for the first time the recurrence of a GIS3-interval (not just of a temporal interval or pitch-class interval). The recurring GIS3-interval is (11,5), recently discussed in connection with the possibility of asserting a "thematic | meter" listening at time-point 5. The sensations prompting such a possible assertion are thereby intensified at time-point 10. Time-point 10 is experienced as an "ictus" in the formal sense of the theory mentioned earlier. Dynamic and registral accents at time-point 10, the loud low C# attack, support the possible hearing of a "strong beat" there, should one want to assert "| meter" beyond purely mensural considerations. We can then expand figure 3.4 to figure 3.5, which portrays a provisional impression one might have while listening just after time-point 10.

FIGURE 3.5

Now let us turn to the sixth row of figure 3.3, investigating our impressions when C is attacked at time-point 11, introducing new manifestations of the GIS3-intervals (9,11), (1,8), (2,7), (10,6), and (11,1). (11,1) here is a recurring GIS3-interval; it confirms our already developed sensations about the beat-defining and other mensural functions of the pitch-class interval 11. The GIS3-interval (2,7) also recurs at time-point 11. This builds up another 5. My analysis of listener psychology just after time-point 5 partly elaborates, partly qualifies, and partly takes issue with the thought-provoking approach to this passage by Christopher Hasty, in his important methodological and analytic study, "Rhythm in Post-Tonal Music: Preliminary Questions of Duration and Motion," Journal of Music Theory vol. 25, no. 2 (Fall 1981), 183-216.

43

3.3.1

Generalized Interval Systems (2)

mensural matrix that tries to expropriate (11,1) for its own purposes, trying to put an ictus at time-point 11, rather than time-point 10. Figure 3.6 sketches this notion.

FIGURE 3.6 The ictus on C in figure 3.6 corresponds to a written barline; this was not the case with the conflicting ictus on C# in figure 3.5. That C# was 5 beats after the D-ictus of figure 3.4; the C of figure 3.6, which picks up the Hauptstimme D in register, is 6 beats after the D-ictus. The mensural conflict of "5 units after" (D-ictus to C#-ictus) and "6 units after" (D-ictus to C-ictus) is highly thematic in op. 27 as a whole.6 The mensural reading of figure 3.6 tries to associate the C#-C event in the music with the "accompaniment figure," the figure that projected the B-B[? event. In contrast, the mensural reading of figure 3.5 tried to associate the D-C# descent with the thematic E[?-D descent. Figure 3.5, of course, "did not know about" the forte and staccato C natural coming up right after the forte and staccato C#. We have already discussed how the recurrent GIS3-interval (2,7) interacts with the accompaniment figure more generally. This completes the exegesis of figure 3.3. The theoretical notion of an "unfolding interval vector," made abstractly available by the temporal aspect of GIS3, was analytically useful for examining our impressions of figure 3.1 as those developed note-by-note, and for discussing to a significant extent our impressions of the music beyond that. GIS3 was particularly useful because it enabled us to consider pitch-class structure and mensural rhythmic structure in conjunction with each other, rather than as independent features of the passage. That is the essence of GIS3 in its capacity as what we shall soon call the direct product of GISt and GIS2.

44

6. A conflict between mensural distances of 5 and 6 units figures in the relation of the rhythmic ostinato to the written meter at the opening of the first movement. A reprise of this rhythmic conflict occurs at the opening of the coda in the last movement (mm. 56-62). The written meters are functional, in ways too complex to indicate here. Special accents attach to the loud and dense trichord-pairs of the middle movement. The lower chords of the trichord-pairs attack at the barlines of measures 4, 9, 4 bis (6 measures after 9), 9 bis, 15, 20, 15 bis (6 measures after 20), and 20 bis. The resulting pattern projects alternating spans of 5 and 6 written measures. Indeed this may well be the strongest mensural function for the written measure as a temporal span in the movement.

Generalized Interval Systems (2)

3.3.3

3.3.2 EXAMPLE: Let GISj be the GIS involving time-points, that just figured as "GIS2" in the preceding example. For the present example, let GIS2 be a GIS of durations as in 2.2.3 earlier: S2 is a certain family of "durations" x, y,... related by certain stipulated proportions; IVLS2 is the multiplicative group of such proportions; int2(x, y) is the quotient y/x. We construct a new GIS, GIS3, as follows. S3 is the Cartesian product S x x S2, that is, the family of pairs (s,x), where s is a time-point and x is a duration. We can conceive (s, x) as modeling an event that begins at time-point s and extends for a duration of x (units) thereafter. IVLS3 is the direct-product group IVLSj (8) IVLS 2. Each member of IVLS3 is a pair (il5i2), where ilis a member of IVLSj (representing a number of beats between time-points) and i2 is a member of IVLS2 (representing a quotient of durations). (i ls i 2 ) and (Ji,j 2 ) combine in the direct-product group IVLS3 according to the rule (i 1 ,i 2 )(Ji»J2) = fli + Ji^zJa)- int3(( s » x )>(t,y)) is defined as (int^s,!), int2(x,y)), that is, loosely speaking, as (t — s,y/x). To put it another way, if int3((s, x), (t, y)) = Oi, i2), then event (t, y) begins i t units of time after event (s, x) and extends for i2 times the extent of event (s, x). The reader can take it on faith that GIS3 as constructed above is indeed a GIS. It will not be necessary to produce an analytic example, I think, in order to convince the reader that this GIS is a useful theoretical tool. It combines two aspects of our mensural rhythmic intuition, as they impinge upon us conjointly, into one compound structure. We are still and again presuming a fixed unit of time here, by which we measure durations and distances-betweentime-points. In both example 3.3.1 and example 3.3.2 we combined a given OK^ with a given GIS2 in a certain manner, obtaining a new GIS that modeled the conjoint action of the two given GIS structures. We can make our procedure formal and general, as a means of combining any given GlSi and any given GIS2 into a "direct-product GIS." The following definition gives the procedure. 3.3.3 DEFINITION: Given G^ = (S l5 IVLS^intJ and GIS2 = (S2, IVLS2, int2), the direct product of GISa and GIS2, denoted GIS1

45

3.4.1

Generalized Interval Systems (2)

It is straightforward to verify that GIS3, as defined above, is indeed a GIS, i.e. that GIS3 satisfies Conditions (A) and (B) of Definition 2.3.1. This finishes our investigation into methods of constructing new GIS structures from old in a general abstract setting. Now we shall see how the notions of "transposing" and "inverting" elements of a space S arise naturally in any GIS; we shall see how operations of transposition and inversion interrelate characteristically with intervallic structure, and we shall explore how the operations combine among themselves. We shall also explore other characteristic operations, the "interval-preserving operations"; these may or may not be the same as the transposition operations, depending upon whether the group of intervals is or is not commutative.7 3.4.1 DEFINITION: Given a GIS; given an interval i of IVLS; then transposition by i, denoted Tj, is defined as a transformation on S as follows. Given a sample element s of S, the i-transpose of s, T;(s), is that unique member of S which lies the interval i from s. That is, T;(s) is well defined by the equation int(s,T,(s)) = i. This definition conforms to our abstract intuition that the i-transpose of a given element s should lie the interval i from s. The definition also conforms to the way in which we already use the word "transposition" in connection with some GIS structures, namely those involving pitches and pitch classes. T;(s) is indeed well defined by the equation within the definition. Condition (B) of 2.3.1 assures us that given i and s, there exists a unique t such that int(s, t) = i; it is precisely this t which we are now calling "the i-transpose of s," Tj(s). 3.4.2 THEOREM: Each Ts is an operation; that is, it is 1-to-l and onto as a transformation on S. The transposition operations form a group of operations on S. That group is anti-isomorphic to the group of intervals. Specifically, let us consider the map f, defined from IVLS onto the family of transpositions by the formula f(i) = T^ then f is an anti-isomorphism. That is, f is 1-to-l as well as onto; and TjTj = T^. Pr00/(optional): We shall prove the assertions of the theorem in an order different from that in which the theorem states them. First we show that f is an anti-homomorphism, i.e. that TjTj = T^. Given intervals i and j, given a sample s in S, then we write int(s, Tj(Tj(s))) as the group product of the two intervals int(s, Tj(s)) and int(Tj(s), T;(Tj(s))), using Condition (A) of 2.3.1. Now int(s, Tj(s)) = j, by the defining equation of Definition 3.4.1. And by the same equation, int(Tj(s), Tj(Tj(s))) = i. Hence 46

7. All the groups in our specific examples of GIS structure so far have been commutative. Later on, in chapter 4, we shall study a non-commutative GIS of musical interest.

Generalized Interval Systems (2)

3.4.4

int(s, Ti(Tj(s))) is expressed as the group product of the two intervals j and i. That is, int(s,Ti(Tj(s))) = ji. Thus Tj(Tj(s)) lies the interval ji from s. So it is equal to TJJ(S). We have shown: For any sample s, TjTj(s) = TJJ(S). So the transformation TjTj has the same effect on S as does the transformation T^; the functional equation TjTj = T^ is true, as claimed. So the map f is an anti-homomorphism. We show now that it is an antiisomorphism. We have only to show that f is a 1-to-l map. Supposing that the functional equation Ts = Tj is true, we wish to prove that i and j must be the same interval. Fix any s. Since Ts = Tj by assumption, Tj(s) = Tj(s). Then, by Definition 3.4.1, i = int(s,Ti(s)) = int(s,Tj(s)) = j, which is what we had to show. It remains only to prove that each Tj is an operation, and that the family of transpositions is a group of operations. To prove all this, it suffices to show that the family of transpositions satisfies Conditions (A) and (B) of 1.3.4 earlier, namely (A) that the family is closed and (B) that for each Tj there is a TJ satisfying TjTj = TjTj = 1. Condition (A) is evident from the formula TjTj = Tjji The composition of two transpositions is a transposition. In order to prove Condition (B), we shall prove a lemma: Te is the identity operation 1 on S. That is true since, given s in S, Te(s) is the member of S which lies the identity interval from s; hence Te(s) = s. Or Te(s) = l(s); that being the case for any sample s, Te = 1 as asserted by the lemma. Now we can prove Condition (B). Given any interval i, take j = i"1. Then ji = ij = e. Tj is the transformation demanded by Condition (B): TjTj = Tjs = Te = 1; likewise TjTj = 1. q.e.d. 3.4.3 THEOREM: Fix some referential member ref of S; then LABEL(Ti(s)) = LABEL(s) • i. Proof: LABEL(Ti(s)) = int(ref, T,(s)) by definition of LABEL in 3.1.1, = int(ref, s)int(s, Ti(s)) by 2.3.1 (A), = LABEL(s)int(s, Tj(s)) by 3.1.1, = LABEL(s)-i by 3.4.1. Theorem 3.4.3 tells us that no matter what ref is chosen for LABELing purposes, the label for the i-transpose of s is the label of s, right-multiplied by i in IVLS. The natural question arises: What happens when we left-multiply ref-LABELs by i? We shall explore that right now. 3.4.4 DEFINITION: Fix some referential member ref of S. Given any interval i, the transformation Pj (more exactly P[ef)is defined on S by the formula LABEL(Pj(s)) = i • LABEL(s), which is to say the relation int(ref, Pi(s)) = i • int(ref, s).

47

3,4.5

Generalized Interval Systems (2)

Given ref, i, and s, Condition (B) of 2.3.1 tells us that a unique member of S, call it t, satisfies the relation int(ref, t) = i • int(ref, s). Definition 3.4.4 takes that unique t, given ref, i, and s, and calls it Pj(s). Note that the specific value of Pj(s) depends on ref as well as on i and s. In contrast, the value of T.(s) was well defined in 3.4.1 independent of any choice of ref. 3.4.5 THEOREM: The transformations P. form a group of operations isomorphic to IVLS under the map f(i) = P}. In particular, the formula PjP. = Pr is valid. Proof (optional): LABEL(Pj(p.(s))) = i • LABEL(P.(s)) = i • (j • LABEL(s)) = (ij) • LABEL(s) = LABEL'S)). The elements Pj(Pj(s)) and Py(s) thus have the same LABEL (lie the same interval from ref). So they are the same: PjPj(s) = Pjj(s). This being the case for any sample s, the functional equation PjP. = P.. is true. The map f of the Theorem is thereby a homomorphism of IVLS onto the family of transformations P.. To prove that f is an isomorphism, it remains only to show that f is 1-to-l. Suppose P. = P.; we are to infer that i = j. Fix any s; then i • LABEL(s) = LABEL(pj(s)) = LABEL(Pj(s)) by supposition; and that = j • LABEL(s). In sum, i • LABEL(s) = j • LABEL(s). Hence, multiplying that equation through on the right by the inverse of LABEL(s), i = j as desired. We can now use arguments just like those we used in the proof of 3.4.2, to show that the family of P. is a group of operations. The operations P. have a special property. They are what we shall call the "interval-preserving" transformations. 3.4.6 and 3.4.7 develop the formalities. 3.4.6 DEFINITION: Given a GIS (S, IVLS, int), a transformation X on S will be called "interval-preserving" if X has this property: For each s and each t, int(X(s), X(t)) = int(s, t). 3.4.7 THEOREM: No matter what ref is chosen, the interval-preserving transformations on S are precisely the P.. Proof (optional): We show first that P. preserves intervals. We can write inters), P,(t))

=LABEL(P i(s))-1LABEL(Pi(t)) by 3.1.2, =(i • LABELS(s))-!(i • LABEL(t)) =(LABEL(s)~1 • i-])(i • LABEL(t)) =LABEL(s)-1LABEL(t) =int(s, t) 48

by 3.4.4,

by 3.1.2.

Thus P. preserves intervals. Now suppose X is any interval-preserving

Generalized Interval Systems (2)

3.4.8

transformation on S. We shall show that there exists some interval i such that X-P,. The i we want here is LABEL(X(ref)) = int(ref, X(ref)). Given any s, we can then write LABEL(X(s)) = int(ref, X(s)) = = = = =

int(ref,X(ref))int(X(ref), X(s)) via 2.3.1 (A), i • int(X(ref), X(s)) by construction of i here, i • int(ref, s) because X is interval-preserving, i • LABEL(s) by the definition of LABEL, LABEL(Pi(s)) via 3.4.4.

In sum, we have LABEL(X(s)) = LABEL(Pj(s)). Since X(s) and Ps(s) have the same LABEL, we infer that X(s) = Pj(s). Since s here is an arbitrary sample member of S, X = Pi as a transformation on S. Thus our intervalpreserving X is in fact this particular P{. q.e.d. Because of 3.4.7 we can conceive the transformations P5 as somewhat less dependent on the choice of ref. It is true that a transformation labeled "Pj" by one choice of ref might be labeled "Pj" by another choice of ref. However the interval-preserving transformations en masse, literally "as a group," remain the same family of transformations en masse, no matter what ref is chosen. The interval-preserving property does not depend on the choice of ref, and that property is sufficient to define the family of transformations as a group. 3.4.8 THEOREM: Given an interval i, Conditions (A) through (D) below are all logically equivalent: If any one of them is true, then they are all true. (A): T; preserves intervals. (B): For some choice of ref, Tj = P}. (C): For any choice of ref, Tj = Pj. (D): i is central in IVLS. That is, i commutes with every j in IVLS (1.8.2). Proof (optional): Suppose (A) is true; we prove that (C) follows. Fix any ref. Since Tt preserves intervals by assumption, 3.4.7 tells us that Tj = Pj for some j. For any s, LABEL(s) • i = LABEL(T;(s)) by 3.4.3 that = LABEL(Pj(s)) since T, = Pjj and that = j • LABEL(s) by 3.4.4. In sum, LABEL(s) • i = j -LABEL(s) for any s. Consider s = ref in particular LABEL(ref) • i = j • LABEL(ref). But LABEL(ref) = int(ref, ref) = e. Henc i = j. Tj = Pj as desired. Clearly the truth of (C) entails the truth of (B). Now suppose (B) is true; we prove that (D) follows. We are supposing Ts = Pj for some ref. Then for every s, LABEL(Tj(s)) = LABEL(Pj(s)). Then for every s, LABEL(s) • i = i • LABEL(s) (3.4.3 and 3.4.4). It follows that i commutes with every j, which is Condition (D) as desired. For given any j, find the s which lies the interval j from ref; then LABEL(s) = j; substituting j for LABEL(s) in the most recent equation involving i, we obtain ji = ij; i commutes with the given j.

49

3.4.9Generalized Interval Systems (2)

Now we close the logical chain by showing that the truth of (D) entails the truth of (A). When we have done this, we shall have shown that (A) implies (C), (C) implies (B), (B) implies (D), and (D) implies (A); hence the truth of any one entails the truth of all the others. Supposing (D) is true, then, we show that (A) will be true. Fix any ref. Given any s and any t, we write int(Ti(s), Tj(t)) = LABEL(Ti(s))-1LABEL(Ti(t)) = (LABEL(s) • i)~1(LABEL(t) • i) = (r1 • LABEL(s)'1) (LABEL(t) • i) = i"1 • (LABEL(s)-1LABEL(t)) • i = i - 1 -mt(s,t)-i = int(s, t)

by 3.1.2 by 3.4.3

by 3.1.2 by the assumption

of Condition (D), that i is central. In sum, assuming Condition (D), then int(T;(s), T;(t)) = int(s, t) for every s and t. Or, assuming Condition (D), Tj will be interval-preserving. Or: (D) implies (A) as desired, q.e.d. 3.4.9 COROLLARIES: (A): In a commutative GIS (a GIS whose group of intervals is commutative), the transposition operations are precisely the interval-preserving operations. (B): In a non-commutative GIS, there exist transpositions that do not preserve intervals, and there exist interval-preserving operations that are not transpositions. 3.4.10 THEOREM: Any transposition operation commutes with any intervalpreserving operation. Proof: Fixing any ref, consider T; and Pj. We apply 3.4.3 and 3.4.4 to various LABELS. Taking any sample s, we have LABEL(Pj(Ti(s))) = j • LABEL(Tj(s)) = j • (LABEL(s) • i) = (j • LABEL(s)) • i = LABEL(Pj(s)) • i = LABEL(Tj(Pj(s))). Thus PjTj(s) and T,Pj(s) have the same LABEL; they lie the same interval from ref; so PjTj(s) = TjPj(s). This being the case for any sample s, the functional equation PjTj = TjPj is true: Ts commutes with Pr q.e.d.

50

We are now ready to define and study "inversion operations" on an abstract GIS. For each u and each v in S (v may possibly equal u), we shall define an operation I*, which we shall call "u/v inversion." Figure 3.7 helps us visualize an appropriate definition for the operation, conforming to our spatial intuitions. The figure shows how we conceive any sample s and its inversion I(s) (short for !„($) here) as balanced about the given u and v in a certain intervallic proportion. I(s) bears to v an intervallic relation which is the inverse of the relation that s bears to u. The inverse proportion is symbolized by the mirror

Generalized Interval Systems (2)

3.5.2

FIGURE 3.7 relation of the two arrows on figure 3.7. The interval from v to I(s), which is the inverse of int(I (s), v), will then be the same as the interval from s to u. That is, we intuit int(v, I(s)) = int(s, u). We can use this equation to define !„ formally in any GIS. 3.5.1 DEFINITION: Given any u in S and any v in S, the operation !„ of u/v inversion is defined by the equation int(v, Iu(s)) = int(s, u) for all s. The operation is well defined by the equation: Given any s, set i = int(s, u) and find the unique t which lies the interval i from v. That t, which satisfies the equation int(v, t) = int(s, u), is precisely the value for IJ^s). We have been referring to the "operation" I* prematurely; so far we have constructed a transformation, but we have not verified that the transformation is indeed an operation, i.e. onto and 1-to-l. The reader may verify, as an exercise, that I* as defined is onto and 1-to-l. (Given t, find an s such that ru(s) = t. Prove that if IJ(s') = IJ(s), then s' = s.) 3.5.2 THEOREM: Fix a referential element ref of S. Set i = LABEL(v) and j = LABEL(u). Then LABELTOs)) - i • LABEL(s)'1 • j. Proof: To save space, we write "I" for "!„" here. int(v, I(s)) = int(s, u) (3.5.1). So int(v, ref )int(ref, I(s)) = int(s, ref )int(ref, u) (2.3.1 (A)). Thence, LABEL(v)'1LABEL(I(s)) = LABEL(s)~1LABEL(u) (3.1.1; 2.3.2) Or: Or:

T1 • LABEL(I(s)) = LABEL(s)'1 • j LABEL(I(s)) = i • LABEL(s)"1 • j

q.e.d.

51

3.5.3

Generalized Interval Systems (2)

The formula of this theorem is very useful despite the dependence of all the LABELs (including i and j) on a possibly arbitrary ref. We shall use the formula to analyze this question: When are lvn and I* the same operation on S? In the familiar GIS of twelve chromatic pitch classes, for instance l£ = l£{ = IA' = ID» = IB'anc* so on- '^iat *s' ^^ mversi°n nas the same effect on any sample pitch class as does F|/F# inversion, or A/Df inversion, or DJt/A inversion, or B/C| inversion, and so on. In this special GIS one can intuit that l£ and I* will be the same operation if and only if the pitch class w is the C/C inversion of the pitch class x. One might thereby conjecture that I* and I* will be the same operation, in a general GIS, if and only if the thing w is the u/v inversion of the thing x. This conjecture is valid if the GIS is commutative. When the GIS is not commutative, we must be content with the broader view provided by the following theorem. 3.5.3 THEOREM: I* = I* as an operation on S if and only if w = I*(x) and the interval int(x, u) is central. Proof (optional): Imagine ref fixed. Let i, j, k, and m be the respective LABELS for v, u, w, and x. Then, via 3.5.2, given any s, LABEL(I^(s)) = i LABEL(s)-1 j, while LABEL(I*(s)) = k LABEL(s)-1 m. Hence the condition that I* be the same operation as I* is equivalent to the condition that i LABEL(s)'1 j = k LABEL(s)-1 m for all s in S. Now as s runs through the various members of S, the inverse of its LABEL runs through all the various intervals in IVLS. Hence the condition under discussion is equivalent to the condition that inj = knm

for every n in IVLS.

And that condition is equivalent, via group theory, to Condition (A) below. (A): (k-1i)n = nOnj"1)

52

for every n in IVLS.

In sum, F[ = I* as an operation if and only if Condition (A) above is satisfied. Now we shall prove that Condition (A) is satisfied if and only if w = I*(x) and int(x, u) is central. That will prove the theorem as stated. Suppose, then, that Condition (A) is true; we shall show that w = IJJ(x) and int(x, u) is central. Condition (A) being true by supposition, it is true for n = e; therefore k~ ! i = mj"1. Let us call this interval c. Condition (A) then tells us that en = nc for every n in IVLS, so c is central. c was taken equal to mj"1. It turns out that c also equals j"1!!!. To see that, we write mj"1 = c; thence m = cj; thence, since c is central, m = jc; thence j-1m = c as asserted.

Generalized Interval Systems (2)

3.5.5

Let us review: Assuming Condition (A), we have so far shown that k 4 = mj-1 = j^m = c is central interval. Now k~M = LABEL(w)~1LABEL(v) = int(v, w), via 3.1.2. Likewise j-1m = int(x, u). Thus we have shown: int(v, w) = int (x, u) and int(x, u) = c is central. Now the relation int(v, w) = int(x, u) is exactly the relation which tells us that w = IJj(x) (Definition 3.5.1). Thus, assuming Condition (A), we have proved that w = I^(x) and int(x, u) is central as desired. Now we prove the converse half of the theorem: Supposing that w = I*(x) and that int(x, u) is central, we prove that Condition (A) is satisfied. w int(v, w) LABEL(w)-1LABEL(v) k~U

= = = =

I*(x) int(x, u) LABEL(u)~1LABEL(x) j-1m

(by supposition). So (3.5.1). Or: (3.1.2). Or: (meaning of i, j, k, m).

Furthermore, we have supposed that j^m, which is LABEL(u)~1LABEL(x), which is int(x, u), is a central interval. Let us call this central interval c. j^m = c; thence m = jc; thence m = cj; thence mj"1 = c. So k H = mj

l

= c, a central interval.

c being central, en = nc for every n in IVLS. Substituting k~!i = mj"1 for c, we obtain (k~ ] i)n = n(mj-1) And that is precisely Condition (A),

for every n in IVLS. q.e.d.

We used ref and LABEL in our proof of Theorem 3.5.3. Note, however, that the statement of the theorem does not depend on a choice of ref. 3.5.4 CORROLARY: I* = IJJ if and only if int(v,u) is central. Proof: Applying the formula of Definition 3.5.1 to the algebraic truism int(v, u) = int(v, u), we infer that u = I*(v). Then, by the logic of Theorem 3.5.3 just proved, 1^ = IJJ if and only if int(v, u) is central. The corollary tells us that in a general GIS, v/u inversion may well not be the same operation as u/v inversion, despite the relations v = I*(u) = I"(u); u = I^(v) = IJJ(v). The operations I* and l^ both transform u to v, and v to u. But there may be some other s, other than u or v, such that IJj(s) is not the same element as I^(s). Indeed the corollary assures us there will be some such s if int(v, u) is not central. These considerations indicate how carefully and rigorously we must proceed hereabouts; intuitions based on our familiarity with a number of commutative GIS structures will not always be reliable. 3.5.5 COROLLARIES: In a commutative GIS, I* always = I|J; generally, I* = I* if and only if w = I»(x).

53

3.5.3

Generalized Interval Systems (2)

In a non-commutative CIS, there will always be some inversion operation Iy which is not the same as 1^. (For there will always be some int(v, u) which is not central.) Now we shall see how inversion operations I* combine with transpositions Tn and interval-preserving operations P. 3.5.6 THEOREM: For any transposition Tn and any inversion I*, (A): T n l^ = I* where x = Tn(u). (B): I v u T n = I* where w = T-^V). (C): Tn commutes with 1^ if and only if n is central and nn = e. Proof (optional): We shall fix some ref and use LABELs to help our computations. Let i = LABEL(v); let j = LABEL(u). Then LABEL(TnIvu(s)) = LABEL(Ivu(s)) - n = i-LABELS)-1-j-n = LABEL(I^(s)),

(3.4.3) (3.5.2)

where x is the member of S whose LABEL is jn (3.5.2). Since LABEL(x) = jn and LABEL(u) = j, x = Tn(u) (3.4.3). Now Tnl*(s) I*(s) have the same LABELs, via the chain of computations above. Hence Tnl*(s) = I*(s). (Both lie the same interval from ref.) Since s was an arbitrary sample member of S, Tnl^ = I* as an operation. This proves (A) of the theorem. To prove (B), we start with a similar chain of computations. LABEL(PuTn(s)) = = = =

54

iCLABELCT^s)))-1] i(LABEL(s) • n)-1] in~1LABEL(s)-1j LABEL(I-(s)),

(3.5.2) (3.4.3)

where w is the member of S whose LABEL is in -1 (3.5.2). Since LABEL(w) = in"1 and LABEL(v) = i, w is the n"1 transpose of v (3.4.3). Since T"1 = T"1, we can write w = T~!(v). (The map of n to Tn is an anti-isomorphism; n"1 maps to T"1.) We go on to infer the operational equality of the operations I*Tn and 1^, exactly in the way we inferred an analogus equation when proving (A) of the theorem. This proves (B) of the theorem. Using the formulas of (A) and (B) that we have now established, we see that Tnl^ = I*Tn if and only if I* = I* x and w being as in (A) and (B) of the theorem. By Theorem 3.5.3, this will be the case if and only if w = I*(u) and the interval int(u, x) is central. That will be so, according to 3.5.1, if and only if int(v, w) = int(u, x) and int(u, x) is central. But int(u, x) = n, since x = Tn(u), and int(v, w) = n"1, since w = T"1^). So, in sum, Tn will commute with I* if and only if n = n"1 and n is central. This proves (C). q.e.d.

Generalized Interval Systems (2)

3.5.7

(C) of the theorem shows that, given any interval n in a general GIS, either Tn commutes with every inversion operation or Tn commutes with no inversion operation. In the familiar GIS of twelve chromatic pitch-classes, T6 commutes with every inversion operation: If you invert and then transpose by a tritone, the net result is the same as if you transpose by a tritone and then invert (about the same center or axis). In that GIS, no other interval of transposition has this property, save for the trivial interval of zero. (T0 is the identity operation.) In fact, no other Tn in that GIS will commute with any inversion operation. (C) of the theorem shows us that this situation is related to the fact that 6 + 6 = Omod 12, while n + n does not = 0 mod 12 for any other non-zero interval mod 12. Theorem 3.5.6 gave us insight into how inversions combine with transpositions. An analogous theorem will give us analogous insight into how inversions combine with the interval-preserving operations P. 3.5.7 THEOREM: For any interval-preserving operation P and any inversion Iu»

(A): PI* = I? where w = P(v). (B): i;;P = I* where x = P~1(u). (C): P commutes with !„ if and only if P = Tc for some transposition Tc such that c is central and cc = e. Proof (optional): We fix a referential element ref. Setting n = int(ref, P(ref)) = LABEL(P(ref)), we write P = Pn in the manner of 3.4.4 earlier. We can then manipulate pertinent LABELs to prove (A) and (B) exactly as we proved (A) and (B) for Theorem 3.5.6 above. To prove (C), we begin in a manner similar to that by which we proved 3.5.6 (C). Using (A) and (B), we note that PI* = I^P if and only if I£ = I*, where w and x are as in (A) and (B). Via 3.5.3, this will be the case if and only if int(v, w) = int(u, x) and int(u, x) is central. Now LABEL(u)=j and, since x = P~'(u) = Pn-»(u), LABEL(x) = n-1 • LABEL(u) = n-1j (3.4.4). Therefore LABEL(x)~1LABEL(u) = (n'M)"1 j = T1 nj. And int(u, x) is precisely LABEL(x)"1 LABEL(u) (3.1.2). So int(u, x) = j"1 nj. We have now showed: P = Pn commutes with I* if and only if int(v, w) = int(u,x) = j-1nj and the interval j-1nj is central. A little group theory provides the proof for the following lemma: The element j-1nj of a group is central if and only if n is central. Thus either n is central, in which case j"1 nj is of course simply n; or else n is not central, in which case j~*nj is not central. We have now proved: P = Pn and I* commute if and onlyifint(v,w) = int(u,x) = n and n is central. The rest follows from 3.4.8 and 3.5.6(C). q.e.d. We have now seen how inversion operations combine with intervalpreserving operations. Earlier we saw how inversions combined with transpositions (3.5.6), and how transpositions commuted with inversion-preserving operations (3.4.10). Earlier still, we noted that the transpositions formed a

55

3.5.8

Generalized Interval Systems (2)

group of operations among themselves (3.4.2), combining according to a certain formula; the interval-preserving operations also formed a group among themselves (3.4.5) combining according to another formula. It remains to explore how the inversion operations combine with each other, and we proceed to do so.

3.5.8 THEOREM: Fix ref, and let the LABELs of v, u, w, and x be respectively i, j, k, and m. Then TVTW = rp— IT~IJ A A x u x

im k

Proof: Given any s, then 3.5.2 tells us that LABEL(I^(s)) = i-LABEL(s)- 1 -] while 1 LABEL(I*(s)) = k • LABEL(s)- • m. Hence LABEL(IvuI-(s)) = KLABELa^s)))-1] = i(k • LABEUXT1 • mr'j = i(m-1LABEL(s)k~1)j = (im-1)LABEL(s)(k-1j) = LABEUPrjT-^s)) (3.4.3; 3.4.4). In sum, I*I*(s) and P^T^s) have the same LABELs, and are therefore the same element of S. Since s was an arbitrary sample member of S, the functional equation of the theorem is true, q.e.d. 3.5.9 COROLLARY: I|J is the inverse operation to I*. Proof: Given u and v, take x = v and w = u in the formula of Theorem 3.5.8 above. Then m = i and k = j, as those intervals are defined in that theorem. So im"1 and k-1j are both equal to e; the formula of the theorem tells us in this special case that !„!" = PeTe, which is of course the identity operation. By the symmetry of the situation (reversing the roles of u and v), PJI* is also the identity operation, q.e.d. 3.5.10 COROLLARIES: Let T and I be any transposition operation and any inversion operation in a commutative GIS. Then

(A): r1 = I and (B): IT = T-'I. Proofs: (A) follows at once from 3.5.9 and the first remark of 3.5.5. To prove (B), set J = IT. Via 3.5.6(B) we know that J is an inversion operation. Then, invoking (A) just proved above, we infer that J = J"1. It follows: IT = J = J-1 = (IT)"1 = T'1!'1 = T~]I; the last step is again a consequence of (A) just proved, q.e.d.

56

We have now explored how various types of operations on the space of an abstract GIS compose among themselves and with each other. Standing back

Generalized Interval Systems (2)

3.5.11

from the niceties of the specific formulas involved, we can get a useful global picture. 3.5.11 THEOREM: Let PETEY be the family of all operations on S that can be expressed as (functionally equal to) something of form PT, where P is some interval-preserving operation and T is some transposition. Let PETINV be the family PETEY plus the family INVS of all inversion operations. Then (A) PETEY is a group of operations and (B) PETINV is also a group of operations. Proof (optional): We already know that PSVS, the family of intervalpreserving operations, and TNSPS, the family of transpositions, are each groups of operations (3.4.5; 3.4.2). Let PT and PT' be any two members of PETEY. Set P" = PP' and T" = TT. Since PSVS and TNSPS are groups, P" is interval-preserving and T" is a transposition; then P"T" is a member of PETEY as PETEY was defined. Furthermore, the composition of the given PT with the given FT' is precisely P"T", this member of PETEY. For (PD(FT') = P(TP')T = P(PT)T' (3.4.10) = P"T". So PETEY is a closed family of transformations. To prove PETEY a group, it suffices via 1.3.4 to show that PETEY-asdefined contains the inverse of each of its member operations. Given PT in PETEY, then P"1 and T-1 are respectively members of PSVS and TNSPS, so that p-1T-1 is a member of PETEY-as-defined. And p^T"1 is the inverse of the given PT: PT = TP (3.4.10), so PT(P~1T~1) = TP(P~1T~1) = 1 and (p-iT-i)PT = (P-1T-1)TP = 1. So (A) of the theorem is proved. We use the criteria of 1.3.4 again to prove (B) of the theorem. We know that PETEY, since it is a group, contains the inverse of each of its members; we also know via 3.5.9 that INVS contains the inverse of each of its members. Hence PETINV, the set-theoretic union of the two families PETEY and INVS, contains the inverse of each of its members. It remains only to prove that PETINV is a closed family of operations. Suppose that X and Y are members of PETINV; we have to show that XY is (operationally equal to) a member of PETINV. We can distinguish four possible cases, which we take up one at a time below. Case 1: X and Y are both members of PETEY. Then XY, being a member of PETEY, is a member of PETINV. Case 2: X is a member of PETEY and Y is a member of INVS. Say X = PT and Y = I. Now TI is some inversion-operation J (3.5.6(A)). And PJ is some inversion-operation K (3.5.7(A)). Then XY = PTI = PJ = K is a member of INVS, and therefore a member of PETINV as desired. Case 3: X is a member of INVS and Y is a member of PETEY. By an argument analogous to that of Case 2, now using 3.5.6(B) and 3.5.7(B), we conclude that XY is a member of INVS, and hence of PETINV as desired. Case 4: X and Y are both members of INVS. Then XY is a member of PETEY (3.5.8). So XY is a member of PETINV as desired, q.e.d.

57

3.6.1

Generalized Interval Systems (2)

We have seen that transpositions are naturally related in a number of ways to interval-preserving operations. We might conjecture that inversions should be naturally related to "interval-reversing" transformations, in the sense of the following definition. 3.6.1 DEFINITION: A transformation Y on the space S of a GIS will be called interval-reversing if int(Y(s),Y(t)) = int(t,s) for all s and all t in S.

There is something to our conjecture above. Specifically, if the GIS is commutative, then the inversions are precisely the interval-reversing operations on S. But if the GIS is now-commutative, then there will not be any intervalreversing transformations at all! We shall now prove these facts, starting with a lemma. 3.6.2 LEMMA (optional): Let Y be an interval-reversing transformation; let ref be fixed; then there is an interval i such that LABEL(Y(t)) = i • (LABEL(t))-1 for every t in S. Proof: int(Y(s), Y(t)) = int(t, s) by supposition. So LABEL(Y(t))~1LABEL(Y(s)) = LABEL(s)~1LABEL(t)

(3.1.2).

Take s = ref in the above equation, so that LABEL(s) = e. Set i = LABEL(Y(ref)) = LABEL(Y(s)) in the above equation. Then LABEL(Y(t))-1 • i = LABEL(t). So LABEL(Y(t))~1 = LABEL(t) • T1 and LABEL(Y(t)) = i • LABEL(t)-1, for any t. 3.6.3 THEOREM (optional): If IVLS is commutative, then the inversion operations reverse intervals, and every interval-reversing transformation is some inversion-operation. Proof: Fix some ref. The result of Lemma 3.6.2, in conjunction with the formula of Theorem 3.5.2, tells us that any interval-reversing transformation must be some inversion, specifically some Vtcf. It remains to show that any I* reverses intervals. Setting i = LABEL(v) and j = LABEL(u), we can write

58

intai(s), W)) = LABELS))-1 LABEL(I^(s)) = (iLABEL(t)~1j)-1(iLABEL(s)-1j) = j~1LABEL(t)i-1iLABEL(s)-1j = LABEL(s)~1LABEL(t),

(3.1.2) (3.5.2)

Generalized Interval Systems (2)

3.6.4

since IVLS is commutative! And that = int(t,s)

via 3.1.2.

q.e.d.

3.6.4 THEOREM (optional): If IVLS is non-commutative, then there exists no interval-reversing transformation on S. Proof: We suppose that Y is an interval-reversing transformation and arrive at a contradiction. For all s and all t, LABEL(Y(t))~1LABEL(Y(s)) = LABEL(s)-1 LABEL(t), as in the proof for Lemma 3.6.2. Having the formula of that lemma at our disposal now, we can substitute for the LABELs of Y(s) and Y(t) in the above equation, using the special interval i of the lemma. We get the new equation (i(LABEL(t)r1)-1 • iLABEL(s)-1 = LABEL(s)~1LABEL(t). Or: (LABELCOr^iLABELts)-1 = LABEL(s)~1LABEL(t). Or: LABEL(t)LABELCs)-1 = LABEL(s)~1LABEL(t). But that equation, holding for all s and t, says that IVLS is a commutative group. And that contradicts the premise of the theorem, q.e.d.

59

4

Generalized Interval Systems (3): A Non-Commutative GIS;

Some Timbral GIS Models

During the discussion of transpositions, interval-preserving operations, and inversions, the reader may have been puzzled by the care with which noncommutative GIS structures were separated from commutative. After all, we have so far not encountered any specimen GIS which is non-commutative. Why should we be concerned at all about the non-commutative case? Why not save ourselves some trouble, and just stipulate in the definition of a GIS that the group IVLS should be commutative? The work of the present chapter will respond to these concerns by exploring a musically significant noncommutative GIS. I have already presented some of the work elsewhere, but it will have a special impact in the present context.1 4.1.1 DEFINITION: By a time span, we will understand an ordered pair (a, x), where a is any real number and x is any positive real number. The pair of numbers is understood to model our sense of location and extension about a musical event that "begins at time a" and "extends x units of time" thereafter. The family of all time spans will be denoted TMSPS. We have already encountered one rhythmic GIS whose objects were certain time spans; that was example 3.3.2. There, we restricted the values for the numbers a to integers, and we restricted the values for the numbers x to certain proportions. Using the same direct-product construction, we could also construct a GIS for time spans in which the number a could assume any rational value, and the number x any positive rational value. Using the direct-

60

1. David Lewin, "On Formal Intervals between Time-Spans," Music Perception vol. 1, no. 4 (Summer 1984), 414-23.

Generalized Interval Systems (3)

4.1.2

product construction, we can also construct a GIS for all time spans in the manner of 4.1.2 following. 4.1.2 EXAMPLE: Take S = TMSPS. Take IVLS to be the direct-product group of the real-numbers-under-addition by the positive-reals-undermultiplication. Define the function int, from S x S into IVLS, by the formula int((a,x),(b,y)) = (b-a,y/x). Then (TMSPS, IVLS, int) is a GIS. It is commutative. The interval (b — a, y/x) measures our presumed sense that time span (b, y) begins b — a units after time span (a, x) and lasts y/x times as long. This commutative GIS is useful and relatively simple, but it is not adequate as a model for the way we perceive time spans interacting under all circumstances. We shall now investigate why that is so. First we shall examine the family of time spans as a conceptual space independent of any particular compositional context; then we shall examine how time spans behave in various specific musical contexts. To begin, then, let us focus on the time span (a, x) as a conceptual object in a conceptual space, modeling our sense that something "begins at time point a" and "extends for x time-units" thereafter. We can ask, what is this absolute conceptual time-unit? In practice, we often proceed as if it were the minute. We do so, that is, when we write metronome marks which reduce various contextual units, in various pieces or passages of music, to fractions of a minute. The minute is not commensurate with our sense of a "beat," but we can use the second for that purpose if we wish, dividing all the metronome numbers by 60. Neither the minute nor the second, though, is very satisfactory as a would-be absolute conceptual time unit; both are derived from certain relative periodic motions of the earth, the sun, and the moon. Scientists today find these motions so erratic and irregular that they use other conceptual units of time for precise measurements. But even those units, deriving from certain sub-atomic motions, are clearly contextual. And that does not even begin to engage other technical problems involving Relativity and quantum mechanics in connection with such sub-atomic "fixed" units of time. In short, if we declare any one time-unit to have absolute conceptual priority, that is a matter of computational convenience, or of scientific, sociological, or religious convention, rather then manifest musical reality. Abandoning this approach, we can make our absolute time-unit a matter of notational structure: We can call it "the brevis" or "the perfection" or "the whole note" or "the notated beat," for instance. But then we are throwing the whole problem back onto some notational convention that is highly restricted socio-historically, a convention that indeed already presupposes a highly structured theory of measuring time by some pre-existing absolute unit. And that will not help us in our inquiry.

61

4.1.2

Generalized Interval Systems (3)

We might try to assert that, though we cannot conceive an absolute timeunit clearly, there will be some clear contextual time-unit, which we can identify and use for theoretical purposes, in any music that we may want to analyze. Such an assertion is reasonable in connection with a large body of music, and not only European music of the Classic-Romantic period. But the assertion is still not valid as a universal proposition about music, unless one is willing to restrict the use of the word "music" circularly, i.e. in precisely this way. Making that restriction would involve at least a broad aesthetic contention. An even then, many critics who might feel no qualms about excluding as "music" say certain improvisations of John Coltrane, or Stockhausen's Aus den sieben Tagen, would feel less comfortable excluding as music certain Tibetan chants, or sections of Elliott Carter's String Quartet no. 1. Figure 4.1 (pp. 64-65) reproduces measures 22-35 of the Carter score. What could one assert as "the" (one clear contextual) time-unit for measures 22-32?2 Figure 4.1 shows that our philosophical musings above do not simply come down to a matter of mensural versus non-mensural perceptions. Measures 22-32 of Carter's piece have a very strong mensural character, despite our difficulty in pinning down "the" beat. The mensural profile of the passage is especially—one might even say unusually—strong within each of the individual instrumental parts. For example, the first violin's A in measure 26 and G in measure 28 each last precisely half as long as every other note in the first violin from measure 22 to measure 30; a player who does not hear this mensural relation will not project the passage effectively. For another example, the B-F|-D-C|-D# of the cello at measure 32 and following is heard not only in syncopation against the otherwise regular half-note beat of measure 33 and following (half-note = MM90), but also as a rubato of the earlier cello melody C-G-E1>-D-E, appearing eleven semitones lower in measures 2729. The earlier melody is presented in notes of constant duration whose beat, every five written eighths, is at MM48, not MM90. A cellist who does not hear the rhythmic relation of the transposed melodies will not project measures 32-35 completely effectively. In sum, the notion of "an" abstract conceptual time-unit, a unit by which we measure the number x of the formal time span (a, x), is a notion fraught with methodological problems. The number a of the time span (a, x), as well as the number x, is measured by our conceptual time-unit. For when we say that we perceive something that "begins at time-point a," we mean implicitly that it

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2. I am grateful to Jonathan W. Bernard for engaging my interest in this passage through his lecture, "The Evolution of Elliott Carter's Rhythmic Practice," delivered to the meeting at Yale of the Society for Music Theory on November 11, 1983. Bernard's observations engaged many of the rhythmic relations I shall be discussing. The uses to which I shall put those observations, as I expound my CIS-theories, are of course my own responsibility.

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begins just that number of conceptual time-units after some referential timepoint, "time-point zero." Having noted this, we see that we must discuss not only the conceptual time-unit in this connection, but also the conceptual "time-point zero." What is this abstractly privileged moment that contributes toward measuring the number a of the time span (a, x)? Is it the moment of the Big Bang, or of the Biblical Creation? Is it a completely arbitrary moment very long ago? (And if so, why should we select an arbitrary moment to play a uniquely referential role?) Should we select "time-point zero" by a notational convention, e.g. as the first vertical line on the score of whatever piece we are analyzing at the moment? Or should we presume to assert, explicitly or implicitly, that there must always be some one uniquely privileged moment, in the score or the performance of any passage we want to discuss, which we can unequivocally label as a contextual zero time-point for the occasion? These methodological expedients involve difficulties similar to those discussed above in connection with the referential time-unit. In one way at least, the choice of a zero time-point is less problematic than the choice of a temporal unit: The former choice does not affect the numbers attached to intervals in the GIS of 4.1.2, while the latter choice does affect those numbers. To see this, first suppose that we move our referential zero time-point back by m units into the past. Then the percept that was formerly manifest over the time span labeled (a, x) in the old scheme will now be manifest over the time span labled (a + m, x) in the new scheme: What used to begin a units after (old) time-point zero will now begin a + m units after (new) time-point zero. Similarly, the time span labeled (b, y) in the old scheme will correspond to the time span labeled (b + m, y) in the new scheme. In the GIS of 4.1.2, the interval between the old labels is int((a, x), (b, y)) = (b — a, y/x). In the same GIS, the interval between the new labels is int((a + m, x), (b + m, y)) = (b + m - (a + m), y/x) = (b - a, y/x). So, in transforming each old time span (a, x) to the new time span (a + m, x), we have not transformed the intervals involved: The interval between a pair of transformed spans is exactly the same as the interval between the corresponding pair of spans prior to transformation. That is, int((a + m, x), (b + m, y)) = int((a, x), (b, y)). Now let us suppose we keep the same referential time-point zero but change the unit of measurement so that what was x old units becomes xu new units, the factor u corresponding to the change of scale in measurement. Then percepts formerly corresponding to the time span (a, x) and (b, y) in the old scheme will now correspond to the time spans (au, xu) and (bu, yu) in the new scheme: What used to begin a old units after time zero and extend x old units therefrom will now begin au new units after time zero and extend xu new units therefrom. We can see that this transformation does change the numbers attached to intervals in GIS 4.1.2: int((a, x), (b, y)) = (b - a, y/x), while

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FIGURE 4.1

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4.1.2

FIGURE 4.1 (continued)

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int((au,xu),(bu,yu)) = (bu — au,yu/(xu)) = ((b — a)u,y/x). Intuitively, the second percept we are discussing used to begin b — a units after the first percept; now it begins (b — a)u units after the first percept, the "unit" having changed. Of course the GIS of 4.1.2 knows nothing of "percepts" or "units"; it simply knows that int((a, x), (b, y)) = (b — a, y/x) is not the same pair of numbers as ((b — a)u, y/x) = int((au, xu), (bu, yu)). Lest the host of methodological problems under discussion appear insuperable, we should recall that we can finesse them all by restricting our attention to music in which we can identify and assert a referential time-unit and a referential zero time-point contextually. Then we can use the GIS of 4.1.2 without problems. There are plenty of pieces and passages for which we can sensibly take this tack. On the other hand, there are also pieces and passages in which we cannot identify such referential entities contextually, music which we would agree nonetheless to consider highly structured mensurally, music within which it seems analytically valid—even necessary—to articulate time spans engaged in mensural interrelationships. We have already begun to explore the Carter example in this connection; we shall continue that analysis soon. Another example is provided by Stockhausen's Klavierstiick XL Stockhausen tells the pianist to look at the sheet of music and begin with any group of notes from among nineteen such groups dispersed over the score, "the first that catches his eye; this he plays, choosing for himself tempo ..., dynamic level and type of attack. At the end of the first group he reads the tempo, dynamic and attack indications that follow, and looks at random to any other group, which he then plays in accordance with the latter indications," and so on and on. "When a group is arrived at for the third time, one possible realization of the piece is completed." 3 Each of the nineteen groups is notated quite traditionally as regards pitches and internal rhythmic proportions. But each group might be played at any of six tempi, ranging from very slow to very fast. Indeed, even during one performed realization, any group might be played two different times at two different tempi. The tempo of each performed group (after the first) depends on the instructions which appear at the end of the group just played, which might itself occur at any of the six tempi. In this context it makes no musical sense to speak of "the" referential time-unit, beyond the interior of each performed group at its performed tempo. And yet, mensural relations among time spans from different groups (especially consecutive groups) are highly audible, and therefore at least perceptually functional in any given realization. We have already mentioned, in footnote 3 of chapter 2, other examples of highly mensural music without a fixed time-unit. Nancarrow's Studies for Player Piano contain many pitch-canons involving elaborately changing 66

3. The cited text is from the Performing Directions by the composer on the score (Universal Edition no. 12654 LW, 3d ed., 1967).

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tempo proportions.4 Ligeti's Poeme symphonique is performed by winding up 100 clockwork metronomes to varying degrees of tension, setting them at a variety of tempi, and releasing them, allowing them to run down over the course of eighteen to twenty minutes.5 The effect includes an ironic poetic commentary, among other things, upon the very issue of the "contextual timeunit." To some extent in all this cited literature, and to a great extent in much of it, any time span has the potential for becoming a local contextual timeunit, setting a local tempo. By "local," I mean here not only over a certain temporally connected section of the total texture, but possibly also within a certain part, instrument, or instrumental group. For example, there is a more than clear mensural structure within the part played by any single metronome within Ligeti's piece. For some less extreme examples let us return once more to figure 4.1, the Carter passage discussed before. The viola moves in notes of constant duration from its entrance at measure 25 up to the middle of measure 35; these local time-units "beat" the tempo of MM 180. The first violin "beats" constant local time-units at MM36 over measures 22-30, except for the A of measure 26 and the G of measure 28 which, as observed earlier, are each half the local time-unit in duration. At measure 33 the first violin starts to project a new constant local time-unit, that beats at MM90. The cello beats constant local time-units at MM 120 over measures 22-26; then over measures 27-31 it beats new constant local time-units at MM48. Finally at measure 32 and following, it stops moving in notes of constant duration as it plays a pitchvariation on measures 27ff., where MM48 began. This variation was discussed earlier. The second violin beats its own constant local time-unit at MM96 from measure 22 through measure 26. Then over measures 27-30 it runs quickly through a number of local time-units at MM 120, MM 160 (m. 27j), MM96 (m. 28|), MM80 (to be discussed later), and MM60 (m. 30). Finally it settles into a more stable local time unit in measure 31, beating at MM90. 4. Some of Nancarrow's recent work involves irrational proportions like n. For the reader who may at first think such an idea is too bizarre to have any musical meaning, I append a brief exercise in conducting the tempo relation of n. Imagine a horizontal line segment at chest height in front of you and somewhat to your right. (I am supposing that you conduct right-handed.) Move your hand (arm) back and forth along the line at a constant speed, beating a horizontal \ allegro vivo. Now imagine the line segment as a radius of a circle whose center is at the leftmost point of your beat. As you reach the rightmost point of your beat, start swinging your hand (arm around and around the circumference of that circle counterclockwise, taking care always to keep your hand moving at the same constant speed. The amount of time it takes you to swing once around the circle is n times the duration of your earlier horizontal \ measure. 5. The composer specifies very precisely how this is to be done. He also specifies that the piece is to be played after an intermission, so that the returning audience finds the metronomes already underway, with no persons on stage. The metronomes are to be arranged on risers, like a chorus; the slow beaters are at the lowest level and the fast beaters at the highest. I am indebted to Martin Bresnick for supplying me with this report, based on a personal communication from the composer.

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Figure 4.2a collates and tabulates these various metronome marks, that reflect the various tempi beat by the various local time-units in the individual instruments over the passage.

FIGURE 4.2b

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Figure 4.2b takes the numbers of figure 4.2a and represents them as pitches. This device will help clarify for musicians the numerical ratios involved among those numbers. The number 180, which labels the fastest beat on figure 4.2a, is represented on figure 4.2b by the highest pitch, high C. Slower tempi, in their numerical ratios to MM 180, are represented on figure 4.2b by lower pitches in the corresponding frequency ratios to high C. For instance, on figure 4.2a the opening tempo of the cello is MM 120, 2/3 of the tempo MM 180 coming up in the viola. On figure 4.2b the tempo MM 120 is

Generalized Interval Systems (3)

4.1.2

represented by the opening pitch F5 in the cello, a pitch whose fundamental is 2/3 the frequency of the high C coming up in the viola, the high C which represents the tempo MM 180. The euphony of figure 4.2b is striking. It makes very clear the network of "numerical consonances" displayed by the tempo relations of figure 4.2a, which are also the proportional relations of the various local contextual timeunits. If we imagine a quartet actually playing figure 4.2b, we can get an even clearer idea of how this numerical network affects interactions among the performers. Here are some, and only some, of the relations the players will heed. (1) The 48 of the cello at measure 27 will lie a good numerical octave below the opening 96 of the second violin. (2) The 120 of the second violin at measure 27 will match the opening 120 of the cello. (3) The 120-to-160 relation in the second violin during measure 27 will match in its ratio the 36-to-48 relation between the first violin and the cello thereabouts. (4) The 80-to-60 relation in the second violin over measures 29-30 will retrograde, an octave lower, the 120-to-160 relation discussed in (3) above. (5) The 60-to-90 relation in the second violin, measures 30-31, will match an octave lower the earlier 120-to-180 relation between the cello at the opening of the passage and the viola entrance. (6) The 90 of the first violin at measure 33 will match the 90 of the second violin at measure 31, which in turn will match, an octave lower, the persistent 180 of the viola. (7) The 36-to-90 profile of the first violin part as a whole will match in transposed retrograde (or inversion) the 120-to-48 profile of the cello part up to measure 33. Exactly these numerical relations, and others of the same sort, must be projected to make the rhythmic structure of Carter's passage come to life and communicate, not only between the players and their listeners, but also among the four players themselves. We shall discuss the above seven performance notes some more later on. On figure 4.2b, it is curious how the D[?4 of the cello can be heard as a root whose major harmony is elaborated by the symbolic pitches of the figure over measures 22-30, along with a major seventh and an added sixth. To be sure, it is doubtful that our tonal perceptions of roots, triads, harmonic sevenths, and added sixths can be used to assert analogous functions in the realm of tempo relations. And yet there is a certain suggestiveness in the idea that the cello part of measure 27 and following has a rhythmically "grounding" function somehow analogous to the tonal root function of the symbolic D|?4 on figure 4.2b. This suggestion is useful for the cellist who wants the MM48 tempo at measure 27 and following to feel stable and referential, rather than syncopated and intrusive. The suggestion is also useful for understanding why just this tempo of MM48 (symbolized by the pitch D|?4 on figure 4.2b) is permitted to launch into a wide and free rubato at measure 32 and following in the cello (symbolized by the "cadenza" on figure 4.2b), just at the time the

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other three instruments are all finally agreeing upon the new referential tempo MM90-or-180 (symbolized by the prominent pitch class C at m. 32ff. on figure 4.2b). Over the second-violin part of figure 4.2b as a whole, D|» moves clearly to C. But the cello is still refusing to change its earlier D\> for C as we leave the passage, though its D\> has been abandoned. All of these remarks amplify the idea suggested above, that the cellist should refer the rhythm of the part over measures 32-35 not just to the beats of the upper instruments at MM90 but also to the earlier beat of the cello itself, MM48 at measure 27 and following. Figure 4.2b shows by analogy how the symbolic pitches of the "cadenza" in the cello at measures 32-35 are heard not only in relation to the Cs of the upper instruments there, but also in relation to the earlier Dl> of the cello itself at measure 27 following. However interesting it may be to think of MM48 as a numerical "ground" organizing the tempi of measures 22-31, it is still clear that the players of the three upper instruments will not treat the cello part of measures 27-31 as a succession of referential time-units. That is, they will not adjust their own beats to conform in proper proportion with the cello beat of measures 27-31. Nor will the cello and the two violins treat the constant beat at MM 180 in the viola from measure 25 on as a succession of referential timeunits for the entire passage; at least they will not do so until after measure 30. During measures 22-24, in particular, the MM36, MM96, and MM 120 of the two violins and the cello will not be referred to a beat of MM 180 in the viola, for the viola has not yet entered there. Once the viola is in, at measure 25 and thereafter, its constant MM 180 will provide a useful check for the ensemble rhythm without necessarily establishing itself as referential, just as the sustained high C on figure 4.2b provides a useful check for the ensemble's intonation without necessarily establishing itself as a root. Of course MM 180 does eventually become much more referential during measures 33-35, along with its lower octave MM90; just so, on figure 4.2b, the pitch class C comes to dominate the tonal texture. The players may decide on purely notational grounds to use MM 120 (changing to MM 180 by measure 33) or MM60 (changing to MM90 by mea sure 33) as a referential tempo for the entire passage. This makes a certain practical sense for early rehearsals, but it can hardly be recommended for an effective performance. We have all heard and seen players fighting their way through slow lyric lines, supposedly tranquillo like that of the first violin in measures 22-32 or sostenuto e cantabile like that of the cello in measures 27-32, all the while jerking their feet up and down spastically in an erratic approximation of some distantly related notational "beat." These lyric lines are not syncopated, as such a method of production makes them sound to both player and listener. Rather, each line has its own autonomous local timeunit, with respect to which it should project an essentially "first-species" character.

Generalized Interval Systems (3)

4.1.2

Indeed, the search for any one overriding referential time-unit, to govern all of measures 22-31, is bound to fail. It must fail because it misconceives the nature of the temporal space at hand. That space comprises a multitude of locally referential time-units, in various more-or-less consonant numerical relationships among themselves. Only by recognizing that character of the space can one hear the music progress over the passage, and not just "be." What I mean will be amplified by the following review of the earlier performance notes, (1) through (7). (1) The MM48 of the cello at measure 27 will lie a good (numerical) octave below the opening MM96 of the second violin. That is, at measure 27 the cellist should hear the cello line moving at half the rate the second violin has been moving so far. This would be easy to hear if the cello came in a sixteenth-note later. In fact there is no problem within measure 27, getting the attack of the pitch C2 at the right time-point: The player need only continue beating MM 120 up to that point. But once the new melody has entered, its tempo may sound arbitrary and its character "syncopated" unless the player hears the melody taking over, albeit out of phase, from the preceding tempo of the second violin, projecting its own local referential time-unit. (2) The MM 120 of the second violin at measure 27 will match the opening MM 120 of the cello. MM 120 is the notated pulse. Still, the second violin should not simply be playing that pulse (with spastic foot-tapping or the like). What makes measure 27 come alive and communicate in this connection is an exchange of local referential beats, between the two instruments. The second violin takes over the preceding beat of the cello, while the cello—as we noted in (1) above—is about to take over the preceding local beat of the second violin, a rhythmic octave lower (and out of phase). The cellist and the second violinist should hear this "voice-exchange with octave transfer," to hear themselves conversing with each other in a quite familiar tradition of chamber music. Figure 4.2b, showing the exchange of the symbolic pitch classes D|? and F in the two instruments at measure 27, serves as a guide to that tradition here. Just as the players would match those pitch classes if playing figure 4.2b, so they should match their exchange of tempi at measure 27 when playing the Carter passage. (3) The MM 120-to-160 relation in the second violin, later in measure 27, will match the MM36-to-48 relation between first violin and cello hereabouts. Assuming that everything else has gone right so far,this will happen automatically if the second violin plays the dotted eighth (beating MM 160) as a precise 3/4 of the quarter note (beating MM 120). That is not so easy to do as it is to say, but the ability should be available to a well-trained player of twentiethcentury music. Here, the two quarter notes of the second violin in measure 27 will function as locally referential timespans for the player. (4) The MM80-to-60 relation in the second violin, measures 29-30, will retrograde the MM120-to-160 relation of (3) above, one rhythmic octave

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lower. Here "MM80" is projected only by the E|?-triad-event in measure 29 of the score; the tempo is not beat by any recurrent durational unit. Nevertheless a conceptual MM80 is useful to the player in a manner indicated by figure 4.3.

FIGURE 4.3

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The figure transcribes the rhythm of the second violin from the middle of measure 26 through measure 30 into a new notation, using MM 160 as a new notational tempo of reference, that is, the tempo at which the new notated quarter beats. This transcription brings out clearly how the rhythmic peripatetics of the second violin are structured by the indicated relationship, that is MM80-to-60 answering MM120-to-160. The transcription "modulates" our rhythmic hearing exactly as the second through sixth notes of the second violin on figure 4.2b would sound "modulated" if we listened to them in B[> minor, rather than D|? major. Just as the Fs of figure 4.2b would sound primarily as fourths below the adjacent locally referential Bjjs, in our modulated pitch-hearing, just so the tempi of MM 120 and MM60 on figure 4.3 sound in rhythmic proportions 3 :4 and 4: 3 to the adjacent locally referential tempi of MM 160 and MM80, that lie alongside them. Figure 4.3 demonstrates a logical internal structure for the second-violin passage as a rhythmic entity in itself; this structure will surely not emerge if the player adjusts each individual tempo of figure 4.3 only to the beats of the viola at MM 180 hereabouts, or to the notated beat of the score at MM 120. It is in order to bring out the quasi-palindromic structure of figure 4.3 that the "tempo" of MM80 is represented, exceptionally, by only one time span. (5) The MM60-to-90 relation in the second violin, measures 30-31, will match, a rhythmic octave lower, the earlier MM120-to-180 relation between the cello at the opening of the passage and the viola entrance. This needs no further discussion; the relation of the relations will emerge without special attention if the players are otherwise temporally "in tune." (6) The MM90 of the first violin at measure 33 will match the MM90 of the second violin at measure 31. Obviously. Here MM90 is a referential local time-unit. Likewise the MM90 of the second violin at measure 31 will have matched the MM 180 of the viola so far, using MM 180 as a referential time unit. (7) The MM36-to-90 profile of the first-violin part as a whole over the passage will match in transposed retrograde (or inversion) the MM120-to-48

Generalized Interval Systems (3)

4.1.2

of the cello part as a whole up to measure 33. It is much harder to hear the large rhythmic proportion here than it is to hear the corresponding symbolic pitch proportion on figure 4.2b. Nevertheless, it will aid communication between cello and first violin, as well as projection between ensemble and audience, if the first violinist hears the MM90 entrance at measure 33 speeding up the earlier MM36 of the instrument in exactly the same ratio as the MM48 of the cello at measure 27 slowed down that instrument's earlier MM 120. The proportion can be sensed when the two instruments rehearse the pertinent music by themselves. Via this proportion, the MM90 of the first violin at measure 33 engages and completes a large mensural structure functioning over measures 22-35; it is not simply a surrender of the first violin passively to the beats of the second violin and viola at measure 31 and following. Of course the first violin will use those beats to find its new tempo at measure 33. To sum up: When performers confront the score of the Carter passage and the numerical network of local tempi or time-units displayed in figure 4.2a, they should not concern themselves with the question, "Which one of these is the overall unifying referential tempo?" That question, a rhythmic analog to the sorts of questions asked about pitch structures by Rameau, Riemann, and Hindemith among others, has no definite answer here. Even if we try to force an answer by selecting MM48, or MM 180, or MM60-then-90 as a "root" tempo on the basis of this or that criterion, we shall still not be engaging thereby the temporal relationships that make this music progress and communicate. Those temporal relationships, some of which were discussed in performance notes (l)-(7) above, involve patterns of local tempo "consonances," patterns in which many different tempi can assume locally referential roles. This attitude toward the numerical network of figure 4.2a, and the symbolic pitch network of figure 4.2b, is more in the spirit of Zarlino: It asks not for one overriding referential unity, but rather for a splendid variety of consonant ratios among the entities involved, as they underlay and succeed one another, projecting a logical compositional idea. In this way of hearing the rhythmic space through which the passage moves, any time span has the potential for becoming locally referential, or behaving as if it were. For example, let time span r be the span covered by the first note of the cello in measure 22 of the score; let s be that time span covered by the F# of the first violin in measures 25-26; let t be that time span covered by the A of the first violin in measure 26. We can say if we-wish that t begins 16j r-spans after the beginning of r, and lasts If times the duration of r. But this way of listening corresponds to the "foot-tapping" performance of the first violin's melody. We can also say that t begins 1 s-span after the beginning of s, and lasts 1/2 the duration of s. And that way of listening corresponds to a much more musical shaping of the melody. Taking s' as the time span covered by the opening D of the first violin in measures 22-23, we could also say that t begins 4 s'-spans after the beginning of s', and lasts 1/2 the duration of s'. This

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way of listening corresponds to an even more musical shaping of the melody. Adopting the above attitude to Carter's rhythmic space, we implicitly deny the relevance of GIS 4.1.2 as a model. Given abstract time spans s and t, we want to be able to conceive t as beginning a certain number of s-beats after s, rather than a certain number of possibly irrelevant "referential units" after s. If s is the span (a, x) and t is the span (b,y), the old GIS of 4.1.2 assigned int(s, t) = (b — a,y/x): t begins b — a referential units after s, and lasts y/x times as long. We want to replace that old notion of time-span interval by a new function: int(s, t) = ((b — a)/x, y/x). The new interval tells us that t begins (b — a)/x x-lengths (s-beats) after s, and lasts y/x times as long. The new interval uses s itself as a measuring rod, to tell us how much later t begins.

FIGURE 4.4

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Figure 4.4 shows how our new "interval" works. On the figure, four numerical time-spans are indicated: Sj = (a^Xj), i1 = (b^y^, s2 = (a 2 ,x 2 ), and t2 = (b2, y 2 ). We shall see later that it does not matter at all, for our new model, what the formal numerical time-point zero is, or what the formal numerical time-unit is. That is, it does not matter to what percept we attach the numerical time-span label (0,1). On the figure, we can imagine an "upper instrument" projecting s x and ^ at a slow tempo, and a "lower instrument" projecting s2 and t2 at a fast tempo. The dotted slurs arching above the upper instrument mark off x1 -lengths, durations that mark a contextual (potential) Sj-beat. The dotted slurs arching below the lower instrument mark off x2lengths, durations that mark a contextual (potential) s2-beat. Using our new interval construct, we write int^,^) = (4,2): t1 begins 4 Sj-beats after s 1? and lasts twice as long. Arithmetically, (b: — a^/Xj = 4 and y!/x x = 2. Using

Generalized Interval Systems (3)

4.1.3.2

the new interval construct, we also write int(s 2 ,t 2 ) = (4,2): t 2 begins 4 s2beats after s2, and lasts twice as long. Arithmetically, (b2 — a2)/x2 = 4 and y 2 /x 2 = 2. Note that our new "interval from s2 to t2" is the same as our new "interval from sl to t^': int(s 2 ,t 2 ) = m^Su^) = (4,2). We shall discuss the implications of this a lot more later on. Note particularly that sx precedes s2 on figure 4.4 in the obvious sense, while t x , the (4,2)-transpose of s^, follows t 2 , the (4,2)-transpose of s 2 . One sees that our intuitions about formal "transposition" will not be completely reliable in our new non-commutative GIS. (Our intuitions about interval-preserving operations will be trustworthy.) We are of course still far from having constructed a formal GIS in which our new notion of "interval" is to work. It is high time to do so now. 4.1.3.1 LEMMA: Let IVLS be the family of pairs (i, p), where i is a real number and p is a positive real number. Then IVLS forms a group under the composition

0,p)(j,q) = (i + pj,pq). In this group, the identity is (0,1) and the inverse of the element (i, p) is the element ( — i/p, 1/p). The group is non-commutative. The proof of the Lemma will be left as an exercise for the interested reader. Do not forget to show that the defined composition is associative: ((i,p)(j,q))(k,r) = (i,p)((j,q)(k,r)). 4.1.3.2 THEOREM: Let int be the function that maps TMSPS x TMSPS into the group IVLS of Lemma 4.1.3.1 according to the formula int((a,x),(b,y)) = ((b-a)/x,y/x). Then (TMSPS, IVLS, int) is a GIS. Proof (optional): We must show that Conditions (A) and (B) of Definition 2.3.1 obtain. (A): Given time spans (a,x), (b,y), and (c, z), we are to show that int((a, x), (b, y))int((b, y), (c, z)) = int((a, x), (c, z)). We write int((a, x), (b, y))int((b, y), (c, z)) = ((b-a)/x,y/x)((c-b)/y,z/y) by the formula defining int in the theorem. This = (((b - a)/x) + (y/x)(c - b)/y, (y/x)(z/y)) by the group composition in IVLS. Canceling factors of y in the numerators and denominators, we see that this = ((b — a + c — b)/x, z/x) which = ((c — a)/x, z/x). And that number pair is indeed int((a, x), (c, z)).

75

4.1.4

Generalized Interval Systems (3)

(B): Given the time span s = (a, x) and the interval (i, p), we are to find a unique time span t = (b, y) which lies the interval (i, p) from the time span s = (a, x). If any such b and y exist, they must satisfy the relation int((a, x), (b, y)) = (i, p), or ((b - a)/x, y/x) = (i, p), or (b — a)/x = i and y/x = p, or b = ix + a and y = xp. So there can be at most one time span t in the desired relation to s and (i, p): That is the time span t = (b, y) = (ix + a, xp). And in fact this particular t is in the desired relation: int(s, t) = int((a, x), (ix + a, xp)); this = (((ix + a) — a)/x, xp/x) by the formula defining int; that = (ix/x, p), which is indeed (i, p) as desired, q.e.d. At long last, we have before us a non-commutative GIS of musical interest. The GIS has important formal properties, which we shall now study. 4.1.4

THEOREM: GIS 4.1.3 has properties (A) and (B) below. (A): For any real number h, the interval from time span (a + h, x) to time span (b + h, y) is the same as the interval from (a, x) to (b, y). (B): For any positive real number u, the interval from time span (au, xu) to time span (bu, yu) is the same as the interval from (a, x) to (b, y). Proof: (A): int((a + M),(b + h,y)) = (((b + h) - (a + h))/x, y/x) (4.1.3.2) = ((b - a)/x, y/x) (algebra) = int((a,x),(b,y)) (4.1.3.2). (B): int((au, xu), (bu, yu)) = ((bu - au)/xu, yu/xu) (4.1.3.2) = ((b - a)/x, y/x) (algebra) = int((a,x),(b,y)) (4.1.3.2). q.e.d.

76

Some commentary on this theorem is in order. The time spans (a, x), (b, y), and so on still rely numerically on an implied referential time-unit and an implied time point zero: (a, x) begins the number a of referential units after the referential zero time-point, and lasts the number x of referential units. The essence of Properties (A) and (B) in the theorem above is that the numerical function int for the GIS under present discussion does not depend at all on the choice of time point zero, or on the choice of referential time-unit. To see this, suppose first that we move the referential zero time-point back h units into the past (= forward ( —h) units into the future). An event originally associated with the time span (a, x) will now be associated with the time span (a + h, x): The event will begin a + h units later than the new zero

Generalized Interval Systems (3)

4.1.5

time-point. Similarly, another event originally associated with the time span (b, y) will now become associated with the time span (b + h, y). Property (A) of the theorem says that in GIS 4.1.3, the formal interval between the time spans associated with the two events is not affected by this transformation. Even though the time spans themselves, as number-pairs, change from (a, x) to (a + h, x) and so on, the interval between transformed spans is the same as the interval between the original spans. Now suppose we change the referential unit of numerical time, so that the old unit is u times the new unit. A duration of x old units is then the same as a duration of xu new units. And the number a of old units after time-point zero is the same as the number au of new units after time-point zero. Hence the events that were associated with time spans (a, x) and (b, y) in the old system will be associated with time spans (au, xu) and (bu, yu) in the new system. Property (B) of the theorem says that in GIS 4.1.3, the formal interval between the time spans associated with the two events is not affected by this transformation. Thus, in the GIS of 4.1.3 the function int(s, t) will always deliver one and the same pair of numbers (i, p), no matter what the referential time-unit and the referential zero time-point by which we calculate numerical durations and distances from time-point zero. To put this intuitively: Given event 1 and event 2 in a piece, we can play the music whenever we want and at any tempo we want, without affecting at all the pair of numbers (i, p) which GIS 4.1.3 will deliver to us as the formal interval between the numerical time spans associated with the two events for any particular analysis. The same can not be said for the commutative GIS of 4.1.2, studied earlier. In that GIS the interval between time spans (a, x) and (b, y) was (b — a, y/x); accordingly, if we replace the referential time-unit so that events once associated with those time spans now become associated with the new spans (au, xu) and (bu, yu), then the interval between the new spans is different. It is not (b — a, y/x), but rather (bu — au, y/x). We noted this earlier. In our present terminology, we can say that GIS 4.1.2 does not enjoy Property (B) of Theorem 4.1.4. In fact, a remarkable theorem is true. Not only does GIS 4.1.3 enjoy the two Properties of Theorem 4.1.4, but it is also essentially the only possible GIS involving time spans as objects that enjoys those two Properties. The meaning of the word "essentially" in the above sentence is made clear by Theorem 4.1.5 following. 4.1.5 THEOREM: Let GIS' = (TMSPS, IVLS', int') be any GIS with time spans for its objects that also enjoys Properties (A) and (B) of Theorem 4.1.4. Then the group IVLS of GIS 4.1.3 and the group IVLS' of the given GIS' are isomorphic via a map f such that, for all time spans s and t, int'(s,t) = f(int(s,t)).

77

4.1.6.1

Generalized Interval Systems (3)

Some commentary is in order before we launch into a proof. The idea of isomorphism between (semi)groups was discussed in 1.11.1 and 1.11.2 earlier. To review here: If G and G' are abstract groups, a function f from G into G' is "an isomorphism of G with G'" if f is 1-to-1, onto, and a homomorphism. f is a homomorphism if f (mn) = f (m)f (n) for every m and every n in G. Supposing f an isomorphism of the abstract groups G and G', then the two abstract groups will have exactly the same algebraic structure under the identification of m in G with its image f(m) in G'. So the first thing Theorem 4.1.5 says is that IVLS and IVLS' have essentially the same algebraic structure, when we identify the member m of IVLS with its image f (m) in IVLS'. Second, Theorem 4.1.5 says that if we take the member m of IVLS to be int(s, t) in particular, then the function f whose existence is asserted makes the image f (m), a member of IVLS', equal precisely to int'(s, t). Thus the function int' is, so to speak, naught but the isomorphic image of the function int under the isomorphism f whose existence the theorem asserts. In this sense, the given GIS' is "essentially" the same as GIS 4.1.3. The (optional) proof of Theorem 4.1.5 is lengthy. To help break it into manageable sections, we shall prove two lemmas. The lemmas appear below as 4.1.6.1 and 4.1.6.2. After that, we shall go on to the proof of the Theorem proper. 4.1.6.1 LEMMA (optional): Let G and G' be abstract groups. Let f be a function from G into G' such that for all m and all n in G, ^m)"1^) = f^n^n). Then f is a homomorphism. Proof of (optional) Lemma: Set m = n = e in the given formula; we get f (e)"1^) = f(e). It follows that f(e)"1 is the identity in G'; hence f(e) is the identity in G', e'. Now set n = e and let m vary in the formula of the Lemma. We get f(m)"1f(e) = fCnT1). Since f(e) = e', we have f(m)-1 = f(m-1) for all m. Then we can rewrite the formula of the Lemma as f(m~ 1 )f(n) = ftm^n)

for all m and all n.

As m runs through the various members of G, m"1 = o runs through the various members of G. Substitute o for m"1 in the rewritten formula; we then obtain the formula f (o)f (n) = f (on) for all o and all n in G.

And thus f is a homomorphism, as claimed. 4.1.6.2 LEMMA (optional): Within the group IVLS of 4.1.3.1,

(i,pr1(j,q) = ((j-i)/P,q/p)78

Proof of (optional) Lemma: We already verified in 4.1.3.1 that (i,p)-1 in

Generalized Interval Systems (3)

4.1.6.2

this group was the element (- i/p, 1 /p). Then (i, p)1 (j, q) = (- i/p, 1 /p) (j, q) = ((-i/p) + (l/p)j,(l/p)q) = ((j - 0/P,q/P) as asserted. Now we are ready for the (optional) Proof of Theorem 4.1.5. We take the time span (0,1) as a referential object within the space of GIS' for purposes of LABELing. That is, we set ref = (0,1). Then the function f for which we are looking, the isomorphism of IVLS with IVLS', is defined by formula (i) below. (i) f(i,p) = LABEL'(i,p) = int'((0, l),(i,p)). On the left of formula (i) the number-pair (i, p) is considered as an interval, a member of IVLS, while in the middle and on the right of the formula, the same number-pair is considered as a time span, a span being LABELed in GIS' by its GIS'-interval from the referential object ref = (0,1). The number-pair (i, p), as a pair of numbers, can be interpreted either way. Now we can write f (i, p)"1 f (j, q) = LABEL'(i, p)'1 LABEL'(j, q), by formula (i). This = int'((i,p),(j,q)) by 3.1.2. This = int'((0, p), (j — i, q)), since GIS' enjoys Property (A) of Theorem 4.1.4 by supposition. And this = int'((0,1), ((j — i)/p, q/p)) since GIS' enjoys Property (B) of Theorem 4.1.4 by supposition. And that = f ((j — i)/p, q/p) by formula (i) above. And that = f((i,p)~ 1 (j,q)) by Lemma 4.1.6.2. Putting together the whole string of equalities we have just noted, substituting m for (i, p) and n for (j, q), we see that we have proved f(m)-1f(n) = f(m-1n) for every m and every n in IVLS. By Lemma 4.1.6.1, we conclude that f is a homomorphism. Since the LABEL' function is 1-to-l from TMSPS onto IVLS' (3.1.2), the function f is 1-to-l from IVLS onto IVLS'. Thus f is an isomorphism of IVLS with IVLS'. It remains to prove that f(int(s,t)) = int'(s,t). Set s = (a, x) and t = (b, y). Then int(s, t) = ((b - a)/x, y/x) and f(int(s, t)) = = = =

int'((0,1), ((b - a)/x, y/x)) (formula (i)) hit' ((0, x), (b - a, y)) (since GIS' enjoys Property (B)) int'((a, x), (b, y)) (since GIS' enjoys Property (A)) int'(s, t). q.e.d.

To recapitulate: Theorem 4.1.5 shows that GIS 4.1.3 is essentially the only possible GIS involving the family TMSPS whose function int is completely independent of the referential time-unit and referential zero time-

79

4.1.7

Generalized In terval Systems (3)

point. To put it more intuitively, GIS 4.1.3 is the .only such GIS, essentially, that will return one and the same element of IVLS as the interval between the numerical time spans associated with two musical events in a piece, regardless of when you play the piece and what tempo you take. GIS 4.1.3 thereby has a privileged theoretical status, as well as a special plausibility for modeling events in the Carter passage and other pertinent music. Since GIS 4.1.3 is non-commutative, it will provide a useful example for illustrating and reviewing the work of sections 3.4 and 3.5 earlier, work that formulated the abstract theory of transpositions, interval-preserving operations, and inversions. 4.1.7 NOTES: Within GIS 4.1.3, the following formulas and facts are true: (A): Given an interval (i, p) and a time span (a, x), the transposition of the given time span by the given interval is T(iip)(a,x) = (a + ix,px). (B): If we fix (0,1) as ref, a referential time span, then the number-pair (a, x), as a member of IVLS, is the LABEL for the time span (a, x): LABEL(a,x) = int((0, l),(a,x)) = (a,x). (C): The (i, p)-transpose of the time span (a, x) is the number-pair given by the composition in IVLS of the two intervals (a, x) and (i, p). T(i(p)(a, x) = (a,x)(i,p). (D): Using the number-pair (a, x) in the same way, as both a time span and an interval, we can show that the interval-preserving operation P(h)U) transforms the time span (a, x) into the time span p

(E): The only central member of IVLS is the identity interval (0,1). (F): No transposition preserves intervals, and no interval-preserving operation is a transposition, the identity operation T(0>1) = P (O ,D excepted. (G): The operation of (c, z)/(d, w) inversion, applied to the time span (a, x), yields the time span I!c;?(a,x) = (d + (c - a)w/x,zw/x) = (d,w)(a,x)~ 1 (c,z).

80

(H): Given time spans s, t, s', and t', then !£'. = I, as an operation on TMSPS if and only if s' = s and t' = t. (I): There are no interval-reversing operations on TMSPS. Proofs and commentary: (A): Via 3.4.1, the transposition of (a, x) by (i, p) is that time span (b, y) which lies the interval (i, p) from the time span (a, x), i.e. which satisfies the equation int((a, x),(b,y)) = (i,p). Thus (b, y) satisfies the equation ((b — a)/x, y/x) = (i, p); whence (b — a)/x = i and y/x = p. So b = a + ix and y = px. The transposed time span (b, y) = (a + ix, px) can be

Generalized Interval Systems (3)

4.L7

described as follows: b lies i x-spans later than a; y lasts p times as long as x. If we turn back to figure 4.4 (p. 74), we will see that the time span tj of the figure is the (4,2)-transpose of s x : ilbegins at bj, 4 xt-spans aftera t ; t tlasts a duration of yt , 2 times the duration Xj of s^^. Likewise, t2 on the figure is the (4,2)-transpose of s2. We noted while studying the figure earlier that s x precedes s2, while tl follows t 2 . We may say that transposition operations, in this GIS, do not only fail to preserve intervals, they even fail to preserve chronology. (B) of the Notes is a straightforward computation: LABEL(a,x) = int((0, l),(a,x) = ((a - 0)/1, Ix) = (a,x). (C) of the Notes applies (B) to the formula of 3.4.3, and (D) applies (B) to the formula of 3.4.4. The interval-preserving operation P(h u) first blows up or shrinks the sample time span (a, x) by a factor of u, transforming (a, x) to (ua, ux), and then moves the latter time span backward or forward in time by h or (—h) numerical units, transforming (ua, ux) to (h + ua, ux) = P(a, x). Remember that these interval-preserving operations are not formal "transpositions" in our non-commutative system! (E) of the Notes is proved as follows. Suppose the interval (i, p) is central in IVLS, that is (i,p)(j,q) = (j,q)(i,p) for all (j,q). Expanding the binary composition on each side of that equation, we infer (i + pj, pq) = (j + qi, qp) for all j and all positive q. Then i + pj = j + qi for all such j and q, whence (p — l)j = (q — l)i for all such j and q. Take j = 1 and q = 1 as one such j and q; then (p — 1)1 = (1 — l)i or p — 1 = 0; we infer that p must be equal to 1. Now we can go back to our general equation, (p — l)j = (q — l)i; substituting p = 1, we infer that (q — l)i = 0 for all positive q. But then i is obviously zero. So p = 1 and i = 0; our given central interval (i, p) must be the identity interval (0,1). (F) of the Notes then follows from Theorem 3.4.8. (G) of the Notes can be computed from 3.5.2 together with (B) of the Notes. Or it can be computed directly from the defining formula of 3.5.1, using the known group structure of IVLS here. (H) follows from 3.5.3, where we proved that l£ = lls if and only if t' = Ig(s') and the interval int(s', s) is central. Via (E) above, this will happen if and only if t' = Ig(s') and s' = s. Since Is(s) = t, this will happen if and only if s' = s and t' = t. (I) of the Notes simply restates 3.6.4 in the present context. We may use figure 4.4 yet once again to picture the effect of an inversion. On that figure, we noted that int(s 2 ,t 2 ) = int^,^). Hence, via Definition 3.5.1, t2 is the tlls2 inversion of s^ that is, Ij^Sj) = t 2 . This concludes our study of a non-commutative GIS which is also a rhythmic GIS of musical interest. We shall now study some timbral GIS structures.

81

4.2.1

82

Generalized Interval Systems (3)

4.2.1 EXAMPLE: Given positive numbers s(l), s(3), and s(5), let the numbertriple s = (s(l), s(3), s(5)) denote the class of all harmonic steady-state sounds (i.e. periodic wave-forms) whose first, third, and fifth partials have respective power s(l), s(3), and s(5). Let S be the family of all such number-triples s, as s(l), s(3), and s(5) range over all positive values. Given positive numbers i(l), i(3), and i(5), let us imagine at hand one or more "devices" (e.g. computer procedures) which have this property: Whenever a harmonic sound is led as input into such a device, the device outputs a harmonic sound whose first, third, and fifth partials have respectively i(l) times, i(3) times, and i(5) times the power of the corresponding input partials. Given i(l), i(3), and i(5), let the number-triple i = (i(l), i(3), i(5)) denote the class of devices that transform harmonic sounds according to these proportions for the first, third, and fifth partials. Let IVLS be the family of all such number-triples i, as i(l), i(3), and i(5) range over all positive values. IVLS is a group under the combination ij = (i(l)j(l),i(3)j(3),i(5)j(5)). Given harmonic class s = (s(l), s(3), s(5)) and harmonic class t = (t(l), t(3), t(5)), take int(s, t) to be that member i of IVLS for which i(l) = t(l)/s(l), i(3) = t(3)/s(3), and i(5) = t(5)/s(5). Then (S, IVLS, int) is a GIS. That is, Conditions (A) and (B) of Definition 2.3.1 obtain. The GIS is commutative. When int(s, t) = i, any sound in class t will have i(l), i(3), and i(5) times the power of any sound in class s, at its first, third, and fifth partials respectively. Another way of regarding the statement "int(s, t) = i" is to think: Any sound of class s, when led as input to any device of class i, will cause a sound of class t to be output. The fundamental frequencies of the sounds are irrelevant here; we are concerned only with certain aspects of their timbral profiles. If a given sound is led through a device of class i, and if the resulting output is then led through a device of class j, the final output will be a sound of the same class as that which would have resulted, had the original sound been led through a device of class ij. Or, more simply, a device of class i concatenated with a device of class j forms a device of class ij. We can make many variations on the specific GIS just discussed. For example, instead of considering partials #1, #3, and #5, we could instead consider partials #1, #2, and #4. Or we could consider partials #1through- # 5, or # 1-through- # 8, or # 1-through- # 8-except-for- # 7, and so on. We can use GIS structures of this sort to build more complex GIS structures of interest. For instance, let GISj be the GIS of the sort just discussed which considers partials # 1-through-#8 of harmonic sounds. We shall call an element s = (s(l),..., s(8)) of GISj a "pertinent spectrum." Now let us take as GIS2 a familiar GIS involving the space S2 of "time points." We imagine a referential zero time-point and a referential time-unit fixed, so that we can label the elements of S2 by real numbers a. IVLS2 is the additive group

Generalized Interval Systems (3)

4.2.1

of real numbers and, given time points a and b, int^a, b) is the number b — a of time units by which b is later than a. (b — a later = a — b earlier.) Let us explore the direct product GIS3 = GISj (x) GIS2. The elements of S3 = Sj X S2 are pairs (s, a), where s = (s(l), ... , s(8)) is a pertinent spectrum and a is a time point. The pair (s, a) models a class of sounds having pertinent spectral profile s at time a. A finite set of such pairs, say the set DVSP = ((Sj, aj), (s2, a-j), ... , (SN, a^), models a class of sounds that have spectrum Sj at time at, spectrum s2 at time a^ ... , and spectrum SN at time a,^. We consider DVSP to be an unordered set of S3-elements, since the chronological order of the time points an imposes a natural ordering on the member pairs of DVSP, no matter in what order we list those pairs. For convenience, we shall assume the members of DVSP to be listed above in chronological order, that is with aj < 82 < ••• < aN. Supposing the time points an to be reasonably close, then DVSP will model a class of sounds with a certain "developing spectrum." Each sound of this class has pertinent spectrum sn at time an.

FIGURE 4.5 Figure 4.5 displays DVSP, with N = 5 in this case, as an array of numbers. If we imagine the plane of the page as a base, and erect at each entry sm(n) a spike of heights sm(n) jutting up from that page, we shall obtain a sort of sketch for a relief map that shows how the spectrum of the sound develops over time. Supposing the time points aj through a5 to be dense enough so as to catch enough salient features of the sound-class involved (e.g. times when some partial has a pronounced local maximum or local minimum value), then we can consider this sketch to be a good approximation for a continuous relief map that characterizes the class of sounds with respect to its developing spectral "signature." This sort of relief-map representation is in common use today as a means for studying various classes of harmonic sounds, including

83

4.2.2

Generalized Interval Systems (3)

familiar instrumental sounds in particular.6 As we have seen, any such relief map can be regarded as approximately a finite (unordered) subset DVSP of S3. The way that DVSP = ((Sj, a x ), (s2, a 2 ),..., (SN, aN)) develops through its own intrinsic chronology can be studied in exactly the same way we earlier studied the unfolding interval vector for a chronologically developing set in another direct-product GIS. That was in section 3.3.1, where we applied our study to the analysis of a passage from Webern's Piano Variations. 4.2.2 EXAMPLE: Fix a lower frequency LO and a higher frequency HI; we shall represent a varying frequency between LO and HI by the variable x. By a "rational spectrum" we shall mean a function s of the variable x, taking on real values, which satisfies Conditions (A) and (B) below. Condition (A): The function s can be written as the quotient of polynomial functions. That is, there exist polynomial functions p and q in x such that s(x) = p(x)/q(x) for every x between LO and HI. Condition (B): s(x) is strictly positive for every x between LO and HI. We shall take as the family S for a GIS the family of all rational spectra s. The rational spectra form a group under multiplication. For if s(x) = Pi(x)Ah(x) and t(x) = p2(x)/q2(x), then p3(x) = p1(x)p2(x) and q3(x) = q1(x)q2(x) are polynomial functions. Hence (st)(x) = s(x)t(x) can be written as a quotient of polynomial functions: (st)(x) = p3(x)/q3(x). Furthermore, (st)(x) is strictly positive since both its factors, s(x) and t(x), are strictly positive. Thus the product of two rational spectra is itself a rational spectrum. The function 1 (x) = 1 is a rational spectrum; it is a multiplicative identity fo the family of rational spectra. If s is a rational spectrum, so is 1/s, where (l/s)(x) = l/s(x); the spectrum 1/s is an inverse for s within the multiplicative system of rational spectra. We shall take as the group IVLS for our GIS the family of rational spectra again, now considered as a multiplicative group. Given rational spectra s and t, considered as members of S, we shall take as int(s, t) for our GIS the rational spectrum t/s, considered as a member of IVLS. It is straightforward to verify that (S, IVLS, int) is a GIS. The GIS models a system of "linear filter classes." With each rational spectrum s we can associate a class of filters. Any filter in this class can be built up from two simple kinds of filters, "all-zero" and "all-pole" filters. Any filte in class s will transform an input sound to an output sound in such a way that the power of frequency x in the output is equal to s(x) times the power of x in the input. The manipulation of sounds and filters in this way is characteristic

84

6. Good examples of the practice can be found in "Lexicon of Analyzed Tones," a series of analysis and plotting programs by James A. Moorer and John Grey published in Computer Music Journal. "Part I: A Violin Tone" appeared in vol. 1, no. 2 (April 1977), 39-45. "Part II: Clarinet and Oboe Tones" appeared in vol. 1, no. 3 (June 1977), 12-29. "Part III: The Trumpet" appeared in vol. 2, no. 2 (September 1978), 23-31. There is also a handsome "relief map" on the cover of that issue.

Generalized Interval Systems (3)

4,3

of certain recent work in computer music.7 We can sensibly talk of "transposing" and "inverting" such filter-classes in our GIS; since the GIS is commutative, these operations will behave in an intuitively familiar way, following the laws developed in sections 3.4 and 3.5 earlier. We can vary our GIS by varying the region within the frequency x varies. We can change the values of LO and HI; we can even consider disconnected regions within which x is to vary. 4.3 METHODOLOGY: In both GIS 4.2.1 and GIS 4.2.2, the formal relations involved match our sonic intuitions only to a certain extent. In either GIS, that is, we may have int(sp tj) = int(s2, tj), while the intuitive proportion between Sj and tj does not much "sound like" the intuitive proportion between s2 and ^ The models suffer here by comparison with the constructs of Wayne Slawson, who has developed an elegant model for an "intuitive" timbral space.8 Yet such considerations should not necessarily lead us to ignore GIS 4.2.1 and GIS 4.2.2. To relate "natural" mathematical structure with intuition is a problem in connection with virtually all theories involving sensory stimuli. For instance, it is mathematically natural to compare the amplitude of two sin waves by saying that one wave has i times the amplitude of the other; this is especially natural if both waves are at the same frequency. Yet if s, and tj = i • Sj are sin waves of the same low frequency, while s2 and t2 = i • s2 are sin waves of the same middle-range frequency, our intuition about the relative loudness of Sj and t, may differ considerably from our intuition about the relative loudness of s2 and t2. Still, nobody would propose that we should feel free to ignore quotients of amplitudes in a study of musical sounds, just because they do not always conform to our intuitions of loudness, give or take some simple transformation. This is the methodological point: It is unfair to demand of a musical theory that it always address our sonic intuitions faithfully in all potentially musical contexts under all circumstances. It is enough to ask that the theory do so in a sufficient number of contexts and circumstances. Perhaps, too, it is fair to ask that the theory be potentially able to address our intuitions in any given musical situation, provided that the situation develops in a suitable musical manner. To support the methodological point, let us explore certain thematic features from the first movement of Chopin's Bl»-Minor Sonata. On figure 4.6, (a) symbolizes aspects of the motive from the opening 7. The techniques are explained lucidly by Richard Cann in "An Analysis/Synthesis Tutorial," Computer Music Journal. Part 1 is in vol. 3, no. 3 (September 1979), 6-11. Part 2 is in vol. 3, no. 4 (December 1979), 9-13. Part 3 is in vol. 4, no. 1 (Spring 1980), 36-42. 8. "The Color of Sound: A Theoretical Study in Musical Timbre," Music Theory Spectrum vol. 3 (1981), 132-41.

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FIGURE 4.6

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Grave measures, written in augmented rhythmic notation to fit the subsequent tempo of the Doppio Movimento. (b) symbolizes the motive of the first theme, (c) symbolizes a motive from the end of the bridge, and (d) symbolizes the motive of the second theme. At the end of the exposition there is a big push to cadence, leading to a big dominant of D[? major. Then there is a repeat sign. Chopin's notation is ambiguous; for this discussion I will assume what I hear strongly in any case, that the repeat goes back to (a), rather than (b). Then th opening note of (a) completes the Dt? cadence; one thereby hears F|? as the second note of (a) all the more strongly, since it is a third of the tonicized Db, despite the notated E natural and the subsequent leading of the note as E natural in the bass line. Here is the assertion I wish to study: The first melodic dyad of (b), marked y on the figure, belongs to the same interval class as x, the first melodic dyad of (a). This relation, like the relation of our four sin tones earlier, is formally "true" but intuitively problematic. At least, the relation of x and y dyads is hard to hear when we first hear the first theme, the first time through the exposition. But, I claim, the asserted relation has the potential for becoming audible, and in fact it does become audible, even highly significant, the second time through the exposition. To hear this one should take motive (b), rather than motive (a), as a poin of departure. By the end of the bridge section, motive (b) has been transformed into motive (c). Rhythmically, (c) augments the durations of (b) by factor of 2, and then augments its own last two notes by yet another factor of 2. Motive (d) introduces the second theme immediately thereafter. Rhythmically, (d) augments (c) by yet another factor of 2, and exchanges the durations of the first two notes (counting the rest as part of the note that precedes it). And then, going around the repeat, motive (a) augments (d) rhythmically by still another factor of 2. Thus the chain of motive-forms (b)-(c)-(d)-(a bis) is generated by a very consistent, indeed relentless, process of rhythmic expansion. After the repeat, when we continue on from (a bis) to (b bis), we a leaping from the end of the chain (b)-(c)-(d)-(a), back to its beginning and

Generalized Interval Systems (3)

4.3

motivic generator. In this larger and later context, we recognize that (b) and (a), the boundary forms of the chain, are rhythmically transformed variants, each of the other. And so we are much more willing to perceive other sorts of relationships linking (b) with (a). In particular it is now much easier, I would say proper and important, to hear dyad y as an ironically scurrying transformation of dyad x, with its portentous weight. We are helped in hearing this relationship by the dyad marked z on figure 4.6, which we have by now heard again and again during the second group. We are also helped by the big D|? cadence prepared at the end of the exposition, which helps us hear F|? at the repeat of (a). The first time around we may have had a certain predilection for E natural because of our associations with the opening of Beethoven's Sonata op. I l l , even specifically with measures 4| to 5^ of that piece. But the second time around, when we clearly hear F(? (as well), it is much easier for us to associate the x dyad, as root-and-minor-third of Dj?, with the y dyad, minorthird-and-root of B(?, particularly since the two dyads both begin on a D(?. To repeat my methodological claim: One should not' ask of a theory, that every formally true statement it can make about musical events be a perception-statement. One can only demand that a preponderance of its true statements be potentially meaningful in sufficiently developed and extended perceptual contexts.

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5

Generalized Set Theory (1):

Interval Functions; Canonical Groups and Canonical Equivalence; Embedding Functions

In this chapter we shall generalize certain aspects of atonal set theory so as to apply in the context of any GIS.1 5.1.1 DEFINITION: Given a GIS(S, IVLS,int), we shall mean by a set in the present chapter a finite unordered subfamily of S. 5.1.2 DEFINITION: If f is any mapping of S into itself and X is any set, we denote by "f (X)" the set of elements f (s) formed as s varies over X. That is, if X = (Sj, s 2 ,..., SN), then f(X) is the set whose members are f(Si), f(s 2 ), ..., and f (SN). If f is not 1-to-l then some of these f-values may not be distinct; then f (X) will have a smaller cardinality than X. If f is 1-to-l then the N f-values listed above will be distinct, and f (X) will have the same cardinality as X. 5.1.3 DEFINITION: Given a GIS, given sets X and Y, then the XjY interval function is a function IFUNC(X,Y) which maps the group IVLS into the family of non-negative integers as follows: For each interval i in IVLS, the value of the function, IFUNC(X, Y) (i), counts the number of distinct pairs (s, t) in S x S such that s is in X, t is in Y, and int(s, t) = i. That is, IFUNC(X, Y) (i) tells us in how many different ways the interval i can be spanned between (members of) X and (members of) Y. Usually, the context will make it clear when we are talking about the

88

1. The agenda, in chapters 5 and 6, parallels and expands upon the presentation of atonal set theory I developed in my article, "Forte's Interval Vector, My Interval Function, and Regener's Common-Note Function," Journal of Music 77ieor>>vol.21,no.2(Fall 1977), 194-237.

Generalized Set Theory (1)

5.13

function int, which maps S x S onto IVLS, and "the interval function" which, given sets X and Y, maps IVLS into the non-negative integers. IFUNC does not figure heavily in the standard literature of atonal set theory. Let us study some examples to see how it applies to that theory; the reader may thereby see the point of the construction, both in that specific application and more generally.

FIGURE 5.1 Figure 5.1 (a) displays the two pitch-class sets Xj = (E, B|?) and Y t = (F, A, C#) as a sort of symbolic melodic antecedent and consequent. Arrows drawn from each note of X t to each note of Y t show which intervals can be spanned, how many ways, between notes of X^ and notes of \l.Here, the even-numbered intervals cannot be spanned at all, and each odd-numbered interval can be spanned in exactly one way. Hence IFUNC(X l5 YJCi) = 0 if i is even, and = 1 if i is odd. Figure 5.1(b) displays the two sets Xj (as before) and Y2 = (G, A, B) in a similar format. We see that IFUNC(X1, Y 2 ) (i) also = 0 if i is even, and = 1 i iisodd.SoIFUNC(X 1 ,Y 2 ) = IFUNC^Y^ as a function on IVLS, even though Y2 is not the same set as Yt—indeed, Y2 is not even a form of Y t . 2 Figure 5.1(c) displays the new sets X 2 and Y3 in the same format. Here X2 is different from X x and Y3 is different from either Y! or Y 2 ; yet IFUNC(X2, Y3) is again the same function of i: Its value is 0 if i is even and 1 if i is odd. Figure 5.1 (d) displays yet another pair of sets, X3 = (E[?, F, G, A, B, C#) and Y4 = (E). The X3/Y4 interval function is still and again the same: IFUNC(X3, Y4)(i) is 0 if i is even, 1 if i is odd. The set X3 here is of different 2. I investigated this sort of phenomenon in an early article, "Intervallic Relations between Two Collections of Notes," Journal of Music Theory, vol. 3, no. 2 (November 1959), 298-301. Its style, unfortunately, makes few concessions to a non-mathematical reader.

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Generalized Set Theory (1)

cardinality from Xj and X 2 ; Y4 is likewise of different cardinality from Y l 5 Y 2 ,andY 3 . Figure 5.1 (d) is displayed in a contrapuntal rather than a melodic format. In general, one could also distinguish a set X from a set Y in other formats: by instrument, by mode of attack (staccato/legato), or by register (as opposed to "voice"), to give only a few examples. The melodic antecedent/consequent format, which seems particularly suggestive, was proposed by Michael Bushnell during his theoretical studies at Stony Brook in the 1970s. Bushnell also initiated and carried through the basic work on the specific analysis following, involving a passage from Webern's op. 7, no. 3, the third of the Four Pieces for Violin and Piano. Figure 5.2(a) reproduces the score through the opening of measure 9. The right hand of the piano over measures 3-8 comprises two melodic phrases, one filling measures 3-4, the other beginning on the B of measure 5 and extending through the F# of measures 7-8. This is all the lyric melody there is in the piece. Figure 5.2(b) displays as X and Y the pitch-class sets that underlie the two melodic phrases. The pitch noteheads representing the pitch classes E, C#, and Eb within Y have been brought down an octave from the music. This is partly for convenience, but partly too because of an idea which will emerge later. Figure 5.2(b) also displays the sets Z0 and Z3 projected by the violin ostinato that accompanies Y. The repeat sign on the figure indicates that we pass from Z3 to Z0, as well as from Z0 to Z3, during the course of this music. Figure 5.2(c) displays some interrelations of X, Y, and the Z-forms, interrelations that all involve the pitch-class interval 3. Each arrow on the figure indicates a T3 relation of one kind or another. The bottommost arrow depicts the T3 relation between Z0 as a whole and Z3 as a whole, a T3-relation of pitches as well as pitch classes. The arrows directly beneath the lower staff of the figure show T3 pitch-relations between the "fourths" of Z0 and those of Z3, and also T3 pitch-class relations between the "fourths" of Z3 and those of Z0, when Z0 is restated directly after Z3. Among all these fourths, Eb-Ab within Z0 and its T3-transform F#-B within Z3 are of special significance. A the crossed lines between the staves of figure 5.2(c) indicate, the Eb-Ab that ends ordered Z0 summarizes in retrograde the Ab-Eb that bounds the total span of ordered melodic phrase X; analogously, the F#-B that ends ordered Z3 summarizes in retrograde the B-F# that bounds the total span of ordered melodic phrase Y. This phenomenon suggests that we explore some sort of T3 relationship between ordered X and ordered Y. On the top staff of figure 5.2(c), the Ab-Eb boundaries of X and the analogous B-F# boundaries of Y are marked by beamed open noteheads. The medial Bb of ordered X is attached to the Xbeam with a stem from a solid notehead; a corresponding C# = T3(Bb) within ordered Y is notated analogously. To hear a function for C#-within-Y analo-

Generalized Set Theory (1)

5.1.3

FIGURE 5.2a

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5.1.3

Generalized Set Theory (1)

FIGURE 5.2b

FIGURE 5.2c

92

gous to the function of B(?-within-X, the reader may listen to the following features of the passage. First, the C# and the F# within Y both receive agogic accents, marked "a.a." on figure 5.2(c). One may query just what this means, in the piano register of figure 5.2(a) at the given tempo. It surely means something, if only something conceptual; the pianist should be thinking like a singing instrument here. Second, the crescendo that begins at the B[? within X, a crescendo reproduced on figure 5.2(c), is analogous in some degree to the crescendo that begins at the C# within Y, even though the latter crescendo does not get all the way to the final note of its phrase. Third, the beamed B, C#, and F# of Y, within figure 5.2(c), occur as every-third-note of ordered Y; there is a serial regularity about their occurrence. Fourth, once the E, C#, and E[> of Y-within-the-music have been brought down an octave to provide the noteheads for Y-within-figure 5.2(c), it is easy to hear the latter structure as a compound gesture, counterpointing the rising beamed B-C#-F# (= T3 (ordered X)) against the falling chromatic counter-gesture-(F-E)-(E[?-D)-. That "falling chromatic line" on figure 5.2(c) fills in the chromatic space between C# and F#, the medial and final beamed notes of the rising gesture.

Generalized Set Theory (1)

5.1.3

The uppermost arrow on figure 5.2(c), then, indicates a structural T3 relation between set X and set T3(X)-embedded-in-Y. It seems secure to assert this relation between the sets, for it seems reasonable to assert much stronger T3 relations, e.g. the structural embedding of T3(ordered X) as every third note of ordered Y, or even the idea that phrase Y as a whole is inter alia a diminuted (ornamented/troped) version of T3 (phrase X). Now we shall inspect some IFUNC values, to see how they interact with these analytic ideas. Figure 5.3(a) tabulates four interval-functions, IFUNC(Z0,Z3), IFUNC(X, Y), IFUNC(X,Z0), and IFUNC(Z3, Y).

FIGURE 5.3

IFUNC (Z0, Z3) tells us that 4 intervals of 3 are spanned between Z0 and Z3; hence Z3 = T3(Z0). Likewise Z3 = T9(Z0) as a pitch-class set, since IFUNC(Z0,Z3)(9) = 4. This T9 relation is concealed, not revealed, by the registration of the pitches involved in the music. Nevertheless the T9 relation has a certain rhythmic effect, as portrayed in figure 5.3(b). The pair "(9,7>)" on that figure is a direct-product interval; it means "a pitch-class interval of 9 is spanned at a distance of 7 sixteenth-notes between attacks."

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Generalized Set Theory (1)

The less prominent values of 2 taken on by IFUNC(Z0, Z3) (less prominent than 4) also have rhythmic implications. Figure 5.3(c) shows how IFUNC(Z0,Z3)(2) = 2 and IFUNC(Z0,Z3)(10) = 2 articulate the rhythm of the violin obbligato. (The sixteenth-note symbols are now omitted from the direct-product intervals.) Figure 5.3(d) shows how IFUNC(Z0,Z3)(4) = 2 and IFUNC(Z0,Z3)(8) = 2 articulate the same obbligato in a different rhythmic way. IFUNC helps us to explore these intervallic/rhythmic substructures more carefully than we otherwise might. (We have plenty of time to do so at the written tempo.) The rhythmic effect of IFUNC(Z0, Z3) (3) = 4 is of course to support the quintuple periodicity of the ostinato, as in figure 5.3(e).

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Now let us return to the IFUNC table of figure 5.3(a) and inspect IFUNC(X,Y). Since X has cardinality 3 and IFUNC(X, Y)(3) = 3, this interval function tells us that T3(X) can be embedded within Y. That is, the function informs us that such an embedding is "true," and would lead us to inspect its potential musical significance in the passage at hand if we had not already done so. The value of 3 is a maximal possible value for IFUNC(X, Y), just as 4 was a maximal possible value for IFUNC(Z 0 ,Z 3 ). Both functions take on their maximal values on the argument i = 3. This sets up a "true" proportion among the four sets involved: X is to subsequent Y just as Z0 is to subsequent Z3, so far as a certain property is concerned (having a maximum IFUNC value at the argument i = 3). Our discussion of figure 5.2(c) has shown that this theoretical truth in fact reflects a musically significant relationship. IFUNC(Z0, Z3) also took on its maximum value at the argument i = 9. But IFUNC(X, Y) does not. Instead, IFUNC(X, Y) has a maximum at i = 8. This tells us that T8(X) as well as T3(X) can be embedded within Y. Is the fact musically significant as well as true? To explore the matter, let us first find the notes of T8(X) as they occur within Y. X = (Ab,B|7,E|7), so T8(X) = (E, Ftf, B). Within ordered-Y, the members of T8(X) appear in the rotated order B-E-F#. Inspecting the score again, one asks if these three notes have any special functions that affect the shaping of phrase Y in the music. I believe they do. Namely, they are the registral and temporal boundary tones for phrase Y. That is, B is at once the first note and the lowest note of the melodic phrase; E is its highest note; F# is its last note. So one can plausibly assert an overall shape for phrase Y that uses T8(X) as a bounding frame, along the lines of figure 5.4. We have been thinking of the set (B, E, F#) as a transposed form of set X, because that transpositional relation is what IFUNC has brought to our attention. As a series, the succession B-E-F# is a rotation of E-F#-B, which is T8 (ordered X). B-E-F# can also be generated as the retrograde series of ordered-X inverted about C#. IFUNC cannot suggest this relation to us; in

Generalized Set Theory (1)

5.1.3

FIGURE 5.4

chapter 6 we shall generalize the interval function to an "injection function" which can. In any case, our noticing that IFUNC(X, Y) (8) = 3 has suggested a search that has led us to the musical relationship of figure 5.4, a relationship that deserves serious aural attention. The relationship engages the register of the high E in the music for phrase Y; we would not hear any boundary function for E if it were an octave lower as in figure 5.2(c).3 Let us return again to the table of figure 5.3(a). We can notice there an interesting resemblance between IFUNC(X,Z0) and IFUNC(Z3,Y). Both functions take on their maximum values at the arguments i = 0, i = 5, i = 6, and i = 11. Figures 5.5(a) through (d) show how the four intervals 0, 5,6, and 11 respectively govern both a model/expansion relation of X to Z0 and an analogous model/expansion relation of (unordered) Z3 to Y. (a) through (d) of the figure show how the two "fourths" of X, transposed variously by i = 0, 5,6, or 11, map into the two "fourths" of Z0; this projects the relation IFUNC(X,Z0)(i) = 2 in each case, (a) through (d) of the figure show analogously how the four possible subtrichords of Z3, all of which are in Forte's set-class 3-5, can each be transposed by one of the key intervals i = 0, 5,6, or 11 so as to map into one of the two subtrichords of Y that lie in the setclass 3-5, both boundary trichords in a certain sense; this projects the analogous relation IFUNC(Z3, Y)(i) = 3 in each case. The latter four mappings, of Z3-trichords into Y-trichords, suggest that Y can be articulated, when heard "against" Z3, as suggested by figure 5.5(e). There we hear Y articulated into a beamed temporal "boundary" comprising the union of its two 3-5 trichords, plus a bracketed temporal "interior" comprising C#, E\>, and D. The articulation of Y in this fashion is reinforced by the coincidence of the interior set (C#, Efr, D) with the cadence set accompanying the melodic Efr that concludes ordered-X in the music. Figure 5.5(f) shows the pitches in the music between the attack of the X-terminating E|? in measure 4 and the comma in measure 5 that separates X from Y in the melody. The set of pitch classes heard in figure 5.5(f) recurs as the interior set of Y, bracketed on figure 5.5(e). A methodological note is in order. Some readers may feel confused rather than enlightened by the variety of ways in which we have shaped and articulated Y during our discussion, specifically in figures 5.2(c), 5.4, and 5.5(e). 3. I am indebted to Taylor Greer for suggesting to me (in another connection) that the "boundary set" of an atonal melodic phrase often has special set-theoretic functions.

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5.1.3

Generalized Set Theory (1)

FIGURE 5.5

96

If Y "really is" an ornamented version of T3(X), then are we not "wrong" to consider Y as a spatio/temporal gesture framed by T8(X), or to consider Y as articulated by figure 5.5(e)? On the other hand, if Y "really is" a spatio/ temporal gesture framed by T8(X), then are we not "wrong" ... (and so on)? One often hears such notions accompanied by a thought like, "after all, the pianist must decide which way to play it." Concerning these issues, the first thing to be said is that the last remark in the above paragraph seriously under-estimates and misapprehends the resources available to a good pianist (or performer in general) even in a context as constrained as that of the Y-phrase in the music. What one can hear, one can play. Let us suppose now that the possibly unquiet reader does hear something significant, or at least engaging, about each of the Yarticulations. (Otherwise there would be no disquiet and no problem.) The disquietude arises intellectually, from considering Y as something which

Generalized Set Theory (1)

5.1.3

"really is," independent of any specified environment. The interpretation of Y manifest in figure 5.2(c) is not an attempt to get at the "real" structure of Y in a context-free environment; rather it asserts that Y, when heard following X and in connection with the many intervals of 3 between notes of X and notes of Y, will tend to be articulated as in that figure. Likewise, figure 5.4 does not pretend to assert something that Y "really is," context-free. Rather, it asserts a structuring for Y that tends to emerge in the environment, "remembering X and hearing the many intervals of 8 between notes of X and notes of Y." Finally, figure 5.5(e) does not try to assert a "real" form for Y either. Rather, it says that Y will tend to articulate in that way when heard in a complicated environment involving Z3, the idea that Y expands Z3 in a manner analogous to the way Z0 expanded X (as in figures 5.5(a)-(d)), and the effect of the cadence harmony displayed in figure 5.5(f), as that event is recalled in its various contexts. The various IFUNC values of figure 5.3(a) are useful tools, as we have seen, for exploring the multifaceted aspects of Y in various of its environments. Figures 5.2(c) and 5.4 explore two different ways in which Yfollowing-X engages IFUNC; these ways will be of practical interest to the pianist shaping the lyric melody of the right hand over measures 3-7. As we noted before, that is all the lyric melody there is in the piece, and since the melody articulates musically into two phrases, the pianist will naturally want to explore listening to the various kinds of "logic" adhering to the way in which Y follows X. Figures 5.2(c) and 5.4 provide that theoretical "logic" in this environment, the one figure in connection with sensitivity to the interval 3, the other in connection with sensitivity to the interval 8. Figure 5.5(e), in contrast, explores Y as it occurs in a different context, that is as it relates to Z3 in a complex theoretical proportion also involving X and Z0 as in figures 5.5(a)-(d). The corresponding musical environment is now not the way the antecedent and consequent phrases of the lyric melody are shaped in the right hand of the piano, but rather the way in which the violin ostinato comments upon that melody, and is commented upon by it. The new musical environment we are now considering involves an interrelation, or rather several interrelations, between the instruments. This will be of particular interest to the violinist, trying to keep the ostinato figure fresh and alive rather than mechanical. In this environment the second appearance of Z3 will sound different from the first. Figure 5.6 shows what I mean. (a) and (b) of the figure show how the first Z3, at measures 6-7, comments upon the opening trichord of Y at the intervals of 5 and 11 respectively, (c) and (d) of the figure, in contrast, show how the second Z3, during measures 7-8, comments (as well) upon the fresh trichord (B, F, F#) within Y, now that F# has appeared on the scene. The new commentary is at the intervals of 0 and 6. When executing these commentaries, it will help the violinist to listen to Y articulated as in figure 5.5(e) earlier.

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FIGURE 5.6

We could of course also consider Y simply in its own environment, spanning various intervals within itself. This would lead us to examine IFUNC(Y, Y), which is essentially Forte's "interval vector of Y." Our study of the Webern passage has made Z0 seem more subordinate than it sounds in the piece; the study so far has also underplayed the role of the 3-note chromatic set (C#, D, E|?), the set which appeared in figure 5.5(e)-(f). A brief discussion of figure 5.7 will attempt to right this balance, without getting fussy about more IFUNC values.

FIGURE 5.7

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The figure transcribes the noteheads of the ensemble up through the cadence at the bar line of measure 5. Over measures 1 through 3 the music slowly exposes the set W0 = (A, 6(7, A(?). (At the written tempo, this takes over 20 seconds.) The boundary tones for the exposition of W0 are shown on the figure with open noteheads: A is lowest and first; Ab is (so far) highest and last. As the piece continues, the rhythm becomes more active and the registral space expands. By the cadence at measure 5 we hear new boundary tones, also shown with open noteheads: E|?5 is a new highest tone; D3 is a new lowest tone and also a new last tone. The four boundary tones on the figure add up precisely to a large-scale projection of the set Z0 = (A, D, Efc>, Aj?). So Z0 has a

Generalized Set Theory (1)

5.1.5

very strong structural meaning in its own right, by the time it enters in the foreground of measure 6 to begin the violin ostinato. The progression of boundary tones on figure 5.7 suggests an inversional relationship: The falling bass, from the opening A3 to the closing D3, sandwiches a rising melody spanned by the local high tones A|?4-then-E|?5. Indeed this rising melody is the X-phrase itself. The injection function, to be discussed in chapter 6, will allow us to engage such inversional relations within and between sets; IFUNC cannot do so. The set W 5 on figure 5.7 is the cadence harmony shown earlier on figure 5.5(f), a set projected in another way by the bracketed "interior of Y" on figure 5.5(e). Figure 5.7 shows how the cadential W5 responds to the opening W0. W5 is of course T5(W0). The bass A of W0 as presented, flanked by its chromatic neighbors B(? and A|? above, progresses over figure 5.7 to the bass D of W 5 , flanked by its chromatic neighbors Eb and C# above. So the T5 relation of W0 as a whole to W 5 as a whole, as that relation moves structurally over figure 5.7, expands upon the T5 relation of A to D in the bass register. This is not lost upon our ears when the violin ostinato begins precisely with A-D- at measure 6, presenting thereby a highly charged T5 relation in the foreground. All these considerations dispose us to bracket off W 5 as the interior of Y with somewhat more aural attention than our earlier discussion could make plausible. Having explored the pertinence of IFUNC in the setting of traditional atonal set theory, let us now return to further study of the formal X/Y interval function in generalized set theory. We recall Definition 5.1.3: Given a GIS, given sets X and Y, then IFUNC(X, Y) is that function which assigns to each i in IVLS the number of ways in which i can be spanned from X to Y, that is the number of pairs (s, t) such that s is in X, t is in Y, and int(s, t) = i. We shall now study how IFUNC is affected as the sets X and Y are manipulated and transformed in various ways. The first theorem to be noted shows that when the roles of sets X and Y are exchanged, IFUNC is in a certain sense "inverted." 5.1.4 THEOREM: IFUNC(Y,X)(i) = IFUNC(X,Y)(i-1). Proof: IFUNC(X, Y) (i-1) is the number of pairs (s, t) such that s is in X, t is in Y, and int(s, t) = i"1. This is the number of pairs (t, s) such that t is in Y, s is in X, and int(t,s) = i. And that number is IFUNC(Y,X)(i). The next theorem shows that IFUNC is not affected when X and Y are both transformed by the same interval-preserving operation P. 5.1.5 THEOREM: Let P be any interval-preserving operation. IFUNC(P(X),P(Y)) = IFUNC(X, Y) as a function on IVLS.

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5.1.6

Generalized Set Theory (1)

Proof (optional): Let PAIRS be the family of pairs (s, t) such that s is in X, t is in Y, and int(s, t) = i. Then IFUNC(X, Y)(i) is the cardinality of PAIRS. Let PAIRS' be the family of pairs (s', t') such that s' is in P(X), t' is in P(Y), and int(s', t') = i. Then IFUNC(P(X), p(Y))(i) is the cardinality of PAIRS'. So, to show that IFUNC(X, Y)(i) = IFUNC(P(X), P(Y))(i), it suffices to show that PAIRS and PAIRS' have the same cardinality. And that will be the case if we can map PAIRS onto PAIRS' by some 1-to-l function f. We shall construct such a function f. Given (s, t) in PAIRS, define f(s, t) = (P(s), P(t)). f(s, t) is indeed a member of PAIRS', for P(s) is in P(X), P(t) is in P(Y), and int(P(s), P(t)) = int(s, t) = i (since P is interval-preserving), f is a 1-to-l map because P is 1-to-l: If (P(s,), P(t,)) = (P(s2), P(g), then P(s,) = P(s2) and P(t,) = P(t,), whence s, = s2 and tj = tj, whence (s,, t, = (s2, tj). It remains only to show that f is onto PAIRS'. The interval-preserving operations form a group of operations on S; hence P"1 exists and is interval-preserving. Given (s', t') in PAIRS', set s = P~V) and t = P"1^'). The reader may verify that (s, t) is in PAIRS, and that the given (s', t') is the image of (s, t) under the map f. q.e.d. Now we shall see how IFUNC is affected when X or Y or both are transposed. 5.1.6 THEOREM: For any transposition operation Tn, the formulas (A), (B), and (C) below obtain. (A): IFUNC(Tn(X), Y)(i) = IFUNC(X, Y)(ni) (B): IFUNC(X, Tn(Y))(i) = IFUNC(X, Y)(in~]) (C): IFUNC(Tn(X), Tn(Y))(i) = IFUNC(X, YXnin'1)

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Proof of (A) (optional): Let PAIRS be the family of pairs (s, t) such that s is in X, t is in Y, and int(s, t) = ni. Then the cardinality of PAIRS is IFUNC(X, Y)(ni), the right side of Formula (A) above. Let PAIRS' be the family of pairs (s', t) such that s' is in Tn(X), t is in Y, and int(s', t) = i. Then the cardinality of PAIRS' is MJNC(Tn(X), Y)(i), the left side of Formula (A). To prove the formula, then, it suffices to show that PAIRS and PAIRS' have the same cardinality. And we can show that by demonstrating a function f which is 1-to-l from PAIRS onto PAIRS'. The desired function is f(s, t) = (Tn(s), t). The reader may verify that f maps PAIRS into PAIRS', that f is 1-to-l, and that f is onto. Proof of (B) (optional): IFUNC(X, Tn(Y))(i) = IFUNC(Tn(Y), XXi'1), via 5.1.4. This = IFUNC(Y, X)(ni-1, via Formula (A) just proved. And that = IFUNC(X, YXiir1), again via 5.1.4. Proof of (C) (optional): IFUNC(Tn(X), Tn(Y))(i) = IFUNC(X, TB(Y))(ni), via Formula (A) above. And that = IFUNC(X, Y)(nin~!), via Formula (B) just proved, q.e.d.

Generalized Set Theory (1)

5.1.8

In contrast to the results of Theorem 5.1.6, there is not much to be said in general about the effect of applying an inversion operation to X, Y, or both, so far as IFUNC is concerned. If we restrict our attention to commutative groups of intervals, though, then we can say something. 5.1.7THEOREM: If I is any inversion operation in a commutative GIS, then IFUNC(I(X),I(Y)) = IFUNC(Y,X). Optional sketch for a proof: In a commutative GIS, the inversions are interval-reversing operations (3.6.3). The present theorem may therefore be proved by the same technique used to prove 5.1.5 above. General questions involving IFUNC and other operations on S will be better pursued using the Injection Function to be developed in chapter 6; that function will generalize IFUNC among other things. The same holds for general questions involving IFUNC and complement relations among sets, where those are relevant. (We have not assumed that S is finite, so the complement of a set need not be a set according to Definition 5.1.1.) IFUNC can be given an interesting interpretation as a probability distribution. 5.1.8 THEOREM: Let X and Y have respective cardinalities M and N. Select a member s of X at random and a member t of Y at random. Then the number IFUNC(X, Y) (i)/(MN) measures the probability that int(s, t) will be found to equal i. Proof: There are MN possible pairs that can be pulled in this way, and IFUNC(X, Y) (i) of these pairs will have the desired property, q.e.d. Theorem 5.1.8 is interesting because we have used IFUNC so far only as a precision tool; the theorem shows that it can also be used to portray a statistical texture. For instance, suppose a clarinet is told to improvise for a time upon the notes of a pitch set X, while a flute is told to improvise for the same span of time upon Y. A statistical field of intervals will result from this improvisation, and that field can be modeled by IFUNC(X, Y) according to the rule of Theorem 5.1.8. Even when we are not applying IFUNC to such "stochastic" compositional settings, it is still sometimes useful to regard it as providing a statistical backdrop for intervallic events. For instance, the notion that a certain interval i appears "often" or "only rarely" between X and Y is implicitly dependent on this backdrop: i appears "only rarely," e.g., compared to how often other intervals appear. We shall look at aspects of the opening from Schoenberg's Violin Fantasy op. 47, taking this point of view. The violin projects the pitch-class set

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5.1.8

Generalized Set Theory (1)

Y = (Bb, A, C#, B, F, G), while the piano accompanies the violin with the set X = (Eb, E, C, D, AJ7, Gb). X is the complement of Y and also an inversion of Y but these relations will not concern us explicitly for present purposes. Figure 5.8(a) displays the values of IFUNC(X, Y).

FIGURE 5.8

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Figure 5.8(a) shows that IFUNC counts "many" odd intervals from X to Y (accompaniment-to-solo, lower-instrument-to-upper-instrument), and "few" even intervals. The theoretical point at hand is that two appearances of an interval is not intrinsically "few"—it would not be few e.g. between two trichords. Rather, the scarce intervals on figure 5.8 (a) are "scarce" only against the statistical backdrop of the table as a whole. The equation IFUNC(X, Y) (0) = 0 expresses the fact that X and Y have no common tones. So the scarce interval 0 does not appear at all between the instruments. The scarce intervals of 4 and 8 each appear in two different ways between piano and violin, and those ways are of analytic interest. Figure 5.8(b) displays the opening noteheads of the piece in order of succession. Figure 5.8(c) shows how the two intervals of 4 between X (piano) and Y (violin) appear. One of those 4-intervals appears between Gb, the last and lowest note, and Bb, the first note and a provisional low note for the violin. The other 4-interval appears between Eb, the first and highest note of the piano, and G, the last and lowest note of the violin. Thus all the notes in figure 5.8(c) are boundary tones of one sort or another for the passage; the figure shows how the scarce interval 4 binds this spatio/temporal frame for the phrase. Figure 5.8(d) shows the scarce interval 8 functioning in a similar way. We have already discussed the Bb and the Eb as boundaries; the B is a high

Generalized Set Theory ( I )

5.1.8

boundary; the low D is a provisional low boundary until the very end of the phrase. Many abstractly interesting questions can be asked about our generalized IFUNC. One family of questions takes the following tack: Given some property that a function from IVLS to the non-negative integers might have, under what conditions on X and Y will IFUNC(X, Y) have that property? For instance, we may ask under just what conditions on X and Y IFUNC(X, Y)^"1) will equal IFUNC(X,Y)(i) for all i. Via 5.1.4, this is the same as asking under what conditions on X and Y IFUNC(Y, X) will be the same function as IFUNC(X,Y). More generally, we can ask under what conditions there will exist intervals m and n such that IFUNC(X,Y)(mi~ 1 n) will equal IFUNC(X,Y)(i) for all i. Satisfactory answers for these questions are not known even in connection with the standard GIS for atonal set theory.4 Another family of questions generalizes in one possible direction the traditional topic of "Z-sets." In Forte's theory, pitch-class sets X t and X2 which are not transposed or inverted forms of each other are Z-related if and only if IFUNCCX^XO = IFUNC(X2,X2), as a function on IVLS. In a general GIS setting we may ask under what conditions on X x and X2 that equation will obtain. We know that the relation will hold if there is an intervalpreserving transformation P such that X2 = PCXJ (5.1.5). The relation will not automatically hold in a non-commutative GIS when X2 is a transposed form of X t : 5.1.6 shows us that if X2 = T^X^, then IFUNC(X2,X2)(i) will equal IFUNCtX^X^nhT1), but not necessarily IFUNCCX^XJCi). Going even further, we may ask under what conditions among the four sets X l 5 Y!, X 2 , and Y2 we will have the relation IFUNCtX^Y^ = IFUNC(X 2 , Y 2 ) (as a function on IVLS). Figure 5.1 earlier provided some examples of this state of affairs, in the relatively well-behaved GIS of traditional atonal set-theory. This is all a vast open ground for mathematical and musical inquiry, even in atonal set-theory. Our questions can be transferred to a more general mathematical setting. Readers who do not have graduate-level mathematical background should skip this paragraph. If we use LABEL to identify S with IVLS, we can see that we are treating S = IVLS as a locally compact group under the discrete topology; our "sets" are the compact subsets of IVLS, and IFUNC(X, Y) (i) is the convolution (f * * g) (i), where f and g are the characteristic functions of the sets X and Y respectively. All of our questions may then be generalized to questions about the interrelations, in a locally compact group, among the characteristic functions of compact subsets. E.g: Given compact subsets X l s 4. Eric Regener, "On Allen Forte's Theory of Chords," Perspectives of New Music vol. 13, no. 1 (Fall-Winter 1974), 191-212. Regener poses essentially equivalent questions in that connection, starting at "Among many other things," on page 204.

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5.2.1

Generalized Set Theory (1)

X 2 , Y t , and Y 2 , with characteristic functions f l s f 2 , g l5 and g2, under what conditions will fr* * gj and f^ * g2 be the same function? As with the special case of IFUNC, the study is much simplified when the group is commutative. Other ways of generalizing our questions about IFUNC will come up later in connection with the Injection Function. We now make a big articulation and turn our attention to generalizing Forte's Interval Vector. To do so, we shall need some further apparatus, in particular the notions of a "Canonical Group" of operations on S and a "Canonical Equivalence Relation" among sets. It is worth noting that IFUNC, and the Injection Function later, can be defined and discussed without invoking those notions. Forte considers pitch-class sets X and Y to be canonically equivalent, by our definition coming up, if Y is a transposition or inversion of X. Here the canonical group comprises the transposition and the inversion operations. (They form a group here because the GIS is commutative.) In other systems of atonal set theory, X and Y are considered canonically equivalent if and only if Y is a transposition of X; then the canonical group comprises transpositions only. In still other systems the canonical group includes not only the transpositions and the inversions but also the circle-of-fifths transformations and possibly other transformations as well.5 The idea of canonical equivalence allows us to speak about "the forms of" a set X; those are the sets X' that can be derived from X by operations in the canonical group, or (what is the same thing) the sets X' which are canonically equivalent to X. The work coming up in section 5.2 generalizes these ideas. 5.2.1 DEFINITIONS: In certain connections we shall fix a group of operations on S and call it "the canonical group." It will be denoted CANON. Sets X and X' will be called "canonically equivalent" if there exists some canonical operation A such that X' = A(X). The defined relation is indeed an equivalence. It is reflexive: X = IDENT(X). It is symmetric: If X' = A(X) then X = A~l(X'). It is transitive: If X' = A(X) and X" = B(X'), then X" = (BA)(X). In any specific context, we suppose there is some good reason for selecting one particular group of operations on the family S as canonical. Generally the good reason will involve intervallic relationships of one sort or another within a GIS for which S is the family of objects. But formally, we do not actually need a GIS structure at all. We could carry through our work if we

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5. An extended discussion of such matters can be found in Robert D. Morris, "Set Groups, Complementation, and Mappings among Pitch-Class Sets," Journal of Music Theory vol. 26, no. 1 (Spring 1982), 101-44.

Generalized Set Theory (1)

5.3.1

just started with some family S of objects and some group CANON of operations on S, not concerning ourselves with formal intervals at all. When we look at things so abstractly, we foreshadow the "transformational" approach to be taken later on in this book. 5.2.2DEFINITIONS: We shall write /X/ to denote the canonical equivalenceclass containing the set X. /X/ will be called, for short, the "set class of X." The term "set class" will grate dreadfully on the ears of any mathematical logician. Still, it is becoming standard usage for atonal theory. In earlier writing I used the term "chord type." But that term loses its intuitive pertinence when we are working with generalized sets of all kinds, including rhythmic sets, timbral sets, sets in direct-product GIS structures, and the like. It is important to understand that the notion of set class depends not only upon the set X at hand but also upon the canonical group CANON selected for the occasion. For example, let us fix the standard GIS of atonal set theory; let us select X = (C, E, G). If we choose CANON to be the group of transposition operations, then /X/, the family of transpositions-of-X, comprises the major triads. But if we choose CANON to be the group of transpositions and inversions, then /X/, the family of transpositions-and-inversions-of-X, comprises all the harmonic triads, major and minor. 5.2.3 LOCUTIONS: "X' is a form of X" means that X' is canonically equivalent to X. /X/ may be referred to as "the forms of X." Given a GIS, it can be a tricky business to decide for any particular theoretical exercise just which operations on S are to be allowed into CANON.6 We shall generally want to include at least the interval-preserving operations in the canonical group. For if P is an interval-preserving operation andX' = P(X),thenIFUNC(X',X') = IFUNC(X,X)(5.1.5).ThusX'-in-itsown-context has the same intervallic structure as X-in-its-own-context; this is a reasonable criterion for wanting X' to be considered "equivalent" to X. When the GIS is commutative, the interval-preserving operations will be exactly the transpositions. When the GIS is non-commutative, we may or may not wish to include the transpositions, as well as the interval-preserving operations, in the canonical group for a given exercise. We can now define the Embedding Number, a construct which generalizes Forte's Interval Vector. 5.3.1 DEFINITION: Given sets X and Y, the embedding number of X in Y, EMB(X, Y), is the number of forms of X (i.e. members of /X/) that are included in Y. 6. Morris (ibid.) discusses this at length, in atonal theory.

705

5.3.2

Generalized Set Theory (1)

The embedding number depends on the notion of set class, which depends in turn upon the canonical group at hand; this cannot be overemphasized. For example, let us work within the standard GIS of atonal theory; let X be some major triad and let Y be some major scale. If CANON consists of the transposition operations only, then EMB(X, Y) = 3: three major triads are embedded in the scale. On the other hand, if CANON consists of both transpositions and inversions, then EMB(X, Y) = 6: six harmonic triads are embedded in the scale. Strictly speaking, we should write EMB(CANON,X, Y) to show that the embedding number varies with the canonical group as well as the sets X and Y. But our notation is already cumbersome enough. If X' is a form of X then /X'/ = /X/; the members of /X'/ are the members of /X/ and therefore, via 5.3.1, EMB(X', Y) = EMB(X,Y). If Y' = A(Y) is a form of Y and EMB(X, Y) = N, let X l f X 2 , . . . , Xn be the distinct forms of X embedded in Y. Then ApCJ, A(X2), ..., A(X n ) are the distinct forms of X embedded in Y'= A(Y). So EMB(X,Y') also = N; i.e. EMB(X, Y') = EMB(X, Y). It follows: If X' is a form of X and Y' is a form of Y then EMB(X', Y') = EMB(X,Y). We have proved that Definitions 5.3.2 following make sense. 5.3.2 DEFINITIONS: EMB(/X/, Y) will mean the value of EMB(X', Y) for any member X' of/X/. EMB(X, /Y/) will mean the value of EMB(X, Y') for any Y in /Y/. EMB(/X/, /Y/) will mean the value of EMB(X', Y') for any X' in /X/ and any Y' in /Y/. Let us consider the standard atonal GIS, and let us fix CANON as either the transpositions, or the transpositions plus the inversions. The various 2note sets will gather into exactly six "2-note set-classes," SC^ SC 2 ,..., SC6. (SC4 for instance contains all the 2-note sets whose notes lie an interval of 4-or-8 from each other.) Given a set Y, we can ask for the values of EMB(SCn, Y) as n runs from 1 through 6, i.e. the values of EMB(/X/, Y) as the variable /X/ runs through the six 2-note set classes. The function giving us those six values is Forte's Interval Vector of Y. From our point of view here, we could call it the "dyad-type vector of Y." By analogy we could study the "trichord-type vector of Y," that is the function which gives us the values of EMB(/X/, Y) as /X/ runs through the various 3-note set classes. (There will be nineteen such classes if CANON contains transpositions only; there will be twelve if CANON contains both transpositions and inversions.) Leaving the GIS of atonal theory now, we can generalize such vectors in an abstract setting by the following definition. 106

5.3.3

DEFINITION: By the "M-class vector of Y," we understand the function

Generalized Set Theory (1)

5.3.5.2

EMB(/X/, Y) as the variable /X/ runs through the various set-classes whose members have cardinality M. In case S is infinite there may be an infinite number of M-member set classes. But since Y is finite it has only a finite number of subsets, which can belong to only a finite number of set classes. Hence EMB(/X/, Y) must be zero for all but a finite number of /X/. 5.3.4 THEOREM : Let the cardinality of Y be N. Let M be a positive integer less than N. Pull M members of Y at random. Then the probability that you have pulled a set of class /X/ is given by the number EMB(/X/, Y)/COMB(M, N). Here COMB(M,N) is the number of combinations of M things that can be extracted from a family of N things; e.g. COMB(13,52) is the number of possible hands at bridge. Proof: There are COMB(M,N) different M-member sets \ve might extract from Y, and EMB(JX|, Y) of those sets will be of class [X|. COMB(M,N) can be calculated to be Nl/(Mi.(N - M)V), where N! is factorial N, etc. Theorem 5.3.4 shows us that EMB, like IFUNC earlier, can be regarded as a statistical measure aside from its uses as a precision tool. The theorem enables us derive a very strong formula interrelating various M-class vectors in a general setting. That formula will be proved in 5.3.5.2 below; we shall first prove a lemma involving some numerical computation. 5.3.5.1 LEMMA: Given positive integers L, M greater than L, and N greater than M, then COMB(L,N)/(COMB(L,M)COMB(M,N)) = 1/COMB(N - M, N - L). Proof (optional): COMB(L, N)/(COMB(L, M)COMB(M, N)) = N!/(L!(N - L)!) divided by the product of M!/(L!(M - L)!) andN!/(M!(N-M)!) = N!/(L!(N - L)!) times L!(M - L)!/M! times M!(N - M)!/N!. In this product the factorials of L, M, and N can each be cancelled from the numerators and the denominators of the participating fractional factors. This leaves the product = 1/(N - L)! times (M - L)! times (N - M)! = (N - M)!(M - L)!/(N - L)!. And that is the multiplicative inverse of COMB(N - M, N - L), as asserted, q.e.d. 5.3.5.2 THEOREM: Let L, M, and N be as in Lemma 5.3.5.1; let ADJUST be the fraction calculated in that lemma. Let Z be a set of cardinality N and let X

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Generalized Set Theory (1)

be a set of cardinality L. Then EMB(X, Z) = ADJUST • SUM(EMB(X, /Y/)EMB(/Y/, Z)), where the SUM is taken over all M-member set-classes /Y/. Proof (optional): Only a finite number of terms in the sum will be nonzero, so summing "over all... /Y/" makes sense. (As /Y/ varies, only a finite number of values EMB(/Y/, Z) can be non-zero.) Let us imagine first pulling an M-member subset from Z, and then pulling an L-member subset from that M-member set. In the first pull, the probability that we have pulled a set of class /Y/ is prob(/Y/) = EMB(/Y/,Z)/ COMB(M,N) (5.3.4). And if we have pulled a set of class /Y/, the probability that our second pull will yield a form of X is prob(/X/-from-/Y/) = EMB(/X/, /Y/)/COMB(L, M) (5.3.4). Probability theory tells us how to manipulate these numbers so as to calculate the chances of ending up with some form of X pulled from Z. Namely: prob(/X/-from-Z) = SUM(prob(/Y/)prob(/X/-from-/Y/)), where the sum is over all possible intermediate pulls /Y/, that is over all the Mmember set-classes /Y/. Now in the probability formula above we can substitute, via 5.3.4, prob(/X/-from-Z) = EMB(X, Z)/COMB(L, N) prob(/Y/) = EMB(/Y/, Z)/COMB(M, N) and prob(/X/-from-/Y/) = EMB(X,/Y/)/COMB(L,M). The probability formula then takes the new form EMB(X, Z) = FACTOR • SUM(EMB(X, /Y/)EMB(/Y/, Z)), where

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FACTOR takes care of the COMB-numbers in the denominators of the various probability values above. Calculating FACTOR out, we see that this fraction specifically equals COMB(L, N)/(COMB(L, M)COMB(M, N)). And that, via the Lemma, is ADJUST, q.e.d. The formula of Lemma 5.3.5.1 is not necessary to prove the formula of the theorem; we could simply define ADJUST to be COMB(L,N)/ (COMB(L,M)COMB(M,N)). But the value 1/COMB(N - M, N - L will often be much easier to compute. For instance try L = 5, M = 9, and N = 11: The value 1/COMB(N - M, N - L) gives us 1/COMB(2,6) = 1/1 very quickly. To give us an intuitive sense of why Theorem 5.3.5.2 is interesting, it will be useful to study a simple example from Fortean set-theory in connection with a topological model. Let Z be the pitch-class tetrachord (A, B, C, D); let X be the dyad (A, C). Here L = 2 and N = 4; we will set M = 3 and examine just what Theorem 5.3.5.2 is telling us. First let us inspect figure 5.9. It represents the tetrachord Z as a tetra-

Generalized Set Theory (1)

5.3.5.2

FIGURE 5.9

hedron in three-dimensional space; the vertices of the tetrahedron are the member pitch-classes A, B, C, and D of the set Z. When we inquire about the 2-note subsets of Z, we are inquiring about the boundary edges of this tetrahedron. The figure lays the boundary edges out for inspection below the tetrahedron. Of the six edges, one belongs to set-class 2-1, two belong to setclass 2-2, two belong to set-class 2-3, and one belongs to set-class 2-5. The embedding numbers at the lower right of the figure express these counts. Now let us inspect figure 5.10. It first analyzes the tetrachord into its four triangular boundary faces, and then analyzes each triangular face into its three boundary edges. Two of the triangles are in Forte-class 3-2; these triangles are labelled Y2 and Y2' on the figure. The other two triangles are in Forte-class 3-7; these triangles are labelled Y7 and Y7'.

709

5.3.5.2

Generalized Set Theory (1)

FIGURE 5.10

110

On figure 5.10 the four triangles are lined up beneath the tetrahedron, and the edges of each triangle are stacked up below that triangle. In the four resulting stacks, each edge of the original Z-tetrahedron appears twice. That is because each edge of the tetrahedron belongs to two of the triangles. (Edge AB, for instance, belongs both to triangle Y2 and to triangle Y7'.) As a result, when we count how many sticks at the bottom of the figure are in Forte-class 2-3, we must divide that count by two, to arrive at the number of sticks in that Forte-class we found on figure 9 earlier. For example, figure 5.9 counted two edges-of-Z lying in Forte-class 2-3, namely AC and BD. EMB(class

Generalized Set Theory (1)

5.3.5.2

2-3, Z) = 2. Figure 5.10 counts as edges-of-faces-of-Z twice as many sticks of class 2-3, namely AC-as-edge-of-Y2, BD-as-edge-of-Y2', AC-as-edge-of-Y7, and AC-as-edge-of-Y7'. What Theorem 5.3.5.2 does in this connection is to ADJUST the count of sticks at the bottom of figure 5.10, dividing it by two to conform with the stick-count at the bottom of figure 5.9. The theorem knows that two is the proper number to divide by here, because 1/2 is the present value of ADJUST = 1/COMB(N - M, N - L) = 1/COMB(1,2) = 1/2. If Z were heptachordal object in six-dimensional space (N = 7) and the Y hyperfaces were pentachordal objects in four-dimensional space (M = 5) and we were again interested in counting edges (L = 2), then we would have to ADJUST our count of sticks analogously by 1/COMB(N - M, N - L) 1/COMB(2,5)= 1/10. The probabilistic method we used to prove Theorem 5.3.5.2 will help us understand figures 5.9 and 5.10 in a somewhat different light. Inspecting figure 5.10, we see that if we peel a triangular face at random off the tetrahedron, the probability is 1/2 that the face will be in Forte-class 3-2 and 1/2 that the face will be in Forte-class 3-7. If we pull an edge at random off a triangle of Forte-class 3-2, our expectation is 1/3 that the edge will be in Forte-class 2-3. And if we pull an edge at random off a triangle of Forte-class 3-7, our expectation is 1/3 that the edge will be in Forte-class 2-3. Hence, according to the theory of probability, our total expectation for pulling an edge of Forte-class 2-3 off the tetrahedron by a random yank is ((1/3) (1/2) + (1/3) (1/2)) = (1/6 + 1/6) = 1/3. And this agrees (as our work says it must) with the probability of that event which we infer from figure 5.9: There we see that of the six tetrahedral edges, two are of Forte-class 2-3; so we infer that our expectation of yanking an edge of that class in a random pull is 2/6, which is 1/3. To pursue farther what Theorem 5.3.5.2 has to do with figures 5.9 and 5.10, and with higher-dimensional analogs of those figures, would lead us deeply into a branch of mathematics called algebraic topology. That pursuit would be very much worth undertaking, but it would be out of place here. In discussing how our generalized embedding number applies to the example of figures 5.9 and 5.10, I have supposed that the reader is already familiar with Forte's use of the interval vector in atonal theory. Now let us see how our generalized theory applies in a very different context, one with which the reader is almost certainly unfamiliar. To that end, we shall study some examples in connection with the non-commutative GIS of time spans which we developed in chapter 4. The first thing we must do is fix the group CANON for our purposes. We shall take CANON to be the group of all interval-preserving operations here. We shall not allow transpositions, much less inversions, as canonical operations for this study. Our reason will become clear.

Ill

5.3.5.2

772

Generalized Set Theory (1)

As we observed in 4.1.7(D), the generic interval-preserving operation P = P (h>u) transforms the sample time span (a,x) into the time span (h + ua, ux). The commentary on 4.1.7(D) elaborated upon this: "The interval-preserving operation P(h u) first blows up or shrinks the sample time span (a, x) by a factor of u, transforming (a, x) to (ua, ux), and then moves the latter time span backward or forward in time by h or ( —h) numerical units, transforming (ua, ux) to (h + ua, ux) = P(a, x)." So if X is a set of time spans (s l 5 s 2 ,...s n ), where sn = (a n ,x n ), then P(X) is the set (s'^s^,.. . SN), wher sj, = P(sn) = (h + ua n ,ux n ). We can imagine the set X here as modeling temporal aspects of a musical "passage" containing N events; then P(X) models analogous aspects of the passage played u times as slowly (1/u times as fast), starting the tempo change from time-point zero, all this played h numerical time-units later ( — h earlier). For example, let us take "the quarter note" as a numerical unit and "the beginning of the piece" as a numerical time-point zero. Imagine a motive consisting of an eighth, a dotted eighth, a sixteenth, and a quarter, played consecutively starting 10 quarters after the beginning of the piece. We could model some temporal aspects of this motive by the set X = ((10,|), (10^,f), (Hi,i), (Hi, !))• Remember that X is formally an unordered set; we have listed its members "in order of appearance" only for convenience here. Let us take h = 1370 and u = 4. Then the transformed set P( h>u) (X) first augments the entire rhythmic setting by a factor of 4, from time-point 0 on; then P(X) plays the augmented motive beginning h = 1370 quarters later, that is beginning 1370 quarters after time point 40, that is beginning at time point 1410. (The augmented motive obtained as an intermediate stage began not at time-point 10, but at time-point u • 10 = 4 • 10 = 40.) So the transformed motive modeled by P(X) consists of a half note, a dotted half, a quarter, and a whole note, played consecutively starting 1410 quarters after the beginning of the piece. Expressing this in numbers, P(X) = ((1410,2), (1412,3), (1415,1), (1416,4)). One can check that each member of P(X) is mathematically related to the corresponding member of X via the transformation P(a, x) = (h + ua, ux), here = (1370 + 4a, 4x). For instance, the third-listed members of the sets X and P(X) are related by the formula P(llii) = (h + u - lli,u-±) = (1370 + 4- lli,4-i) = (1370 + 45,1) = (1415,1). There is no need to restrict our attention to sets modeling consecutive events, as X and P(X) did in the preceding example. We could for instance consider a passage in which a violin plays four consecutive quarters, while a viola plays three triplet halves, while a cello rests for an eighth and then plays two consecutive quarters followed by a dotted quarter. We could model temporal aspects of this passage by a time-span set Y. Supposing that the onset of the passage comes 16 quarters into the piece, we can write Y = ((16,1), (17,1), (18,1), (19,1), (16, f), (17if), (18f,f), (16i 1), (17*. 1), (18-|, l£)). As P varies over the interval-preserving operations, P(Y) models

Generalized Set Theory (1)

5.3.5.2

the ensemble passage, played at (all) different tempos and at (all) different times. The elements of the unordered set Y above are listed, not "in order of appearance," but "by parts," as they were described in the text. Suppose the numerical time-span set Y t models the above passage for string trio at the precise time the music was first imagined clearly by the composer. Suppose the different numerical set Y2 models the passage at the precise time it was played during the first performance. Suppose the still different numerical set Y3 models the passage at the precise time my trio played it yesterday, taking a considerably faster tempo. Given one fixed referential time-point zero and one fixed referential time unit, the numbers denoting the members of the three sets Y l 9 Y 2 , and Y3 will be very different. Our formalism, though, enables us to say that the three sets are all (approximately) canonically equivalent. That is one powerful methodological reason for choosing CANON here to be the group of interval-preserving operations. Another good reason for the choice is provided by the way in which this group relates dyad structure to interval structure in the GIS at hand. We shall now explore that topic. By a "dyad" we understand a set containing two distinct members s and t. By an attack-ordereddyad(AOD) we shall mean a dyad containing say s and t, ordered in the following way: If s begins before t (as a time span), the order is (s, t); if t begins before s, the order is (t, s); if both time spans begin at the same time, the shorter of the two spans is listed first. Since s and t are distinct time spans, these criteria are sufficient to order the dyad. Given an AOD D = (s, t), let (i, p) = int(s, t). Then t begins i s-durations after s begins, and t lasts p times as long as s. Because of the ordering criteria on D, the number i must be non-negative, and ifi = 0 then the number p must be greater than 1. Let us call an interval (i,p) of this form a forwards-oriented interval. We have seen that if D = (s, t) is an AOD, then int(s, t) is forwardsoriented. The converse is also easily seen: If s and t are time spans such that int(s, t) is forwards-oriented, then D = (s, t) is an AOD. We can define (j, q) to be a "backwards-oriented interval" in an analogous way: j must be non-positive, and if j = 0 the number q must be less than 1. Now in the group IVLS the inverse of the interval (i,p) is ( — i/p, 1/p). It follows that the inverse of a forwards-oriented interval is backwards-oriented, and vice-versa. One sees quickly that the members of IVLS can be partitioned into three categories: the forwards-oriented intervals, the backwards-oriented intervals, and the identity interval (0,1). Here now is the crucial manner in which our stipulated canonical group comes into play: Given AODsDj = (s 1 ,t 1 )andD 2 = (s2,t2), then Dj and D2 are canonically equivalent if and only if int(s l5 t t ) = int(s2, t2). It would take too long to include here a formal proof of that theorem; such a proof is appended to the end of the chapter as section 5.6. The theorem is by no means obvious or trivial. Once we have proved it, we can note that the 2-element set

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classes correspond 1-to-l with the forwards-oriented intervals. If D = (s, t) is an AOD, then the set class /D/ corresponds to the forwards-oriented interval (i, p) = int(s, t): Every member D' = (s', t') of /D/, when attack-ordered, has int(s', t') = (i, p); furthermore, if s' and t' are any time spans such that int(s', t') = (i, p), then the AOD D' = (s', t') is a member of the set class /D/. The forwards-oriented intervals thus play exactly the same role here that Forte's "interval classes" play in his atonal theory: They can be used to label the distinct set-classes of dyads. They can be so used, that is, z/we take the interval-preserving operations as CANONical in constructing those setclasses. As a result of this structure, we can develop a very strong formal analog for Forte's interval vector in this particular system (NB). Let X be a set containing more than two members; let D be a dyad; then EMB(D, X), the number of forms of D embedded within X, is equal to the number of ways the forwards-oriented interval (i, p) can be spanned between members of X, where (i, p) is the interval spanning the attack-ordered members of D. In other words, EMB(D,X) = IFUNC(X,X)(i,p). In this sense we can speak of EMB(/D/, X), when /D/ varies over the dyad-classes, as an "interval vector;" (i,p) will vary concomitantly over the forwards-oriented intervals. Let us study some actual interval vectors in this system by way of example.

FIGURE 5.11

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5.4.1 EXAMPLE: Figure 5.11 shows the mensural skeletons for motives (b), (c), and (d) from the Chopin sonata studied earlier (in section 4.3). The rhythmic motives are modeled by sets of time spans, and their interval vectors are tabulated on figure 5.12. Forming and reading these interval vectors becomes easy with practice. The forwards-oriented interval (1,1) labels the set-class of AODs D = (s,t) such that t begins right after s (1 s-length after s begins) and extends the same duration as s (1 times the length of s). Within set (b) we count three instances of such AODs. The AODs are formed by the first-and-second notes of the motive, its third-and-fourth notes, and its fourth-and-fifth notes. Thus the number 3 is entered on the table of figure 5.12, in the row of the table headed by the interval (1,1) and in the column of the table headed "vector of (b)." Set (c) includes only two AODs in the set-class (1,1): the first two notes of motive (c), and the last two notes of the motive. Remember: A pair of successive quarter notes in any tempo at any point in the piece (or any other piece any

Generalized Set Theory (1)

5.4.1

FIGURE 5.12

time) is canonically equivalent to such a pair of successive eighths or successive halves or successive quintuplet sixteenths. All the AODs just indicated belong to the same set-class, the set-class determined by the interval (1, 1) between the first and second members of each AOD. The two AODs of class (1,1) embedded in set (c) are counted on figure 5.12 by the number 2, entered to the right of the interval (1, 1) and in the column headed "vector of (c)." Set (d) also embeds two AODs of class (1, 1), namely the pair of half notes and the pair of whole notes at the end of the motive. Let us now consider the set-class corresponding to the forwards-oriented interval (3, 2). An AOD D = (s, t) belongs to this class if t begins 3 s-spans later than s begins, and lasts twice as long as s. The second quarter of (c) and the first half-note of (c) form such an AOD. So do the last quarter of (c) and the second half-note of (c). These two dyads are tabulated by the entry 2 in the (3, 2)-row and the (c)-vector-column of figure 5.12. The entry of 1 in the (11/2, !/2)-row arises from the AOD formed by the first and third notes of motive (d). In section 4.3 we noted a progressive "expansion" from motive (b), through motives (c) and (d). The progressive broadening of note values, from eighths to quarters to halves to whole notes, is obviously crucial. Our model does not address this aspect of the progression. But it does address and analyze well another aspect of interest, something we might call the "progres-

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sive diversification" of the motives in their internal rhythmic structures. As one sees from the first column of figure 5.12, motive (b) concentrates on only a few intervals, most of which appear more than once. Motive (c) projects only two intervals that appear more than once (in the second column of the figure); no interval appears thrice (in that column). Motive (c) projects more intervals, and more diverse intervals, than (b). Motive (d) projects only one interval that appears more than once; that interval appears only twice. Motive (d) thus continues the process of diversification. The insensitivity of our interval vector to changes in tempo, a defect in some ways, is useful here: It enables us to compare motives (b), (c), and (d), each in its own intrinsic context. We touched on this idea earlier in connection with IFUNC(Y, Y). Motive (a), the motive of the opening Grave, does not appear on figure 5.11 or figure 5.12. If one ignores the anticipation of F(? = E natural in the music, then motive (a) is canonically equivalent to motive (d). 5.4.2 In connection with figure 3.3 (page 41), we earlier studied an "unrolling interval vector" for a set in a different GIS, a set pertinent to Webern's Piano Variations. The present GIS, like the earlier one, has an intrinsic chronology, so we can "unroll" its interval vectors too. The abstract method of doing so will involve a number of technical finesses. To begin the abstract study let us consider the imaginary string trio we discussed a short time ago, and let us imagine another of its passages, which we can symbolize as in figure 5.13.

FIGURE 5.13

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We can model certain temporal aspects of this passage, as we did with the last one, by a set Y of time spans. The violin projects four time spans, (16,1), (17,1), (18,1), and (19,1); let us call these spans vnl, vn2, vn3, and vn4 respectively. The viola projects the two time spans (16,|) and (18f,f); let us call these spans val and va2. The cello projects the two time spans (16j, 2) and (18^, 1^); let us call these spans vcl and vc2. We can list the members of Y "in parts" as vnl, vn2, vn3, vn4, val, va2, vcl, vc2. Of course that is not their "order of appearance" in the music. But what isl

Generalized Set Theory (1)

5.4.2

We might try attack-ordering to list the members of Y "in order of appearance." Then we would list them as vnl, val, vcl, vn2, vn3, vc2, va2, vn4. For many purposes attack-ordering is natural, and we have seen how cogent it is in connection with dyads, intervals, and the canonical group. But we shall not want to use attack-ordering in connection with the way we perceive the members of Y "appearing." To see why not, imagine that we stop the music of figure 5.13 just after the attack of vn2, that is, just after time-point 17, and suppose that we ask just which time spans we have perceived up through that time. Obviously we have perceived vnl. But we have not yet perceived any other spans. True, we have heard both the other instruments attack other spans. But we do not yet know how long those spans are going to be, so we cannot claim to have perceived them as spans, using them e.g. to form intervals in an unrolling interval vector. For all we know as we listen at time-point 17, the viola may be intending to hold onto its note for 24 quarters. Now when we unroll the interval vector for Y in connection with figure 5.13, we are going to want precisely to "stop the music" of the figure at various stages, asking at each stage what intervals we have heard so far. As we have just seen, when we stop the music at time-point 17, we can be sure of having perceived only one span, namely vnl; hence we cannot say we have perceived any proper intervals at all so far. The attack-ordering for Y is deceptive in this connection. That ordering, beginning vnl, val, vcl, vn2,..., makes it seem as if the spans val and vcl have "already occurred" by the time vn2 occurs, attacking at time-point 17; hence it seems (wrongly) as if we ought to count the forwards-oriented intervals int(vnl, val) and int(vnl,vcl) and int(val,vcl) as "having already occurred" by the time we "get to" vn2. But this inference is wrong. As we have seen, no intervals have yet "occurred" by time-point 17, so far as our perceptions of spans and their interrelations are concerned. To reflect the true order of our span-perceptions, we shall want to use a different system of ordering, not attack-ordering but release-ordering. Given distinct spans s and t, s precedes t in the release-ordering if s ends before t ends, or if they end simultaneously and s is longer. If s and t correspond to musical events event 1 and event2, then s precedes t in the release-ordering when we perceive the time span during which event 1 has happened before we perceive the time span during which event2 has happened, or if we perceive both spans simultaneously and recall that event 1 began first. Release-ordering for the set Y thus enables us to articulate the music of figure 5.13 into stages that correspond to our evolving perceptions of time spans "having happened" as we listen. The members of Y in the releaseordering are vnl, vn2, vcl, val, vn3, vc2, va2, vn4. Furthermore, when we articulate the music into such stages, we shall want to demarcate the stages by the time points at which various spans are released, not at which they are attacked. Thus the finest possible articulation of the set Y = (vnl, vn2, vcl, val, vn3, vc2, va2, vn4) into stages for our purposes can be realized as follows.

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5.4.2

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Stage 1: We have heard Y t = (vnl, vn2) at time-point 18, the release of vn2. Stage 2: We have heard Y2 = (vnl,vn2,vcl) at time-point 18^, the releasepoint of vcl. Stage 3: We have heard Y3 = (vnl, vn2,vcl,val) at time-point 18f, the release-point of val. Stage 4: We have heard Y4 = all of Y at timepoint 20, the simultaneous release for vc2, va2, and vn4. By calculating how the interval vectors for Y 1} Y 2 , Y3, and Y4 develop, each expanding the counts of the last among the various intervals counted, we shall be able to model how our sense of intervallic structure evolves as we listen to the musical passage. We shall be able to use our formal model analytically, just as we used analogous machinery earlier in connection with the Webern passage and the expanding interval-counts of figure 3.3. Our work above can now be generalized. Given any set Y of time spans, first list Y as (s:, s 2 , . . . , SN) in the release-ordering. Next identify N or fewer "stages" associated with certain subsets of Y as follows. Stage 1 is articulated at the release of s2; it is associated with a certain subset Y x of Y. Y t = ( s t , s 2 ) unless s3 releases simultaneously with s2; in that case Y t = (s r , s2, s3) unless s4 also releases simultaneously at that time; in that case ... (etc. etc.). After Y! = ($!, s 2 , . . . , SM) has been found, Stage 2 is articulated by the release point of s M+1 . Stage 2 is associated with a certain subset Y2 of Y. Y2 = Y1 + (sM+1) unless s M+2 releases simultaneously with s M+1 (etc. etc.). And so on. Eventually one attains the release point of SN and exhausts the set Y. We can regard the stages as developing in a simple serial rhythm as stage 1, stage 2, stage 3, and so forth. Or we can regard them as developing in a "perceptual rhythm," the rhythm of the various release-points at which the stages articulate. (This is interesting but it oversimplifies the psychology of what is going on.) As the stages develop rhythmically, the evolving interval vectors of Y 1} Y 2 , etc. can be studied. Care must be taken here because the release-ordering of Y does not necessarily coincide with the attack-ordering. It is possible for sm to precede sn in the release-ordering, but to follow sn in the attack-ordering. (The different listings of Y in connection with figure 5.13 illustrate the possibility.) Should this happen, when we get to the stage that notices (the release of) sn in the release-ordering, we shall want to tabulate the forwards-oriented interval int(s n , sm) in our updated interval vector, not the backwards-oriented interval int(s m ,s n ). The reader who likes to fool with computer programming and who has a home computer with a color monitor will enjoy writing an "unrolling interval vector" program. The program will take a set Y of time spans, arrange it in release-ordering, determine the articulation-points of the various stages, and find the corresponding subsets Yj, Y 2 ,..., Y. The program will then compute the interval vector for Y^ and display it on the screen as follows. For each forwards-oriented interval (i, p) that is counted, a colored dot appears at the point (i, logp) on a half-plane grid, (i is always non-negative; log p is positive, zero, or negative.) If the interval appears only once in the set, the dot is violet;

Generalized Set Theory (1)

5.4.3

the more times the interval appears, the more the color of the dot moves toward the red end of the spectrum. (The background of the screen is either white or black.) After the program has computed the interval vector for Y x , it will update the count of various intervals so as to obtain the interval vector for Y 2 , changing the color of some dots on the screen as pertinent. Then it will update the count of various intervals to obtain the interval vector for Y3, and so on. The updating can be done quickly following the method of figure 3.3. (Remember that you may have to adjoin more than one releasing time-span at any new stage. Also remember to adjust for any new dyads that may be release-ordered but not attack-ordered.) The rhythmic updating of the screen can follow either the serial rhythm of the stages or their "perceptual rhythm" as discussed above, either in real time or suitably scaled for visual effect. 5.4.3 EXAMPLE: The technique of unrolling can be applied to EMB-related functions beyond the interval vector.

FIGURE 5.14

The "set" Y of figure 5.14, for example, is articulated into four stages. (We could articulate it farther, but we shall not do so here. Since neither the time unit nor the point zero is specified, Y is not strictly a numerical "set" within TMSPS, but I am assuming the reader will not mind a certain looseness in discourse at this point.) Figure 5.14 also displays "sets" X l 5 X 2 , X'l5 and X'2, all of which can be found embedded within Y. X\ is a canonical form of Xj and X'2 is a canonical form of X 2 . Figure 5.15 shows how the embedding numbers of the set-classes /XJ and /X 2 / within Y develop, as Y develops over the four stages. The values rise at Stage 4 because the dotted half releases there and the appearances of X\ and X'2, augmented (canonical) forms ofXtandX 2 ,can now be counted as "embedded" in Y. Figure 5.15 shows us how the set-class /X 2 / comes on late and strong, pulling ahead of (XJ at Stage 3 and then decisively ahead at Stage 4.

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5.4.4

Generalized Set Theory (1)

FIGURE 5.15

No doubt the reader has recognized Y as interpreting the opening of Brahms's G-Minor Rhapsody. The idea that "/X2/ comes on late and strong" is reinforced by the end of the closing group in the music, where the closing theme is liquidated rhythmically down to a succession of X 2 -forms alternating on the tonic and dominant of D minor. The rhythmic interpretation of figure 5.14 does not exclude other possible rhythmic readings of this music. E.g. one could read triplet eighths where there are triplet rests on figure 5.14; then /Xl/ and /X 2 / would come out in a tie on figure 5.15. But such other interpretations just as clearly do not exclude the reading of figure 5.14. The reader who consults the score will find ways enough in which the relation of quarter note to accompaniment changes after Stage 2 so as to support the reading of the figure. 5.4.4 NOTE: Through section 5.4 so far, we have focused upon the interval vector and more generally the EMB function, in connection with our non-commutative GIS of time spans. The technique of "unrolling in stages" which we applied to this study could also be applied in connection with IFUNC(X, Y), as we unroll either X or Y or both in stages. 5.5 NOTES: Let us return now to the most general abstract setting, that of a family S and a group CANON of operations on S. Following the suggestions of my writings elsewhere, we can explore numbers of interest beyond EMB(X, Y).7 We may define COV(X, Y), for example, the coverfng number of X in Y, as the number of forms of Y that include X. This is not necessarily the same number as EMB(X, Y), the number of forms of X that are embedded in Y. E.g. in atonal theory take X = (C, E) and Y = (C, E, G#); then EMB(X,Y) = 3 but COV(X,Y)=1. If S is finite then COV(X,Y) = EMB(Y,X), where Y and X are the complements of Y and X.

720

7. "Some New Constructions Involving Abstract Pcsets, and Probabilistic Applications," Perspectives of New Music vol. 18, nos. 1-2 (Fall-Winter 1979 and Spring-Summer 1980), 433-44.

Generalized Set Theory (1)

5.6

We may also consider SNDW(X,Y,Z), the sandwich number of Y between X and Z; this is the number of forms of Y that both include X and are included in Z. If we write 0 for the empty set, then SNDW(0,Y,Z) = EMB(Y, Z); if S is finite then SNDW(X, Y, S) = COV(X, Y). If Y' is a form of Y then SNDW(X,Y,Z) = SNDW(X, Y',Z); hence we can write SNDW(X,/Y/, Z) without ambiguity. But we cannot use /X/ or /Z/ for sandwich arguments in this way; SNDW(X,Y, Z) depends very much on the specific forms of X and Z being used as arguments. For example let Z be the C-major scale; let /Y/ be Forte-class 3-4, which we could write as /(B,C,E)/. Let Xj = (C, E). Then, allowing both transpositions and inversions as canonical, SNDW(X l5 /Y/,Z) = 2: There are 2 forms of (B,C,E) that can be sandwiched between (C, E) and the scale, namely (B, C, E) and (C, E, F). Now let X2 = (F, A). X2 and Xj belong to the same set-class, but SNDW(X 2 ,/Y/,Z) = 1, not 2: Only 1 form of (B,C,E) can be sandwiched between (F, A) and the scale, namely (E, F, A). Another interesting number is ADJOIN(X, Y, Z). This is the number of forms Y' of Y satisfying both (A) and (B) following. (A): Y' is disjoint from X. (B): There is some form of Z that includes both X and Y'. To illustrate what this number is inspecting, let X = (C, E), Y = (D, G), Z = the C-major scale. (C, E) + (D, G) lies within some major scale; so does (C, E) + (F, 6(7); so does (C, E) + (F#, B); so does (C, E) + (A, D). Exactly the 4 fourths (forms of Y) metioned in the preceding sentence have both the desired properties (A) and (B); other fourths (forms of Y) either contain C or contain E or do not add up together with (C, E) to lie within any major scale. So ADJOIN((C, E), (D, G), C-major scale) = 4. Inspecting properties (A) and (B), one sees that we can write ADJOIN(X,/Y/,/Z/); it follows that we can even write ADJOIN(/X/,/Y/,/Z/). 5.6 APPENDIX (optional): We prove here the crucial theorem stated earlier, on the relation of dyads and intervals in our non-commutative GIS for TMSPS, using the group of interval-preserving operations as CANONical. Here is that theorem stated again: Given attack-ordered dyads D x = (s^tj) and D2 = (s 2 ,t 2 ), then Dj and D2 are canonically equivalent if and only if imXs^tj) = int(s 2 ,t 2 ). The proof follows. Set Si = (a^xj, t t = (b^yO, s2 = (a 2 ,x 2 ), t2 = (b 2 ,y 2 ). Suppose first that D! and D2 are canonically equivalent; we shall show that int(s 1 ,t 1 ) = int(s2, t 2 ). Say that P = P (h%u) is the canonical operation mapping D x onto D 2 . P maps the members of the first dyad somehow onto the members of the second; conceivably the operation might transform s t into t2 and s2 into t t . But in fact that cannot happen in this situation: P transforms st into s2 and t x into t 2 . To see that, we use the fact that both dyads are attack-ordered. Since D! is attack-ordered, either ^ < bt or (at = b x and x t < y x ). If a1 < bl then ua1 < ubi and h + ua x < h + ub^ hence the P-transform of Sj begins before

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Generalized Set Theory (1)

the P-transform of il. Since D2 is also attack-ordered, that means that the Ptransform of s t must be s2 and the P-transform of tl must be t 2 . (Otherwise we would have b2 < a 2 , contradicting the attack-ordering of D 2 .) So the case a x < bi leads to the desired result; let us investigate the case (a^ = b x and xi < vi)- In tnat case» h + uai — h + ubi and ux1again we infer that t < uy^ (P^), P(tj)) is the attack-ordering for D 2 , and so P(S!) = SjandP^) = t 2 as desired. Now that we have established the relations P^) = s2 and P(ti) = t 2 , the rest is easy: int(s 2 ,t 2 ) = int(P(s1),?(!!)) = int^,^) as claimed, because P is interval-preserving. Now we shall prove the converse. Supposing that int^, tj) = int(s 2 , t 2 ), we shall prove that D x and D2 are canonically equivalent. int(s 1 ,t 1 ) = ((bi — aj/x^yjxi) and int(s 2 ,t 2 ) = ((b2 — a 2 )/x 2 ,y 2 /x 2 ). So what we are assuming can be expressed by equations (A) below. (A): (bt - aj/xj = (b2 - a 2 )/x 2 ; y1/xl = y 2 /x 2 . Using some algebra applied to equation (A), we can derive (B); manipulation of (B) produces (C). (B): x ^ - x ^ = x 1 b 2 - x 1 a 2 ; yiX 2 = x t y 2 . (C): x^ - x 2aj = Xib2 - Xjb^ x^ = y2l Let g be the number such thatx t a 2 — x 2 aj = g = Xjb 2 — x 2 b 1 . Let u be the number such that x 2 /Xj = u = y 2 /y 1 . From the equation x x a 2 — x 2 a t = g we infer x^ = g + x 2 a x ; thence we infer a 2 = (g/x t ) -f (x 2 /x 1 )a 1 , or a2 = (g/Xj) + uaj. In similar fashion, we derive the other equations of (D) below.

(D): a 2 = (g/xO + ua f ; x 2 = ux x b2 = (g/x1) + ub1; y 2 = uy 1 . Seth = (g/x 1 ).Then(a 2 ,x 2 ) = (h + ua^uxj,while(b 2 ,y 2 ) = (h 4- ub 1? uyj. Thus, taking P = P (hiU) , we have s2 = P(s^ and ta = P(tj). Since Pis an interval-preserving operation and D2 = PCDJ, the dyads D! and D2 are canonically equivalent, q.e.d

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6

Generalized Set Theory (2): The Injection Function

This chapter is a pivot in a sort of modulation from the study of Generalized Intervals to the study of Generalized Transformations. The Injection Function can be defined, discussed, and applied to musical analysis without invoking the notion of interval or canonical operation at all. In that regard the Injection Function looks forward to the work lying beyond chapter 6. At the same time the Injection Function is strongly suggested by IFUNC and EMB, either of which it can be used to generalize when there is a GIS or a CANONical group at hand. In that regard the Injection Function logically continues the work of chapter 5; indeed it elucidates some problems that chapter 5 left hanging. It was hard for me to decide, as I pondered this dichotomy, how best to arrange my exposition of the material for this chapter. At first it seemed natural to emphasize aspects of continuity, first showing how the Injection Function grows out of IFUNC and EMB, and then moving gradually to higher and higher levels of abstraction. But when I drafted an exposition in that spirit, I did not like the effect. Looking back at IFUNC and EMB gave rise to so much interesting discussion that an endless time seemed to go by before I could get to discuss the more radical abstract features of the new construction. So I decided upon a different method of exposition. That is why I shall begin by discussing the Injection Function at a high level of abstraction, emphasizing its novelty in contrast to the material of chapter 5. Then, once the reader has become familiar with the new construct, I shall begin stitching in threads from here and there in chapter 5, gradually sewing together the seam I have created. At any rate, that is my plan. The reader who is not happy with it will I hope be able to follow along nevertheless without too much discomfort, once aware of it.

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6.1

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6.1 CONVENTIONS: We shall be concerned with a family S of objects, and with various transformations f that map S into itself. We do not assume that the transformations are necessarily operations (1-to-l and onto S). Operations are capable of entering into groups, e.g. canonical groups, groups of interval-preserving operations and/or transpositions in a GIS, and the like. A transformation that is not an operation can have no inverse transformation on S, and so cannot belong to any group of operations on S. As in chapter 5, we shall use the word "set" to denote a finite subset X of S; more exactly, we shall do so until explicit notice to the contrary. (Toward the end of chapter 6 we shall indicate how the work can be generalized to deal with infinite subsets of an infinite S.) 6.2.1 DEFINITION: Given sets X and Y, given a transformation f on S, then the injection number of X into Y for f, denoted LNJ(X, Y)(f), is the number of elements s in X such that f(s) is a member of Y. INJ(X, Y)(f) answers the question: "If I apply the transformation f to the set X, how many members of X will I thereby map into members of Y?" If f is 1-to-l, then those distinct members of X will map into distinct members of Y, so that INJ(X, Y)(f) will also be the cardinality of f(X) D Y, that is the number of elements that the sets f(X) and Y have in common. But when f is not 1-to-l, this is not necessarily the case. We might e.g. have 5,273,647 distinct members of X all mapping into one member of Y; in this case we might have INJ(X, Y)(f) = 5,273,647, while f(X) and Y might have only 1 common member. 6.2.2 EXAMPLE: Take S to be the twelve chromatic pitch classes. Define f as follows: f(s) is the pitch class C when s is any "white note"; f(s) is the pitch class F| when s is any "black note." Fix X = (C, C|, D, El», E). Take Y = (B, C#, D, E, F, F|). Then INJ(X, Y)(f) = 2. The 2 black notes of X map via f into the member F# of Y; the other notes of X, being white, do not map into any member of Y. They do not because the pitch class C, the image of all white notes under f, is not a member of Y. Now take Y = (F, F#). Then INJ(X, Y)(f) = 2 still and again, for the same reasons. Now take Y = X. Then INJ(X, Y)(f) = 3. The 3 white notes of X all map via f into the member C of Y = X; the other notes of X, being black, do not map into any member of this Y. (Ff, the image of all black notes under f, is not a member of this Y.)

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6.2.3 EXAMPLE: Take S to be the twelve chromatic pitch classes. We define a transformation called wedging-to-E, which we shall denote as WE. The transformation maps E into E and B\> into B\>. Every other pitch class when wedged

Generalized Set Theory (2)

6.2.3

advances one hour towards E along the clock of pitch classes, clockwise or counterclockwise by whichever route is shorter. The effect of W E on the clock of pitch classes is portrayed by figure 6.1.

FIGURE 6.1 W E is not an operation; it is neither 1-to-l nor onto. Nevertheless it is a useful transformation to have at hand for analyzing "Angst und Hoffen," the seventh song from Schoenberg's Book of the Hanging Gardens op. 15. Figure 6.2 shows the pitches in the two chords that form the opening "Angst und Hoffen" motto, chords X and Y.

FIGURE 6.2 Here INJ(X, Y)(wE) = 2: When the transformation W E is applied to the notes of X, 2 of those notes map into notes of Y. The D of chord X, specifically, maps into the E|? of chord Y and the B|? of chord X maps into the Bb of chord Y. Thus "two-thirds of X" is mapped into Y by the wedge. If only the G(? of X wedged into Y then all of X would wedge into Y. That would happen if the F|j of Y were an F natural. So we can consider the F|j of Y (in this context!) as a "wrong note" or a "blue note"; it substitutes for the F natural

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that "should" be there. Indeed, there is a lot of musical action later on in the song that involves the idea of "getting F|? (or E) to move to F." The F(? of Y is not just a blue note, though; it is also the point to which the wedge converges. In that connection it has a tonic character as a potential point of repose or arrival. We shall see later how this potentiality is realized in the music. The operation I = l|j£ = I| is also heavily involved in the effect of the music sketched by figure 6.2. Inversional symmetry about B|? is very strongly projected not only by sonic features but also by the visual layout of the chords, symmetrical about the middle line of the staff. In our new terminology, we can write INJ(X, X) (I) = 3: f or all of X maps into X under the transformation I. We can also observe that INJ(Y, Y)(I) = 2: f of Y maps into Y under that transformation. Once more we can point out that if only F-flat were Fnatural, and so on. (We can presume that F|?-for-F is the pertinent substitution here, rather than E|?-for-E in the upper register. Replacing E[? by E in the music here would result in the syntactically implausible chord (F|?, B[7, E).) Neither IFUNC nor EMB can adequately engage the ideas we have been considering in connection with INJ here. The injection function enables us to discuss several thematic functions for the F|? of chord Y: That note is at one and the same time a substitute for F natural, and the convergence-point of the wedge, and one center for the operation I. INJ also enables us to distinguish very different structural functions for the transformations I and W E . Specifically, I transforms each chord of figure 6.2 into something very like itself"; WE, in contrast, transforms an antecedent chord into something very like a consequent chord. We may think of I as an "internal" transformation and WE as a "progressive" transformation in this musical context. We shall pick up the theoretical implications of that notion later on. Figure 6.3 helps us hear how the transformational ideas under discussion persist, develop, and resolve over the last third of the piece. The figure comprises mainly the notes of the piano over this section, notes which carry the harmony in a homophonic texture. Figure 6.3(a) labels the chords involved. The last three notes of the vocal part are also included, beamed as an arpeggiated chord Z5 that belongs in this progression. The progression leads back to X and Y, "Angst" and "Hoffen," now as an outcome rather than a point of departure. Figure 6.3(b) indicates those notes of (a) which participate in E-wedging activity. Two notes of Z1 wedge into Z2 along the beams, two notes of Z2 wedge into Z3, and two notes (pitch classes) of Z3 wedge into Z4. (Bb of Z3 wedges as a pitch class into Bb of Z4; the registral symmetry of the high and low Bb pitches about the pitch E4 is nice.) Thus INJ(Z n ,Z n+1 )(w E ) is consistently = 2 for n = 1,2, and 3. All of Zj-as-2-note-pcset is projected into its successor set Z2 by the wedge; f of Z2 is projected into its successor, and |ofZ 3 .

Generalized Set Theory (2)

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FIGURE 6.3

The F-with-a-question-mark, notated above the upper beam at Z4, indicates a "missing" F that breaks an otherwise consistent pattern of wedging and inversional balance. In this respect the missing F is exactly like the missing F on figure 6.2 earlier. It is like that F in breaking an E-wedge; it is like that F, too, in leaving the Eb of chord Z4 bereft of its I-partner, just as the E|? of chord Y on figure 6.2 was bereft of its I-partner. The note-against-note relations of figure 6.3(b) show that INJ(Z n ,Z n )(I) = 2 for n = 1 and 2; INJ(Z 3 ,Z 3 )(I) actually = 3. (Within the set Z3 of pitch classes, D and F# invert into each other, while B(? inverts into itself.) Hence the abrupt inversional imbalance at Z4 is all the more strongly felt, since INJ(Z 4 ,Z 4 )(I) = only 1. The metaphor of "imbalance" caused by "something missing" interacts well with the text: The singer is in a state of emotional discombobulation caused by the lover's absence. Continuing along figure 6.3(b), we hear how, when the progression Zt etc. recurs as the progression Z\ etc., the singer's final sign-off supplies the required F natural at the right moment, within the arpeggiated set Z5. The required Z5, which continues the patterns of wedging and inversion consistently, sets the word begehre = require. (The singer claims not to require the consolation of any friend.) Once the begehrte Z5 has appeared, the wedge can

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and does converge all the way to the E of Z6, across the pickup chord Z'4 in the music. The level of progressive wedge-projection and internal I-projection is thus restored: INJ(Z'3,Z5)(wE) = 2, INJ(Z 5 ,Z 6 )(w E ) = 3; INJ(Z5, Z5)(I) = 2,INJ(Z 6 ,Z 6 )(I) = 3. The convergence of the wedge to its focal point E within Z6 coincides with and supports a big structural downbeat. The sonority of Z6 has earlier been associated with the word Seufzer, during the text, "meine Worte sich in Seufzer dehnen (my words trail off in sighs (or groans))." Figure 6.3.(b) portrays the sighing and trailing-off ideas very well. It also shows how the beamed sighing-progressions are framed by the elements (C, A[?) and (E) of the Seufzer-chord, which (for this reason and for others too) takes a big downbeat when it appears as Z6. Figure 6.3(b) shows how the wedging commences yet once more after Z6, continuing through X to Y, which is the end of the piece. Within X and Y, the pitches D5 and Et>5 of the music are brought down an octave, to be displayed as D4 and E|?4 noteheads on the figure; this shows clearly how the pitch classes D and E[? contribute to the final wedge. In particular, it brings out strongly how the final progression, Z6-Z7-X-Y, recapitulates the initial wedgestructure of the opening progression Z1-Z2-Z3-Z4 on figure 6.3(b). The "blue note" Fb of the final Y, shown on figure 6.3(a), takes on added significance in its "tonic" function, as it prolongs the downbeat E from the Seufzer-Z6(Dehnung). Figure 6.3(c) sketches in a format similar to (b) the influence of a subordinate wedge and inversion over this passage. That is wedging-to-F# and the inversion-operation J = Ipj| = l£. The symbols "A|?-G-F#//," at the beginning at bottom of the figure, show how the inner voice of the chords attains the goal and center F# of the wedge, getting there "from above" over chords Zi through Z3. Then from Z\ right on, all the way to X, the outer voices almost succeed in converging to F#(G[?), except that F is missing in the lower voice. Again we run into the thematic and structural "missing F!" The missing F now has a new structural function: It is missing as a semitone neighbor to Gj? in a wedge converging to G(?, the bass of the Angst-chord X. The earlier missing F was missing (inter alia) as a semitone neighbor to E in wedges converging to E, the bass of the Hoffen-chord Y. The T10-relation between the bass note of X and the bass note of Y is thus expanded into a larger T10-relation, a relation involving the respective wedges and inversions about those notes. The larger relation can be expressed by the

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1. I discuss these ideas and others of the same sort elsewhere, exploring more systematically rhythm, meter, text setting, registers, doublings, and other features of the music. The reader who would like to go deeper into the piece itself will be interested by that article, which is less preoccupied than we must be here by theoretical constructions of various sorts. The article is "A Way Into Schoenberg's Opus 15, Number 7," In Theory Only vol. 6, no. 1 (November 1981), 3-24.

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transformational equations WE = T^w^T^1; I = T10JT101. Essentially, these equations "modulate by T10" the wedging and inversional transformations centering on F#, to obtain the analogous transformations centering on E. We shall explore the idea of such "modulation" more generally and more formally later on. The present remarks are meant to prepare that later exploration. The reader can hear in connection with the example at hand that the relation T10(Gb) = E has to do both with the bass line in the chord-succession X-Y, and also with the relation of WF* and J, in figure 6.3(c), to W E and I, in figure 6.3(b). The reader will then be able to summon that musical experience to mind during the abstract discussion of transformational "modulation" later on (in connection with Formula 6.7.2(c)). The formalities of INJ, applied to the transformations W E and I on S, have engaged figure 6.3(b) only partially. The visual layout of that figure conveys a good deal more than our formal machinery has so far described. For instance, the figure shows by its note-against-note layout that we do not simply have an Ab in Zt wedging to a G in Z2 and, independently, a C in Zl wedging to a C# in Z2; rather the "I-partnership" of (Ab, C) within Z^ wedges as a whole to the "I-partnership" of (G, C#) within Z2. In a similar sense, the missing F of Z4 is not just a missing note; it is a missing I-partner for Eb, without which the wedge cannot converge; as a missing partner it symbolizes the absent lover. And in a similar sense, the last line of text, "(that) I require the consolation of no friend," is symbolized exquisitely by the pitch class E as goal of the wedge and center of inversion: The pitch class has and needs no partner; it gets along by itself, perfectly self-centered in its Seufzer. The visual layout of the figure brings out such ideas; our transformational machinery has not as yet adequately engaged them. But it can be formally developed so as to do so. The operation I partitions the family S into distinct "transitivity sets" (Bb), (A,B), (Ab,C), (G,C#), (Gb,D), (F, Eb), and (E). I transforms the members of each transitivity set among themselves: I(Bb) = Bb; I(B) = A and I(A) = B; and so on. Such transitivity sets enable us to engage the notion of "I-partnerships" in our formal machinery. Many of the chords under consideration embed an entire transitivity set; some chords even embed two (e.g. Z3 = X = Angst and Z6 = Seufzer). Frequently a transitivity set embedded in one chord is transformed as a whole by W E into a transitivity set embedded in the next chord. Such ideas can be developed very abstractly in connection with the generalized INJ function. Here, it is formally important that the transformations I and W E commute. It is not remarkable that the visual aspects of figure 6.3 can be described by formal aspects of the INJ machinery when suitably extended. After all, one could hardly conceive the layout of the figure without some prior intuitions of W E and I as transformations; the formal extensions of the machinery, in that connection, amount only to making the relevant intuitions explicit.

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Our investigations so far have focused on harmony and voice leading, as we explored the structural functions of various transformations in this song, using INJ. Now let us use INJ to explore some melodic functions of various transformations therein.

FIGURE 6.4

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Figure 6.4 will help us in this endeavor. The figure transcribes the pitches from the opening of the voice part, where they set the first line of text. Schoenberg's spelling projects throughout the phrase a strong visual inversional symmetry about the third line of the staff, where BW would appear as a center of I. Ordinal numbers 1, 2, ... , 10 appear under the first, second, ... tenth notes of the figure. The events of the melody are modeled here in a space S whose members are pairs (n, p), n being an ordinal number and p a pitch class. As a serial structure, the "melody" is modeled by an unordered set of ten such pairs; the elements of this set are the pairs (2, G!>), (1, D), (10, El>), (3, Et), and so on. "(2, G!>)" can be interpreted as saying "The second note is Gk" Arrows on the figure indicate transformational relations that will interest us here. Each arrow is labeled by a pair of symbols comprising a number (1, 2, 3, 5, or 6) and a letter (I or w). The number indicates how many ordinals later the transformed pitch class appears. The letter indicates a pitch-class transformation, w standing for WE. Thus the arrow labeled "6, w" which issues from the third note of the series, E\>, indicates that the note is transformed into a note appearing 6 order positions later, via the wedge transformation. The arrow from A to Cl>, labeled "2, I," indicates that the A is transformed into a note appearing 2 order positions later, via the / transformation. More formally, the transformation 6, w maps the element (3, E\>) of the melody into the element (3 + 6, w(EI>)) = (9, Fl>); the transformation 2, I maps the element (6, A) into the element (6 + 2, I(A)) = (8, Cl»). The transformations (6, w) and (2, I) are well defined by this formal method on the space S of pairs (n, p). (6, w) maps the pair (n, p) into the pair (n + 6, w(p)); (2, I) maps the pair (n, p) into the pair (n + 2, I(p)). The transformationsare not operations, w itself is not an operation on the twelve pitch

Generalized Set Theory (2)

6.2.3

classes; even beyond that, the ordinal aspect of the mappings prevents the transformations from mapping S onto itself. E.g. there is no (n, p) in S (with n a positive integer) such that (2,I)(n,p) = (1, Ab). The dotted arrows and question marks on figure 6.4 arise from the possibilities of considering the F|?s as substitutes for F naturals. If the flats were naturals, the dotted arrows would be solid and the question marks would disappear. We shall denote by XJJ, the set comprising the mth through nth events of the series (i.e. melody-set). Using the subsets XJJ,, we can apply familiar "unrolling" techniques to the situation, now using our injection numbers. For instance INJ(X*, X*) (f) = 2, where f is either (2, w) or (1,1), assuming we allow the dotted lines. The equation states: The set comprising (1, D), (2, Gb), (3, Eb), and (4, F(b)) contains "2" members that transform into the set under the transformation (2, w), and "2" members that transform into the set under the transformation (1,1), supposing that Fb is read "as if" F natural. This equation engages a significant "unrolling" when compared to the equation INJ(Xi,Xi)(f) = (only)l for the same transformations f. We can compare these internal transformations of X| with the internal transformations of Xf, the next 4-element subset of the melody. Xf comprises (5, C), (6, A), (7, Ab), and (8, Cb). INJ(Xf, Xf) (2,1) = 2: With respect to the new tetrad Xf, the transformation (2,1) plays the same internal role that (1,1) did in connection with the first tetrad X*. And, as the figure shows, (3,w) plays the same role with respect to Xf that (2,w) played with respect to X^. That is so even though INJ(Xf, Xf) (3, w) is only 1, not 2. Our arrow diagrams capture a certain picture of X* on the figure, as it appears bound together internally in a certain way by (1,1) and (2, w) arrows. The same kinds of arrow shapes capture a similar picture of Xf on the figure, as it appears bound together in a similar way by (2,1) and (3, w) arrows. Our model enables us to observe an interesting augmentation of ordinal distances, from the arrow transforms binding X| to the arrow transforms binding Xf. That is, within X* I-relations occur 1 note apart and w-relations occur 2 notes apart; within Xf these ordinal distances are expanded: Irelations occur 2 notes apart and a w-relation occurs 3 notes apart. This serial augmentation is particularly interesting because Xf takes only half the time to sing as did X?, in the clock time of the music. Our discussion of X*, Xf, I, and w is enhanced by the observation that no I-arrows and no w-arrows lead events of the first tetrad to events of the second, on figure 6.4. In our terminology, INJ(Xf, Xf) (f) = 0 when f is either (n, I) or (n, w), for any n. This observation specifically enhances our sense that the melody articulates into X* + Xf + Xg°, when heard in the context ofw and I relations. The italicized phrase is meant to recall our earlier discussion in connection with the various contexts of a melodic phrase within the Webern violin piece. Our sense of X* + Xf + X^0 in this context is further enhanced

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by the (6, w) arrow on figure 6.4, extending from inside X* over X| to Xg°. INJ(X}, X*°)(6, w) = 1, after INJ(Xf, X|)(n, w) had been zero for all n. The ordinal distance over which w functions continues to grow: The first w-arrow(s) had ordinal span 2 within Xf; the next w-arrow had ordinal span 3 within X|; now a w-arrow has ordinal span 6, between X* and Xg°. In this context the F flat of (9, F|?) in the melody is "correct"; it is in fact the goal of the wedge. One notes the care with which the high F[? is distinguished by the composer from the low F[?. When we hear the penultimate F|? as "correct" and ignore the last dotted arrow on figure 6.4, we get a sense of "ordinal expansion" over the phrase as regards not just w spans but also I spans. First I projects at ordinal distance 1 and w at distance 2; next I projects at distance 2 and w at distance 3; finally I and w both project at distance 5, and w projects at distance 6.

FIGURE 6.5

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The injection function, like IFUNC earlier, enables us to discover and explore relationships our ears might not otherwise pick up quickly. Figure 6.5 (a), for instance, elaborates on the observation that there are 4 transformations f of form (n,T6) such that INJpC|,Xg)(f) is non-zero, allowing Ffr to represent F natural. So there are four T6 arrows on the figure, projecting al of X* progessively into all of Xf at a variety of ordinal distances. This contrasts sharply with the absence of any I or w arrows on figure 6.4 that led from anywhere within X* to anywhere within Xf. So far as the two tetrads in the vocal melody are concerned, we may put it that I and w are "internal" transformations, while T6 is "progressive." The structure of figure 6.5(a) is hard to pick up by ear alone because its predominant ordinal rhythm of "3 later" conflicts both with the motivic rhythm of the music, and with the 2-later rhythm established in figure 6.4 within X*.

Generalized Set Theory (2)

6.2.3

The structure of figure 6.5(b) is also easy to pick up by inspecting INJ^X^X*), hard to pick up at first by ear alone. It is reasonable to pay some attention to the hexad X* here because that set spans the transfer of the low F[? to the high Ffj in the melody, the first repeated pitch-class of the series. Inspecting the internal structure of the hexad with our machinery, one notes that there are relatively many transformations f of form (n, T8)-for-some-n such that INJ(X4,X4)(f) is high or positive. The arrows on the figure show how this works out analytically. For transformations f involving other T,, not as many analogous arrows would appear. The strong ordinal rhythm of the arrows on figure 6.5(b) is supported by the contour of the pitches—C, Ab, and both the F|?s are all turning points—and also by the rhythm and meter of the music. In this context, the melodic rise over the octave F[? takes place through an ornamented arpeggiation of an augmented triad F|?, C, A^, Ffr. One recognizes the triadic set of pitch classes from earlier discussion. It is Z 6 , the Seufzer triad into which the E-wedge will converge at the big downbeat near the end of the piece. And so on. One could combine the pitch classes or the pitches of the melody into formal pairs not only with the ordinal numbers 1 through 10, but also with the durations of the written notes, or with the time points at which the notes are attacked, or with the time spans of the written notes in our noncommutative GIS, and so on. One would get interesting results in each case. I have used ordinal numbers here because they furnish some new kinds of ideas about non-mensural rhythm, and because they guarantee that none of our transformations on the space of elements (n, p) can be operations. Even if OP is an operation on pitch classes, like I, the transformation (5, OP) cannot map our pair-space onto itself: There is no (n, p) such that (n + 5, OP(p)) = (2, q). So using the ordinal-number model gives INJ another opportunity to show how smoothly it handles transformations that are not operations. So far as the "new kinds of ideas about non-mensural rhythm" are concerned, we can take note that our pair-space is one useful way to model serial melody. Later we shall explore other interesting models for representing series of pitches or pitch classes (or anything else). As I mentioned earlier, the reader who is interested can find a more ample analysis of "Angst und Hoffen" for its own sake elsewhere in my writings. The interested reader might also wish to consult my analytic remarks elsewhere on "Die Kreuze," Number 14 from Pierrot Lunaire.2 Ideas of wedging and inversion are also engaged there. The pertinent wedge transformations are wedging-to-(C/C#) and wedging-to-(F#/G). wc/c# maps F#-to-F-to-E-to-Ebto-D-to-C#-to-C# and G-to-A|?-to-A-to-B|Ho-B-to-C-to-C. WF*/G transforms the pitch classes in analogous wise with respect to the focal goal-dyad 2. "Inversional Balance as an Organizing Force in Schoenberg's Music and Thought," Perspectives of New Music vol. 6, no. 2 (Spring-Summer 1968), 1-21. The discussion of "Die Kreuze" is on pages 4-8.

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F#/G. Figure 6.6 sketches a sense of how these transformations pertain to the opening of "Die Kreuze."

FIGURE 6.6 No pitch class p satisfies wc/c*(p) = F#. Likewise no p satisfies wF*/G(p) = D[?. This feature of the transformations is actually projected by the music, as the figure shows: The first F# and the last Db sit out their respective wedgegames. Iptf = Ic* naturally functions prominently in connection with the two wedges. l£ also figures in the music; it structures the second chord of figure 6.6 as an "internal" transformation. Let us stand back for a moment and think about the analytic uses to which we have put INJ so far. Nowhere in the discussion of the Schoenberg pieces have we used the word "interval" or even invoked the concept, except so far as it is implicit when we label certain operations as T 10 , T8, and so on. Nowhere, therefore, have we needed to use the fact that the family of pitch classes is a GIS. Nor did we need to suppose that our melodic space of elements (n, p) was a GIS, which in fact it was not. We have nowhere needed to suppose that the transformations we were inspecting were 1-to-l or onto; many in fact were not. From all this we get some idea of how generally the INJ construct can be applied in how great a variety of situations. We shall increase our sense of that variety now by studying another application of INJ to a situation not directly involving a GIS for the space S of elements. 6.2.4 EXAMPLE: Our space of elements for this study will be the family PROT of protocol pairs. A protocol pair is an ordered pair (p, q) of distinct (NB) chromatic pitch classes.3 There are thus 132 = 12 times 11 protocol pairs. A twelve-tone row can be regarded as a certain set within PROT: The pair (p, q)

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3. More generally, we could consider protocol pairs of distinct objects from any finite family, and the mechanics of our discussion coming up would obtain, so far as the theory was concerned.

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6.2.4

is in the set if and only if p precedes q in the row. Note that while the row imposes a certain ordering on the twelve pitch classes, the set under consideration is an w«ordered subset of PROT, i.e. an unordered collection of pitchclass pairs. Besides rows, we can consider other subsets of PROT that are consistent with our intuitions of "ordering pitch classes." To be consistent in this way, a set X must satisfy two conditions. First, we cannot intuit both p-preceding-q and q-preceding-p. Second, if we intuit p-preceding-q and q-preceding-r, then we intuit p-preceding-r. These two conditions translate into the two formal properties following, PO1 and PO2, which X must satisfy as a collection of pairs. (PO1): There is no (p, q) in PROT such that X contains both (p, q) and

(q,p).

(PO2): If (p, q) and (q, r) are members of X, then so is (p, r). Mathematically, a subset X of PROT that satisfies (PO1) and (PO2) is called a (strict) partial ordering of the pitch classes. The special partial orderings that correspond to rows are the "linear" or the "simple" orderings L; these subsets of PROT satisfy in addition the condition (SIMP) below. (SIMP): For any (p, q) in PROT, either (p, q) or (q, p) belongs to L. The set-theoretic condition matches our intuition that either p will precede q in the row, or q will precede p. Representing twelve-tone rows as linear orderings is attractive in many ways. For one thing it makes all rows conceptually equal. That is, it does not assign explicit or implicit priority to one row (e.g. the chromatic-scale row), from which other rows are explicitly or implicitly derived. The model assumes no a priori ordering of the pitch classes; any row orders them as well as any other row. This is very much in the spirit of the classical twelve-tone method. Other attractive features of the model will become apparent presently. In connection with the melody from Schoenberg's "Angst und Hoffen" a little while ago, we brought attention to the way in which series of pitches, pitch classes, and the like could be represented by pairs (n, p) consisting of ordinal numbers n and objects p. Now we have a different way of representing such series, provided they are non-repeating (NB). Our new representation allows us to apply set theory to a linear ordering L, together with its various transforms and other partial orderings X of interest as subsets of PROT. The old model represents the row of Schoenberg's Fourth Quartet by a family of pairs (1, D), (4, B|?), (3, A), (2, C#),... and so on: The first note of the row is D, the fourth note is 6)7, the third note is A, the second note is C#, and so on. The new model represents the same row by the family of pairs (A, B(?), (D, B[>), (C#, A), (D, C#),... and so on: A precedes B^, D precedes Bj?, C# precedes A, D precedes C#, and so on.

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No matter which formal model we use, it will still be convenient to use the notation D-C#-A-B[?-... for quick perusal. Partial orderings that are not rows can model many structures of interest in twelve-tone theory. The partial orderings X: and X 2 on figure 6.7 exemplify only two such types of structure from among many.

FIGURE 6.7 Xj models the small linear motive E-A-Bb. As a subset of PROT, X! contains the three protocol pairs (E, A), (E, 8(7), and (A, 6(7). The reader may check formally that this 3-element set satisfies conditions (PO1) and (PO2). X t models our being sure of the three cited precedence relations, and unsure of or indifferent to any other precedence relations. Ll is what I take to be "the row" of Schoenberg's Moses undAron; as a subset of PROT, Lj contains the pairs (A,Bb), (A,E),..., (A,C); (Bb,E), (Bb,D),..., (Bb,C), (E,D),..., (E,C);...;(B,C). X2 models an aggregate governing the soprano, alto, tenor, and bass voices of the four-part texture at the beginning of Variation 3 in Babbitt's Semi-Simple Variations* X2 contains the twelve pairs (B, D), (B, Eb), (D, Eb); (G, Bb), (G, F), (Bb, F); and so on. L2 models what I take as the row of the piece, that is, the succession of pitch classes formed by the first twelve notes in the soprano voice. We shall be examining various numbers INJ(L 1 ,X 1 )(f) and INJ(L2, X 2 )(f) in connection with a few analytic observations on the two pieces. The transformations f on PROT which will attract our interest are of types (l)-(4) following. (1) Transpositions of pairs, T;(p, q) = (Tj(p), Tj(q)); (2) in versions of pairs, I(p,q) = (I(p),I(q)); (3) retrogression of pairs, R(p,q) = (q, p); (4) combinations of types (1) through (3). These transformations are all well defined on PROT; they do not map any protocol pair into some pair (q,q) whose members are not distinct. The transformations are in fact oper-

136

4. Christopher Wintle provides a very useful analytic study of the piece in "Milton Babbitt's Semi-Simple Variations" Perspectives of New Music vol. 14, no. 2 and vol. 15, no. 1 (Spring-Summer/Fall-Winter 1976), 111 -54.

Generalized Set Theory (2)

6.2.4

aliens on PROT. They form a group isomorphic to the twelve-tone group of operations on rows. Furthermore, when we identify rows with subsets L of PROT, the sense of any row-operation coincides with the sense of the corresponding set-operation. E.g. if the set L corresponds to a certain row, then the set T3RI(L), as determined by the operations T3, R, and I on PROT, corresponds to the row formed by inverting, retrograding, and T3-ing the given row. Let J = 1^, the inversion operation that maps the pitch class A to the pitch class E. Then the small linear motive X, can be extracted from the row J(L,). We can see this by writing out the inverted row and italicizing the entries E, A, and Bl> of the motive Xt as they come along in X,-order: J(Lj) = E-E\>-A-B-B\>-C-F%-.... Another way of expressing the phenomenon is to point out that all three of the member pairs of Xj are member pairs of the inverted row: E precedes A in X} and also in J(Lj); A precedes Bl> both in Xl and in J(Lj); E precedes Bl> both in Xj and in J(Lj). That is, the three members of Xj, (E, A), (A, Bl>), and (E, Bl>), are also all members of J(Lr We reformulate this observation once more: There are three members of Lj whose J-transforms are members of Xr And finally, we can reformulate the observation into our present terminology most concisely: INJ(Lj, Xj)(J) = 3. Are there other inversion operations I such that INJ(Lj, Xj)(I) = 3? As it turns out here, there are not: J is the only one of the twelve inversion operations with that property. Working backwards through the semantic equivalencies of our observations in the paragraph above, we can interpret our most recent observation as telling us that J(Lj) is the only inverted form of Lj that serially embeds the small linear motive Xr We might say that Xj has a high "signature value" for J(Lj) among the twelve inverted forms of Lt: If we sense that an inverted form is at hand and we intuit the three protocol pairs of Xj clearly, that is enough information, abstractly, to identify J(Lj) as the specific form at hand. This property of Xj was noted by Michael Cherlin in connection with events near the opening of act 1, scene 2 in Moses und Aron.5 The scene portrays the brothers meeting in the desert; it begins with a lot of Aron music, light-textured, grazioso, piano and pianissimo, scored for solo flute accompanied by violins, harp and pianissimo horns. Then, just before Aron starts to sing, there is one measure of Moses music, scored for loud trombone and string bass. The trombone plays the short linear motive Xp extracted from the row-form J(Lj). The "signature motive" Xj is here attached to Moses as he steps forth on stage. Aron immediately thereafter begins to address Moses, singing the prime row-form combinatorial to J(Lj) and then J(L,) itself. Xl sounds particularly powerful here because it rearranges the opening trichord 5. The Formal and Dramatic Organization of Schoenberg's Moses und Aron (Ph.D. diss., Yale University, 1983).

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6.2.4

Generalized Set Theory (2)

of L! , a trichord which had a strong tonic character as a harmony during the preceding (opening) scene of the drama. Babbitt has published a discussion of just such small linear signature motives in his own music.6 He gives the row of his compositionReflectionsas L = C-B-D-A-Db-Bb-E-F-G-Eb-Gb-Ab. Then he points out that the ordered trichord X = B-D-A is "uniquely characteristic [of the row] t within the transpositional sub-array, and of [one inverted form] to within the inversional sub-array." In our terminology, INJ(L, X)(Tj) is less than 3 unless i = 0; also INJ(L,X)(I) is less than 3 unless I is the one specific inversion J which Babbitt singles out. He then goes on to discuss the ordered tetrachor Y = B-D-A-Db. He notes that Y "is unique ... for the total array." In our terminology, INJ(L, Y) (f) is less than 6 for the forty-seven twelve-tone operations f other than f = T0; L itself is the only twelve-tone form of L that contains all of the six protocol pairs of Y. Y is a signature for L among its forty-eight forms; X is a signature for L among its twelve transposed forms; X is also a signature for J(L) among the twelve inverted forms of the row. Now let us turn out attention back to X 2 and L2 on figure 6.7. Th aggregate X 2 , considered as a subset of PROT, contains 12 member pairs. Hence INJ(L 2 , X 2 ) (f) can be at most 12, ifis an operation. (In that ca map N distinct members of L2 1-to-l into N distinct members of X 2 , so that N must be 12 or less.) The forty-eight specific transformations f that interest us here are in fact operations. As it turns out, none of our forty-eight operations f actually embed X 2 in some form of L 2 , satisfying INJ(L 2 ,X 2 )(f) = 12. However there are operations f that do {£ of the job, satisfying INJ(L 2 ,X 2 )(f) = 11. These operations are f = T! , f = RT7, f = J, and f = RT6 J, where J is the inversion l£ . Since the row is its own retrograde at the tritone, these four operations generate only two forms of L 2 , namely T t (L 2 ) = RT 7 (L 2 ) and J(L 2 ) = RT6J(L2). Figure 6.8 shows how well the ordering of the aggregate X 2 fits into each of these row-forms.

FIGURE 6.8 138

6. "Responses: A First Approximation," Perspectives of New Music vol. 14, no. 2 and vol. 15, no. 1 (Spring-Summer/Fall-Winter 1976), 3-23. The discussion is on page 10.

Generalized Set Theory (2)

6.2.4

The figure makes it visually clear how X2 fits '4i within" either row. Figure 6.8 (b) shows how only the pair (C#, F#) of X2 is not within the row T1(L2); the row contains instead the protocol pair (F#, C#)- Figure 6.8(c shows how only the pair (Bb, F) of X2 is not within the row J(L2); the row contains instead the protocol pair (F, B|?). If only the tenor voice of X2 went E-F#-C# instead of E-C#-F#, then the embedding of figure 6.8(b) would be perfect. Or, if only the alto voice of X2 went G-F-Bj? instead of G-B[?-F, then the embedding of figure 6.8(c) would be perfect. Or, yet again, if only the tenth and eleventh notes of all the rows involved were exchanged, then both embeddings would be perfect. This urge to "make small adjustments" with one set or another will be further discussed later. It is a typical feature of situations in which an INJ function almost attains a theoretical maximum possible value. X 2 , as mentioned before, is the aggregate governing soprano, alto, tenor, and bass voices at the opening of Variation 3 in the Semi-Simple Variations. Various (12-tone) forms of X2 govern SATB relations throughout Variation 3. SATB aggregates of similar format govern other variations, but none of those fit more than "{§ within" any form of the row L 2 . To put it in our terminology, if X is an aggregate governing SATB anywhere in the piece outside Variation 3, the INJ(L 2 ,X)(f) is at most 10, for each operation f we are considering. Thus we can say that the SATB-aggregates of Variation 3 are maximally compatible with forms of L 2 , compared to such aggregates from other variations. Statements of this sort are very useful to express structural differencesamong sections of a piece that sounds at first extremel geneous in texture throughout. That is particularly so when the statements can be backed up by precise measurements like -fj, y§, and the like. INJ helps us pinpoint and explore precisely other structural differences among sections of the composition. For example, let V and V be SATBaggregates from any one variation; then INJ(V, V) (T0) = either 0 or 4. That is, V and V will either have no common pairs or exactly 4 common pairs. Let V l 5 V 2 , ..., V5 be SATB-aggregates from the first, second, ..., fifth variations; then with two exceptions INJ(V m , V n )(T 0 ) is less than or equal to 2. That is, with two exceptions, aggregates Vn, and Vn from different variations will have only 2 or fewer common pairs. Since two distinct aggregates in this format could theoretically share as many as 11 common pairs, we can say that the level of "ordering cross-talk" between variations is very low, half as low as the level of cross-talk within each variation (INJ(V, V')(T0) = 4). Indeed that latter level (4 pairs out of a possible 11) is itself none too high. The two exceptions are these: INJ(V5, V 2 )(T 0 ) = 4 and INJ(V4, V 2 )(T 0 ) = 5. Wecan say that Variations 5 and 4 thus "talk with" Variation 2, so far as SATBaggregate ordering goes, at a level equalling or even surpassing the level of cross-talk within each individual variation. INJ(V4, V 2 )(T 0 ) = 5 is a maximum compared with other values of

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6.3

Generalized Set Theory (2)

INJ(Vm, Vn)(T0). This observation suggests we devote special attention to aggregate-relations between V4 and V 2 . And when we do so, we shall notice a feature of the composition we might not quickly have come to notice otherwise. Whichever V4 and V2 we select to represent Variations 4 and 2 respectively, the two SATB-aggregates will share exactly one 3-note linear segment from among the four segments D#-B-E, Ab-C-G, C#-F#-D, and B[?-F-A. If Vm and Vn come from variations other than 4 and 2, the two SATB-aggregates will not share any 3-note linear segments. Let us call the family of four segments listed above the "pivot aggregate." Figure 6.9 shows how the pivot aggregate controls the tenor and bass voices of Variation 2, and the soprano and alto voices of Variation 4. The bar lines on the figure mark off SATB-aggregates within each variation.

FIGURE 6.9

INJ numbers bring quickly and effortlessly to our attention the fact that the relationship of figure 6.9 between Variation 4 and Variation 2 is a unique relationship between variations in the piece; it is not a ubiquitous feature of a large-scale design. As noted before, this sort of observation is very useful in bringing to our attention special discriminations within a composition that sounds at first extremely homogeneous.

140

6.3 In this section we shall explore further the notion of "if-only adjustments" in connection with INJ. The abstract notion can be formulated as follows. Suppose INJ(X, Y)(f) is near its theoretical possible maximum in a

Generalized Set Theory (2)

6.4

certain situation. The number might be close to the cardinality of X for instance, so that almost all of X is mapped into Y by f. Or f might be 1-to-l and the injection number might be very close to the cardinality of Y, so that the transformed set f(X) comes close to embedding Y. In such cases, a small adjustment in X or in Y might enable us to remove the "almost" component of the situation, bringing the injection number up to its theoretical maximum value. The parts of X or Y that do not quite fit may come under pressure to conform, giving rise in the music to urges for generating new material. We encountered an if-only situation in discussing the Semi-Simple Variations. Figure 6.8 earlier showed how the SATB-aggregate X2 was almost embedded in the rows Ti(L 2 ) and J(L2); more precisely "{i embedded." While discussing the figure, we mused about making if-only adjustments in either the aggregate or the row, to make the embedding work completely. So far as I can tell, the speculative adjustments do not correspond with musical pressures in Babbitt's piece. But in the Schoenberg song we examined earlier, similar speculative adjustments do correspond with strong musical pressures. In examining the progression of the 3-note chord X = Angst to the 3-note chord Y = Hoffen, we observed that INJ(X, Y)(wE) = 2 out of a maximum possible 3, and that INJ(X, Y)(I) also = 2 out of a maximum possible 3, "I" here meaning I| = ^Bb- "If only" the F|? of chord Y were adjusted to F natural, we noted, both the injection values of 2 above would rise to the maximum 3. In this connection the if-only speculation led to fruitful analytic ideas about the "missing F natural," the "missing I-partner," the missing lover, begehren, F[? as functional substitute for F natural, and so on. The interested reader will find an extensive treatment of if-only adjustment and its analytic implications elsewhere in my writings, in connection with the opening two chords of Schoenberg's piano piece op. 19, no. 6.7 I call the chords "rh" and "Ih" for right hand and left hand. TjL

6.4 The cited article goes on to discuss "progressive" and "internal" transformations in connection with the succession rh-lh. "Progressive" transformations make rh into something much like Ih; "internal" transformations make rh into something much like itself, or Ih into something much like itself, or both. The reader will recall that we used this nomenclature earlier, in connection with various transformations pertaining to harmony and melody in the song "Angst und Hoffen." We shall now extend the nomenclature and put it into a completely 7. "Transformational Techniques in Atonal and Other Music Theories," Perspectives of New Music, vol. 21, nos. 1-2 (Fall-Winter 1982/Spring-Summer 1983), 312-71, especially 33642.1 am indebted to Michael Bushnell for having observed the rewards that this sort of approach brings in analyzing the music. He worked with transposition operations only; I enlarge his aural field to include inversions as well.

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general theoretical setting, invoking only a family S of elements, certain transformations f on S, and the INJ function. Given sets X and Y, suppose we are inspecting the values of INJ(X, Y)(f), MT(X, X)(f), and INJ(Y, Y)(f) as f varies over a certain family INSPECT of transformations. For certain transformations f within the family INSPECT, the value of INJ(X, Y)(f) will be maximal, or at least relatively high subject to the constraints of the situation. We shall call these transformations progressive. They map a lot of X into Y. For certain transformations f within the family INSPECT, the value of MT(X, X)(f) or MT(Y, Y)(f) will be high. We shall call these transformations X-internal or Y-internal accordingly. A transformation which is both Xinternal and Y-internal can be called "internal (for the progression X-Y)." An X-internal transformation maps a lot of X into X. Intuitively, an X-internal transformation tends to extend/elaborate/ develop/prolong X in the music, while a progressive transformation tends to urge X onwards, to become something else (like Y). Progressive and internal transformations will tend to combine mathematically in certain interrelated ways, by their very natures. If I transform X to be much like itself, and then transform the result to be much like itself, it is likely that the composition of the two gestures will make X much like itself. That is, the composition of two X-internal transformations will tend to be X-internal. Similarly, the inverse of an X-internal operation will tend to be X-internal. Similarly, an X-internal transformation followed by an X-Yprogressive transformation will tend to be an X-Y-progressive transformation; and an X-Y-progressive transformation followed by a Y-internal transformation will tend to be X-Y-progressive. As a result, when we inspect the families of progressive and internal transformations pertinent to a given X-Y situation, we shall find those families tending to interrelate algebraically according to the considerations just surveyed. We can introduce other useful nomenclature. An f such that INJ(X, X)(f) is minimal or at least relatively small, given the constraints of X and INSPECT, can be called X-external. Such an f maps X largely outside itself. We can also define a dispersive transformation to be one that maps X largely outside Y, makeing the value of INJ(X, Y)(f) minimal or relatively small. These definitions avoid mentioning the complements of the sets X and Y in S, which may not be "sets" according to our definition if S is infinite. External and dispersive transformations tend to enter into typical algebraic relations with themselves, with each other, and with internal and progressive transformations. An X-internal transformation followed by an Xexternal one will tend to be X-external; a progressive transformation followed by a Y-external one will tend to be dispersive; and so on. A good example of dispersive transformations is furnished by measure 8

Generalized Set Theory (2)

6.4

of Schoenberg's op. 19, no. 6. This is the cryptic, very dense measure that precedes the final return of the chords rh and Ih in measure 9. Figure 6.10(a) reproduces measure 8.

FIGURE 6.10

Forte has noted that the music embeds many forms of the rh chord.8 For our purposes, we can observe that the music embeds four transposed forms of rh. Those are the four shown in figure 6.10(b): T2(rh), T5(rh), T7(rh), and T9(rh). Now if we let the interval i range from 0 through 11, we shall find that there are six values of i for which INJ(rh, Ih) (Tj) = 0; these values are i = 2,4, 5, 7, 9, and 0. That is, T2, T4, T5, T7, T9, and T0 are the six dispersive transposition operations, given the progression rh-to-lh. To put it another way, T2(rh), T4(rh), T5(rh), T7(rh), T9(rh), and T0(rh) = rh are the transposed forms of rh that have no common tones with Ih. As figure 6.10(b) shows us, four of these six forms are embedded within the music of figure 6.10(a). (The rest of that music does contain common tones with Ih.) And, as figure 6.10(c) shows us, a fifth dispersive form, T0(rh) = rh, ensues immediately thereafter, joining the parade of dispersive forms and thereby linking measure 8 to the downbeat of the final reprise at measure 9. The relevance of external transformations to traditional theory is illustrated by the "semi-combinatorial hexachord." If X is a 6-note set of pitch classes and I is an inversion operation that transforms X into its complement X, then I is an X-external transformation: INJ(X,X)(I) = 0. This is a good place to think about exploring how the injection function relates with set-conplementation when S is finite. We shall soon carry out that exploration, in section 6.6. Before that, it will be helpful to prove a theorem and a corollary about INJ(X, Y)(f) when f is an operation OP. 8. The Structure of Atonal Music, example 102 (p. 99).

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6.5.1 THEOREM: If f is an operation OP, then INJ(X, Y)(OP) is the cardinality of OP(X) n Y, that is the number of common members shared by the sets OP(X) and Y. We took note of this theorem informally in the commentary following Definition 6.2.1 earlier. The fact is important enough to warrant formal verification. Proof: Let M = INJ(X, Y)(OP); let N = card(OP(X) n Y). Let x t , x 2 , . . . , X M be the distinct members of X that map into Y via OP. Since OP is 1-to-l, the elements OP(Xi), OP(x2), ..., OP(xM) are distinct members of OP(X) n Y. Thus OP(X) n Y contains at least M distinct members; N is greater than or equal to M. Now let y l 5 y 2 , ..., yN be the distinct members of OP(X)nY. Then OP^iX OP"1^)* ..., OP"1^) are N distinct members of X, each of which maps into Y under OP. So there are at least N distinct members of X that map into Y under OP; M is greater than or equal to N. q.e.d. The injection function applied to operations thus generalizes Regener's Common-Note Function, which was developed for the special case in which X and Y are sets of pitch classes and OP runs through the twelve transposition operations.9 6.5.2 COROLLARY: If f is an operation OP, then INJ(Y,X)(OP) = INJP^YXOP"1). Proof: An element z is a member of the set OP~ 1 (X)n Y if an only if OP(z) is in X and z is in Y; this is the case if and only if OP(z) is in X and OP(z) is in OP(Y), that is if and only if OP(z) is a member of the set X n OP(Y). So the transformation OP maps the set (OP~1(X)n Y) 1-to-l onto the set (XnOP(Y)). Therefore the two sets have the same cardinality: card(OP(Y)nX) = card(OP~ 1 (X)nY). Applying Theorem 6.5.1 to both sides of this equation, we infer the formula of the Corollary. Now we are ready to explore set-complementation in connection with INJ. It will simplify matters greatly to restrict our attention to transformations f that are operations OP in this context. We shall then be able to use the theorem and the corollary we have just proved. We mustjestrict the abstract setting by supposing S to be finite; then the complement X of a set X will be a formal "set" by our criterion, i.e. a finite subfamily of S. 6.6.1 THEOREM: Suppose S is finite. Given sets X and Y with complements X and Y; given any operation OP; then formulas (A) through (E) below obtain. 144

9. Eric Regener, "On Allen Forte's Theory of Chords," Perspectives of New Music vol. 13, no. 1 (Fall-Winter 1974), 191-212. The Common-Note Function is defined on page 202.

Generalized Set Theory (2)

6.6.2

(A): (B): (C): (D):

INJ(X, Y)(OP) = cardX - INJ(X, Y)(OP). INJ(X, Y) (OP) = cardY - INJ(X, Y) (OP). INJ(X,_Y)(OP) = cardY - cardX + INJ(X, Y)(OP). IfcardY_=cardX, then INJ(X, Y)(OP) = INJ(X,Y)(OP). (E): (Generalized Babbitt Hexachord Theorem) If cardX_= ^cardS, then INJ(X,X)(OP) = INJ(X,X)(OP).

Proofs:_ (A): The operation OP maps each member of X either into Y or into Y. So cardX = INJ(X, Y)(OP) (the number of X-members mapped into Y) plus INJ(X, Y)(OP) (the number of X-members mapped into Y). The formula follows. This argument works just as well for any transformation f. (B): Here it is essential that OP be an operation, so that we can apply Corollary 6.5.2. We write INJ(X, Y)(OP) = INJ(Y,X)(OP-1), via 6.5.2. This, via formula (A) just proved, = cardY - INJ(Y, X)(OP-1). And, applying 6.5.2 again, we infer that this number is indeed cardY — INJ(X, Y)(OP), as claimed. (C): INJ(X,Y)(OP) = cardY = INJ(X,Y)(OP), via (B); this is cardY - (cardX - INJ(X, Y) (OP)), via (A); this is cardY - cardX + INJ(X,Y) (OP), as desired. (D) is an obvious corollary of (C). And (E) is an obvious corollary of (D), setting Y = X. q.e.d. The methods of proof I have used are essentially Regener's. I have called formula (E) the Generalized Babbitt Hexachord Theorem because Babbitt's theorem, somewhat disguised, is a special case of this formula. For readers who may feel the disguise is perfect, I shall show the connection. Let X be a sixnote pitch-class set; let OP be a transposition operation Tj. Theorem 6.6.1 (E) above tells us that INJ(X,X)(TJ = INJ(X,X)(Ti). Theorem 6.5.1 then tells us that the cardinality of (Tj(X) n X) is the same as the cardinality of (T;(X) n X). Now the cardinality of (T;(X) n X) is the number of members of X that lie the interval i from some member of X. In other words, the number is IFUNC(X, X) (i). Similarly, the cardinalitypf (T}(X) n X) is IFUNC(X, X) (i). We have shown, then, that IFUNC(X,X)(i) = IFUNC(X,X)(i) for every i: the complementary hexachords contain the same number of i-dyads for each i. This is Babbitt's theorem. One sees that 6.6.1(E) is a very broad generalization. 6.6.2 EXAMPLE: Let us see how the Generalized Hexachord Theorem applies in another specific context. The reader will recall the space PROT of protocol pairs which we constructed earlier (6.2.4) in examining some topics from serial

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theory. We noted that the various twelve-tone rows can be regarded as those subsets L of PROT that are linear orderings on the pitch classes. In this model, the retrograde of the row L corresponds to the set-theoretic complement L of the set L in PROT. For a pair (p, q) is in the complement of the set if and only if p does not precede q in the row, which is the case if and only if p precedes q in the retrograde of the row, which is the case if and only if (p, q) lies in that portion of PROT corresponding to the retrograde row. The family PROT has 132 = 12 times 11 members. And any row, as a subset of PROT, contains exactly 66 protocol pairs. To see that, suppose the pitch classes of the row come in the serial order p l 9 p 2 , ..., p 12 . Then the subset L of PROT contains 11 pairs of form (Pi,p n ), and 10 pairs of form (p 2 ,p n ), ..., and 1 pair of form (p u , pn). The cardinality of the set L is thus 11 + 10 H + 1, which is 66. We are thus in a setting to which 6.6.1(E) pertains. card(S) = card(PROT) = 132; card(X) = card(L) = 66 = |card(S). In this setting, a row and its retrograde (complement) play a set-theoretic role formally analogous to that of a hexachord and its complement in traditional atonal theory. Here, 6.6.1 (E) tells us the following. Let OP be any operation on PROT. Given any row L with retrograde L, let N be the number of pairs (p, q) in L such that the pair OP(p, q) is also a precedence relation in L; let N' be the number of pairs (p', q') in L such that OP(p', q') is also a precedence relation in L. Then N = N'. In this connection, OP(L) need not itself be a row. Indeed L itself can be replaced by any set of cardinality 66 within PROT and the theorem remains ture, "L" now being simply the set-theoretic complement of L. But the application is of particular interest when we interpret the complementary sets L-and-L as row-and-retrograde. The twelve-tone operations Tj and I are induced on PROT by corresponding operations on individual pitch classes: Tj(p, q) = (Tj(p),Tj(q)); I(p,q) = (I(p), I(q)). But in general there need not be any operation op on pitch classes such that OP(p, q) = (op(p), op(q)). There is no such op, for instance, in the case of the retrograde operation R on PROT: R(p, q) = (q, p). Nor is there such an op for any of the operations to which R contributes, e.g. RTj and RI. One can also construct fancier operations on PROT not induced by pitch-class operations op. For example: If p and q are in the same wholetone scale, OP(p, q) = (p, q); if p and q are in opposite whole-tone scales, OP(p > q) = (T3(q),T9(p)). When L is a row and OP(L) is also a row, then INJ(L, L) (OP) measures the size of the largest partial ordering on the pitch classes which can be embedded in both OP(L) and L, i.e. whose protocol pairs are compatible with both those rows. That is because INJ(L, L)(OP) is the cardinality of OP(L) n L, as we know from 6.5.1.

Generalized Set Theory (2)

6.7.2

Now we shall show formally how INJ completely generalizes IFUNC when there is a GIS at hand. 6.7.1 THEOREM: Let (S, IVLS, int) be a GIS. Then for each interval i and for all sets X and Y, IFUNC(X, Y)(i) = INJ(X, Y)(T,). Proof: Let IFUNC(X, Y)(i) = M; let INJ(X, Y)(Tj) = N. We shall see that N must be at least as big as M, and that M must be at least as big as N. Since IFUNC(X,Y)(i) = M, there are M distinct pairs (Xj.yj), (x2, y2), ..., (XM, yM) such that xm lies in X, ym lies in Y, and int(xm, ym) = i. For each such pair, ym = Ti(xm). For m and n distinct, x m and x n are distinct members of X. (Otherwise we would have ym = Ti(xm) = Tj(xn) = yn, whence the pairs (xm, ym) and (xn, y n ) would not be distinct, contrary to supposition.) Thus X has at least M distinct members whose i-transposes lie in Y. That is, INJ(X, Y)(Tj) is at least as big as M. Or: N is at least as big as M. Now let Zi, z 2 , . . . , ZN be the N distinct members of X whose i-transposes lie within Y. (There are N such, since INJ(X, Y)(T;) = N.) For each such z n ,z n is in X, u = Tj(z n ) is in Y, and int(zn, u) = i. Therefore the pair (z n ,u) was counted as some (xm, ym) above. So every one of the N elements z 1 ,..., ZN is one of the M elements x l 5 . . . , X M . Hence M is at least as big as N. q.e.d. The logic of Theorem 6.7.1 can be visualized through the following aid. Imagine X and Y as two finite configurations of points in the Euclidean plane. Suppose i is the vector (directed distance) "to the right and up 30 degrees for a distance of 5 inches." We can ask: "From points of X to points of Y, how many distinct arrows can I draw that go to the right and up 30 degrees for a distance of 5 inches?" The answer to this question is IFUNC(X, Y)(i). We can also ask: "If I move the whole X-configuration to the right and up 30 degrees for a distance of 5 inches, how many points of the displaced configuration will then coincide with points of Y?" The answer to that question is INJ(X, Y)(Tj). One intuits easily that the two questions are logically equivalent. Next we shall explore what happens to INJ when the sets X and/or Y are transformed by some operation A. 6.7.2 THEOREM: Given a family S of objects, given sets X and Y, given a transformation f on S and an operation A on S, then formulas (A), (B), and (C) below obtain. (A): INJ(A(X),Y)(f) = INJ(X, Y)(fA). (B): INJ(X, A(Y))(f) = INJ(X, Y)(A~ 1 f). (C): INJ(A(X), A(Y)) (f) = INJ(X, Y) (A'1 f A).

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Proofs: (A): INJ(A(X), Y)(f) is the number of t within A(X) such that f(t) is a member of Y. Set t = A(s); then the family of such t is in 1-to-l correspondence, via A, with the family of s in X such that fA(s) is a member of Y. And the number of such s is exactly INJ(X, Y)(fA). (B): INJ(X, A(Y)) (f) is the number of s in X such that f (s) is a member of A(Y). Now f (s) is a member of A(Y) if and only if A"1 f (s) is a member of Y. So the number at issue is the number of s in X such that A~ 1 f(s) belongs to Y. And that is exactly INJ(X, YXA^f). (C): The formula follows at once from (A) and (B). q.e.d.

148

Formula 6.7.2(C) is of particular abstract interest. We can imagine that the shift from X-and-Y to A(X)-and-A(Y) reflects a "modulation" of the system by the operation A. For instance, if we are in a GIS and A is a transposition or an inversion, we are transposing or inverting (the sets of) the system accordingly. It is natural to ask: "If we modulate the system by A, what effect does that have on the INJ function?" At first one might suppose that INJ would remain unaffected by the modulation: INJ(A(X), A(Y)) = INJ(X, Y). But, as formula 6.7.2(C) shows, that is not in fact the case. Rather the INJ function is itself "modulated" according to the formula. We noted a specific instance of this general phenomenon earlier, during our analysis of "Angst und Hoffen." The wedge-structure of figure 6.3(b) converged on E; it was studied in connection with the transformations W E = wedging-to-E and I = inversion about E. The wedge-structure of figure 6.3(c) converged on F#; it was studied in connection with the transformations WF* = wedging-to-F# and J = inversion about F#. To get from the situation of (b) to the situation of (c), one "modulates the system by T2." The following equations obtain: F# = T2(E); wF* = T 2 w E T 2 " 1 ; J = T2IT2"1. The first equation relates the focal points of the two wedges, also the bass notes of the Angst and Hoffen chords. The second equation leads, via 6.7.2(C), to the relationship INJ(T 2 (X),T 2 (Y))(w F «) = INJ(X, YXT^w1^) = INJ(X, Y) (WE). The third equation leads via the same formula to the relationship INJ(T2(X),T2(Y))(J) = INJ(X, Y)(T 2 1 JT 2 ) = INJ(X, Y)(I). Thus, when we modulate the system from E-centricity to F#-centricity via the operation T 2 , then the wedge WF* plays the role, with respect to the modulated sets T2(X) and T2(Y), that the wedge W E originally played with respect to the sets X and Y. Similarly, the inversion J plays the role, with respect to the modulated sets, that the inversion I originally played with respect to the unmodulated sets. The reader will recall, perhaps, our earlier remarks on this subject by way of preparation (pp. 128-129). Here is another, more abstract, example of system-modulation. Let X be an atonal hexachord that inverts into its complement via the inversion I = IB. Then INJ(X,X)(I) = 0. Suppose some music projecting X-andits-complement "modulates" to a new section projecting T5(X)-and-//j-

Generalized Set Theory (2)

6.7.3

complement. Here I plays the role of "f" and T5 plays the role of "A" in Formula 6.7.2(C). We cannot suppose that INJ(T5(X), T5(X))(I) = 0: the new hexachord T5(X) will not invert into its complement via the inversion I, B/C inversion. Rather, T5(X), the modulated hexachord, inverts into its complement by the inversion J = T5IT~^. Formula 6.7.2(C) tells us this: INJ(T5(X), T5(X))(J) = INJ(X, X)(T-i JT5) = INJ(X, X)(I) = 0. Using formulas 3.5.6 (A) and (B), we can compute J = T 5 l£T 7 = I|T7 = Ij:. Thus T5(X) inverts into its complement about E and F (or about B!> and B). The system having modulated by T5, the transformation J = T5IT~* now plays the role that the transformation I originally played. Formula 6.7.2(C) tells us this sort of thing in great generality: When a system modulates by an operation A, the transformation f = AfA"1 plays the structural role in the modulated system that f played in the original system, in the sense that INJ(modulated X, modulated Y(f) = MT(X, Y)(f). 6.7.3 Theorems 6.7.1 and 6.7.2 enable us to generalize the abstract questions about IFUNC we asked earlier, toward the end of the section 5.1. We asked, for instance, under what circumstances in a GIS we would have IFUNCCXj, Xj) = IFUNC(X2, XJ. Via 6.7.1, we can rephrase the question, asking under what circumstances in a GIS we shall have INJ(Xj, Xj)(Tj) = INJ^Xj, XjXTj) for every transposition Tr And that question can easily be generalized: Given any family S of objects and any family INSPECT of transformation on S, under what circumstances shall we have INJ(Xj, Xj)(f) = IN](X2, XjXf) for every member f of INSPECT? We do not have to demand that S be in a GIS, or that INSPECT be a group; indeed, the question makes sense even if INSPECT is not a closed family (semigroup) of transformations. Likewise, we earlier asked under what conditions in a GIS we would have IFUNC(X,, XJ = IFUNC(Y,, Y2). We can generalize that question analogouly: Given any family INSPECT of transformations f on a family S of objects, under what conditions shall we have INJ(X,, X^f) = MT(Y,, Y2)(f) for every member f of INSPECT? The ideas of 6.7.2 enable us to expand that question even farther. Suppose we have a family S of objects, a family INSPECT of transformations f on S, and a group MDLT of "modulating operations" A on S. Under what conditions, given sets Xj, X^ Y,, and Y2, will there exist some modulation A such that when we modulate Yj and Y2 by A, obtaining Y', and Y'2, we shall have DSfJCX,, X^f) = INKY',, Y'2)(f) for every f in INSPECT? Via 6.7.2(C) this amounts to demanding that ESTJ(X,, X^f) and INJ(Yt, Y2(A~1fA) be equal numbers, for each member f of INSPECT. In the special case where we are in a GIS, where INSPECT is the family of transpositions, and where MDLT is the group of transpositions, the question asks under what conditions, given the four sets, there will exist some interval

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j such that for every interval i, IFUNCCX^XjXi) = IFUNC(Tj(Y1), Tj(Y2))(i) = IFUNCCY^Y^Gij- 1 ). (5.1.6 gives us the last equality. It is consistent with 6.7.2(C) because T^-i = T^TjTji The group of transpositions is aAm'-isomorphic to the group of intervals.) Besides using 6.7.1 and 6.7.2 to generalize earlier questions, we can also use them to help out with computations we earlier found difficult to execute. The formula following is a good example. 6.7.4 THEOREM: In any GIS let I = !„. Fix any referential element and set j = LABEL(u). Then for any sets X and Y, and for each interval i, IFUNC(I(X),I(Y))(i) = INJ(X, Y)(Pk) where Pk is the interval-preserving operation labeled by k = ji"1]"1. Proof (optional): IFUNC(I(X), I(Y)) (i) = INJ(I(X), I(Y)) (Tj), via 6.7.1. This, via 6.7.2, = INJ(X, Y)^"1^!). Set I^T,! = OP. It now suffices to prove that OP = P k , where k = ji"1.)""1. r1 = 1^(3.5.9). So OP = I^Tji;;. We can compute T^ by 3.5.6(A); the result is I*, where x = Tj(u). So OP = I"I*. And we can compute the composition of the two inversions by 3.5.8. It is PmTn, where m = LABEL(u)LABEL(x)-1 and n = LABELOO^LABELCv). Here n = e, so we have computed that OP = Pm, where m = LABEL(u)LABEL(x)~1. Since x = T,(u), LABEL(x) = LABEL(u)-i(3.4.3). Replacing LABEL(u) by j, we then have m = j(ji)"1 = ji-1j-1 = k. Thus OP = Pk as desired.

750

6.8 To demonstrate further the generalizing powers of INJ, we shall now use it to generalize Forte's K and Kh relations. For our generalization we suppose only a family S of objects and a group of operations on S which we shall call "canonical" for reasons known only to us. We shall denote the group by CANON. The complement of the set X will be denoted X. If S is infinite, X will not be finite, hence not a "set" in our terminology. Nevertheless we shall speak of its "cardinality," understanding the value infinity for the expression cardX. If S is finite, cardX will mean the finite cardinality of the set X. We shall restrict our attention to sets X and Y such that cardX < cardX and cardY < cardY. We can do this because Forte's K and Kh relations are not affected by the restriction. If cardX should be less than cardX, we can simply exchange the roles of the sets X and X as they do or do not enter into K or Kh relations with other sets. Having made that restriction, we may also suppose that cardX < cardY. Otherwise, we can simply exchange the roles of the sets X and Y in the arguments coming up. So our restrictions, in sum, are these: cardX < cardX, cardX < cardY < cardY. Given those restrictions, Forte's K and Kh relations hinge on the

Generalized Set Theory (2)

6.8

logical combination of two more primitive relations, which we can call K! and K2. (K x ): Some (canonical) form of X is embedded in Y. (K2): Some form of X is disjoint from Y (and hence embedded in Y). X and Y, subject to our restrictions, enjoy Forte's K relation if they enjoy either Kj or K 2 ; they enjoy his Kh relation if they enjoy both K^ and K 2 . Now the relations Kj and K 2 correspond to very natural aspects of INJ(X, Y), as that function takes arguments from the canonical group. Specifically, K t and K 2 are respectively equivalent to K\ and K 2 below. (K;): For some A in CANON, INJ(X, Y)(A) = cardX. (K'2): For some B in CANON, INJ(X, Y)(B) = 0. Now we can express Forte's relations, generalized, as follows: (K): INJ(X, Y) (A) takes on either its theoretical maximum value cardX (subject to our restrictions), or its theoretical minimum value 0, as A varies over CANON. (Kh): INJ(X, Y)(A) takes on both its theoretical maximum and its theoretical minimum values, as A varies over CANON. We can use the ideas of "progressive" and "dispersive" transformations (6.4), to rephrase K\ and K 2 yet once more, into the respective forms of K/{ and K 2 below. (K'i): CANON contains some maximally progressive transformation with regard to INJ(X, Y). (K2): CANON contains some maximally dispersive transformation with regard to INJ(X,Y). Let us return to relation K\ above. K't says there is some A in CANON such that INJ(X, Y)(A) = cardX, but it does not tell us how many such A there are. In case CANON is infinite, that may or may not be a meaningful question. When CANON is finite, the question is definitely interesting. By asking it, we can distinguish multiplicities of K! -ness about the K x relations of various sets: Some pairs of sets are "more K^-related" than others, Likewise, when CANON is finite, we can distinguish multiplicities of K2-ness about K2 relations, counting how many members B of CANON satisfy condition K 2 above. We can then attach pairs of multiplicity numbers to sets in the Kh relation, to indicate the multiplicities of K!-ness and K2-ness involved. For a specific example, take S to be the twelve chromatic pitch classes, and take CANON to comprise both transpositions and inversions. Take Y to be the black-note pentatonic scale. Take X to be the F#-major triad. Then we may say that X and Y are Kh-related with multiplicity (2,6): There are two members A of CANON embedding A(X) in Y, and there are six members B of CANON such that B(X) is disjoint from Y. Now, holding onto Y, take X to be the set (Ajj, B|?, Dfc>). The new X is Kh-related to Y with multiplicity (4,8). 6.9 Coming back from the specific example, let us return to the general

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situation with CANON finite, still subject to our conditions on the cardinalities of X and Y. For each integer N between 0 and cardX inclusive, we can ask how many members A of CANON satisfy the equation INJ(X, Y)(A) = N. Our multiplicity-values aoove asked that question for N = cardX and for N = 0, but we can just as well ask it for every N in between. The answer we get, i.e. how many such A there are in CANON, is the value for the argument N ofRegener's Partition Function, here generalized.10 We shall denote that value byRGNPF(X,Y)(N). In this special case (CANON being finite), the Partition Function enables us to derive the EMB function by the formula below. RGNPF(X.Y)(cardX) EMB(X, Y) -

RGNpF(X)X)(cardx)-

That is, EMB(X, Y) is the multiplicity of Kj-ness about X-in-Y, divided by the multiplicity of Kj-ness about X-in-itself. The formula is given without proof. If X is symmetrical in some way, there may be several distinct members of CANON, other than the identity, that map X onto itself. If M members of CANON map X onto itself, then there will be M times as many distinct operations embedding X in Y, as there are forms of X embedded in Y. RGNPF counts operations; EMB counts forms. This accounts for the denominator of the fraction in the formula above. 6.10 OPTIONAL: In all our work so far with set theory, we have supposed the "sets" under scrutiny to be finite. This section will outline briefly the work needed to extend our results so that they can be applied to sets "of finite measure" in certain "measure-spaces." During this discussion we shall relax the use of the word "set" so as to conform to standard mathematical usage. Roughly speaking, that usage makes the word a synonym for "family" or "ensemble" of things. Given a family S of objects, a family FLD of subsets of S is called a fieldof subsets if it satisfies (1), (2), and (3) below. (1): 0 (the empty set) and S are members of FLD. (2): If X is a member of FLD, then so is its complement X. (3): If X and Y are members of FLD, then so is X u Y, their set-theoretic union. It follows deductively that whenever X and Y are in FLD, so are X n Y (which is the complement of X u Y), Y — X (defined as Y n X), and so on. A field of sets is called asigma-fieldif (4) below is also tr (4): Whenever Xj, X 2 ,... is a countable family of members of FLD, so is the countable union X = XiuX2u — X here is the set of s such that s is a member of at least one of the sets X n . "Countable" means "capable of being put into 1-to-l correspondence with the natural integers." 752

10. Ibid., p. 206ff.

Generalized Set Theory (2)

6.10

A measure on the field FLD is a function mes that assigns to each member X of FLD a value which is either a non-negative real number or infinity, in such fashion that mes(X u Y) = mesX + mesY whenever X and Y are disjoint. If FLD is a sigma-field, one supposes mes to be sigma-finite unless it is specified as not so. The condition for sigma-finiteness is that mes(UNION X n ) = SUM(mesXn) whenever X 1? X 2 ,... is a countable family of mutually disjoint members of FLD. The countably infinite sum is understood in the usual sense, as the limit of its partial sums; that limit may be infinity. We can deduce that if X is included in Y, then mesX < mesY. For mesX < mesX + mes(Y — X), mes being non-negative, and that number is mesY since X and (Y — X) are disjoint. We shall need one more concept. Given a field FLD of subsets of a family S, a transformation f on S is called measurable when, given any member Y of FLD, the set of s mapped by f into Y is also a member of FLD. Now we can generalize INJ. We suppose a family S, a field FLD of subsets, and a measure mes on FLD. By a "set of finite measure" we mean just that, a member X of FLD such that mesX is finite. If X and Y are sets of finite measure, and if f is a measurable transformation on S, then the family F of s such that f (s) belongs to Y is a member of FLD, so the family X n F is also a member of FLD. Furthermore, mes(X n F) < mesX, so X n F is a set of finit measure. We then define INJ(X, Y) (f) = mes(X n F). This number measures "how much of X" (according to mes) is mapped into Y by f. Using this definition for INJ and restricting X, Y, etc. to be sets of finite measure, we can generalize much of the machinery in chapter 6 as it stands. Theorem 6.5.1 and the formulas that follow from it must be restricted so that the operations at hand are "measure-preserving," satisfying mesOP(X) = mesX. We could derive more flexible and much more complicated formulas by allowing "measure-scaling" operations; OP is such if for some constant number "scale," mesOP(X) = scale • mesX. Our work on the Partition Func tion (6.9) may not generalize easily, since CANON is not likely to be finite if the sets X, Y, etc. are not finite. But often it will be possible to establish a "good" field of subsets within CANON, and a "good" measure on that field; then for any real numbers a and b with 0 < a < b, we can define RGNPF(X, Y)(a, b) to be the measure, within CANON, of the set of operations A satisfying a < INJ(X, Y)(A) < b. The mathematical ramifications of all this lie far beyond the scope of the present book. The work of chapter 6 up to the present is just one special case of the general system now being sketched. In that special case, FLD contains all subsets of S, so all transformations fare measurable; mesX is the cardinality of X, allowing infinity as a possible value. The sets of finite measure are exactly the finite sets, what we have been calling "sets" up until now. For another special case we can consider a non-musical setting. Let S be

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a Euclidean plane containing a Seurat painting. We take FLD to consist of all regions in the plane with a well-defined total area, and we use area as our measure. Let X be the region of the plane consisting of all animals in the painting; let Y be the region consisting of all plants. Let f be the transformation on S: move up and to the right at a 45-degree angle for 3 centimeters. Then INJ(X, Y)(f)/areaX answers the question: To what extent are the animals of the painting to be found 3-cm.-below-and-at-a-45-degree-angle-to-the-left-of plants? We now adopt a new measure, redmes. This measure assigns to each member of FLD the number of red dots it contains completely. (red)INJ(X, Y) (f )/redmesX now answers the question: To what extent do red dots within the animals of the painting lie 3-cm.-below-and-at-a-45-degreeangle-to-the-left-of plants? If we want to shift our attention to yellow dots, then (yellow)INJ(X, Y) (f )/yellowmesX answers the analogous question with respect to yellow dots. For our next example suppose we are analyzing a certain piece and we want to ask questions like this: Of the amount of time the violin is playing above high C during the piece, how much of that time will the clarinet be playing pianissimo or softer five seconds later? We can use a special case of the general model at hand. Fixing a referential unit of time and a referential timepoint zero, we take S = the real numbers, as modeling the continuum of timepoints within which our piece occurs. For each real number a, let Za be the set of numbers s satisfying s > a, and let Za be the set of numbers s satisfying s < a. Take FLD to be the smallest sigma-field that contains every Za and every Za. As it happens, there is essentially only one well-behaved measure on FLD that satisfies mes(Za n Zb) = (b — a) for every pair of numbers a < b. We shall use this measure; we can think of it as measuring the "amount of time" in a set. Take X to be the set of time-points in the piece during which the violin is playing above high C; take Y to be the set of time-points in the piece during which the clarinet is playing pianissimo or softer. Take f to be timepoint transposition by five seconds: The time-point f (s) occurs five seconds after s. Then the number we asked for above is INJ(X, Y)(f)/mesX. For a final example we shall explore the space S of time spans. We represent S by the Euclidean half-plane of number-pairs (a, x), where x is positive. FLD will be the smallest sigma-field containing all square-shaped regions within the half-plane. FLD contains all the sets we shall want to discuss here; e.g. it would contain the region comprising all animals in the Seurat painting. There is a unique measure on FLD that makes the measure of each square region its area; we shall call this measure "area." We may or may not want to use area as a measure for our musical purposes. There are other measures that have other desirable properties. For instance, in connection with our non-commutative GIS, there is an essentially unique measure that makes every interval-preserving operation measure-preserving: mesP(X) =

Generalized Set Theory (2)

6.10

mesX. We can call this measure "P-invariant measure." It is essentially unique in this sense: If mes' is any other P-invariant measure, then there is some positive number "scale" such that mes'(X) = scale • mesX for every member X of FLD. There is also adifferentmeasure on S, also essentially un makes every transposition-operation measure-preserving: mesT(X) = mesX. We can call this measure "T-invariant measure." If we are analyzing a specific piece, we may also want to use measures like the Seurat measure earlier that counted red dots. The following discussion will help develop a "Seurat model" for our analysis. Suppose the piece begins at the time point BEGIN and extends for the duration EXTENT. Then every time span that pertains to an event within the piece must satisfy the two constraints BEGIN < a and a + x < BEGIN + EXTENT. The set of time spans (a, x) satisfying those constraints forms a triangular region in the halfplane. This triangle plays a role in our model analogous to that of the rectangle on which Seurat painted, a bounded region within its plane. In applying time-span analysis to the piece, we are hearing events with well-articulated beginnings and durations, events whose temporal location and extent can be sensibly modeled by time spans. We may then fill in the triangle-of-the-piece by dots. A dot at the point (a, x) within the triangle models the location and extent of an event in the piece which begins at a and lasts x units of time. We can color certain dots red. Say these are the dots pertaining to stringed-instrument events. We can color certain dots yellow. Say these are the dots pertaining to events "above middle C." Orange dots will then pertain to events involving some stringed instrument (s) playing above middle C. Our model assumes the format of a triangular Seurat painting, and we can apply the ideas of "red measure," and so on. Given a time span (b, y), we can construct the family SHADOW(b, y) of all time spans (a, x) that "happen within" the time of (b, y). We constructed the shadow of (BEGIN, EXTENT) above; that was the triangle-of-the-piece. SHADOW (b, y) in general is the set of all spans (a, x) satisfying the two constraints b < a and a + x < b + y. This set forms a triangular region in the half-plane. Given two events in the piece, event 1 and event2, let (a, x) and (b, y) be the respectively pertinent time spans. Then (a, x) is in the SHADOW of (b,y) if and only if event 1 happens during the time of event2. Let us imagine a certain section of the piece which begins at the time point BEGSEC and lasts for a duration of DURSEC time units. Take X = SHADOW (BEGSEC, DURSEC). X is the set of all time spans pertaining to potential or actual events within the given section of the piece. Let BRASS be the family of time spans which articulate events played by brass instruments. (Those spans might be the green dots on our "painting.") Take Y = SHADOW(BRASS). A time span belongs to the set Y if and only if an actual or potential event to which it pertains happens or would happen completely during some brass event, e.g. a sustained brass note or chord. Take

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f to be transposition-by-the-interval-(4,2) in our non-commutative GIS. Thus the span f (a, x) begins 4 x-lengths after a, and lasts for a duration of 2x time units. We may then ask for the value of orange INJ(X, Y) (f). When we do so we are asking the following question. How many string events pitched above middle C are there within our given section of the piece such that if you start at the event, counting "ONE," and then beat three more event-durations, counting "-two-three-four," and then beat two more event-durations yet, counting "ONE-two," your new count of "ONE-two" will measure a span of time lying completely within the extent of some sustained brass note or chord?

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We saw in Theorem 6.7.1 how INJ generalizes IFUNC when there is a GIS at hand: IFUNC(X, Y) (i) = INJ(X, YXT,). This relationship enables us not only to generalize IFUNC, but also to replace entirely the concept of intervalin-a-GIS by the concept of transposition-operation-on-a-space. Instead of thinking: "i is the intervallic distance from s to t," we can think: "Tj is the unique transposition operation on this space that maps s into t." We can even shift our attention, if we wish, from the atomic "points" s and t to the oneelement "Gestalts" X and Y, X being the set that contains the unique member s and Y being the set that contains the unique member t; then there is a unique member T; of the transposition-group satisfying INJ(X, Y) (Tj) > 0; the label i for this unique transformation is i = int(s, t). 7.1.1 By such thinking, we can replace the idea of GIS structure by the idea of a space S together with a special sort of operation-group on S. This special sort of group is what mathematicians call simply transitive on S. The group STRANS of operations on S is simply transitive when the following condition is satisfied: Given any elements s and t of S, then there exists a unique member OP of STRANS such that OP(s) = t. Given any s and any t in the space of a GIS, then there is a unique transposition-operation T satisfying T(s) = t, namely T = Tint(M). So the group of transpositions is simply transitive on the space of any GIS. Conversely, the following theorem is also true: Let S be a family of objects and let STRANS be a simply transitive group of operations on S; then there exists a GIS having S for its space and STRANS for its group of transpositions. We shall now prove that theorem. Given S and STRANS as described, let IVLS be an "index family," that is a family of elements i, j,... which can be put into 1-to-l correspondence with the family STRANS. We shall write "OP;"

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for that operation within STRANS that corresponds to the member i of the index family IVLS. Now we turn IVLS into a group by defining the binary combination ij = k in IVLS when (OPj)(OPj) = OPk in STRANS. The group IVLS, by this construction, is anti-isomorphic to the group STRANS. We have a space S and a group IVLS; next we define a function int from S x S into IVLS. Given r and s in the space S, we take int(r, s) to be that unique member i of IVLS such that OPj(r) = s. i is unique because STRANS is simply transitive. Holding onto r, s, and i above, suppose that int(s, t) = j in the same sense. That means OPj(s) = t. Then (OPj)(OPi)(r) = OPj(s) = t. By the group structure defined for IVLS, (OPj)(OP,) is OP(ij). Hence OP^r) = t. Then, by the construction of the function int, int(r, t) = ij. Thus int(r, t) = int(r, s) int(s, t); Condition (A) of 2.3.1 is satisfied. Now we shall show that Condition (B) of that definition is also satisfied, so that (S, IVLS, int) is a GIS. Given s and i, set t = OPj(s). By definition of int, int(s,t) = i. Iff is any member of S satisfying int(s, t') = i, then OPj(s) = t'. But OPj(s) is precisely t. So in this case t' = t. We have shown that, given s and i, there is a unique t satisfying int(s, t) = i. So we have shown that Condition 2.3.1(B) is satisfied. Thus (S, IVLS, int) is a GIS. And STRANS is the group of transpositions for this GIS; indeed Tj = OPi for every i in IVLS. To see this, we recall that T;(s), in any GIS, is the unique member of S that lies the interval i from the element s. Now in this particular GIS, the member of S that lies the interval i from s is OPj(s). Hence in this GIS Tj(s) = OP^s). That being the case for every sample s, T; = OPj as an operation on S. q.e.d.

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7.1.2 By virtue of 7.1.1, all the work we have done with GIS structures since chapter 2 can be regarded as a special branch of transformational theory, namely that branch in which we study a space S and a simply transitive group STRANS of operations on S. From a strictly mathematical point of view, this would have been a more elegant way to develop the material of chapters 2 through 5. We could even have deferred the study of GIS structure until much later, after a more general exploration of transformations on musical spaces using CANON, INJ, and other such constructions. Yet there are also advantages to the order of presentation we have adopted in this book. By starting with a study of GIS structure, we have built a link between the historically central concept of "interval" and our present transformational machinery. To some extent for cultural-historical reasons, it is easier for us to hear "intervals" between individual objects than to hear transpositional relations between them; we are more used to conceiving transpositions as affecting Gestalts built up from individual objects. As this way of talking suggests, we are very much under the influence of Cartesian thinking in such matters. We tend to conceive the primary objects in our

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musical spaces as atomic individual "elements" rather than contextually articulated phenomena like sets, melodic series, and the like. And we tend to imagine ourselves in the position of observers when we theorize about musical space; the space is "out there," away from our dancing bodies or singing voices. "The interval from s to t" is thereby conceived as modeling a relation of extension, observed in that space external to ourselves; we "see" it out there just as we see distances between holes in a flute, or points along a stretched string. The reader may recall our touching on these matters in 2.1.5, where we pointed out how the historical development of harmonic theory has depended on such a projection of our intuitions into a geometric space outside our bodies, that is, the "line" of the stretched string, a space to which we can relate as detached observers. In contrast, the transformational attitude is much less Cartesian. Given locations s and t in our space, this attitude does not ask for some observed measure of extension between reified "points"; rather it asks: "If I am at s and wish to get to t, what characteristic gesture (e.g. member of STRANS) should I perform in order to arrive there?" The question generalizes in several important respects: "If I want to change Gestalt 1 into Gestalt 2 (as regards content, or location, or anything else), what sorts of admissible transformations in my space (members of STRANS or otherwise) will do the best job?" Perhaps none will work completely, but "if only ...," etc. This attitude is by and large the attitude of someone inside the music, as idealized dancer and/or singer. No external observer (analyst, listener) is needed. In 7.1.1 above we sketched a mathematical dichotomy between intervals in a GIS and transposition-operations on a space: Either can be generated formally from the characteristic properties of the other. More significant than this dichotomy, I believe, is the generalizing power of the transformational attitude: It enables us to subsume the theory of GIS structure, along with the theory of simply transitive groups, into a broader theory of transformations. This enables us to consider intervals-between-things and transpositionalrelations-between-Gestalts not as alternatives, but as the same phenomenon manifested in different ways. Consider figure 7.1 in this connection.

FIGURE 7.1

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The figure shows how the melodic motif of the "falling minor ninth (sixteenth)" is developed over Schoenberg's piano piece op. 19, no. 6, out from the intervallic structure of the opening chord in the right hand, a chord we have earlier called "rh." Figure 7.1 (a) displays rh, along with the three falling pitch-intervals that can be heard within it; they are notated on the figure as — 5, — 9, and —14 semitones. Figure 7.1(b) shows how the same network of intervals governs the scheme by which the falling-ninth motif is transposed over the course of the piece. This is interval-language. Alternatively, we could use transposition-language to put the matter as follows: The three transposition-operations T_ 5 , T_ 9 , and T_ 14 , which move the falling-ninth motif forwards in time over figure 7.1(b), are exactly those members of STRANS which move the individual pitches of rh downwards in space, as shown on figure 7.1 (a). "Forwards in time" and "downwards in space" are phenomena that work together in many ways over the course of the piece.1 But we do not have to choose either interval-language or transpositionlanguage; the generalizing power of transformational theory enables us to consider them as two aspects of one phenomenon, manifest in two different aspects of this musical composition: Intervals structure the referential sonority rh as an Unterklang; transpositions make the falling-ninth motif move forward through the piece. This, I think, is the sense in which we accept the symbol " — 5" on figure 7.1 (a) and the symbol " — 5" on figure 7.1(b) a legitimately the same, finding the usage suggestive rather than confusing. The two symbols are pointing at the same phenomenon, not at different phenomena. Before we leave figure 7.1, let us note that the chord of (a) and the various falling ninths of (b) have only one common pitch-class. This emphasizes that in comparing (a) to (b), we are talking about intervals and transpositions, not about pitches and pitch-elaboration (diminution). If figure 7.1 (b) were to have appeared five semitones lower, then one could argue that the basic phenomenon involved was that of the pitches B, F#, and A in the chord, elaborated to become BBt>, F#F, and AG# in the new figure 7.1(b). But this is not the case; the phenomenon under discussion involves intervals and transpositions, not pitches or pitch-classes and their structural ornamentation. The remainder of chapter 7 comprises further examples demonstrating various interrelationships of intervallic structures with transpositional progressions in the manner of figure 7.1(a)-(b), over a variety of musical styles. All these examples will involve intervals among pitches or pitch-classes. The distinction between pitch or pitch-class elaboration and intervallic/ transpositional interrelationship will not always be so clear, through the examples, as it was in figure 7.1. We shall concentrate mainly upon the

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1. This aspect of the music is brought out very well in the analysis by Robert Cogan and Pozzi Escot, Sonic Design (Englewood Cliffs, N. J.: Prentice-Hall, 1976), 50-59.

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intervallic/transpositional interrelationships, for those are our current focus of study. But we shall also discuss from time to time how those interrelationships interact with pitch or pitch-class elaborations.

FIGURE 7.2 7.2 EXAMPLE: Figure 7.2 shows the beginning of the Zauber motive from Wagner's Parsifal, so far as pitch classes are concerned. The motive appears with a variety of rhythms in the music drama. I shall call the serial network of figure 7.2 "Zauber" or "Z" for present purposes. Forms of the motive do not appear in the foreground of the music until Kundry's first-act entrance (ride); figure 7.2 shows Z as its pitch classes appear during the kiss in the middle of act 2. Figure 7.3 sketches melody and harmony for the phrase that introduces the Motive of Faith in the Prelude to act 1. This is long before Z has appeared in the foreground. Yet the intervallic structure of the Z motive governs the plan of modulations for the phrase, as the arrows on the figure show us.

FIGURE 7.3

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In figure 7.3 one hears not only the intervals of modulation but also the specific pitch classes Ab-Cb-Ebb~Eb being tonicized; these are the pitch classes for Z which were displayed in figure 7.2 above. Of course we hear the music of figure 7.3 long before the music of figure 7.2 (second-act kiss). On the other hand, we do hear the local keys of figure 7.3 elaborating a pitch-class variation on a structure related to Z, namely the Liebesmahl motive that opened the opera: Ab-C-Eb-F-(etc). In this context, hearing the successive tonics Ab-Cb-Ebb-Eb of figure 7.3 as a variation on the Liebesmahl helps us hear Zauber itself, when it appears in the foreground later on, as a variant of the Liebesmahl.

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Figure 7.4 shows the succession of local tonics during the transformation music of the first act. The principal local tonics are represented by open noteheads; measure numbers above them indicate where the cadential tonicizations discharge. Usually these discharges coincide with entries of important motives; the names of the motives appear on the figure preceding the measure numbers: BELL, GRAIL, AG = Agony, and LM = Liebesmahl. Some filledin noteheads also appear on the figure; these represent local tonics which are subordinated to the principal tonics in various ways. The B tonic at measure 1084 is a structural dominant preparation for the E tonic at measure 1092. The B tonic takes the Grail Motive, not the Bell Motive like other tonics from measure 1074 to measure 1106. The C tonic at measure 1115 is supported by no special motivic entry; it is a structural neighbor to the Db = C# tonics that surround it by open noteheads on the figure. After the neighboring event and the enharmonic shift, the predominant motive on the figure changes, from BELL to AG (measure 1123 and following). Then, after measure 1140, AG disappears and BELL returns. The cadence at measure 1140 is deceptive. The

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7.2

implied tonic is A|>, which appears as an open notehead on the figure; the substitute root that actually carries the Liebesmahl entry is F(?. The situation at measure 1148 is similar, in fact sequential. Up to measure 1140 the open noteheads are organized by serial forms of the Zauber motive. These forms are beamed on figure 7.4 and labeled as Z t , Z 2 , Z3, and Z4, meaning the first Z-form, the second Z-form, and so on.2 Each Z-form is a retrograde inversion of the Z-form that precedes it, specifically that RI form which uses for its opening two notes the final two notes of the preceding form. Thus Z2 begins E(?-E-, taking its point of departure from the end of Z^-Eb-E. Likewise Z3 begins G-B|?-, taking its point of departure from the end of Z2,-G-B|?. It does not much matter whether we call figure 7.4 up to measure 1140 a pattern of "intervals" among tonics, or of "transpositions" among tonics, or even of "modulations" among keys: We are talking about the motivic exfoliation of one phenomenon in various ways. "Modulations" is only awkward if we try to associate the term with Schenkerian notions of pitch or pitch-class prolongation. That is clearly not happening here (up to measure 1140), and since Schenker himself rejected the word for his discourse, I feel free to use it in this non-Schenkerian connection. Indeed I prefer it (for that reason) to the word "tonicizations." Continuing along figure 7.4, we note that the open noteheads followin measure 1140 build up to one beamed form of the BELL motive. It is very significant that this structural Bell Motive appears "in A|j." The deceptive cadence at measure 1140, featuring an entry of the Liebesmahl in the trombones, forcefully recalls the same event right at the curtain-rise of act 1. In this connection, pitch-class relations clearly are important. We are to understand the Ab of measure 1140 as the tonic of the opera; we are to understand the structural Bell Motive of measures 1140-50, beamed on figure 7.4, as a prolongation of that tonic; and we are thereby to understand the mammoth local tonic C of measures 1150-62ff. as the third degree of that Ab. It is just this section of act 1 that enables us (with the help of the Parsifal bells) to be sure that the C tonic which ends the act is indeed a functional III of A(? (inter alia). To support this hearing, we can note that the four pitch-classes of the Bell Motive in Ab, portrayed by the beamed noteheads of measures 1140-50 on figure 7.4, are in fact a serial permutation of the four pitch-classes that begin the Liebesmahl (and the opera): Ab-C-Eb-F. The pitch-class relation is very striking, yet Wagner apparently goes to great lengths to conceal it. The deceptive cadences at measure 1140 and measure 1148 help to do so; so does the absence of the Bell Motive in Ab from the foreground of the music here (and so far as I can recall anywhere else in the music). Whatever private games 2. I am grateful to Thomas Christensen for calling Zj and Z3 to my attention.

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Wagner may have been playing, it is safer for us as analysts to treat the permutational relation between the Bell Motive and the Liebesmahl as intervallic, rather than pitch-class prolongational. Figure 7.4 enables us to explore another motivic relationship involving BELL. This motive has the same organizing function after measure 1140 that Zauber had before measure 1140, that is the function symbolized by grouping open noteheads on the figure with beams. The compositional relationship invites us to explore the intervals to which those beams give rise in each case, and indeed we can hear an interesting intervallic relationship now that we are primed to listen for it.

FIGURE 7.5 Figure 7.5 shows how we can hear BELL as an overall progression of - 3, elaborated by two subprogressions of 7. Compare this network of intervals to that of figure 7.2, which displayed Zauber as an overall progesssion of 7, elaborated by two subprogressions of 3 (inter alia). Let us now return our attention to the network of Z-forms displayed by figure 7.4 up to measure 1140. We shall have much more to say later about the serial technique of Rl-chaining that links successive Z-forms. When this technique is applied to any given serial motive over and over, alternate forms of the motive-chain will be transposed forms of each other, the interval of transposition depending upon the serial structure of the motive. In this specific case, Z3 = T 10 (Z 1 )andZ 4 = T10(Z2). The repeated Rl-chaining thus gives rise to "structural sequencing" on figure 7.4: Measures 1096-1140 (with A[?) "sequence" measures 1074-1100 structurally. (The musical foregrounds of the two passages are not related sequentially.) The sequencing-interval of 10 in this particular case is a dispersive interval for Z as an unordered set: INJ(Z, Z)(T10) = 0; T10(Z) has no common notes with Z. The dispersive function is clear on figure 7.4, where the sequencing forms of Z fill up chromatic pitch-class space very diligently. Indeed, the open noteheads of the figure up through the A|? of measure 1140 constitute a non-repeating ten-note series. (F# and B are missing. It is amusing, if far-fetched, to imagine them as representing the absent Klingsor.) 164

7.3 EXAMPLE: Figure 7.6 graphs an intervallic/transpositional structure which we shall call "the nuclear gesture with pickup." The nuclear gesture

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7.3

FIGURE 7.6

comprises the pitch-class interval (transposition) 4, subarticulated into two intervals (transpositions) of 2. The pickup gesture consists in approaching something by pitch-class interval/transposition 5. Figure 7.7 shows through a variety of examples how the nuclear gesture dominates intervallic/transpositional configurations over the last movement of Brahms's Horn Trio op. 40. It may seem awkward to be using pitch-class intervals in this context, since the piece is so highly structured registrally and is so diatonic. Our reasons will become clear by the end of this discussion. For other purposes it would make more sense to use pitch intervals in semitones up, or diatonic intervals in scale steps up, or degree intervals in scale steps up modulo 7. (a) of the figure displays the basic motive of the movement. The nuclear gesture with pickup governs the first half of the motive; the second half is governed by what we shall call the "complementary gesture," which articulates 8 into 10 + 10. (b) of the figure sketches the bass line for the opening phrase. We hear how it is governed as a whole by the nuclear gesture. Now each stage of that gesture is inflected by its own pickup. The octave leap on B|?, which appears in parentheses at the end of (b), is an important motive of the piece. Here it interrelates with the octave B[? that delimits the ambitus of motive (a). (c) of the figure shows the melody beginning to descend from its first climax. The complementary gesture governs the melodic structure as a whole; each stage of that gesture is inflected by a pickup. The pickup interval of 5 does not get complemented; it always remains 5, never becoming 7. (d) shows how the network of (c) gets tightened rhythmically at the approach to the last recurrence of motive (a) within the first group. Over (a) through (d), the intervallic/transpositional interrelationships can be analyzed as fallout from the pitch and pitch-class motives. The nuclear and complementary gestures are always applied to Ejj-F-G and G-F-Efr. functioning as degrees 1-2-3 and 3-2-1 in E(? major. The pitch and pitchclass motives, with their degree functions, can claim priority here: In (a)-(b)(c)-(d) we can deduce the intervallic relations of interest from stronger pitch and pitch-class relations, but we cannot run such deductions in the other direction. So a traditional Schenkerian approach to (a)-(b)-(c)-(d) reveals more than does our intervallic/transpositional approach. The same can still be said of (e), which shows the melody at the opening of

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FIGURE 7.7 166

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FIGURE 7.7 (continued)

7.3

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an episode that begins toward the end of the second group and eventually leads to the closing theme. The principal pitch-classes that carry the structure of our (e)-network are still G and F, even though they are now degrees 6 and 5 in Bk (The whole passage takes place over a dominant pedal in that key.) Intervallically and transpositionally, (e) shows how transposition-by-10, which governs the sequence, exfoliates from the interval-of-10 between G and F; this interval of 10 develops the "subinterval of 10" from the complementary gesture, blowing it up rhythmically. Each stage of the 10-gesture is inflected by a pickup. In passages (f) through (h), the intervallic/transpositional networks take on a life of their own. They become autonomous structures; no longer subordinated to concomitant local pitch or pitch-class events, they rather interact with such events or perhaps even determine them, (f) shows an extended sequential passage from the development section. The model for the sequence is sketched on (f) over measures' 103-05 "etc." Over measures 104-05, the pedal note B of the ostinato figure combines with the opening two notes of the legato theme that follows, projecting the complementary gesture. (At "etc." the legato theme stops moving stepwise.) The pedal pitch-class B is inflected by a pickup. To hear the gestures here, one must listen to pitch classes, not pitches, (f) continues on, showing how the sequence from measure 103 etc, through measure 113 etc., to measure 123 etc., projects the nuclear gesture. The mobile harmony of (f) achieves as its goal a reattained dominant of El> major. Thereupon the horn launches an unusually extended solo passage, (g) shows first how the opening eight measures of the solo, measures 137-44, are structured by the complementary gesture with pickup. By projecting that gesture at this pitch-class level, Brahms finally gets his "octave-Bt" idea to interact with the system of nuclear and complementary gestures. (The octaveBl> idea stood apart from those gestures in passages (a) and (b) above.) As a result of this interaction Brahms generates a Gl> at the end of the complementary gesture, a G!> standing in an 8-relation to the initial Bl> of that gesture. We are aware of the 8-relation from B\> to Gl> as it carries over into the music of measures 145-47. And that gives us a clue as to what is going on over measures 145-49 more broadly. Another clue is furnished by rhythmic augmentation: The pickup motive of measure 137 etc. is augmented rhythmically to measure 145 etc., and the complementary-gesture motive of measure 141 is blown up rhythmically to measure 149 etc. These clues explain the new intervallic/transpositional gesture graphed in (g) over measures 145-49 (with pickup): An overall interval of 4 is subarticulated into 8 + 8. The subinterval 8 of the new gesture identifies with the overall interval 8 of the complementary gesture—both span Bl>-to-GI> during passage (g). And, just as rhythmic value are multiplied by 2 in passing from measures 137-44 to measures 145-56, so intervallic values are also multiplied by 2. Instead of the complementary gesture 8 = 10 + 10, we now have an expanded gesture involving twice those

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7.4

numbers: 2 - 8 = 2-10 + 2-10or, modulo 12,4 = 8 + 8. The new gesture will therefore be called "the complementary gesture times 2." Our ears follow the intervallic idea "times 2" here because of the rhythmic augmentation. Each stage of the new gesture takes its own pickup. (h) shows how the new gesture was already presented in the first movement, derived in a musical manner that was almost the equivalent of an algebraic demonstration. While the horn melody in the treble clef chains 10intervals into suspension-patterns sequencing through 8-intervals, the concomitant bass line indicates how those 8-intervals in turn chain up to form the complementary-gesture-times-2. Each stage of the times-2 gesture, in the bass of (h), takes its own pickup; we must use the fundamental bass in this connection for the first stage. I have chosen for example (h) the reprise-form of this passage rather than its form in the exposition. That is because the pitch classes in the bass line of (h) as given make it easy for the reader to hear the relation of (h) to measures 145-49 in (g).

FIGURE 7.8 7.4 EXAMPLE: Figure 7.8(a) sketches the opening of the Minuet from Beethoven's First Symphony. The C pedal bass is omitted from the first complete measure, signifying that we are to understand an F root and a G root for the two harmonies of that measure in the present context. This is an oldfashioned way of hearing, especially at the usual tempo for the piece, but we shall find that the old-fashioned hearing is of interest here. Those who have trouble hearing an F root, given the C bass, will be helped to do so by summoning up the memory of the slow movement. That movement, the last music we have heard before figure 7.8(a) begins, has just ended in F major. Figure 7.8(b) aligns beneath (a) a sketch for the opening of the entire

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symphony. In the deceptive cadence of measure 2, a C harmony is understood as interpolated between the G7 and the A-minor harmonies. This too is an oldfashioned but consequential hearing in the context. The progression of G to (interpolated) C is bracketed. On (a) above, the repeated progression of G to C harmony is also bracketed. Ignoring the repetition of the bracketed harmonic progression in (a), counting the theoretically interpolated C harmony in (b), and considering the first complete measure of (a) to contain F and G harmonies, we can then refer both passages to a common network governing the progression of roots, a network suggested by figure 7.9.

FIGURE 7.9

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The intervals on the arrows are familiar ratios from fundamental bass theory. They are expressed here as fractions between 1/2 and 1, to capture the "falling" sense of the root progressions. The representation of chord-changes by "intervals" between roots is not quite adequate. We shall improve the model in chapter 8; meanwhile the somewhat inadequate representation will serve. The root-intervals have an impressive tradition behind them, exemplifying the desire of earlier theorists to subsume intervals and transformations (here harmonic changes) into one general phenomenon (the fundamental bass). Traditional criticism would describe the relation between the Adagio and the Minuet by saying that the latter parodies the former, as a passage in a satyr-play may parody a dramatic theme treated majestically in an earlier drama of a tetralogy. Our "common network" of figure 7.9 enables us to propose a different relation between the passages, one that does not infer so much structural priority for the Adagio from its temporal priority in the composition. Namely, we can conceive both (a) and (b) of figure 7.8 as different realizations, in different environments, of one underlying abstract gesture, a gesture symbolized by figure 7.9. The mere possibility of this shift in

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7.4

critical stance is important, no matter which stance one prefers in viewing the particular art-work at hand. Consider this analogy. When one first encounters Mr. X, one sees him in formal attire, discharging an important and solemn public professional duty. When one next encounters him, he is surrounded by screaming children and friends in a park, his mouth filled with potato chips, rushing down a hill carrying a football in one hand and an open can of beer in the other. Here we would surely be reluctant to adopt a view analogous to that of the traditional criticism, claiming that the first X is a norm for X-ness and the second X is a parody of the first. Rather, our attitude would be analogous to the alternative view. We might notice in X's body, features, movements, voice, and the like certain things common to both occasions; from that we would infer a certain abstract "X-ness," and we would say that this X-ness was being manifested on both occasions, albeit differently in different environments. A dramatist or novelist might first introduce us to X either at the public occasion or in the park. One could also imagine an open-form play or novel, in which the author allowed either scene to precede the other at the choice of the performers or the reader. Traditional criticism would attack this idea on the grounds that we cannot separate our concept of X-ness from the particular way in which we have built up that concept through time. And so on; having suggested the philosophical and aesthetic issues our investigations engage, I shall not pursue the matter farther here. I do, nevertheless, want to work out some specific critical consequences that emerge when we take figure 7.9 as a norm for both figure 7.8(a) and figure 7.8(b). First, let us analyze the "tail" on our normative figure 7.9, that goes from the C harmony to the A-minor harmony, as a secondary feature of the structure. (The minor harmony is a "secondary triad.") Imagine the node containing the A-minor harmony, then, as erased from figure 7.9, along with the arrow labeled "5/6." It is then possible to analyze the rest of the figure as a succession of three V-I cadences moving along the circle of fifths. This interpretation of the graph is projected by the music of figure 7.8(b), the opening Adagio. There we hear a motive, the motive repeated with harmonic variation, and the motive repeated again with rhythmic variation. The local tonics for the motive-forms move along nodes of figure 7.9 located at successive arrowheads, first F, then C, then G. (We have dropped the tail from the figure.) This music, in sum, projects a "progressive" reading of figure 7.9-as-norm. The opening of the Minuet, in contrast, projects a different way of reading figure 7.9-as-norm. Here the figure is interpreted as manifesting a sense of balance in its cadence structure, not a progressive chaining of tonics. To explore the cadential sense of balance, let us construct an intervallic graph called CADENCE, shown in figure 7.10. Now let us apply CADENCE to the analysis of figure 7.9-as-norm,

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FIGURE 7.10 always ignoring the tail. The leftmost side of figure 7.9 exhibits CADENCE formatting the subnetwork C-F; G-C. The rightmost side of figure 7.9 exhibits CADENCE formatting the subnetwork G-C; D-G. This suggests hearing the entire normative network of figure 7.9 (without tail) as comprising two CADENCES, the right side of the first CADENCE being elided into the left side of the second CADENCE. According to this reading, figure 7.9 without tail is to be understood as a contraction of figure 7.11.

FIGURE7.il

772

This interpretation of figure 7.9 illuminates the repeated G-C progression within the Minuet theme, marked by the brackets on figure 7.8(a): That projects the normative repeated G-C progression which we see on figure 7.11. The pitch classes that fill the nodes of figure 7.11 are the roots of various harmonies. Some harmonies are (local) tonics; others are dominants. The tonic pitch classes fill nodes that have arrowheads pointing to them. Figure 7.12 isolates the tonics of figure 7.11 and organizes them in a network of their own. This network in fact fills the nodes of the CADENCE graph (figure 7.10) yet again. The relation is hard to see because the diagonal arrows of figure 7.10 go to the right while those of figure 7.12 go to the left. But the CADENCE

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7.4

FIGURE 7.12

graph, as a configuration of nodes and labeled arrows, knows no "right" and "left." We are using those visual distinctions here to indicate musical chronology, not graph-structure as such. Musical chronology is naturally crucial. We may hear a dominant precede its tonic, symbolized e.g. by the rightpointing diagonal arrows on figure 7.10; we may hear a dominant follow and inflect its tonic, symbolized e.g. by the left-pointing arrows on figure 7.12. But in this particular system of root-relations, the normative graph itself displays only a dominant node and a tonic node connected by a labeled arrow; chronological distinctions in this system function not as norms but as different interpretations of the normative graphs. That is a feature of classical fundamental-bass theory, not of our graphic structures per se. We have many systems of rhythmic "intervals" at hand within which we could construct analogous graphs that would enforce musical chronology. The opening of the Minuet interprets the basic norm of figure 7.9 in the manner of figures 7.11 and 7.12. The music specifically reads the normative network as a backwards-laid-out CADENCE of local tonics (figure 7.12), diminuted into two balanced fowards-laid-out CADENCEs of roots (figure 7.11). We have already noted how the repeated G-C progression within the Minuet theme helps to bring out this balanced structure. The idea of balance is also projected melodically to some extent: The first CADENCE of roots harmonizes the tetrachord G-A-B-C in the principal melodic line; the second CADENCE harmonizes the answering similar tetrachord D-E-F#-G. Of course the irrepressible thrust of the rising scale works against this feeling of balance, along with other features of the passage. It is curious that figure 7.8(a), so ebullient in its texture, should project a reading of our underlying harmonic norm as balanced, while figure 7.8(b), so stately and measured in its texture, should project a reading of the same norm as progressive. Figure 7.13 continues from the end of figure 7.8, asserting a structural correspondence between the next four measures of the Minuet, arriving at a structural dominant, and the continuation of the first movement all the way up to the dominant which prepares the first big tonic tutti in the Allegro. The correspondences displayed here are mostly melodic, serial, and motivic, involving particularly the thematic leading-tone-to-tonic idea. Fundamental-bass

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FIGURE 7.13

structure is not involved to the extent it was in connection with figure 7.8. Figure 7.13 is really not our business here, but I think the reader will be glad to hear it anyway. One would not easily conceive listening for its correspondences without having first noted the correspondences of figure 7.8. Then too, to the extent one assents to the analytic implications of figure 7.13, that confirms the propriety of figure 7.8 and our work that issued therefrom. And of course figure 7.13 is interesting in its own right. It owes an enormous debt to the theoretical ideas of Schenker. 7.5 The interested reader will find in an article by John Peel a sensitive and analytically revealing use of small transposition-graphs and networks for discussing a passage from Schoenberg's String Trio.3 John Rahn has published a discussion of the theme from the second movement of Webern's Symphony op. 21 that bears very suggestively on ideas we have been considering throughout this chapter.4 Casting his discussion in the form of an ear-training exercise, Rahn directs students' attention to networks of tritones and networks of semitones; he also directs their attention to concomitant rhythmic structures. One could perhaps construct an appropriate GIS for his discourse, a GIS involving pitch and rhythm together in a direct-product system. Aspects of his analysis might then be recast and extended, to involve networks of direct-product intervals. 3. John Peel, "On Some Celebrated Measures of the Schoenberg String Trio," Perspectives of New Music vol. 14, no. 2 and vol. 15, no. 1 (Spring-Summer/Fall-Winter 1976), 260-79. 4. John Rahn, Basic Atonal Theory (New York: Longman, 1980), 4-17.

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Transformations

Figure 7.9, the normative fundamental-bass network we constructed for the Beethoven symphony, exhibited a feature we discussed early in chapter 7. On the one hand, we could conceive of its arrows as signifying intervals between individual elements, in this case roots. On the other hand, we could also conceive of the arrows as signifying transformational relations between Gestalts, in this case chords or harmonies, or even potential keys. Resolving the ambiguity, we could also conceive of the arrows as denoting some tertium quid, a phenomenon whose manifestations include both harmonic intervals between pitch classes and transformational relations between chords. To conceive such a tertium quid, in the form of the fundamental bass and its progression, was Rameau's supreme inspiration. And some of the problems to which that conception gives rise are neatly pinpointed by figure 7.9 as well. That is particularly the case as regards its A-minor "tail," the tail we conveniently docked before starting our earlier discussion. The interval of 5/6, which labeled that tail, does inform us correctly that the root A, as a pitch class in just intonation, is 5/6 of the root C (modulo the pertinent congruence relation). But the numerical ratio does not tell us that the A harmony is minor rather than major. If we transpose a C-major chord by the interval 5/6 we obtain an A-major chord, not an A-minor chord. Thus, when we pass from the C-node to the A-node on figure 7.9, we are really applying some transformation other than harmonic-transposition-by-(5/6), some transformation which is more than a synonym or isomorphic image for that interval. 8.1.1 We can solve this problem very elegantly by adopting and modifying some ideas from the function theories of Hugo Riemann. Our space will consist not of pitch classes but of objects we shall call "Klangs" Each Klang

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is an ordered pair (p, sign), where p is a pitch class and sign takes on the values + and — for major and minor respectively. The Klang models a harmonic object with p as root or tonic, an object whose modality is determined by the sign. We can transpose a Klang by transposing its pitch class while preserving its sign; thus (C, +) transposed by 5/6 is (A, +). Rather than writing T(C, +) = (A, +) here, we shall write the symbol for the transformation to the right of its argument: (C, +)T = (A, +). The reader will recall our having discussed such "right orthography" a long time ago, in section 1.2.4. Right orthography will conform much better than left orthography to our intuitions in the contexts we shall be exploring just here. The one special thing we have to watch is that the order of composing transformations is reversed under right orthography: If f and g are transformations, then (Klang)fg = ((Klang)f)g denotes Klang-transformed-by-f, all transformed by g; this was denoted by "gf(Klang)" in left orthography. We define the operation DOM on Klangs: DOM is transposition by the inverse of the dominant interval. Thus (p,sign)DOM = (q, sign), where q is that pitch class of which p is the dominant. We can read this equation as telling us that (p, sign) becomes the dominant 0/(q, sign). On a graph we could have a (C, +) node, an (F, +) node, and an arrow labeled DOM from the first to the second. Right orthography conforms nicely to the visual layout of the graph here: Being at (C, +) and following an arrow labeled DOM we arrive at (F, +); that is, (C, +)DOM = (F, +). In a similar spirit we define the operation MED: (p, sign) becomes the mediant of its MED-transform. For example, (C, + )MED = (A, —), and (C, —)MED = (Ab, +). If we are at (C, + ) on a graph and follow an arrow labeled MED, we arrive at (C, + )MED = (A, -). Now we can rewrite the network of figure 7.9 as a network of Klangtransformations, rather than fundamental-bass intervals. Figure 8.1 is the result. The transformations DOM and MED drive the network of figure 8.1 in a

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FIGURE 8.1

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8.1.1

natural musical way. In general, DOM and MED will always drive similar Klang-networks in the same way. That would happen even where an instance of (F, +) occurred in some music followed by an instance of a dominantrelated (C, +). We could use the visual layout of an analytic network to reflect the musical chronology, having a DOM arrow pointing to the left from a (C, 4-) node to an (F, +) node. In that case we would be saying: "(C, +) refers (back), as dominant, to (the preceding) (F, +)." The DOM arrow, here as before, makes a dependent Klang point at the local tonic Klang to which it refers. For the configuration of nodes and arrows as a configuration, it is immaterial whether the DOM arrows point right or left (or up or down). Our unusual definition of DOM is what makes the graphs move naturally in this way. The orientation of all the DOM arrows would be reversed if we had chosen the more usual alternative idea, in defining a "dominant" Klang-transformation. The usual idea is represented by the transformation DOM', which takes a Klang and transforms it into its own dominant. E.g. (F, +)DOM' = (C, +): "Being at (F, + ), take its dominant and obtain (C, 4-)." Observe how poorly figure 8.1 would fit our kinetic intuitions about the music under study if we reversed all the DOM and MED arrows, using DOM' and the analogous MED' instead. This elucidates one problem with the conceptual structure of Riemann's function theories. His dominants, other than secondary dominants, do not point to their tonics via implicit DOM arrows. Rather the tonics point to their dominants, generating them by implicit DOM' arrows. Then the dominants just sit around, not going anywhere. This conceptual flaw in Riemann's approach makes his musical analyses subject to inertia and lifelessness, seldom doing justice to the power and originality of his theoretical ideas. An even more basic problem for Riemann was that he never quite worked through in his own mind the transformational character of his theories. He did not quite ever realize that he was conceiving "dominant" (whether DOM or DOM') as something one does to a Klang, to obtain another Klang. Here, I conjecture, he was unduly influenced by a desire to promote his notation as a substitute for Roman-Numeral notation; I think it was this desire that led him to conceive "dominant" and the like as labels for Klangs in a key, rather than as labels for transformations that generate Klangs from a local tonic (along the lines of DOM'), or that urge the Klangs of a key towards their tonics (along the lines of DOM). We may continue to explore other transformations on the family of Klangs, following or modifying Riemann. We can define SUBD, the formal inverse of DOM. "(F, +)SUBD = (C, +)" means that F major becomes the subdominant of C major. Even though SUBD = DOM"1 in a group of operations, the arrows on the graphic format enable us to distinguish a SUBD arrow read forwards from a DOM arrow read backwards. We also have left and right directionality at our disposal in this connection, to represent musical chronology. Thus, analyzing a plagal cadence in C major, we would draw a

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SUBD arrow pointing from (F, +) on the left to (C, +) on the right. Analyzing a tonic-dominant progression in F major, say an opening phrase terminating with a half-cadence, we would still put (F, + ) on the left and (C, -1-) on the right, but now we would draw a DOM arrow pointing leftwards, from (C, +) to(F, +). In similar vein, we can define and explore SUBM, the formal inverse of MED. We can consider other sorts of transformations too. For example, we can define REL, the operation that takes any Klang into its relative minor/ major. (C, +)REL = (A, -); (C, -)REL = (Eb, +). REL is not the same operation as MED or SUBM: (C, -)REL = (Eb, +) but (C, -)MED = (Ab, +); (C, +)REL = (A, -) but (C, +)SUBM = (E, -). We can also define PAR, the operation that takes any Klang into its parallel minor/major, (p, sign)PAR = (p, —sign). We can define Riemann's "leading-tone exchange" as an operation LT: (C, +)LT = (E, -); (E, -)LT = (C, +). We can also define more exotic operations on Klangs. For instance we can define an operation SLIDE that preserves the third of a triad while changing its mode: (F, +)SLIDE = (F#, -);(F#, -)SLIDE = (F, +). The SLIDE relations between (F, +) and (F#, —) can be heard in the last movement of Beethoven's Eighth Symphony, where the F-major theme that begins on the note A, the third of the triad, is transformed at measures 379-91 into F# minor, where it begins on the same A; the theme slides back into F major at measure 392. A SLIDE relation between (C, +) and (C#, —) can be heard over measures 103-10 in the slow movement of Schubert's posthumous Bb-Major Piano Sonata. Over those measures, thematic material which we expect to hear in C# minor is presented in C major instead. Using such transformations to label arrows, we can construct networks that could not be conceived using only intervals-and-transpositions. For example, figure 8.2 displays interesting relations between a "Tarnhelm network" (a) and a "Valhalla network" (b). The Tarnhelm network of (a) takes (B, +) as a tonic for the motive in its own context; it asserts structural relaxation on that Klang. This seems legitimate; besides, Wagner not infrequently interprets the motive in a larger context of B minor or B major, e.g. at the end of G otterdammerung I, or at the end of Tristan. (E, + ) on network (a) is bracketed to indicate that this Klang functions by implication only; (E, —) substitutes for it in the music. The Valhalla network, figure 8.2(b), asserts the indicated relations among the principal Klangs over the first presentation of the Valhalla theme in Das Rheingold, during measures 1-20 of scene 2. We shall be concerned with the "modulating" part of the network, the part that extends from measure 7 onwards. That is why the events of measures 1 -6 are in parentheses. Graphs (a) and (b) make visually clear a strong functional relationship between the Tarnhelm progression and the modulating portion of the Valhalla theme, a relationship which it is difficult to express in words.

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8.1.2

FIGURE 8.2

8.1.2 Obviously, we could not construct the graphs of figure 8.2 using intervals-cum-transpositions. The fact is obvious if by "intervals" we mean intervals between the notes of a fundamental bass. However, might we not be able to think of "intervals" here in some more extended formal sense? The matter bears theoretical examination, if only for the sake of review. The reader will recall how we showed in section 7.1.1 that the entire notion of a GIS can be developed formally from a given family S of objects and a given simply transitive group STRANS of operations on S. When a GIS is developed therefrom, STRANS becomes the group of transposition operations for that GIS. Let us take S to be a family comprising a given Klang and all other Klangs that can be derived from it via any chains of DOM, MED, SUBD, and SUBM transformations. Let STRANS be the group generated by the four cited operations. Since SUBD = DOM"1 and SUBM = MED"1, STRANS is generated by DOM and MED. The reader can now verify that DOM itself is generable by MED: DOM = (MED)(MED). That is, given any Klang, if it becomes the mediant Klang of a second Klang which in turn becomes the mediant Klang of a third Klang, then the first Klang is the dominant of the third Klang. E.g. ((C, +)MED)MED = (A, -)MED = (F, +) =

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(C, +)DOM;((C, -)MED)MED = (Ab, + )MED = (F, -) = (C, -)DOM. Since DOM = MED2, the group STRANS here is generated by MED alone: The operations of STRANS are precisely the powers of MED, including MED° = IDENT and MED~ n = (MED"1)" = SUBMn. Once we have this insight into the structure of our group STRANS here, it is not difficult to show that the group is in fact simply transitive on the defined family of Klangs, whatever the system of intonation we are using. Then we can regard the powers of MED as formal "intervals" on that family, in the sense of 7.1.1. We may then regard figure 8.1 as a formal "intervallic" network among Klangs, since it involves among its transformations only powers of MED. In this system, the Klang (C, —) is derived from (C, +) by 7 applications of MED: (C, +)MED7 = (A, -)MED6 = (F, +)MED5 = (Bb, +)MED3 = (Eb, +)MED = (C, —). Thus MED7, applied to a major Klang in some systems of intonation, will yield the parallel minor Klang. However, MED7 applied to a minor Klang will not yield the parallel major Klang, no matter what the system of intonation. E.g. (C, -)MED7 = (F, -)MED5 = (Bb, -)MED 3 = (Eb, -)MED = (Cb, +). Thus MED7 is not the same operation as PAR on the family of Klangs at hand. Now the graphs of figure 8.2(a) and (b) reference both the operations SUBM and PAR. It follows that we shall not be able to find a simply transitive group on a suitable family of Klangs that enables us to consider figure 8.2(a) and (b) as formally "intervallic" graphs. Our group would have to include both SUBM and PAR; containing SUBM, it would contain MED = SUBM'1; then the group would contain both MED7 and PAR. Then the group could not be simply transitive. E.g. given elements (C, +) and (C, —) in our family of elements, there would not be one unique member of the group transforming the former Klang into the latter; both MED7 and PAR would do the job. We could only salvage the formalities of this situation by cleaving so firmly to just intonation that we were willing to admit an infinite number of distinct Klangs (C0, +), (Clf +), (C2, +),..., (C0, -), (C,, -), (C2, -),... and so on, whose roots lay syntonic commas apart. Then (C0, +)MED 7 and (C0, +)PAR would have different formal values: The former would be (C_ 1? —) and the latter would be (C0, —). We would end up with a game board like that of figure 2.2 earlier, only with yet a third dimension reversing the modality of each Klang on the board. Clearly, this model would be at least awkward for analyzing Wagner's music. (It does engage some of the performance problems, especially for the singers.)

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8.2.1 We have seen that non-intervallic transformation-networks can be revealing in connection with Klangs. They can also be revealing in connection with serial transformations. Let us consider a series s of pitches or pitchclasses s 1? s 2 , ..., SN. We can apply to s the Rl-chaining operation RICH. RICH(s) is that retrograde-inverted form of s whose first two elements are SN-! and S N , in that order. Thus if s = A-C-Eb-E (a form of Zauber), then

Transformation Graphs and Networks (2)

8.2.2

RICH(s) is Eb-E-G-Bb, and RICH(RICH(s)) is G-Bb-Db-D. The RICH transform of RICH(s), being a retrograde-inverted form of a retrogradeinverted form of s, must always be some transposed form of s. In the Zauber example above, the interval of transposition is 10: G-Bb-Db-D is the 10transpose of A-C-Eb-E. For another example, let us examine s = Eb-B-Bb~D-C#-C-F#-EG-F-A-G#. This is the prime row of Webern's Piano Variations op. 27. RICH(s) is A-G#-C-Bb-Qf-B-F-E-Eb-G-F#-D, and RICH(RICH(s)) is then F#-D-Db~F- etc. As before, the RICH transform of RICH(s) is a transposed form of s, but now the specific interval of transposition is different. F#-D-Db-F-etc. is the 3-transpose, not the 10-transpose, of Webern's row. In general, when we RICH the RICH-transform of an abstract pitch or pitch-class series s = s t , s 2 ,..., SN, the transposed form of s that results will be TJ(S), where the interval of transposition is i = int(s l5 s N ) + int(s 2 ,s N _ 1 ). (We are supposing that N > 2. The formula is given without proof.) Thus the interval of transposition for the Zauber series A-C-Eb-E was 10 = 7 + 3 = int(A, E) + int(C, Eb), while the interval of transposition for Webern's row Eb-BA-G# was 3 = 5 + 10 = int(Eb,G#) + int(B, A). It follows: When we define the operation TCH as (RICH) (RICH), then TCH(s) is always some transposed form of s, but just which transposed form depends on the internal structure of any given argument s upon which TCH is operating. Specifically, if i = int(s l5 s N ) + int(s 2 ,s N _ 1 ), then TCH(s) = Tj(s). We shall call i here "the TCH-interval for s." The TCH interval for a retrograde or an inverted form of s will be the negative (group inverse) of this i; the TCH interval for a retrograde-inverted form of s will be the same as the TCH interval for s itself. (These facts follow from the formula defining i above.) 8.2.2 EXAMPLE: Let us turn our attention once more to figure 7.4, the earlier example from Parsifal, inspecting the first, second, third, and fourth forms of Zauber that are beamed thereon and labeled as Z t , Z 2 , Z3, and Z4. Considering Z as a serial motive, we can graph Wagner's transformational technique here by figure 8.3.

FIGURE 8.3

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We have already discussed how the TCH operation makes Z3 the 10transpose of Z x and Z4 the 10-transpose of Z 2 . During our earlier discussion of the passage, we pointed out how the chaining technique creates structural sequencing as a result. We can "hear" the structural sequence in the note heads of figure 7.4, even though the foreground events of the music over measures 1074-1100 are quite different from those of the music over measures 1096-1140. 8.2.3 EXAMPLE: Webern is fond of using RICH and TCH, especially as rowtransformations in his serial works. Figure 8.4 graphs two instances from the Piano Variations, (a) of the figure coincides with the thematic middle section of the first movement; (b) coincides with the frantically syncopated variation in the last movement. The nodes on figure 8.4 are understood to contain various forms of the row. On (a) of the figure, T5 is not TCH. The row-form that fills both the leftmost and the rightmost nodes of (b) is the prime row-form cited earlier, the form which opens the third movement. The same form also fills the rightmost node of (a). This is the first occurrence of the prime form in the piece; after

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FIGURE 8.4

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8.2.5

measure 37, the music continues to project the prime form of the row as it launches the big thematic reprise of the movement. The prime form that fills the rightmost node of (b) launches the coda of the third movement and the entire piece. The total rhythm of this coda matches that of the first-movement theme reprised at 1,37: In both cases we hear a rhythmic ostinato whose repeated units project four attacks and a rest in steady note-values. 8.2.4 If we excise any four consecutive nodes from figure 8.4(b), along with the arrows that connect them, we shall have essentially the same graph as that of figure 8.3. The same transformations are arranged and combined in the same structure of nodes and arrows, even though the contents of the nodes are Wagnerian in one case and Webernian in the other. We shall say that the two networks-of-series are isographic. The isography would not obtain if we wrote "T10" for TCH on figure 8.3 and "T3" for TCH on figure 8.4(b): T10 and T3 are not the same transformation. 8.2.5 EXAMPLE: Let us define another operation on series, an operation called MUCH. MUCH(s) is that retrograde-inverted form of s whose beginning overlaps the ending of s to the maximum possible extent. Figure 8.5(a) shows how Bach chains MUCH and RICH in the first Two-Part Invention. Figure 8.5(b) shows Bach's transformational technique in a graph which is very similar to the earlier graphs involving RICH. OP is the transformation RICH-after-MUCH; OP' is the transformation MUCH-after-RICH. MUCH and RICH do not commute on the family of all series, but the OP-interval-oftransposition for any given s is always the same as the OP'-interval for the

FIGURE 8.5

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retrograde-inversion of s. In the specific example at hand, that interval is 2diatonic-steps-down. Bach's foreground sequence is constructed by a method very similar to that used by Wagner and Webern, in building their structural sequences. Figure 8.5 shows how artfully Bach's transformational technique uses the characteristics of his motive to fit his meter. 8.2.6 EXAMPLE: Wagner also uses RICH in the foreground. Figure 8.6 sketches the opening of the "Todesverkiindigung," Die Walkure, act 2, scene 4, starting from three measures before the scene begins.

FIGURE 8.6

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The scene is dominated by the Fate motive, whose melodic component FATE is bracketed over measures 1-2 on the figure. FATE functions both as a 3-note series and as an intervallic network, 2 = (— 1) + 3 semitones. The interval 2 of that network is originally spanned from A to B, and the thematic pitch-idea of "A to B" thereby goes along with the prime form of the FATE motive/series/network. Locally, "A to B" helps us hear FATE as related to the

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8.2.6

(head of the) LOVE motive, which has just been repeated over and over in the melody preceding measure 1. The repeated LOVE fragment also projects an overall sense of "A to B"; as it repeats, it inflects A by a pickup F# and thereby defines a total ambitus of F#-to-B. These features enable us to extract the intervallic network of figure 8.7(a).

FIGURE 8.7

Figure 8.7(b) displays the FATE network for purposes of comparison. A 3-arrow and a 2-arrow are common to both networks. Another way of relating the motivic fragments is to regard LOVE as an essential A-to-B elaborated by an F# pickup 3 below A, while regarding FATE as an essential A-to-B elaborated by a G# pickup 3 below B. This view attributes emphatic priority to pitch relations and even pitch hierarchies, at the expense of intervallic relationships per se. We shall see that the intervallic structures of the "Todesverkiindigung," like those of the Brahms Horn Trio studied earlier, take on a more autonomous role as the music develops. We can hear another strong relation of LOVE and FATE in the music through the harmonization of LOVE that immediately precedes FATE: A-toB within LOVE is supported by C-to-B in the bass; the total matrix thereby projects a pitch-class form of FATE. Figure 8.6 indicates the pitch-class intervals of the FATE network on the LOVE matrix accordingly. A 2-arrow connects A to B within the melody of LOVE; that 2-arrow is subarticulated into a "3"-arrow, from A to the C below, and a (— l)-arrow, from C in the bass to B in the bass. The bass B supports and doubles the B of the LOVE melody as a pitch class. Supposing a fundamental bass for the A minor harmony of LOVE, an A that goes under and hence conceptually "before" the C of the basso continue, we can even identify a serial form of FATE embedded in this LOVE music, namely A-C-B. This is the unique form of the FATE series, other than the prime form, that embeds the essential A-to-B gesture. Starting from that A-C-B, we can hear a RICH chain of FATE forms proceeding along the bass line of figure 8.6. A-C-B before measure 1 chains

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#.2.6

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into C-B-D across measure 1; C-B-D in turn chains into B-D-C#, which carries the harmony through the bass of the FATE motive proper. The chaining continues on in the bass line, through D-C#-E, C#-E-D#, and even farther (across the DEATH music). We recognize the sequencing of the FATE bass as a typical TCH sequence, created by Rl-chainmg. The interval of the sequence, that is the TCH interval for the FATE bass, is 2. Computing the interval specifically from the series A-C-B, our point of departure in the bass, we have i = int(A, B) + int(C, C) = int(A, B), which is 2. So we see that 2-as-TCH-interval, governing the sequence, is the "same" 2 as the 2 that spans A-to-B, that is int(A, B). We do not need such heavy transformational machinery, just to hear that the 2-sequences of the music are related to the A-B gesture of the motive. But the transformational machinery clarifies just how the relation is worked out, and it attributes thereby a special and characteristic formal function to the compositionally prominent interval in this serial context. The FATE series is Rl-chained in the melody, too. Just as the bass chain began with A-C-B, one form of FATE that embeds A-to-B, so the melodic chain begins with A-G#-B (measures 1-2), the other form of FATE that embeds A-to-B. Again the TCH interval is 2 = int(A, B) + int(G#, G#) = int(A, B). Again the musical sequence from measures 1-2 to measures 5-6 is carried by a TCH sequence, now in the melody. The melodic chain of measures 1-8, covered by a slur on figure 8.6, is recalled in summary at the end of the DEATH motive, covered by another slur over measures 10-12 on the figure. The last stage of FATE under the slur, B-A#-C#, recurs yet again at measure 13, to launch the transposed return of all the FATE music, which starts the chaining underway all over again. Via the relationships just mentioned, we can hear that the melodic B-A#-C# of measures 13-14 is in the TCH-relation to the A-G#-B of measures 1-2. Hence the large sequence, the sequence that carries all of measures 1-12 into all of measures 13-24, is itself a TCHsequence, a product of Rl-chaining. It would be possible and even legitimate to draw many 2-arrows upon figure 8.6, e.g. transposing measures 1-4 into measures 5-8, transposing measures 10-llj of the melody into measures 11-12 of the melody, and transposing all of measures 1-12 into all of measures 13-24. We are certainly strongly aware of the interval 2 in these connections. The resulting graph would be inadequate, though, so far as it suggested that the relations involved were only transpositional, the results of expanding the interval 2 = int(A, B) into T2-relations between larger Gestalts. We would miss the function of 2 as a TCH interval in two independent FATE chains, one in the bass and one in the melody, each chain launched by one of the two FATE-forms that embed A-to-B. We would miss hearing how the sequences in the music hang on the pertinent stages of RI chaining in the outer voices. Figure 8.8 shows how a motive I shall call FATE' is Rl-chained over

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8.2.6

FIGURE 8.8

measures 56 and following. This is where Siegmund "looks into the eyes of the Valkyrie," as Brunnhilde puts it later on. The subinterval 3 of the FATE network 2 = (—1) + 3 now becomes the overall interval of the FATE' network 3 = (— 1) + 4; (—1) remains a subinterval of FATE'. The pitch class A of FATE' is bereft of its FATE-partner B, just as Siegmund will be lonely in Valhalla, bereft of his sister/wife. The comparison is suggestive because the F# and A of FATE' recall the F# and A of LOVE, displayed earlier in figure 8.7(a). Interval 3, the TCH-interval for FATE', is precisely int(F#, A)(= int(F#, A) + int(E#, E#)). On figure 8.8, the FATE' series goes through four TCH-sequences. Its course is therefore isographic to the bassline chain of figure 8.6, which also went through four TCH-sequences. RICH and TCH thus enable us to hear a way in which the bass line of figure 8.6 and all of figure 8.8 project the same overall transformational gesture. Obviously we could not hear such a relation using T2 in one case and T3 in the other; those are different transformations. After Siegmund has looked into Brimnhilde's eyes he goes into an extended Wagnerian stichomythy, asking a series of questions about Valhalla which Brunnhilde answers in turn. What really concerns Siegmund, we discover, is the idea that Sieglinde may not be able to accompany him to Valhalla. His concern is well founded. As he asks each question, he sings or sings along with an entry of the DEATH motive. Figure 8.9 tabulates his questions in brief, along with the keys in which the DEATH melody enters therewith, during measures 70-133.

FIGURE 8.9

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We are intensely aware of Siegmund's concern for Sieglinde even before his final questions, because the keys of figure 8.9 and the intervals among their tonics project the pitch classes and intervals of the LOVE motive, as diagrammed in figure 8.7 (a) earlier. During the repeated LOVE music just before the scene change, Siegmund had calmed Sieglinde, and she is lying asleep in his arms during the whole exchange with Brunnhilde. 8.3.1 EXAMPLE: RICH and TCH have provided good examples of serial transformations that are not intervallic/transpositional but are nonetheless very suggestive in connection with node/arrow analytic graphs. The retrograde operation on series is another such example. So are the various operations RT and RI, T being some transposition and I some inversion. (For a given u and a given v, RI* and RICH are not the same operation, as they operate upon a variety of series.) Webern's piece for string quartet, op. 5, no. 4, demonstrates other suggestive serial operations. The operation TLAST transposes a series by its last interval; the operation TFIRST transposes a series by its first interval. Hence TFIRST"1 transposes a series by the complement of its first interval. TLAST makes the last note of a given series the next-to-last note of the transformed series; TFIRST"1 has a sort of "dual" effect, in that it makes the first note of a given series the second note of the transformed series. Figure 8.10 shows three series of pitch classes, arranged in a network that involves TFIRST'1 and TLAST in this "dual" relationship.

FIGURE 8,10

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The three series cited by the figure are projected by the three appearances of a prominent unaccompanied rising motive FLYAWAY in Webern's piece. Each of these unaccompanied appearances ends a major section of the work. The order in which figure 8.10 cites the three series is not the chronological order in which the music presents the forms of FLYAWAY. In the music, the form that begins on A(7 occurs last; indeed it is the last event of the piece. The visual "centrality" of this form on figure 8.10 portrays a cadential function for the event, transformationally "balanced" as it is between the other two forms. Just so, the formal "centrality" of a tonic Klang, balanced between its

Transformation Graphs and Networks (2)

8.4

dominant and subdominant, is often projected temporally by a final cadence in which the tonic is the last event. While it would be perfectly possible to label the arrows of figure 8.10 as "T8" and "T3" rather than TFIRST'1 and TLAST, the transpositional labels would conceal, not reveal, the balancing centrality of the form beginning on A|?, the form that ends the piece. The intervals of transposition are of course very important in other connections. 8.3.2 EXAMPLE: We define operations FLIPEND and FLIPSTART on series of three pitches or three pitch-classes. FLIPEND transforms the series S1-s2-s3 into the series s^Sj-a, where a is the inversion-about-s3 of s2. (a = Is*(s2); int(s3,a) = int(s2, s3).) The inverse operation FLIPEND"1 then transforms the series ti-t2-t3 into the series ti-b-t^ where b is the inversionabout-t2 of t3. (To verify this, given t l s t 2 , and t3, set Si = t l 5 s2 = b-as-b-isdefined, and s3 = t 2 . Then observe that FLIPEND transforms the s series into the given t series.) Dually, FLIPSTART transforms S1-s2-s3 into a-s1-s3, where a is the inversion-about-Si of s2; then FLIPSTART"1 transforms ti~t2-^-3 into t2-b-t3, where b is the inversion-about-t2oftl. Figure 8.11 shows what happens when FLIPEND and FLIPSTART"1 are chained in alternation, starting with one series of three pitches in (a) and another in (b). The arrows above the staff indicate applications of FLIPEND; the arrows below the staff indicate applications of FLIPSTART"1. FLIPEND and FLIPSTART are my own names for transformations used by Jonathan W. Bernard in studying how Varese's music expands, contracts, and displaces registral space.1 8.4 The serial transformations just studied, i.e. RICH, TCH, MUCH, TLAST, TFIRST, FLIPEND, and FLIPSTART, are all easily generalized to operate on series whose elements are members of an abstract commutative GIS. In the non-commutative case, it is not clear just how some of the operations are to be defined; different possibilities are equally plausible. For instance, given the abstract series s = s l5 s2, ..., a, b within some abstract GIS, let us try to define RICH(s) abstractly. We can consider three possibly different series, t, u, and v, as plausible candidates for "RICH(s)." t is the retrograde of I(s), where I is (a/b)-inversion l£. u is the retrograde of J(s), where J is (b/a)-inversion I£ = (I*)"1, v is a series constructed as follows. Let the serial intervals of s be i t = int(s1, s2), i2 = int(s2, s 3 ),..., i N _j = int(a, b). The series v starts with the element a, and then proceeds according to the succession of intervals i N _ 1} i N _2 ,... ,i2,i!. Elements a and b will be the first and second elements of all three series t, u, and v. t and u are both retrograde1. Jonathan W. Bernard, The Music of Edgard Varese (New Haven and London: Yale University Press, forthcoming).

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8.4

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FIGURE 8.11 inverted forms of s. The serial intervals of v are the same as the intervals of s in reverse order. If the GIS is commutative, t, u, and v will all be the same series. If the GIS is not commutative, t, u, and v may be three distinct series.

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8.5 In section 8.1 we studied Klang-transformations as potential labels for arrows on graphs, arrows that could not be analyzed adequately by intervallic/ transpositional ideas alone. In sections 8.2 and 8.3 we studied a number of serial transformations in the same connection. Inversional relations between elements in a GIS, even a well-behaved commutative GIS, may also give rise to non-transpositional arrows on graphs, that is I-arrows. Consider figures 8.12 (a) and (b). The pitch classes on the networks stand for the pitches presented at the opening of the second movement in Webern's Piano Variations. There, the pitches come in pairs articulated by a rhythmic motive that remains essentially constant over the events portrayed in the figure. Graph (a) analyzes the pitches

Transformation Graphs and Networks (2)

8.4

FIGURE 8.12

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8.4

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of the phrase in an intervallic/transpositional network. One could add more interval-arrows connecting certain pitches by criteria other than immediate temporal succession. Graph (b) analyzes the same pitches using not only some intervallic/transpositional arrows (shown as curved) but also some inversional arrows (shown as straight). "I" is inversion about the pitch A4. Graph (b) reflects the rhythmic motif of the music by a visual motif, the vertical I-arrow. It represents by a visual symmetry the mirror symmetry of the pitch registration. It makes manifest the row-structure of the passage, and the way in which that structure interacts with the dux/comes structure of the musical canon. Graph (b) thus reflects more clearly than graph (a) not only the compositional structure of the passage but also our foreground perception of its shapes. Let us denote by IPAIR the graph consisting of two nodes connected by a two-way I-arrow. Now "I" will mean pitch-class inversion about A. As a configuration of pitch classes, network 8.12(b) has four subnetworks manifesting the graph IPAIR. Those are the subnetworks relating Bt?-and-G#, A-and-A, F-and-C#, and D-and-E; the subnetworks are all isographic. The network of row-forms being used is also isographic: Two row-forms are being used, each of which is the I-inversion of the other. Also manifesting IPAIR isographically is a network interrelating the "antecedent" and "consequent" row forms that respectively control the first twelve and the second twelve notes of the third movement. Antecedent and consequent forms there are I-inversions, each of the other. The consequent is presented in the music isorhythmically to the antecedent. IPAIR, in short, is a thematic graph.

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9

Transformation Graphs and

Networks (3): Formalities

It is time now to become formal about what we are doing when we draw nodes, arrows connecting some pairs of nodes, and names of transformations labeling those arrows, sometimes also putting pitches, Klangs, series, row-forms, or other objects "into" the nodes. I have already worked out most of these formalities elsewhere.1 Their treatment here differs somewhat, mainly in that the present exposition is more general. 9.1.1 DEFINITION: By a node I arrow system we shall mean an ordered pair (NODES, ARROW), where NODES is a family (i.e. set in the mathematical sense), and ARROW is a subfamily of NODES x NODES, i.e. a collection containing some ordered pairs (N l9 N 2 ) of NODES. We say that nodes N t and N2 are "in the arrow relation" if the pair (N l 5 N 2 ) is a member of the collection ARROW. For present purposes, we shall stipulate that every node is in the arrow relation with itself. That is, we assume that (N, N) is a member of ARROW for every node N. On figure 9.1, the nodes Mt and M 2 are not in the arrow relation; neither are M 2 and Mj. Nodes M4 and M 5 are in the arrow relation; so are M 5 and M4. Nodes M 5 and M6 are in the arrow relation; M6 and M 5 are not. Arrows from M! to M! , from M 2 to M 2 , and so on, are all understood on the figure. 9.1.2 DEFINITION: Nodes N and N' in a node/arrow system communicate if 1. David Lewin, "Transformational Techniques in Atonal and Other Music Theories," Perspectives of New Music vol. 21, nos. 1-2 (Fall-Winter 1982/Spring-Summer 1983), 312-71. The formalism there is confined to pages 360-66. The article consists mostly of analyses, together with exemplary graphs and networks, that will interest the reader who has enjoyed chapters 7 and 8 of the present book. The music studied there overlaps that studied here only slightly.

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9.1.3

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FIGURE 9.1

there exist nodes N 0 , N t , ..., Nj which satisfy criteria (A), (B), and (C) following. (A): N0 = N. (B): For each j between 1 and J inclusive, either (Nj.^Nj) is in the ARROW relation, or else (NJ5 Nj.J is. (C): Nj = N'. The criteria demand a finite unbroken path of forwards-or-backwards arrows which starts at N and ends at N'. The nodes M t and M 2 of figure 9.1 communicate. But neither Mj nor M 2 communicates with M4. The communication relation among nodes is easily proved to be reflexive, symmetric, and transitive. Hence "communication" is an equivalence relation on the NODES of a node/arrow system. As figure 9.1 suggests, the nodes within any equivalence class all communicate, each with any other, while nodes in different equivalence classes do not communicate. 9.1.3 DEFINITION: A node/arrow system is connected if any two nodes communicate. The system displayed in figure 9.1 is not connected. It can be analyzed into two component subsystems, each of which is connected; there is no communication between any node of one subsystem and any node of the other. This sort of structure is typical of disconnected node/arrow systems. Any disconnected system can be analyzed into component connected subsystems that do not communicate with one another. Each such subsystem is a pair (NODESa, ARROWJ, where NODESa is an equivalence-class of NODES under the communication relation, and ARROWa is that subcollection of ARROW comprising pairs of nodes from NODEa. As we proceed, we shall find the following construct necessary and useful.

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9.1.4 DEFINITION: An arrow chain from node N to node N' in a node/arrow system is a finite series of nodes N0, N x , . . . , N, satisfying criteria (A), (B), and (C) below.

Transformation Graphs and Networks (3)

9.2.2

(A): NO = N. (B): For each j between 1 and J inclusive, (N^, Nj) is in the ARROW relation. (C): Nj = N'. The criteria demand a finite unbroken path of forwards-orientedarrows, starting at N and ending at N'. We are now ready to "label" our arrows formally with symbolic transformations. At this stage of our work, the nodes do not yet "contain" any objects to be transformed, so we shall represent the eventual transformations abstractly, as members of some abstract semigroup. 9.2.1 DEFINITION: A transformation graph is an ordered quadruple (NODES, ARROW, SGP, TRANSIT) satisfying criteria (A), (B), (C), and (D) below. (A): (NODES, ARROW) is a node/arrow system. (B): SGP is a semigroup. (C): TRANSIT is a function mapping ARROW into SGP. (D): Given nodes N and N', suppose that N0, N1? ..., Nj is an arrow chain from N to N'. Suppose that M 0 , M 1 } ..., MK is also an arrow chain from N to N'. For each j between 1 and J inclusive, let Xj = TRANSIT(Nj.j^Nj). For each k between 1 and K inclusive, let yk = TRANSIT(Mk_1, Mk). Then the semigroup product X j . . . x2xi is equal to the semigroup product yK ... y 2 yi-

FIGURE 9.2 The setting in which criterion (D) holds sway is illustrated by figure 9.2. Symbolically, the eventual "contents" of N' will be the Xj-transform of the... of the x2-transform of the x^transform of the eventual "contents" of N. That is, the contents of N' will be the (Xj... x 2 x ^-transform of the contents of N, in the sense of our left-orthographic convention. Likewise, the contents of N' must also be the (yK ... yjy^-transform of the contents of N. To ensure that no contradiction can possibly arise, we must ensure that the semigroup products are equal. And that is what Criterion (D) of the definition does. 9.2.2 OPTIONAL: Criterion (D) also enables us to prove that for each node N,

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TRANSIT(N, N) must be some idempotent member of SGP. Tp prove this, consider the following two formal arrow-chains from N to N: N0 = N t = Nj = N; M0 = M j = M 2 = MK = N. Set Xj = TRANSITCN^NO, y t = TRANSITCM^Mi), y 2 = TRANSIT^, M 2 ). Then, via the criterion, x =V v i 2 i- But x1? y1} and y2 are all the same member of SGP, namely TRANSIT (N, N). So that element is idempotent, as claimed. Criterion (D) then ensures that whenever M0, M l 5 . . . , MK is an arrow chain from N back to N, the product yK ... y^ (as in 9.2.1) is equal to the idempotent TRANSIT(N, N). The only idempotent member of a group is the identity element. (If zz = z, then zzz"1 = zz"1 and z = e.) Hence, when SGP is a group, TRANSIT (N, N) must be the identity. 9.2.3 DEFINITION: An operation graph is a transformation graph in which SGP is a group. Now we shall render formal the idea of "putting objects into" the nodes of a transformation graph, and transforming the objects about as indicated by the graph and by some appropriate semigroup of transformations. That is the idea of a transformation network, which we shall now define, distinguishing it formally from the transformation graph whose nodes its objects fill. 9.3.1 DEFINITION: A transformation network is an ordered sextuple (S, NODES, ARROW, SGP, TRANSIT, CONTENTS) having the features (A) through (D) below. (A): S is a family of objects (that are to be transformed in various ways). (B): (NODES, ARROW, SGP, TRANSIT) is a transformation graph such that SGP is a semigroup of transformations on S. (C): CONTENTS is a function mapping NODES into S. "CONTENTS(N)" can be read: "the contents of node N." (D): Given nodes N t and N2 in the ARROW relation; if f = TRANSIT(N1,N2), then f (CONTENTS^)) = CONTENTS(N2). To see how feature (D) of the construction reflects our intuition, we can inspect figure 9.3. The figure depicts nodes N t and N2 in the arrow relation. On the graph, the member f of SGP labels the arrow from N! to N2; that is, f =

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FIGURE 9.3

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9.3.3

TRANSIT(N1,N2). f is some transformation on S, according to feature (B) of 9.3.1. Inspecting figure 9.3 further, we see that $! is the CONTENTS of N! and s2 is the CONTENTS of N 2 . In this situation, (D) of the definition assures us that we shall have f^) = s 2 . 9.3.2 DEFINITION: An operation network is a transformation network for which SGP is a group of operations on S. 9.3.3 THEOREM: Let S be a family of objects. Let GP be a group of operations on S. Let (NODES, ARROW, GP, TRANSIT) be an operation graph whose node/arrow system is connected. Let N0 be a node; let s0 be a member of S. Then there exists a unique operation-network having S for its objects and (NODES, ARROW, GP, TRANSIT) for its graph such that s0 is the CONTENTS of N0. The theorem is given without proof. It says that all the contents of all the nodes in a connected operation-network are uniquely determined, once we know the contents of any one node. Intuitively, we can follow some path of forwards-or-backwards arrows from the given N0 to any other node, since the system is connected. As we go along that path, we can fill in the CONTENTS of each node by applying the indicated TRANSIT operation when we traverse an arrow forwards, or the inverse (NB) of that operation when we traverse an arrow backwards. Criterion (D) of 9.2.1 enables us to infer that it does not matter which path we might follow from N0 to a specific other node in this connection; the end result will be the same as regards the necessary CONTENTS of that other node. To get some sense of why this works, let us look at the graph displayed in figure 9.4.

FIGURE 9.4 Here, we are supposing A, B, C, and D to be operations upon some family S. Fixing some member s0 of S, let us construct an operation-network having S for its objects and figure 9.4 for its graph, such that s0 is the CONTENTS of node N0 on the figure. Since s0 is the CONTENTS of N0 and A is the

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TRANSIT-operation from N0 to N3, the CONTENTS of N3 on the figure will have to be A(s0). (We know this by 9.3.1(D).) Let us write "s3" for "the necessary CONTENTS of node N3." Then we shall have to have s3 = A(s0). Now what about s l5 the necessary CONTENTS of node Nj? Since B is the TRANSIT-operation from N x to N3, we shall have to have s3 = 6(5!). Since we have already determined what s3 must be, and we are searching for s1, we can infer that the Si we are looking for is derived as s t = B"1 (s3). Note how we have leaned on the fact that B is an operation at this point in our construction. When we write "Sj = B"1^)," we are using the idea that a unique s t is well defined by the relation s3 = B^); that idea in turn rests on the supposition that B is 1-to-l and onto. So, given our initial s0, we have derived the necessary CONTENTS s3 and Sj for the nodes N3 and N t in the operation network we are constructing: s3 = A(s0); Si = B'^SS). What about s2, the necessary CONTENTS of N2? Here, it seems at first that we have two different choices for s2. Since an arrow labeled D points from N x to N 2 , we must have s2 = D^). But also, since an arrow labeled C points from N2 to N3, we must have s2 = C"1^). It is just here that 9.2.1 (D) comes to our rescue. The criterion tells us in this situation that, in fact, D(SJ) = C~1(s3). Therefore it only seems that we have "two" choices for S 2 ; in fact the value of s2 is well determined. Let us see just how we can infer from Criterion 9.2.1(D) that D(SJ) = C~1(s3). The Criterion notices an arrow-chain going from N x directly to N3; it also notices an arrow-chain going from N x through N2 to N3. It informs us that a certain algebraic relation must therefore obtain among the TRANSIToperations B, D, and C that link the nodes along those arrow-chains. The relation must obtain, that is, for us to have spoken at all of a well-formed "operation graph." The relation given by the Criterion here is B = CD. Now Si was chosen to satisfy the relation B(Si) = s3. So we can infer that CD(S!) = s3. And thence we can infer that D^J = C"1^), which is just what we wanted to show. Obviously, the sort of situation we have just examined in connection with the arrow-chains of figure 9.4 can get very complicated on a general operation-graph. The way in which Criterion 9.2.1 (D) helps us out is basically the same, though.

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During the musical discussions of chapters 7 and 8 we invoked from time to time the concept of "isography." It would seem that we can now define that concept rigorously: Two transformation networks are isographic if they "have the same graph." But that definition is not yet formal enough. For suppose we want to assert an isography between the networks (S, NODES, ARROW, SGP, TRANSIT, CONTENTS) and (S', NODES, ARROW, SGP', TRANSIT', CONTENTS'), where S and S' are different families of objects. Those two networks have the same node/arrow system, but they

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9.4.3

cannot have the same graph. The graph of one is (NODES, ARROW, SGP, TRANSIT); the graph of the other is (NODES, ARROW, SGF, TRANSIT'). SGP', a semigroup of transformations on S', cannot be "the same as" SGP, a semigroup of transformations on S. And therefore TRANSIT', a function taking on values in SGP', cannot be "the same as" TRANSIT, a function taking on values in SGP. Moreover, the concept of isography is particularly suggestive exactly when S' and S are different families of objects, as above. S for example might be the twelve chromatic pitch classes, while S' might be a family of set-forms, motive-forms, or row-forms; SGP could comprise transpositions and inversions on pitch classes, while SGP' comprised transpositions and inversions of set/motive/row forms. To make our intuitions about isography work out formally here, we need the concept of isomorphism between one transformation graph and another. We can then define two networks to be isographic if their graphs are isomorphic. To define the isomorphism of graphs, we shall in turn have to define the isomorphism of node/arrow systems. 9.4.1 DEFINITION: Two node/arrow systems, (NODES, ARROW) and (NODES', ARROW), are isomorphic if there exists a 1-to-l map NODEMAP of NODES onto NODES' such that for every pair (Ni,N 2 ) of NODES, (N 1} N 2 ) is in the ARROW relation if and only if (NODEMAP(N1), NODEMAP(N2)) is in the ARROW' relation. A function NODEMAP having the indicated property will be called an isomorphism of (NODES, ARROW) with (NODES', ARROW'). 9.4.2 DEFINITION: Given two transformation graphs (NODES, ARROW, SGP, TRANSIT) and (NODES', ARROW, SGP', TRANSIT'), the two will be called isomorphic if there exists a pair (NODEMAP, SGMAP) having properties (A), (B), and (C) following. (A): NODEMAP is an isomorphism of (NODES, ARROW) with (NODES', ARROW'). (B) SGMAP is an isomorphism of SGP with SGP'. (C): For every pair (N x , N2) in ARROW, TRANSIT'(NODEMAP(N1), NODEMAP(N2)) = SGMAP(TRANSIT(Nt, N2)). The pair (NODEMAP, SGMAP) will be called an isomorphism of the first graph with the second. 9.4.3 DEFINITION: The transformation networks (S, NODES, ARROW, SGP, TRANSIT, CONTENTS) and (S', NODES', ARROW, SGP', TRANSIT', CONTENTS') are isographic if the transformation graphs (NODES, ARROW, SGP, TRANSIT) and (NODES', ARROW, SGP',

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TRANSIT') are isomorphic. If (NODEMAP, SGMAP) is an isomorphism of the first graph with the second, then that pair is an isography of the first network with the second. 9.4.4 EXAMPLE: Figure 9.5 shows some transformation networks, all of which are isographic. (a), (b), and (c) will be familiar from our recent discussion (in section 8.5) of Webern's Piano Variations.

FIGURE 9.5

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Let us work out the isographies of the figure very formally. We can fix the node/arrow system that underlies all the graphs and networks: NODES is a two-element family; every pair of NODES is in the ARROW relation. Graph (a) has as its SGP the group of operations on pitch classes that contains the identity E and the operation I = l£. (This is a group, since II = E.) Graph (a) has as its TRANSIT function the function TRANSIT^, N x ) = TRANSIT(N2, N 2 ) = E; TRANSIT^, N2) = TRANSIT(N2, Nj) = I. Graph (b) has the same SGP and the same TRANSIT function as graph (a). So in this special case, graph (b) is in fact literally "the same as" graph (a). Graph (c) however is "different" (though isomorphic). Its semigroup comprises two operations on twelve-tone rows, not two operations on pitch classes. The row-operations are E (which leaves any row alone) and I (which inverts any row about the pitch class A). The TRANSIT function for graph (c) maps ARROW into this new semigroup of row-operations. The semigroup of row-operations, while "new," is isomorphic with the old semigroup of pitchclass operations, under the correspondence of pitch-class-operation E with row-operation E and pitch-class-operation I with row-operation I. We can take this map of the old semigroup into the new one as our formal SGMAP. And we can take as NODEMAP the identity map of our fixed NODES onto itself. The (NODEMAP, SGMAP) is a formal isomorphism of graph (a) (or graph (b)) with graph (c). Therefore (NODEMAP, SGMAP) is a formal isography of network (a) (or network (b)) with network (c).

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9.5.2

Networks (a) and (b) are derived from the opening of the second movement in Webern's piece. Suppose we played the music a semitone higher; then we could derive networks (d) and (e) of the figure instead. Here J is inversion-about-B)?. For graphs (d) and (e) the semigroup consists of the identity operation E on pitch classes and the inversion-operation J on pitch classes. The new semigroup is isomorphic with semigroup (a) under the map SGMAP(E) = E; SGMAP(I) = J. Using this SGMAP and the identity map on NODES as NODEM AP, we establish a formal isomorphism of graph (a) (or graph (b)) with graph (d) (or graph (e)). Networks (a) (or (b)) and (d) (or (e)) are thereby isographic. The notions of isomorphism we have just been exploring can be extended suggestively to more general notions of "homomorphism." 9.5.1 DEFINITION: Given node/arrow systems (NODES, ARROW) and (NODES', ARROW), a mapping NODEMAP of NODES into NODES' is a homomorphism of the first system into the second if (NODEMAP^j), NODEMAP(N2)) is in the ARROW' relation whenever (N l 5 N 2 ) is in the ARROW relation. NODEMAP is a homomorphism onto if it maps NODES onto NODES' in a special way: Whenever N't and N'2 are in the ARROW relation, there exist N t and N 2 in the ARROW relation such that N\ = NODEMAP(Ni) and N'2 = NODEMAP(N2). A homomorphism NODEMAP is 1-to-l as a homomorphism between systems if it is 1-to-l as a map of NODES into NODES'. Under these definitions, a 1-to-l homomorphism of one system onto another is an isomorphism in the sense of 9.4.1, and an isomorphism in that sense is a 1-to-l homomorphism of the first system onto the second. Here, the special definition of "homomorphism onto" in 9.5.1 above is crucial. It is possible for NODEMAP to be a homomorphism of (NODES, ARROW) into (NODES', ARROW) and also a 1-to-l map of the family NODES onto the family NODES', without being an isomorphism of the two systems. That is so because the second system may "have more arrows." For example we could simply take NODES' = NODES and add more arrows to ARROW for ARROW'. Then the identity map of NODES into NODES' is a homomorphism of (NODES, ARROW) into (NODES', ARROW') and also a 1-to-l map of NODES onto NODES', but it is clearly not an isomorphism of the two systems. It is not a homomorphism of the first system onto the second, in the full sense of 9.5.1. 9.5.2 DEFINITION: Given transformation graphs (NODES, ARROW, SGP, TRANSIT) and (NODES', ARROW', SGP', TRANSIT'), a homomorphism of the first graph into/onto the second is a pair (NODEMAP, SGMAP) having features (A), (B), and (C) below.

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(A): NODEMAP is a homomorphism of the node/arrow system (NODES, ARROW) into/onto the system (NODES', ARROW). (B): SGMAP is a homomorphism of the semigroup SGP into/onto the semigroup SGP'. (C): For every pair of nodes (N l 5 N 2 ) in the ARROW relation, TRANSIT'(NODEMAP(N1), NODEMAP(N2)) = SGMAP(TRANSIT(N15 N2)). The graph homomorphism (NODEMAP, SGMAP) is defined to be 1-to1 if both NODEMAP and SGMAP are 1-to-l maps, of NODES into NODES' and SGP into SGP' respectively. According to these definitions, an isomorphism of the first graph with the second (in the sense of 9.4.2 earlier) is precisely a 1-to-l homomorphism of the first graph onto the second.

FIGURE 9.6

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9.5.3 EXAMPLE: Earlier (section 7.3), we studied graphs (a) and (b) of figure 9.6 in connection with Brahms's Horn Trio. We called graph (a) the "complementary gesture," and graph (b) the "complementary gesture times 2." The intervallic augmentation that transforms graph (a) into graph (b) is in fact a formal homomorphism. To verify this, let us begin by attaching the name (NODES, ARROW) to the three-node node/arrow system common for both graphs. Take NODEMAP to be the identity map on NODES. NODEMAP is then, trivially, an isomorphism of the node/arrow systems involved for the two graphs. Take SGPa, the semigroup for graph (a), to be the group of the twelve chromatic pitch-class intervals. Take SGPb, the semigroup for graph (b), to be (provisionally) the same group. The values of TRANSITa and TRANSIT,, are as indicated on graphs (a) and (b) of the figure. Take SGMAP to be the mapping of the interval i into the interval 2i, a map that transforms SGPa into SGPb. That is, take SGMAP(i) = 2i for each interval i. Then SGMAP is a homomorphism of SGPa into SGPb: SGMAP (i + j) = SGMAP(i) + SGMAP(j) (mod 12). As defined, SGMAP is neither 1-to-l nor onto. Requirements (A) and (B) for Definition 9.5.2 are now verified, as regards a potential homomorphism of graph (a) into graph (b). It remains to verify requirement (C) of the definition. This is easily done by inspecting the

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9.5.4

numbers labeling the three arrows on each graph: If interval i labels any arrow on graph (a), then interval 2i labels the corresponding arrow on graph (b). That is, if i = TRANSIT^, N2), then 2i = TRANSIT'(N^Nj). Or, yet more formally, TRANSIT'(NODEMAPCNO, NODEMAP(N2)) = TRANSIT'(N15 N2) = 2i = SGMAP(i) = SGMAP(TRANSIT(Nl5 N2)), as demanded by requirement (C) of the definition. The requirement is also satisfied in case N2 = N t = N: TRANSIT'(NODEMAP(N), NODEMAP(N)) = TRANSIT'(N, N) = 0 = 2 - 0 = SGMAP(O) = SGM AP(TRANSIT(N, N)). The case N! = N2 = N is trivial here, as it always will be when the semigroups involved are groups; when the semigroups are not groups, TRANSIT'(N', N') and TRANSIT(N, N) will be idempotents within semigroups that may have many idempotents, and the requirement of the definition is not trivially satisfied. To make our graph homomorphism here a "homomorphism onto," we need only redefine SGPb as the group of all even intervals. Then SGMAP takes SGPa onto SGPb, and the graph homomorphism thereby becomes "onto" in the sense of 9.5.2. It is not, of course, 1-to-l. (SGMAP is still not 1-to-l.) 9.5.4 EXAMPLE: For this example we shall use the word "tritone" to mean a collection of two pitch-classes spanning that interval. There are six tritones in that sense: (C, F#), (C#, G), (D, Ab), (E[>, A), (E, B[?), and (F, B). Transposing a tritone by a pitch-class interval i has the same effect on the set as transposing it by interval i + 6. So for instance transposing (C,F#) by 5 yields (F, B); transposing (C, F#) by 11 also yields the unordered set (F, B). Accordingly, we can define six formal "transposition operations" on the family of tritones. We shall call the six operations O-or-6, l-or-7,2-or-8,3-or-9,4-or-10, and 5-or-l 1. The six operations form a simply transitive group on the family of tritones. We can therefore construct a GIS having the tritones for its objects and the six operations for its formal intervals. We write "int((C, F#), (F, B)) = 5-or-l 1," and so on. Consider the map SGMAP that takes pitch-class interval i into tritoneinterval i-or-(i + 6). This map is a homomorphism from the group of pitchclass intervals, onto the group of tritone intervals. That is, if we transpose a given tritone by i-or-(i + 6), and then transpose the resulting tritone by j-or(j + 6), we shall end up having transposed the original tritone by (i + j)-or(i + j + 6), all mod 12 of course. SGMAP is part of a homomorphism that transforms the graph of network (a), in figure 9.7, onto the graph of network (b). We imagine arrows labeled "0" from each node of (a) to itself, and arrows labeled "O-or-6" from each node of (b) to itself. The NODEMAP for the graph homomorphism takes the two top nodes of (a) into the top node of (b), and the two bottom nodes of (a) into the bottom node of (b). The CONTENTS of various nodes on networks(a) and (b) in the figure help us see why

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FIGURE 9.7

NODEMAP is a musically plausible function here. But the homomorphism of the graphs does not depend formally upon the CONTENTS with which the nodes are filled in the networks. The graphs as such know nothing of these contents; they know only the node/arrow systems, the semigroups involved in labeling the arrows, and the TRANSIT functions that provide those labels. 9.5.5 EXAMPLE: Figure 9.8(a) transcribes the opening phrase from the first example in the Scholica Enchiriadis that shows the Symphony of the Diatesseron.2 (b) of the figure graphs the melody "Nos qui vivimus"; the numbers measure steps up or down in the mode, (c) of the figure is a network whose node/arrow system is disconnected; the network exhibits the vocal lines of Principalis and Organalis separately. Graph (b) is a homomorphic image of graph (c). The homomorphism works as follows: NODEMAP takes the first Principalis node of (c) and the first Organalis node of (c) both into the first node of (b); NODEMAP takes the second Principalis node and the second Organalis node of (c) both into the second node of (b); and so on; SGMAP is the identity map of SGP onto itself, where SGP is the pertinent group of intervals, that is the group of distances in steps up a scale. Graph (b) is however not a homomorphic image of graph (d) under the analogous NODEMAP. For there is no possible SGMAP, mapping our intervals homomorphically into themselves, that satisfies both SGMAP(l) = 1 and SGMAP(3) = 0. Any homomorphism SGMAP that satisfies SGMAP(l) = 1 must satisfy SGMAP(3) = SGMAP(1 + 1 + 1) = SGMAP(l) + SGMAP(l) 4- SGMAP(l) = 1 + 1 + 1 = 3 . The graph of (d) can be constructed as a formal "product" of graph (b) with graph (e). But that is quite another matter. Network (d) must be distin204

2. The transcription is taken from Oliver Strunk, Source Readings in Music History (New York: W. W. Norton, 1950), p. 130.

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FIGURE 9.8

9.5.5

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guished not only from (c) but also from (f) and (g). (f) is a network whose graph is (b); each node of (f) contains a network whose graph is (e). (f) is thus a network-of-networks; the arrows on (f) labeled 1, — 1, and 0 transpose entire (e)-networks. (f) models the thought, "We are singing (the graph of) 'Nos qui vivimus,' singing diatessera ((e)-networks) as we go." (g) is a network whose graph is (e); each node of (g) contains a network whose graph is (b). (g) is thus, like (f), a network-of-networks. It reflects a way in which Organalis might think: "Principalis is singing 'Nos qui vivimus'; I too am singing 'Nos qui vivimus,' and my relation to Principalis is governed by the Symphony of the Diatesseron (the symbol '3' on the graph)." Our ability to form the "product network" of (d), and the networks-ofnetworks displayed in (f) and (g), is heavily dependent on the following aspects of the situation: The transformations involved in (b), that is T15 T_ l 5 and T0, are all operations; also the transformation T3 involved in (e) is an operation; also T3 commutes with T l 9 with T_ 15 and with T0. We shall not pursue the abstract theory of such matters any farther here.3 Earlier, in connection with the vocal line of the song "Angst und Hoffen" (figure 6.4), and again in connection with our study of twelve-tone rows as families of "protocol pairs," (section 6.2.4), we discussed various techniques for modeling series of pitches, pitch classes, or other objects. Figure 9.8(b) suggests yet another technique: We can regard a series as a certain type of transformation-network. Just what type is a matter we shall clarify and make formal later on (in section 9.7.7). The terminology we shall develop there will tell us that a network can model a series if the node/arrow system is "precedence-ordered and linearly ordered under that ordering." The three examples we have just studied, 9.5.3, 9.5.4, and 9.5.5, show us that there is a lot of variety in the forms that graph-homomorphisms can assume. In 9.5.3, NODEMAP was an isomorphism of the node/arrow system and SGMAP was a proper homomorphism of the semigroup, i.e. not an isomorphism. In 9.5.4, NODEMAP was onto but not 1-to-l, while SGMAP was again a proper homomorphism. In 9.5.5, NODEMAP was not 1-to-l, while SGMAP was an isomorphism. We now turn away from isographies, isomorphisms, and homomorphisms, to explore some different matters. When we use a transformation-

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3. T3, a gesture which we can call "climb three rungs up on the modal ladder," corresponds to the concept of dia + tesseron, given the difference in the manner of counting rungs. T3 is not the same gesture as RISE(4/3), meaning "rise so as to get higher in the harmonic pitch-ratio of 4-to3." RISE(4/3) does not commute with the transformations TI and T_j of the example. That is of course a salient problem of the style. And that is how our machinery views the problem. Our machinery also provides us with the formally different models of (d), (f), and (g), which give us interestingly different ways of thinking about what is going on in the Symphony.

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9.6.3

network to discuss events in a musical passage, the configuration of its node/arrow system may allow us to isolate and discuss formal properties of certain nodes that implicitly assert corresponding formal functions for the CONTENTS of those nodes. In this connection, it is interesting to study "input nodes" and "output nodes" for node/arrow systems, and hence for graphs and networks that involve those systems. 9.6.1 DEFINITION: An input node for a node/arrow system is a node IN to which no proper arrows point. That is, if (N, IN) is in the ARROW relation, we must have N = IN. Analogously, an output node is a node OUT from which no proper arrows issue. That is, if (OUT, N) is in the ARROW relation, we must have N = OUT. 9.6.2 EXAMPLE: In the network of figure 9.9, the node on the left is an input node and the node on the right is an output node.

FIGURE 9.9

The reader will recognize the graph as the "complementary gesture" from the Brahms Horn Trio. In discussing figure 7.7(g) earlier, we noted how the motif of the Bb octave-leap, when led into the complementary gesture, generates the pitch class G[?. And we explored some consequences of that generation. Here we can note that our intuitions about "putting in" Bj? and "getting out" G[? correspond nicely to the formal input and output functions of the left hand and right hand nodes on figure 9.9. 9.6.3 EXAMPLE: Figure 9.10 shows a network of Klangs whose graph we

FIGURE 9.10

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9.6.4

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earlier called CADENCE, when we were studying passages from Beethoven's First Symphony (in section 7.4). The Klang (C, +) that fills the left node of the figure is the same as the Klang that fills the right node. But the function of the Klang as CONTENTS(left node) is different from '^function as CONTENTS (right node). In the former capacity, the Klang is an input; in the latter capacity it is an output. The input and output functions for (C, +) in the network reflect very well in this setting two of the three principal ideas about tonicity that have governed most theories of tonality since the eighteenth century. The input function reflects the idea of tonic-as-generator, a tonic that asserts itself in the very act of sounding a tone, setting a musical process in action, a tonic which generates other tones through that action. The output function reflects the idea of tonic-as-goal, a tonic that appears at the end of a completed gesture as a point of repose towards which events have been moving. The third principal idea asserts as tonic a center of balance in a well-balanced structure. That idea, too, is manifest in the visual aspect of figure 9.10: The figure balances about the two nodes containing (C, +). We cannot completely translate out input/output formalities into ideas about tonicity. For instance we do not want to assert (G, +) or (F, +) on figure 9.10 as "tonics," even though the former Klang fills an input node and the latter Klang an output node. Nevertheless the input/output formalities are suggestive in connection with tonal theory. 9.6.4 EXAMPLE: Figure 9.11 shows a network whose nodes are filled by forms of the FATE motive from the opening of Die Walkiire, act 2, scene 4, the "Todesverkiindigung." We discussed this network in connection with figure 8.6 earlier. The horizontal arrows on the figure are labeled by the TRANSIToperation RICH; the curved arrows are labeled by TCH. The transformation

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FIGURE9.il

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9.7.1

BIND which labels the diagonal arrows takes a series of pitch classes and transforms it into that one of its retrograde-inverted forms which has the same first and last notes. BIND commutes with TCH. Figure 9.11 has precisely one input node, the node indicated at the lower left. The input node is filled by the FATE-form A-C-B, which thereby acquires a special generative function for the network. This formal status of A-C-B as network-generator corresponds very well with the musical priority we assigned the motive-form in our earlier analysis. There we noted how the entire opening of the "Todesverkiindigung" grows out of the measures immediately preceding the scene, measures in which the A-B gesture of the LOVE motive is harmonized with C-B in the bass, A functioning also as fundamental bass below C and hence implicitly asserting a relation of Abefore-C. The three protocol pairs (A, B), (C, B), and (A, C) determine the motive A-C-B as a partial ordering. Dramatically, the (A, B), (C, B) and (A, C) protocols in the music just before scene 4 represent Siegmund's relation to the sleeping Sieglinde, a relation which metaphorically lies under the entire Siegmund/Brunnhilde scene just as Sieglinde lies under it literally. The input and output functions we have been exploring are essentially rhythmic aspects of node/arrow systems, in a certain structural sense: Input nodes "happen before" other nodes with which they communicate; output nodes "happen after" others with which they communicate. More generally, one observes that the arrows of any node/arrow system have a formal rhythmic structure of their own, a structure which can engage musical rhythm in varied and sometimes complicated ways. Our practice of laying out graphs visually so that most arrows go from left to right on the page has made it easy for us to put off investigating the issues that arise when we try to match the internal arrow-flows of a network with the temporal flow of the music upon which the network comments. Here, now, we shall attempt to explore some of those issues, though we can hardly do them justice in one section of one chapter. We recall the definition of an "arrow chain" from node N to node N' in a node/arrow system; it is a finite series of nodes N0, N t , . . . , Nj such that N0 = N, Nj = N', and (N^, N3) is in the ARROW relation for each j between 1 and J inclusive (9.1.4). 9.7.1 DEFINITION: An arrow chain (as above) is proper if there is at least one j between 1 and J inclusive such that (Njs N^) is not in the ARROW relation. Intuitively, the definition demands at least one "one-way arrow" along the chain. (The implicit arrow between any node and itself counts as "twoway" in this connection.) Another way of intuiting the definition is to think of a proper arrow-chain as one that cannot be "walked backwards."

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9.7.2 DEFINITION: In a node/arrow system, node N precedes node N', and N' follows N, if there exists some proper arrow-chain from N to N'. One must be careful to distinguish the relation "N precedes N'" from the relation "N is in the ARROW relation to N'."

FIGURE 9.12 On figure 9.12, for instance, M t precedes M3 because M1-M2-M3 is an arrow-chain from M^ to M 3 that involves a one-way arrow (from M 2 to M3). But M! is not in the ARROW relation to M 3 . M t is, on the other hand, in the ARROW relation to M 2 . But Mt does not precede M2: There is no arrowchain from M! to M 2 which involves any one-way arrow. 9.7.3 DEFINITION: A node/arrow system is precedence-ordered if there is no pair of nodes (N, N') such that N both precedes and follows N'. The reason why we speak of a precedence-ordered system as "ordered" is implicit in the following theorem.

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9.7.4 THEOREM: Let (NODES, ARROW) be a precedence-ordered node/ arrow system. Let PRECEDENCE be the family of node-pairs (Ni, N 2 ) such that N: precedes N 2 . Then PRECEDENCE is a (strict) partial ordering on NODES. That is, PRECEDENCE satisfies conditions (PO1) and (PO2) below. (PO1): There is no pair (N l 9 N 2 ) such that both (N 1 ,N 2 ) and (N^NJ are members of PRECEDENCE. (PO2): If (N! , N 2 ) and (N 2 , N3) are both members of PRECEDENCE, then so is (N 1? N 3 ). Proof: (PO1) for PRECEDENCE is equivalent to the condition of 9.7.3, which is true in a precedence-ordered system. (PO2) is obvious: If there is a "good" arrow-chain from N x to N2 and a "good" arrow-chain from N2 to N 3 , then there will be a "good" arrow-chain from N! to N3. q.e.d. The reader will recall (PO1) and (PO2) from section 6.2.4 earlier, where we invoked them to characterize collections of protocol-pairs that were (strict) partial orderings on the twelve pitch-classes. Here we use the same mathematical conditions to characterize (strict) partial orderings on NODES. A precedence-ordered system is at least potentially compatible with our naive sense of chronology. When used for analytic purposes, that system will

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9.7.5

not have to assert that one musical event both "precedes" and "follows" another, in the strictly formal sense of 9.7.2. There is nothing intrinsically correct or good about avoiding such assertions, but it is useful to have at hand a formal criterion that characterizes those particular node/arrow systems which enable us to avoid them.4 9.7.5 OPTIONAL: We can be quite precise mathematically about what we mean in saying that a precedence-ordered system is "potentially compatible with our naive sense of chronology." Readers who do not care can move on from here to section 9.7.6. Given the family of NODES, let us review what we mean when we speak of a (strict) partial ordering on NODES. We mean a collection P of nodepairs, a collection that satisfies conditions (PO1) and (PO2) following. (PO1): There is no pair (N t , N 2 ) such that both (N t , N2) and (N2, NJ are members of P. (PO2): If (N l 5 N 2 ) and (N 2 ,N 3 ) are both members or P, then so is (Ni, N3). Theorem 9.7.4 told us that P = PRECEDENCE is a (strict) partial ordering on NODES in a precedence-ordered node/arrow system. A partial ordering L on NODES is "linear" or "simple" when, given any distinct nodes N and N', either (N, N') or (N', N) is a member of L. The following theorem can be proved: If NODES is finite, containing J nodes, and if L is a linear ordering on NODES, then the members of NODES can be arranged in a series Nj,, N 2 , . . . , N, such that (NJ5 N k ) is a member of L if and only if j is less than k. The partial ordering P is "weaker than" the partial ordering Q, and Q is "stronger than" P, when every node-pair that is a member of P is a member of Q, and Q contains some pair that P does not contain. That is, this relation between P and Q obtains when P is strictly included in Q as a set of node-pairs. A partial ordering P is "maximally strong" when there is no partial ordering Q stronger than P. The following theorem can be proved: Every maximally strong partial ordering is linear, and every linear ordering is maximally strong. When NODES is finite, the following theorem can also be proved: Given any partial ordering P, there exists some maximally strong (i.e. linear) ordering L which is either equal to P or stronger than P. When NODES is not finite, the same theorem can be proved if one makes an additional logical assumption which need not concern us here. We can apply all these theorems as follows: Given a precedence-ordered node/arrow system, there exists a linear ordering L of NODES which is either 4. Jonathan Kramer develops a very interesting sense in which he claims that the first movement of Beethoven's F-major String Quartet op. 135 begins with its ending. Kramer does this in the article, "Multiple and Non-Linear Time in Beethoven's Opus 135," Perspectives of New Music vol. 11, no. 2 (Spring-Summer 1973), 122-45. The last movement of Haydn's D-major String Quartet op. 76, no. 5, might serve as another example.

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stronger than PRECEDENCE or equal to PRECEDENCE. Assuming that NODES is finite (which we shall assume from now on), we can then L-order the notes of this precedence-ordered system as a series Nj, N2, ... , N;, in such wise that (N, Nk) is a pair within L if and only if j is less than k. Since PRECEDENCE is weaker than or equal to L, it follows that j must be less than k whenever N. precedes Nk in the system. So if we imagine a chronology in which Nj "happens first," N2 "happens second, ". . . , and N; "happens last," this chronology cannot be violated by the precedence relation of the system. That is, whenever N. formally precedes Nk, N. will "happen before" Nk in the L-chronology. If PRECEDENCE is not itself linear, there will be more than one linear ordering stronger than PRECEDENCE; accordingly there will be more than one "linear chronology" of the above sort with which the precedence relation is compatible.

FIGURE 9.13 For example, in the precedence-ordered system of figure 9.13 we can take either of the two left-hand nodes as "N," and the other one as "N2," in imposing a linear chronology; we can similarly take either of the two righthand nodes as "N3" and the other as "N4." This reflects the structure of PRECEDENCE here, which makes each left-hand node precede each righthand node, while neither left-hand node precedes the other and neither righthand node precedes the other. The groupings of segments within the various possible linear chronologies are typical.

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9.7.6 The gist of section 9.7.5 may be summarized as follows: When a finite node/arrow system is precedence-ordered, its J nodes can be labeled by the numbers 1 through J in such fashion that when j is less than k, it is possible for the j* node N. to precede the k* node Nk, but impossible for Nk to precede N. This means that we can always display the system visually on a page (in theory) using a format in which all one-way arrows go from left to right. We must be very careful to recognize that the words "precedence" and "precede" in the paragraph above refer to formal aspects of the node/arrow configuration, and not necessarily to the musical chronology of any passage upon which a network using that node/arrow system may be commenting. Even when the node/arrow system is precedence-ordered, it is perfectly pos-

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9.7.6

sible for a node N to precede a node N' in a network, while the contents of N are heard after the contents of N' in the pertinent music. Figure 9.14 will help us explore the possibility.

FIGURE 9.14 (a) of the figure shows a network of Klangs. All the Klangs are understood to be major, except for the lower-case e|?-Klang, which is minor. The network models the harmonic progression at the opening of the slow movement in Beethoven's Appassionato Sonata. The Ej?-major Klang is bracketed to indicate that the Klang is not actually sounded but is theoretically understood. The fourth sonority heard in the music is modeled by two Klangs. It is first understood as a G(?-major Klang (with added sixth); then it is understood as an eb-minor Klang (with minor seventh, inverted). This is Rameau's double emploi. The arrow goes only one way, from G[? to e(? but not back. The operation REL takes a Klang into its relative minor/major. The left-to-right format of (a) arranges the nodes in an order that corresponds to the order of events in the music. This order is not compatible with the precedence relation of the node/arrow system, even though that

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system is precedence-ordered. We see this incompatibility on (a) in the form of some one-way arrows that point from right to left, (b) of the figure rearranges the nodes on the page, in a new visual format compatible with the precedenceordering; on (b), all one-way arrows point from left to right. We shall discuss later the box containing the word START and the arrow issuing from that box; the reader should ignore them for the time being. The left-to-right format of (b), while respecting the one-way arrows, violates the musical chronology of the passage. A special aspect of this violation is the clarity with which it accents the input function of the two Gb nodes. The input function violates musical chronology: Db, not Gb, is what gets this music under way. The input function also violates our sense of structural "priority": Db, not Gb, is the Klang structurally prior to all others here. The Gb Klangs should be manifest as inflecting such a Db "point of departure" for the tonal structure; the Gb Klangs should not themselves appear as structural "points of departure," which figure 9.14(b) seems to make them. These concerns are very well addressed by Schenkerian theory, which provides us with an apparatus of hierarchical levels, voice-leading events, and subordinate Klangs harmonizing voice-leading events. With such a model in mind, we can easily see that the configuration of Klangs displayed in figure 9.14((a) or (b)) is not an adequate representation for the way tonality controls this passage. The representation fails to model the middleground progression Db-Ab-Db supporting a sustained melodic fifth degree; it fails to model the inflection of that fifth degree by its upper neighbor twice, once within the opening Db Klang of the middleground and once on the way from Db to Ab Klangs within the middleground; it fails to show how the Gb Klangs support that neighboring inflection of the principal tone 5. Later on, we shall construct a "Schenkerian network" that can address these matters within our network format. Meanwhile, we can observe that figure 9.14, incomplete as it is for analytic purposes, still does represent a foreground configuration of Klangs that engages a valid part of our musical experience. It would impoverish, not refine, that experience to explain away the peripatetics of the configuration as "merely" incidental, as "naught but" the harmonization of voice leading, as "only" preparing the eventual dominant, and the like. More specifically, even though the formal Gb inputs on figure 9.14(b) must yield somehow to overall D(? priority in a complete analysis, the formal Gb inputs still reflect an interesting feature of the music, a feature it would be easy to overlook if one were to plow through the foreground to a middleground level prematurely. One hears this feature upon singing over the music mentally while looking at network-format (b). The input Gb nodes are then sensed strongly as "carriage returns," especially the second one. This feeling of "carriage return" on the Gb harmonies, sensed as one reads figure 9.14(b) in the musical chronology, interacts very effectively with the phrasing of the passage.

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9.7.6

It would be hard to express the carriage-return function so precisely in any other theoretical vocabulary. The function marks those precise moments in the listening experience at which we shoot back from right to left on the figure, violating the sense of the one-way arrows. More formally: The carriage-return moments are precisely those moments in the listening chronology at which that chronology violates precedence-ordering. At all other moments, listening chronology is compatible with precedence-ordering. By using the expression "precedence-ordering," we are implicitly supposing that the node/arrow system to which we are applying this concept is precedenceordered, with all that this entails about left-to-right, and so on. Our theoretical machinery enables us to pinpoint carriage-return moments, define them precisely, and attribute a special theoretical function to them. All this duly noted, we have still not resolved the theoretical problems raised by the formal priority our model assigns to Gj? nodes over D[? nodes as "input" to the flow of events, and to the tonal structure. One line of attack on the problems is suggested by the box on figure 9.14(b) containing the word START, and by the arrow from that box to the indicated D[? node. We may formally adjoin the box and the arrow to the node/arrow system, and the contents START to the network, so as to help ourselves along. The START node is an input node, and we can declare a formal convention that it supersedes all other input nodes in function. When we start at the START node, we cannot reach the G|? nodes without traversing some arrows backwards. We might use just that feature of the system as a formal criterion for assigning a special sort of subordinate status to those nodes. There is no problem walking arrows backwards here because our transformations are all operations. Our analytic criterion for pointing the START arrow at the indicated D[? node could be diachronic (because the music starts there) ar synchronic (because that node begins the foreground elaboration of a higherlevel tonic function). Another formal device for modeling a discrepancy between precedenceordering and musical chronology is indicated by figure 9.15. The figure

FIGURE 9.15

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attaches to each node of figure 9.14(b) a certain time span; the musical event corresponding to the contents of that node occurs over that time span. Formally, we are constructing a new apparatus which we might call a "time-spanning network." That is a transformation network together with a function TIMESPAN that maps each node into a certain time span. In the case of a network whose contents are already time spans, we could take TIMESPAN(N) = CONTENTS(N). A time-spanning network could model via TIMESPAN the exact time spans over which its events occur; that is the case with figure 9.15. Or TIMESPAN(N) could model a certain range of time during which CONTENTS (N) might occur. Instead of attaching time spans to the nodes of a network in this way, we could also attach time spans to the contents of those nodes. On figure 9.15, for example, instead of a node N with CONTENTS(N) = Gb and TIMESPAN(N) = (3.5, .5), we could have a node N whose CONTENTS are the ordered pair (Gb,(3.5, .5)). On the revised figure 9.15, the family of transformations would have to be more complicated. Each graphtransformation would be an ordered pair comprising both a Klangtransformation and a time-span-transformation. Yet another formal device at our disposal is to incorporate Schenkerian transformations into a network format. Figure 9.16 indicates one way in which this might be done. The contents of the nodes in this network are ordered triples (Klang, degree, level). Thus (Ab, 5, 2) denotes an Ab-major Klang supporting a fifth degree in the structural melodic voice at level 2. The operation PROJ + incre-

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FIGURE 9.16

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9.7.6

ments the level of its operand, transforming (Kng, deg, lev) into (Kng, deg, lev + 1). Klang and degree are thereby PROJected one level closer to the foreground. The operation PROJ— is the inverse of PROJ + ; it decrements the level of (Kng, deg, lev), transforming it into (Kng, deg, lev — 1). Since the objects (Kng, deg, lev) are purely formal, we can always reference levels "lev + 1" and "lev — 1" formally, even when they have no analytic pertinence to a given situation. That is desirable here, in order to make PROJ-f and PROJ — well-behaved context-free operations. To save space on figure 9.16, the PROJ arrows have all been drawn as two-way, signifying PROJ + or PROJ — as appropriate. Within each level on the figure, each transformation is specified by a pair (Klangtrans, degtrans). Klangtrans is the pertinent Klang transformation, and degtrans is the pertinent degree transformation. Thus an arrow labeled (DOM, SUST) from (Kng, deg, lev) to (Kng', deg', lev) indicates that the Klang Kng is the dominant of the Klang Kng', while degree deg sustains to become degree deg' = deg. An arrow labeled (SUED, N +) from (Kng, deg, lev) to (Kng', deg', lev) indicates that Kng is the subdominant of Kng', while degree deg is the upper neighbor to degree deg'. Distinguishing levels in the manner of figure 9.16 enables us to make input terminology conform better to our intuitions. The Gb nodes of figure 9.16 are indeed still input nodes, but we can now say that they are "input at level 3. In the same sense, we can say that the node containing (Ab, 5,2) is "input at level 2," distinguishing it in this capacity from the node containing (Ab, 5,3). The Db nodes of level 2 are both output nodes at level 2; the Db nodes of level 3 are all output nodes at level 3. The Db node of level 1 is both input and output at that level, according to our definition (9.6.1). Figure 9.16 may be made to engage rhythmic mensuration by attaching time spans to its nodes along one of the lines suggested earlier. Determining where to end the time span for (Db, 5,2) and where to begin the time span for (A)?, 5,2) is an interesting methodological and phenomenological problem.5 a number of assertions seem plausible. My own preference is to carry the Db time span right up to the Ab Klang, and also to begin the At? time span right after the Db Klang stops sounding. The two time spans would then overlap on level 2, and the Gb-eb-Eb part of level 3 would all occur during the time span of the overlap. This satisfies my hearing, and it is also an elegant way to elaborate the theoretical idea behind the double emploi. Figure 9.16 as it stands is not equivalent to a Schenkerian reading, which would devote less attention to Klangs that do not project Stufen, more attention to the bass line and to the essential counterpoint between the outer 5. Fred Lerdahl and Ray Jackendoff discuss pertinent matters at length in their important book, A Generative Theory of Tonal Music (Cambridge, Mass, and London: MIT Press, 1983). The interested reader can explore various ways in which their tonal tree-structures resemble and differ from transformation-networks of the sort displayed by figure 9.16.

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FIGURE 9.17

voices.6 I think it likely, though I am not certain, that actual Schenkerian graphs could be represented in network formats of the sort under present consideration, when suitably extended. 9.7.7 A short time ago, in connection with our study of "Nos qui vivimus," we observed that a series of objects could be modeled by a certain type of transformation network. Now we are in a position to specify formally just

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6. A good Schenkerian analysis of the theme as a whole is presented by Allen Forte and Steven E. Gilbert in their Introduction to Schenkerian Analysis (New York: W. W. Norton, 1982), 154-56.

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9.7.7

what type. Specifically, a formal "melody" can be defined as a transformation network whose node/arrow system is precedence-ordered, and linearly ordered under that ordering. According to our work so far, that means there will be one and only one way of labeling the J nodes with the numbers 1 through J so as to be compatible with the one-way arrows of the system. This concept of "melody" is very elaborate, for it carries within it the idea of transforming earlier events to later ones, along the arrows of the network, by transformations from a specified semigroup. That idea was not implicit in our earlier models for series. Different configurations of arrows can give rise to one and the same precedence-ordering on a given family of NODES. When that ordering is linear, corresponding networks whose nodes have the same contents will nevertheless be formally different "melodies" by our definition. We should find a better word than "melody" if we want to continue to work along these lines. For instance, the two node/arrow systems of figure 9.17(a) and (b) give rise to one and the same precedence-ordering. Networks (a) and (b) are formally different "melodies."7

7. "Melody," however, is exactly the proper term if we want to follow and extend the usage of Ernst Kurth, when he claims that "the basis of melody is, in the psychological sense, not a succession of tones... but rather the impetus of transition between the tones." Figures 9.17 (a) and (b) depict different transition-structures. The cited text appears in Grundlagen des linearen Kontrapunkts (Bern: Drechsel, 1917), p. 2. ("Der Grundinhalt des Melodischen ist im psychologischen Sinne nicht eine Folge von Tonen... sondern das Moment des tfbergangs zwischen den Tonen.")

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10

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FIGURE 10.1

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10.1 EXAMPLE: Figure 10.1 sketches some motivic work from the passage opening the development section in the last movement of Mozart's G-Minor Symphony, K.550. Up to measure 133, the entire orchestra except horns is playing the indicated line, allegro assai and essentially staccato. There is one exception: The quarter note B which appears on the figure at the end of measure 128 is actually an eighth rest and a sixteenth triplet, filling in the

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10.1

diminished fourth scalewise under a slur. I am supposing that it is legitimate for us to consider that gesture a variation on the motivic model of the figure, inter alia. From the pickup of measure 127 until measure 133, the passage projects forms of a pitch motive PM, comprising a diminished fourth up followed by a diminished seventh down. This motive is Rl-chained in the manner familiar to us by now; the transformation RICH takes us along the chain of PMforms (E-Ab-B), (Ab-B-Eb), (B-Eb-Ffl), (Eb-Ffr-Bb), (FJ-Bb-Ctf), (Bb-Ctf-F), and (C#-F-G#). The TCH interval is a falling fourth. On figure 10.1, the four prime forms of PM are indicated by the brackets numbered 1,2, 3, and 4a/b. The form (C#-F-G#) is bracketed twice, by brackets 4a and 4b. This reflects an interesting ambiguity about its rhythmic location, an ambiguity we shall soon investigate. Beneath the staff on the figure a series of numerical durations appears. The numbers label the distances in quarter-note beats between the time points at which successive notes are attacked. I am supposing that we hear an ictus at the barline of measure 128. (Later on, we shall hear how the music of measures 125-27 prepares this ictus.) The barline of measure 128 thereby articulates the duration of five quarters, between the two B naturals, into (2 + 3) quarters. The three attacks within PM, together with the ictus, define the durational series 1 + 2 + 2 (quarters) as a rhythmic setting for the pitch idea. We shall call 1 + 2 + 2 the "durational motive" DM. On the figure, bracket 1 is placed around PM so as to articulate DM; the corresponding durationnumbers 1, 2, and 2 below the staff are also bracketed. Bracket 2 also articulates DM, now as the rhythmic setting for the next TCH-form of PM. Bracket 3 is placed around the next TCH-form of the pitch motive, that is around F#-Bb-C#. Here the rhythmic setting is no longer DM itself, but rather an augmented (rhythmically transposed) form of DM: 2 + 4 + 4. The rhythmic transposition will be denoted as T2, multiplying all durations by 2. The next TCH stage of the pitch motive is C#-F-G#. Bracket 4a gives this stage the durational setting 4 + 4 + 2; bracket 4b gives it the setting 4 + 2 + 2. Both these durational settings are serial transformations of DM. The series of 4a retrogrades the elements of series 3: 4 + 4 + 2 retrogrades 2 + 4 + 4. The series of 4b inverts the elements of series 3: 4 + 2 + 2 inverts 2 + 4 + 4. We can regard the inversion as multiplicative, about the numerical product 8: 8 divided by 2,4, and 4 (series 3) yields 4,2, and 2 (series 4b). Or we can regard the inversion as additive, about the numerical sum 6:6 take away 2, 4, and 4 (series 3) yields 4,2, and 2 (series 4b). Whichever way we think of the inversion, durational series 4b is a retrograde-inversion of durational series 4a. In fact, series 4b is precisely RICH(series 4a). Thus the transformational motif of Rl-chaining, very audible in the pitch structure of the passage, is also projected in the durational structure. The

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musical effect is bewildering on first hearing, because both the durational series involved in the RICH relation, that is both 4a and 4b, are heard as alternative rhythmic settings for one and the same pitch motive. And this effect strikes our ears just as we are beginning to adapt to a heavy reliance on motivic listening, having temporarily lost the local tonic. Bracket 5 shows how the play of durational motives continues on through the change of texture at measure 133. The durational series for bracket 5, 2 + 2 + 4, is a retrograde-inversion (multiplicative or additive) of series 3, just as series 4b was a retrograde-inversion of series 4a. Since series 3 was a multiplicative transposition of DM, durational series 5 is also a multiplicative retrograde-inversion of DM itself. In fact series 5 is precisely the RICH-transform of DM, using multiplicative inversion: (multiplicative) RICH(1 +2 + 2 ) - 2 + 2 +4.

FIGURE 10.2

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Figure 10.2 summarizes in a network the transformational interrelations of the durational motive-forms so far surveyed. On the figure, T is multiplicative T 2 ,1 is the pertinent multiplicative inversion, and RICH is considered to be defined multiplicatively also. The RICH-arrow between the node marked "DM = 1; 2" and the node marked "5" suggests some aural explorations. Going back to figure 10.1, we can hear the rhythmic RICH-relation between bracket 1 and bracket 5 by focusing on a linking rhythmic element, that is, the rhythmic identity of measure 127 (at the end of bracket 1) with measure 132 (at the beginning of bracket 5). This is exactly the linking aspect of the Rl-chain involved. We hear that the high Ab of measure 127 and the low G# of measure 132 are the registral boundaries for figure 10.1. The enharmonically equivalent A(? and G# are tied together, too, by the diminished-seventh harmony implied over measure 127; this harmony implicitly recurs under bracket 5. We can hear the rhythmic RICH-relation between bracket 2 and bracket 5 by focusing on measures 129-30 as an intermediating stage. That pair of measures spans the same durational series as series 5: 2 + 2 + 4. And in

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10.1

measures 129-30 the series 2 + 2 + 4 is explicitly linked to the end of bracket 2, as the RICH-transform of series 2. This rhythmic RICH-transformation is easy to pick up aurally because it coincides with a RICH-transformation of the pitch motives that correspond: B-E|?-F#, with rhythm 1 + 2 + 2, is transformed into RICH(B-Eb-F#) = Eb~F#-Bb, with rhythm RICH(1 + 2 + 2) = 2 + 2 + 4. It remains then only to hear that measures 132-33, under bracket 5, reproduce the attack rhythm of measures 129-30, the intermediating stage. And this is quite possible if one hears measures 127-29 and measures 130-32 as a pair of three-measure groups, or if one hears measures 129-31 and 132-34 as such a pair. Let us return now to brackets 6 and 7 on figure 10.1. Bracket 6 applies to the winds only, whose durational series is 4 + 2 -I- 2 over this span. The series recapitulates the series of 4b. Despite the radical change of texture, we are aided in hearing this relation by the boundary tones of measures 133-35, specifically by the opening high F of measure 133, the final C# at measure 135, and the low G# at the end of measure 134. Those three notes keep alive in permuted order the pitch classes C#, F, and G# from the PM-form of measures 131-32, spanned by bracket 4b. Bracket 7 shows how the rhythm of the winds during bracket 6 becomes diminuted approaching the barline of measure 135, into the diminuted series 2 + 1 + 1. This diminution (multiplicative transposition by ^) undoes the effect of the earlier augmentation (transposition by 2), the augmentation we underwent in passing from bracket 2 to bracket 3. Accordingly, durational series 7 bears to series 1 = 2 = DM the same relation that series 6 = 4b, the augmentation of series 7, bore to series 3, the augmentation of series 1 = 2 = DM. Thus series 7 inverts series 1 either multiplicatively or additively: 2 divided by 1,2, and 2 (series 1) yields 2,1, and 1 (series 7); alternatively, 3 take away 1,2, and 2 also yields 2, 1, and 1. Figure 10.3 extends figure 10.2 to display some of the new relationships involving series 6 and series 7. J is the pertinent multiplicative inversion \\. Our study has shown us how the straightforward Rl-chaining in one dimension of this music both suggests and conceals a very elaborate transformational network, also involving RICH-relations, in another dimension. The whole discussion is grossly oversimplified as regards both dimensions. We have not, for instance, considered the rhythmic implications of the PM-form that goes from the G# of measure 132 to the F of measure 133 in the bassoon (an octave below figure 10.1), and thence through the stepwise filler down to the C# of measure 135 (again an octave below the figure). This form retrogrades in register the PM-form of measures 131-32. The stepwise filling-in of the diminished fourth F-to-C# is a huge rhythmic expansion, in retrograde, of the tiny sixteenth-note triplet figure that filled in from B to Eb in measures 128-29. We have also not considered the crucial way in which the music gets to figure 10.1. The Rl-chaining of PM actually begins earlier; it involves the

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FIGURE 10.3 all-important high D|? of the rocket theme, the main theme of the movement and the source of the diminished-seventh leaps. Figure 10.4 helps us explore this a bit.

FIGURE 10.4

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The beamed arrows on the figure pick out a PM-form that extends the RI chain of pitch motives one TCH stage backwards from our earlier analysis. The rhythmic setting of the PM-form A-D[?-E is not related to DM by any standard serial transformation. It does, however, interact with the ictus at the barline of measure 128. Figure 10.4 allows us to hear the source of the 3 quarters' duration between the ictus and the following B. That duration is in rhythmic sequence with the 3 quarters' duration between the high D[? and the E natural of measure 126. The 3 quarters' sequence causes the rhythmic identity of bracket 1 with bracket 2 to extend backwards. Not only do EAJ7-B-ictus and B-Eb-Fft-Bb demarcate the same durational series DM = 1 + 2 + 2, but also Dj?-E-A[7-B-ictus and ictus-B-Eb-F#-Bb demarcate

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70.2

the same series 3 + 1 + 2 + 2. In fact, the rhythmic identity extends back even farther: A-Db-E-Ab-B-ictus and B-ictus-B-Eb-F#-Bb demarcate the same series 2 + 3 + 1+2 + 2. Only after the Bb of measure 130 does the rhythmic parallelism break off: After that Bb the first duration of 4 appears, launching us into the transformational complexities of figure 10.2.

FIGURE 10.5

10.2 EXAMPLE: Figure 10.5 sketches some structural aspects of Bartok's "Syncopation," no. 133 from Mikrokosmos, vol. 5. Our ability to hear prolongation in this sort of context is the subject of an important study by Roy Travis.1 Travis tries to hear (GBD) + (Eb F#) as a tonic chord for the piece. It works better to hear (G) + (A#C#D#F#) in that capacity. This is the final sonority of the piece and the only complete harmony attacked by both hands simultaneously, an event which occurs at the final barline. That is, it is the only harmony which does not involve a "Syncopation." The syncopations of the music bring out a broad variety of structural contrasts between the two hands. The left hand plays white notes; the right hand plays black notes. The left hand plays "down" both metrically and sotto; the right hand plays "up" both metrically and sopra (with its black notes). The white notes in the left hand arrange themselves into triadic formations within a G Mixolydian mode; the black notes in the right hand arrange themselves within a pentatonic F# mode. The left hand's chord of reference is a G-major |, a chord with G on the bottom; the right hand's chord of reference is an F# added-sixth harmony in 3 inversion, a chord with F# on the top. Over measures 1-10 of figure 10.5, one sees how these elements contend during the opening section of the piece. The black notes win out. The opening (D#F#) of the figure is spelled (EbF#) by the composer, to emphasize its local dependency on the left-hand G tonic. But as the section progresses, (EbF#) becomes (EbGb) and shows that it can move down an octave to displace the 1. "Toward a New Concept of Tonality?" Journal of Music Theory, vol. 3, no. 2 (November 1959), 257-84.

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left-hand G at measure 9. The black dyad celebrates the success of its incursion by bobbing triumphantly back up to its original register, with a crescendo to sf; now the composer spells it (D#F#), reflecting its forthcoming orientation towards the right-hand tonic F#. The next section of the piece, measures 11-25, shows (D#F#) continuing to exercise its registral mobility: It moves up an octave over that span of the piece, along with C# and eventually A#. The octave transfer, represented by a slur on figure 10.5, plows through the climactic D of measure 18. This D, as fifth of G, shows that left-hand, G-oriented material can also become registrally mobile. Ds and Gs continue to move downwards in register from measure 25 to the end, while the right-hand chord of measure 25 moves up another octave during that time. Bartok puts (D#F#) back into its original spelling, as (E|?F#), from measure 25 on. In the right-hand scheme of things, as displayed on figure 10.5, the climactic D5 of measure 18 is apparently "only" a chromatic passing tone. But D5 is also the pitch about which the sonority A#4, C#5, D#5, F#5 is inversionally symmetrical. That sonority, as it appears with a structural downbeat at measure 25, projects the chord-of-reference for the right hand. Also, the pitch class D is a center of inversion for both the blacknote collection and the white-note collection. In both the left hand and the right hand over measures 11-25, RIchaining provides the means of harmonic progression. The left hand produces the Rl-chain of registrally ordered trichords A-C-E, C-E-G, E-G-B, G-B-D. There is even a hint that this process continues on in register, as the right hand takes over the climactic D5 at measure 18 and leads it on up to F#5. But the black tonic F# is too clearly foreign to the white-note chain, for this hint to develop further. Indeed the left hand emphasizes just that point, at measure 25, by providing the white F natural (not the black F#) as a chordal third above D, within the sonority G-B-D-F. The F natural breaks the RIchain of the left hand, which cannot pass the color barrier. Just as the Rl-chain of white trichords in the left hand appears beamed over measures 11-25 on figure 10.5, so do two analogous Rl-chains of black trichords in the right hand. The first chain breaks at F#-G#-B; rather than continuing on to the next RICH-stage G#-B-C#, the right hand substitutes a different form of the same trichord, namely G#-A$-C#, and then starts a new Rl-chain therefrom. The color barrier is again involved: Once the white B has been generated by the Rl-chain in the right hand, it must be replaced by the black A# = Bj?. So, just when the Rl-chaining is about to carry F#-G#-B on to G#-B-C#, A# substitutes for B and the next trichord is G#-A#-C# instead. The substitution preserves the outer voices of the trichord involved, and also preserves its set-class. The replacement of B by A# recapitulates in a setting of F# tonicity the replacement of B by B[? that we heard earlier, at measure 3, in a setting of G tonicity. The double Rl-chaining of the right hand over measures 11 -25 is also

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10.3

FIGURE 10.6

involved with a cadential proportioning (balance) of the trichord forms. Figure 10.6 displays this proportioning by a pertinent transformationnetwork. The straight diagonal arrows are RICH transformations. Among its input and output chords, figure 10.6 manifests a classical "triple proportion": C#-D#-F# (input) is to Ftf-G#-B (output) as G#-A#-C# (input) is to C#-D#-F# (output). The triple proportion also involves an isography of the left side, on figure 10.6, with the right side, on the same figure. That is, the relation of the material grouped by the two right-hand beams, over measures 11-25 on figure 10.5, is an isography. Intervals of 5 span the trichords and (therefore) measure the TCH-transpositions. They make the triple proportion here sound almost like I-IV-V-I, especially since the pitch class F#, appearing at the bottom of the first "I" and at the top of the final "I" in the I-IV-V-I, has a certain tonicity about it as regards the pitch structure of the right hand. Perhaps this sound led Travis to hear a functional I-IV-V-I governing the tonal structure of the piece as a whole. 10.3 EXAMPLE: Figure 10.7 sketches aspects of the opening seven measures from the first of Prokofieff's Melodies op. 35.1 am indebted to Neil Minturn for bringing this passage to my attention. The four-fiat signature on the figure is mine; Prokofieff writes no signature. The harmony in the music is far from traditional. Yet it is diatonic enough, and the outer voices are diatonic enough, so that some harmonic events stand out as "strange." Foremost among these are the cadence harmonies, E minor over the last half of measure 2 and Eb minor over the last half of measure 5. At both these cadences we expect Eb-major harmony. We can analyze the cadential substitutions by using the terminology of Klangs and Klang-transformations. The network of figure 10.8 (a) does so, and also brings the D-major Klang of measure 6 into the picture: D major, the SLIDE transform of Eb minor, bears to the latter Klang the same relation

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FIGURE 10.8

228

which E|j major, a Klang heard in the upper voices of measure 1 and expected at measure 2^, bears to E minor, the Klang actually heard at measure 2j. Figure 10.8 (a) does not attempt to engage the C-major and F-minor triads of measure 6 in its Klang-network. The proportion among the four

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10.3

triads of measure 6 (with pickup) is clear enough aurally, supported as it is by a rhythm-and-contour motive: E[? minor is to C major as F minor is to D major. But the proportion seems hard to portray by Klang-relations that mesh with our aural intuitions. Formally, we can of course note that F minor is the minor subdominant of C major, and that C major is the major dominant of F minor: (F, -) (PAR) (SUED) = (C,+) and (C, +) (DOM) (PAR) = (F, -). But how functional are these relations in a context that emphasizes the proportion (E(? minor)-is-to-(C major) as (F minor)-is-to-(D major), all after a cadence in Et>? And what sort of Klang relation might we hear in this tempo and rhythm governing the terms of that proportion? Can we hear, for instance, (Eb, -)(PAR)(REL)(PAR) = (C, +) here? Our attempt to hear any functional Klang-relation is bound to be hindered by the parallel voice-leading of the chords involved here. It seems more fruitful to analyze the triadic formations of measure 6 using the discourse of pitch-class sets and pitch-class transformations. Indeed we can fruitfully study the triadic formations of the entire passage using that discourse. Figure 10.8(b) shows the result. To save space, major and minor triads are denoted there by upper- and lower-case letter names. The new analysis describes the triadic structure of the passage as follows, ignoring sevenths and added sixths. First, E minor substitutes for an expected Efc> major in the cadence at measure 2^; the transformation involved is inversion about G, as depicted by the leftmost vertical arrow on figure 10.8(b). (Since we are now talking about pitch-class sets and not Klangs, IQ is a pertinent transformation while SLIDE, which inverts a Klang about its mediant, is no longer pertinent.) The pitch class G is a plausible center of inversion; it is highly accented by its boundary functions within the opening melodic phrase of measures 1-3. In particular, the substitution of E-minor harmony for E[? major occurs exactly when the melody reaches its climax G, and the E-minor harmony remains around while the melody drops back to the low G, where it sits until its next phrase begins in measure 4 (on that G). The next vertical arrow on figure 10.8(b) shows inversion-about-F#-andG relating the E|?-major harmony we expect to the Ej?-minor harmony we hear, when the next cadence arrives at measure 5j. This inversion, which replaces the pitch class G of the E(?-major harmony by the pitch class F# = G[? of Eb minor, thereby develops the chromatic relation of the pitch classes G and F#, the relation which governed the first chromatic event of the melody (and of the music as a whole), when F# appeared on the second beat of measure 1. The structuring power of that F# is then developed as shown by the next vertical arrows on figure 10.8(b). F# becomes a new center of inversion: Inversion-about-F# relates E^-minor harmony to D-major harmony, and also (NB) C-major harmony to F-minor harmony, all during the new thematic material of measure 6 and following. E[? -minor and D-major harmonies mark

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the temporal boundaries of the little phrase, while C-major and F-minor harmonies mark its registral boundaries. The repeated rhythm-and-contour motif within the phrase (of measure 6 with pickup) groups E[j minor with C major, and F minor with D major. Since all the minor/major relations of figure 10.8(b) so far have been analyzed as pitch-class inversions, the figure also explores hearing E[? minor become C major by inversion (about F), and hearing F minor become D major by inversion (about G). Those inversions are depicted by horizontal arrows on the figure. Ip and I % thereby inflect the central inversion, Ip"j|, for the little phrase of measure 6. The structural centrality of F# between F and G in this arrangement recalls the melodic position of F# in measure 1, where it mediated between a preceding melodic G and a subsequent melodic F. In connection with the vertical arrows of figure 10.8(b), we have noted how the progressing centers of inversion develop the pitch classes G and F# which are so characteristic of the incipit motive in the melody. We can also adopt a purely intervallic stance toward the progression of inversional centers, and toward the structure of the incipit motive. Figure 10.9 elaborates that idea.

FIGURE 10.9

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(a) of the figure is a network whose nodes contain the inversion operations associated with the vertical arrows of figure 10.8(b). The T M arrows mean T n Ig = Ip^ and T n Ip# = Ip*. The family S of operands here is the family of inversion-operations, and SGP is the group of left-multiplicationsby-transposition-operations, as that group operates on the stipulated S. Figure 10.9(b) is essentially extracted from the accompaniment of measure 1, where it supports the thematic G-F#-F gesture of the melody. The network implied by figure 10.9(b) is isographic to the network of figure 10.9(a). The networks discussed in connection with figures 10.8 and 10.9 are musically compelling at least to the extent that one would want to show how

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they relate to other aspects of the music, as one goes on to explore those other aspects. These topics, for instance, would repay such a study: the "minor-third root relations" within measure 6 (with pickup); the chromatic line, rising in parallel minor thirds, that starts with Eb-and-Gb at the pickup to measure 6 and ends with F#-and-A in the middle of measure 6; the tonal structure of the bass line, either in Ab major or in a Mixolydian Eb. 10.4 EXAMPLE: The opening section of Debussy's Reflets dans I'eau, up to the reprise at measure 35, is rich in interrelations between transformational networks and other sorts of musical structures. A sketch for the passage appears as figure 10.10. The opening motive X plays a strong generative role. Particularly full of import are the transpositions T and T' that respectively take Db to Eb and Ab to Eb within the motive. Figure 10.11 (a) isolates this structure for study. Depending on various contexts to come, we sometimes hear the interval associated with T as one diatonic step up, sometimes as a major second up, sometimes as two semitones up, and sometimes as the pitch-class interval 2. The interval associated with T' varies similarly depending on the GIS supplied by the context. Figure 10.11 (b) shows how motive Y, which follows X in the music, can be derived from X. Ab and Eb, connected by the T'-arrow, remain within Y. The T-relation of 10.11 (a), between Db and Eb within X, is "folded in" both temporally (serially) and in registral space, to become the T"1 relation displayed in 10.1 l(b), between F and Eb within Y.2 The mathematical logic of the X-to-Y transformation has the following implication: If X should become T-transposed into T(X), then the new Trelation between the pitches in order positions 1 and 3 of T(X) will specifically engage the relation Eb-to-F, thereby retrograding the last two pitches of Y. And in fact X is T-transposed into T(X) right after Y has sounded. The reader can follow these events along on figure 10.10. The new pitch Bb4 of T(X) is thereby generated in the principal melodic line. Within T(X), Bb-to-F is in the T'-relation. And Bb, as the high point of T(X), is the T-transpose of Ab, the high point of X. Measures 5-8 repeat measures 1-4. As a result of the repeat, we hear T(X) return to X via T"1; Bb also returns to Ab via T"1; and so on. Thus the change from T to T"1, a prominent characteristic of the change from the internal structure of X to the internal structure of Y, now characterizes on a larger rhythmic level the change from the progression X-T(X) (measures 1-3), to the progression T(X)-X (measures 3-5). 2. The reader will recall our discussing earlier just such kinds of "folding" transformations in connection with serial trichords. That was in section 8.3.2, where we examined the transformations FLIPEND and FLIPSTART. The transformation taking X to Y here is neither of those, but it is of the same genre.

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FIGURE 10.10

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FIGURE 10.11 At measure 9 the new motive Zj appears in the melody. The way in which Zj pulls together the pitches and intervals of X, Y, T(X), and the repeat of those cells is aurally clear. The transformational situation is already so complex that it would require an inordinate amount of discussion to describe this synthesis adequately in words. Let us just explore a few of its features, as displayed by figure 10.12.

FIGURE 10.12 The T"1 relation from F to E|>, which we heard within Y, is recapitulated at the opening of Z t . The T"1 relation from Bfc> to At?, which we heard when T(X) returned to X at measure 5, is recapitulated by the last two notes of Z t . And the relation between those two T"1 relations, within Zj, is an inverse Trelation. Figure 10.12 shows how this recalls the T'-features of X and T(X). Beneath Zj in the accompaniment, a new idea makes its appearance. This idea involves a continuous chromatic rise of a certain object (here a certain 3-note chord), over a total span of three semitones. The graph of this idea will be called CHR; it is depicted in figure 10.13.

FIGURE 10.13

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On Figure 10.10 we see how a CHR network fits homophonically beneath Z^ The first, second, and third of the resulting 4-note chords in measure 9 all have different structures. The fourth and last 4-note chord has the same structure as the first. That is because the CHR network in the accompaniment has risen 3 semitones overall, while Zl in the melody has also risen 3 semitones overall, from its opening F to its closing Ab- The homophony thus identifies the 3-arrow on the CHR graph with the interval that spans F-to-Ab across all of Z x . Because of the ways in which Z t synthesizes earlier motives, the F-to-Ab is heard in turn as part of a permuted Y embedded within Zj. So the 3-arrow of the CHR graph can ultimately be traced back to the Ab-F dyad within Y. The homophony between Zj^ and its CHR accompaniment is marked by a crescendo. The crescendo is to become a significant thematic element. In measure 10 the music of measure 9 is repeated and extended. The crescendo recurs. In the melody the repetition gives rise to a rotated form of Z l 5 marked "rot Z:" on figure 10.10. Rot Zl is Bb-Ab~F-Eb; it embeds serially the original form of Y, Ab-F-Eb, and precedes this Y by its overlapping inverse-RI-chained form Bb-Ab-F. (Bb-Ab-F is RICH'^Ab-F-Eb).) This relationship is more or less inherent in the derivations of X, Y, T(X), their repetitions, and Z:. When the music of measure 9 is extended during measure 10, a new motive Z2 arises as shown on figure 10.10. Z2 is articulated, and associated with Z l s by its contour and its rhythm. Figure 10.12 earlier analyzed Z x as a pair of T"1-related dyads in an inverse-T' relation. Z2 can be similarly analyzed as a pair of T"1-related dyads in a T-relation. Figure 10.14 displays that analysis.

FIGURE 10.14

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The accompaniment below Z2 projects a new network whose graph is CHR, a new network isographic to the network of measure 9 in the accompaniment. The new network is the T-transpose of the old, as indicated on figure 10.10 by the T arrow leading from under measure 9 to under measure 10. The extended accompaniment within measure 10 rises 5 semitones from its point of departure, in contrast to the accompaniment of measure 9, which rose

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FIGURE 10.15

only 3 semitones from the same point of departure. Figure 10.15(a) is a graph conveying this idea. Graph (b) of Figure 10.15 is a homomorphic image of graph (a). The two lower nodes of (a) each map into the one low node of (b), and the semigroups (groups) of intervals for the graphs are isomorphic. Figure 10.15(c) is a network whose graph is (b). We recognize that the pitches of (c) build a serial form of Y in the precedence ordering, namely the series F4-AJ?4-8^4. This is Y inverted about F4-and-A|?4. The retrograde of the form was Rl-chained into Y within the rotated Zl motive we recently examined. We earlier noted that F4 and A[?4 within Y were identified with the 3-semitone rise of the chromatic accompaniment during measure 9. During measure 10, the F4 and B[?4 of inverted-Y (that is, of figure 10.15(c)) are similarly identified with the 5-semitone rise of the extended chromatic accompaniment. The inverted Yform F4-A(74-B|?4, and the network of figure 10.15(c), are brought out by the homophony in the same way as was the F-A[> dyad during measure 9: F4, Aj?4, and Bfc>4 are the three notes of the melody in measure 10 that are supported by "dominant-thirteenth" harmonies; no other harmony appears more than twice during measure 10. Figure 10.16 is a "product network." It adjoins beneath a copy of figure 10.15(c) the bass notes of the dominant-thirteenth chords in an isographic network. These are exactly the bass notes of measures 9-10 which are in pitchclass relation 3 to the melody above them. Beyond participating in that relation, the pitches A|?2 and D|?3 on figure 10.16 also function as temporal and registral boundaries for the bass line over measures 9-10. The tonality of the piece further reinforces their structural significance.

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FIGURE 10.16

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Figure 10.16 shows how the pitch and pitch-class interval 3 is proliferating as a constructive element of the composition. The interval began as a secondary phenomenon; its complement spanned Ab-to-F within Y, which appeared as the difference between the T'-related Ab-to-Eb and the T"1related F-to-Eb- F-to-Ab then assumed greater prominence in the melody as the temporal boundary for Z x , supported by the concomitant 3-semitone rise of CHR in the accompaniment. Figure 10.16 shows how the melodic F-to-Ab is verticalized in the harmony, and how the interplay of horizontal and vertical 3-intervals next generates a structural Cb in the bass. That Cb is the first "middleground" chromaticism of the piece; it will take on formidable proportions hereafter, especially after measure 18. In that connection, Cb will often be heard in conjunction with F and Ab, following its prototypic generation on figure 10.16. Returning again to figure 10.10, let us now examine the large structure of the principal melodic line over measures 1-17. The pitch C5 of measure 10 is the climax of this line, which has been using the rising T and T'-inverse transformations to ascend up to that point from the initial Db4. The line is completely diatonic, so the leading tone C5 makes a strong effect as a provisional climax. The effect is somewhat concealed by the chromatic harmonization and by the possibility of hearing C5 as a neighbor to Bb4 on a subordinate level, even though the harmony does not support the neighboring function. Still, as one listens to the top staff of figure 10.10 by itself through measure 17, it is clear that there is unfinished business for the principal melodic voice in its upper register. That business will not be fully discharged until well beyond measure 35, where our present analysis will stop. Nevertheless, we shall hear before measure 35 further important developments engaging C5 and Db5 in the principal melodic voice. After the provisional climax on C5 in measure 10, the principal melody

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sits for a while on B|?4 and then returns over measure 14 back down to Dj?4, its original point of departure. The melodic descent embeds rotated Z l f Bb-Ab-F-Eb, on the strong sixteenths of measure 14. Figure 10.17(a) shows this and also shows how a rotated form of T(X) within the figuration proceeds back down via T"1 to the correspondingly rotated form of X.

FIGURE 10.17

The sixteenth-note appoggiaturas, all by descending whole tones, give the T"1 idea a heavy workout in the forefront of the melodic texture. Within measure 14, the second half of the melody, F-Eb-Eb-Db, is in T'-relation to the first half of the melody, Bb-Ab~Ab~Gb. The T' relations of Bb to F, and of Ab to Eb, are very familiar by now. Figure 10.17(b) shows a further T'relation, one that involves the retrograde CHR gestures in the alto voice of the same measure. Indeed, the entire second half of measure 14 is in T'-relation to the entire first half of the measure. Figure 10.10 brackets a pentachord called MAGIC which it asserts as controlling the first half of measure 14; T' (MAGIC) then controls the second half. Tonally, the music has progressed from the tonic pedal of measures 1-8, through the dominant that opens measure 9, to the dominant-of-the-subdominant that ends measure 10 under the melodic Bb4. After the peripatetic harmonies that prolong the melodic Bb4, the harmony discharges its subdominant obligation with T' (MAGIC) in the second half of measure 14, and the tonal idea that carries measures 15-17 is a plagal cadence supporting a prolonged Db4 in the melody. Figure 10.10 shows how T' (MAGIC) leads into that cadence. I forego with great reluctance analyzing the music hereabouts in greater detail, particularly the ingenious chaining of Y, rotated X, and rotated Zl series in the outer voices of measures 16-17, and the set-theoretic relations of those formations to the vertical sonorities there. At measure 17^ and following in the music (represented as measure 18 on figure 10.10), the cadential F of the plagal 4-3 gesture is confirmed by its own verticalized Y-form F-D(natural)-C. The first interval of Y thereby expands in structural power, transposing Y = Ab4-F4-Eb4 into T_3(Y) = F4-D4-C4. Earlier, figure 10.16 showed us how the original Ab-F dyad

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FIGURE 10.18

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within Y generated Cb, the minor third above Ab; just so, the same Ab-F now generates D natural, the minor third below F. Figure 10.18 shows how this idea is expressed very precisely by the three verticalized Y-forms within measure 18. The music ties T_3(Y) and its D natural together with Y and with J(Y) = A\?-C\?-D\>, the inverted form which originally generated Cb in the bass of figure 10.16. J(Y) appeared there as T3I(Y), where I is inversion about F-and-Ab: J(Y) appeared in the bass of figure 10.16 coupled at the 3-interval beneath a structural melodic I(Y) = F-Ab-Bb. Within the soprano line of measure 18 there are also references to permuted prime forms of melodic Y and melodic X, as indicated on figure 10.10. These recollections help get the new large section of the piece underway, by recalling material associated with the opening. Figure 10.10 also draws attention to the chromatic voice-leadings D-Eb and C-Cb during measure 18; when the material repeats the voice leadings are reversed and then repeated. The chromatic notes D natural and Cb involved in the voice-leading gestures arise as already discussed in connection with figure 10.18. I shall call the characteristic rhythm and contour that govern the second half of measure 18 the "ruffling motive"; here the wind first ruffles the surface of the pond. The ruffling motive is bound together with the Cb events we have just explored, including the C-Cb voice leading, the vertical Y of figure 10.18, and the vertical J(Y) of the same figure. The motive arpeggiates a Tristan chord upwards and then partially arpeggiates a £5 harmony downwards, all within the registral confines of the bass F3 and the upper note Ab4. The Tristan chord is in the correct spacing at the right pitch-level, once the doubling Ab3 of the ascending ruffle is removed. Over measures 20-21, the Tristan-harmony-cum-ruffle-motive moves up quasi cadenza in literal sequences, 3 semitones per stage, until it gets essentially two octaves higher. The Tristan chord no longer contains any doublings. Figure 10.19 graphs the beginning of the cadenza sequence. Now that D natural, as well as Cb, is on the scene along with F and Ab, the minor third or 3-semitone interval can be fully unleashed; the sequence at hand unleashes it. Figure 10.19 has striking features in common with figure 10.16 earlier, where the 3-interval first began to flex its muscles in a chromatic

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FIGURE 10.19

context. Both the figures start with their outer voices on the basic 3-dyad Fand-Ab. In both, the outer voices then move by the melodic 3-interval, to project Ab-and-Cb (G#-and-B) in the homophony. Both figures project forms of the Y motive: In figure 10.161(Y) = F-Ab-Bb above is coupled to J(Y) = T3I(Y) = Ab-Cb-Db below; figure 10.19 embeds Y and J(Y) subnetworks in the ruffling as shown, and then sequences all Y-forms indefinitely. I have no idea what private commentary on Wagner Debussy may have intended by his use of the Tristan chord and his continued sequencing of it by 3 semitones, a sequence which Wagner artfully only suggests in the first-act Prelude. Debussy's sequential cadenza, unleashing the hitherto restrained 3interval, makes perfect sense in his own composition, of course. The first transposition of the Tristan chord, which Debussy spells as G#-D-F#-B, is a subset of the MAGIC pentachord D-Cb~Gb-(Bb)-Ab. Ruffle-cum-Tristan sequences eight times in the music, arriving at measure 22 essentially two octaves above its point of departure. Figure 10.10 shows the Tristan chord at measure 22 only one octave higher, with its upper voices rearranged. That is to accommodate the subsequent voice-leading into measure 24, and thence into measure 27 and measure 30; those events are pretty clearly in the correct structural registers where figure 10.10 portrays them.

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For the music between measure 22 and measure 24, where the cadenza again becomes mesure and a new theme appears, I hear two possible readings. One of them, marked "CHR and ret CHR?" on figure 10.10, is reflected in other notational aspects of the figure hereabouts. This reading takes the descending minor third from F to D, first introduced on the figure at measure 18, and fills it in by a retrograde CHR line; in counterpoint to this, an ascending (prime) CHR line connects the Ej? of Tristan to the G|? of MAGIC; MAGIC then saturates the music from measure 23^ to measure 27. The ascending CHR line of this reading, from E|? to G^-within-MAGIC, retrogrades the opening gesture of figure 10.17(b) earlier, which showed a retrograde CHR line proceeding from Gb-within-MAGIC to E[?. On figure 10.10, brackets demarcate the CHR groups of four events into which the voice leading is articulated by this reading. Since the earlier CHR networks of measures 9 and 10 were associated with thematic crescendi, and since the retrograde CHR events of measure 14 (figure 10.17(b)) could easily take a diminuendo, it would be useful to explore in connection with this reading how best to structure the "poco a poco cresc. e stringendo" that begins at measure 20. Does it stop at measure 22? Does it intensify there and continue on right up to the arrival of the D natural at measure 23|? Does it continue even past that, right up to the (subito?) ppp mesure at measure 24, where the low Ab pedal comes in? I have put parentheses and a question mark on figure 10.10, together with the crescendo sign under measure 22 there, to suggest and emphasize these questions. The other possible reading I hear is indicated on the figure by the annotation "Voice-exchange Eb/F?" According to this reading, the basic grouping is not of four CHR events, followed by ppp mesure; rather the basic grouping is of three events within the Tristan harmony, followed by the arrival of MAGIC harmony at the D natural that launches the uwmeasured section of the cadenza, at measure 23^. This is the point where the progressively more agitated ripples of the water turn into turbulence; at the (subito) ppp mesure of measure 24 we presume that the burst of wind abruptly stops. The idea of putting the structural downbeat for the MAGIC arrival not at the obvious measure 24, but rather just at the moment where periodic wave motion turns to "aperiodic" turbulence, in measure 23^, is extraordinarily poetic. I would enjoy trying to play the passage this way for a small group of close friends; I am not sure how well I could project it to a large public audience. Again, it would be very helpful to have more indications for tempo and dynamics between "pp, poco a poco cresc. e stringendo" at measure 20 and "(subito?) ppp, mesure" at measure 24. As it is, the pianist's decisions about tempo and dynamics will very much affect the sense of our alternate readings, and viceversa. I write "(subito?) ppp" because I can conceive that Debussy heard a diminuendo from a MAGIC arrival at measure 23^ to ppp at measure 24, even though he did not write such a diminuendo. In that case, one imagines the

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wind dying down more gradually, making the turbulence subside gradually into regular ripples again. The rippling figuration over measures 24-29 ornaments a basic succession of two harmonies: The MAGIC harmony just under discussion moves to a dominant-ninth structure at measure 27, as shown on figure 10.10. The figure also shows, by notations under the staff, how the progression involves precisely the two chromatic voice-leading gestures already exposed during measures 18-19, that is D-to-Eb and Ct>-to-C. Within the overall progression, each of the two harmonies is prolonged by (or accompanies) a form of a new motive I call V, marked in parentheses on figure 10.10. V stands for "variable"; the motive always proceeds by step up, third-leap up, step down, step down, and step down, ending on the same note with which it started, but the sizes of the "steps" and the "third" vary considerably, sometimes even within one V-statement, and the rhythm of the motive is also extremely plastic. These features of V presumably represent the perpetual mutations of things seen through reflets dans I'eau. (One thinks of Monet's pond at Giverny.) The descending part of the second V-statement here hooks up with the rippling figuration just before measure 30, to project the descending hexachordal line BJ7-A[7-G(?; F-Et?-D[?. We have heard this descending hexachord before, namely in the melody of measure 14; there the B|?-A[7-Gb segment was also launched by MAGIC harmony. At measure 30 the V motive carries the final Dfr of the hexachord on down to C5. We noted earlier that C5 was an interesting climax for the principal melodic line over measures 1-17, and that the melody seemed to have unfinished business in its upper register. Debussy is now, at measure 30, turning his attention to some of that business. All at once, we hear bursting forth from the ripples a Z-form in a principal melodic line, a CHR-related network of trichords in parallel motion, and a crescendo; these were all aspects of the music during its earlier rise over measures 9-10. The Z-form Z 3 , articulated like Z2 earlier by rhythm and contour, circles around the critical pitch C5. It does touch D[?5, but there is some question as to how essential D[?5 sounds in the melodic line. Hitherto, all motivic Trelations have been both by one diatonic scale degree and by two semitones (or by major second). Now we have to respond motivically to a relation that is and is not the same, between C5 and D|?5. This relation is quintessential^ by one diatonic scale degree, but it is just as quintessentially by only one semitone (or by minor second). Is this, or is this not, a bona-fide "T-relation?" The answer, of course, is "Yes and no." We shall not get far arguing the question in that form; but it will be very much worth our while to explore the ways in which the answer is ambivalent. More generally, let us explore further ways (beyond the issue of T-relation) in which we feel ambivalent about the idea that the line might rise from C5 to D|?5 here. The sense of structural rise seems strongly contradicted by the retrograde

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forms for Z3 and CHR. Since we associated prime Z-forms with rising, over measures 9-10, we naturally feel that a retrograde Z-form is falling. And of course the retrograde CHR network is falling very decisively. But the crescendo is motivically important too. No matter how much the retrograde motives sound falling, the music is still rising in dynamic, like the music of measures 9-10 that first led the melody up to C5. Debussy would surely have written a diminuendo from mf, not a crescendo, had he heard a completely subsiding effect here. It is difficult but crucial for the pianist to husband the dynamics scrupulously, so the crescendo can proceed past mf and on to f without actually attaining ff. (Think of narrow-bore brass!) To support further the idea that the line might be rising to D[?5, we can hear that D[?5 gets strong, if fleeting, harmonic support within the Z3-form. D[?5 is specifically supported by T' (MAGIC) harmony, continuing to follow the precedent of measure 14. Figure 10.20 shows what I mean.

FIGURE 10.20

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Now the Dj? at the end of measure 14, the model for the D|?5 of figure 10.20 in the relation just pointed out, is (or becomes) a very strong cadence tone; its function as an essential tone is not in doubt. By analogy, the D^S of measure 30 sounds that much more essential. But now a new consideration arises. The very strength of the fit between figure 10.20 and measure 14 reminds us that everything shown on figure 10.10 from measure 22 through measure 30 has in effect been transferred up an octave from the Tristan chord of measure 20, which served as a launching pad. Figure 10.20, in particular, is still "an octave up" from measure 14. According to this reading, the retrograde Z3 at measure 30 is an octave transfer of a form that actually belongs an octave lower in the large melodic structure. C5 and Dt? 5 within ret Z3 are to that extent not in the structural climax register at all; they are octave transfers of C4 and D[?4, from the cadence register. That view is afforded support by the melodic doublings hereabouts: Both the second Vstatement and the retrograde Z3 motive are doubled in the music an octave below where they appear on figure 10.10.

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10.4

Still, no matter what the pianist does with the balance of these doublings, there has been no written dynamic heretofore as loud as mf, let alone mf with a crescendo to f, and that aspect of measure 30 by itself makes it hard to deny that we should hear C5 and D[?5 as climactic events to some extent, and not just as doublings or transfers of C4 and D|?4. In sum, the C5 and D(?5 of the retrograde Z3 at measure 30 are ambivalent in many ways: It is unclear to what extent they address the issue of unfinished business in the climax register of the principal line, and to what extent they represent C4 and D[?4 transferred temporarily up an octave; it is also unclear to what extent ret Z3 is "rising," and to what extent it is "falling." The music plays with these ambiguities in the immediate sequel, via the little descending arpeggios that alternate C with Db first in register 6, then in register 5, and then in register 4. (The arpeggios are not shown on figure 10.10.) Eventually the ret Z3 figure returns down the octave, as shown on figure 10.10. Or one could put it that the doubling in the upper octave drops away. In any case, the registral issue is temporarily resolved in favor of the lower register. We become particularly convinced of this at measure 35, where the reprise begins. There we recognize the notes of Z3 in the lower register as the notes of an alto voice that fits underneath the X motive of the soprano. We might say that Z3 belonged in the lower octave all the time; it was only tossed up an octave higher at measure 30 by the agitated activity of the waters, so to speak, as a rare and curious submarine object that flashed momentarily into view and then sank again. (One supposes that D|?4 is the surface of the pond.) As Z3 subsides under the surface at the reprise, the unfinished business in the climax register remains unfinished. But the discussion of figure 10.10 has ended. I should discuss, if only briefly, significant later events in the climactic register of C5 and above. The key to this analysis is the passage sketched in figure 10.21.

FIGURE 10.21 The tune is supported by a cadential progression in Efr major over a tonic pedal. The dynamic, ff and crescendo, is uniquely climactic for the composition. (The unexpected bursting forth of Eb-major harmony a measure before figure 10.21 is only f, with a hairpin. Again the pianist must husband the dynamics very meticulously.) Here B[?4 and C5, that could not rise farther at

243

10.4

Transformation Graphs and Networks (4)

measure 10 and rose to Db5 only problematically at measure 30, finally make their breakthrough. Indeed, they get as far as £[7 5, with no question about it. Db 5, significantly, is not on the scene now that we are in the key of Eb. And C5 is not a leading tone in the new key. The "breakthrough" form of V at measures 57-58 directly follows two Vessays of a very different character. Debussy tells us that the Bb4 and C5 of the breakthrough are to be compared with the same pitches in measure 9-10, by his use of the retrograde T(Z X ) motive. Whereas in measure 10 only the Bb-Ab of Z x moved up to T(Bb-Ab) = C-Bb within Z2, now all of Zj = F-Eb-Bb-Ab has moved up to T(Z1) = G-F-C-Bb, which is then retrograded in measures 58-59 so as to descend from the climactic V-statement. Further, the whole tonality of the piece has temporarily moved up from Db to Eb = T(Db), to accommodate and support the T-transformation of Zl. This is indeed an extraordinary expansion of the original T-relation which, the reader will recall, obtained precisely between the notes Db and Eb = T(Db), the notes which started and ended the initial motive X. The potential for the change of tonality was perhaps already latent in the transformation of X to T(X) during measure 3. To the extent that T means "two semitones," T(X) would have had to be supported by Eb harmony in measure 3 to have the same "meaning" as X in measure 1. We suggested in discussing measure 30 that the rise from C5 to Db5 within a Z-form could not be completely convincing because the minor second (or distance of one semitone) is not completely T-ish in character. Now, over measures 56 and following, the change of tonality from Db to Eb provides an appropriate and enormous T-ish boost, enabling the melody to rise definitively beyond C5.3

244

3. For a sensitive appreciation of dynamics and contours rising and falling over this piece on a large scale, and also for a convincing view of the E|? major breakthrough as it articulates a large rhythmic design, the reader is referred to Roy Howat, Debussy in Proportion (Cambridge, England: Cambridge University Press, 1983), 23-29.

Appendix A:

Melodic and harmonic GIS Structures; Some Notes on the History of Tonal Theory

Chapter 2 surveyed a variety of musical spaces pertinent to theories of Western tonality. In some of these spaces, our intuitions of directed distance or motion from one position to another were measured by steps along some melodic scale, diatonic or chromatic, linear or modular. In other spaces, our intuitions were measured by numbers reflecting harmonic relationships of various kinds, or by moves on a game board derived from harmonic relationships. The richness of tonal music, and of music in related idioms, is much enhanced by the ways in which a variety of such intuitions come into play. We can review in this connection the Protean meanings of "the interval from F to At?" in Reflets dans I'eau. We sense a harmonic interval within the Dt?-major triad and the Tristan Chord; we sense also a melodic interval moving two steps along a diatonic scale in D(? major, or along a diatonic hexachord (the cadential descending hexachord B^AbGbFEbDj?); to some extent we can even hear F-to-At? as one melodic step along the pentatonic scale DbE^FA^Bt?, as we listen to the melody at the beginning of the piece; finally we also hear F-to-Aj? as spanning three semitones along a chromatic scale, once the CHROM figure comes onto the scene. A transformational approach enabled us to sidestep these ambiguities in chapter 10, referring there to transformations T and T' that mapped D[> to E|? and Aj? to £[7 respectively; we could conceive T and T' as transpositions by "intervals" in any-or-all of the conceptual spaces involved; then we could compute a corresponding transformation T-1T' which mapped A(? to F in any-or-all of the spaces, a transformation worked out musically in the change from motive X to motive Y. Such transformational discourse is particularly useful to discuss architectural features of Reflets that obtain no matter what sorts of intervallic

245

Appendix A

intuitions one considers. For instance, the tonic of the climactic fortissimo E|? major is in a T-relation to the tonic of D[? major no matter what kind of interval, in what kind of melodic or harmonic space, we consider T to be transposing D(? by. On the other hand, transformational discourse is correspondingly impoverished when it conies to exploring the varieties of spatial and intervallic intuitions at hand, and the ways in which the music brings those intuitions into play, each with the others. In connection with the big climax of Reflets, for instance, the major mode of the fortissimo music in E[? favors certain kinds of intervallic intuitions over others, when we hear Ej? major, not E|? minor, in relation to D[? major. The reader may also recall our discussion of the high C-Db in the principal melodic line, and our question: "Is this, or is this not, a T-relation?" That question implicitly involves very broad questions about the premises of the composition: To what extent is the piece diatonic-melodic, so that one scale-step is one scale-step, regardless of its acoustical size? To what extent is the melos chromatic, so that one semitone is something necessarily very different from two semitones, even if both are spanning one diatonic step? To what extent is "the interval" attached to T heard in a harmonic context that gives it a size somewhere between 10/9 and 9/8, but not as small as 16/15? To what extent can techniques of melodic motivic transformation, involving the rhythm and contour of the Z motive in particular, alter our impressions in any or all of these respects? and so on. Exactly these ambiguities must be appreciated, if we are adequately to appreciate the conceptual tensions of the local climax involving C5 and D|?5, beyond its high register and relatively high dynamic level. We return, then, to the variety of intervallic intuitions surveyed in chapter 2. Pertinent syntheses of these intuitions are essential not only for many occasions in critical listening and analysis, but also for many abstract theoretical purposes. Indeed, such syntheses are among the greatest triumphs in the history of Western music theory, and their neglect or failure has led to some of the more embarrassing moments in that history. Among the latter, we may cite Rameau's argument that the harmonic intervals of 5/4 and 6/5 may be exchanged in relative register within the harmonic triad, so as to derive the minor triad from the major. This may be allowed, he says, since 5/4 and 6/5 are both "thirds."* But at the time he says this, he has not as yet presented us with any scale along which we can measure distances of "three" degrees, and he has assured us very strongly that melody is in any case thoroughly subordinate to harmony.2

246

1. Jean-Philippe Rameau, Traite de I'harmonie reduite a ses principes naturels (Paris: Ballard, 1722). "... la difference du majeur au mineur qui s'y rencontre n'en cause aucune dans le genre de 1'intervale qui est toujours une Tierce de part & d'autre;..." (p. 13). 2. Ibid. "On divise ordinairement la Musique en Harmonic & en Melodic, quoique celle-cy ne soil qu'une partie de 1'autre, & qu'il suffise de connoitre I'Harmonie, pour etre parfaitement instruit de toutes les proprietez de la Musique,..." (p. 1).

Appendix A

One might try to replace Rameau's implicitly melodic argument about the "thirds" by a suitable harmonic argument: The intervals 5/4 and 6/5 are adjacent within the senario; they divide the interval 3/2 = 6/4 harmonically, and so arithmetically when reversed; therefore such a reversal is logical. But one could argue in exactly the same fashion for the intervals 4/3 and 5/4, as they divide 5/3. Would Rameau, then, have accepted the argument that we ought to consider the harmony G4-C5-E5 as functionally equivalent to the harmony G4-B4-E5, by the analogous reasoning? Obviously, he would not have; such reasoning there would violate the principle of the Fundamental Bass. Helmholtz, however, might have been willing to entertain the argument; indeed he actually asserts harmonic equivalence of a certain sort between the six-four position of the major triad and the six-three position of the minor triad. Both those positions comprise the highly consonant verticalities of a fourth, a major third, and a major sixth; having the same vertical-interval content, they are thereby the "most consonant" close positions for their respective pitch-class sets.3 Among the triumphal syntheses mentioned earlier a high position must be reserved for Zarlino's Istitutioni harmoniche.* Book 1 discusses intervals as phenomena in a harmonic space. Book 3 discusses intervals all over again as phenomena in melodic space, and synthesizes that approach with the mathematical ideas of book K Abstract harmonic ratios are accessible to our perception (as well as our intellect) because they can be filled in by notes of a diatonic series in melodic space; conversely, articulated segments of a unidirectional diatonic series make sense to our understanding (as well as our perception) because of the harmonic relations obtaining between the boundaries of the segments. This way of interrelating harmonic and melodic space has much in common with central aspects of Schenker's theories, in particular with Schenker's understanding of the Zug, and even specifically of the Urlinie. Schenker's mature theory contains another triumphal synthesis of harmonic and melodic space, understood now in the context of functional tonality.5 Even Zarlino has an embarrassing moment, confusing melodic with harmonic space, when he comes to discuss the minor sixth. He wants to analyze the major and the minor sixths as analogous structures. Specifically, he says that they "are composed ... from the fourth plus the major third, or 3. Hermann Helmholtz, Die Lehre von den Tonempflndungen als physiologische Grundlage fur die Theorie der Musik, 2d ed. (Brunswick: Friedrich Vieweg und Sohn, 1865). "..., so folgt hieraus, dass die Quartsextenlage des Disaccords wohllautender ist als die fundamental, und diese besser als die Sextenlage. Umgekehrt ist die Sextenlage beim Mollaccord besser als die fundamentale, und diese besser als die Quartsextenlage." (p. 325). 4. Gioseffo Zarlino, Istitutioni harmoniche, 2d ed. (Venice: Senese, 1573). Facsimile republication (Ridgewood, N. J.: Gregg Press, 1966). 5. Heinrich Schenker, Neue musikalische Theorien und Phantasien, vol. 3, Derfreie Satz (Vienna: Universal Edition, 1935).

247

Appendix A

the minor third." 6 He might have continued: So, in a major mode such as Ionian, the fourth G3-C4 plus the modal major third C4-E4 yields the major sixth G3-E4, while in a minor mode such as Dorian, the fourth A3-D4 plus the modal minor third D4-F4 yields the minor sixth A3-F4. The modal idea is perfectly clear. Elsewhere, too, Zarlino makes a great point of the modal relation between major and minor thirds in harmonic contexts; indeed he even points to this as a specific resource for harmonic variety, beyond the resources of the senario itself: Some chords have a major third or tenth over the bass, others a minor third or tenth.7 One wishes, then, that he would have produced G3-(C4)-E4 and A3-(D4)-F4, to illustrate a modal analogy between the two sixths as being "composed ... from the fourth plus the major third, or the minor third." Unfortunately he does not do so. Probably he was not as sensitive as we are to the thirds above the modal tonics C4 and D4 in the harmonic structures above; those thirds are not over the bass notes of the structures. Whatever his motivation, he attempts to realize the analogy of the sixths as a feature of his harmonic space rather than his modal theory, and that leads him into confusion. He has to adjoin the number 8 to the senario in order to get the harmonic ratio 8:5 at hand for the minor sixth, and then he has to argue that the proportion 8:6:5 is somehow analogous to the proportion 5:4:3 in his harmonic world. He even claims that 6 is a "harmonic mean term" between 8 and 5; this is simply false if "harmonic" is to mean anything at all in the context.8 We may fairly put his argument into modern dress by regarding it as an attempt to draw a direct analogy between the major sixth G3-E4, as divided by C4, and the minor sixth E4-C5, as divided by G5. Of course this does not work. In particular, the conjuction of the minor third E4-G4 below with the fourth G4-C5 above is not at all the same thing as the conjunction of some fourth below with some minor third above, as in the Dorian modal sixth A3-(D4)-F4. Zarlino could also, of course, have analyzed the sixths as arising by inversion from the thirds. But this approach would have been foreign to his purpose, for then the sixths would not have been primary features of his harmonic space, somehow embedded within the senario. Besides, the sixths that arise from inverting thirds have a very different modal character from the sixths that interest (or should interest) Zarlino. That is exactly the problem

248

6. Zarlino, 1st. harm., book 3, chapter 21. "... sono composte ... della Diatesseron et del Ditono, over del Semiditono;..." (p. 193). 7. Ibid.,book3,chapter31. "... lavarietadell'Harmonia...nonconsistesolamentenella varieta delle Consonanze, che si trova tra due parti; ma nella varieta anco delle Harmonic, la quale consiste nella positione della chorda, che fa la Terza, over la Decima sopra la parte grave ..., overo che sono minori ...; overo sono maggiori..." (p. 210). 8. Ibid., book 1, chapter 16. "... tal proportione tra 8 & 5 termini son capaci di unmezzano termine harmonico, che e il 6;..." (p. 33).

Aooendix A

with his "minor" sixth 8:5 (E4-C5). To the extent that we hear it in a "Cmajor" modal context as third-degree-to-octave, inverting tonic-to-thirddegree = C4-E4 = 5:4, a major third, the sixth itself has a "major" modal character about it, despite its small size. Contrast that with the modal character of A3-F4 in a D-Dorian context, as fifth-degree-(through-tonic-)to-thirddegree: This sixth, in its context, has a "minor" modal character as well as a minor absolute size. Similarly, F4-D5 in D-Dorian, inverting D4-F4, has a "minor" modal character despite its large size, while G3-E4 in C-Ionian has both a "major" modal character and a large size. Zarlino had no Stufen theory that could enable him to make such discriminations. And yet it is quite possible that, even if one had been available to him, he might have rejected it. He would have been uncomfortable making the meaning of his harmonic intervals so dependent on the contextual assignment of a modal tonic. For him this would have weakened the context-free universality of his harmonic theory. Schenker, quite willing to assign structural priority to contextual modal tonics inter alia, uses his Stufen theory to powerful effect in related connections. On the other hand, he finesses certain problems about the universality of minor harmonic structures which Zarlino attempts to confront, and succeeds in confronting to a remarkable extent. Hindemith makes an interesting synthesis of melodic and harmonic spaces.9 He tries to show that a chromatic scale from C2 to C3 is filled by those pitches, and only those pitches, which lie in "closest" harmonic relation to C2 within a certain harmonic space. We ignore the overtones of C2; then G2, within the desired scale-segment, is harmonically "close" to C2 because the second partial of G2 is the third partial of C2. F2, within the desired scalesegment, is "close" to C2 since the third partial of F2 is the fourth partial of C2. And so on, casting away harmonic octave-replicates of pitches already generated (which, happily, do not lie within the desired scale-segment). For the most part, this works quite well, though a bit of strain is perceptible in the construction of certain secondary relationships. The essence of Hindemith's achievement was not just to find pitch classes that can be represented by pitches within a chromatic scale. After all, Zarlino and his forerunners could do that well enough and better. Rather, the achievement was to have shown how pitches within a melodically well-packed region, a chromatic scale from C2 to C3, could be regarded as pitches also within a harmonically well-packed region around the tonic, harmonically well-packed according to Hindemith's special algorithms for generating harmonic pitch-space. The one pitch with which Hindemith has trouble is A[?2, the minor sixth above the tonic C2. Curiously enough, his troubles resemble Zarlino's troubles with the minor sixth. Hindemith can generate E2, E[?2, and A2 without using partials of those 9. Paul Hindemith, Unterweisung im Tonsatz: Theoretischer Teil (Mainz: B. Schott's Sonne, 1937), 47-61.

249

Appendix A

pitches, or of C2, that involve numbers greater than 6. E.g.: The fifth partial of Eb2 is the sixth partial of C2; the third partial of A2 is the fifth partial of C2. But in order to find A[?2 by this method, he would have to have used an eighthpartial relationship: The fifth partial of A|?2 is the eighth partial of C2. This relation would presumably have made A(?2 too "remote" in harmonic space; besides, it might have given rise to awkward questions about seventh-partial relationships. (Zarlino has to deal with the analogs of such questions, when he adjoins the number 8, but not the number 7, to his senario.) Presumably for reasons of these sorts, Hindemith produces not A[?2 but Afc> 1 by his algorithm; A)?! is a unique pitch which he generates in this way outside the octave C2-C3. Then, without much explanation, he brings A|? 1 up an octave, so that it will lie within his desired scale-segment.10 The foregoing discussion of ways in which some theorists have attempted to integrate harmonic and melodic tonal spaces, or have failed to integrate them, is not meant to be exhaustive or even representative. It is rather intended to show that we do not really have one intuition of something called "musical space." Instead, we intuit several or many musical spaces at once. GIS structures and transformational systems can help us to explore each one of these intuitions, and to investigate the ways in which they interact, both logically and inside specific musical compositions.

250

10. Ibid. "The frequency ... of the fourth overtone [of C2] is now divided by 5 ... and so generates ... Ab 1 ..., whose second overtone ... is inserted in our store of pitches. (Die Schwingungzahl ... des vierten Obertones c1 wird nunmehr noch durch 5 ... geteilt ... und erzeugt so ... das lAs ..., dessen zweiter Oberton ... in unseren Tonvorrat eingereiht wird.)" (p. 54).

Appendix B:

Non-Commutative Octatonic GIS Structures; More on

Simply Transitive Groups

Let S be the octatonic family of pitch classes comprising C, Cft, Dft, E, Fft, G, A, and Aft. Eight of the standard "atonal" operations on the twelve pitchclasses transform S into itself; these operations are T0, T3, T6, T9, l£*, IG, 1°» and l£*. The eight operations form a group on the twelve pitch-classes and therefore, mapping S into itself, induce a group of corresponding operations on S; we shall call those corresponding operations RO, R3, R6, R9, K, L, M, and N respectively. It is not hard to verify that the latter group is simply transitive on S: Given members s and t of S, there is a unique OP, among the eight cited operations on S, satisfying OP(s) = t. (If t is in the same diminished-seventh chord as s, OP will be RO, R3, R6, or R9; if t is in the opposite diminished-seventh chord from s, OP will be K, L, M, or N.) We shall call this simply transitive group of operations STRANS1. The operations RO, R3, R6, and R9 may be thought of as "rotations," to justify the use of the letter R in their names. We can define another group of operations on S, STRANS2, as follows. RO and R6 (as above) are members of the group; so are two "queer" operations Q3 and Q9. Q3 rotates each of the diminished-seventh chords within S, but in opposite directions; it maps C to Dft, Dft to Fft, Fft to A, A to C, and also Cft to Aft (not to E), Aft to G, G to E, and E to Cft. Q9 is the inverse operation to Q3; it maps C to A , . . . , Dft to C, and also Cft to E,..., and Aft to Cft. Besides RO, Q3, R6, and Q9, STRANS2 also contains four "exchanging" operations XI, X2, X4, and X5. XI exchanges pitch classes within S that lie one semitone apart; it thus maps C to Cft, Cft to C, Dft to E, E to Dft, Fft to G, G to Fft, A to A#, and Aft to A. X2 exchanges pitch classes that lie two semitones apart; it maps C to Aft, Aft to C, Cft to Dft, Fft to E, and so on. X4

257

Appendix B

252

exchanges pitch classes that lie four semitones apart; it maps F# to A#, E to C, G to D#, and so on. X5 exchanges pitch classes that lie five semitones apart; it maps A to E, D# to A#, F# to C#, and so on. It can be verified that STRANS2 is a group of operations on S, and that it is simply transitive. Using the method discussed in 7.1.1, we can develop a GIS structure for S in which the members of STRANS1 are exactly the formal transposition operations. We can call this structure GIS1 = (S, IVLS1, intl). In GIS1, then, applying any one of the operations RO, R3, R6, R9, K, L, M, or N to a member s of S amounts formally precisely to "transposing" the given s by a suitable corresponding interval of IVLS1. We must be careful to distinguish the operations K, L, M, and N, which are "GIS 1-transpositions" under this formalism, from the operations l£* etc. that gave rise to them; !£* etc. are inversion-operations in a different GIS, a GIS involving a different family of (twelve not eight) objects, a different group of (twelve not eight) formal intervals, and a different function int. Likewise, and more subtly, we must distinguish the octatonic GIS 1-transpositions RO, R3, R6, and R9 from the dodecaphonic atonal-GIS-transpositions T0, T3, T6, and T9. As it turns out, the members of STRANS2 are exactly the intervalpreserving operations for GIS1. Every member of STRANS2 commutes with every member of STRANS1. In fact, the members of STRANS2 are precisely those transformations on S that commute with every member of STRANS1. Using the method of 7.1.1, we can develop another GIS involving the family S, a GIS for which the members of STRANS2 are exactly the formal transposition operations. We can call this structure GIS2 = (S, IVLS2, int2). In this GIS, applying any of the operations RO, Q3, R6, Q9, XI, X2, X4, or X5 to a member s of S amounts precisely to transposing s, formally, by a suitable corresponding interval of GIS2. The interval-preserving operations for GIS2 are exactly the members of STRANS1; those are in fact precisely the transformations on S that commute with every member of STRANS2. Either GIS1, or GIS2, or both, might lead to results of interest in analyzing a variety of octatonic musics. STRANS2 and STRANS1, which figure as groups of interval-preserving operations in those respective GIS structures, are thereby also likely candidates for CANONical groups of operations in a variety of set-theoretical studies. The STRANS1-forms of a set within S are exactly the dodecaphonically transposed and inverted forms of the set that lie within S. The STRANS2-forms of a set within S are in general a more novel sort of family. Taking (C, E, G), for instance, we apply to it in turn the operations RO, Q3, R6, Q9, XI, X2, X4, and X5; its STRANS2forms are thereby computed as RO(C, E, G) = (C, E, G), Q3(C, E, G) = (D#, Qf, E), R6(C, E, G) = (F#, A*, C|), Q9(C, E, G) = (A, G, AJ), XI (C, E, G) = (Cfl,DJF,FJF) f X2(C,E,G) = (A#, Ffl, A), X4(C,E,G) = (E,C,D#), and X5(C, E, G) = (G, A, C). If Y is any one of those eight sets, and Y' is any other one, and f is any one of the eight operations in STR ANSI, then the number of

Appendix B

members of Y whose f-transforms lie within Y is the same as the number of members of Y' whose f-transforms lie within Y': JNJ(Y, Y)(f) = INJ(Y', Y')(f). More generally, if f is any one of the eight operations in STRANS 1, and A is any one of the eight operations in STRANS2, and Y and Z are any sets whatsoever within S, then INJ(Y, Z)(f) = INJ(A(Y), A(Z))(f): the number of members of Y whose f-transforms lie within Z is the same as the number of members of A(Y) whose f-transforms lie within A(Z). As an exercise, the reader may consider a new family of pitch classes, S = (C, C|, E, F, Gf, A), and develop on the new S two analogous simply transitive groups of operations. More generally, suppose now that S is any family of objects and that STRANS is any simply transitive group of operations on S. Consider the family STRANS' of transformations f on S such that f commutes with every member of the given group STRANS. It can be proved that STRANS' is itself a simply transitive group of operations on S, and that every transformation A which commutes with every member of STRANS' is (already) a member of the given group STRANS. When S is considered as a GIS whose formal transpositions are the members of STRANS, then the members of STRANS' will be the interval-preserving operations. Dually, when S is considered as a GIS whose formal transpositions are the members of STRANS', then the members of STRANS will be the interval-preserving operations. If STRANS is commutative, then STRANS' will be precisely STRANS itself.

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Index

ADJOIN, 121 Anti-homomorphism, 14 Anti-isomorphism: 14; i-to-Tj, 46, 47, 150 AOD: 113-15; and interval-preserving operations, 121-22 Argument, of function, 1 ARROW. See Node/arrow system Arrow chain: 194-95; proper, 209 Associative: law, 5; binary composition, 5 Attack-ordered dyad, 113, 121-22. See also AOD Attack ordering, 117 Babbitt, Milton: and beat classes, 23; SemiSimple Variations, 136, 138-40, 141, figs. 6.7-6.9; Reflections, 138; hexachord theorem, 145 Bach, Johann Sebastian, Two-Part Invention #1, 183-84, fig. 8.5 Bart6k, B61a, "Syncopation," 225-27, figs. 10.5-10.6 Beat class, 23 Beethoven, Ludwig van: First Symphony, 169-74, figs. 7.8-7.13, 8.1; Quartet op. 135, 2lln; Sonata Appassionata, 213-17, figs. 9.14-9.16 Bernard, Jonathan W, 62/», 189 Binary composition, 5 BIND. See Serial transformations Boundary tones, 94, 95, 98, 102-03, 235 Brahms, Johannes: G-Minor Rhapsody, 119-21, figs. 5.14-5.15; Horn Trio, 16569, 202-03, 207, figs. 7.6-7.7, 9.6 Bresnick, Martin, 67n5 Bushnell, Michael, 90, 141n Cann, Richard, 85n7 Canonical: group, 104, 106, 111, 113, 15052; equivalence, 104, 113, 121-22; forms, 105, 119 Carriage return, 214-15 Carter, Elliott, First String Quartet, 23n3, 62, 67-74, figs. 4.2-4.3 Cartesian: product, 1; thinking, 158

Central: defined, 7; interval, 49, 52-55 passim Cherlin, Michael, 137 Chopin, Fr£d6ric, B-Flat-Minor Sonata, 8587, 114, figs. 4.6, 5.11 Christensen, Thomas, 163n Chronology: 173, 177; and precedence, 210-18. See also Precedence Closed, family of transformations, 4 Cogan, Robert, and Escot, Pozzi, 160n Collection, of objects, defined, 1 Coltrane, John, 62 Common-note function, 144 Commutative: defined, 6; GIS, 50, 53, 56, 58 Complement, of set, 144-45 Composition: of functions, 2; binary, 5 Cone, Edward X, 39n Congruence, defined, 10 CONTENTS. See Network, transformation COV, 122 Debussy, Claude, Reflets dans I'eau, 231-44, 245-46, figs. 10.10-10.21 Diatesseron, symphony of, 204-06, fig. 9.8 Direct product: of semigroups, 15; of groups, 15, 61; of intervals, 93-94; of GISs, 37-46, 174 Dispersive. See Transformations DOM, 176-78, 229 Double emploi. See Rameau Duration classes, 24, 25 EMB: defined for sets, 105; for set classes, 106; as probability, 107, 111; properties, 107-08; topological model, 108-11, figs. 5.9-5.10; and Partition Function, 152 Embedding number. See EMB Environment, of a set, 97-98 Equivalence classes: defined, 8; canonical, 104 Equivalence relation: defined, 7; arising from function, 7; and partitioning, 8;

255

Index Equivalence relation (continued) examples, 9; in quotient GIS, 33-35; canonical, 104 External. See Transformations

256

If-only adjustments, 140-41 IFUNC: defined, 90; examples, 89-99, figs. 5.1-5.7; maximum values, 94, 95; properties, 99-101; as probability, 101; and Zsets, 103; questions about, 103-04; unFamily: of objects, denned, 1; finite, 144rolling, 120; generalized by INJ, 147; ef45 fect of inversion on, 150 FLIPEND. See Serial transformations Induced map, of quotient family, 10 FLIPSTART. See Serial transformations INJ: defined, 124; free of GIS structure, Forms, of a set, 105 134; and operations, 144-45; and setForte, Allen: 9n, 103, 104, 143, 150; and complements, 144-45; as generalized Gilbert, Steven E., 218/1 IFUNC, 147; for transformed X and Y, Function: defined, 1; onto, 3; one-to-one, 3; 147-49; questions about, 149-50; generalinverse of, 3; and equivalence relation, 7 izing K and Kh, 150-51; for infinite sets, Functional: orthography, 2; equality, 2; 153, 154, 156 equations, 2-3 Injection function, 95, 99. See also INJ Fundamental bass, 169, 170, 173, 175, 185, Injection number. See INJ 209, 247 Input: node, 207, fig. 9.9; Klang, 208, 215, fig. 9.10; motive, 209; trichords, 227 Int, 26 Generalized Interval System (GIS): defined and discussed, 26-28; review of prelimiInternal. See Transformations nary examples, 28-30; and intuitions, Interval: intuitions about, xi-xii, 16, 17-20, 250. See also Commutative; Direct prod25-26, 74-75, 245-46, figs. 0.1, 4.4; reuct; IFUNC; Interval-preserving operaplaced conceptually by transposition, xiii, tions; Interval-reversing transformations; 157-60, 245-46; scarce, 102-03; forwards Inversions; LABEL; Non-commutative (backwards) oriented in time-span GIS, GIS; Octatonic; Quotient; Simply transi113-14; between roots, 170; for TCH, tive group; Timbral GIS; Time spans; 181; varying in Debussy, 231, 245-46; in Transpositions Rameau, 246-47; in Zarlino, 247-49. See also Central; GIS; IFUNC; Vector Graph, 171, 190-92; transformation, deInterval-preserving operations: and fined, 195; operation, defined, 196; isoLABEL, 47; defined, 47-48; form a morphism, 199; homomorphism, 201-06; product, 204, fig. 9.8(d) group, 48; characteristic behavior, 48-49; are sometimes transpositions, 49; comGreer, Taylor, 95n mute with transpositions, 50; combined Group: of operations, 4; abstract, 6; direct product, 15; homomorphism, 14-15; quowith inversions, 55; effect on IFUNC, 99; in time-span GIS, 111-13; and AODs, tient, 15; of intervals, 25-26; of transpositions, 46-47; of interval-preserving 121-24; octatonic, 252 operations, 48; locally compact, 103; can- Interval-reversing transformations, 58-59 onical, defined, 104. See also Canonical; Inverse: function, 3; element of semigroup, 6; operation, 56 Simply transitive group Inversions: defined in a GIS, 50-51, fig. 3.7; and LABEL, 51; when functionally Harvey, Jonathan, 24n equal, 52, 53; combined with transposiHasty, Christopher, 43n tions, 54; combined with interval-preservHaydn, Joseph, Quartet op. 76, no. 5, 211/j ing operations, 55; combined one with Helmholtz, Hermann, 247 another, 56; inverses of, 56; effect on Hexachord: semi-combinatorial, 143, 148IFUNC, 101, 150; as graph transforma49; Babbitt Theorem, 145 tions, 190-92, fig. 8.12 Hindemith, Paul, 75, 249-50 Homomorphism: of semigroup, 13; and nat- Isography: informally mentioned, 183, 187, 192; defined, 198-200; further examples, ural map, 14; of group, 14-15; of node/ 227, 230, 234 arrow system, 201; of graph, 201-06, 235 Isomorphism: of semigroups, 13; i-to-Pj, 48; Howat, Roy, 244n of node/arrow systems and graphs, 199 IVLS, 26 Ictus, 42-44, figs. 3.4-3.6 Identity: operation, 4; element for semiJones, James Rives, 39« group, 5

Index K, and Kh, 150-51 Klang: 175-81, 213-14, 216-17, 227-29; input and output, 207-08, fig. 9.10 Kramer, Jonathan, 211n Kurth, Ernst, 219/t LABEL: defined and discussed, 31-32; for T,(s), 47; for Pi(s), 47; for Itfs), 51 Lerdahl, Fred, and Jackendoff, Ray, 217« Lewin, David, 32n, 42«, 60n, 88n, 89n, 120n, 128n, 141n, 193n Ligeti, Gyorgy, Poeme symphonique, 23/i3, 67 Linear ordering: 135,146; and precedence, 211-12, 219 LT, 178 Mapping: defined, 1; of a set, 88 Maximum values: of IFUNC, 94; of INJ, 140-41, 142 Measurable transformations, 153 Measure: abstract, 153; on time-span GIS, 154-55 MED, 176-77 Melody, models for, 133, 219, fig. 9.17 Minturn, Neil, 227 Modular musical spaces, 17, 20-25 passim, 36. See also Quotient, GIS Modulation of transformational system, 129, 148-49 Moorer, James A., and Grey, John, 84n Morris, Robert D., 104n, 105/j Mozart, Wolfgang Amadeus, G-Minor Symphony, 220-25, figs. 10.1-10.4 MUCH. See Serial transformations Nancarrow, Conlon, Studies for Player Piano, 23n3, 66-67 Natural map: onto quotient family, 8; onto quotient semigroup, 12 Network, 164, 165, 168, 170, 172, 176, 177, 178, 192; transformation, defined, 196; connected operation, 197; operation defined, 197; isography, 198-200; product, 206, 235, figs. 9.8(d), 10.16; as model for series, 206, 218; Schenkerian, 214, 21618, fig. 9.16; and time spans, 215-16, 217, fig. 9.15; and melody, 219, fig. 9.17 Node/arrow system: defined, 193; communication in, 193-94; connected, 194; arrow chain in, 194-95; isomorphism of, 199; homomorphism of, 201. See also Precedence NODEMAP. See Node/arrow system, isomorphism of, homomorphism of NODES. See Node/arrow system

Non-commutative GIS, 50, 58. See also Octatonic; Time spans Octatonic: scale, 17; non-commutative GISs, 251-53 One-to-one. See Function Onto. See Function; Graph, homomorphism Operations: defined, 3; group of, 4; on a GIS, 46-59; and INJ, 144-45. See also DOM; Graph; LT; MED; PAR; REL; Serial transformations; SLIDE; SUED; SUBM Ordering. See Attack ordering; Linear ordering; Partial ordering; Precedence; Release ordering Orthography, left and right, 2, 176 Output: node, 207, fig. 9.9; Klang, 208, fig. 9.10; trichords, 227

PAR, 178, 229 Partial ordering: of pitch classes, 135-40 passim, figs. 6.7-6.9; of nodes, 209, 211 Partition Function, 152 Peel, John, 174 Pitch notation, xiii Precedence: relation, 210; ordering, 210, 219; and chronology, 210-18 Probability: and IFUNC, 101; and EMB, 107, 111 Product: of graphs, 204, fig. 9.8(d); network, 206, 235, figs. 9.8(d), 10.16. See also Cartesian; Direct product Progressive. See Transformations Prokofieff, Serge, Melodies op. 35, 227, figs. 10.7-10.9 PROT: defined, 134; and rows, 134-35, 146 Protocol pairs, 134, 211. See also PROT Quotient: family, 8; semigroup, 10-12; group, 15, 29; GIS, 29, 32-37 Rahn,John, 174 Rameau, Jean-Philippe: 73, 175, 246-47; double emploi, 213, 217 Reflexive property, 7 Regener, Eric, 103n, 144, 145, 152 REL, 178, 213, 229 Release ordering, 117 RICH. See Serial transformations Rl-chaining: 164, 235; in both pitch and rhythm, 221-22; in 1. and r. hands, 22627. See also Serial transformations, RICH and TCH Riemann, Hugo, 22n, 73, 175, 177 Row, as set in PROT, 134-35

257

Index Schenker, Heinrich: 165, 174, 214, 247, 249; and networks, 216-18, fig. 9.16 Schoenberg, Arnold: Violin Fantasy, 10103, fig. 5.8; "Angst und Hoffen," 125-33, 148, figs. 6.1-6.5; "Die Kreuze," 133-34, fig. 6.6; Moses und Aron, 136, 137-38, fig. 6.7; Piano Piece op. 19, no. 6, 141, 143, 160, figs. 6.10, 7.1 Scholica Enchiriadis, 204 Semigroup: of transformations, 4; abstract, 5; multiplicative notation, 5; congruences and quotients, 10-12; homomorphism and isomorphism, 13-15; direct product, 15 Serial transformations: RICH and TCH, 180-88 passim, 221-24, 226-27; MUCH, 183, fig. 8.5; RT and RI, 188; TLAST and TFIRST, 188-89, fig. 8.10; FLIPEND and FLIPSTART, 189, fig. 8.11; and commutative GIS, 189-90; I, 192; BIND, 208-09 Set: mathematical, 1; in a GIS, 88; environments of, 97-98; class, 105; in a family S, 124; transitivity, 129; theory in infinite case, 152-56 SGMAP. See Graph, isomorphism, homomorphism SGP. See Graph, transformation Signature motive, 137-38 Simple ordering. See Linear ordering Simply transitive group, 157; as transposition group, 157-58; on Klangs, 179-80; on octatonic family, 251-53; and its commuting group, 253 Slawson, Wayne, 85 SLIDE, 178, 227 SNDW, 121 Stockhausen, Karlheinz: rhythmic theory, 24n; Gruppen, 24n; Aus den sieben Tagen, 62; Klavierstuck XI, 66 Strunk, Oliver, 2Q4n SUED, 177-78, 229 SUBM, 178 Symmetric property, 7 TCH. See Serial transformations TFIRST. See Serial transformations Timbral GIS, 82-84, 84-85, fig. 4.5 Time spans: defined, 60; commutative GIS for, 61; non-commutative GIS for, 74-81,

258

112, 154-56, fig. 4.4; and networks, 21516, 217, fig. 9.15 Time unit: discussed, 61; contextual, 67 TLAST. See Serial transformations TMSPS, 60 Transformations: and intuition of musical space, xiii; defined, 3; internal and progressive, 126, 132, 134, 141, fig. 6.6; external and dispersive, 142-43, 164; modulated, 148-49; measurable, 153. See also Graph; Network; Operations; Serial transformations Transitive property, 7 Transpositions: can replace intervals conceptually, xiii, 157-60, 245-46; defined in GIS, 46; form a group, 46-47; and LABEL, 47; sometimes preserve intervals, 49; commute with interval-preserving operations, 50; combined with inversions, 54; effect on IFUNC, 100; form simply transitive group, 157; in octatonic GISs, 252 Travis, Roy, 225, 227 Tristan chord, 238-40, 242 Unfolding. See Unrolling; Vector Unrolling: interval vector, 116-19; EMB, 119-20; IFUNC, 120; INJ, 131 Value, of function, 1 Varese, Edgard, 189 Vector: unfolding interval, 44; interval, 98, 104, 114, fig. 5.12; M-class, 106-07 Wagner, Richard: Parsifal, 161-64, 181-82, figs. 7.2-7.5, 8.3; Tarnhelm and Valhalla networks, 179, fig. 8.2; Todesverkiindigung, 183-88, 208-09, figs. 8.6-8.9, 9.11. See also Tristan chord Webern, Anton: Piano Variations, 38-44, 181, 182-83, 190-92, 200, figs. 3.1-3.6, 8.4, 8.12, 9.5; Pieces for Violin and Piano, no. 3, 90-99, figs. 5.2-5.7; Pieces for String Quartet, op. 5, no. 4, 188-89, fig. 8.10 Wedging, 124-32 passim, figs. 6.1-6.4, 6.6 Wintle, Christopher, 136n Zarlino, Gioseffo, 73, 247-49, 250 Zero time point, 63 Z-sets: generalized by IFUNC, 103-04; generalized further by INJ, 149-50

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Generalized Musical Intervals and Transformations

David Lewin

OXFORD UNIVERSITY PRESS

2007

OXFORD UNIVERSITY PRESS

Oxford University Press, Inc., publishes works that further Oxford University's objective of excellence in research, scholarship, and education. Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam

Copyright © 2007 by Oxford University Press, Inc. Originally published 1987 by Yale University Press Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Lewin, David, 1933-2003. Generalized musical intervals and transformations / David Lewin. p. cm. Originally published: New Haven: Yale University Press, c1987. Includes bibliographical references and index. ISBN 978-0-19-531713-8 1. Music intervals and scales. 2. Music theory. 3. Title. ML3809.L39 2007 781.2'37—dc22 2006051121 1 3 5 7 9 8 6 4 2 Printed in the United States of America on acid-free paper

For June and Alex

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Contents

Foreword by Edward Gollin Preface

xiii

Acknowledgments Introduction 1. 2.

ix

xxvii

xxix

Mathematical Preliminaries

1

Generalized Interval Systems (1): Preliminary Examples and Definition

16

3.

Generalized Interval Systems (2): Formal Features

31

4.

Generalized Interval Systems (3): A Non-Commutative GIS; Some Timbral GIS models

60

5. Generalized Set Theory (1): Interval Functions; Canonical Groups and Canonical Equivalence; Embedding Functions

88

6. 7. Transformation Graphs and Networks (1):

Generalized Set Theory (2): The Injection Function

Intervals and Transpositions

123

157

8. Transformation Graphs and Networks (2): Non-Intervallic Transformations

175

193 9. Transformation Graphs and Networks (3): Formalities 10. Transformation Graphs and Networks (4): Some Further Analyses 11. Appendix A: Melodic and Harmonic GIS Structures; Some Notes on the History of Tonal Theory

245

12. Appendix B: Non-Commutative Octatonic GIS Structures; More on Simply Transitive Groups Index

255

251

220

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Foreword to the Oxford Edition Edward Gollin

It has been nearly twenty years since the initial publication of David Lewin's Generalized Musical Intervals and Transformations (GMIT), and the work has aged well. This is due in part to the foundational nature of the book's subject matter. The work, a methodical examination of the concept of a musical interval, explores how the familiar notion of interval as "a distance extended between pitches in a Cartesian space" is merely one specific case of a more general idea, one that can embrace different kinds of musical objects (durations, meters, Klangs, timbres, and so on), different (i.e. non-Euclidean) geometries, and different orientational perspectives (interval as action or gesture rather than as simply measurement of distance between things). Along the way, the work recasts set theory, the concepts of transposition and inversion, and notions of musical time in this generalized image. But the work has maintained its relevance and importance as well because of the brilliance and musicality of its author. David had a gift for finding musically significant examples for his sometimes abstract concepts, and a gifted musical imagination that delighted in finding new ways to hear and understand familiar musical passages. While GMIT does not offer the extended musical analyses of his later books, Musical Form and TransformationorStudies in Music with Text, the work is nonetheless rich with smaller analytical gems. To be sure, transformational theory has evolved in the years sinceGMITfirst appeared—the analytical use of Klumpenhouwer networks, the development of neo-Riemannian theory, and the resurgence of spatial methodologies and metaphors in analysis all postdate David's seminal study. But each of these subsequent developments can find its basis in the framework David sets forth in GMIT: Klumpenhouwer networks apply the Generalized Interval System (GIS) concept recursively to create networks of networks; neo-Riemannian theory, which emerged from explorations begun in chapter 8 of GMIT, takes families of contextual transforma-

ix

Foreword to the Oxford Edition

tions to be the formal intervals between the familiar set of harmonic triads or seventh chords; spatial methodologies simply extend the idea of transformational networks to create graphs that embrace all members of a family of objects (pitches, pitch sets, rhythmic durations, and so on) related by certain contextually significant intervals.1 One notable new feature of this edition is an author's addendum (the preface), drawn from a previously unpublished typescript titled "Updating GMIT," which presents, in a sometimes synoptic form, concepts or musical examples David had planned for a future edition of GMIT. The document was likely written in the summer of 1987 and was used as the handout for a talk given at the Eastman School of Music in the fall of that same year. It should not be surprising to those who knew David's incredible industry and the speed with which he could read and suggest revisions to others' work that David would have been drafting plans for a new edition of GMIT so soon after its publication—for David, it was often difficult to stop thinking about a project, or tinkering with its ideas, once begun, and the document clearly represents David's residual energy following the writing of GMIT. The examples explored in the addendum are diverse, although certain themes recur. For one, David seems to have been particularly concerned with examples that involve non-commutative groups of operations, no doubt because such groups often defy our accustomed and familiar intuitions about the way intervals work. For another, David seems to have been interested in finding examples that do not simply involve individual pitch classes (transformations of melodies, of Lagen in triple counterpoint, of ordered hexachords), again because these are less familiar, and often reveal less intuitive aspects of interval. Although the document is perfectly intelligible, some sections of "Updating GMIT" deserve additional comment. 1. The error in figure 8.2 (g minor instead of g# minor) that prompted David's commentary in section I has been corrected in this edition. The first section of David's notes was expanded to become his article "Some Notes on Analyzing Wagner: The Ring and Parsifal" (19th-century Music 16.1, 1992, reprinted in David Lewin, Studies in Music with Text [Oxford University Press, 2006]). 2. David developed and expanded section IV into a pair of unpublished exercises for his math and music course at Harvard University. Exercise 5 (2 pages) directs the student to discover the elements of the Q-X group acting on the augmented triads of sc (014589) and then find transformations of the "rapture of the

x

1. David has written articles on each of these topics subsequent to the publication of GMIT. Klumpenhouwer networks are the topic of two articles: "Klumpenhouwer Networks and Some Isographies that Involve Them," Music Theory Spectrum 12.1 (1990): 83-120, and "A Tutorial on Klumpenhouwer Networks, Using the Chorale in Schoenberg's op. 11, no. 2," Journal of Music Theory 38.1 (1994): 79-101. David's most significant post-GMIT contribution to neo-Riemannian theory is the article "Cohn Functions," Journal of Music Theory 40.2 (1996): 181-216. Two of David's contributions to graphical methods of analysis are "The D-major Fugue Subject from WTCII: Spatial Saturation?" Music Theory Online 4.4 (1998), and "Notes on the Opening of the F# Minor Fugue from WTC I," Journal of Music Theory 42.2 (1998): 235-239.

Foreword to the Oxford Edition

strife" figure under Q4, Q8, and X5 in Schoenberg's Ode to Napoleon (as in David's example 2 from the addendum). An optional part of that exercise encourages students to explore transformations of characteristic tetrachords in Schoenberg's Ode using the members of the same Q-X group. Exercise 8 (3 pages) explores the simple transitivity of the Q-X group and has the student find the (interval-preserving) elements of the commuting group, {T0, T4, T8, I,, I5, I9). David's article "Generalized Interval Systems for Babbitt's Lists, and for Schoenberg's String Trio" (Music Theory Spectrum 17.1 [1995]: 81-118), in particular "Part 5: Background on Non-Commutative GISs," explores the relationship between non-commutative GISs and their commuting groups. 3. David similarly developed and extended section V into an exercise for his math and music course (exercise 9,4 pages). The Daniel Harrison article to which David refers was published as "Some Group Properties of Triple Counterpoint and Their Influence on Compositions by J. S. Bach" (Journal of Music Theory 32.1 [1988]: 23-49). David inserted a manuscript page into the "Updating GMIT" typescript that presents a TPERM and VPERM analysis of Bach's c-minor fugue from the Well-Tempered Clavier, Book I. The manuscript notes that the diagram is modeled after Schenker's "Table of Voices" from "Das Organische der Fuge" in Das Meisterwerk in derMusik, Band II, p. 59, and further observes that the Lagen symbol "'A' can mean 'Subject,' 'B' can mean 'Countersubject' and 'C' can mean 'any third part of roughly characteristic rhythm'" (emphasis Lewin's), suggesting that the methodology is not bound to works in strict triple counterpoint. David's diagram, however, has not been incorporated into the author's addendum of this volume because David wrote no accompanying text for it—creating new text would have adversely disrupted David's prose in the rest of the section. David, however, did use the c-minor fugue analysis as part of exercise 9 in his math and music course, which I present below for interested readers to explore if they wish (terminology has been adapted to conform to the text of "Updating GMIT"): PART I OF EXERCISE 9: (a) Complete the partially-filled diagram below, which pertains to the c-minor fugue in Book I: Meas.

Stufe

1

i

11 15 20 26.5

m V

i i

Lage

TPERM interval

VPERM interval

(b) Discuss features of the construction which you find revealed by the double intervallic analysis. For instance, does the use of 3-cycles bring out any aspect of the structure? Do the TPERM and VPERM

xi

Foreword to the Oxford Edition

analyses coincide as they did [in the A-major Prelude]? What aspects of the piece are bound together by repetition of TPERM intervals? By repetition of VPERM intervals? 4. Section VI considers the GIS structure of a family of 12-tone-row transformations that David first explored in his article "On Certain Techniques of ReOrdering in Serial Music" (Journal of Music Theory 10.2 [1966]: 276-287). David refers in the section to "an excellent work, as yet unpublished" by Andrew Mead. That work was published in two parts as "Some Implications of the PitchClass/Order-Number Isomorphism Inherent in the Twelve-Tone System: Part One" (Perspectives of New Music 26.2 [1988]: 96-163) and, more pertinent to Lewin's addendum, "Some Implications of the Pitch-Class/Order-Number Isomorphism Inherent in the Twelve-Tone System Part Two: The Mallalieu Complex: Its Extensions and Related Rows" (Perspectives of New Music 27.1[1989]: 180-233). David, of course, never created a second edition ofGMIT, an undertaking that, he wrote, would have involved "[fixing] a lot of errata & corrigenda; some major rewrites here and there; a reasonable amount of bibliographic updating."2 This edition ofGMIT, while retaining the text of the original, does incorporate the corrections indicated by David's errata list. Moreover, while it does not attempt to identify or alter passages that David felt needed rewriting, the articles cited in this foreword give a picture of David's evolving ideas about transformational theory. And while David may have wanted a new edition of GMIT, rather than a second printing, he was also eager to make GMIT available to students and scholars. In these respects, this Oxford edition fulfills David's wishes—that his ideas be available to all who seek them, so that they may grow, evolve and multiply. 2. 1995 e-mail correspondence, recipient unknown.

xii

Preface

I. The following figures redo those of figure 8.2 on p. 179. Music examples la and b present scores of the relevant passages.

L = LEITTONWECHSEL; +- = MAJOR-MINOR; S = "BECOMES SUBDOMINANTOF".

EXAMPLE la

xiii

Preface b) Modulating section of Valhalla, Rheingold II,5ff.

EXAMPLE Ib

The analysis is better than that in the book. It brings out a clear isography between the passages. Figure 8.2a in the book is not a well-formed "graph" by the later definition. (SUBM is not = LT SUED on major as well as minor Klangs: (C,+) SUBM = (e,-) but (C,+)LT SUED = (e,-)SUBD = (b,-).) The symbol "(G,-)" on figure 8.2a is a misprint for (Gt,-).1 The discussion of section 8.1.2, pages 179-180, still applies: a group that contains L, S, and +— operations on Klangs will not be simply transitive in equal temperament. (For instance, (C,+)SSSS =(E,+), but (C,+)L +- also = (E,+).) xiv

1. See item 1 in the foreword, p. x.

Preface

Later in the Ring, Wagner develops the relationship of Valhalla and Tarnhelm themes very ambitiously. Figures c) through f) below analyze a transformation that occurs at the climax of Walkure 11,2: Woton, coming to realize the full implications of Valhallagate, ironically gives his blessing to Hagen ("So nimm meinen Segen, Niblungen Sohn!"). Music examples Ic through le are coordinated with the figures.

EXAMPLE Ic-e Figure Ic) shows the Valhalla Kopf put into At major and 4/4 meter, with the original harmonization. Figure d) is the +- transform of c). Figure e) transforms d) so that the subdominant inflection of c)—d) is applied not to the tonic but to the Leittonwechsel of the tonic; also the inflected Klangs change mode as they go, via + — . Music example le is essentially the upper part of the accompaniment for Wotan's pronouncement (there is more beneath!). The Tarnhelm network infects the diatonic aspect of Valhalla here. Figure f) brings that out by rewriting e) in a format that suggests a). In the Waltraute scene of Gotterdammerung, the idea gets even more overloaded ... rather like the picture of Dorian Grey.

JCV

Preface

II. An interesting transformation network is used by Lora L. Gingerich, ". . . Melodic Motivic Analysis in ... Charles Ives," MTS 8 (1986), 75-93. The network appears as her example 24, page 90.

III. Let s be the twelve-tone row of Schonberg's Fourth Quartet. Let S be the family comprising the 48 forms of s. Let TTO be the group of forty-eight twelve-tone operations. TTO is simply transitive on S (given forms s and t, there exists a unique member OP of TTO such that OP(s) = t.) It follows that we can develop a CIS structure for S in such wise that the members of TTO are exactly the formal transposition operations for the GIS (GMIT1A. 1, pp. 157-58). The standard practice, in which forms of the row are labeled by their TTO-intervals from a "tonic" referential row-form—as "RI3," "17," etc.—instances the LABELing practice discussed in chapter 3 of GMIT. If s is any one of the 48 forms, then there exists a unique inverted form of s (in this case) which shares the same three tetrachordal segments with s. Define a transformation TETRA on S: given a sample s, TETRA transforms s into this inverted tetrachordal associate. For instance: TETRA (Ob78 312a 6549) = 780b 4659 123a; TETRA (5012 6a9b 4378) = 6ba9 5120 7843. The transformation TETRA is a formal interval-preserving operation of the GIS under discussion (GMIT 3.4.6, p. 48). Similar operations for this particular row, like TRI and HEXA, are also interval-preserving operations. In his dissertation on Moses undAron (Yale, 1983), Michael Cherlin argues that transformations of this sort, engaging the forms of the Moses row, are highly constructive features of Schonberg's compositional method in the opera.

xvi

IV. Appendix B in GMIT outlines two possible non-commutative GIS structures for the octatonic set. It develops two simply transitive groups of operations on that set; either may be taken as the group of formal transpositions for a GIS; the other then becomes the group of formal interval-preserving operations. A similar situation obtains for set-class 6-20. Taking S as [modeled by] the six numbers 0,1,4,5,8, and 9 mod 12, two simply transitive groups of operations may be defined on S as follows. The group Gl comprises the operations R0= identity, R4 = pc transposition by 4, R8 = pc transposition by 8, Jl = pc inversion with index number 1, J5 = pc inversion with index number 5, and J9 = pc inversion with index number 9. (In the GIS determined by this simply transitive group, all the six operations are formal "transpositions" for that GIS.) The group G2 comprises the six operations RO, Q4, Q8, XI, X5, and X9, defined as follows:

Preface

RO = identity operation. Q4 takes pcs 0,4, and 8 to pcs 4, 8, and 0 resp.; takes pcs 1,5, and 9 topes 9,1, and 5 resp. Q8 takes pcs 0,4, and 8 to pcs 8,0, and 4 resp.; takes pcs 1,5, and 9 to pcs 5, 9, and 1 resp. The Qs are "queer" operations, as opposed to the "rotations" R. XI exchanges each pc of 6-20 with that pc which lies ic 7 away. Thus XI maps 0 to 1,1 to 0,4 to 5, 5 to 4, 8 to 9, and 9 to 8. X5 exchanges each pc with the pc that lies ic 5 away. X9 exchanges each pc with the pc that lies ic 3 away. Both the groups Gl and G2 are simply transitive on S. Either group may be taken as the group of formal transpositions for a formal GIS involving S; the other group thereupon becomes the group of interval-preserving transformations. The pertinence of G2 is manifest in Schonberg's Ode to Napoleon. Music example 2 shows some prominent thematic motives of the piece, all interrelated by operations of G2. Example 2a projects a six-note series that is mapped into example 2b by Q8. 2b' retrogrades 2b; 2c shows the series of 2b' in action. Example 2d is the Q4-transform of series 2a; 2e shows series 2d in action. Example 2f is the

EXAMPLE 2

xvii

Preface

X5-transform of series 2a; 2g is a combinatorial inversion of series 2f, and 2h shows series 2g in action. The motives of 2a, 2c, 2e, and 2h appear frequently in the work, at a variety of pitch levels, retrograded (= inverted), etc.; the various sixnote series generate characteristic tetrachordal segments that are ubiquitous motivic germs in the music. (These tetrachords are all G2-forms of one another).

V. Daniel Harrison, in a recent study of triple counterpoint, has made interesting analytic use of GIS structures.2 I adapt his procedures to my terminology here. Let us suppose three tunes, A, B, and C, that work in triple counterpoint. Let us suppose three voices, 1, 2, and 3, in which the tunes can appear. We can consider the six various possible dispositions of the three tunes in the three voices; let us call each such disposition a "Lage." We can model each Lage by a three-element series: thus the series

xviii

2. See item 3 in the foreword, p. xi.

Preface

family is LAGEN; the other group thereupon becomes the group of formal intervalpreserving operations for the GIS. This situation is as in the last paragraph of appendix B, GMIT. Harrison analyzes the D-major 3-part invention, observing most of the following structure. "A" is the lead-off theme in the rh; "B" is the counterpoint that runs along in sixteenths; "C" is the counterpoint which steps down in leisurely suspensions. Meas.

Stufe

Lage

TPERM interval

VPERM interval

(ACB)

(123)

(ACB)

(123)

3.5

V

6

I

10

vi

19

IV

21.5

I

23.5

I

(132) (132)

The columns headed "TPERM interval" and "VPERM interval" are read as follows: from Lage

xix

Preface Meas.

Stufe

Lage

1

I

4

V

8.5

I

12

vi

16.5

I

19

I

TPERM interval

VPERM interval

(ABC)

(132)

(ABC)

(132)

(ABC)

(132)

(ABC)

(132)

(AB)

(13)

This analysis is useful to contrast to the D-major invention. Here only four Lagen are used. The idea seems to be that the final tonic Lage has a special function here: it breaks the otherwise incessant chain of (ABC) or (132) intervals. Harrison's analysis of the f-minor invention provides still a different idea, for laying out various Lagen. The whole enterprise smells of Marpurg; perhaps the way in which he formulated "Rameau's" (i.e. his) theories of chord inversion might bear similar updating, perhaps even in a somewhat isomorphic vein. VI. Let us consider the family SPECIAL of 12-tone rows whose order-rotation beginning on order-number 4 is the same as their T4-transpose. An example of a SPECIAL row is Ob56439a8712: starting the row at order-number 4 and proceeding therefrom, we derive 439a8712 [and around the end to] Ob56; this order-rotation is the same as T4 of the original row. To fix a notation, we consider each SPECIAL row as a function s mapping the order number ord [mod 12] into the pc number s(ord) [mod 12]. The SPECIAL row of the above paragraph is thus conceived as a function s: s(0) = 0, s(/) = b, s(2) = 5, ...,s(a) = l,s(fc) = 2. Using this notation, we can write out the algebraic property that characterizes SPECIAL rows: SPECIAL PROPERTY: for all ord, s(ord + 4 = s(ord) + 4. [all addition mod 12]

xx

Of interest to us here is the fact that the family of SPECIAL rows admits a simply transitive group-of-operations G in a natural way. Therefore, according to the discussion of GMIT, the family of SPECIAL rows has a natural GIS structure, a structure in which the operations of G play the role of formal transpositions. What follows is a semi-formal development of the group G, and a semi-formal indication that G is simply transitive on SPECIAL.

Preface

To begin with we consider certain operations ADD{j,m}, where j is some multiple of 4 mod 12 and m is some multiple of 3 mod 12; i.e. j = 0, 4, or 8, and m = 0,3,6, or 9. The operation ADD{j,m}, when applied to the SPECIAL row s, adds the pc interval j to the mm, the (m+4)th, and the (m+S)th notes of s. For example, let us take the SPECIAL row of the first paragraph above and apply the operation A{8,3} to it, adding the pc interval 8 to its 3rd, 7th, and b\h notes: o r dnumbers: 0 1 2 3 4 5 6 7 8 9 a b pc numbers of s: O b 5 6 4 3 9 a 8 7 1 2 We add 8 to 3rd, 7th, and Mi, +8 +8 +8 obtaining pc numbers of ADD{8,3}(s): O b 5 2 4 3 9 6 8 7 1 a In the example, we note that ADD {8,3}(s) is still a row. That is because s is SPECIAL: since s(ord + 4) = s(ord) + 4, it follows that the pc numbers s(3), s(7), and s(&)—that is the 3rd, 7th, and 6th notes of s—form an augmented triad. In the example above the augmented triad comprises the pc numbers 6, a, and 2. When we add the interval j = 8 to each of these pc numbers, we simply permute the members of that augmented triad among themselves, without disturbing the other pcs in the other order-positions of the row. Thus, in the above example,

order-positions contain pcs of the row s; when 8 is added to each of those pc numbers, the same order-positions then contain pcs of the row ADD{ 8,3}(s), while the other pcs of s "carry on down" to ADD{8,3}(s), unchanged in their order-positions.

3, 6,

7, a,

b 2

2,

6,

a

This observation can be made rigorous and general, to show that each operation ADD{j,m}, when applied to any SPECIAL row s, yields a row. Furthermore, it can be proved what is intuitively obvious: the new row ADD{j,m}(s) will itself be SPECIAL. The following formulas are easily verified, for j and k any multiples of 4 mod 12, and for m and n any multiples of 3 mod 12: FORMULA 1: ADD{j,m} ADD{k,m} = ADD{j+k,m} FORMULA 2: ADD{j,m} ADD{k,n} = ADD{k,n}ADD{j,m} ADD{0,m} is the identity operation, for each m: it leaves [the pcs of] any sample SPECIAL row unchanged. It follows, via formulas 1 and 2, that the collection of all operations that can be written in form ADD{jO,0} ADD{j3,3} ADD{j6,<5} ADD{j9,9} is a group of operations. We will call this group "ADDINGS." The group is commutative. It has 3-times-3-times-3-times-3 members, ie 81 members.

xxi

Preface

Now we shall develop another group of operations on SPECIAL rows, a group we shall call "PERM." A PERM operation X{p} is defined by any permutation p that acts upon the four symbols 0,3, (5, and 9. Here the permutation p is to be considered as any 1-to-l function that maps the family of four symbols onto itself. The PERM operation X{p} is determined by the PERM DEFINITION: X{p}(s)(m +/) = s(p(m) + j ) where m symbolizes a multiple of 3 mod 12 and; symbolizes a multiple of 4 mod 12. Fix m = 0 in the formula of the definition, and let; run through the values 0, 4, and & The formula tells us that the Oth, 4th, and 8th notes of the X{p}(s) will be respectively the p(0)th, (p(0)+4)th, and (p(0)+5)th notes of s. Similarly [for m = 3] the 3rd, 7th, and £th notes of X{p}(s) will be respectively the p(3)rd, (p(3)+4)th, and (p(3)+S)th notes of s. And so forth [for m = 6 and m = 9]. For an example, fix p to be the permutation p(0) = 3, p(3) = 0, p(6) = 6, p(9) = 9. Then, according to the work we have just gone through, X{p}(s) will have in its Oth, 4th, and 5th order-positions the 3rd, 7th, and bth notes of s respectively, while X{p}(s) will have in its 3rd, 7th, and bth order-positions the Oth, 4th, and 5th notes of s respectively; otherwise X{p}(s) will maintain the [other] notes of s in their respective order positions. The diagram below shows this X{p} applied to the specimen special row used before. o r dnumbers: 0 1 2 3 4 5 6 7 8 9 a b pc numbers of s: 0 4 8 b 5 39 71 6 a 2 pc numbers of X{p}(s): 6 a 2 b 5 39 71 0 4 8 For any SPECIAL row s, and any permutation p, X{p}(s) is a SPECIAL row. If p and q are permutations, then we have FORMULA 3: X{p} X{q) = X{qp}. The PERM operations on SPECIAL rows form a group (anti)-isomorphic to the group of permutations on the four symbols 0,3,6,9. PERM therefore has 4! = 24 members. The group is not commutative. The following formula can be proved: FORMULA 4: ADD{j,m} X{p) = X{p} ADD{j,p(m)}]. In general, therefore, members of PERM do not commute with members of ADDINGS. However, formula 4 tells us that the collection of all operations which can be expressed as some-ADDING-following-some-PERM is a closed family of xxii

Preface

EXAMPLE 3

xxiii

Preface

operations: following one such by another such will generate a third such. It follows that this collection of operations is a group of operations. It is our desired group G. A specimen member of G can be written in CANONICAL FORM: ADD{jO,0} ADD{j3,3} ADD{j6,<5} ADD{J9,P} X{p}. The chromatic scale is one SPECIAL row. It is straightforward, if tedious, to show that given any SPECIAL row s, there is a unique member of our group G which transforms s into the chromatic scale. (Set kO = s(0), k3 = s(3), k6 = s(6), k9 = s(9); since s is SPECIAL, each of the k's must lie within a different augmented triad; apply four appropriate ADDs to obtain a new s' in which the set of k'-values is 0,3,6, and 9; permute the row s' into the chromatic scale. Etc. etc.) It follows that the group G is simply transitive on SPECIAL rows: given any two SPECIAL rows s and t, there is a unique member of G, in the canonical form above, which transformes s to t. (Transform s into the chromatic scale; then transform the chromatic scale into t.) Thus the family of SPECIAL rows has a natural GIS structure, as discussed above. The group G has cardinality 81-times-24 = 1944; that then is also the number of SPECIAL rows. SPECIAL rows become more interesting when one notes their relation to "semi-Mallalieu" rows. Andrew Mead, in excellent work as yet unpublished, has investigated semi-Mallalieu rows exhaustively; some interesting insight can be shed on his work by placing it in a GIS setting.3 Pertaining to our SPECIAL rows are those semi-Mallalieu rows whose every-third-note transform is identical with their T4-transposition. Every-ninth-note of such a row will then be its T8-transposition. (Every-ninth-note = retrograde-of-every-fourth-note.) Such a row, for example, is a premise of my piano piece Just a Minute, Roger [PNM 16.2 (SS 1978), 143-45]: Ob45732681a9. Right at the opening of the piece, one hears quite clearly that every-third-note of this row is its T4-transpose: in meas. 1-4 (music example 3a), the total texture is governed by the row, while the right hand picks out every-third-note, thereby projecting T4-of-the-row. Later on, in meas. 30-35 (example 3b), the total texture is governed by the T4-form of the row, while the right hand picks out every-fourth-note-of-T4, thereby projecting the retrograde of every-ninth-note-of-T4 = the retrograde of T8(T4) = the retrograde of the original row. We shall focus in on these rows for the nonce, calling them SEMI-MALLALIEU. The point is, that the family of SEMI-MALLALIEU rows and the family of SPECIAL rows, along with their characteristic properties, are mathematically equivalent in structure under a transformation that makes each row in one family correspond

xxiv

3. See item 4 in the foreword, p. xii.

Preface

uniquely to a row in the other. I discuss that transformation in ITO 2.7 (October 1976), 8. The upshot of this is that the SEMI-MALLALIEU rows also form a natural GIS, whose formal transposition operations are the members of a natural simply transitive group G' that corresponds to the group G we have just explored. Mead's procedures enable one to find the equivalent of our G or G' for any operation OP on order numbers that can be written as four permutation-cycles of order 3 on the order numbers. OP generalizes "rotation-by-4" in the case of SPECIAL rows, or "every-third-note" in the case of SEMI-MALLALIEU rows.

VII. The injection function, discussed in chapter 6 of GMIT, can be generalized even farther. Let S and S' be any two families of objects—we allow here for the possibility that S' may be a different family from S. Let X be a (finite sub)set of S; let Y' be a (finite sub)set of S'; let f be any function from S into S'. Then the injection number into Y' for f, denoted INJ(X,Y')(f), is the number of elements s in X such that f(s) is a member of Y'. We can also develop a twin concept, not developed in GMIT: the surjection number of X into Y' for f, denoted SURJ (X,Y')(f), is the number of elements s' in Y' such that s' = f(s) for some member s of X. If f is not oneto-one or onto, SURJ(X,Y')(f) may be a very different number from INJ(X,Y')(f). EXAMPLE: Take S to be the family of numbers {0,1,2,4,6,8,10,14,16,18,19,20,22,23,24,26,27,28}. Take S' to be the family of pitches {A3,CH,D4,E4,F4,G4,A4,BI4,D5}. Take the function f to map S into S' according to the following table: s= 0 1 2 4 6 8 10 14 16 18 19 20 22 23 24 26 27 28 f(s)= D4 E4 F4 E4 D4 A4 D5 A4 BW G4 E4 A4 F4 D4 G4 E4 C#4 A3

The function f is onto but not one-to-one, f models certain aspects of the theme from Bach's d-minor Concerto. Take X to be the set of all numbers in S divisible by 4; thus X = {0,4,8,16,20,24,28}. Take Y' to be the triad {D4,F4,A4} within S'.

XXV

Preface

We have f(0) = D4, f(8) = A4, and f(20) = A4. Otherwise f(s) is not a member of the set Y' when s is a member of the set X. The three members 0,8, and 20 of X are mapped by f into members of Y'. Hence INJ(X,Y')(f) = 3. The two members D4 and A4 of Y' are attacked, during this passage, at time-points that are multiples of 4. Hence SURJ(X,Y')(f) = 2.

xxvi

A cknowledgmen ts

The time and leisure I needed to do this work were afforded by a Senior Faculty Fellowship from Yale University and by a Guggenheim Fellowship. The Guggenheim Foundation also provided a subvention toward publication. The excerpt from the score of Elliott Carter's String Quartet No, 1 is used by permission of Associated Music Publishers, New York. Material from Arnold Schoenberg's songs, Opus 15, from his Six Little Piano Pieces, Opus 19, and from Pierrot Lunaire, Opus 21, is used by permission of Belmont Music Publishers, Los Angeles, California 90049. Arnold Schoenberg's Phantasiefor Violin with Piano Accompaniment, Opus 47, is copyright 1952 by Henmar Press Inc. It has been used by permission of C. F. Peters Corporation. The excerpt from the score of Anton Webern's Four Pieces for Violin and Piano, Opus 7, and material from the third movement of his Variations for Piano, Opus 27, are used with these permissions: "Anton Webern—Four Piecesfor Violin and Piano, Op. 7. Copyright 1922 by Universal Edition. Copyright renewed 1950. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. agent for Universal Edition" and "Anton Webern—Variations for Piano, Op. 27. Copyright 1937 by Universal Edition. Copyright Renewed 1965. All Rights Reserved. Used by permission of European American Music Distributors Corporation, sole U.S. agent for Universal Edition." Analytic sketches for works by Bartok and Prokofieff appear with the following permissions: "Syncopation # 133 from Mikrokosomos (volumes 1-6) Bela Bartok © copyright 1940 by Hawkes & Son (London) Ltd.; Renewed 1967. Reprinted by permission of Boosey and Hawkes, Inc." and "Prokofieff: Melody # 1 from Melodies Op. 35. Reprinted by permission of Boosey & Hawkes, Inc. Copyright Owner."

xxvii

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Introduction

The following overview of the book will provide a good point of departure. Chapter 1 is purely mathematical; it presents terminology and notation that will be needed later, along with a few important theorems. I am not happy to begin a book about music with a mathematical essay. On the other hand, I do feel that it is helpful for the reader to have this material collated and isolated from the rest of the book. Chapter 1 can be used for quick reference where it stands, and the material obtrudes only minimally into musical discussions later on. Readers who find themselves put off or fatigued in the middle of this chapter are urged to move on into the rest of the book; they can return to chapter 1 later, when later applications of the material make the reference back seem natural or desirable. Chapter 2 takes as its point of departure the general situation portrayed schematically by figure 0.1.

FIGURE 0.1 The figure shows two points s and t in a symbolic musical space. The arrow marked i symbolizes a characteristic directed measurement, distance, or motion from s to t. We intuit such situations in many musical spaces, and we are used to calling i "the interval from s to t" when the symbolic points are

xxix

Introduction

xxx

pitches or pitch classes. Chapter 2 begins by running through twelve examples of musical spaces for which we have the intuition of figure 0.1. Six involve pitches or pitch classes in melodic or harmonic relations; six involve aspects of measured rhythm. The general intuition at hand is then made formal by a mathematical model which I call a Generalized Interval System, GIS for short. A few basic formal properties of the model are explored. Then the twelve examples are reviewed to see how each (with one exception) instances the generalized structure. Chapter 3 concerns itself with further formal properties of the GIS model. In that model, the points of the space may be labeled by their intervals from one referential point; this has advantages and disadvantages. New GIS structures may be constructed from old in various ways. A passage from Webern is examined in connection with a combined pitch-and-rhythm GIS constructed in one such way. Generalized analogs of transposition and inversion operations are explored. So are "interval-preserving operations"; these coincide with transpositions in some GIS models but not in others, specifically not in GISs that are "non-commutative." The bulk of chapter 4 explores one non-commutative GIS of musical interest. The elements of the system are formal time-spans. Extended discussion of a passage from Carter's First Quartet demonstrates the pertinence of this GIS to exploring music in which there are functional measured relations among time spans, but no one overriding time span that acts as a unit to measure all others. After that, chapter 4 presents two examples of timbral GISs, and ends with a methodological note on the relations of music theory, perception, and the intuitions of a listener. Some motivic work by Chopin is considered in this connection. Chapter 5 begins a study of generalized set theory, that is, the interrelationships among finite sets of objects in musical spaces. The first construction studied is the Interval Function between sets X and Y; this function assigns to each interval i in a GIS the number of ways i can be spanned between a member of X and a member of Y. Then the Embedding Number of X in Y is studied; this is the number of distinct forms of X that are subsets of Y. To study that number, we have to establish what we mean by a "form" of the set X, a notion that involves stipulating a Canonical Group of operations. Both the Interval Function and the Embedding Number generalize Forte's Interval Vector. Passages from Webern, Chopin, and Brahms illustrate applications of the constructs. Chapter 6 continues the study of set theory, generalizing the work of chapter 5 even farther. The basic construction is now the Injection Function: Given a space S, finite subsets X and Y of S, and a transformation f mapping S into itself, INJ(X, Y) (f) counts how many members of X are mapped by f into members of Y. This number is meaningful even when S does not have a GIS structure, and even when the transformation f js not so well behaved as are

Introduction

transpositions, inversions, and the like. Passages from Schoenberg and from Babbitt are studied by way of illustration. Instead of starting with a GIS and deriving certain characteristic transformations therefrom, it is possible to start with a family of characteristic transformations on a musical space and derive a GIS structure therefrom. That is, instead of regarding the i-arrow on figure 0.1 as a measurement of extension between points s and t observed passively "out there" in a Cartesian res extensa, one can regard the situation actively, like a singer, player, or composer, thinking: "I am at s; what characteristic transformation do I perform in order to arrive at t?" Chapter 7 explores this conceptual interrelation between interval-as-extension and transposition-as-characteristicmotion-through-space. After developing the mathematics that shows a logical equivalence between GIS structures and certain structures of transformations on spaces, the work proceeds by example. Passages from Schoenberg, Wagner, Brahms, and Beethoven indicate how suggestive it can be to consider networks of "intervals" and networks of "transpositions" (modulations, and so forth) as various aspects of the same basic phenomenon. The morphology of such networks can be carried over to that of networks involving other sorts of transformations. Chapter 8 studies networks involving transformations of Klangs in the sense of Riemann, networks involving serial transformations of various sorts, and networks involving inversional transformations. The Beethoven example from chapter 7 is reconsidered, and there are further examples from Wagner, Webern, and Bach. Chapter 9 develops the formalities of transformation networks in a rigorous way. The structure of a network allows us to assign a formal "input" function to some things and a formal "output" function to other things; these functions seem of considerable musical interest in some cases. The networks have intrinsic rhythmic properties which can also be studied formally. Network structure can accommodate hierarchic levels in a quasi-Schenkerian setting, as an example shows. Chapter 10 applies the network concept in a variety of ways to passages from Mozart, Bartok, Prokofieff, and Debussy.

Note on Musical Terminology All references to specific pitches in this book will be made according to the notation suggested by the Acoustical Society of America: The pitch class is symbolized by an upper-case letter and its specific octave placement by a number following the letter. An octave number refers to pitches from a given C through the B a major seventh above it. Cello C is C2, viola C is C3, middle C is C4, and so on. Any B# gets the same octave number as the B just below it; thus B#3 is enharmonically C4. Likewise, any Cb gets the same octave number as the C just above it; thus Q?4 is enharmonically B3.

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1

Mathematical

Preliminaries

A mathematician would begin by saying, "Let S be a set." Unfortunately, music theory today has expropriated the word "set" to denote special musictheoretical things in a few special contexts. So I shall avoid the word here. Instead I shall speak of a "family" or a "collection" of objects or members. When I do so, I mean just what mathematicians mean by a "set." For present purposes, it will be safe to leave the sense of that concept to the reader's intuition. 1.1 DEFINITION: Let S and S' be families of objects. The Cartesian product S x S' is the family of all ordered pairs (s, s') such that s is a member of S and s' is a member of S'. 1.2.1 DEFINITION: A function or mapping from S into S' is a subfamily f of S x S' which has this property: Given any s in S, there is exactly one pair (s, s') within the family f which has the given s as the first entry of the pair. We say that s', in this situation, is the value of the function f for the argument s; we shall write f (s) = s'. If we think of fas a table, listing members of S (arguments) in a column on the left and corresponding members of S' (values) in a column on the right, then the defining property for functionhood stipulates that each member of S appear once and only once in the left-hand column. (Some members of S' may appear more than once in the right-hand column. Some members of S' may not appear at all in the right-hand column.)

1

1.2.2

Mathematical Preliminaries

1.2.2 DEFINITION: Given families S and S', we shall say that the functions f and g from S into S' are the same, writing f = g, if f and g are the same subsets of S x S', that is if they produce the same table. This special definition of functional equality is worth stressing. We shall soon see why. 1.2.3 DEFINITION: Let f be a function from S into S', and let f' be a function from S' into S". Then the composition function f'f is defined from S into S" as follows: Given an argument s in S, the value (f'f)(s) is f'(f(s)). 1.2.4 Let me draw special attention to the orthographic convention whereby f' appears to the left of fin the notation for the composition function f'f. That convention follows logically from another orthographic convention, the convention of writing the function name to the left of the argument in the expression "f (s)." The reader is no doubt used to this convention. One can read "f (s)" as "the resulting value, when function f is applied to argument s." Then "f'f(s)" is "the result when f is applied to the result of applying f to s." These conventions will be called left (functional) orthography. Right functional orthography is preferred by some mathematicians for all contexts and by most mathematicians for some contexts. In right orthography, one writes "sf" or "(s)f" for "the operand s, transformed by the function f." This value is what was written "f(s)" in left orthography. The composition function which we called "f'f" in left orthography is called "ff " in right orthography, so as to be consistent: "(s)ff'" in right orthography is "s-transformed-by-f, all transformed by f'." This is what was notated "f f (s)" in left orthography. In the following work we shall use left orthography almost exclusively. We shall use right orthography only once, when its intuitive pertinence seems overwhelming. At that point in the text, the reader will be reminded of this discussion. Right orthography would abstractly be more suitable for our eventual purposes, but the reader's presumed familiarity with left orthography seemed decisive to me in making my choice.

2

1.2.5 Suppose that f t and f 2 are functions from S to S'; suppose that f{ and f'2 are functions from S' to S"; suppose that f" is a function from S to S". We can consider the truth or falsity of functional equations like f^ = f", fif t = f 2 f 2 , and so on. Our discussion of "functional equality" in 1.2.2 tells us how to understand these equations, in evaluating their truth or falsity. The first equation above asserts, "for any sample s, the result of applying f[ to ^(s) is the same as the result of applying f" to the given s." The second equation above asserts, "for any sample s, applying f{ to f^ (s) yields the same result as applying f 2 to f2(s)."

Mathematical Preliminaries

1.3.1

For an example, let us take S, S', and S" all to be the family of positive integers. Let f^s) = s + 3, f{(s) = 2s, f2(s) = 2s, and f^s) = s + 6. The four specified functions satisfy the functional equation f'1f1= f^. That is, given any integer s, if we compute fjfi(s), multiplying by two the result of adding 3 to s, we obtain the same net result as we do when we compute f^Cs), adding 6 to the result of multiplying s by two. For another example, let us take S, S', and S" all to be the family of the twelve pitch-classes. Let f(s) = s transposed by 2, f(s) = s inverted with respect to the pitch class C, and f"(s) = s inverted with respect to the pitch class B. The three specified functions satisfy the functional equation f'f = f". That is, given any pitch class s, if we compute f T(s), inverting about C the result of transposing s by 2, we obtain the same net result as we do when we compute f"(s), inverting the given s about B. 1.2.6.1 DEFINITION: The function f from S into S' is onto S' if every member of S' is the value of some argument. (Every member of S' appears at least once in the right-hand column of the function table.) 1.2.6.2 DEFINITION: The function f from S into S' is 1-to-l if no two distinct arguments share the same value. (No member of S' appears more than once in the right-hand column of the function table.) 1.2.6.3 DEFINITION: Let f be a 1-to-l function from S onto S'. Then f"1, the inverse function off, is defined as the family of pairs (s', s) within S' x S such that (s, s') is a member of f. 1.2.6.4 THEOREMS: Given the situation as in 1.2.6.3 above, then f ~* is indeed a function in the sense of 1.2.1. f - 1 is in fact a 1-to-l function from S' onto S. The inverse function off" 1 is, of course, f. The theorems are stated without proof. 1.2.6.5 THEOREM: Let f and f' be functions from S into S' and from S' into S respectively. Suppose that the functions satisfy the two conditions (A) and (B) following. (A): for every s in S, f'f(s) = s. (B): for every s' in S', ff'(s') = s'. Then f and f' are both 1 -to-1; they are respectively onto S' and onto S; and they are inverse functions, each of the other. The theorem is given without proof. 1.3.1 DEFINITION: A function from a family S into S itself will be called a transformation on S. If the function is 1-to-l and onto, it will be called an operation on S.

3

1.3.2

Mathematical Preliminaries

1.3.2 DEFINITION: Given a family S, a collection F of transformations on S is called closed if, given any members f and g of F, the composition fg is a member of F. A closed collection of transformations on S will also be called a semigroup of transformations on S. 1.3.3.1 DEFINITION: The identity operation on a family S is that operation 1 on S which assigns the value 1 (s) = s to any argument s. 1.3.3.2 THEOREM: For any transformation f on S, the functional equations If = f and fl = fare true (in the sense of 1.2.5 above). 1.3.3.3 THEOREM: A transformation f on S is an operation (i.e., 1-to-l and onto) if and only if there exists a transformation f on S satisfying the functional equations f T = 1; ff' = 1. If this be the case then f' is the inverse operation of f. The theorem follows from the various matters studied over section 1.2.6. 1.3.4 DEFINITION: By a group of operations on S we shall mean a family (i.e. collection) G of transformations on S which satisfies conditions (A) and (B) following. (A): G is a closed family, a semigroup of transformations in the sense of 1.3.2. (B): Given any member f of G, there exists a member f of G satisfying f'f = ff = 1. Condition (B) guarantees that the members of G are indeed operations, via 1.3.3.3. (B) also guarantees that G contains the inverse operation for each of its member operations. (A) and (B) together imply that G contains the identity operation 1, provided that G contains any members. Whether we call G a "collection" or a "family" is immaterial; for us the terms are synonymous with each other as they also are with the terms "ensemble" and "set-in-themathematical-sense."

4

1.3.5 The work of section 1.3 so far has explored certain algebraic behavior characteristic of transformations on S. The transformations compose one with another, f with g to form the transformation fg. There is an identity transformation 1, which composes left or right with any f to yield f itself: If = fl = f. Certain transformations, the operations, have inverses; if f is such then f"1 is characterized by the algebraic relations f -1 f = ff" 1 = 1. These algebraic features of the situation are abstracted and generalized by the study of "abstract" semigroups and groups, a study we shall shortly commence. Before we do so, we should note one more aspect of transformation algebra which the abstract study will generalize. This is the associativity

Mathematical Preliminaries

1.5.1

of transformational composition. That is, the composition of transformations obeys the Associative Law f(gh) = (fg)h: Given any sample s, the result of applying f to the (gh)-transform of s is the same as the result of applying (fg) to the h-transform of the given s. 1.4 Now we begin the abstract study. We fix a family (i.e. collection) X of abstract objects x, y, z,..., and develop abstract algebraic systems that model the behavior of transformational algebra. First we must specify how the objects of X are to "compose" one with another. 1.4.1 DEFINITION: A binary composition on X is a function BIN that maps X x X into X. We write BIN(x, y) for the value of BIN on the pair (x, y). 1.4.2 DEFINITION: A binary composition on X is associative if BIN(x, BIN(y, z)) = BIN(BIN(x, y), z) for all x, y, and z. A familiar non-associative binary composition on the natural numbers is exponentiation: BIN(x, y) = x-to-the-y-power. For example BIN(3, BIN (2,3)) = 3-to-the-(2-cubed)-power, or 3-to-the-eighth-power, while BIN(BIN(3,2), 3) = (3-squared)-to-the-third-power, or 9-cubed. Nine-cubed is 3-to-the-sixth, not 3-to-the-eighth. 1.4.3 DEFINITION: A semigroup is an ordered pair (X, BIN) comprising a family X and an associative binary composition BIN on X. It is traditional to write the binary composition for a semigroup using multiplicative notation when there is no reason to use some specific other notation. Thus we shall generally write "xy" to signify BIN(x, y) in a semigroup, failing some reason to write "x + y" or "x * y" and the like. The Associative Law for BIN then reads "x(yz) = (xy)z." This notational convention simplifies the look of the page. It is important, though, not to carry over into our general study intuitions about numerical multiplication which may not be valid within a specific semigroup at hand. It is also important to remember that in order to define a particular semigroup, we must specify not only the family X of elements but also the composition BIN under which the elements combine. Despite this, it is customary to refer (improperly) to "the semigroup X" when the binary composition is clearly understood in a given context. 1.5.1 DEFINITION: A left identity for a semigroup is an element 1 such that for every x, Ix = x. A right identity is defined dually: For every x, xr = x. An identity is an element e which is both a left identity and a right identity.

5

7.5.2

Mathematical Preliminaries

1.5.2 THEOREM: If a semigroup has both a left identity 1 and a right identity r, then 1 and r must be equal. Hence there can be at most one identity for a semigroup. If a semigroup has one, we can therefore speak of "the" identity element. Proof:lr must equal r since 1 is a left identity. Ir must also equal 1 since r is a right identity. There are, incidentally, semigroups that have an infinite number of left identities. (By the theorem above, a semigroup that has more than one left identity cannot have any right identities.) There are, in fact, both finite and infinite semigroups in which every element is a left identity. To illustrate this, take any family X and define on X the composition BIN(x, y) = y for all x and all y; (X, BIN) is such a semigroup. 1.6.1 DEFINITION: Given a semigroup with identity e; given an element x, a left inverse for x is an element 1 satisfying Ix = e. A right inverse for x is an element r satisfying xr = e. An inverse for x is an x' which is both a left inverse and a right inverse. 1.6.2 THEOREM: If an element x of a semigroup with identity has both a left inverse 1 and a right inverse r, then 1 = r. Hence x can have at most one inverse. If x has one, we can therefore call it "the" inverse of x. Proof:1 = le = l(xr) = (lx)r = er = r. 1.6.3 In multiplicative notation for a semigroup with identity, the inverse of an element x that has one is denoted x"1. 1.7 DEFINITION: A group is a semigroup with identity in which every element has an, inverse. The abstract definitions of "semigroup" and "group" (1.4.3; 1.7) are consistent with the earlier use of those terms in connection with families of transformations (1.3.2; 1.3.4). 1.8.1 DEFINITION: Given a binary composition BIN on a family X, elements x and y commute if BIN(y, x) = BIN(x, y), that is, if yx = xy in multiplicative notation. The composition BIN is commutative if every pair of elements commutes. A semigroup or group is commutative if its binary composition is commutative.

6

The group of transposition and inversion operations on the twelve pitchclasses is non-commutative. To illustrate this, let T2 be the operation of

Mathematical Preliminaries

1.9.2

transposing-by-2; let I, J, and K be the respective operations of invertingabout-C, inverting-about-B, and inverting-about-Cft. Then, as we observed earlier, IT2 = J (1.2.5). On the other hand, T2I = K. Thus the operations T2 and I do not commute. (Remember that we are using left orthography. "IT2 = J" means: "Given any sample pitch-class s, if you invert-about-C the 2-transpose of s, you will obtain the inversion-about-B of the given s." "T2I = K" means: "Given any sample pitch-class s, if you transpose-by-2 the inversion-about-C of s, you will obtain the inversion-about-C# of the given s.") 1.8.2 DEFINITION: Given a binary composition BIN on a family X, an element c of X is central if c commutes with every x in X. The family of all central c is the center of the system (X, BIN). 1.9 In this section we shall develop the conceptual structure and terminology for equivalence relations on a family S (not necessarily a semigroup). We shall see in particular how the notion of an equivalence relation is intimately connected with the idea of mapping S onto another family S' by some function f. 1.9.1 DEFINITION: Given a family S, an equivalence relation on S is a subfamily EQUIV of S x S that satisfies conditions (A), (B), and (C) following. (A): For every s in S, (s, s) is in EQUIV. (B): If (s, t) is in EQUIV, then so is (t, s). (C): If (r, s) and (s, t) are in EQUIV, then so is (r, t). The three conditions are called the "reflexive," "symmetric," and "transitive" properties of the relation. The conditions express formally some of our intuitions about things that are "equivalent." (A) matches our intuition that any object s should be equivalent to itself. (B) matches our intuition that if s is equivalent to t, then t should be equivalent to s. (C) matches our intuition that if r is equivalent to s and s is equivalent to t, then r should be equivalent to t. 1.9.2 THEOREM: Let f be a function from S onto S'. Define a relation EQUIV on S by putting the pair (s, t) in the relation if and only if f (s) = f (t). Then EQUIV is an equivalence relation. Proof: (A) f (s) = f (s), so (s, s) is in the defined relation. (B) If (s, t) is in the defined relation, f (s) = f (t). Then f (t) = f (s), so that (t, s) is in the defined relation. (C) If f(r) = f(s) and f(s) = f(t) then f(r) = f(t). We shall see soon that every equivalence relation on S can be regarded as being generated in precisely the above fashion, for some suitable choice of S' andf.

7

1.9.3

Mathematical Preliminaries

1.9.3 THEOREM: Let EQUIV be an equivalence relation on a family S. For each s in S let E(s) be the subfamily of S comprising exactly those members of S which are in the EQUIV relation to s, i.e. those t such that (s, t) is in the EQUIV relation. Then, giveu any s and any t in S, either (A) or (B) below will be true. (A): s and t are equivalent; E(s) and E(t) are the same collection. (B): s and t are not equivalent; E(s) and E(t) are disjoint (have no common member). Proof: Suppose first that s and t are equivalent. Then, by the symmetric and transitive laws, r is equivalent to s if and only if r is equivalent to t. In other words, r is a member of E(s) if and only if r is a member of E(t). Thus E(s) and E(t) are the same collection. (A) of the theorem obtains. Now suppose that s and t are not equivalent. Then there can be no r which is both a member of E(s) and a member of E(t). For if there were such an r, then r would be equivalent to both s and t; by the symmetric and transitive laws, we could infer that s was equivalent to t, which we have supposed is not the case. Thus E(s) and E(t) are disjoint. (B) of the theorem obtains, q.e.d. 1.9.4 By virtue of Theorem 1.9.3, an equivalence relation partitions S into the set-theoretic union of mutually disjoint subfamilies E^,.. .,En,... These subfamilies are called the equivalence classes of the relation. For each s in S, there is precisely one equivalence class En to which s belongs, s is a member of the class En if and only if En = E(s), where E(s) is the family defined in 1.9.3, the family of objects equivalent to s. Indeed, it would be possible to define any equivalence relation by partitioning S into mutually disjoint subfamilies S l f ..., S n , ... One could then define the pair (s, t) to be in a relation REL if both s and t lie in the same subfamily of the partition. One could show that REL is an equivalence relation, and that the members S j , . . . , S n ,... of the given partition are exactly the equivalence classes for that equivalence relation. 1.9.5 DEFINITION: Given an equivalence relation EQUIV on a family S, the family of equivalence classes is called the quotient family of S modulo EQUIV. We shall denote it symbolically by S/EQUIV. The function E of 1.9.3 maps S onto S/EQUIV, mapping each argument s to the value E(s), the member of the quotient family that contains s. The function E is called the natural map of S onto S/EQUIV.

8

We may now regard every equivalence relation as potentially generated in the manner of 1.9.2. Given EQUIV on S, take S' = S/EQUIV and take f = E, the natural map of S onto S'. s and t are then equivalent under the given relation if and only if f (s) = f (t).

Mathematical Preliminaries

1.9.7

1.9.6.1 EXAMPLE: Let S be the family of all pitches under twelve-tone equal temperament. Define EQUIV by putting (s, t) in EQUIV if s and t have the same letter name, give or take enharmonic equivalence. The quotient family S/EQUIV comprises the twelve pitch classes. The natural map E takes each pitch s into its pitch class E(s). 1.9.6.2 EXAMPLE: Let S be the family of all beats in a certain waltz. Define a function f from S into the numbers 1,2, and 3: f (s) = 1 if s is the first beat of its measure; f(s) = 2 if s is the second beat of its measure; f(s) = 3 if s is the third beat of its measure. A dancing master might construct this function by calling "one-two-three," over and over again as the beats go by. The function f induces an equivalence relation on S by the method of 1.9.2: s and t are EQUIValent if they share the same f-value. The three equivalence classes can be called the "beat classes" of the relation; they comprise the first beats, the second beats, and the third beats of the waltz. 1.9.6.3 EXAMPLE: Let S be the family of all collections of pitch classes. Put the pair (s, t) into the relation SAMETYPE if the collection t is a transposed or inverted form of the collection s. (Transposition-by-zero is considered a formal transposition here.) SAMETYPE is an equivalence relation. The equivalence classes are Forte's set-types.1 The class 3-11, in Forte's nomenclature, contains the twenty-four major and minor triads. The class 3-12 contains the four augmented triads. And so on. 1.9.7 OPTIONAL: This section of the work is for those who are curious to explore the material in a bit more depth. Given a function f from S onto S', define EQUIV as in 1.9.2; i.e. put (s, t) into the EQUIV relation if and only if f(s) = f(t). Given any member s' of S'; since f is onto, there is some s in S satisfying f(s) = s'. The family of all s satisfying f(s) = s' is an equivalence class En; En contains just those arguments for f having the given s' as their f-value. We write En = ARGS(s'). ARCS is a function from S' into S/EQUIV. ARGS maps S' onto S/EQUIV: Given any equivalence class En, let s be a member of En and let s' = f(s); then ARGS(s') = En; the given En is an ARGS-value. The function ARGS is also 1-to-l: If s' and t' are distinct members of S', then the equivalence class ARGS(s'), comprising those s such that f(s) = s', is obviously different from the equivalence class ARGS(t'), comprising those t such that f (t) = t'. By the method of its construction above, the function ARGS satisfies 1. Allen Forte, The Structure of Atonal Music (New Haven and London: Yale University Press, 1973).

9

1.10

Mathematical Preliminaries

formula (A) below. (A): ARGS(f (s)) = E(s)

for all s in S.

We have observed that the function ARGS is 1-to-l and onto. Hence it has an inverse function, which we shall call f/EQUIV. ARGS maps S' 1-to-l onto S/EQUIV as in formula (A) above. f/EQUIV maps S/EQUIV 1-to-l onto S'. Applying f/EQUIV to both sides of formula (A), we obtain formula (B). (B): f (s) = (f/EQUIV) (E(s)) for all s in S. f/EQUIV is called the induced map on S/EQUIV. While f may map many of its arguments onto the single value f (s) in S', f/EQUIV maps exactly one of its arguments onto that value. Via formulas (A) and (B), the mutually inverse functions ARGS and f/EQUIV set up a 1-to-l correspondence between the members f (s) of the image family S' and the members E(s) of the quotient family S/EQUIV, the family of equivalence classes. 1.10 When we shift our attention from an arbitrary family S to a semigroup (X, BIN), certain sorts of equivalence relations on X are of special interest because of the ways they interact with the algebraic structure of the semigroup. We shall study here some special equivalence relations called congruences. They interrelate with special sorts of functions on semigroups, functions called homomorphisms. Homomorphisms map semigroups one into another in a special way that engages algebraic structure. 1.10.1 DEFINITION: An equivalence relation on a semigroup is a congruence if it has this property: Given xt equivalent to y x and x2 equivalent to y 2 , then \1x2 is equivalent to y^a1.10.2 THEOREM: Given a congruence on a semigroup, let C^ and C2 be any congruence classes (equivalence classes for the congruence). Then there is a unique congruence class C3 such that whenever X A and x2 are members of C t and C2 respectively, the composition x t x 2 is a member of C3. Proof: Take any specimen y: in Q and any specimen y 2 in C 2 . Let C3 be the congruence class containing y^. C3 is the class whose existence the theorem asserts. To see this, suppose that x t and x 2 are any members of C x and C 2 respectively. Since Xj is congruent to y : and x 2 is congruent to y 2 , x t x 2 will be equivalent to y^ (1.10.1). Hence XjX 2 will lie within the same congruence class as y x y 2 . That is, x x x 2 will lie within the constructed C3. q.e.d.

w

1.10.3 THEOREM: Let CONG be a congruence on the semigroup (X, BIN). Then the quotient family X/CONG (i.e. the family of congruence classes) becomes a semigroup itself under the binary composition BIN/CONG defined as follows. Given congruence classes Cj and C2 (members of X/CONG),

Mathematical Preliminaries

J.JO.4.2

the composition (BIN/CONG) (C 1 ,C 2 ) is the congruence class C3 of Theorem 1.10.2, that is the unique congruence class which contains BIN(x 1 ,x 2 ) whenever x1 belongs to C t and x2 belongs to C 2 . Sketch of proof: The heart of the theorem is that the binary composition BIN/CONG is well defined. Given Cj and C 2 , the value of C3 does not depend at all on the specimen \v and x2 we may select to represent C t and C2. C3 depends only upon the classes C1 and C2 themselves. Having noted that, it is not hard to prove that BIN/CONG is associative. 1.10.4.1 EXAMPLE: Let (X, BIN) be the group of all integers, positive, negative, or zero, under addition. Define a relation CONG: the pair of integers (x, y) is in this relation if the difference y — x is an integral multiple of 12. CONG is reflexive: (x, x) is in the relation since x — x = 0 = 0-times-12 is an integral multiple of 12. CONG is symmetric: If (x,y) is in the relation, then there is some integer n such that y — x = n-times-12; then there is some integer m such that x — y = m-times-12 (take m = — n); then (y, x) is in the relation. CONG is transitive: If y — x = m-times-12 and z — y = n-times-12, then z — x = (z — y) + (y — x) = (n + m)-times-12. So CONG is an equivalence relation. It is in fact a congruence, for it satisfies the requirement of 1.10.1: If y t — x t is a multiple of 12 and y2 — x2 is a multiple of 12, then (yt + y 2 ) — (xj + x 2 ) is a multiple of 12. We write C(x) for the congruence class containing x. Since C(x) = C(xplus-or-minus-any-multiple-of-12), every congruence class is one of the twelve classes C(0), C(l), ..., C(ll). The quotient semigroup, then, contains just those twelve members. For each i between 0 and 11 inclusive, the class C(i) contains exactly those integers that can be written as i-plus-some-multiple-of12. Composition of the twelve congruence classes within the quotient semigroup follows the rule of addition modulo 12. That is, C(i) + C(j) = C(i + j) if i + j is less than 12; otherwise C(i) + C(j) = C(i + j - 12). Thus C(5) + C(8) = C(l). According to 1.10.2, we can read this as stating correctly: "Any number divisible by 12 with a remainder of 5, added to any number divisible by 12 with a remainder of 8, produces some number divisible by 12 with a remainder of 1." The equation "C(5) + C(8) = C(l)" in the above context is customarily abbreviated: "5 + 8 = 1 (mod 12)." The quotient semigroup is called "the integers modulo 12." It is in fact a group. We shall see later that the quotient semigroup of any group must itself be a group. If we replace the modulus 12 in the above construction by an arbitrary integer N greater than 1, we obtain "the integers modulo N" as a quotient group. 1.10.4.2 EXAMPLE: Let (X, BIN) be the group of all rational numbers that can be expressed as x = 2a3b5c, where a, b, and c are integers (positive, negative, or zero); BIN is multiplication. We can consider these numbers to model all possible ratios of pitches in just intonation.

11

1.11

Mathematical Preliminaries

Define a relation CONG: The pair (x, y) is in this relation if the number y is some power of 2 (positive, negative, or zero) times the number x. In our intervallic model, this will be the case when the intervals x and y differ by some number of octaves. For example, any one of the numbers 12/5, 6/5, 3/5, and 3/10 is in this relation to itself or to any other one. The four numbers model the four intervals of a minor tenth up, a minor third up, a major sixth down, and a major thirteenth down. As an exercise, using the procedure of 1.10.4.1 as a guide, the reader may verify that CONG is a congruence. (Remember to verify first that it is an equivalence relation!) The quotient group models all pitch-class intervals in just intonation. That is, each congruence class consists of one interval, give or take any number of octaves. Mathematically, C(x) = C(2x) = C(4x) = • • • = C(x/2) = C(x/4) = . . From this, it can be proved: Given any x, there is a unique member x' of C(x) which lies between 1 (inclusive) and 2 (exclusive). In this way, the members x' of X that lie between 1 and 2 provide a plausible system of labels for the congruence classes C(x'). (The various pitch-intervals between the unison and the rising octave can be used to label the various intervals-modulo-theoctave.) It can also be proved: Given any x, there is a unique member x" of C(x) which can be expressed as x" = 3b5c. So the numbers x" that have factors of 3 and 5 only in their rational expressions provide another plausible system of labels for the congruence classes C(x"). (x" = 3b5c labels the pitch-class interval of "b dominants and c mediants, modulo the octave.") 1.11 When we studied an equivalence relation on a family S, we made a number of observations about the natural map E, the function that maps each element s of S into the equivalence class E(s) of which s is a member. When the family S is a semigroup X and the equivalence relation is a congruence, we shall replace the name "E" of this natural map by the name "C": C maps each element x of the semigroup X into the congruence class C(x) of which x is a member. We have already used this nomenclature in examples 1.10.4.1 and 1.10.4.2 above. Everything that we observed earlier about the natural map E (1.9.3,1.9.4, 1.9.5,1.9.7) is true for the natural map C, which is only a special notation for E in the particular event that S is a semigroup and the equivalence relation is a congruence. Beyond that, C has special properties that engage the algebraic structure of the semigroup X and the quotient semigroup X/CONG. Specifically, the natural map C of X onto X/CONG satisfies law (A) below. (A): C(x!)C(x2) = C(XiX 2 ) for all x x and x 2 . 72

Indeed we defined the binary composition "C(x1)C(x2)" in X/CONG

Mathematical Preliminaries

1.11.3

precisely so as to satisfy this law. That was the work of 1.10.2 and 1.10.3. Mathematicians express property (A) above by saying, "C is a homomorphism of X onto X/CONG." The crucial term "homomorphism" is defined in 1.11.1 below. 1.11.1 DEFINITION: A function f from a semigroup (X, BIN) into a semigroup (X', BIN') is a homomorphism if it satisfies the law: BIN'(f( Xl ),f(x 2 )) = f(BIN( Xl ,x 2 )) for all Xi and all x2 in X. One can express this law colloquially by saying, "The combination of the images is the image of the combination." Using multiplicative notation for both semigroups, the law looks simpler: f( Xl )f(x 2 ) = f( Xl x 2 ). Certain sorts of homomorphisms are of special interest. 1.11.2 DEFINITION: A homomorphism is an isomorphism (into) if it is 1 -to-1 If f is an isomorphism of (X, BIN) onto (X', BIN'), we say the two semigroups are isomorphic (via f). In that case the inverse map f -1 is an isomorphism of (X', BIN') onto (X, BIN). 1.11.3 OPTIONAL: Let f be a homomorphism of a semigroup (X, BIN) onto a semigroup (X', BIN'). We have already seen (in 1.9.2) that an equivalence relation is defined if we select as equivalent just those pairs (x, y) satisfying f (x) = f (y). We can show that the relation in this case is in fact a congruence CONG. From earlier work (1.9.7) we know that the mapping ARGS of X' into X/CONG is 1-to-l and onto. (ARGS(x') is the congruence class comprising exactly those x such that f (x) = x'.) In this case, f being a homomorphism, we can show that ARGS is a homomorphism of the semigroup (X', BIN') into the quotient semigroup (X, BIN)/CONG. Here is a sketch for the proof of that. We want to show that for all \l and for allx 2 ,ARGS(f(x 1 ))ARGS(f(x 2 )) = ARGS(f(x1)f(x2)). In this equation, the symbolic product of the two ARGS-values on the left means the binary composition of those values in the quotient semigroup; the symbolic product f(Xi)f(x 2 ) within the equation means the binary composition of those two f-values in the semigroup (X',BIN'). Now f(x t )f(x 2 ) = f(x 1 x 2 ), since f is a homomorphism, and ARGS (f (any thing)) = C(that thing), as per 1.9.7(A). Hence the equation we have to show reduces to the equation C(x1)C(x2) = C(\1x2). And the latter equation is indeed true, since CONG is a congruence (1.11 (A)). Since ARGS is 1-to-l, onto, and a homomorphism, it is an isomorphism of the two semigroups (X', BIN') and (X, BIN)/CONG. Colloquially speak-

13

1.11.4

Mathematical Preliminaries

ing, we can say that the image semigroup is isomorphic with the quotient semigroup in this context. f/CONG, the inverse map of ARGS, the induced map of the quotient semigroup onto the image semigroup, is therefore also an isomorphism. This is very significant. It means that any homomorphic image (X', BIN') of a semigroup (X, BIN) "is essentially" some quotient semigroup of (X, BIN), and the generic homomorphism f of (X, BIN) onto that image "is essentially" the natural map of (X, BIN) onto that quotient. The words "is essentially" here must be interpreted with some care. They express the idea of identification up to within isomorphism of the image semigroups. With that understanding, we can say that it suffices to study the possible congruence relations on (X, BIN), in order to know all possible homomorphisms which can map (X, BIN) onto other semigroups, and all possible other semigroups which are homomorphic images of (X, BIN). 1.11.4 One more term should be introduced here. An anti-homomorphism of one semigroup into another is a function f satisfying f(x!)f(x 2 ) = f^x^. Given a semigroup (X, BIN) we can define another binary composition ANTIBIN on the family X: ANTIBIN (x l 5 x 2 ) = BIN(x 2 ,x 1 ). ANTIBIN is associative, so (X, ANTIBIN) is a semigroup. (X, ANTIBIN) is antiisomorphic to (X, BIN) under the map f (x) = x. In the obvious sense, every anti-homomorphism of (X, BIN) is a homomorphism of (X, ANTIBIN) and vice-versa. Thus we will not normally have to concern ourselves with antihomomorphisms. We will only have to do so when we have to deal with both homomorphisms and anti-homomorphisms of the same semigroup at the same time. Such a situation will in fact arise later on. We shall be studying a certain group whose elements are i, j, k ...; we shall also be studying various families of transformations on a certain family of objects. One such family will be called "transpositions"; for each i there will be a corresponding transpositionoperation TJ. Another such family will be called "interval-preserving operations"; for each i there will be a corresponding interval-preserving operation PJ. The P-operations will combine according to the rule PjPj = P^; the Toperations will combine according to the rule T;Tj = T^. The map of i to P; will be an isomorphism, while the map of i to T, will be an anti-isomorphism of the same group. In such a situation we must perforce deal with the concept of antihomomorphism. We could change BIN to ANTIBIN in the index group i, j, k . . . so as to make the mapping of i to Tj an isomorphism, but then the mapping of i to PJ would become an anti-isomorphism. Using right orthography for the operations T; and P; would have the same effect. 14

1.12.1 THEOREM: Let f be a homomorphism of the semigroup (X, BIN) onto the semigroup (X', BIN'). If e is an identity for (X, BIN) then f (e) is an identity

Mathematical Preliminaries

1.13

for (X', BIN'). In that case, if x has an inverse x * in (X, BIN) then f(x-1) is an inverse for f(x) in (X', BIN'). Proof: f(e)f(x) = f(ex) = f(x); f(x)f(e) = f(xe) = f(x). Thus f(e) is an identity for the family of all f(x). Since f is onto, every member of X' can be written as the value f (x) of some argument x. Hence f (e) is an identity for all of X'. If x has an inverse then f(x -1 )f(x) = f(x-1x) = f(e) = the identity in X'; likewise f(x)f(x-1) is the identity in X'. So f(x-1) is the inverse in X' for f(x). Thatis, f(x~ 1 ) = (f(x)r1. 1.12.2 THEOREM: A homomorphic image of a group is a group. The theorem follows at once from 1.12.1. 1.12.3 THEOREM: Any quotient semigroup of a group is a group. The theorem follows from 1.12.2, since the natural map of the given group onto its quotient semigroup is a homomorphism (1.11 (A)). The quotient construction is one common way to derive new semigroups or groups from old. Another way is to form "direct products" as sketched below. 1.13 Let SGPi = (X t , BINO and SGP2 = (X2, BIN2) be semigroups. The direct product of SGPj and SGP2 is a semigroup SGP3 = (X3,BIN3) constructed as follows. X3 is the Cartesian product Xt x X 2 . Given (x ls x 2 ) and (y1}y2) in X3, BIN3((xl5x2), (y l5 y 2 )) is defined as the element (BIN^x^yj), BIN 2 (x 2 ,y 2 )) of X3. In multiplicative notation, (Xi,x 2 )(y 1 ,y 2 ) is defined = (x 1 y 1 ,x 2 y 2 ). BIN3 as defined is associative, so that SGP3 is indeed a semigroup. To symbolize that SGP3 is the direct product of SCPj and SGP2 we write SGP3 = SGP 1 ®SGP 2 . If ej and e2 are identities for SGPj and SGP2, then e3 = (e ls e 2 ) is an identity for SGP3. If xl in Xt and x2 in X2 have inverses in their respective semigroups, then (x^SxJ 1 ) is an inverse for the element (Xj,x 2 ) of SGP3. It follows: If SGPj and SGP2 are both groups, then so is their direct product SGP3.

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2

Generalized Interval Systems (1): Preliminary Examples and Definition

In conceptualizing a particular musical space, it often happens that we conceptualize along with it, as one of its characteristic textural features, a family of directed measurements, distances, or motions of some sort. Contemplating elements s and t of such a musical space, we are characteristically aware of the particular directed measurement, distance, or motion that proceeds "from s to t." Figure 0.1 on page xi earlier symbolized this awareness, using an arrow marked i extending or moving from a point marked s to a point marked t to help us render visible our intuition. When s and t are pitches or pitch classes we are comfortable with the word "interval" as a term to use in connection with the i-arrow. That is why I have decided to keep using the word "interval" when generalizing our intuitions about the i-arrow to musical spaces whose objects s, t, and the like are not necessarily pitches or pitch classes. It will be helpful to explore some specific musical spaces informally in this connection, before proceeding more formally later on. "int(s, t)" will provisionally denote our intuition of a directed measurement or motion behaving like an "interval from s to t." Later on we shall attach a more formal significance to the expression "int(s, t)."

16

2.1.1 EXAMPLE: The musical space is a diatonic gamut of pitches arranged in scalar order. Given pitches s and t, int(s, t) is the number of scale steps one must move in an upwards-oriented sense to get from s to t. Thus int(C4, C4) = 0, int(C4,D4) = 1, int(C4,E4) = 2, and int(C4,C5) = 7. Int(C4,A3) = — 2, since moving " — 2 steps up" amounts to moving 2 steps down. Using these measurements, if we take 2 steps up (e.g. from C4 to E4) and then take 2 more steps up (in this case, from E4 to G4), we have taken 4 steps up in all (in this case, from C4 to G4). Symbolically, int(C4, E4) = 2,

Generalized Interval Systems (1)

2.1.5

int(E4, G4) = 2, int(C4, G4) = 4, and 2 + 2 = 4. The intervallic measurements of the model thus interact effectively with ordinary arithmetic. This obviates a defect in the traditional measurements which tell us, for example, that a "3rd" and another "3rd" compose to form a "5th." (3 + 3 = 5 ???) 2.1.2 EXAMPLE: The musical space is a gamut of chromatic pitches under twelve-tone equal temperament. Given pitches s and t, int(s, t) is the number of semitones one must move in an upwards-oriented sense to get from s to t, not counting s itself. Thus int(C4, D4) = 2, int(C4, G4) = 7, int(C4, C5) = 12, int(C4, F3) = -7, and int(C4, F2) = -19. 2.1.3 EXAMPLE: The musical space comprises the twelve pitch-classes under equal temperament. If we arrange the pitch classes around the face of a clock following the order of a chromatic scale, then int(s, t) is the number of hours that we traverse in proceeding clockwise from s to t. For instance, if s is at 8 o'clock and t is at 1 o'clock, int(s, t) = 5. Note that the number int(s, t) does not depend on which pitch class is positioned at 12 o'clock. In any case, int(E, E) = 0, int(E, F) = 1, and int(F, E) = 11. 2.1.4 EXAMPLE: The musical space comprises seven pitch-classes, corresponding to the seven mode degrees of system 2.1.1. If we wrap the scale around the face of a seven-hour clock, then int(s, t) is the number of hours that we traverse on that clock, in proceeding clockwise from s to t. Thus int(D, D) = 0, int(D, E) = 1, and int(D, C) = 6. We could produce analogs for the linear spaces of examples 2.1.1 and 2.1.2, using other sorts of scales. And, for systems in which octave equivalence is functional, we could derive analogs for the modular spaces of examples 2.1.3 and 2.1.4. For example, we could investigate octatonic-scale space in the manner of 2.1.1 and 2.1.2; we could derive there from a modular space of eight pitch-classes, wrapping the octatonic scale around an eight-hour clock and measuring intervals modulo 8. 2.1.5 EXAMPLE: This musical space, harmonic rather than melodic, comprises pitches available from a given pitch using just intonation. If we write FQ(s) to denote the fundamental frequency of the pitch s, then int(s, t) is the quotient FQ(t)/FQ(s). That quotient will be some number of the form 2a3b5c, where a, b, and c are integers, positive, negative, or zero. It is not immediately clear what intuitions of "distance" or "motion" we are measuring by these intervals. Personally, I am convinced that our intuitions are highly conditioned by cultural factors. In particular, I do not think that the acoustics of harmonically vibrating bodies provide in themselves an adequate basis for grounding those intuitions. For instance, when we

17

2.7.5

Generalized Interval Systems (1)

write int(C4, F#4) = 45/32 (= 2~ 5 32 5), I do not believe that we are intuiting a common partial frequency F#9 for both C4 and F#4, a partial which is intuited forthwith in some harmonic space as both the 32nd partial of F#4 and the 45th partial of C4. Nor do I believe that we intuit a path in harmonic space which corresponds directly to a compound series of individual multiplications and divisions by 2,3, and 5. That is, if we take the 5th partial of the 3rd partial of the 3rd partial of C4, and then find the frequency of which that is the 2nd partial, and then find the frequency of which that is the 2nd partial, continuing on in this way and so arriving eventually at F#4,1 do not believe that the way of getting from C4 to F#4 which we have intellectually reconstructed in harmonic space is in any sense an intuition of distance or gesture being measured by the composite ratio 45/32 = 5 times 3 times 3 divided by 2 divided by 2 and so on. In order to describe an actual harmonic intuition, I would rather proceed as follows. When we hear or imagine the succession C4-F#4 in its own context and try to intuit a harmonic sensation, we intuit a tonic followed by the leading tone of its dominant. And we intuit the secondary leading tone harmonically as the third of a harmony whose root is the dominant of the dominant. Constructing a fundamental bass representative for that root, i.e. some D below the F$4, we will locate that D in register as D3, to keep it completely beneath the "soprano line" C4-F#4. For the same reason, in constructing a fundamental bass for the note G4 that we imagine following F#4 in the soprano, a fundamental bass representing the implicit role of dominant harmony in the context, we will locate the bass G as G3. In this way we intuit the enlarged harmonic context of figure 2.1 (a) from the given stimulus C4-F#4.

FIGURE 2.1

18

The arrows on figure 2.1(b) show the path in harmonic space which I believe we actually intuit in this enlarged context. Starting at C4 as a local tonic, the first arrow takes us to G3, a fundamental bass for the dominant harmony where we hear the implicit enlarged context closing. The second arrow shows G3 inflected by its own dominant immediately preceding; the arrow points to the fundamental bass D3 for that event. The third arrow points to F#3, the major third of the harmony over D3. The fourth arrow points to F#4, the octave above F#3. Collating the entire path, we can retrace it and express it in prose: F#4 lies an octave above the major third of that

Generalized In terval Systems (1)

2.1.5

dominant which lies a fourth below that dominant which lies a fourth below C4. Now we can finally see in what sense the number 45/32 is a valid measurement for some intuition of a characteristic way from C4 to F#4 in harmonic space. We do have clear intuitions for a number of basic harmonic moves; we can measure those moves, and we have intuited (not just constructed) a chain of them. Our belief in the validity of mathematics carries us the rest of the way. Specifically, we intuit clearly the relation "t lies an octave above s," and we accept empirically the measurement FQ(t) = 2FQ(s) as a valid reflection of that intuition. We also intuit clearly the relation "t is the major third of the s harmony," and we accept the measurement FQ(t) = (5/4)FQ(s) as valid in connection with that intuition. Finally, we intuit clearly the relation "t is that dominant which lies a fourth below s," and we accept the measurement FQ(t) = (3/4)FQ(s) as valid in that connection. Applying those basic measurements to the arrows of figure 2.1(b), we get FQ(F#4) = 2FQ(F#3), FQ(F#3) = (5/4)FQ(D3), FQ(D3) = (3/4)FQ(G3), and FQ(G3) = (3/4)FQ(C4). Applying mathematics to this chain of measurements, we infer that the equation FQ(F#4) = 2FQ(F#3) = 2(5/4)FQ(D3) = 2(5/4)(3/4)FQ(G3) = 2(5/4)(3/4)(3/4)FQ(C4) is valid as measuring an intuited chain of intuitions, that is, not simply as an empirical fact. Observe that the number int(C4, F#4) = FQ(F#4)/FQ(C4) = 45/32 arises here not as the product of 2~ 5 and 32 and 5, which is its most "natural" mathematical factorization. Rather, 45/32 arises as the product of the four factors 2, (5/4), (3/4), and (3/4), reflecting its "natural" way of measuring an intuited chain of intuitions in the given situation. I should stress again not only the sophistication and complexity of this system (compared, for example, to the melodic system of example 2.1.1) but also its heavy reliance on cultural conditioning. A brief review of just how the noteheads got onto figure 2.1 will emphasize the point: Cultural conditioning is obviously important in our construction of the extended mental/aural context, given only the acoustical stimulus C4-F#4 and the intent to think/hear "harmonically." That intent, in turn, is itself a cultural phenomenon. Imagine a culture whose members, when they hear the notes of figure 2.1, are able to override the melodic relation of F#4 to the G4 which follows, as a primary structural determinant for a system of music theory that addresses such passages! The amount of time it takes the reader to discover that the last sentence is ironic indicates the greater or lesser extent to which we are all still within the grip of that culture. Then too, our interest in the harmonic system under study depends to a considerable extent on a cultural predilection for harmonically resonating instruments (with one degree of freedom) producing sustained (steady-state) sounds. In fact, much of the traditional theory pertinent to the system was developed under the supposition that the domain of investigation could be

19

2.1.6

Generalized In terval Systems (1)

adequately represented by aspects of one or more stretched strings. Despite its problems, that representation had the great advantage of enabling theorists to connect their intuitions of musical intervals with the measurement of visible, tangible material distances along the strings. This Cartesian modelling of harmonic intervals as res extensae, in a space outside the minds and bodies of the musicians, enabled harmonic theorists of earlier times to avoid having to confront such complex and problematic gestural intuitions as those of figure 2.1.

20

2.1.6 EXAMPLE: The musical space comprises the pitch classes generated by the space of example 2.1.5 above. Given pitch classes s and t, int(s, t) is the ordered pair of integers (b, c) such that t lies b dominants and c mediants from s. Here, if p and q are pitches belonging to the classes s and t respectively, then there is some integral power of 2, say 2a, such that the interval from p to q in the system of 2.1.5 above is 2a3b5°. We have int(C, G) = (1,0), int(G, D) = (1,0), int(D, F#) = (0,1), int(C, F#) = (2,l), int(C,F) = (-l,0), int(C,Ab) = (0,-l), and int(C,Db) = (—1, —1). (Since we are in just intonation, there are many distinct pitch classes that have any given letter name C, G, D, F#, F, A(7, D|?, and so on. I am supposing above that we are considering the simplest possible harmonic relationship in each case. Later discussion will clarify the exact issues involved.) This system modularizes the system of 2.1.5 by reducing out all octave relationships among both pitches and intervals. Its utility is clear in connection with figure 2.1. The most salient aspect of our harmonic intuition there was that the pitch class F# represented the mediant of the dominant of that dominant of the pitch class C. Our having to manipulate precise registers in the figure, particularly registers for a fundamental bass, made the working out of that intuition more complicated and problematical than necessary in this case. Now that we have available the modular harmonic system of the present example, we could get from C4 to F#4 more simply by a new chain of intuitions. Contemplating the succession of pitches C4-F#4 as before, we arrive as before at the intuition that F#4 is a leading tone to a dominant of C, and therefore functions harmonically as a mediant for a dominant of the dominant. We have now intuited int(C, F#) = (2,1) in modular harmonic space. We know that the pitch interval between two pitches whose pitch classes are in a tonic/dominant relation is validly measured by some-powerof-2 times 3/2. We also know that the pitch interval between two pitches whose pitch classes are in a root/mediant relation is validly measured by somepower-of-2 times 5/4. Hence we infer from our intuition of the pitch-clas interval int(C, F#) = (2,1) the validity of measuring the pitch interval int(C4, F#4) as = some-power-of-2 times (3/2)2 times (5/4). We know that a rising interval between two pitches lying closer than an octave is validly

Generalized Interval Systems (1)

2.1.6

measured by a number between 1 and 2. So we infer that int(C4, F|4) is validly measured by the number 2a times (3/2)2 times (5/4), where the integer a makes the product greater than 1 and less than 2. We can visualize modular harmonic space as a two-dimensional map or game board in the format of figure 2.2.

FIGURE 2.2 On this map, if int(s, t) = (b, c) then the pitch class t lies b places to the right of s and c places above s. (—b places to the right is b places to the left.) The subscripts help us keep track of the mediant dimension: If s has subscript m and t has subscript n then int(s, t) will be of form (something, n - m). The subscripts and the visual format more generally clarify the functional distinctions in this space between pitch classes with the same letter-name but different subscripts; they are at different places on the map. Since we are assuming just intonation, the frequencies involved will differ by factors involving the syntonic comma and the pitch classes C_,, C0, C,, ... will be distinct acoustically. But even in equal temperament, the visual format of figure 2.2 portrays a conceptually infinite harmonic space, whose distinct places correspond to subscripted pitch-classes. (Subscripted pitch-classes can be expressed formally as ordered pairs; for example A\>3 can be expressed as the pair (A|>, 3).) We can conceptualize the intervals of modular harmonic space as characteristic "moves" on the game board of figure 2.2. For example, in going from CQ to Ffj, we move 2 squares to the right and 1 square up. The number-pair (2, 1), which is int(C0, F|j), can thereby be conceptualized as a label for this particular form of the "knight's move" on the game board, i.e. the knight's move east by northeast. The same move (interval) takes us from Al to D#2, or from DL, to G0.

27

2.2.7

Generalized Interval Systems (1)

We could reduce the system of 2.1.6 farther if we considered pitch classes to be equivalent when they shared the same letter name, differing only by subscript. Then C_ l 5 C0, Cj, C 2 ,... C n ,... would all mean the same thing; so would £_!, E0, E l 5 E 2 , . . . , E n . In this case, moving one square north on the game board of figure 2.2 would be functionally equivalent to moving four squares east. The north/south dimension of the board would functionally disappear, and we could reduce our map to a one-dimensional east/west succession of dominant-related pitch classes... E|?0, 6(70, F0, C0, G0, D0, A 0 , E0, ... Because of the equivalence relation that led to this series, we may as well consider the reduced pitch classes to represent pitches in quarter-comma mean-tone temperament: Four new "fifths" (that were steps east on figure 2.2) are pitch-class equivalent to a "major third" (that was a step north on figure 2.2). There is no reason to keep the zero subscripts in the series of mean-tone pitch classes, so we can just write... E[?, 8(7, F, C, G, D, A, E,... The intervals of this reduced system are integers measuring steps "east" on that chain; since we have lost the north/south dimension of our earlier figure, we may as well say "to the right" rather than "east." We could reduce the mean-tone system even farther by declaring the enharmonic equivalence of Gl? with F#, of Dfr with C#, and so on. The infinite series of mean-tone pitch classes thereby gets wrapped around the face of a clock, and we find ourselves back at the system of 2.1.3, only now measuring intervals-modulo-the-octave by (equally tempered) fifths rather than by semitones.1 We have explored six examples, and suggested some further examples, of musical spaces in connection with which we traditionally use the word "interval" to denote a directed measurement, distance, or motion. All six of these musical spaces, melodic or harmonic, had pitches or pitch classes for their elements. Now we transfer our attention to some musical spaces whose elements are measured rhythmic entities of various sorts. We presuppose a context that makes us sensitive to time in segments that can be measured by some temporal unit, whether this unit is some local pulse within a piece or some conceptual span, like the minute that underlies metronome markings. 2.2.1 EXAMPLE: The musical space is a succession of time points pulsing at regular temporal distances one time unit apart. Given time points s and t,

22

1. Maps like figure 2.2 have been especially common in German theories of tonality since the eighteenth century, generally in connection with key relationships rather than root relationships (though some theories do not dwell on such a distinction). The closest precedent I can find for the actual configuration of figure 2.2 itself appears in Hugo Riemann, Grosse Kompositionslehre, vol. 1, Der homophone Satz (Melodielehre und Harmonielehre) (Berlin and Stuttgart: W. Spemann, 1902). Riemann's map is on page 479. He illustrates intervals as moves on the board, on page 480.

Generalized Interval Systems (1)

2.2.3

int(s, t) is the number of temporal units by which t is later than s. (—x units later is x units earlier.) 2.2.2EXAMPLE: The musical space is the preceding one, wrapped around the face of an N-hour clock. We can imagine this as modeling the imposition of an N-unit meter on the earlier space, so that barlines appear regularly every N pulses. The present space has N members, which we shall call "beat classes," labeling them by numbers from O through N — 1. Beat-class 0 comprises all the pulses of 2.2.1 that occur at some bar-line; beat-class 1 comprises all the pulses of 2.2.1 that^ occur one unit after some barline;...; beat-class (N — 1) comprises all the pulses of 2.2.1 that occur one unit before some barline. If s and t are beat classes, int(s, t) is the number of hours clockwise that t lies from s on the N-hour clock. Thus, in twelve-eighths meter (N = 12) the interval from beat-class 10 to beat-class 5 is 7. We discussed the notion of beat classes earlier (1.9.6.2), as exemplifying the concept of equivalence classes. We observed there that a dancing master often calls out beat classes over and over as the pulses go by, using numbers 1 through N rather than 0 through N — 1, e.g. "ONE-two-three, ONE-twothree,..." Conductors and conducting students will also be familiar with the notion kinetically. For them, beat classes are associated with definite spatial positions of the hand, positions which are numbered on pedagogical diagrams. Intervals of 1 between beat classes correspond to minimal unbroken hand gestures for the conductor, gestures that proceed from each beat class to the next along smooth arrows on the diagrams, tracing a characteristic gestural path through this modular space over and over again. This is the path along which we ride "from s to t," making int(s, t) gestural articulations along the way. Milton Babbitt has worked with a system of 12 beat classes that behaves formally exactly like the traditional 12-tone system for pitch classes.2 2.2.3 EXAMPLE: The musical space is a family of durations, each duration measuring a temporal span in time units. And int(s, t) is the quotient of the t and s measurements, t/s. If s spans 4 time units and t spans 3 time units, then int(s, t) = 3/4. t is "3/4 the length of" s. We may, if we wish, identify each duration with the beat for a certain tempo. The numerical quotients of our durations then measure the inverse quotients of the corresponding tempi in tempo-space.3 2. He describes the system in "Twelve-Tone Rhythmic Structure and the Electronic Medium," Perspectives of New Music vol. 1, no. 1 (Fall 1962), 49-79. 3. Influential compositions whose rhythmic textures involve such proportions include Elliott Carter's String Quartet no. 1 (1950-51), Conlon Nancarrow's Studies for Player Piano (1951-), and Gyorgy Ligeti's Poeme symphonique for 100 metronomes (1962).

23

2.2.4

Generalized Interval Systems (I)

2.2A EXAMPLE: We reduce the system of 2.2.3 by a durational modulus M greater than 1. Two durations are conceived as equivalent if one is some integral power of M times the other. This leads us to a modular musical space whose elements are duration-classes (i.e. equivalence classes of durations under the defined equivalence relation). The intervals of 2.2.3 are reduced in the same manner: Two numerical quotients or proportions are conceived as equivalent if one is some power of M times the other. The ratio-classes can be used as formal intervals in the reduced system. Mathematically, the reduction from 2.2.3 to 2.2.4 is exactly the same as the reduction from a system of pitches and pitch-ratios, to a system of pitch classes and ratio-classes modulo powers of M = 2, that is pitch classes and intervals-modulo-the-octave. To illustrate, let us take M = 2 for a rhythmic modulus: Two durations, or two numerical proportions, are conceived as equivalent if one is twice the other, or four times the other, or eight times the other, or half the other, or one-quarter of the other, and so on. One equivalence class of durations is then the family r = (...,5/32,5/16,5/8,5/4,5/2,5,10,20,...). Another equivalence class is the family s = (..., 1/96,1/48,1/24,1/12,1/6,1/3,2/3,4/3, 8/3,...). Yet another equivalence class is the family t = (..., 7/80,7/40,7/20, 7/10,7/5,14/5,28/5,...). With reference to these particular classes r, s, and t, int(r, s) is the ratio-class that contains the number 16/15. The members of this ratio-class are exactly the numbers of form (some-power-of-2)-times-(16/15). If p is any member of class r and q is any member of class s, then the ratio q/p is (some-power-of-2)-times-(16/15). In similar wise, int(s, t) is the ratio-class that contains 21/20; the members of this class are exactly the numbers of form (some-power-of-2)-times-(21 /20). If we allow irrational durations (or tempi with respect to the time-unit), we can consider the equivalence class u = (..., n/4, n/2, n, 2n, 4rc,...). Int(s, u) is the ratio-class that contains the number 3?r/8; the members of this class are exactly the numbers of form (some-power-of-2)-times-(37c/8).4 2.2.5 EXAMPLE: The musical space is a family of durations. Int(s, t) is the difference (NB not the quotient) of time units between s and t: Int(s, t) = (t — s) units. So if r, s, and t are respectively 3, 4, and 8 units long, then int(r, s) = (4 — 3)units = 1 unit, int(s, t) = (8 — 4)units = 4 units, and int(t,r) = (3 — 8)units = — 5 units. In the earlier system of quotients (2.2.3), the corresponding intervals would have been 4/3, 2, and 3/8.

24

4. Karlheinz Stockhausen argues the plausibility of system 2.2.4, with M = 2, in "... how time passes ...," trans. C. Cardew, Die Reihe (English version) vol. 3, pp. 10-40. He also argues there the interrelatedness of his rhythmic system with traditional pitch systems. A clear view of how Stockhausen uses these ideas for the tempi of Gruppen (1955-57) is provided by Jonathan Harvey, The Music of Stockhausen (Berkeley and Los Angeles: University of California Press, 1975), 55-76.

Generalized Interval Systems (1)

2.2.6

The additive system now under study measures intervals in units of time; the earlier system of quotients measured intervals as pure numbers (ratios). The difference between the systems here becomes striking if we set the time unit as "one sixteenth note." Then the durations r = 3, s = 4, and t = 8 can be symbolized respectively by a dotted eighth, a quarter note, and a half note. The additive intervals int(r, s), int(s, t), and int(t, r), computed above as 1 unit 4 units, and — 5 units, can be expressed as "plus a sixteenth," "plus a quarter," and "minus a-quarter-tied-to-a-sixteenth." The corresponding multiplicative intervals are simply the numbers 4/3, 2, and 3/8, numbers that express ratios involving the durations. 2.2.6 EXAMPLE: To simplify matters, we restrict our attention to the durations of 2.2.5 that are exactly the positive integral multiples of some basic small duration, which we take as the temporal unit. We wrap these durations around an M-hour clock, accordingly reducing the system to a modular system. The modular space comprises M duration-classes: Two durations belong to the same duration-class if their lengths differ by some integral multiple of M. The interval between duration classes s-units-mod-M and tunits-mod-M is (t — s)-units-mod-M. t — s is the number of hours clockwise from s to t on the M-hour clock. The duration t is int(s, t) units longer than s, give or take any number of M-unit "measures". For example take M = 16; take s = 8 and t = 4 units mod 16. Then int(s,t) = 4 — 8= — 4 = 12 units mod 16. If we represent the unit as a sixteenth note, then the M-unit "measure" lasts a whole note. The duration-class s = 8 is represented by a half note, give or take any number of whole notes tied on. The duration-class t = 4 is represented by a quarter note, give or take the same. The interval int(s, t) = 12 is represented by "plus a dotted half," give or take the same. Our arithmetic mod 16 above reflects this observation: A quarter note, tied to an extra whole note for free, is a dotted half longer than a half note. We have now explored six rhythmic spaces as well as six tonal ones. To the extent we intuit these spaces, we intuit "intervals" in connection with them. Later on we shall explore yet other spaces, including some more rhythmic ones and some timbral ones. At this point, though, it will be helpful to stop and develop some formal generalities. All of the examples in this chapter so far have certain structural features in common. Foremost among these is our intuition in each case of a group, explicitly or implicitly defined, within which the intervals lie. If i and j are intervals (characteristic measurements, distances, motions, or the like) we intuit being able to compose them in some characteristic way (e.g. by addition, addition mod 12, multiplication, multiplication mod powers of 2, concatena-

25

2.3.1

Generalized Interval Systems (1)

tion of moves on a game board, and so on). And we intuit the composition ij of the intervals i and j to be itself an interval of the system (characteristic measurement, distance, motion, and the like). Indeed, we intuit that for any elements r, s, and t of the musical space, the interval-from-r-to-s composes with the interval-from-s-to-t to yield the interval-from-r-to-t. Symbolically: int(r, s)int(s, t) = int(r, t). We intuit the composition of intervals to be associative: i(jk) = (ij)k. We intuit an identity interval e, that composes with any interval j to yield j: ej = je = j. Indeed, we intuit that each object s of the space lies the identity interval from itself: int(s, s) = e. We intuit that each interval has an "inverse interval" in the sense of measurement, distance or motion: i""1 measures, extends, or moves in the reverse sense from i. We intuit that this intuitive inverse is also a group inverse: 'i-1i = ii"1 = e. Indeed, we intuit that if i is the interval from s to t, then i"1 will be the interval from t to s. Symbolically: int(t, s) = int(s, t)"1. We can collate all these intuitions to construct a formal generalized system. As we shall see, all our examples so far with one exception suggest specific instances of the generalized system. 2.3.1 DEFINITION: A Generalized Interval System (GIS) is an ordered triple (S, IVLS, int), where S, the space of the GIS, is a family of elements, IVLS, the group of intervals for the GIS, is a mathematical group, and int is a function mapping S x S into IVLS, all subject to the two conditions (A) and (B) following. (A): For all r, s, and t in S, int(r, s)int(s, t) = int(r, t). (B): For every s in S and every i in IVLS, there is a unique t in S which lies the interval i from s, that is a unique t which satisfies the equation int(s, t) = i. Condition (B) of the definition is a new idea. We shall discuss it shortly. Condition (A) has already been discussed. But what about the other equations involving the function int, equations we also discussed above? Should we not also stipulate these other equations in defining a GIS? It turns out that we do not have to, because they are logically implied by the group structure and Condition (A). We demonstrate that in the form of a theorem. 2.3.2 THEOREM: In any GIS, int(s, s) = e and int(t, s) = int(s, t)"1 for every s and t in S. Proof: int(s, s)int(s, s) = int(s, s), via Condition (A). Multiply both sides of that equation by int(s, s)"1; we obtain int(s, s) = e as asserted.

26

int(s, t)int(t, s) = int(s, s) via Condition (A). We have just proved that int(s, s) = e; hence int(s, t)int(t, s) = e. Multiply both sides of that equation on the left by int(s, t)"1; we obtain int(t, s) = int(s, t)"1 as asserted.

Generalized Interval Systems (1)

2.3.2

Now let us turn our attention to Condition (B) of the definition. Essentially, it guarantees that the space S is large enough to contain all the elements we could conceive of in theory. The idea is: If we can conceive of an element s and if we can conceive of a characteristic measurement, distance, or motion i, then we can conceive of an element t which lies the interval i from s. In certain specific cases, application of this idea may require enlarging practical families of musical elements, to become larger formal spaces that are theoretically conceivable while musically impractical. For instance, we shall need to conceive supersonic and subsonic "pitches" in order to accommodate the idea of being able to go up or down one scale degree from any note, in connection with example 2.1.1. Figure 2.2 affords another good example: Obviously no finite musical context can explore the entire extent of this map, which accommodates the idea of being able to conceive the dominant, mediant, subdominant, and submediant of any pitch class. This is the methodological point: We must conceive the formal space of a GIS as a space of theoretical potentialities, rather than as a compendium of musical practicalities. In a specific compositional or theoretical context, the space S of a GIS might be perfectly accessible in practice. Such is the case, for example, with the "twelve-tone" GIS pertaining to example 2.1.3: Every one of its twelve pitch-classes is easily referenced by any pertinent music. On the other hand, in other compositional or theoretical contexts, the space S of a GIS might be pertinent as an entirety only to the extent it is suggested or implied by the actually stated musical material, plus the characteristic relationships actually employed. In just this way a painting or statue might suggest or imply the entire extension of Euclidean two-or-three-dimensional space, or some other geometrical space. (I am thinking in particular of the parabolic space in some of Van Gogh's late work.) Let us consider figure 2.2 yet again in this connection. In order to conceive the extension of the entire map, we need only three things: one (tonic) place on the map, the characteristic idea of a "just dominant" relation involving pitch classes, and the characteristic idea of an independent "just mediant" relation. In other words, we need only the pitch classes and the intervals we can infer from one tonic triad (!) in order to generate the entire conceptual group of intervals, and thereby to infer the conceptual extension of the entire map, as a terrain within which a particular composition or theory may occupy some particular region. Another feature of Condition (B) also requires discussion. Given s and i, the condition demands not just some t that satisfies int(s, t) = i, but a unique such t. We might consider weakening the condition, replacing the words "a unique," where they appear in 2.3.1(B), with the word "some." Let us call the weakened condition "(weak B)". Could we gain even greater generality by using (weak B) instead of (B)? Not really. Under condition (weak B), the space S would be partitioned into equivalence classes: s and s' would be equivalent if and only if int(s, s') = e. Given s' equivalent to s and t' equivalent to t, it would

27

2.4

Generalized Interval Systems (1)

be true that int(s', t') = int(s, t). We could thus think of the intervals as being from one equivalence class to another. We could replace S by the quotient family S/EQUIV, the family of equivalence classes, and obtain a GIS thereby. (That is, Condition (B) would apply to the function int on equivalence classes.) It is hard to see what we could possibly want to do with S that we could not do as well or better with the reduced space S/EQUIV of equivalence classes.

28

2.4 At this point, let us briefly review all our examples, noting the various GIS structures which they suggest. 2.1.1 suggests a GIS in which S is the indicated gamut, extended indefinitely both up and down. In this GIS, IVLS is the group of integers under addition and int(s, t) is the number of steps up from s to t. — i steps up is i steps down. The reader may verify that Conditions (A) and (B) of Definition 2.3.1 are satisfied. As we observed informally during the earlier discussion of this example, the non-traditional numbering of the scalar intervals is necessary so that the algebra of Condition (A) can obtain. The GIS suggested by example 2.1.2 consists of the indicated space S, a chromatic scale extended indefinitely up and down, the group IVLS = integers under addition, and the function int(s, t) = number of semitones up from s to t, not counting s. Again, — i up is i down. The reader may verify that Conditions (A) and (B) of 2.3.1 are satisfied. The GIS for example 2.1.3 consists of the space S = the twelve pitch classes as indicated, the group IVLS = the integers under addition modulo 12, and the function int(s, t) = number of hours clockwise from s to t on a 12-hour clock. The GIS for example 2.1.4 consists of the space S = the seven mode degrees as indicated, the group IVLS = the integers under addition modulo 7, and the function int(s, t) = number of hours clockwise from s to t on a 7-hour clock. The GIS for example 2.1.5 has for its space S the extended family of all "pitches" conceptually available from a given pitch using just intonation. The group IVLS here is the multiplicative group comprising all rational numbers that can be expressed in the form 2a3b5c, where a, b, and c are integers. The function int here is the quotient int(s, t) = FQ(t)/FQ(s). To conceive the GIS for example 2.1.6, we take as our space S the "game board" of figure 2.2. IVLS here is the group of ordered pairs of integers (b,c) under componentwise addition: (b^c^ + (b 2 ,c 2 ) = (bt + b 2 ,C! + c2). (IVLS is thus the direct product of the-integers-under-addition with itself.) The function int is as discussed: int(s, t) = (b, c), where t lies b squares east and c squares north of s on the game board. The GIS for example 2.2.1 has as its space S the indicated succession of time points, conceptually extending indefinitely both backwards and forwards in time. IVLS here is the integers under addition, and int(s, t) is the number of time units by which t is later than s, — i later meaning i earlier. For example 2.2.2, the GIS consists of S = the N beat-classes, IVLS = the integers

Generalized Interval Systems (1)

2.4

under addition modulo N, and int(s, t) = the number of hours clockwise that t lies from s on an N-hour clock. The GIS for example 2.2.3 depends upon just what proportions among durations we wish to allow. IVLS in any case will be some multiplicative group of positive numbers. If we allow only the basic proportions of 2 and 3, then IVLS will comprise those numbers of form 2a3b, where a and b are integers; S will then comprise exactly such durations as are in those proportions to any given duration. S is conceptually extended to allow indefinitely short and indefinitely long "durations." If we allow basic rhythmic proportions of 5 and 7, as well as 2 and 3, then IVLS will comprise all numbers of form 2a3b5c7d, and S will contain the corresponding extra members. Or we might take, as basic rhythmic proportions, the square root of 2 (2(1/2)) and the square root of 3 (3(1/2)); then IVLS will contain exactly those numbers of form 2(a/2)3(b/2), where a and b are integers, and S will contain the corresponding durations in proportion to any one given duration. And so on. In each case, the function int(s, t) is given by the quotient t/s = (t units)/(s units). When we reduce the space of 2.2.3 to the space of 2.2.4 by the modulus M, we also reduce the group of intervals for 2.2.3 to a quotient group, the new group of intervals for 2.2.4. Specifically, intervals i' and i for 2.2.3 are declared congruent if one is the other multiplied by some integral power of M: i' = iMa. This relation is indeed a congruence in the group-theoretic sense (1.10.1): It is an equivalence relation, and if j' is congruent to j (f =jM b ), then i'j' is congruent to ij (i'j' = (ij)M(a+b)). The congruence generates a quotient group, imposing a group structure on the family of congruence-classes. Those congruence-classes are the "ratio-classes" of example 2.2.4. The function int for 2.2.4 can be derived mathematically from the function int for 2.2.3, along with the declared equivalence and congruence relations on the space and group of 2.2.3. We shall explore this derivation formally in chapter 3, where we shall study the general construction of a "quotient GIS" from a given GIS and a given congruence on its group of intervals. Example 2.2.5, exceptionally, does not lead at once to a GIS structure. If we try to find a pertinent GIS, we will take IVLS to be some additive group of numbers i, j, ..., for as durations s and t vary in the family S of durations, t may be ±i, ±j,... units longer or shorter than s. And int(s, t) = (t — s)units But S, IVLS, and int here cannot satisfy Condition (B) of Definition 2.3.1. For instance, try s = 3 units and i = — 8 units; then there is no duration t in S satisfying int(s, t) = i. If there were such a t, then (t — s)units = (t — 3)units would have to equal i units = — 8 units, t — 3 would equal — 8, and t would be — 5 units in duration. But S does not contain "negative durations," and failing some convention not yet specified, it is not clear what intuition we could possibly be modeling, when we stipulate a duration t that lasts not only less than no time at all, but also measurably less than no time at all. The modeling problem here is different in kind from those involved in other spatial exten-

29

2.4

Generalized In terval Systems (1)

sions we have made conceptually. E.g., while we cannot hear a "pitch" of .001 Hertz fundamental, or of five trillion Hz fundamental, we can conceive such pitches. That is, we are at ease with the notion of periodic vibration at those rates, and we can imagine that other creatures might be sensibly aware of them. In the same sense, we can conceive indefinitely short and indefinitely long "durations," and we can conceive "time points" that lie indefinitely far back in the past or ahead in the future. But we can not conceive, in such a sense, a duration lasting precisely 5 units less than no time at all, which is thereby precisely 2 units longer than a duration lasting precisely 7 units less than no time at all. So example 2.2.5 does not lead to a GIS. Example 2.2.6 can be regarded as one means of salvaging example 2.2.5 in this connection, by providing a convention that attaches meaning to the concept of a negative duration-class. E.g. we can think of duration-class " — 5" as that class containing all durations lasting just 5 units less than some multiple of the modulus duration. " — 5" thus means the same as "M — 5," or the same as " — 5, modulo M." Example 2.2.6 has a GIS structure. The space S comprises the M duration-classes. The group IVLS is the additive group of integers modulo M. int(s, t) is the number of hours t lies clockwise from s when the duration-classes are wrapped around an M-hour clock. The reader can verify that (S, IVLS, int) is indeed a GIS. In particular, Condition (B) of the definition, which failed for example 2.2.5, obtains for example 2.2.6.

30

3

Generalized Interval Systems (2): Formal Features

There is a familiar convention whereby the twelve pitch classes in equal temperament are labeled by their intervals from a referential pitch class C. C, C#, D, ..., Bb, B are thereby labeled 0, 1, 2, ..., 10, 11 (mod 12). This convention can be generalized. That is, in any GIS we can always use the intervals of the group IVLS to label the members of the space S by their respective intervals from an assumed referential object in S. We can make the notion formal by introducing some new terminology. 3.1.1 DEFINITION: Given a GIS (S, IVLS, int) and a fixed referential member "ref" of S, the function LABEL, mapping S into IVLS, is defined by the equation LABEL(s) = int(ref, s). 3.1.2THEOREM: Whatever the element ref, the function LABEL maps S 1-to-l onto IVLS, and it satisfies the formula int(s,t) = LABEL(s)~1LABEL(t). Proof: Given the element ref and any interval i, there is one and only one s in S satisfying int(ref,s) = i, by Condition (B) of Definition 2.3.1 for a GIS. Since int(ref, s) is LABEL(s), we have observed the following: Given any i, there is some s satisfying LABEL (s) = i; furthermore, there is only one such s. Thus the function LABEL is onto; furthermore it is 1-to-l. Now we prove the formula of the theorem. By definition 3.1.1, LABEL(s)"1LABEL(t) = int(ref,s)~1int(ref,t). In that equation we can substitute int(ref, s)"1 = int(s, ref), via 2.3.2. The equation then states: LABEL(s)~1LABEL(t) = int(s, ref)int(ref, t). And the expression on the

31

3.1.2

Generalized Interval Systems (2)

right side of that equation equals int(s, t) as desired, via Condition (A) of Definition 2.3.1. q.e.d. The LABEL function can be very useful, particularly for computations involving members of S. On the other hand, its use in some contexts can be problematic, both conceptually and computationally. Conceptually, there may not always be adequate musical reasons for assigning a special referential status to ref. Why, for example, should I assign this status to the pitch class C a priori? Perhaps, in a certain context, I hear no referential pitch class at all, but rather notes related only to each other, and not to any one given note. Or perhaps I hear E as referential in this piece; then why should I call E "4" in a CLABELing system, rather than "0" in an E-LABELing system? But the violins and string basses have actually decided how E will sound by tuning their Es a certain interval from A; then would it not be methodologically most accurate to use an A-LABELing system? And so on. Some of the conceptual issues will be familiar from similar issues in fixed-do and movable-do systems of solfege. Conceptual difficulties aside, computations themselves can be muddled, when we use a LABELing system, by the algebraic influence of irrelevant intervals, intervals arising from irrelevant relations of ref to the objects s, t , . . . , whose interrelations we actually wish to compute.1 Now we shall explore some ways in which we can formally construct new GIS structures from old. We have observed one such construction used informally again and again during our survey of examples 2.1 and 2.2 earlier; that is the construction of a "quotient GIS" from a given GIS and a stipulated congruence relation on IVLS. Whenever we spoke informally about "modularizing" one system to obtain another, we invoked such a notion. Soon we shall make the notion formal. By way of preparation before doing so, we shall examine a bit more carefully just how it applies in one specific and familiar case. We consider for this purpose how the "chromatic scale" GIS of 2.1.2 gives rise to the "twelve-tone" GIS of 2.1.3. Let us call the two systems respectively GISj = (S^IVLS^intO and GIS2 = (S 2 ,IVLS 2 ,int 2 ). S t is a chromatic scale extended conceptually up and down indefinitely. IVLSi is the integers under addition. int: (s, t) is the number of semitones up from s to t. When "modularizing" GISX to obtain GIS2, we use a certain congruence on the group IVLSi. Intervals i and i' are congruent if i' = i plus or minus some integral multiple of 12. This relation is indeed a congruence in the grouptheoretic sense of 1.10.1. That is, it is an equivalence relation; furthermore, i' + j' is congruent to i + j whenever i' is congruent to i and j' is congruent to

32

1. I discuss these matters further in "A Label-Free Development for 12-Pitch-Class Systems," Journal of Music Theory vol. 21, no. 1 (Spring 1977), 29-48.

Generalized Interval Systems (2)

3.1.2

j. As we saw in chapter 1, the congruence gives rise to a quotient group IVLSJCONG. The elements of this group are the congruence classes, and the binary combination of these elements in the quotient group is well defined (!) by the formula (class-containing-i) 4- (class-containing-j) = (classcontaining-(i + j)). In this case the quotient group is the integers modulo 12. And that is IVLS2, the group of intervals for GIS2. In modularizing GlSi to GIS2 we also invoked a certain equivalence relation on S^ Pitches s and s' were declared equivalent if they differed by some integral number of octaves. The condition for equivalence of s and s' can be expressed by using the congruence relation on IVLSj (and that is important): s and s' are equivalent as defined if and only if int^s, s') is divisible by 12, which is the case if and only if intl (s, s') is congruent to the identity interval oflVLSj. The equivalence relation reduces the family S! of pitches into the family S2 of pitch classes. S2 = S1/EQUIV: The 12 pitch classes are precisely the 12 equivalence classes of pitches, under the constructed equivalence relation. Finally, in modularizing GlSt to GIS2, we implicitly invoked a significant interrelation between the congruence on IVLS^ and the equivalence relation on Sv: If pitches s and s' are equivalent (different by some number of octaves), and if pitches t and t' are also equivalent, then the intervals int^s.t) and int^s'jt') are congruent (different as integers by some multiple of 12). We can put this another way, significant for our purposes: The congruence class to which intj(s, t) belongs depends only on the equivalence class to which s belongs and the equivalence class to which t belongs, not on any specific s' and t' chosen to represent those equivalence classes. This feature of the situation enables us to see how the function int2 works for our present example: Given pitch classes (equivalence classes) p and q, int2(p, q) in IVLS2 is the congruence class (interval mod 12) to which int1(s, t) belongs, whenever s and t are any pitches belonging to the pitch classes (equivalence classes) p and q respectively. For example, let p and q be the pitch classes C and F. If s and t are middle C and Queen-of-the-Night F, then int^s.t) = 29 (semitones). If s' and t' are high C and contra F, then intjCs'.t') = -55 (semitones). The integers 29 = 24 + 5 and -55 = — 60 + 5 belong to the same congruence class mod 12, "congruence-class 5." If s" and t" are any other pitches belonging to p and q (i.e. named C and F), then intj (s", t") also belongs to congruence-class 5. That is what gives rigorous meaning to our saying "int2(C, F) = 5." Otherwise, our clock-face model for GIS2 would remain only arbitrarily or vaguely related to GISj. Indeed, we could use the relationships under present discussion to define int2'. int 2 (p,q) is that unique congruence class which contains any and all values of inti(s, t), s being any member of p and t being any member of q. Everything noted about OK^ and GIS2 in the example just studied can be generalized so as to define a "quotient GIS," given any GIS and any con-

33

3.2.1

Generalized Interval Systems (2)

gruence relation on its group of intervals. The work of section 3.2 following will be devoted to this generalization. 3.2.1 THEOREM AND DEFINITION: Let (S, IVLS, int) be a CIS; let CONG be a congruence on the group IVLS. Then an equivalence relation EQUIV is induced on S by declaring s and s' to be equivalent whenever int(s, s') is congruent to the identity e in IVLS. EQUIV will be called the induced equivalence on S. Proof: int(s, s) = e, so EQUIV as defined is reflexive. If int(s, s') is congruent to e, then int(s', s) = int(s, s')"1 is congruent to e; thus EQUIV as defined is symmetric.2 Finally, if int(s, s') and int(s', s") are both congruent to e, then int(s, s") = int(s, s')int(s', s") is congruent to e-times-e = e. Thus EQUIV is transitive. 3.2.2 LEMMA: Let (S, IVLS, int) be a GIS; let CONG be a congruence on IVLS. Then the following is true. Suppose s and s' are equivalent in S under the equivalence induced by CONG; suppose t and t' are also equivalent under that equivalence relation; then int(s, t) and int(s', t') are congruent members of IVLS. Proof: int(s,t) = int(s,s')int(s',t')int(t',t). Now both int(s,s') and int(t', t), by supposition, are congruent to e. Hence int(s, t) is congruent to e • int(s', t') • e. That is, int(s, t) is congruent to int(s', t'). q.e.d. 3.2.3 THEOREM: Let (S^IVLS^inti) be a GIS. Let CONG be any congruence on IVLSj; let EQUIV be the induced equivalence relation on S^ Let S2 be the quotient space Sj/EQUIV, the family of equivalence classes within §! under the induced equivalence. Let IVLS2 be the quotient group IVLS ! /CONG, whose members are the congruence classes within IVLS^ Then a function int2 from S2 x S2 into IVLS2 is well defined by the following method: Given equivalence classes p and q (members of S2), the value int 2 (p, q) is that congruence class (member of IVLS2) to which int1(s, t) belongs, whenever s and t are members of p and q respectively. Furthermore, (S 2 ,IVLS 2 ,int 2 ) is itself a GIS. Proof (optional): Lemma 3.2.2 assures us that int2 is well defined by the indicated procedure: Given p and q, if s and s' are any members of p, and if t and t' are any members of q, then int^s', t') belongs to the same congruence class as int^s, t). That congruence class thus depends only on the equivalence classes p and q, and not on the particular s-and-t, or s'-and-t', which we choose

34

2. The reader may prove as an exercise the necessary lemma: If x is congruent to e in a group, then x"1 is also congruent to e. After that, prove this: If x is congruent to y, then x"1 is congruent to y-1. (Hint: x"1 y is congruent to x"1 x; apply the preceding lemma. Alternatively, but less elegantly, one can bludgeon out all the desired results as corollaries of 1.11, 1.12.1, and related results already established.)

Generalized Interval Systems (2)

3.2.4

to represent p and q. And it is that congruence class which is thus well defined, asint 2 (p,q). We now prove that (S2, IVLS 2 ,int 2 ) is a GIS. To do so, we must show that Conditions (A) and (B) of Definition 2.3.1 are satisfied. We show first that (A) is satisfied. Given equivalence classes o, p, and q in S 2 , we want to show that int2(o, p)int2(p, q) = int2(o, q). Let r, s, and t be elements of Sj that are members of classes o, p, and q respectively. The congruence class containing inti(r, s) combines in the quotient group with the congruence class containing int^s, t), to yield the congruence class containing int^r, s)int1(s, t) (1.11). And, since (S^IVLS^intJ is a GIS, int^r, 8)111^(8,1)1 = int^M). Hence the congruence class containing int^r, s) combines in the quotient group with the congruence class containing int^s, t), to yield the congruence class containing intj(r, t). Or, by the definition of int 2 in the theorem, int 2 (o,p)int 2 (p,q) = int 2 (o,q), as desired. So the system (S 2 ,IVLS 2 ,int 2 ) satisfies Condition (A) of 2.3.1. It remains to show that the system also satisfies Condition (B) of that definition. Given any equivalence class p (member of S 2 ) and any congruence class J (member of IVLS2), we must show that there is a unique equivalence class q that satisfies the equation int2(p, q) = J. Let s be a member of the given p; let j be a member of the given J. Since (S:, IVLSi, intl) is a GIS, we can find a (unique) t in Sj which satisfies the equation int t (s, t) = j. Let q be the equivalence class containing this t. q is that class (member of S 2 ) for which we are searching. First of all, q satisfies the equation int2(p, q) = J. That is so because int^s, t) = j, s belongs to p, t belongs to q, and j belongs to J; int2 was defined precisely so as to make this happen. Furthermore, q is a unique solution for the equation int2(p, q) = J. To see this, let us suppose that int 2 (p,q') = J; we shall show that q' must equal q. Let t' be a member of q'; then by the nature of int 2 , in^ (s, t') lies in the congruence-class J. So intj (s, t') is congruent to int^s, t). Thence, applying the second lemma of footnote 2, we infer that int^s, t')"1 is congruent to int^s, t)"1. Or: int^t', s) is congruent to int t (t,s). Then int^t', s^nt^s,t) is congruent to int^t,s)int 1 (s,t). Or: inti(t', t) is congruent to int^t, t) = e. But then t' is equivalent to t under the induced equivalence relation: The interval between them is congruent to e. Hence t' and t lie in the same equivalence class; that is, q' = q as asserted, q.e.d. 3.2.4 DEFINITION: Given the situation as in 3.2.3, the GIS(S 2 ,IVLS 2 ,int 2 ) will be called the quotient GIS of (S^IVLS^intJ modulo CONG. We write GIS2 = GIS1/CONG. Thus, to review yet once more the specific example discussed just preceding 3.2.1, let (S l5 IVLS15 int x ) be the GIS such that S t is an infinite chromatic scale, IVLSj is the additive group of integers, and int x (s, t) is the number of

35

3.2.4

36

Generalized Interval Systems (2)

semitones up from s to t. Let CONG be the relation on IVLSj that makes i congruent to i' when the intervals differ by any integral multiple of 12 semitones. Then the quotient GIS(S2, IVLS2, intj), constructed by the method of section 3.2, has these components: S2 is the family of twelve pitch-classes, IVLS2 is the integers modulo 12, and int^p, q) is the reduction modulo 12 of the integer intj(s, t), where s and t are any pitches belonging to the pitch classes p and q respectively. Here is another specific example. S, is the family of all pitches that can be conceptually generated from a given pitch using just intonation. IVLSj is the group under multiplication of all rational numbers that can be written in the form 2a3b5c, where a, b, and c are integers. int,(s, t) is the fundamental frequency of t divided by the fundamental frequency of s. (Sp FVLSj, intj) is a GIS. It was studied earlier in example 2.1.5. When we informally "modularized" that GIS to obtain the "modular harmonic space" of example 2.1.6, we were actually constructing a quotient GIS, using a certain congruence relation on IVLSj. Specifically, we declared intervals i and i' within IVLSj to be congruent if i' was some-power-of-2 times i. As we noted earlier (in example 1.10.4.2), this relation is indeed a congruence on IVLSj: It is an equivalence relation; further, if I' is some-power-of-2 times I and j' is some-power-of-2 times j, then i'j' is some-power-of-2 times ij. Using this congruence, we can note how the formal constructions of section 3.2 go through for the GIS at hand, producing the GIS for example 2.1.6 as a quotient GIS. By Definition 3.2.1, the induced equivalence on Sj makes s equivalent to s' if and only if intj(s,s') is some-power-of-2. Thus s and s' enjoy the induced equivalence relation here if and only if the pitches lie some number of octaves apart. The equivalence classes (members of S2) are thus precisely the pitch classes determined by the pitches of S,. And, given pitch classes p and q, int^p^) in the quotient group is well defined as the congruence class to which intj(s, t) belongs, whenever s and t are members of p and q. If the intervals, i, i', i", . . . all belong to this congruence class, then we can write i = 2a3b5c, i' = 2a'3b5c, i" = 2a"3b5c, ... and so on. So we may identify the congruence class with the number-pair (b, c). When s and t are members of p and q, then t, give or take some number of octaves, lies b twelfths and c majorseventeenths from s. Taking some octaves, we can say that t, give or take some number of octaves, lies b fifths and c major-thirds from s. Returning our attention to the pitch classes p and q, of which s and t are members, we can say that q lies b dominants and c mediants from p. That is, on the game board of figure 2.2, q lies b steps east and c steps north from p. The quotient GIS here is thus the GIS associated with that game board in example 2.1.6. As a further specific example, the reader can work out how the formal constructions of section 3.2 apply to the "diatonic scale" GIS associated with example 2.1.1, together with the congruence which collects octave-related intervals (those here which differ by multiples of 7 scale steps), to give rise to a

Generalized Interval Systems (2)

3.3.1

quotient GIS which is in fact the GIS of example 2.1.4, a GIS of seven scaledegrees. In similar fashion the GIS associated with the time-point space of example 2.2.1, together with the congruence which collects time intervals differing by multiples of "N beats," gives rise to a quotient GIS which is in fact the GIS of example 2.2.2, the GIS associated with the space of N beat-classes. Likewise, the GIS of example 2.2.3, together with a suitable congruence, gives rise to the GIS of example 2.2.4 as a quotient GIS; the discussion of example 2.2.4 may be reviewed in this connection. The reader may also review further the discussion of example 2.1.6, to see how the GIS of just pitch-classes, corresponding to the game board of figure 2.2, can give rise to a quotient GIS whose space is an infinite one-dimensional chain of dominant-related pitch classes in quarter-comma mean-tone temperament. The congruence at issue declares two "moves" (b, c) and (b', c') on the game board of figure 2.2 to be congruent if there is some integer N such that b' = b + 4N while c' = c — N, that is, if b' steps east is the same as b steps east plus 4N steps east, while c' steps north is the same as c steps north and then N steps south. ( — N east, north, etc. means N west, south, etc.) If we take b = 0, c = 0, and N = 1 in the above arithmetic, then b' = 4 and c' = — 1. Thus going four squares east and one square south on the game board is a move congruent to "staying still." The reader may verify that the defined relation is a congruence, and that the induced equivalence relation on the game board renders equivalent just those pitch classes that share the same letter-name (differing, if at all, only by subscript). Each of the specific examples above instances, in one setting or another, the general and abstract relation of a given GIS, modulo a given congruence, to a quotient GIS. We can see thereby how ubiquitous the quotient construction really is, as a method of generating new GIS structures from old. Another useful method is the construction of a "direct-product GIS." Before proceeding to an abstract discussion of that idea, let us study some examples of it. 3.3.1 EXAMPLE: Let GISj be the GIS of example 2.1.3: Sj is the twelve equally-tempered pitch classes; IVLSi is the integers modulo 12; int^p, q) is the number of semitones up, mod 12, from any pitch within class p to any pitch within class q. Let GIS2 be the GIS of example 2.2.1: S2 is an indefinite series of equally spaced time-points; IVLS2 is the integers under addition; int2(s, t) is the number of beats from time-point s to time-point t. We construct a new GIS, GIS3, as follows. S3 is the Cartesian product of S x and S2, that is, S3 = Si x S2. S3 is thus the family of ordered pairs (p, s), where p is a pitch class in Sj and s is a time point in S2. IVLS3 is the direct-product group of the groups IVLS! and IVLS2; that is, IVLS3 = IVLSj (g) IVLS2. A sample member of IVLS3 thus has the form (ij, i 2 ), where ij is an integer mod 12 (member of IVLSj) and i 2 is an integer (member of IVLS2). The members of IVLS3 combine under the law of 1.13 for direct-

37

3.3.1

Generalized Interval Systems (2)

product groups: (ilt\2) + (JiJ 2 ) = Oi + Ji,i 2 + J2>; here the sum h + Ji is computed mod 12 in IVLS l9 while the sum i 2 + J 2 is the ordinary integer sum in IVLS2. The function int3 works as follows. Given two members (p, s) and (q, t) of S3, each a pitch-class-cum-time-point pair, the interval int3((p, s),(q, t)) is taken as the pair (int 1 (p,q),int 2 (s,t)) within the group IVLS 1 (g)IVLS 2 = IVLS3. (S3, IVLS3, int 3 ) as constructed is in fact a GIS. The reader may take this on faith or verify it as an exercise. Later on we shall call this GIS the direct product GIS of GlSi and GIS2, writing GIS3 = GISj ® GIS2. Right now, let us explore what GIS3 models. One sample member of S3 is the pair (C#, 35), which models a reference to pitch class C# at time 35. Another member of S3 is the pair (F,46), which models a reference to pitch class F at time 46. The interval between the two cited objects is (int^Cft, F),int 2 (35,46)) = (4,11). The compound interval (4,11) models the spanning of pitch-class interval 4 between events happening at time-points 11 beats apart.

FIGURE 3.1

38

Figure 3.1 will be used to suggest the relevance of GIS3 for musical analysis and theory. It transcribes the opening of the third movement from Webern's Piano Variations op. 27. The brackets on the figure display some GIS3-intervals of interest, using the written quarter note as the temporal unit. Temporal intervals between time-points are calculated between attacks of the notes. Thus the bracket from E|? to D on figure 3.1 is labeled by the GIS3interval (11,5): A pitch-class interval of 11 is spanned between the attack of the Eb and the attack of the D, 5 beats later. Similarly, the bracket on the figure between D and C# is labeled by the GIS3-interval (11, 5): A pitch-class interval of 11 is again spanned between two time-points 5 beats apart, here between the attack of the D and the attack of the C#. The GIS3-interval (11,1) labels the brackets extending from B to B[? attacks, from C# to C attacks, and from A to G# attacks; in each case, the pitch-class interval 11 is spanned between events happening 1 beat apart. The recurrences of GIS3-intervals on figure 3.1 are of analytic interest,

Generalized Interval Systems (2)

3.3.1

for not many GIS3-intervals occur more than once in this passage. The recurrent GIS3-interval (11, 1) is associated with a quiet slurred figure in the left hand which we shall call "the accompaniment figure"; this figure occurs in the music at B-B|> and again A-G#. However the musical presentation of C#-C, also associated with the GIS3 interval (11, 1), does not project the accompaniment figure; rather, it is loud and staccato. The recurrence of the GIS3 interval (11, 1) on figure 3.1 imbues the pitch-class interval 11 with a special function: As the passage unfolds in time, pitch-class interval 11 becomes bound up with defining the beat. That is, pitch-class interval 11 recurs significantly in connection with beat-interval 1, forming the GIS3-interval (11, 1). And no other pitch-class interval recurs in conjunction with beat-interval 1. This special beat-defining function for pitch-class interval 11 gives special meaning in turn to the recurrence of GIS3-interval (11, 5) on figure 3.1, a GIS3-interval which also involves pitch-class interval 11. Via the recurrent GIS3-interval (11, 5), the temporal interval of 5-beats-later becomes associated with the pitch-class interval 11, a pitch-class interval of special mensural status. "5 beats later" thereby acquires a special mensural status itself, linking it with "1 beat later" via the pitch-class interval 11 that is shared by the recurrent GIS3-intervals (11, 5) and (11, 1). This special mensural status for "5 beats later" is not the only reason many analysts hear the music "in * meter," but it does endow the possibility of such a hearing with a special meaning and thematic richness.3 I myself believe that pertinent statements involving GIS3-intervals provide a more exact and less problematic account of the mensural structures at issue here, then does the notion of "^ meter." Figure 3.1 also shows the recurrence of the GIS3-interval (3, 2), spanning B-attack to D-attack and also E-attack to G-attack. The recurrent (3, 2) seems to have a cadential function in the music, to my ear, "2 beats later," the "2" of the GIS3-interval (3, 2), engages the notated meter of the music, which "5 beats later" does not. In these connections, the pitch-class interval 3, as it recurs within the GIS3-interval (3, 2), has its own special function, to my ear a cadential function. Hearing a cadence on the D and another on the G is consistent with the maximal amount of time we wait, after the attack of the D and again after the attack of the G, before hearing the next attack in each case. It is curious that the next attack after this maximal wait comes precisely "5 beats later" in each case. Indeed if we start measuring spans of 5 beats starting 3. A \ "cross-meter" is heard as one of several contending temporal patterns by Edward T Cone, "Analysis Today," Problems of Modem Music, ed. Paul Henry Lang (New York: W. W. Norton, 1962), p. 44. Elsewhere, we can read that the first three notes we hear form a rhythmic group which "is clearly *; the motion of the two quarter notes into the next downbeat defines the meter precisely." This is the hearing of James Rives Jones, "Some Aspects of Rhythm and Meter in Webem's Opus 27," Perspectives of New Music vol. 7, no. 1 (Fall-Winter 1968), p. 103. Later, still on page 103, he refers to "the \ meter" as "already ... established" by the cited criterion.

39

3.3.1

Generalized Interval System (2)

at the attack of the opening El., then the D and the G on figure 3.1, with the rests that follow them, each fill one such span. I do not know what to make of this. The idea of quintuple "perfections" seems a better metaphor for my hearing this aspect of the music than does the idea of quintuple "meter." Figure 3.1 also shows the recurrent GIS3-interval (2, 7), interlocking the mensural function of "7 beats later" with the pitch-class interval 2. The first (2, 7) recurrence links the two notes of the B-B[> accompaniment figure to the corresponding two notes of the C|-C figure, 7 beats later. The second recurrence (third occurrence) of (2, 7) links the attack of the cadential G to the beginning of the accompaniment figure A-G| 7 beats later. In sum, the recurrent GIS3-interval (2, 7) links aspects of the recurrent accompaniment figure with other events of the music.

FIGURE 3.2

40

Figure 3.2 summarizes the discussion of figure 3.1 so far, in the form of a table. It shows how the recurrence of GIS3-intervals gives special meanings to the pitch-class intervals 11, 3, and 2, as those interrelate with the temporal

Generalized Interval Systems (2)

3.3.1

intervals 1,5,2, and 7, all in connection with various compositional features of the music. 83-elements

GIS3-interval vectors of S3 sets

FIGURE3.3 Figure 3.3 applies to the opening of the passage a theoretical construction suggested by the temporal aspect of GIS3. The left-hand column of the figure lists the first six members of S3 in their order of appearance during the passage. First comes pitch-class Eb at time-point 0, instancing element (Eb,0) of S3. Next comes pitch-class B at time-point 3, instancing element (B, 3) of S3. At this time (i.e. just after time-point 3), we become aware of a 2-element S3-set, that is, the set ((Eb, 0), (B, 3)). The elements of this set form one GIS3-interval with a positive time-component, that is, the interval (8,3) from (Eb,0) to (B, 3). The GIS3-interval (8,3) is listed in the second row of the second column in figure 3.3; that interval is the sole constituent within the interval vector of the 2-element S3-set. Now the pitch class 8(7 occurs at time-point 4, providing the new S3 element (Bb,4) for the third row of column 1 on figure 3.3. At this time (i.e. just after time-point 4), we become aware of a 3-element S3-set, ((Eb,0),(B,3),(Bb,4)). Besides the GIS3-interval of (8,3) already listed in row 2, the elements of the 3-member set produce new GIS3-intervals of (7,4) (= int 3 ((Eb,0),(Bb,4))) and (11, !)(= int 3 ((B,3),(Bb,4))). These new constituents for the interval vector of the expanded 3-element S3-set are listed in the third row of column 2 on figure 3.3; the old interval vector expands to include the occurrences of the two new intervals. When pitch class D enters at time-point 5, the 3-element S3-set expands to a 4-element S3-set ((Eb,0),(B,3),(Bb,4),(D,5)), and the interval vector of the 3-element set expands to adjoin new occurrences of the GIS3-intervals (11,5)(= int 3 ((Eb,0),(D,5))), (3,2)(= int 3 ((B,3),(D,5))), and (4,1) (= int 3 ((Bb,4),(D, 5))). The new intervals are listed in the fourth row of column 2, on figure 3.3. At this time (i.e. just after time-point 5), we become

41

3.3.1

Generalized Interval Systems (2)

aware for the first time that some pitch-class intervals are predominating over others, and that some temporal intervals are predominating over others, as we note the various intervals going by. In earlier work I have suggested that our becoming aware of such predominances is associated with our marking such a time-point as an "ictus."4 The present analysis supports that theoretical idea, since time-point 5 is both a notated barline, indeed the first notated barline, and also audible to some extent as marking the attack of an intuitively "strong" quarter. These considerations lie behind my bracketing the first four entries of column 1 on figure 3.3, as belonging together in a special way. According to the theory just sketched, it is only when the 4-element S3-set has been completely exposed, i.e. it is at and only at a moment just after time-point 5, as we listen along, that we first become sensitive to any regular mensural structuring in the passage. Just after time-point 5 we become aware that temporal interval 1 is predominating over other temporal intervals; we can then (and only then) hear temporal interval 1 as a beat with which to measure other temporal intervals. In that sense the GIS3-structure of figures 3.1-3.3 really "begins" for a listener at (and only at) time-point 5, the first written barline and the first perceived ictus; any GIS3-structure preceding time-point 5, according to this theory, is reconstructed^ a listener at (or following) time-point 5. Not only is the beat established at time-point 5, the pitch-class interval 11 is also established at the same time, as a predominating pitch-class interval. Pitch-class interval 11 is thus bound psychologically to the establishment of mensural structure in the piece, as part of one and the same Gestaltist experience that a listener will be having just after time-point 5.

FIGURE 3.4

Figure 3.4 symbolically collates the ideas discussed just above. The figure also suggests how the temporal interval of "5 beats later" has already acquired a special significance at the moment the listener hears time-point 5. "5 beats later," namely, spans the temporal distance from the opening attack to the first ictus. Figure 3.4 shows how that temporal distance is already clearly associated with the structural function of the recurrent pitch-class interval 11,

42

4. David Lewin, "Some Investigations into Foreground Rhythmic and Metric Patterning," Music Theory, Special Topics, ed. Richmond Browne (New York: Academic Press, 1981), 101-37.

Generalized Interval Systems (2)

3.3.1

the pitch-class interval between the opening Eb and the D of the first ictus, 5 beats later. These considerations enable us to analyze with greater precision how the idea of "being in 4" might arise, and how it would become associated with the compositionally thematic E(?-D gesture.5 Let us now inspect the fifth row of figure 3.3, investigating how our impressions develop when C# is attacked at time-point 10, introducing new manifestations of the GIS3-intervals (10,10), (2,7), (3,6), and (11,5). We now hear a second pitch-class interval of 3, but simultaneously we also hear a third pitch-class interval of 11. The latter interval, by recurring yet again, continues to predominate over other pitch-class intervals. Indeed, its predomination itself recurs. Furthermore, we now (i.e. just after time-point 10) hear for the first time the recurrence of a GIS3-interval (not just of a temporal interval or pitch-class interval). The recurring GIS3-interval is (11,5), recently discussed in connection with the possibility of asserting a "thematic | meter" listening at time-point 5. The sensations prompting such a possible assertion are thereby intensified at time-point 10. Time-point 10 is experienced as an "ictus" in the formal sense of the theory mentioned earlier. Dynamic and registral accents at time-point 10, the loud low C# attack, support the possible hearing of a "strong beat" there, should one want to assert "| meter" beyond purely mensural considerations. We can then expand figure 3.4 to figure 3.5, which portrays a provisional impression one might have while listening just after time-point 10.

FIGURE 3.5

Now let us turn to the sixth row of figure 3.3, investigating our impressions when C is attacked at time-point 11, introducing new manifestations of the GIS3-intervals (9,11), (1,8), (2,7), (10,6), and (11,1). (11,1) here is a recurring GIS3-interval; it confirms our already developed sensations about the beat-defining and other mensural functions of the pitch-class interval 11. The GIS3-interval (2,7) also recurs at time-point 11. This builds up another 5. My analysis of listener psychology just after time-point 5 partly elaborates, partly qualifies, and partly takes issue with the thought-provoking approach to this passage by Christopher Hasty, in his important methodological and analytic study, "Rhythm in Post-Tonal Music: Preliminary Questions of Duration and Motion," Journal of Music Theory vol. 25, no. 2 (Fall 1981), 183-216.

43

3.3.1

Generalized Interval Systems (2)

mensural matrix that tries to expropriate (11,1) for its own purposes, trying to put an ictus at time-point 11, rather than time-point 10. Figure 3.6 sketches this notion.

FIGURE 3.6 The ictus on C in figure 3.6 corresponds to a written barline; this was not the case with the conflicting ictus on C# in figure 3.5. That C# was 5 beats after the D-ictus of figure 3.4; the C of figure 3.6, which picks up the Hauptstimme D in register, is 6 beats after the D-ictus. The mensural conflict of "5 units after" (D-ictus to C#-ictus) and "6 units after" (D-ictus to C-ictus) is highly thematic in op. 27 as a whole.6 The mensural reading of figure 3.6 tries to associate the C#-C event in the music with the "accompaniment figure," the figure that projected the B-B[? event. In contrast, the mensural reading of figure 3.5 tried to associate the D-C# descent with the thematic E[?-D descent. Figure 3.5, of course, "did not know about" the forte and staccato C natural coming up right after the forte and staccato C#. We have already discussed how the recurrent GIS3-interval (2,7) interacts with the accompaniment figure more generally. This completes the exegesis of figure 3.3. The theoretical notion of an "unfolding interval vector," made abstractly available by the temporal aspect of GIS3, was analytically useful for examining our impressions of figure 3.1 as those developed note-by-note, and for discussing to a significant extent our impressions of the music beyond that. GIS3 was particularly useful because it enabled us to consider pitch-class structure and mensural rhythmic structure in conjunction with each other, rather than as independent features of the passage. That is the essence of GIS3 in its capacity as what we shall soon call the direct product of GISt and GIS2.

44

6. A conflict between mensural distances of 5 and 6 units figures in the relation of the rhythmic ostinato to the written meter at the opening of the first movement. A reprise of this rhythmic conflict occurs at the opening of the coda in the last movement (mm. 56-62). The written meters are functional, in ways too complex to indicate here. Special accents attach to the loud and dense trichord-pairs of the middle movement. The lower chords of the trichord-pairs attack at the barlines of measures 4, 9, 4 bis (6 measures after 9), 9 bis, 15, 20, 15 bis (6 measures after 20), and 20 bis. The resulting pattern projects alternating spans of 5 and 6 written measures. Indeed this may well be the strongest mensural function for the written measure as a temporal span in the movement.

Generalized Interval Systems (2)

3.3.3

3.3.2 EXAMPLE: Let GISj be the GIS involving time-points, that just figured as "GIS2" in the preceding example. For the present example, let GIS2 be a GIS of durations as in 2.2.3 earlier: S2 is a certain family of "durations" x, y,... related by certain stipulated proportions; IVLS2 is the multiplicative group of such proportions; int2(x, y) is the quotient y/x. We construct a new GIS, GIS3, as follows. S3 is the Cartesian product S x x S2, that is, the family of pairs (s,x), where s is a time-point and x is a duration. We can conceive (s, x) as modeling an event that begins at time-point s and extends for a duration of x (units) thereafter. IVLS3 is the direct-product group IVLSj (8) IVLS 2. Each member of IVLS3 is a pair (il5i2), where ilis a member of IVLSj (representing a number of beats between time-points) and i2 is a member of IVLS2 (representing a quotient of durations). (i ls i 2 ) and (Ji,j 2 ) combine in the direct-product group IVLS3 according to the rule (i 1 ,i 2 )(Ji»J2) = fli + Ji^zJa)- int3(( s » x )>(t,y)) is defined as (int^s,!), int2(x,y)), that is, loosely speaking, as (t — s,y/x). To put it another way, if int3((s, x), (t, y)) = Oi, i2), then event (t, y) begins i t units of time after event (s, x) and extends for i2 times the extent of event (s, x). The reader can take it on faith that GIS3 as constructed above is indeed a GIS. It will not be necessary to produce an analytic example, I think, in order to convince the reader that this GIS is a useful theoretical tool. It combines two aspects of our mensural rhythmic intuition, as they impinge upon us conjointly, into one compound structure. We are still and again presuming a fixed unit of time here, by which we measure durations and distances-betweentime-points. In both example 3.3.1 and example 3.3.2 we combined a given OK^ with a given GIS2 in a certain manner, obtaining a new GIS that modeled the conjoint action of the two given GIS structures. We can make our procedure formal and general, as a means of combining any given GlSi and any given GIS2 into a "direct-product GIS." The following definition gives the procedure. 3.3.3 DEFINITION: Given G^ = (S l5 IVLS^intJ and GIS2 = (S2, IVLS2, int2), the direct product of GISa and GIS2, denoted GIS1

45

3.4.1

Generalized Interval Systems (2)

It is straightforward to verify that GIS3, as defined above, is indeed a GIS, i.e. that GIS3 satisfies Conditions (A) and (B) of Definition 2.3.1. This finishes our investigation into methods of constructing new GIS structures from old in a general abstract setting. Now we shall see how the notions of "transposing" and "inverting" elements of a space S arise naturally in any GIS; we shall see how operations of transposition and inversion interrelate characteristically with intervallic structure, and we shall explore how the operations combine among themselves. We shall also explore other characteristic operations, the "interval-preserving operations"; these may or may not be the same as the transposition operations, depending upon whether the group of intervals is or is not commutative.7 3.4.1 DEFINITION: Given a GIS; given an interval i of IVLS; then transposition by i, denoted Tj, is defined as a transformation on S as follows. Given a sample element s of S, the i-transpose of s, T;(s), is that unique member of S which lies the interval i from s. That is, T;(s) is well defined by the equation int(s,T,(s)) = i. This definition conforms to our abstract intuition that the i-transpose of a given element s should lie the interval i from s. The definition also conforms to the way in which we already use the word "transposition" in connection with some GIS structures, namely those involving pitches and pitch classes. T;(s) is indeed well defined by the equation within the definition. Condition (B) of 2.3.1 assures us that given i and s, there exists a unique t such that int(s, t) = i; it is precisely this t which we are now calling "the i-transpose of s," Tj(s). 3.4.2 THEOREM: Each Ts is an operation; that is, it is 1-to-l and onto as a transformation on S. The transposition operations form a group of operations on S. That group is anti-isomorphic to the group of intervals. Specifically, let us consider the map f, defined from IVLS onto the family of transpositions by the formula f(i) = T^ then f is an anti-isomorphism. That is, f is 1-to-l as well as onto; and TjTj = T^. Pr00/(optional): We shall prove the assertions of the theorem in an order different from that in which the theorem states them. First we show that f is an anti-homomorphism, i.e. that TjTj = T^. Given intervals i and j, given a sample s in S, then we write int(s, Tj(Tj(s))) as the group product of the two intervals int(s, Tj(s)) and int(Tj(s), T;(Tj(s))), using Condition (A) of 2.3.1. Now int(s, Tj(s)) = j, by the defining equation of Definition 3.4.1. And by the same equation, int(Tj(s), Tj(Tj(s))) = i. Hence 46

7. All the groups in our specific examples of GIS structure so far have been commutative. Later on, in chapter 4, we shall study a non-commutative GIS of musical interest.

Generalized Interval Systems (2)

3.4.4

int(s, Ti(Tj(s))) is expressed as the group product of the two intervals j and i. That is, int(s,Ti(Tj(s))) = ji. Thus Tj(Tj(s)) lies the interval ji from s. So it is equal to TJJ(S). We have shown: For any sample s, TjTj(s) = TJJ(S). So the transformation TjTj has the same effect on S as does the transformation T^; the functional equation TjTj = T^ is true, as claimed. So the map f is an anti-homomorphism. We show now that it is an antiisomorphism. We have only to show that f is a 1-to-l map. Supposing that the functional equation Ts = Tj is true, we wish to prove that i and j must be the same interval. Fix any s. Since Ts = Tj by assumption, Tj(s) = Tj(s). Then, by Definition 3.4.1, i = int(s,Ti(s)) = int(s,Tj(s)) = j, which is what we had to show. It remains only to prove that each Tj is an operation, and that the family of transpositions is a group of operations. To prove all this, it suffices to show that the family of transpositions satisfies Conditions (A) and (B) of 1.3.4 earlier, namely (A) that the family is closed and (B) that for each Tj there is a TJ satisfying TjTj = TjTj = 1. Condition (A) is evident from the formula TjTj = Tjji The composition of two transpositions is a transposition. In order to prove Condition (B), we shall prove a lemma: Te is the identity operation 1 on S. That is true since, given s in S, Te(s) is the member of S which lies the identity interval from s; hence Te(s) = s. Or Te(s) = l(s); that being the case for any sample s, Te = 1 as asserted by the lemma. Now we can prove Condition (B). Given any interval i, take j = i"1. Then ji = ij = e. Tj is the transformation demanded by Condition (B): TjTj = Tjs = Te = 1; likewise TjTj = 1. q.e.d. 3.4.3 THEOREM: Fix some referential member ref of S; then LABEL(Ti(s)) = LABEL(s) • i. Proof: LABEL(Ti(s)) = int(ref, T,(s)) by definition of LABEL in 3.1.1, = int(ref, s)int(s, Ti(s)) by 2.3.1 (A), = LABEL(s)int(s, Tj(s)) by 3.1.1, = LABEL(s)-i by 3.4.1. Theorem 3.4.3 tells us that no matter what ref is chosen for LABELing purposes, the label for the i-transpose of s is the label of s, right-multiplied by i in IVLS. The natural question arises: What happens when we left-multiply ref-LABELs by i? We shall explore that right now. 3.4.4 DEFINITION: Fix some referential member ref of S. Given any interval i, the transformation Pj (more exactly P[ef)is defined on S by the formula LABEL(Pj(s)) = i • LABEL(s), which is to say the relation int(ref, Pi(s)) = i • int(ref, s).

47

3,4.5

Generalized Interval Systems (2)

Given ref, i, and s, Condition (B) of 2.3.1 tells us that a unique member of S, call it t, satisfies the relation int(ref, t) = i • int(ref, s). Definition 3.4.4 takes that unique t, given ref, i, and s, and calls it Pj(s). Note that the specific value of Pj(s) depends on ref as well as on i and s. In contrast, the value of T.(s) was well defined in 3.4.1 independent of any choice of ref. 3.4.5 THEOREM: The transformations P. form a group of operations isomorphic to IVLS under the map f(i) = P}. In particular, the formula PjP. = Pr is valid. Proof (optional): LABEL(Pj(p.(s))) = i • LABEL(P.(s)) = i • (j • LABEL(s)) = (ij) • LABEL(s) = LABEL'S)). The elements Pj(Pj(s)) and Py(s) thus have the same LABEL (lie the same interval from ref). So they are the same: PjPj(s) = Pjj(s). This being the case for any sample s, the functional equation PjP. = P.. is true. The map f of the Theorem is thereby a homomorphism of IVLS onto the family of transformations P.. To prove that f is an isomorphism, it remains only to show that f is 1-to-l. Suppose P. = P.; we are to infer that i = j. Fix any s; then i • LABEL(s) = LABEL(pj(s)) = LABEL(Pj(s)) by supposition; and that = j • LABEL(s). In sum, i • LABEL(s) = j • LABEL(s). Hence, multiplying that equation through on the right by the inverse of LABEL(s), i = j as desired. We can now use arguments just like those we used in the proof of 3.4.2, to show that the family of P. is a group of operations. The operations P. have a special property. They are what we shall call the "interval-preserving" transformations. 3.4.6 and 3.4.7 develop the formalities. 3.4.6 DEFINITION: Given a GIS (S, IVLS, int), a transformation X on S will be called "interval-preserving" if X has this property: For each s and each t, int(X(s), X(t)) = int(s, t). 3.4.7 THEOREM: No matter what ref is chosen, the interval-preserving transformations on S are precisely the P.. Proof (optional): We show first that P. preserves intervals. We can write inters), P,(t))

=LABEL(P i(s))-1LABEL(Pi(t)) by 3.1.2, =(i • LABELS(s))-!(i • LABEL(t)) =(LABEL(s)~1 • i-])(i • LABEL(t)) =LABEL(s)-1LABEL(t) =int(s, t) 48

by 3.4.4,

by 3.1.2.

Thus P. preserves intervals. Now suppose X is any interval-preserving

Generalized Interval Systems (2)

3.4.8

transformation on S. We shall show that there exists some interval i such that X-P,. The i we want here is LABEL(X(ref)) = int(ref, X(ref)). Given any s, we can then write LABEL(X(s)) = int(ref, X(s)) = = = = =

int(ref,X(ref))int(X(ref), X(s)) via 2.3.1 (A), i • int(X(ref), X(s)) by construction of i here, i • int(ref, s) because X is interval-preserving, i • LABEL(s) by the definition of LABEL, LABEL(Pi(s)) via 3.4.4.

In sum, we have LABEL(X(s)) = LABEL(Pj(s)). Since X(s) and Ps(s) have the same LABEL, we infer that X(s) = Pj(s). Since s here is an arbitrary sample member of S, X = Pi as a transformation on S. Thus our intervalpreserving X is in fact this particular P{. q.e.d. Because of 3.4.7 we can conceive the transformations P5 as somewhat less dependent on the choice of ref. It is true that a transformation labeled "Pj" by one choice of ref might be labeled "Pj" by another choice of ref. However the interval-preserving transformations en masse, literally "as a group," remain the same family of transformations en masse, no matter what ref is chosen. The interval-preserving property does not depend on the choice of ref, and that property is sufficient to define the family of transformations as a group. 3.4.8 THEOREM: Given an interval i, Conditions (A) through (D) below are all logically equivalent: If any one of them is true, then they are all true. (A): T; preserves intervals. (B): For some choice of ref, Tj = P}. (C): For any choice of ref, Tj = Pj. (D): i is central in IVLS. That is, i commutes with every j in IVLS (1.8.2). Proof (optional): Suppose (A) is true; we prove that (C) follows. Fix any ref. Since Tt preserves intervals by assumption, 3.4.7 tells us that Tj = Pj for some j. For any s, LABEL(s) • i = LABEL(T;(s)) by 3.4.3 that = LABEL(Pj(s)) since T, = Pjj and that = j • LABEL(s) by 3.4.4. In sum, LABEL(s) • i = j -LABEL(s) for any s. Consider s = ref in particular LABEL(ref) • i = j • LABEL(ref). But LABEL(ref) = int(ref, ref) = e. Henc i = j. Tj = Pj as desired. Clearly the truth of (C) entails the truth of (B). Now suppose (B) is true; we prove that (D) follows. We are supposing Ts = Pj for some ref. Then for every s, LABEL(Tj(s)) = LABEL(Pj(s)). Then for every s, LABEL(s) • i = i • LABEL(s) (3.4.3 and 3.4.4). It follows that i commutes with every j, which is Condition (D) as desired. For given any j, find the s which lies the interval j from ref; then LABEL(s) = j; substituting j for LABEL(s) in the most recent equation involving i, we obtain ji = ij; i commutes with the given j.

49

3.4.9Generalized Interval Systems (2)

Now we close the logical chain by showing that the truth of (D) entails the truth of (A). When we have done this, we shall have shown that (A) implies (C), (C) implies (B), (B) implies (D), and (D) implies (A); hence the truth of any one entails the truth of all the others. Supposing (D) is true, then, we show that (A) will be true. Fix any ref. Given any s and any t, we write int(Ti(s), Tj(t)) = LABEL(Ti(s))-1LABEL(Ti(t)) = (LABEL(s) • i)~1(LABEL(t) • i) = (r1 • LABEL(s)'1) (LABEL(t) • i) = i"1 • (LABEL(s)-1LABEL(t)) • i = i - 1 -mt(s,t)-i = int(s, t)

by 3.1.2 by 3.4.3

by 3.1.2 by the assumption

of Condition (D), that i is central. In sum, assuming Condition (D), then int(T;(s), T;(t)) = int(s, t) for every s and t. Or, assuming Condition (D), Tj will be interval-preserving. Or: (D) implies (A) as desired, q.e.d. 3.4.9 COROLLARIES: (A): In a commutative GIS (a GIS whose group of intervals is commutative), the transposition operations are precisely the interval-preserving operations. (B): In a non-commutative GIS, there exist transpositions that do not preserve intervals, and there exist interval-preserving operations that are not transpositions. 3.4.10 THEOREM: Any transposition operation commutes with any intervalpreserving operation. Proof: Fixing any ref, consider T; and Pj. We apply 3.4.3 and 3.4.4 to various LABELS. Taking any sample s, we have LABEL(Pj(Ti(s))) = j • LABEL(Tj(s)) = j • (LABEL(s) • i) = (j • LABEL(s)) • i = LABEL(Pj(s)) • i = LABEL(Tj(Pj(s))). Thus PjTj(s) and T,Pj(s) have the same LABEL; they lie the same interval from ref; so PjTj(s) = TjPj(s). This being the case for any sample s, the functional equation PjTj = TjPj is true: Ts commutes with Pr q.e.d.

50

We are now ready to define and study "inversion operations" on an abstract GIS. For each u and each v in S (v may possibly equal u), we shall define an operation I*, which we shall call "u/v inversion." Figure 3.7 helps us visualize an appropriate definition for the operation, conforming to our spatial intuitions. The figure shows how we conceive any sample s and its inversion I(s) (short for !„($) here) as balanced about the given u and v in a certain intervallic proportion. I(s) bears to v an intervallic relation which is the inverse of the relation that s bears to u. The inverse proportion is symbolized by the mirror

Generalized Interval Systems (2)

3.5.2

FIGURE 3.7 relation of the two arrows on figure 3.7. The interval from v to I(s), which is the inverse of int(I (s), v), will then be the same as the interval from s to u. That is, we intuit int(v, I(s)) = int(s, u). We can use this equation to define !„ formally in any GIS. 3.5.1 DEFINITION: Given any u in S and any v in S, the operation !„ of u/v inversion is defined by the equation int(v, Iu(s)) = int(s, u) for all s. The operation is well defined by the equation: Given any s, set i = int(s, u) and find the unique t which lies the interval i from v. That t, which satisfies the equation int(v, t) = int(s, u), is precisely the value for IJ^s). We have been referring to the "operation" I* prematurely; so far we have constructed a transformation, but we have not verified that the transformation is indeed an operation, i.e. onto and 1-to-l. The reader may verify, as an exercise, that I* as defined is onto and 1-to-l. (Given t, find an s such that ru(s) = t. Prove that if IJ(s') = IJ(s), then s' = s.) 3.5.2 THEOREM: Fix a referential element ref of S. Set i = LABEL(v) and j = LABEL(u). Then LABELTOs)) - i • LABEL(s)'1 • j. Proof: To save space, we write "I" for "!„" here. int(v, I(s)) = int(s, u) (3.5.1). So int(v, ref )int(ref, I(s)) = int(s, ref )int(ref, u) (2.3.1 (A)). Thence, LABEL(v)'1LABEL(I(s)) = LABEL(s)~1LABEL(u) (3.1.1; 2.3.2) Or: Or:

T1 • LABEL(I(s)) = LABEL(s)'1 • j LABEL(I(s)) = i • LABEL(s)"1 • j

q.e.d.

51

3.5.3

Generalized Interval Systems (2)

The formula of this theorem is very useful despite the dependence of all the LABELs (including i and j) on a possibly arbitrary ref. We shall use the formula to analyze this question: When are lvn and I* the same operation on S? In the familiar GIS of twelve chromatic pitch classes, for instance l£ = l£{ = IA' = ID» = IB'anc* so on- '^iat *s' ^^ mversi°n nas the same effect on any sample pitch class as does F|/F# inversion, or A/Df inversion, or DJt/A inversion, or B/C| inversion, and so on. In this special GIS one can intuit that l£ and I* will be the same operation if and only if the pitch class w is the C/C inversion of the pitch class x. One might thereby conjecture that I* and I* will be the same operation, in a general GIS, if and only if the thing w is the u/v inversion of the thing x. This conjecture is valid if the GIS is commutative. When the GIS is not commutative, we must be content with the broader view provided by the following theorem. 3.5.3 THEOREM: I* = I* as an operation on S if and only if w = I*(x) and the interval int(x, u) is central. Proof (optional): Imagine ref fixed. Let i, j, k, and m be the respective LABELS for v, u, w, and x. Then, via 3.5.2, given any s, LABEL(I^(s)) = i LABEL(s)-1 j, while LABEL(I*(s)) = k LABEL(s)-1 m. Hence the condition that I* be the same operation as I* is equivalent to the condition that i LABEL(s)'1 j = k LABEL(s)-1 m for all s in S. Now as s runs through the various members of S, the inverse of its LABEL runs through all the various intervals in IVLS. Hence the condition under discussion is equivalent to the condition that inj = knm

for every n in IVLS.

And that condition is equivalent, via group theory, to Condition (A) below. (A): (k-1i)n = nOnj"1)

52

for every n in IVLS.

In sum, F[ = I* as an operation if and only if Condition (A) above is satisfied. Now we shall prove that Condition (A) is satisfied if and only if w = I*(x) and int(x, u) is central. That will prove the theorem as stated. Suppose, then, that Condition (A) is true; we shall show that w = IJJ(x) and int(x, u) is central. Condition (A) being true by supposition, it is true for n = e; therefore k~ ! i = mj"1. Let us call this interval c. Condition (A) then tells us that en = nc for every n in IVLS, so c is central. c was taken equal to mj"1. It turns out that c also equals j"1!!!. To see that, we write mj"1 = c; thence m = cj; thence, since c is central, m = jc; thence j-1m = c as asserted.

Generalized Interval Systems (2)

3.5.5

Let us review: Assuming Condition (A), we have so far shown that k 4 = mj-1 = j^m = c is central interval. Now k~M = LABEL(w)~1LABEL(v) = int(v, w), via 3.1.2. Likewise j-1m = int(x, u). Thus we have shown: int(v, w) = int (x, u) and int(x, u) = c is central. Now the relation int(v, w) = int(x, u) is exactly the relation which tells us that w = IJj(x) (Definition 3.5.1). Thus, assuming Condition (A), we have proved that w = I^(x) and int(x, u) is central as desired. Now we prove the converse half of the theorem: Supposing that w = I*(x) and that int(x, u) is central, we prove that Condition (A) is satisfied. w int(v, w) LABEL(w)-1LABEL(v) k~U

= = = =

I*(x) int(x, u) LABEL(u)~1LABEL(x) j-1m

(by supposition). So (3.5.1). Or: (3.1.2). Or: (meaning of i, j, k, m).

Furthermore, we have supposed that j^m, which is LABEL(u)~1LABEL(x), which is int(x, u), is a central interval. Let us call this central interval c. j^m = c; thence m = jc; thence m = cj; thence mj"1 = c. So k H = mj

l

= c, a central interval.

c being central, en = nc for every n in IVLS. Substituting k~!i = mj"1 for c, we obtain (k~ ] i)n = n(mj-1) And that is precisely Condition (A),

for every n in IVLS. q.e.d.

We used ref and LABEL in our proof of Theorem 3.5.3. Note, however, that the statement of the theorem does not depend on a choice of ref. 3.5.4 CORROLARY: I* = IJJ if and only if int(v,u) is central. Proof: Applying the formula of Definition 3.5.1 to the algebraic truism int(v, u) = int(v, u), we infer that u = I*(v). Then, by the logic of Theorem 3.5.3 just proved, 1^ = IJJ if and only if int(v, u) is central. The corollary tells us that in a general GIS, v/u inversion may well not be the same operation as u/v inversion, despite the relations v = I*(u) = I"(u); u = I^(v) = IJJ(v). The operations I* and l^ both transform u to v, and v to u. But there may be some other s, other than u or v, such that IJj(s) is not the same element as I^(s). Indeed the corollary assures us there will be some such s if int(v, u) is not central. These considerations indicate how carefully and rigorously we must proceed hereabouts; intuitions based on our familiarity with a number of commutative GIS structures will not always be reliable. 3.5.5 COROLLARIES: In a commutative GIS, I* always = I|J; generally, I* = I* if and only if w = I»(x).

53

3.5.3

Generalized Interval Systems (2)

In a non-commutative CIS, there will always be some inversion operation Iy which is not the same as 1^. (For there will always be some int(v, u) which is not central.) Now we shall see how inversion operations I* combine with transpositions Tn and interval-preserving operations P. 3.5.6 THEOREM: For any transposition Tn and any inversion I*, (A): T n l^ = I* where x = Tn(u). (B): I v u T n = I* where w = T-^V). (C): Tn commutes with 1^ if and only if n is central and nn = e. Proof (optional): We shall fix some ref and use LABELs to help our computations. Let i = LABEL(v); let j = LABEL(u). Then LABEL(TnIvu(s)) = LABEL(Ivu(s)) - n = i-LABELS)-1-j-n = LABEL(I^(s)),

(3.4.3) (3.5.2)

where x is the member of S whose LABEL is jn (3.5.2). Since LABEL(x) = jn and LABEL(u) = j, x = Tn(u) (3.4.3). Now Tnl*(s) I*(s) have the same LABELs, via the chain of computations above. Hence Tnl*(s) = I*(s). (Both lie the same interval from ref.) Since s was an arbitrary sample member of S, Tnl^ = I* as an operation. This proves (A) of the theorem. To prove (B), we start with a similar chain of computations. LABEL(PuTn(s)) = = = =

54

iCLABELCT^s)))-1] i(LABEL(s) • n)-1] in~1LABEL(s)-1j LABEL(I-(s)),

(3.5.2) (3.4.3)

where w is the member of S whose LABEL is in -1 (3.5.2). Since LABEL(w) = in"1 and LABEL(v) = i, w is the n"1 transpose of v (3.4.3). Since T"1 = T"1, we can write w = T~!(v). (The map of n to Tn is an anti-isomorphism; n"1 maps to T"1.) We go on to infer the operational equality of the operations I*Tn and 1^, exactly in the way we inferred an analogus equation when proving (A) of the theorem. This proves (B) of the theorem. Using the formulas of (A) and (B) that we have now established, we see that Tnl^ = I*Tn if and only if I* = I* x and w being as in (A) and (B) of the theorem. By Theorem 3.5.3, this will be the case if and only if w = I*(u) and the interval int(u, x) is central. That will be so, according to 3.5.1, if and only if int(v, w) = int(u, x) and int(u, x) is central. But int(u, x) = n, since x = Tn(u), and int(v, w) = n"1, since w = T"1^). So, in sum, Tn will commute with I* if and only if n = n"1 and n is central. This proves (C). q.e.d.

Generalized Interval Systems (2)

3.5.7

(C) of the theorem shows that, given any interval n in a general GIS, either Tn commutes with every inversion operation or Tn commutes with no inversion operation. In the familiar GIS of twelve chromatic pitch-classes, T6 commutes with every inversion operation: If you invert and then transpose by a tritone, the net result is the same as if you transpose by a tritone and then invert (about the same center or axis). In that GIS, no other interval of transposition has this property, save for the trivial interval of zero. (T0 is the identity operation.) In fact, no other Tn in that GIS will commute with any inversion operation. (C) of the theorem shows us that this situation is related to the fact that 6 + 6 = Omod 12, while n + n does not = 0 mod 12 for any other non-zero interval mod 12. Theorem 3.5.6 gave us insight into how inversions combine with transpositions. An analogous theorem will give us analogous insight into how inversions combine with the interval-preserving operations P. 3.5.7 THEOREM: For any interval-preserving operation P and any inversion Iu»

(A): PI* = I? where w = P(v). (B): i;;P = I* where x = P~1(u). (C): P commutes with !„ if and only if P = Tc for some transposition Tc such that c is central and cc = e. Proof (optional): We fix a referential element ref. Setting n = int(ref, P(ref)) = LABEL(P(ref)), we write P = Pn in the manner of 3.4.4 earlier. We can then manipulate pertinent LABELs to prove (A) and (B) exactly as we proved (A) and (B) for Theorem 3.5.6 above. To prove (C), we begin in a manner similar to that by which we proved 3.5.6 (C). Using (A) and (B), we note that PI* = I^P if and only if I£ = I*, where w and x are as in (A) and (B). Via 3.5.3, this will be the case if and only if int(v, w) = int(u, x) and int(u, x) is central. Now LABEL(u)=j and, since x = P~'(u) = Pn-»(u), LABEL(x) = n-1 • LABEL(u) = n-1j (3.4.4). Therefore LABEL(x)~1LABEL(u) = (n'M)"1 j = T1 nj. And int(u, x) is precisely LABEL(x)"1 LABEL(u) (3.1.2). So int(u, x) = j"1 nj. We have now showed: P = Pn commutes with I* if and only if int(v, w) = int(u,x) = j-1nj and the interval j-1nj is central. A little group theory provides the proof for the following lemma: The element j-1nj of a group is central if and only if n is central. Thus either n is central, in which case j"1 nj is of course simply n; or else n is not central, in which case j~*nj is not central. We have now proved: P = Pn and I* commute if and onlyifint(v,w) = int(u,x) = n and n is central. The rest follows from 3.4.8 and 3.5.6(C). q.e.d. We have now seen how inversion operations combine with intervalpreserving operations. Earlier we saw how inversions combined with transpositions (3.5.6), and how transpositions commuted with inversion-preserving operations (3.4.10). Earlier still, we noted that the transpositions formed a

55

3.5.8

Generalized Interval Systems (2)

group of operations among themselves (3.4.2), combining according to a certain formula; the interval-preserving operations also formed a group among themselves (3.4.5) combining according to another formula. It remains to explore how the inversion operations combine with each other, and we proceed to do so.

3.5.8 THEOREM: Fix ref, and let the LABELs of v, u, w, and x be respectively i, j, k, and m. Then TVTW = rp— IT~IJ A A x u x

im k

Proof: Given any s, then 3.5.2 tells us that LABEL(I^(s)) = i-LABEL(s)- 1 -] while 1 LABEL(I*(s)) = k • LABEL(s)- • m. Hence LABEL(IvuI-(s)) = KLABELa^s)))-1] = i(k • LABEUXT1 • mr'j = i(m-1LABEL(s)k~1)j = (im-1)LABEL(s)(k-1j) = LABEUPrjT-^s)) (3.4.3; 3.4.4). In sum, I*I*(s) and P^T^s) have the same LABELs, and are therefore the same element of S. Since s was an arbitrary sample member of S, the functional equation of the theorem is true, q.e.d. 3.5.9 COROLLARY: I|J is the inverse operation to I*. Proof: Given u and v, take x = v and w = u in the formula of Theorem 3.5.8 above. Then m = i and k = j, as those intervals are defined in that theorem. So im"1 and k-1j are both equal to e; the formula of the theorem tells us in this special case that !„!" = PeTe, which is of course the identity operation. By the symmetry of the situation (reversing the roles of u and v), PJI* is also the identity operation, q.e.d. 3.5.10 COROLLARIES: Let T and I be any transposition operation and any inversion operation in a commutative GIS. Then

(A): r1 = I and (B): IT = T-'I. Proofs: (A) follows at once from 3.5.9 and the first remark of 3.5.5. To prove (B), set J = IT. Via 3.5.6(B) we know that J is an inversion operation. Then, invoking (A) just proved above, we infer that J = J"1. It follows: IT = J = J-1 = (IT)"1 = T'1!'1 = T~]I; the last step is again a consequence of (A) just proved, q.e.d.

56

We have now explored how various types of operations on the space of an abstract GIS compose among themselves and with each other. Standing back

Generalized Interval Systems (2)

3.5.11

from the niceties of the specific formulas involved, we can get a useful global picture. 3.5.11 THEOREM: Let PETEY be the family of all operations on S that can be expressed as (functionally equal to) something of form PT, where P is some interval-preserving operation and T is some transposition. Let PETINV be the family PETEY plus the family INVS of all inversion operations. Then (A) PETEY is a group of operations and (B) PETINV is also a group of operations. Proof (optional): We already know that PSVS, the family of intervalpreserving operations, and TNSPS, the family of transpositions, are each groups of operations (3.4.5; 3.4.2). Let PT and PT' be any two members of PETEY. Set P" = PP' and T" = TT. Since PSVS and TNSPS are groups, P" is interval-preserving and T" is a transposition; then P"T" is a member of PETEY as PETEY was defined. Furthermore, the composition of the given PT with the given FT' is precisely P"T", this member of PETEY. For (PD(FT') = P(TP')T = P(PT)T' (3.4.10) = P"T". So PETEY is a closed family of transformations. To prove PETEY a group, it suffices via 1.3.4 to show that PETEY-asdefined contains the inverse of each of its member operations. Given PT in PETEY, then P"1 and T-1 are respectively members of PSVS and TNSPS, so that p-1T-1 is a member of PETEY-as-defined. And p^T"1 is the inverse of the given PT: PT = TP (3.4.10), so PT(P~1T~1) = TP(P~1T~1) = 1 and (p-iT-i)PT = (P-1T-1)TP = 1. So (A) of the theorem is proved. We use the criteria of 1.3.4 again to prove (B) of the theorem. We know that PETEY, since it is a group, contains the inverse of each of its members; we also know via 3.5.9 that INVS contains the inverse of each of its members. Hence PETINV, the set-theoretic union of the two families PETEY and INVS, contains the inverse of each of its members. It remains only to prove that PETINV is a closed family of operations. Suppose that X and Y are members of PETINV; we have to show that XY is (operationally equal to) a member of PETINV. We can distinguish four possible cases, which we take up one at a time below. Case 1: X and Y are both members of PETEY. Then XY, being a member of PETEY, is a member of PETINV. Case 2: X is a member of PETEY and Y is a member of INVS. Say X = PT and Y = I. Now TI is some inversion-operation J (3.5.6(A)). And PJ is some inversion-operation K (3.5.7(A)). Then XY = PTI = PJ = K is a member of INVS, and therefore a member of PETINV as desired. Case 3: X is a member of INVS and Y is a member of PETEY. By an argument analogous to that of Case 2, now using 3.5.6(B) and 3.5.7(B), we conclude that XY is a member of INVS, and hence of PETINV as desired. Case 4: X and Y are both members of INVS. Then XY is a member of PETEY (3.5.8). So XY is a member of PETINV as desired, q.e.d.

57

3.6.1

Generalized Interval Systems (2)

We have seen that transpositions are naturally related in a number of ways to interval-preserving operations. We might conjecture that inversions should be naturally related to "interval-reversing" transformations, in the sense of the following definition. 3.6.1 DEFINITION: A transformation Y on the space S of a GIS will be called interval-reversing if int(Y(s),Y(t)) = int(t,s) for all s and all t in S.

There is something to our conjecture above. Specifically, if the GIS is commutative, then the inversions are precisely the interval-reversing operations on S. But if the GIS is now-commutative, then there will not be any intervalreversing transformations at all! We shall now prove these facts, starting with a lemma. 3.6.2 LEMMA (optional): Let Y be an interval-reversing transformation; let ref be fixed; then there is an interval i such that LABEL(Y(t)) = i • (LABEL(t))-1 for every t in S. Proof: int(Y(s), Y(t)) = int(t, s) by supposition. So LABEL(Y(t))~1LABEL(Y(s)) = LABEL(s)~1LABEL(t)

(3.1.2).

Take s = ref in the above equation, so that LABEL(s) = e. Set i = LABEL(Y(ref)) = LABEL(Y(s)) in the above equation. Then LABEL(Y(t))-1 • i = LABEL(t). So LABEL(Y(t))~1 = LABEL(t) • T1 and LABEL(Y(t)) = i • LABEL(t)-1, for any t. 3.6.3 THEOREM (optional): If IVLS is commutative, then the inversion operations reverse intervals, and every interval-reversing transformation is some inversion-operation. Proof: Fix some ref. The result of Lemma 3.6.2, in conjunction with the formula of Theorem 3.5.2, tells us that any interval-reversing transformation must be some inversion, specifically some Vtcf. It remains to show that any I* reverses intervals. Setting i = LABEL(v) and j = LABEL(u), we can write

58

intai(s), W)) = LABELS))-1 LABEL(I^(s)) = (iLABEL(t)~1j)-1(iLABEL(s)-1j) = j~1LABEL(t)i-1iLABEL(s)-1j = LABEL(s)~1LABEL(t),

(3.1.2) (3.5.2)

Generalized Interval Systems (2)

3.6.4

since IVLS is commutative! And that = int(t,s)

via 3.1.2.

q.e.d.

3.6.4 THEOREM (optional): If IVLS is non-commutative, then there exists no interval-reversing transformation on S. Proof: We suppose that Y is an interval-reversing transformation and arrive at a contradiction. For all s and all t, LABEL(Y(t))~1LABEL(Y(s)) = LABEL(s)-1 LABEL(t), as in the proof for Lemma 3.6.2. Having the formula of that lemma at our disposal now, we can substitute for the LABELs of Y(s) and Y(t) in the above equation, using the special interval i of the lemma. We get the new equation (i(LABEL(t)r1)-1 • iLABEL(s)-1 = LABEL(s)~1LABEL(t). Or: (LABELCOr^iLABELts)-1 = LABEL(s)~1LABEL(t). Or: LABEL(t)LABELCs)-1 = LABEL(s)~1LABEL(t). But that equation, holding for all s and t, says that IVLS is a commutative group. And that contradicts the premise of the theorem, q.e.d.

59

4

Generalized Interval Systems (3): A Non-Commutative GIS;

Some Timbral GIS Models

During the discussion of transpositions, interval-preserving operations, and inversions, the reader may have been puzzled by the care with which noncommutative GIS structures were separated from commutative. After all, we have so far not encountered any specimen GIS which is non-commutative. Why should we be concerned at all about the non-commutative case? Why not save ourselves some trouble, and just stipulate in the definition of a GIS that the group IVLS should be commutative? The work of the present chapter will respond to these concerns by exploring a musically significant noncommutative GIS. I have already presented some of the work elsewhere, but it will have a special impact in the present context.1 4.1.1 DEFINITION: By a time span, we will understand an ordered pair (a, x), where a is any real number and x is any positive real number. The pair of numbers is understood to model our sense of location and extension about a musical event that "begins at time a" and "extends x units of time" thereafter. The family of all time spans will be denoted TMSPS. We have already encountered one rhythmic GIS whose objects were certain time spans; that was example 3.3.2. There, we restricted the values for the numbers a to integers, and we restricted the values for the numbers x to certain proportions. Using the same direct-product construction, we could also construct a GIS for time spans in which the number a could assume any rational value, and the number x any positive rational value. Using the direct-

60

1. David Lewin, "On Formal Intervals between Time-Spans," Music Perception vol. 1, no. 4 (Summer 1984), 414-23.

Generalized Interval Systems (3)

4.1.2

product construction, we can also construct a GIS for all time spans in the manner of 4.1.2 following. 4.1.2 EXAMPLE: Take S = TMSPS. Take IVLS to be the direct-product group of the real-numbers-under-addition by the positive-reals-undermultiplication. Define the function int, from S x S into IVLS, by the formula int((a,x),(b,y)) = (b-a,y/x). Then (TMSPS, IVLS, int) is a GIS. It is commutative. The interval (b — a, y/x) measures our presumed sense that time span (b, y) begins b — a units after time span (a, x) and lasts y/x times as long. This commutative GIS is useful and relatively simple, but it is not adequate as a model for the way we perceive time spans interacting under all circumstances. We shall now investigate why that is so. First we shall examine the family of time spans as a conceptual space independent of any particular compositional context; then we shall examine how time spans behave in various specific musical contexts. To begin, then, let us focus on the time span (a, x) as a conceptual object in a conceptual space, modeling our sense that something "begins at time point a" and "extends for x time-units" thereafter. We can ask, what is this absolute conceptual time-unit? In practice, we often proceed as if it were the minute. We do so, that is, when we write metronome marks which reduce various contextual units, in various pieces or passages of music, to fractions of a minute. The minute is not commensurate with our sense of a "beat," but we can use the second for that purpose if we wish, dividing all the metronome numbers by 60. Neither the minute nor the second, though, is very satisfactory as a would-be absolute conceptual time unit; both are derived from certain relative periodic motions of the earth, the sun, and the moon. Scientists today find these motions so erratic and irregular that they use other conceptual units of time for precise measurements. But even those units, deriving from certain sub-atomic motions, are clearly contextual. And that does not even begin to engage other technical problems involving Relativity and quantum mechanics in connection with such sub-atomic "fixed" units of time. In short, if we declare any one time-unit to have absolute conceptual priority, that is a matter of computational convenience, or of scientific, sociological, or religious convention, rather then manifest musical reality. Abandoning this approach, we can make our absolute time-unit a matter of notational structure: We can call it "the brevis" or "the perfection" or "the whole note" or "the notated beat," for instance. But then we are throwing the whole problem back onto some notational convention that is highly restricted socio-historically, a convention that indeed already presupposes a highly structured theory of measuring time by some pre-existing absolute unit. And that will not help us in our inquiry.

61

4.1.2

Generalized Interval Systems (3)

We might try to assert that, though we cannot conceive an absolute timeunit clearly, there will be some clear contextual time-unit, which we can identify and use for theoretical purposes, in any music that we may want to analyze. Such an assertion is reasonable in connection with a large body of music, and not only European music of the Classic-Romantic period. But the assertion is still not valid as a universal proposition about music, unless one is willing to restrict the use of the word "music" circularly, i.e. in precisely this way. Making that restriction would involve at least a broad aesthetic contention. An even then, many critics who might feel no qualms about excluding as "music" say certain improvisations of John Coltrane, or Stockhausen's Aus den sieben Tagen, would feel less comfortable excluding as music certain Tibetan chants, or sections of Elliott Carter's String Quartet no. 1. Figure 4.1 (pp. 64-65) reproduces measures 22-35 of the Carter score. What could one assert as "the" (one clear contextual) time-unit for measures 22-32?2 Figure 4.1 shows that our philosophical musings above do not simply come down to a matter of mensural versus non-mensural perceptions. Measures 22-32 of Carter's piece have a very strong mensural character, despite our difficulty in pinning down "the" beat. The mensural profile of the passage is especially—one might even say unusually—strong within each of the individual instrumental parts. For example, the first violin's A in measure 26 and G in measure 28 each last precisely half as long as every other note in the first violin from measure 22 to measure 30; a player who does not hear this mensural relation will not project the passage effectively. For another example, the B-F|-D-C|-D# of the cello at measure 32 and following is heard not only in syncopation against the otherwise regular half-note beat of measure 33 and following (half-note = MM90), but also as a rubato of the earlier cello melody C-G-E1>-D-E, appearing eleven semitones lower in measures 2729. The earlier melody is presented in notes of constant duration whose beat, every five written eighths, is at MM48, not MM90. A cellist who does not hear the rhythmic relation of the transposed melodies will not project measures 32-35 completely effectively. In sum, the notion of "an" abstract conceptual time-unit, a unit by which we measure the number x of the formal time span (a, x), is a notion fraught with methodological problems. The number a of the time span (a, x), as well as the number x, is measured by our conceptual time-unit. For when we say that we perceive something that "begins at time-point a," we mean implicitly that it

62

2. I am grateful to Jonathan W. Bernard for engaging my interest in this passage through his lecture, "The Evolution of Elliott Carter's Rhythmic Practice," delivered to the meeting at Yale of the Society for Music Theory on November 11, 1983. Bernard's observations engaged many of the rhythmic relations I shall be discussing. The uses to which I shall put those observations, as I expound my CIS-theories, are of course my own responsibility.

Generalized Interval Systems (3)

4.1.2

begins just that number of conceptual time-units after some referential timepoint, "time-point zero." Having noted this, we see that we must discuss not only the conceptual time-unit in this connection, but also the conceptual "time-point zero." What is this abstractly privileged moment that contributes toward measuring the number a of the time span (a, x)? Is it the moment of the Big Bang, or of the Biblical Creation? Is it a completely arbitrary moment very long ago? (And if so, why should we select an arbitrary moment to play a uniquely referential role?) Should we select "time-point zero" by a notational convention, e.g. as the first vertical line on the score of whatever piece we are analyzing at the moment? Or should we presume to assert, explicitly or implicitly, that there must always be some one uniquely privileged moment, in the score or the performance of any passage we want to discuss, which we can unequivocally label as a contextual zero time-point for the occasion? These methodological expedients involve difficulties similar to those discussed above in connection with the referential time-unit. In one way at least, the choice of a zero time-point is less problematic than the choice of a temporal unit: The former choice does not affect the numbers attached to intervals in the GIS of 4.1.2, while the latter choice does affect those numbers. To see this, first suppose that we move our referential zero time-point back by m units into the past. Then the percept that was formerly manifest over the time span labeled (a, x) in the old scheme will now be manifest over the time span labled (a + m, x) in the new scheme: What used to begin a units after (old) time-point zero will now begin a + m units after (new) time-point zero. Similarly, the time span labeled (b, y) in the old scheme will correspond to the time span labeled (b + m, y) in the new scheme. In the GIS of 4.1.2, the interval between the old labels is int((a, x), (b, y)) = (b — a, y/x). In the same GIS, the interval between the new labels is int((a + m, x), (b + m, y)) = (b + m - (a + m), y/x) = (b - a, y/x). So, in transforming each old time span (a, x) to the new time span (a + m, x), we have not transformed the intervals involved: The interval between a pair of transformed spans is exactly the same as the interval between the corresponding pair of spans prior to transformation. That is, int((a + m, x), (b + m, y)) = int((a, x), (b, y)). Now let us suppose we keep the same referential time-point zero but change the unit of measurement so that what was x old units becomes xu new units, the factor u corresponding to the change of scale in measurement. Then percepts formerly corresponding to the time span (a, x) and (b, y) in the old scheme will now correspond to the time spans (au, xu) and (bu, yu) in the new scheme: What used to begin a old units after time zero and extend x old units therefrom will now begin au new units after time zero and extend xu new units therefrom. We can see that this transformation does change the numbers attached to intervals in GIS 4.1.2: int((a, x), (b, y)) = (b - a, y/x), while

63

4.1.2

Generalized Interval Systems (3)

FIGURE 4.1

64

Generalized Interval Systems (3)

4.1.2

FIGURE 4.1 (continued)

65

4.1.2

Generalized Interval Systems (3)

int((au,xu),(bu,yu)) = (bu — au,yu/(xu)) = ((b — a)u,y/x). Intuitively, the second percept we are discussing used to begin b — a units after the first percept; now it begins (b — a)u units after the first percept, the "unit" having changed. Of course the GIS of 4.1.2 knows nothing of "percepts" or "units"; it simply knows that int((a, x), (b, y)) = (b — a, y/x) is not the same pair of numbers as ((b — a)u, y/x) = int((au, xu), (bu, yu)). Lest the host of methodological problems under discussion appear insuperable, we should recall that we can finesse them all by restricting our attention to music in which we can identify and assert a referential time-unit and a referential zero time-point contextually. Then we can use the GIS of 4.1.2 without problems. There are plenty of pieces and passages for which we can sensibly take this tack. On the other hand, there are also pieces and passages in which we cannot identify such referential entities contextually, music which we would agree nonetheless to consider highly structured mensurally, music within which it seems analytically valid—even necessary—to articulate time spans engaged in mensural interrelationships. We have already begun to explore the Carter example in this connection; we shall continue that analysis soon. Another example is provided by Stockhausen's Klavierstiick XL Stockhausen tells the pianist to look at the sheet of music and begin with any group of notes from among nineteen such groups dispersed over the score, "the first that catches his eye; this he plays, choosing for himself tempo ..., dynamic level and type of attack. At the end of the first group he reads the tempo, dynamic and attack indications that follow, and looks at random to any other group, which he then plays in accordance with the latter indications," and so on and on. "When a group is arrived at for the third time, one possible realization of the piece is completed." 3 Each of the nineteen groups is notated quite traditionally as regards pitches and internal rhythmic proportions. But each group might be played at any of six tempi, ranging from very slow to very fast. Indeed, even during one performed realization, any group might be played two different times at two different tempi. The tempo of each performed group (after the first) depends on the instructions which appear at the end of the group just played, which might itself occur at any of the six tempi. In this context it makes no musical sense to speak of "the" referential time-unit, beyond the interior of each performed group at its performed tempo. And yet, mensural relations among time spans from different groups (especially consecutive groups) are highly audible, and therefore at least perceptually functional in any given realization. We have already mentioned, in footnote 3 of chapter 2, other examples of highly mensural music without a fixed time-unit. Nancarrow's Studies for Player Piano contain many pitch-canons involving elaborately changing 66

3. The cited text is from the Performing Directions by the composer on the score (Universal Edition no. 12654 LW, 3d ed., 1967).

Generalized Interval Systems (3)

4.1.2

tempo proportions.4 Ligeti's Poeme symphonique is performed by winding up 100 clockwork metronomes to varying degrees of tension, setting them at a variety of tempi, and releasing them, allowing them to run down over the course of eighteen to twenty minutes.5 The effect includes an ironic poetic commentary, among other things, upon the very issue of the "contextual timeunit." To some extent in all this cited literature, and to a great extent in much of it, any time span has the potential for becoming a local contextual timeunit, setting a local tempo. By "local," I mean here not only over a certain temporally connected section of the total texture, but possibly also within a certain part, instrument, or instrumental group. For example, there is a more than clear mensural structure within the part played by any single metronome within Ligeti's piece. For some less extreme examples let us return once more to figure 4.1, the Carter passage discussed before. The viola moves in notes of constant duration from its entrance at measure 25 up to the middle of measure 35; these local time-units "beat" the tempo of MM 180. The first violin "beats" constant local time-units at MM36 over measures 22-30, except for the A of measure 26 and the G of measure 28 which, as observed earlier, are each half the local time-unit in duration. At measure 33 the first violin starts to project a new constant local time-unit, that beats at MM90. The cello beats constant local time-units at MM 120 over measures 22-26; then over measures 27-31 it beats new constant local time-units at MM48. Finally at measure 32 and following, it stops moving in notes of constant duration as it plays a pitchvariation on measures 27ff., where MM48 began. This variation was discussed earlier. The second violin beats its own constant local time-unit at MM96 from measure 22 through measure 26. Then over measures 27-30 it runs quickly through a number of local time-units at MM 120, MM 160 (m. 27j), MM96 (m. 28|), MM80 (to be discussed later), and MM60 (m. 30). Finally it settles into a more stable local time unit in measure 31, beating at MM90. 4. Some of Nancarrow's recent work involves irrational proportions like n. For the reader who may at first think such an idea is too bizarre to have any musical meaning, I append a brief exercise in conducting the tempo relation of n. Imagine a horizontal line segment at chest height in front of you and somewhat to your right. (I am supposing that you conduct right-handed.) Move your hand (arm) back and forth along the line at a constant speed, beating a horizontal \ allegro vivo. Now imagine the line segment as a radius of a circle whose center is at the leftmost point of your beat. As you reach the rightmost point of your beat, start swinging your hand (arm around and around the circumference of that circle counterclockwise, taking care always to keep your hand moving at the same constant speed. The amount of time it takes you to swing once around the circle is n times the duration of your earlier horizontal \ measure. 5. The composer specifies very precisely how this is to be done. He also specifies that the piece is to be played after an intermission, so that the returning audience finds the metronomes already underway, with no persons on stage. The metronomes are to be arranged on risers, like a chorus; the slow beaters are at the lowest level and the fast beaters at the highest. I am indebted to Martin Bresnick for supplying me with this report, based on a personal communication from the composer.

67

4.1.2

Generalized Interval Systems (3)

Figure 4.2a collates and tabulates these various metronome marks, that reflect the various tempi beat by the various local time-units in the individual instruments over the passage.

FIGURE 4.2b

68

Figure 4.2b takes the numbers of figure 4.2a and represents them as pitches. This device will help clarify for musicians the numerical ratios involved among those numbers. The number 180, which labels the fastest beat on figure 4.2a, is represented on figure 4.2b by the highest pitch, high C. Slower tempi, in their numerical ratios to MM 180, are represented on figure 4.2b by lower pitches in the corresponding frequency ratios to high C. For instance, on figure 4.2a the opening tempo of the cello is MM 120, 2/3 of the tempo MM 180 coming up in the viola. On figure 4.2b the tempo MM 120 is

Generalized Interval Systems (3)

4.1.2

represented by the opening pitch F5 in the cello, a pitch whose fundamental is 2/3 the frequency of the high C coming up in the viola, the high C which represents the tempo MM 180. The euphony of figure 4.2b is striking. It makes very clear the network of "numerical consonances" displayed by the tempo relations of figure 4.2a, which are also the proportional relations of the various local contextual timeunits. If we imagine a quartet actually playing figure 4.2b, we can get an even clearer idea of how this numerical network affects interactions among the performers. Here are some, and only some, of the relations the players will heed. (1) The 48 of the cello at measure 27 will lie a good numerical octave below the opening 96 of the second violin. (2) The 120 of the second violin at measure 27 will match the opening 120 of the cello. (3) The 120-to-160 relation in the second violin during measure 27 will match in its ratio the 36-to-48 relation between the first violin and the cello thereabouts. (4) The 80-to-60 relation in the second violin over measures 29-30 will retrograde, an octave lower, the 120-to-160 relation discussed in (3) above. (5) The 60-to-90 relation in the second violin, measures 30-31, will match an octave lower the earlier 120-to-180 relation between the cello at the opening of the passage and the viola entrance. (6) The 90 of the first violin at measure 33 will match the 90 of the second violin at measure 31, which in turn will match, an octave lower, the persistent 180 of the viola. (7) The 36-to-90 profile of the first violin part as a whole will match in transposed retrograde (or inversion) the 120-to-48 profile of the cello part up to measure 33. Exactly these numerical relations, and others of the same sort, must be projected to make the rhythmic structure of Carter's passage come to life and communicate, not only between the players and their listeners, but also among the four players themselves. We shall discuss the above seven performance notes some more later on. On figure 4.2b, it is curious how the D[?4 of the cello can be heard as a root whose major harmony is elaborated by the symbolic pitches of the figure over measures 22-30, along with a major seventh and an added sixth. To be sure, it is doubtful that our tonal perceptions of roots, triads, harmonic sevenths, and added sixths can be used to assert analogous functions in the realm of tempo relations. And yet there is a certain suggestiveness in the idea that the cello part of measure 27 and following has a rhythmically "grounding" function somehow analogous to the tonal root function of the symbolic D|?4 on figure 4.2b. This suggestion is useful for the cellist who wants the MM48 tempo at measure 27 and following to feel stable and referential, rather than syncopated and intrusive. The suggestion is also useful for understanding why just this tempo of MM48 (symbolized by the pitch D|?4 on figure 4.2b) is permitted to launch into a wide and free rubato at measure 32 and following in the cello (symbolized by the "cadenza" on figure 4.2b), just at the time the

69

4.1.2

70

Generalized Interval Systems (3)

other three instruments are all finally agreeing upon the new referential tempo MM90-or-180 (symbolized by the prominent pitch class C at m. 32ff. on figure 4.2b). Over the second-violin part of figure 4.2b as a whole, D|» moves clearly to C. But the cello is still refusing to change its earlier D\> for C as we leave the passage, though its D\> has been abandoned. All of these remarks amplify the idea suggested above, that the cellist should refer the rhythm of the part over measures 32-35 not just to the beats of the upper instruments at MM90 but also to the earlier beat of the cello itself, MM48 at measure 27 and following. Figure 4.2b shows by analogy how the symbolic pitches of the "cadenza" in the cello at measures 32-35 are heard not only in relation to the Cs of the upper instruments there, but also in relation to the earlier Dl> of the cello itself at measure 27 following. However interesting it may be to think of MM48 as a numerical "ground" organizing the tempi of measures 22-31, it is still clear that the players of the three upper instruments will not treat the cello part of measures 27-31 as a succession of referential time-units. That is, they will not adjust their own beats to conform in proper proportion with the cello beat of measures 27-31. Nor will the cello and the two violins treat the constant beat at MM 180 in the viola from measure 25 on as a succession of referential timeunits for the entire passage; at least they will not do so until after measure 30. During measures 22-24, in particular, the MM36, MM96, and MM 120 of the two violins and the cello will not be referred to a beat of MM 180 in the viola, for the viola has not yet entered there. Once the viola is in, at measure 25 and thereafter, its constant MM 180 will provide a useful check for the ensemble rhythm without necessarily establishing itself as referential, just as the sustained high C on figure 4.2b provides a useful check for the ensemble's intonation without necessarily establishing itself as a root. Of course MM 180 does eventually become much more referential during measures 33-35, along with its lower octave MM90; just so, on figure 4.2b, the pitch class C comes to dominate the tonal texture. The players may decide on purely notational grounds to use MM 120 (changing to MM 180 by measure 33) or MM60 (changing to MM90 by mea sure 33) as a referential tempo for the entire passage. This makes a certain practical sense for early rehearsals, but it can hardly be recommended for an effective performance. We have all heard and seen players fighting their way through slow lyric lines, supposedly tranquillo like that of the first violin in measures 22-32 or sostenuto e cantabile like that of the cello in measures 27-32, all the while jerking their feet up and down spastically in an erratic approximation of some distantly related notational "beat." These lyric lines are not syncopated, as such a method of production makes them sound to both player and listener. Rather, each line has its own autonomous local timeunit, with respect to which it should project an essentially "first-species" character.

Generalized Interval Systems (3)

4.1.2

Indeed, the search for any one overriding referential time-unit, to govern all of measures 22-31, is bound to fail. It must fail because it misconceives the nature of the temporal space at hand. That space comprises a multitude of locally referential time-units, in various more-or-less consonant numerical relationships among themselves. Only by recognizing that character of the space can one hear the music progress over the passage, and not just "be." What I mean will be amplified by the following review of the earlier performance notes, (1) through (7). (1) The MM48 of the cello at measure 27 will lie a good (numerical) octave below the opening MM96 of the second violin. That is, at measure 27 the cellist should hear the cello line moving at half the rate the second violin has been moving so far. This would be easy to hear if the cello came in a sixteenth-note later. In fact there is no problem within measure 27, getting the attack of the pitch C2 at the right time-point: The player need only continue beating MM 120 up to that point. But once the new melody has entered, its tempo may sound arbitrary and its character "syncopated" unless the player hears the melody taking over, albeit out of phase, from the preceding tempo of the second violin, projecting its own local referential time-unit. (2) The MM 120 of the second violin at measure 27 will match the opening MM 120 of the cello. MM 120 is the notated pulse. Still, the second violin should not simply be playing that pulse (with spastic foot-tapping or the like). What makes measure 27 come alive and communicate in this connection is an exchange of local referential beats, between the two instruments. The second violin takes over the preceding beat of the cello, while the cello—as we noted in (1) above—is about to take over the preceding local beat of the second violin, a rhythmic octave lower (and out of phase). The cellist and the second violinist should hear this "voice-exchange with octave transfer," to hear themselves conversing with each other in a quite familiar tradition of chamber music. Figure 4.2b, showing the exchange of the symbolic pitch classes D|? and F in the two instruments at measure 27, serves as a guide to that tradition here. Just as the players would match those pitch classes if playing figure 4.2b, so they should match their exchange of tempi at measure 27 when playing the Carter passage. (3) The MM 120-to-160 relation in the second violin, later in measure 27, will match the MM36-to-48 relation between first violin and cello hereabouts. Assuming that everything else has gone right so far,this will happen automatically if the second violin plays the dotted eighth (beating MM 160) as a precise 3/4 of the quarter note (beating MM 120). That is not so easy to do as it is to say, but the ability should be available to a well-trained player of twentiethcentury music. Here, the two quarter notes of the second violin in measure 27 will function as locally referential timespans for the player. (4) The MM80-to-60 relation in the second violin, measures 29-30, will retrograde the MM120-to-160 relation of (3) above, one rhythmic octave

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lower. Here "MM80" is projected only by the E|?-triad-event in measure 29 of the score; the tempo is not beat by any recurrent durational unit. Nevertheless a conceptual MM80 is useful to the player in a manner indicated by figure 4.3.

FIGURE 4.3

72

The figure transcribes the rhythm of the second violin from the middle of measure 26 through measure 30 into a new notation, using MM 160 as a new notational tempo of reference, that is, the tempo at which the new notated quarter beats. This transcription brings out clearly how the rhythmic peripatetics of the second violin are structured by the indicated relationship, that is MM80-to-60 answering MM120-to-160. The transcription "modulates" our rhythmic hearing exactly as the second through sixth notes of the second violin on figure 4.2b would sound "modulated" if we listened to them in B[> minor, rather than D|? major. Just as the Fs of figure 4.2b would sound primarily as fourths below the adjacent locally referential Bjjs, in our modulated pitch-hearing, just so the tempi of MM 120 and MM60 on figure 4.3 sound in rhythmic proportions 3 :4 and 4: 3 to the adjacent locally referential tempi of MM 160 and MM80, that lie alongside them. Figure 4.3 demonstrates a logical internal structure for the second-violin passage as a rhythmic entity in itself; this structure will surely not emerge if the player adjusts each individual tempo of figure 4.3 only to the beats of the viola at MM 180 hereabouts, or to the notated beat of the score at MM 120. It is in order to bring out the quasi-palindromic structure of figure 4.3 that the "tempo" of MM80 is represented, exceptionally, by only one time span. (5) The MM60-to-90 relation in the second violin, measures 30-31, will match, a rhythmic octave lower, the earlier MM120-to-180 relation between the cello at the opening of the passage and the viola entrance. This needs no further discussion; the relation of the relations will emerge without special attention if the players are otherwise temporally "in tune." (6) The MM90 of the first violin at measure 33 will match the MM90 of the second violin at measure 31. Obviously. Here MM90 is a referential local time-unit. Likewise the MM90 of the second violin at measure 31 will have matched the MM 180 of the viola so far, using MM 180 as a referential time unit. (7) The MM36-to-90 profile of the first-violin part as a whole over the passage will match in transposed retrograde (or inversion) the MM120-to-48

Generalized Interval Systems (3)

4.1.2

of the cello part as a whole up to measure 33. It is much harder to hear the large rhythmic proportion here than it is to hear the corresponding symbolic pitch proportion on figure 4.2b. Nevertheless, it will aid communication between cello and first violin, as well as projection between ensemble and audience, if the first violinist hears the MM90 entrance at measure 33 speeding up the earlier MM36 of the instrument in exactly the same ratio as the MM48 of the cello at measure 27 slowed down that instrument's earlier MM 120. The proportion can be sensed when the two instruments rehearse the pertinent music by themselves. Via this proportion, the MM90 of the first violin at measure 33 engages and completes a large mensural structure functioning over measures 22-35; it is not simply a surrender of the first violin passively to the beats of the second violin and viola at measure 31 and following. Of course the first violin will use those beats to find its new tempo at measure 33. To sum up: When performers confront the score of the Carter passage and the numerical network of local tempi or time-units displayed in figure 4.2a, they should not concern themselves with the question, "Which one of these is the overall unifying referential tempo?" That question, a rhythmic analog to the sorts of questions asked about pitch structures by Rameau, Riemann, and Hindemith among others, has no definite answer here. Even if we try to force an answer by selecting MM48, or MM 180, or MM60-then-90 as a "root" tempo on the basis of this or that criterion, we shall still not be engaging thereby the temporal relationships that make this music progress and communicate. Those temporal relationships, some of which were discussed in performance notes (l)-(7) above, involve patterns of local tempo "consonances," patterns in which many different tempi can assume locally referential roles. This attitude toward the numerical network of figure 4.2a, and the symbolic pitch network of figure 4.2b, is more in the spirit of Zarlino: It asks not for one overriding referential unity, but rather for a splendid variety of consonant ratios among the entities involved, as they underlay and succeed one another, projecting a logical compositional idea. In this way of hearing the rhythmic space through which the passage moves, any time span has the potential for becoming locally referential, or behaving as if it were. For example, let time span r be the span covered by the first note of the cello in measure 22 of the score; let s be that time span covered by the F# of the first violin in measures 25-26; let t be that time span covered by the A of the first violin in measure 26. We can say if we-wish that t begins 16j r-spans after the beginning of r, and lasts If times the duration of r. But this way of listening corresponds to the "foot-tapping" performance of the first violin's melody. We can also say that t begins 1 s-span after the beginning of s, and lasts 1/2 the duration of s. And that way of listening corresponds to a much more musical shaping of the melody. Taking s' as the time span covered by the opening D of the first violin in measures 22-23, we could also say that t begins 4 s'-spans after the beginning of s', and lasts 1/2 the duration of s'. This

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way of listening corresponds to an even more musical shaping of the melody. Adopting the above attitude to Carter's rhythmic space, we implicitly deny the relevance of GIS 4.1.2 as a model. Given abstract time spans s and t, we want to be able to conceive t as beginning a certain number of s-beats after s, rather than a certain number of possibly irrelevant "referential units" after s. If s is the span (a, x) and t is the span (b,y), the old GIS of 4.1.2 assigned int(s, t) = (b — a,y/x): t begins b — a referential units after s, and lasts y/x times as long. We want to replace that old notion of time-span interval by a new function: int(s, t) = ((b — a)/x, y/x). The new interval tells us that t begins (b — a)/x x-lengths (s-beats) after s, and lasts y/x times as long. The new interval uses s itself as a measuring rod, to tell us how much later t begins.

FIGURE 4.4

74

Figure 4.4 shows how our new "interval" works. On the figure, four numerical time-spans are indicated: Sj = (a^Xj), i1 = (b^y^, s2 = (a 2 ,x 2 ), and t2 = (b2, y 2 ). We shall see later that it does not matter at all, for our new model, what the formal numerical time-point zero is, or what the formal numerical time-unit is. That is, it does not matter to what percept we attach the numerical time-span label (0,1). On the figure, we can imagine an "upper instrument" projecting s x and ^ at a slow tempo, and a "lower instrument" projecting s2 and t2 at a fast tempo. The dotted slurs arching above the upper instrument mark off x1 -lengths, durations that mark a contextual (potential) Sj-beat. The dotted slurs arching below the lower instrument mark off x2lengths, durations that mark a contextual (potential) s2-beat. Using our new interval construct, we write int^,^) = (4,2): t1 begins 4 Sj-beats after s 1? and lasts twice as long. Arithmetically, (b: — a^/Xj = 4 and y!/x x = 2. Using

Generalized Interval Systems (3)

4.1.3.2

the new interval construct, we also write int(s 2 ,t 2 ) = (4,2): t 2 begins 4 s2beats after s2, and lasts twice as long. Arithmetically, (b2 — a2)/x2 = 4 and y 2 /x 2 = 2. Note that our new "interval from s2 to t2" is the same as our new "interval from sl to t^': int(s 2 ,t 2 ) = m^Su^) = (4,2). We shall discuss the implications of this a lot more later on. Note particularly that sx precedes s2 on figure 4.4 in the obvious sense, while t x , the (4,2)-transpose of s^, follows t 2 , the (4,2)-transpose of s 2 . One sees that our intuitions about formal "transposition" will not be completely reliable in our new non-commutative GIS. (Our intuitions about interval-preserving operations will be trustworthy.) We are of course still far from having constructed a formal GIS in which our new notion of "interval" is to work. It is high time to do so now. 4.1.3.1 LEMMA: Let IVLS be the family of pairs (i, p), where i is a real number and p is a positive real number. Then IVLS forms a group under the composition

0,p)(j,q) = (i + pj,pq). In this group, the identity is (0,1) and the inverse of the element (i, p) is the element ( — i/p, 1/p). The group is non-commutative. The proof of the Lemma will be left as an exercise for the interested reader. Do not forget to show that the defined composition is associative: ((i,p)(j,q))(k,r) = (i,p)((j,q)(k,r)). 4.1.3.2 THEOREM: Let int be the function that maps TMSPS x TMSPS into the group IVLS of Lemma 4.1.3.1 according to the formula int((a,x),(b,y)) = ((b-a)/x,y/x). Then (TMSPS, IVLS, int) is a GIS. Proof (optional): We must show that Conditions (A) and (B) of Definition 2.3.1 obtain. (A): Given time spans (a,x), (b,y), and (c, z), we are to show that int((a, x), (b, y))int((b, y), (c, z)) = int((a, x), (c, z)). We write int((a, x), (b, y))int((b, y), (c, z)) = ((b-a)/x,y/x)((c-b)/y,z/y) by the formula defining int in the theorem. This = (((b - a)/x) + (y/x)(c - b)/y, (y/x)(z/y)) by the group composition in IVLS. Canceling factors of y in the numerators and denominators, we see that this = ((b — a + c — b)/x, z/x) which = ((c — a)/x, z/x). And that number pair is indeed int((a, x), (c, z)).

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(B): Given the time span s = (a, x) and the interval (i, p), we are to find a unique time span t = (b, y) which lies the interval (i, p) from the time span s = (a, x). If any such b and y exist, they must satisfy the relation int((a, x), (b, y)) = (i, p), or ((b - a)/x, y/x) = (i, p), or (b — a)/x = i and y/x = p, or b = ix + a and y = xp. So there can be at most one time span t in the desired relation to s and (i, p): That is the time span t = (b, y) = (ix + a, xp). And in fact this particular t is in the desired relation: int(s, t) = int((a, x), (ix + a, xp)); this = (((ix + a) — a)/x, xp/x) by the formula defining int; that = (ix/x, p), which is indeed (i, p) as desired, q.e.d. At long last, we have before us a non-commutative GIS of musical interest. The GIS has important formal properties, which we shall now study. 4.1.4

THEOREM: GIS 4.1.3 has properties (A) and (B) below. (A): For any real number h, the interval from time span (a + h, x) to time span (b + h, y) is the same as the interval from (a, x) to (b, y). (B): For any positive real number u, the interval from time span (au, xu) to time span (bu, yu) is the same as the interval from (a, x) to (b, y). Proof: (A): int((a + M),(b + h,y)) = (((b + h) - (a + h))/x, y/x) (4.1.3.2) = ((b - a)/x, y/x) (algebra) = int((a,x),(b,y)) (4.1.3.2). (B): int((au, xu), (bu, yu)) = ((bu - au)/xu, yu/xu) (4.1.3.2) = ((b - a)/x, y/x) (algebra) = int((a,x),(b,y)) (4.1.3.2). q.e.d.

76

Some commentary on this theorem is in order. The time spans (a, x), (b, y), and so on still rely numerically on an implied referential time-unit and an implied time point zero: (a, x) begins the number a of referential units after the referential zero time-point, and lasts the number x of referential units. The essence of Properties (A) and (B) in the theorem above is that the numerical function int for the GIS under present discussion does not depend at all on the choice of time point zero, or on the choice of referential time-unit. To see this, suppose first that we move the referential zero time-point back h units into the past (= forward ( —h) units into the future). An event originally associated with the time span (a, x) will now be associated with the time span (a + h, x): The event will begin a + h units later than the new zero

Generalized Interval Systems (3)

4.1.5

time-point. Similarly, another event originally associated with the time span (b, y) will now become associated with the time span (b + h, y). Property (A) of the theorem says that in GIS 4.1.3, the formal interval between the time spans associated with the two events is not affected by this transformation. Even though the time spans themselves, as number-pairs, change from (a, x) to (a + h, x) and so on, the interval between transformed spans is the same as the interval between the original spans. Now suppose we change the referential unit of numerical time, so that the old unit is u times the new unit. A duration of x old units is then the same as a duration of xu new units. And the number a of old units after time-point zero is the same as the number au of new units after time-point zero. Hence the events that were associated with time spans (a, x) and (b, y) in the old system will be associated with time spans (au, xu) and (bu, yu) in the new system. Property (B) of the theorem says that in GIS 4.1.3, the formal interval between the time spans associated with the two events is not affected by this transformation. Thus, in the GIS of 4.1.3 the function int(s, t) will always deliver one and the same pair of numbers (i, p), no matter what the referential time-unit and the referential zero time-point by which we calculate numerical durations and distances from time-point zero. To put this intuitively: Given event 1 and event 2 in a piece, we can play the music whenever we want and at any tempo we want, without affecting at all the pair of numbers (i, p) which GIS 4.1.3 will deliver to us as the formal interval between the numerical time spans associated with the two events for any particular analysis. The same can not be said for the commutative GIS of 4.1.2, studied earlier. In that GIS the interval between time spans (a, x) and (b, y) was (b — a, y/x); accordingly, if we replace the referential time-unit so that events once associated with those time spans now become associated with the new spans (au, xu) and (bu, yu), then the interval between the new spans is different. It is not (b — a, y/x), but rather (bu — au, y/x). We noted this earlier. In our present terminology, we can say that GIS 4.1.2 does not enjoy Property (B) of Theorem 4.1.4. In fact, a remarkable theorem is true. Not only does GIS 4.1.3 enjoy the two Properties of Theorem 4.1.4, but it is also essentially the only possible GIS involving time spans as objects that enjoys those two Properties. The meaning of the word "essentially" in the above sentence is made clear by Theorem 4.1.5 following. 4.1.5 THEOREM: Let GIS' = (TMSPS, IVLS', int') be any GIS with time spans for its objects that also enjoys Properties (A) and (B) of Theorem 4.1.4. Then the group IVLS of GIS 4.1.3 and the group IVLS' of the given GIS' are isomorphic via a map f such that, for all time spans s and t, int'(s,t) = f(int(s,t)).

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Some commentary is in order before we launch into a proof. The idea of isomorphism between (semi)groups was discussed in 1.11.1 and 1.11.2 earlier. To review here: If G and G' are abstract groups, a function f from G into G' is "an isomorphism of G with G'" if f is 1-to-1, onto, and a homomorphism. f is a homomorphism if f (mn) = f (m)f (n) for every m and every n in G. Supposing f an isomorphism of the abstract groups G and G', then the two abstract groups will have exactly the same algebraic structure under the identification of m in G with its image f(m) in G'. So the first thing Theorem 4.1.5 says is that IVLS and IVLS' have essentially the same algebraic structure, when we identify the member m of IVLS with its image f (m) in IVLS'. Second, Theorem 4.1.5 says that if we take the member m of IVLS to be int(s, t) in particular, then the function f whose existence is asserted makes the image f (m), a member of IVLS', equal precisely to int'(s, t). Thus the function int' is, so to speak, naught but the isomorphic image of the function int under the isomorphism f whose existence the theorem asserts. In this sense, the given GIS' is "essentially" the same as GIS 4.1.3. The (optional) proof of Theorem 4.1.5 is lengthy. To help break it into manageable sections, we shall prove two lemmas. The lemmas appear below as 4.1.6.1 and 4.1.6.2. After that, we shall go on to the proof of the Theorem proper. 4.1.6.1 LEMMA (optional): Let G and G' be abstract groups. Let f be a function from G into G' such that for all m and all n in G, ^m)"1^) = f^n^n). Then f is a homomorphism. Proof of (optional) Lemma: Set m = n = e in the given formula; we get f (e)"1^) = f(e). It follows that f(e)"1 is the identity in G'; hence f(e) is the identity in G', e'. Now set n = e and let m vary in the formula of the Lemma. We get f(m)"1f(e) = fCnT1). Since f(e) = e', we have f(m)-1 = f(m-1) for all m. Then we can rewrite the formula of the Lemma as f(m~ 1 )f(n) = ftm^n)

for all m and all n.

As m runs through the various members of G, m"1 = o runs through the various members of G. Substitute o for m"1 in the rewritten formula; we then obtain the formula f (o)f (n) = f (on) for all o and all n in G.

And thus f is a homomorphism, as claimed. 4.1.6.2 LEMMA (optional): Within the group IVLS of 4.1.3.1,

(i,pr1(j,q) = ((j-i)/P,q/p)78

Proof of (optional) Lemma: We already verified in 4.1.3.1 that (i,p)-1 in

Generalized Interval Systems (3)

4.1.6.2

this group was the element (- i/p, 1 /p). Then (i, p)1 (j, q) = (- i/p, 1 /p) (j, q) = ((-i/p) + (l/p)j,(l/p)q) = ((j - 0/P,q/P) as asserted. Now we are ready for the (optional) Proof of Theorem 4.1.5. We take the time span (0,1) as a referential object within the space of GIS' for purposes of LABELing. That is, we set ref = (0,1). Then the function f for which we are looking, the isomorphism of IVLS with IVLS', is defined by formula (i) below. (i) f(i,p) = LABEL'(i,p) = int'((0, l),(i,p)). On the left of formula (i) the number-pair (i, p) is considered as an interval, a member of IVLS, while in the middle and on the right of the formula, the same number-pair is considered as a time span, a span being LABELed in GIS' by its GIS'-interval from the referential object ref = (0,1). The number-pair (i, p), as a pair of numbers, can be interpreted either way. Now we can write f (i, p)"1 f (j, q) = LABEL'(i, p)'1 LABEL'(j, q), by formula (i). This = int'((i,p),(j,q)) by 3.1.2. This = int'((0, p), (j — i, q)), since GIS' enjoys Property (A) of Theorem 4.1.4 by supposition. And this = int'((0,1), ((j — i)/p, q/p)) since GIS' enjoys Property (B) of Theorem 4.1.4 by supposition. And that = f ((j — i)/p, q/p) by formula (i) above. And that = f((i,p)~ 1 (j,q)) by Lemma 4.1.6.2. Putting together the whole string of equalities we have just noted, substituting m for (i, p) and n for (j, q), we see that we have proved f(m)-1f(n) = f(m-1n) for every m and every n in IVLS. By Lemma 4.1.6.1, we conclude that f is a homomorphism. Since the LABEL' function is 1-to-l from TMSPS onto IVLS' (3.1.2), the function f is 1-to-l from IVLS onto IVLS'. Thus f is an isomorphism of IVLS with IVLS'. It remains to prove that f(int(s,t)) = int'(s,t). Set s = (a, x) and t = (b, y). Then int(s, t) = ((b - a)/x, y/x) and f(int(s, t)) = = = =

int'((0,1), ((b - a)/x, y/x)) (formula (i)) hit' ((0, x), (b - a, y)) (since GIS' enjoys Property (B)) int'((a, x), (b, y)) (since GIS' enjoys Property (A)) int'(s, t). q.e.d.

To recapitulate: Theorem 4.1.5 shows that GIS 4.1.3 is essentially the only possible GIS involving the family TMSPS whose function int is completely independent of the referential time-unit and referential zero time-

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Generalized In terval Systems (3)

point. To put it more intuitively, GIS 4.1.3 is the .only such GIS, essentially, that will return one and the same element of IVLS as the interval between the numerical time spans associated with two musical events in a piece, regardless of when you play the piece and what tempo you take. GIS 4.1.3 thereby has a privileged theoretical status, as well as a special plausibility for modeling events in the Carter passage and other pertinent music. Since GIS 4.1.3 is non-commutative, it will provide a useful example for illustrating and reviewing the work of sections 3.4 and 3.5 earlier, work that formulated the abstract theory of transpositions, interval-preserving operations, and inversions. 4.1.7 NOTES: Within GIS 4.1.3, the following formulas and facts are true: (A): Given an interval (i, p) and a time span (a, x), the transposition of the given time span by the given interval is T(iip)(a,x) = (a + ix,px). (B): If we fix (0,1) as ref, a referential time span, then the number-pair (a, x), as a member of IVLS, is the LABEL for the time span (a, x): LABEL(a,x) = int((0, l),(a,x)) = (a,x). (C): The (i, p)-transpose of the time span (a, x) is the number-pair given by the composition in IVLS of the two intervals (a, x) and (i, p). T(i(p)(a, x) = (a,x)(i,p). (D): Using the number-pair (a, x) in the same way, as both a time span and an interval, we can show that the interval-preserving operation P(h)U) transforms the time span (a, x) into the time span p

(E): The only central member of IVLS is the identity interval (0,1). (F): No transposition preserves intervals, and no interval-preserving operation is a transposition, the identity operation T(0>1) = P (O ,D excepted. (G): The operation of (c, z)/(d, w) inversion, applied to the time span (a, x), yields the time span I!c;?(a,x) = (d + (c - a)w/x,zw/x) = (d,w)(a,x)~ 1 (c,z).

80

(H): Given time spans s, t, s', and t', then !£'. = I, as an operation on TMSPS if and only if s' = s and t' = t. (I): There are no interval-reversing operations on TMSPS. Proofs and commentary: (A): Via 3.4.1, the transposition of (a, x) by (i, p) is that time span (b, y) which lies the interval (i, p) from the time span (a, x), i.e. which satisfies the equation int((a, x),(b,y)) = (i,p). Thus (b, y) satisfies the equation ((b — a)/x, y/x) = (i, p); whence (b — a)/x = i and y/x = p. So b = a + ix and y = px. The transposed time span (b, y) = (a + ix, px) can be

Generalized Interval Systems (3)

4.L7

described as follows: b lies i x-spans later than a; y lasts p times as long as x. If we turn back to figure 4.4 (p. 74), we will see that the time span tj of the figure is the (4,2)-transpose of s x : ilbegins at bj, 4 xt-spans aftera t ; t tlasts a duration of yt , 2 times the duration Xj of s^^. Likewise, t2 on the figure is the (4,2)-transpose of s2. We noted while studying the figure earlier that s x precedes s2, while tl follows t 2 . We may say that transposition operations, in this GIS, do not only fail to preserve intervals, they even fail to preserve chronology. (B) of the Notes is a straightforward computation: LABEL(a,x) = int((0, l),(a,x) = ((a - 0)/1, Ix) = (a,x). (C) of the Notes applies (B) to the formula of 3.4.3, and (D) applies (B) to the formula of 3.4.4. The interval-preserving operation P(h u) first blows up or shrinks the sample time span (a, x) by a factor of u, transforming (a, x) to (ua, ux), and then moves the latter time span backward or forward in time by h or (—h) numerical units, transforming (ua, ux) to (h + ua, ux) = P(a, x). Remember that these interval-preserving operations are not formal "transpositions" in our non-commutative system! (E) of the Notes is proved as follows. Suppose the interval (i, p) is central in IVLS, that is (i,p)(j,q) = (j,q)(i,p) for all (j,q). Expanding the binary composition on each side of that equation, we infer (i + pj, pq) = (j + qi, qp) for all j and all positive q. Then i + pj = j + qi for all such j and q, whence (p — l)j = (q — l)i for all such j and q. Take j = 1 and q = 1 as one such j and q; then (p — 1)1 = (1 — l)i or p — 1 = 0; we infer that p must be equal to 1. Now we can go back to our general equation, (p — l)j = (q — l)i; substituting p = 1, we infer that (q — l)i = 0 for all positive q. But then i is obviously zero. So p = 1 and i = 0; our given central interval (i, p) must be the identity interval (0,1). (F) of the Notes then follows from Theorem 3.4.8. (G) of the Notes can be computed from 3.5.2 together with (B) of the Notes. Or it can be computed directly from the defining formula of 3.5.1, using the known group structure of IVLS here. (H) follows from 3.5.3, where we proved that l£ = lls if and only if t' = Ig(s') and the interval int(s', s) is central. Via (E) above, this will happen if and only if t' = Ig(s') and s' = s. Since Is(s) = t, this will happen if and only if s' = s and t' = t. (I) of the Notes simply restates 3.6.4 in the present context. We may use figure 4.4 yet once again to picture the effect of an inversion. On that figure, we noted that int(s 2 ,t 2 ) = int^,^). Hence, via Definition 3.5.1, t2 is the tlls2 inversion of s^ that is, Ij^Sj) = t 2 . This concludes our study of a non-commutative GIS which is also a rhythmic GIS of musical interest. We shall now study some timbral GIS structures.

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4.2.1

82

Generalized Interval Systems (3)

4.2.1 EXAMPLE: Given positive numbers s(l), s(3), and s(5), let the numbertriple s = (s(l), s(3), s(5)) denote the class of all harmonic steady-state sounds (i.e. periodic wave-forms) whose first, third, and fifth partials have respective power s(l), s(3), and s(5). Let S be the family of all such number-triples s, as s(l), s(3), and s(5) range over all positive values. Given positive numbers i(l), i(3), and i(5), let us imagine at hand one or more "devices" (e.g. computer procedures) which have this property: Whenever a harmonic sound is led as input into such a device, the device outputs a harmonic sound whose first, third, and fifth partials have respectively i(l) times, i(3) times, and i(5) times the power of the corresponding input partials. Given i(l), i(3), and i(5), let the number-triple i = (i(l), i(3), i(5)) denote the class of devices that transform harmonic sounds according to these proportions for the first, third, and fifth partials. Let IVLS be the family of all such number-triples i, as i(l), i(3), and i(5) range over all positive values. IVLS is a group under the combination ij = (i(l)j(l),i(3)j(3),i(5)j(5)). Given harmonic class s = (s(l), s(3), s(5)) and harmonic class t = (t(l), t(3), t(5)), take int(s, t) to be that member i of IVLS for which i(l) = t(l)/s(l), i(3) = t(3)/s(3), and i(5) = t(5)/s(5). Then (S, IVLS, int) is a GIS. That is, Conditions (A) and (B) of Definition 2.3.1 obtain. The GIS is commutative. When int(s, t) = i, any sound in class t will have i(l), i(3), and i(5) times the power of any sound in class s, at its first, third, and fifth partials respectively. Another way of regarding the statement "int(s, t) = i" is to think: Any sound of class s, when led as input to any device of class i, will cause a sound of class t to be output. The fundamental frequencies of the sounds are irrelevant here; we are concerned only with certain aspects of their timbral profiles. If a given sound is led through a device of class i, and if the resulting output is then led through a device of class j, the final output will be a sound of the same class as that which would have resulted, had the original sound been led through a device of class ij. Or, more simply, a device of class i concatenated with a device of class j forms a device of class ij. We can make many variations on the specific GIS just discussed. For example, instead of considering partials #1, #3, and #5, we could instead consider partials #1, #2, and #4. Or we could consider partials #1through- # 5, or # 1-through- # 8, or # 1-through- # 8-except-for- # 7, and so on. We can use GIS structures of this sort to build more complex GIS structures of interest. For instance, let GISj be the GIS of the sort just discussed which considers partials # 1-through-#8 of harmonic sounds. We shall call an element s = (s(l),..., s(8)) of GISj a "pertinent spectrum." Now let us take as GIS2 a familiar GIS involving the space S2 of "time points." We imagine a referential zero time-point and a referential time-unit fixed, so that we can label the elements of S2 by real numbers a. IVLS2 is the additive group

Generalized Interval Systems (3)

4.2.1

of real numbers and, given time points a and b, int^a, b) is the number b — a of time units by which b is later than a. (b — a later = a — b earlier.) Let us explore the direct product GIS3 = GISj (x) GIS2. The elements of S3 = Sj X S2 are pairs (s, a), where s = (s(l), ... , s(8)) is a pertinent spectrum and a is a time point. The pair (s, a) models a class of sounds having pertinent spectral profile s at time a. A finite set of such pairs, say the set DVSP = ((Sj, aj), (s2, a-j), ... , (SN, a^), models a class of sounds that have spectrum Sj at time at, spectrum s2 at time a^ ... , and spectrum SN at time a,^. We consider DVSP to be an unordered set of S3-elements, since the chronological order of the time points an imposes a natural ordering on the member pairs of DVSP, no matter in what order we list those pairs. For convenience, we shall assume the members of DVSP to be listed above in chronological order, that is with aj < 82 < ••• < aN. Supposing the time points an to be reasonably close, then DVSP will model a class of sounds with a certain "developing spectrum." Each sound of this class has pertinent spectrum sn at time an.

FIGURE 4.5 Figure 4.5 displays DVSP, with N = 5 in this case, as an array of numbers. If we imagine the plane of the page as a base, and erect at each entry sm(n) a spike of heights sm(n) jutting up from that page, we shall obtain a sort of sketch for a relief map that shows how the spectrum of the sound develops over time. Supposing the time points aj through a5 to be dense enough so as to catch enough salient features of the sound-class involved (e.g. times when some partial has a pronounced local maximum or local minimum value), then we can consider this sketch to be a good approximation for a continuous relief map that characterizes the class of sounds with respect to its developing spectral "signature." This sort of relief-map representation is in common use today as a means for studying various classes of harmonic sounds, including

83

4.2.2

Generalized Interval Systems (3)

familiar instrumental sounds in particular.6 As we have seen, any such relief map can be regarded as approximately a finite (unordered) subset DVSP of S3. The way that DVSP = ((Sj, a x ), (s2, a 2 ),..., (SN, aN)) develops through its own intrinsic chronology can be studied in exactly the same way we earlier studied the unfolding interval vector for a chronologically developing set in another direct-product GIS. That was in section 3.3.1, where we applied our study to the analysis of a passage from Webern's Piano Variations. 4.2.2 EXAMPLE: Fix a lower frequency LO and a higher frequency HI; we shall represent a varying frequency between LO and HI by the variable x. By a "rational spectrum" we shall mean a function s of the variable x, taking on real values, which satisfies Conditions (A) and (B) below. Condition (A): The function s can be written as the quotient of polynomial functions. That is, there exist polynomial functions p and q in x such that s(x) = p(x)/q(x) for every x between LO and HI. Condition (B): s(x) is strictly positive for every x between LO and HI. We shall take as the family S for a GIS the family of all rational spectra s. The rational spectra form a group under multiplication. For if s(x) = Pi(x)Ah(x) and t(x) = p2(x)/q2(x), then p3(x) = p1(x)p2(x) and q3(x) = q1(x)q2(x) are polynomial functions. Hence (st)(x) = s(x)t(x) can be written as a quotient of polynomial functions: (st)(x) = p3(x)/q3(x). Furthermore, (st)(x) is strictly positive since both its factors, s(x) and t(x), are strictly positive. Thus the product of two rational spectra is itself a rational spectrum. The function 1 (x) = 1 is a rational spectrum; it is a multiplicative identity fo the family of rational spectra. If s is a rational spectrum, so is 1/s, where (l/s)(x) = l/s(x); the spectrum 1/s is an inverse for s within the multiplicative system of rational spectra. We shall take as the group IVLS for our GIS the family of rational spectra again, now considered as a multiplicative group. Given rational spectra s and t, considered as members of S, we shall take as int(s, t) for our GIS the rational spectrum t/s, considered as a member of IVLS. It is straightforward to verify that (S, IVLS, int) is a GIS. The GIS models a system of "linear filter classes." With each rational spectrum s we can associate a class of filters. Any filter in this class can be built up from two simple kinds of filters, "all-zero" and "all-pole" filters. Any filte in class s will transform an input sound to an output sound in such a way that the power of frequency x in the output is equal to s(x) times the power of x in the input. The manipulation of sounds and filters in this way is characteristic

84

6. Good examples of the practice can be found in "Lexicon of Analyzed Tones," a series of analysis and plotting programs by James A. Moorer and John Grey published in Computer Music Journal. "Part I: A Violin Tone" appeared in vol. 1, no. 2 (April 1977), 39-45. "Part II: Clarinet and Oboe Tones" appeared in vol. 1, no. 3 (June 1977), 12-29. "Part III: The Trumpet" appeared in vol. 2, no. 2 (September 1978), 23-31. There is also a handsome "relief map" on the cover of that issue.

Generalized Interval Systems (3)

4,3

of certain recent work in computer music.7 We can sensibly talk of "transposing" and "inverting" such filter-classes in our GIS; since the GIS is commutative, these operations will behave in an intuitively familiar way, following the laws developed in sections 3.4 and 3.5 earlier. We can vary our GIS by varying the region within the frequency x varies. We can change the values of LO and HI; we can even consider disconnected regions within which x is to vary. 4.3 METHODOLOGY: In both GIS 4.2.1 and GIS 4.2.2, the formal relations involved match our sonic intuitions only to a certain extent. In either GIS, that is, we may have int(sp tj) = int(s2, tj), while the intuitive proportion between Sj and tj does not much "sound like" the intuitive proportion between s2 and ^ The models suffer here by comparison with the constructs of Wayne Slawson, who has developed an elegant model for an "intuitive" timbral space.8 Yet such considerations should not necessarily lead us to ignore GIS 4.2.1 and GIS 4.2.2. To relate "natural" mathematical structure with intuition is a problem in connection with virtually all theories involving sensory stimuli. For instance, it is mathematically natural to compare the amplitude of two sin waves by saying that one wave has i times the amplitude of the other; this is especially natural if both waves are at the same frequency. Yet if s, and tj = i • Sj are sin waves of the same low frequency, while s2 and t2 = i • s2 are sin waves of the same middle-range frequency, our intuition about the relative loudness of Sj and t, may differ considerably from our intuition about the relative loudness of s2 and t2. Still, nobody would propose that we should feel free to ignore quotients of amplitudes in a study of musical sounds, just because they do not always conform to our intuitions of loudness, give or take some simple transformation. This is the methodological point: It is unfair to demand of a musical theory that it always address our sonic intuitions faithfully in all potentially musical contexts under all circumstances. It is enough to ask that the theory do so in a sufficient number of contexts and circumstances. Perhaps, too, it is fair to ask that the theory be potentially able to address our intuitions in any given musical situation, provided that the situation develops in a suitable musical manner. To support the methodological point, let us explore certain thematic features from the first movement of Chopin's Bl»-Minor Sonata. On figure 4.6, (a) symbolizes aspects of the motive from the opening 7. The techniques are explained lucidly by Richard Cann in "An Analysis/Synthesis Tutorial," Computer Music Journal. Part 1 is in vol. 3, no. 3 (September 1979), 6-11. Part 2 is in vol. 3, no. 4 (December 1979), 9-13. Part 3 is in vol. 4, no. 1 (Spring 1980), 36-42. 8. "The Color of Sound: A Theoretical Study in Musical Timbre," Music Theory Spectrum vol. 3 (1981), 132-41.

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4.3

Generalized Interval Systems (3)

FIGURE 4.6

86

Grave measures, written in augmented rhythmic notation to fit the subsequent tempo of the Doppio Movimento. (b) symbolizes the motive of the first theme, (c) symbolizes a motive from the end of the bridge, and (d) symbolizes the motive of the second theme. At the end of the exposition there is a big push to cadence, leading to a big dominant of D[? major. Then there is a repeat sign. Chopin's notation is ambiguous; for this discussion I will assume what I hear strongly in any case, that the repeat goes back to (a), rather than (b). Then th opening note of (a) completes the Dt? cadence; one thereby hears F|? as the second note of (a) all the more strongly, since it is a third of the tonicized Db, despite the notated E natural and the subsequent leading of the note as E natural in the bass line. Here is the assertion I wish to study: The first melodic dyad of (b), marked y on the figure, belongs to the same interval class as x, the first melodic dyad of (a). This relation, like the relation of our four sin tones earlier, is formally "true" but intuitively problematic. At least, the relation of x and y dyads is hard to hear when we first hear the first theme, the first time through the exposition. But, I claim, the asserted relation has the potential for becoming audible, and in fact it does become audible, even highly significant, the second time through the exposition. To hear this one should take motive (b), rather than motive (a), as a poin of departure. By the end of the bridge section, motive (b) has been transformed into motive (c). Rhythmically, (c) augments the durations of (b) by factor of 2, and then augments its own last two notes by yet another factor of 2. Motive (d) introduces the second theme immediately thereafter. Rhythmically, (d) augments (c) by yet another factor of 2, and exchanges the durations of the first two notes (counting the rest as part of the note that precedes it). And then, going around the repeat, motive (a) augments (d) rhythmically by still another factor of 2. Thus the chain of motive-forms (b)-(c)-(d)-(a bis) is generated by a very consistent, indeed relentless, process of rhythmic expansion. After the repeat, when we continue on from (a bis) to (b bis), we a leaping from the end of the chain (b)-(c)-(d)-(a), back to its beginning and

Generalized Interval Systems (3)

4.3

motivic generator. In this larger and later context, we recognize that (b) and (a), the boundary forms of the chain, are rhythmically transformed variants, each of the other. And so we are much more willing to perceive other sorts of relationships linking (b) with (a). In particular it is now much easier, I would say proper and important, to hear dyad y as an ironically scurrying transformation of dyad x, with its portentous weight. We are helped in hearing this relationship by the dyad marked z on figure 4.6, which we have by now heard again and again during the second group. We are also helped by the big D|? cadence prepared at the end of the exposition, which helps us hear F|? at the repeat of (a). The first time around we may have had a certain predilection for E natural because of our associations with the opening of Beethoven's Sonata op. I l l , even specifically with measures 4| to 5^ of that piece. But the second time around, when we clearly hear F(? (as well), it is much easier for us to associate the x dyad, as root-and-minor-third of Dj?, with the y dyad, minorthird-and-root of B(?, particularly since the two dyads both begin on a D(?. To repeat my methodological claim: One should not' ask of a theory, that every formally true statement it can make about musical events be a perception-statement. One can only demand that a preponderance of its true statements be potentially meaningful in sufficiently developed and extended perceptual contexts.

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5

Generalized Set Theory (1):

Interval Functions; Canonical Groups and Canonical Equivalence; Embedding Functions

In this chapter we shall generalize certain aspects of atonal set theory so as to apply in the context of any GIS.1 5.1.1 DEFINITION: Given a GIS(S, IVLS,int), we shall mean by a set in the present chapter a finite unordered subfamily of S. 5.1.2 DEFINITION: If f is any mapping of S into itself and X is any set, we denote by "f (X)" the set of elements f (s) formed as s varies over X. That is, if X = (Sj, s 2 ,..., SN), then f(X) is the set whose members are f(Si), f(s 2 ), ..., and f (SN). If f is not 1-to-l then some of these f-values may not be distinct; then f (X) will have a smaller cardinality than X. If f is 1-to-l then the N f-values listed above will be distinct, and f (X) will have the same cardinality as X. 5.1.3 DEFINITION: Given a GIS, given sets X and Y, then the XjY interval function is a function IFUNC(X,Y) which maps the group IVLS into the family of non-negative integers as follows: For each interval i in IVLS, the value of the function, IFUNC(X, Y) (i), counts the number of distinct pairs (s, t) in S x S such that s is in X, t is in Y, and int(s, t) = i. That is, IFUNC(X, Y) (i) tells us in how many different ways the interval i can be spanned between (members of) X and (members of) Y. Usually, the context will make it clear when we are talking about the

88

1. The agenda, in chapters 5 and 6, parallels and expands upon the presentation of atonal set theory I developed in my article, "Forte's Interval Vector, My Interval Function, and Regener's Common-Note Function," Journal of Music 77ieor>>vol.21,no.2(Fall 1977), 194-237.

Generalized Set Theory (1)

5.13

function int, which maps S x S onto IVLS, and "the interval function" which, given sets X and Y, maps IVLS into the non-negative integers. IFUNC does not figure heavily in the standard literature of atonal set theory. Let us study some examples to see how it applies to that theory; the reader may thereby see the point of the construction, both in that specific application and more generally.

FIGURE 5.1 Figure 5.1 (a) displays the two pitch-class sets Xj = (E, B|?) and Y t = (F, A, C#) as a sort of symbolic melodic antecedent and consequent. Arrows drawn from each note of X t to each note of Y t show which intervals can be spanned, how many ways, between notes of X^ and notes of \l.Here, the even-numbered intervals cannot be spanned at all, and each odd-numbered interval can be spanned in exactly one way. Hence IFUNC(X l5 YJCi) = 0 if i is even, and = 1 if i is odd. Figure 5.1(b) displays the two sets Xj (as before) and Y2 = (G, A, B) in a similar format. We see that IFUNC(X1, Y 2 ) (i) also = 0 if i is even, and = 1 i iisodd.SoIFUNC(X 1 ,Y 2 ) = IFUNC^Y^ as a function on IVLS, even though Y2 is not the same set as Yt—indeed, Y2 is not even a form of Y t . 2 Figure 5.1(c) displays the new sets X 2 and Y3 in the same format. Here X2 is different from X x and Y3 is different from either Y! or Y 2 ; yet IFUNC(X2, Y3) is again the same function of i: Its value is 0 if i is even and 1 if i is odd. Figure 5.1 (d) displays yet another pair of sets, X3 = (E[?, F, G, A, B, C#) and Y4 = (E). The X3/Y4 interval function is still and again the same: IFUNC(X3, Y4)(i) is 0 if i is even, 1 if i is odd. The set X3 here is of different 2. I investigated this sort of phenomenon in an early article, "Intervallic Relations between Two Collections of Notes," Journal of Music Theory, vol. 3, no. 2 (November 1959), 298-301. Its style, unfortunately, makes few concessions to a non-mathematical reader.

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5.1.3

90

Generalized Set Theory (1)

cardinality from Xj and X 2 ; Y4 is likewise of different cardinality from Y l 5 Y 2 ,andY 3 . Figure 5.1 (d) is displayed in a contrapuntal rather than a melodic format. In general, one could also distinguish a set X from a set Y in other formats: by instrument, by mode of attack (staccato/legato), or by register (as opposed to "voice"), to give only a few examples. The melodic antecedent/consequent format, which seems particularly suggestive, was proposed by Michael Bushnell during his theoretical studies at Stony Brook in the 1970s. Bushnell also initiated and carried through the basic work on the specific analysis following, involving a passage from Webern's op. 7, no. 3, the third of the Four Pieces for Violin and Piano. Figure 5.2(a) reproduces the score through the opening of measure 9. The right hand of the piano over measures 3-8 comprises two melodic phrases, one filling measures 3-4, the other beginning on the B of measure 5 and extending through the F# of measures 7-8. This is all the lyric melody there is in the piece. Figure 5.2(b) displays as X and Y the pitch-class sets that underlie the two melodic phrases. The pitch noteheads representing the pitch classes E, C#, and Eb within Y have been brought down an octave from the music. This is partly for convenience, but partly too because of an idea which will emerge later. Figure 5.2(b) also displays the sets Z0 and Z3 projected by the violin ostinato that accompanies Y. The repeat sign on the figure indicates that we pass from Z3 to Z0, as well as from Z0 to Z3, during the course of this music. Figure 5.2(c) displays some interrelations of X, Y, and the Z-forms, interrelations that all involve the pitch-class interval 3. Each arrow on the figure indicates a T3 relation of one kind or another. The bottommost arrow depicts the T3 relation between Z0 as a whole and Z3 as a whole, a T3-relation of pitches as well as pitch classes. The arrows directly beneath the lower staff of the figure show T3 pitch-relations between the "fourths" of Z0 and those of Z3, and also T3 pitch-class relations between the "fourths" of Z3 and those of Z0, when Z0 is restated directly after Z3. Among all these fourths, Eb-Ab within Z0 and its T3-transform F#-B within Z3 are of special significance. A the crossed lines between the staves of figure 5.2(c) indicate, the Eb-Ab that ends ordered Z0 summarizes in retrograde the Ab-Eb that bounds the total span of ordered melodic phrase X; analogously, the F#-B that ends ordered Z3 summarizes in retrograde the B-F# that bounds the total span of ordered melodic phrase Y. This phenomenon suggests that we explore some sort of T3 relationship between ordered X and ordered Y. On the top staff of figure 5.2(c), the Ab-Eb boundaries of X and the analogous B-F# boundaries of Y are marked by beamed open noteheads. The medial Bb of ordered X is attached to the Xbeam with a stem from a solid notehead; a corresponding C# = T3(Bb) within ordered Y is notated analogously. To hear a function for C#-within-Y analo-

Generalized Set Theory (1)

5.1.3

FIGURE 5.2a

91

5.1.3

Generalized Set Theory (1)

FIGURE 5.2b

FIGURE 5.2c

92

gous to the function of B(?-within-X, the reader may listen to the following features of the passage. First, the C# and the F# within Y both receive agogic accents, marked "a.a." on figure 5.2(c). One may query just what this means, in the piano register of figure 5.2(a) at the given tempo. It surely means something, if only something conceptual; the pianist should be thinking like a singing instrument here. Second, the crescendo that begins at the B[? within X, a crescendo reproduced on figure 5.2(c), is analogous in some degree to the crescendo that begins at the C# within Y, even though the latter crescendo does not get all the way to the final note of its phrase. Third, the beamed B, C#, and F# of Y, within figure 5.2(c), occur as every-third-note of ordered Y; there is a serial regularity about their occurrence. Fourth, once the E, C#, and E[> of Y-within-the-music have been brought down an octave to provide the noteheads for Y-within-figure 5.2(c), it is easy to hear the latter structure as a compound gesture, counterpointing the rising beamed B-C#-F# (= T3 (ordered X)) against the falling chromatic counter-gesture-(F-E)-(E[?-D)-. That "falling chromatic line" on figure 5.2(c) fills in the chromatic space between C# and F#, the medial and final beamed notes of the rising gesture.

Generalized Set Theory (1)

5.1.3

The uppermost arrow on figure 5.2(c), then, indicates a structural T3 relation between set X and set T3(X)-embedded-in-Y. It seems secure to assert this relation between the sets, for it seems reasonable to assert much stronger T3 relations, e.g. the structural embedding of T3(ordered X) as every third note of ordered Y, or even the idea that phrase Y as a whole is inter alia a diminuted (ornamented/troped) version of T3 (phrase X). Now we shall inspect some IFUNC values, to see how they interact with these analytic ideas. Figure 5.3(a) tabulates four interval-functions, IFUNC(Z0,Z3), IFUNC(X, Y), IFUNC(X,Z0), and IFUNC(Z3, Y).

FIGURE 5.3

IFUNC (Z0, Z3) tells us that 4 intervals of 3 are spanned between Z0 and Z3; hence Z3 = T3(Z0). Likewise Z3 = T9(Z0) as a pitch-class set, since IFUNC(Z0,Z3)(9) = 4. This T9 relation is concealed, not revealed, by the registration of the pitches involved in the music. Nevertheless the T9 relation has a certain rhythmic effect, as portrayed in figure 5.3(b). The pair "(9,7>)" on that figure is a direct-product interval; it means "a pitch-class interval of 9 is spanned at a distance of 7 sixteenth-notes between attacks."

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5.1.3

Generalized Set Theory (1)

The less prominent values of 2 taken on by IFUNC(Z0, Z3) (less prominent than 4) also have rhythmic implications. Figure 5.3(c) shows how IFUNC(Z0,Z3)(2) = 2 and IFUNC(Z0,Z3)(10) = 2 articulate the rhythm of the violin obbligato. (The sixteenth-note symbols are now omitted from the direct-product intervals.) Figure 5.3(d) shows how IFUNC(Z0,Z3)(4) = 2 and IFUNC(Z0,Z3)(8) = 2 articulate the same obbligato in a different rhythmic way. IFUNC helps us to explore these intervallic/rhythmic substructures more carefully than we otherwise might. (We have plenty of time to do so at the written tempo.) The rhythmic effect of IFUNC(Z0, Z3) (3) = 4 is of course to support the quintuple periodicity of the ostinato, as in figure 5.3(e).

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Now let us return to the IFUNC table of figure 5.3(a) and inspect IFUNC(X,Y). Since X has cardinality 3 and IFUNC(X, Y)(3) = 3, this interval function tells us that T3(X) can be embedded within Y. That is, the function informs us that such an embedding is "true," and would lead us to inspect its potential musical significance in the passage at hand if we had not already done so. The value of 3 is a maximal possible value for IFUNC(X, Y), just as 4 was a maximal possible value for IFUNC(Z 0 ,Z 3 ). Both functions take on their maximal values on the argument i = 3. This sets up a "true" proportion among the four sets involved: X is to subsequent Y just as Z0 is to subsequent Z3, so far as a certain property is concerned (having a maximum IFUNC value at the argument i = 3). Our discussion of figure 5.2(c) has shown that this theoretical truth in fact reflects a musically significant relationship. IFUNC(Z0, Z3) also took on its maximum value at the argument i = 9. But IFUNC(X, Y) does not. Instead, IFUNC(X, Y) has a maximum at i = 8. This tells us that T8(X) as well as T3(X) can be embedded within Y. Is the fact musically significant as well as true? To explore the matter, let us first find the notes of T8(X) as they occur within Y. X = (Ab,B|7,E|7), so T8(X) = (E, Ftf, B). Within ordered-Y, the members of T8(X) appear in the rotated order B-E-F#. Inspecting the score again, one asks if these three notes have any special functions that affect the shaping of phrase Y in the music. I believe they do. Namely, they are the registral and temporal boundary tones for phrase Y. That is, B is at once the first note and the lowest note of the melodic phrase; E is its highest note; F# is its last note. So one can plausibly assert an overall shape for phrase Y that uses T8(X) as a bounding frame, along the lines of figure 5.4. We have been thinking of the set (B, E, F#) as a transposed form of set X, because that transpositional relation is what IFUNC has brought to our attention. As a series, the succession B-E-F# is a rotation of E-F#-B, which is T8 (ordered X). B-E-F# can also be generated as the retrograde series of ordered-X inverted about C#. IFUNC cannot suggest this relation to us; in

Generalized Set Theory (1)

5.1.3

FIGURE 5.4

chapter 6 we shall generalize the interval function to an "injection function" which can. In any case, our noticing that IFUNC(X, Y) (8) = 3 has suggested a search that has led us to the musical relationship of figure 5.4, a relationship that deserves serious aural attention. The relationship engages the register of the high E in the music for phrase Y; we would not hear any boundary function for E if it were an octave lower as in figure 5.2(c).3 Let us return again to the table of figure 5.3(a). We can notice there an interesting resemblance between IFUNC(X,Z0) and IFUNC(Z3,Y). Both functions take on their maximum values at the arguments i = 0, i = 5, i = 6, and i = 11. Figures 5.5(a) through (d) show how the four intervals 0, 5,6, and 11 respectively govern both a model/expansion relation of X to Z0 and an analogous model/expansion relation of (unordered) Z3 to Y. (a) through (d) of the figure show how the two "fourths" of X, transposed variously by i = 0, 5,6, or 11, map into the two "fourths" of Z0; this projects the relation IFUNC(X,Z0)(i) = 2 in each case, (a) through (d) of the figure show analogously how the four possible subtrichords of Z3, all of which are in Forte's set-class 3-5, can each be transposed by one of the key intervals i = 0, 5,6, or 11 so as to map into one of the two subtrichords of Y that lie in the setclass 3-5, both boundary trichords in a certain sense; this projects the analogous relation IFUNC(Z3, Y)(i) = 3 in each case. The latter four mappings, of Z3-trichords into Y-trichords, suggest that Y can be articulated, when heard "against" Z3, as suggested by figure 5.5(e). There we hear Y articulated into a beamed temporal "boundary" comprising the union of its two 3-5 trichords, plus a bracketed temporal "interior" comprising C#, E\>, and D. The articulation of Y in this fashion is reinforced by the coincidence of the interior set (C#, Efr, D) with the cadence set accompanying the melodic Efr that concludes ordered-X in the music. Figure 5.5(f) shows the pitches in the music between the attack of the X-terminating E|? in measure 4 and the comma in measure 5 that separates X from Y in the melody. The set of pitch classes heard in figure 5.5(f) recurs as the interior set of Y, bracketed on figure 5.5(e). A methodological note is in order. Some readers may feel confused rather than enlightened by the variety of ways in which we have shaped and articulated Y during our discussion, specifically in figures 5.2(c), 5.4, and 5.5(e). 3. I am indebted to Taylor Greer for suggesting to me (in another connection) that the "boundary set" of an atonal melodic phrase often has special set-theoretic functions.

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FIGURE 5.5

96

If Y "really is" an ornamented version of T3(X), then are we not "wrong" to consider Y as a spatio/temporal gesture framed by T8(X), or to consider Y as articulated by figure 5.5(e)? On the other hand, if Y "really is" a spatio/ temporal gesture framed by T8(X), then are we not "wrong" ... (and so on)? One often hears such notions accompanied by a thought like, "after all, the pianist must decide which way to play it." Concerning these issues, the first thing to be said is that the last remark in the above paragraph seriously under-estimates and misapprehends the resources available to a good pianist (or performer in general) even in a context as constrained as that of the Y-phrase in the music. What one can hear, one can play. Let us suppose now that the possibly unquiet reader does hear something significant, or at least engaging, about each of the Yarticulations. (Otherwise there would be no disquiet and no problem.) The disquietude arises intellectually, from considering Y as something which

Generalized Set Theory (1)

5.1.3

"really is," independent of any specified environment. The interpretation of Y manifest in figure 5.2(c) is not an attempt to get at the "real" structure of Y in a context-free environment; rather it asserts that Y, when heard following X and in connection with the many intervals of 3 between notes of X and notes of Y, will tend to be articulated as in that figure. Likewise, figure 5.4 does not pretend to assert something that Y "really is," context-free. Rather, it asserts a structuring for Y that tends to emerge in the environment, "remembering X and hearing the many intervals of 8 between notes of X and notes of Y." Finally, figure 5.5(e) does not try to assert a "real" form for Y either. Rather, it says that Y will tend to articulate in that way when heard in a complicated environment involving Z3, the idea that Y expands Z3 in a manner analogous to the way Z0 expanded X (as in figures 5.5(a)-(d)), and the effect of the cadence harmony displayed in figure 5.5(f), as that event is recalled in its various contexts. The various IFUNC values of figure 5.3(a) are useful tools, as we have seen, for exploring the multifaceted aspects of Y in various of its environments. Figures 5.2(c) and 5.4 explore two different ways in which Yfollowing-X engages IFUNC; these ways will be of practical interest to the pianist shaping the lyric melody of the right hand over measures 3-7. As we noted before, that is all the lyric melody there is in the piece, and since the melody articulates musically into two phrases, the pianist will naturally want to explore listening to the various kinds of "logic" adhering to the way in which Y follows X. Figures 5.2(c) and 5.4 provide that theoretical "logic" in this environment, the one figure in connection with sensitivity to the interval 3, the other in connection with sensitivity to the interval 8. Figure 5.5(e), in contrast, explores Y as it occurs in a different context, that is as it relates to Z3 in a complex theoretical proportion also involving X and Z0 as in figures 5.5(a)-(d). The corresponding musical environment is now not the way the antecedent and consequent phrases of the lyric melody are shaped in the right hand of the piano, but rather the way in which the violin ostinato comments upon that melody, and is commented upon by it. The new musical environment we are now considering involves an interrelation, or rather several interrelations, between the instruments. This will be of particular interest to the violinist, trying to keep the ostinato figure fresh and alive rather than mechanical. In this environment the second appearance of Z3 will sound different from the first. Figure 5.6 shows what I mean. (a) and (b) of the figure show how the first Z3, at measures 6-7, comments upon the opening trichord of Y at the intervals of 5 and 11 respectively, (c) and (d) of the figure, in contrast, show how the second Z3, during measures 7-8, comments (as well) upon the fresh trichord (B, F, F#) within Y, now that F# has appeared on the scene. The new commentary is at the intervals of 0 and 6. When executing these commentaries, it will help the violinist to listen to Y articulated as in figure 5.5(e) earlier.

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FIGURE 5.6

We could of course also consider Y simply in its own environment, spanning various intervals within itself. This would lead us to examine IFUNC(Y, Y), which is essentially Forte's "interval vector of Y." Our study of the Webern passage has made Z0 seem more subordinate than it sounds in the piece; the study so far has also underplayed the role of the 3-note chromatic set (C#, D, E|?), the set which appeared in figure 5.5(e)-(f). A brief discussion of figure 5.7 will attempt to right this balance, without getting fussy about more IFUNC values.

FIGURE 5.7

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The figure transcribes the noteheads of the ensemble up through the cadence at the bar line of measure 5. Over measures 1 through 3 the music slowly exposes the set W0 = (A, 6(7, A(?). (At the written tempo, this takes over 20 seconds.) The boundary tones for the exposition of W0 are shown on the figure with open noteheads: A is lowest and first; Ab is (so far) highest and last. As the piece continues, the rhythm becomes more active and the registral space expands. By the cadence at measure 5 we hear new boundary tones, also shown with open noteheads: E|?5 is a new highest tone; D3 is a new lowest tone and also a new last tone. The four boundary tones on the figure add up precisely to a large-scale projection of the set Z0 = (A, D, Efc>, Aj?). So Z0 has a

Generalized Set Theory (1)

5.1.5

very strong structural meaning in its own right, by the time it enters in the foreground of measure 6 to begin the violin ostinato. The progression of boundary tones on figure 5.7 suggests an inversional relationship: The falling bass, from the opening A3 to the closing D3, sandwiches a rising melody spanned by the local high tones A|?4-then-E|?5. Indeed this rising melody is the X-phrase itself. The injection function, to be discussed in chapter 6, will allow us to engage such inversional relations within and between sets; IFUNC cannot do so. The set W 5 on figure 5.7 is the cadence harmony shown earlier on figure 5.5(f), a set projected in another way by the bracketed "interior of Y" on figure 5.5(e). Figure 5.7 shows how the cadential W5 responds to the opening W0. W5 is of course T5(W0). The bass A of W0 as presented, flanked by its chromatic neighbors B(? and A|? above, progresses over figure 5.7 to the bass D of W 5 , flanked by its chromatic neighbors Eb and C# above. So the T5 relation of W0 as a whole to W 5 as a whole, as that relation moves structurally over figure 5.7, expands upon the T5 relation of A to D in the bass register. This is not lost upon our ears when the violin ostinato begins precisely with A-D- at measure 6, presenting thereby a highly charged T5 relation in the foreground. All these considerations dispose us to bracket off W 5 as the interior of Y with somewhat more aural attention than our earlier discussion could make plausible. Having explored the pertinence of IFUNC in the setting of traditional atonal set theory, let us now return to further study of the formal X/Y interval function in generalized set theory. We recall Definition 5.1.3: Given a GIS, given sets X and Y, then IFUNC(X, Y) is that function which assigns to each i in IVLS the number of ways in which i can be spanned from X to Y, that is the number of pairs (s, t) such that s is in X, t is in Y, and int(s, t) = i. We shall now study how IFUNC is affected as the sets X and Y are manipulated and transformed in various ways. The first theorem to be noted shows that when the roles of sets X and Y are exchanged, IFUNC is in a certain sense "inverted." 5.1.4 THEOREM: IFUNC(Y,X)(i) = IFUNC(X,Y)(i-1). Proof: IFUNC(X, Y) (i-1) is the number of pairs (s, t) such that s is in X, t is in Y, and int(s, t) = i"1. This is the number of pairs (t, s) such that t is in Y, s is in X, and int(t,s) = i. And that number is IFUNC(Y,X)(i). The next theorem shows that IFUNC is not affected when X and Y are both transformed by the same interval-preserving operation P. 5.1.5 THEOREM: Let P be any interval-preserving operation. IFUNC(P(X),P(Y)) = IFUNC(X, Y) as a function on IVLS.

Then 99

5.1.6

Generalized Set Theory (1)

Proof (optional): Let PAIRS be the family of pairs (s, t) such that s is in X, t is in Y, and int(s, t) = i. Then IFUNC(X, Y)(i) is the cardinality of PAIRS. Let PAIRS' be the family of pairs (s', t') such that s' is in P(X), t' is in P(Y), and int(s', t') = i. Then IFUNC(P(X), p(Y))(i) is the cardinality of PAIRS'. So, to show that IFUNC(X, Y)(i) = IFUNC(P(X), P(Y))(i), it suffices to show that PAIRS and PAIRS' have the same cardinality. And that will be the case if we can map PAIRS onto PAIRS' by some 1-to-l function f. We shall construct such a function f. Given (s, t) in PAIRS, define f(s, t) = (P(s), P(t)). f(s, t) is indeed a member of PAIRS', for P(s) is in P(X), P(t) is in P(Y), and int(P(s), P(t)) = int(s, t) = i (since P is interval-preserving), f is a 1-to-l map because P is 1-to-l: If (P(s,), P(t,)) = (P(s2), P(g), then P(s,) = P(s2) and P(t,) = P(t,), whence s, = s2 and tj = tj, whence (s,, t, = (s2, tj). It remains only to show that f is onto PAIRS'. The interval-preserving operations form a group of operations on S; hence P"1 exists and is interval-preserving. Given (s', t') in PAIRS', set s = P~V) and t = P"1^'). The reader may verify that (s, t) is in PAIRS, and that the given (s', t') is the image of (s, t) under the map f. q.e.d. Now we shall see how IFUNC is affected when X or Y or both are transposed. 5.1.6 THEOREM: For any transposition operation Tn, the formulas (A), (B), and (C) below obtain. (A): IFUNC(Tn(X), Y)(i) = IFUNC(X, Y)(ni) (B): IFUNC(X, Tn(Y))(i) = IFUNC(X, Y)(in~]) (C): IFUNC(Tn(X), Tn(Y))(i) = IFUNC(X, YXnin'1)

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Proof of (A) (optional): Let PAIRS be the family of pairs (s, t) such that s is in X, t is in Y, and int(s, t) = ni. Then the cardinality of PAIRS is IFUNC(X, Y)(ni), the right side of Formula (A) above. Let PAIRS' be the family of pairs (s', t) such that s' is in Tn(X), t is in Y, and int(s', t) = i. Then the cardinality of PAIRS' is MJNC(Tn(X), Y)(i), the left side of Formula (A). To prove the formula, then, it suffices to show that PAIRS and PAIRS' have the same cardinality. And we can show that by demonstrating a function f which is 1-to-l from PAIRS onto PAIRS'. The desired function is f(s, t) = (Tn(s), t). The reader may verify that f maps PAIRS into PAIRS', that f is 1-to-l, and that f is onto. Proof of (B) (optional): IFUNC(X, Tn(Y))(i) = IFUNC(Tn(Y), XXi'1), via 5.1.4. This = IFUNC(Y, X)(ni-1, via Formula (A) just proved. And that = IFUNC(X, YXiir1), again via 5.1.4. Proof of (C) (optional): IFUNC(Tn(X), Tn(Y))(i) = IFUNC(X, TB(Y))(ni), via Formula (A) above. And that = IFUNC(X, Y)(nin~!), via Formula (B) just proved, q.e.d.

Generalized Set Theory (1)

5.1.8

In contrast to the results of Theorem 5.1.6, there is not much to be said in general about the effect of applying an inversion operation to X, Y, or both, so far as IFUNC is concerned. If we restrict our attention to commutative groups of intervals, though, then we can say something. 5.1.7THEOREM: If I is any inversion operation in a commutative GIS, then IFUNC(I(X),I(Y)) = IFUNC(Y,X). Optional sketch for a proof: In a commutative GIS, the inversions are interval-reversing operations (3.6.3). The present theorem may therefore be proved by the same technique used to prove 5.1.5 above. General questions involving IFUNC and other operations on S will be better pursued using the Injection Function to be developed in chapter 6; that function will generalize IFUNC among other things. The same holds for general questions involving IFUNC and complement relations among sets, where those are relevant. (We have not assumed that S is finite, so the complement of a set need not be a set according to Definition 5.1.1.) IFUNC can be given an interesting interpretation as a probability distribution. 5.1.8 THEOREM: Let X and Y have respective cardinalities M and N. Select a member s of X at random and a member t of Y at random. Then the number IFUNC(X, Y) (i)/(MN) measures the probability that int(s, t) will be found to equal i. Proof: There are MN possible pairs that can be pulled in this way, and IFUNC(X, Y) (i) of these pairs will have the desired property, q.e.d. Theorem 5.1.8 is interesting because we have used IFUNC so far only as a precision tool; the theorem shows that it can also be used to portray a statistical texture. For instance, suppose a clarinet is told to improvise for a time upon the notes of a pitch set X, while a flute is told to improvise for the same span of time upon Y. A statistical field of intervals will result from this improvisation, and that field can be modeled by IFUNC(X, Y) according to the rule of Theorem 5.1.8. Even when we are not applying IFUNC to such "stochastic" compositional settings, it is still sometimes useful to regard it as providing a statistical backdrop for intervallic events. For instance, the notion that a certain interval i appears "often" or "only rarely" between X and Y is implicitly dependent on this backdrop: i appears "only rarely," e.g., compared to how often other intervals appear. We shall look at aspects of the opening from Schoenberg's Violin Fantasy op. 47, taking this point of view. The violin projects the pitch-class set

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5.1.8

Generalized Set Theory (1)

Y = (Bb, A, C#, B, F, G), while the piano accompanies the violin with the set X = (Eb, E, C, D, AJ7, Gb). X is the complement of Y and also an inversion of Y but these relations will not concern us explicitly for present purposes. Figure 5.8(a) displays the values of IFUNC(X, Y).

FIGURE 5.8

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Figure 5.8(a) shows that IFUNC counts "many" odd intervals from X to Y (accompaniment-to-solo, lower-instrument-to-upper-instrument), and "few" even intervals. The theoretical point at hand is that two appearances of an interval is not intrinsically "few"—it would not be few e.g. between two trichords. Rather, the scarce intervals on figure 5.8 (a) are "scarce" only against the statistical backdrop of the table as a whole. The equation IFUNC(X, Y) (0) = 0 expresses the fact that X and Y have no common tones. So the scarce interval 0 does not appear at all between the instruments. The scarce intervals of 4 and 8 each appear in two different ways between piano and violin, and those ways are of analytic interest. Figure 5.8(b) displays the opening noteheads of the piece in order of succession. Figure 5.8(c) shows how the two intervals of 4 between X (piano) and Y (violin) appear. One of those 4-intervals appears between Gb, the last and lowest note, and Bb, the first note and a provisional low note for the violin. The other 4-interval appears between Eb, the first and highest note of the piano, and G, the last and lowest note of the violin. Thus all the notes in figure 5.8(c) are boundary tones of one sort or another for the passage; the figure shows how the scarce interval 4 binds this spatio/temporal frame for the phrase. Figure 5.8(d) shows the scarce interval 8 functioning in a similar way. We have already discussed the Bb and the Eb as boundaries; the B is a high

Generalized Set Theory ( I )

5.1.8

boundary; the low D is a provisional low boundary until the very end of the phrase. Many abstractly interesting questions can be asked about our generalized IFUNC. One family of questions takes the following tack: Given some property that a function from IVLS to the non-negative integers might have, under what conditions on X and Y will IFUNC(X, Y) have that property? For instance, we may ask under just what conditions on X and Y IFUNC(X, Y)^"1) will equal IFUNC(X,Y)(i) for all i. Via 5.1.4, this is the same as asking under what conditions on X and Y IFUNC(Y, X) will be the same function as IFUNC(X,Y). More generally, we can ask under what conditions there will exist intervals m and n such that IFUNC(X,Y)(mi~ 1 n) will equal IFUNC(X,Y)(i) for all i. Satisfactory answers for these questions are not known even in connection with the standard GIS for atonal set theory.4 Another family of questions generalizes in one possible direction the traditional topic of "Z-sets." In Forte's theory, pitch-class sets X t and X2 which are not transposed or inverted forms of each other are Z-related if and only if IFUNCCX^XO = IFUNC(X2,X2), as a function on IVLS. In a general GIS setting we may ask under what conditions on X x and X2 that equation will obtain. We know that the relation will hold if there is an intervalpreserving transformation P such that X2 = PCXJ (5.1.5). The relation will not automatically hold in a non-commutative GIS when X2 is a transposed form of X t : 5.1.6 shows us that if X2 = T^X^, then IFUNC(X2,X2)(i) will equal IFUNCtX^X^nhT1), but not necessarily IFUNCCX^XJCi). Going even further, we may ask under what conditions among the four sets X l 5 Y!, X 2 , and Y2 we will have the relation IFUNCtX^Y^ = IFUNC(X 2 , Y 2 ) (as a function on IVLS). Figure 5.1 earlier provided some examples of this state of affairs, in the relatively well-behaved GIS of traditional atonal set-theory. This is all a vast open ground for mathematical and musical inquiry, even in atonal set-theory. Our questions can be transferred to a more general mathematical setting. Readers who do not have graduate-level mathematical background should skip this paragraph. If we use LABEL to identify S with IVLS, we can see that we are treating S = IVLS as a locally compact group under the discrete topology; our "sets" are the compact subsets of IVLS, and IFUNC(X, Y) (i) is the convolution (f * * g) (i), where f and g are the characteristic functions of the sets X and Y respectively. All of our questions may then be generalized to questions about the interrelations, in a locally compact group, among the characteristic functions of compact subsets. E.g: Given compact subsets X l s 4. Eric Regener, "On Allen Forte's Theory of Chords," Perspectives of New Music vol. 13, no. 1 (Fall-Winter 1974), 191-212. Regener poses essentially equivalent questions in that connection, starting at "Among many other things," on page 204.

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5.2.1

Generalized Set Theory (1)

X 2 , Y t , and Y 2 , with characteristic functions f l s f 2 , g l5 and g2, under what conditions will fr* * gj and f^ * g2 be the same function? As with the special case of IFUNC, the study is much simplified when the group is commutative. Other ways of generalizing our questions about IFUNC will come up later in connection with the Injection Function. We now make a big articulation and turn our attention to generalizing Forte's Interval Vector. To do so, we shall need some further apparatus, in particular the notions of a "Canonical Group" of operations on S and a "Canonical Equivalence Relation" among sets. It is worth noting that IFUNC, and the Injection Function later, can be defined and discussed without invoking those notions. Forte considers pitch-class sets X and Y to be canonically equivalent, by our definition coming up, if Y is a transposition or inversion of X. Here the canonical group comprises the transposition and the inversion operations. (They form a group here because the GIS is commutative.) In other systems of atonal set theory, X and Y are considered canonically equivalent if and only if Y is a transposition of X; then the canonical group comprises transpositions only. In still other systems the canonical group includes not only the transpositions and the inversions but also the circle-of-fifths transformations and possibly other transformations as well.5 The idea of canonical equivalence allows us to speak about "the forms of" a set X; those are the sets X' that can be derived from X by operations in the canonical group, or (what is the same thing) the sets X' which are canonically equivalent to X. The work coming up in section 5.2 generalizes these ideas. 5.2.1 DEFINITIONS: In certain connections we shall fix a group of operations on S and call it "the canonical group." It will be denoted CANON. Sets X and X' will be called "canonically equivalent" if there exists some canonical operation A such that X' = A(X). The defined relation is indeed an equivalence. It is reflexive: X = IDENT(X). It is symmetric: If X' = A(X) then X = A~l(X'). It is transitive: If X' = A(X) and X" = B(X'), then X" = (BA)(X). In any specific context, we suppose there is some good reason for selecting one particular group of operations on the family S as canonical. Generally the good reason will involve intervallic relationships of one sort or another within a GIS for which S is the family of objects. But formally, we do not actually need a GIS structure at all. We could carry through our work if we

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5. An extended discussion of such matters can be found in Robert D. Morris, "Set Groups, Complementation, and Mappings among Pitch-Class Sets," Journal of Music Theory vol. 26, no. 1 (Spring 1982), 101-44.

Generalized Set Theory (1)

5.3.1

just started with some family S of objects and some group CANON of operations on S, not concerning ourselves with formal intervals at all. When we look at things so abstractly, we foreshadow the "transformational" approach to be taken later on in this book. 5.2.2DEFINITIONS: We shall write /X/ to denote the canonical equivalenceclass containing the set X. /X/ will be called, for short, the "set class of X." The term "set class" will grate dreadfully on the ears of any mathematical logician. Still, it is becoming standard usage for atonal theory. In earlier writing I used the term "chord type." But that term loses its intuitive pertinence when we are working with generalized sets of all kinds, including rhythmic sets, timbral sets, sets in direct-product GIS structures, and the like. It is important to understand that the notion of set class depends not only upon the set X at hand but also upon the canonical group CANON selected for the occasion. For example, let us fix the standard GIS of atonal set theory; let us select X = (C, E, G). If we choose CANON to be the group of transposition operations, then /X/, the family of transpositions-of-X, comprises the major triads. But if we choose CANON to be the group of transpositions and inversions, then /X/, the family of transpositions-and-inversions-of-X, comprises all the harmonic triads, major and minor. 5.2.3 LOCUTIONS: "X' is a form of X" means that X' is canonically equivalent to X. /X/ may be referred to as "the forms of X." Given a GIS, it can be a tricky business to decide for any particular theoretical exercise just which operations on S are to be allowed into CANON.6 We shall generally want to include at least the interval-preserving operations in the canonical group. For if P is an interval-preserving operation andX' = P(X),thenIFUNC(X',X') = IFUNC(X,X)(5.1.5).ThusX'-in-itsown-context has the same intervallic structure as X-in-its-own-context; this is a reasonable criterion for wanting X' to be considered "equivalent" to X. When the GIS is commutative, the interval-preserving operations will be exactly the transpositions. When the GIS is non-commutative, we may or may not wish to include the transpositions, as well as the interval-preserving operations, in the canonical group for a given exercise. We can now define the Embedding Number, a construct which generalizes Forte's Interval Vector. 5.3.1 DEFINITION: Given sets X and Y, the embedding number of X in Y, EMB(X, Y), is the number of forms of X (i.e. members of /X/) that are included in Y. 6. Morris (ibid.) discusses this at length, in atonal theory.

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5.3.2

Generalized Set Theory (1)

The embedding number depends on the notion of set class, which depends in turn upon the canonical group at hand; this cannot be overemphasized. For example, let us work within the standard GIS of atonal theory; let X be some major triad and let Y be some major scale. If CANON consists of the transposition operations only, then EMB(X, Y) = 3: three major triads are embedded in the scale. On the other hand, if CANON consists of both transpositions and inversions, then EMB(X, Y) = 6: six harmonic triads are embedded in the scale. Strictly speaking, we should write EMB(CANON,X, Y) to show that the embedding number varies with the canonical group as well as the sets X and Y. But our notation is already cumbersome enough. If X' is a form of X then /X'/ = /X/; the members of /X'/ are the members of /X/ and therefore, via 5.3.1, EMB(X', Y) = EMB(X,Y). If Y' = A(Y) is a form of Y and EMB(X, Y) = N, let X l f X 2 , . . . , Xn be the distinct forms of X embedded in Y. Then ApCJ, A(X2), ..., A(X n ) are the distinct forms of X embedded in Y'= A(Y). So EMB(X,Y') also = N; i.e. EMB(X, Y') = EMB(X, Y). It follows: If X' is a form of X and Y' is a form of Y then EMB(X', Y') = EMB(X,Y). We have proved that Definitions 5.3.2 following make sense. 5.3.2 DEFINITIONS: EMB(/X/, Y) will mean the value of EMB(X', Y) for any member X' of/X/. EMB(X, /Y/) will mean the value of EMB(X, Y') for any Y in /Y/. EMB(/X/, /Y/) will mean the value of EMB(X', Y') for any X' in /X/ and any Y' in /Y/. Let us consider the standard atonal GIS, and let us fix CANON as either the transpositions, or the transpositions plus the inversions. The various 2note sets will gather into exactly six "2-note set-classes," SC^ SC 2 ,..., SC6. (SC4 for instance contains all the 2-note sets whose notes lie an interval of 4-or-8 from each other.) Given a set Y, we can ask for the values of EMB(SCn, Y) as n runs from 1 through 6, i.e. the values of EMB(/X/, Y) as the variable /X/ runs through the six 2-note set classes. The function giving us those six values is Forte's Interval Vector of Y. From our point of view here, we could call it the "dyad-type vector of Y." By analogy we could study the "trichord-type vector of Y," that is the function which gives us the values of EMB(/X/, Y) as /X/ runs through the various 3-note set classes. (There will be nineteen such classes if CANON contains transpositions only; there will be twelve if CANON contains both transpositions and inversions.) Leaving the GIS of atonal theory now, we can generalize such vectors in an abstract setting by the following definition. 106

5.3.3

DEFINITION: By the "M-class vector of Y," we understand the function

Generalized Set Theory (1)

5.3.5.2

EMB(/X/, Y) as the variable /X/ runs through the various set-classes whose members have cardinality M. In case S is infinite there may be an infinite number of M-member set classes. But since Y is finite it has only a finite number of subsets, which can belong to only a finite number of set classes. Hence EMB(/X/, Y) must be zero for all but a finite number of /X/. 5.3.4 THEOREM : Let the cardinality of Y be N. Let M be a positive integer less than N. Pull M members of Y at random. Then the probability that you have pulled a set of class /X/ is given by the number EMB(/X/, Y)/COMB(M, N). Here COMB(M,N) is the number of combinations of M things that can be extracted from a family of N things; e.g. COMB(13,52) is the number of possible hands at bridge. Proof: There are COMB(M,N) different M-member sets \ve might extract from Y, and EMB(JX|, Y) of those sets will be of class [X|. COMB(M,N) can be calculated to be Nl/(Mi.(N - M)V), where N! is factorial N, etc. Theorem 5.3.4 shows us that EMB, like IFUNC earlier, can be regarded as a statistical measure aside from its uses as a precision tool. The theorem enables us derive a very strong formula interrelating various M-class vectors in a general setting. That formula will be proved in 5.3.5.2 below; we shall first prove a lemma involving some numerical computation. 5.3.5.1 LEMMA: Given positive integers L, M greater than L, and N greater than M, then COMB(L,N)/(COMB(L,M)COMB(M,N)) = 1/COMB(N - M, N - L). Proof (optional): COMB(L, N)/(COMB(L, M)COMB(M, N)) = N!/(L!(N - L)!) divided by the product of M!/(L!(M - L)!) andN!/(M!(N-M)!) = N!/(L!(N - L)!) times L!(M - L)!/M! times M!(N - M)!/N!. In this product the factorials of L, M, and N can each be cancelled from the numerators and the denominators of the participating fractional factors. This leaves the product = 1/(N - L)! times (M - L)! times (N - M)! = (N - M)!(M - L)!/(N - L)!. And that is the multiplicative inverse of COMB(N - M, N - L), as asserted, q.e.d. 5.3.5.2 THEOREM: Let L, M, and N be as in Lemma 5.3.5.1; let ADJUST be the fraction calculated in that lemma. Let Z be a set of cardinality N and let X

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be a set of cardinality L. Then EMB(X, Z) = ADJUST • SUM(EMB(X, /Y/)EMB(/Y/, Z)), where the SUM is taken over all M-member set-classes /Y/. Proof (optional): Only a finite number of terms in the sum will be nonzero, so summing "over all... /Y/" makes sense. (As /Y/ varies, only a finite number of values EMB(/Y/, Z) can be non-zero.) Let us imagine first pulling an M-member subset from Z, and then pulling an L-member subset from that M-member set. In the first pull, the probability that we have pulled a set of class /Y/ is prob(/Y/) = EMB(/Y/,Z)/ COMB(M,N) (5.3.4). And if we have pulled a set of class /Y/, the probability that our second pull will yield a form of X is prob(/X/-from-/Y/) = EMB(/X/, /Y/)/COMB(L, M) (5.3.4). Probability theory tells us how to manipulate these numbers so as to calculate the chances of ending up with some form of X pulled from Z. Namely: prob(/X/-from-Z) = SUM(prob(/Y/)prob(/X/-from-/Y/)), where the sum is over all possible intermediate pulls /Y/, that is over all the Mmember set-classes /Y/. Now in the probability formula above we can substitute, via 5.3.4, prob(/X/-from-Z) = EMB(X, Z)/COMB(L, N) prob(/Y/) = EMB(/Y/, Z)/COMB(M, N) and prob(/X/-from-/Y/) = EMB(X,/Y/)/COMB(L,M). The probability formula then takes the new form EMB(X, Z) = FACTOR • SUM(EMB(X, /Y/)EMB(/Y/, Z)), where

108

FACTOR takes care of the COMB-numbers in the denominators of the various probability values above. Calculating FACTOR out, we see that this fraction specifically equals COMB(L, N)/(COMB(L, M)COMB(M, N)). And that, via the Lemma, is ADJUST, q.e.d. The formula of Lemma 5.3.5.1 is not necessary to prove the formula of the theorem; we could simply define ADJUST to be COMB(L,N)/ (COMB(L,M)COMB(M,N)). But the value 1/COMB(N - M, N - L will often be much easier to compute. For instance try L = 5, M = 9, and N = 11: The value 1/COMB(N - M, N - L) gives us 1/COMB(2,6) = 1/1 very quickly. To give us an intuitive sense of why Theorem 5.3.5.2 is interesting, it will be useful to study a simple example from Fortean set-theory in connection with a topological model. Let Z be the pitch-class tetrachord (A, B, C, D); let X be the dyad (A, C). Here L = 2 and N = 4; we will set M = 3 and examine just what Theorem 5.3.5.2 is telling us. First let us inspect figure 5.9. It represents the tetrachord Z as a tetra-

Generalized Set Theory (1)

5.3.5.2

FIGURE 5.9

hedron in three-dimensional space; the vertices of the tetrahedron are the member pitch-classes A, B, C, and D of the set Z. When we inquire about the 2-note subsets of Z, we are inquiring about the boundary edges of this tetrahedron. The figure lays the boundary edges out for inspection below the tetrahedron. Of the six edges, one belongs to set-class 2-1, two belong to setclass 2-2, two belong to set-class 2-3, and one belongs to set-class 2-5. The embedding numbers at the lower right of the figure express these counts. Now let us inspect figure 5.10. It first analyzes the tetrachord into its four triangular boundary faces, and then analyzes each triangular face into its three boundary edges. Two of the triangles are in Forte-class 3-2; these triangles are labelled Y2 and Y2' on the figure. The other two triangles are in Forte-class 3-7; these triangles are labelled Y7 and Y7'.

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FIGURE 5.10

110

On figure 5.10 the four triangles are lined up beneath the tetrahedron, and the edges of each triangle are stacked up below that triangle. In the four resulting stacks, each edge of the original Z-tetrahedron appears twice. That is because each edge of the tetrahedron belongs to two of the triangles. (Edge AB, for instance, belongs both to triangle Y2 and to triangle Y7'.) As a result, when we count how many sticks at the bottom of the figure are in Forte-class 2-3, we must divide that count by two, to arrive at the number of sticks in that Forte-class we found on figure 9 earlier. For example, figure 5.9 counted two edges-of-Z lying in Forte-class 2-3, namely AC and BD. EMB(class

Generalized Set Theory (1)

5.3.5.2

2-3, Z) = 2. Figure 5.10 counts as edges-of-faces-of-Z twice as many sticks of class 2-3, namely AC-as-edge-of-Y2, BD-as-edge-of-Y2', AC-as-edge-of-Y7, and AC-as-edge-of-Y7'. What Theorem 5.3.5.2 does in this connection is to ADJUST the count of sticks at the bottom of figure 5.10, dividing it by two to conform with the stick-count at the bottom of figure 5.9. The theorem knows that two is the proper number to divide by here, because 1/2 is the present value of ADJUST = 1/COMB(N - M, N - L) = 1/COMB(1,2) = 1/2. If Z were heptachordal object in six-dimensional space (N = 7) and the Y hyperfaces were pentachordal objects in four-dimensional space (M = 5) and we were again interested in counting edges (L = 2), then we would have to ADJUST our count of sticks analogously by 1/COMB(N - M, N - L) 1/COMB(2,5)= 1/10. The probabilistic method we used to prove Theorem 5.3.5.2 will help us understand figures 5.9 and 5.10 in a somewhat different light. Inspecting figure 5.10, we see that if we peel a triangular face at random off the tetrahedron, the probability is 1/2 that the face will be in Forte-class 3-2 and 1/2 that the face will be in Forte-class 3-7. If we pull an edge at random off a triangle of Forte-class 3-2, our expectation is 1/3 that the edge will be in Forte-class 2-3. And if we pull an edge at random off a triangle of Forte-class 3-7, our expectation is 1/3 that the edge will be in Forte-class 2-3. Hence, according to the theory of probability, our total expectation for pulling an edge of Forte-class 2-3 off the tetrahedron by a random yank is ((1/3) (1/2) + (1/3) (1/2)) = (1/6 + 1/6) = 1/3. And this agrees (as our work says it must) with the probability of that event which we infer from figure 5.9: There we see that of the six tetrahedral edges, two are of Forte-class 2-3; so we infer that our expectation of yanking an edge of that class in a random pull is 2/6, which is 1/3. To pursue farther what Theorem 5.3.5.2 has to do with figures 5.9 and 5.10, and with higher-dimensional analogs of those figures, would lead us deeply into a branch of mathematics called algebraic topology. That pursuit would be very much worth undertaking, but it would be out of place here. In discussing how our generalized embedding number applies to the example of figures 5.9 and 5.10, I have supposed that the reader is already familiar with Forte's use of the interval vector in atonal theory. Now let us see how our generalized theory applies in a very different context, one with which the reader is almost certainly unfamiliar. To that end, we shall study some examples in connection with the non-commutative GIS of time spans which we developed in chapter 4. The first thing we must do is fix the group CANON for our purposes. We shall take CANON to be the group of all interval-preserving operations here. We shall not allow transpositions, much less inversions, as canonical operations for this study. Our reason will become clear.

Ill

5.3.5.2

772

Generalized Set Theory (1)

As we observed in 4.1.7(D), the generic interval-preserving operation P = P (h>u) transforms the sample time span (a,x) into the time span (h + ua, ux). The commentary on 4.1.7(D) elaborated upon this: "The interval-preserving operation P(h u) first blows up or shrinks the sample time span (a, x) by a factor of u, transforming (a, x) to (ua, ux), and then moves the latter time span backward or forward in time by h or ( —h) numerical units, transforming (ua, ux) to (h + ua, ux) = P(a, x)." So if X is a set of time spans (s l 5 s 2 ,...s n ), where sn = (a n ,x n ), then P(X) is the set (s'^s^,.. . SN), wher sj, = P(sn) = (h + ua n ,ux n ). We can imagine the set X here as modeling temporal aspects of a musical "passage" containing N events; then P(X) models analogous aspects of the passage played u times as slowly (1/u times as fast), starting the tempo change from time-point zero, all this played h numerical time-units later ( — h earlier). For example, let us take "the quarter note" as a numerical unit and "the beginning of the piece" as a numerical time-point zero. Imagine a motive consisting of an eighth, a dotted eighth, a sixteenth, and a quarter, played consecutively starting 10 quarters after the beginning of the piece. We could model some temporal aspects of this motive by the set X = ((10,|), (10^,f), (Hi,i), (Hi, !))• Remember that X is formally an unordered set; we have listed its members "in order of appearance" only for convenience here. Let us take h = 1370 and u = 4. Then the transformed set P( h>u) (X) first augments the entire rhythmic setting by a factor of 4, from time-point 0 on; then P(X) plays the augmented motive beginning h = 1370 quarters later, that is beginning 1370 quarters after time point 40, that is beginning at time point 1410. (The augmented motive obtained as an intermediate stage began not at time-point 10, but at time-point u • 10 = 4 • 10 = 40.) So the transformed motive modeled by P(X) consists of a half note, a dotted half, a quarter, and a whole note, played consecutively starting 1410 quarters after the beginning of the piece. Expressing this in numbers, P(X) = ((1410,2), (1412,3), (1415,1), (1416,4)). One can check that each member of P(X) is mathematically related to the corresponding member of X via the transformation P(a, x) = (h + ua, ux), here = (1370 + 4a, 4x). For instance, the third-listed members of the sets X and P(X) are related by the formula P(llii) = (h + u - lli,u-±) = (1370 + 4- lli,4-i) = (1370 + 45,1) = (1415,1). There is no need to restrict our attention to sets modeling consecutive events, as X and P(X) did in the preceding example. We could for instance consider a passage in which a violin plays four consecutive quarters, while a viola plays three triplet halves, while a cello rests for an eighth and then plays two consecutive quarters followed by a dotted quarter. We could model temporal aspects of this passage by a time-span set Y. Supposing that the onset of the passage comes 16 quarters into the piece, we can write Y = ((16,1), (17,1), (18,1), (19,1), (16, f), (17if), (18f,f), (16i 1), (17*. 1), (18-|, l£)). As P varies over the interval-preserving operations, P(Y) models

Generalized Set Theory (1)

5.3.5.2

the ensemble passage, played at (all) different tempos and at (all) different times. The elements of the unordered set Y above are listed, not "in order of appearance," but "by parts," as they were described in the text. Suppose the numerical time-span set Y t models the above passage for string trio at the precise time the music was first imagined clearly by the composer. Suppose the different numerical set Y2 models the passage at the precise time it was played during the first performance. Suppose the still different numerical set Y3 models the passage at the precise time my trio played it yesterday, taking a considerably faster tempo. Given one fixed referential time-point zero and one fixed referential time unit, the numbers denoting the members of the three sets Y l 9 Y 2 , and Y3 will be very different. Our formalism, though, enables us to say that the three sets are all (approximately) canonically equivalent. That is one powerful methodological reason for choosing CANON here to be the group of interval-preserving operations. Another good reason for the choice is provided by the way in which this group relates dyad structure to interval structure in the GIS at hand. We shall now explore that topic. By a "dyad" we understand a set containing two distinct members s and t. By an attack-ordereddyad(AOD) we shall mean a dyad containing say s and t, ordered in the following way: If s begins before t (as a time span), the order is (s, t); if t begins before s, the order is (t, s); if both time spans begin at the same time, the shorter of the two spans is listed first. Since s and t are distinct time spans, these criteria are sufficient to order the dyad. Given an AOD D = (s, t), let (i, p) = int(s, t). Then t begins i s-durations after s begins, and t lasts p times as long as s. Because of the ordering criteria on D, the number i must be non-negative, and ifi = 0 then the number p must be greater than 1. Let us call an interval (i,p) of this form a forwards-oriented interval. We have seen that if D = (s, t) is an AOD, then int(s, t) is forwardsoriented. The converse is also easily seen: If s and t are time spans such that int(s, t) is forwards-oriented, then D = (s, t) is an AOD. We can define (j, q) to be a "backwards-oriented interval" in an analogous way: j must be non-positive, and if j = 0 the number q must be less than 1. Now in the group IVLS the inverse of the interval (i,p) is ( — i/p, 1/p). It follows that the inverse of a forwards-oriented interval is backwards-oriented, and vice-versa. One sees quickly that the members of IVLS can be partitioned into three categories: the forwards-oriented intervals, the backwards-oriented intervals, and the identity interval (0,1). Here now is the crucial manner in which our stipulated canonical group comes into play: Given AODsDj = (s 1 ,t 1 )andD 2 = (s2,t2), then Dj and D2 are canonically equivalent if and only if int(s l5 t t ) = int(s2, t2). It would take too long to include here a formal proof of that theorem; such a proof is appended to the end of the chapter as section 5.6. The theorem is by no means obvious or trivial. Once we have proved it, we can note that the 2-element set

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classes correspond 1-to-l with the forwards-oriented intervals. If D = (s, t) is an AOD, then the set class /D/ corresponds to the forwards-oriented interval (i, p) = int(s, t): Every member D' = (s', t') of /D/, when attack-ordered, has int(s', t') = (i, p); furthermore, if s' and t' are any time spans such that int(s', t') = (i, p), then the AOD D' = (s', t') is a member of the set class /D/. The forwards-oriented intervals thus play exactly the same role here that Forte's "interval classes" play in his atonal theory: They can be used to label the distinct set-classes of dyads. They can be so used, that is, z/we take the interval-preserving operations as CANONical in constructing those setclasses. As a result of this structure, we can develop a very strong formal analog for Forte's interval vector in this particular system (NB). Let X be a set containing more than two members; let D be a dyad; then EMB(D, X), the number of forms of D embedded within X, is equal to the number of ways the forwards-oriented interval (i, p) can be spanned between members of X, where (i, p) is the interval spanning the attack-ordered members of D. In other words, EMB(D,X) = IFUNC(X,X)(i,p). In this sense we can speak of EMB(/D/, X), when /D/ varies over the dyad-classes, as an "interval vector;" (i,p) will vary concomitantly over the forwards-oriented intervals. Let us study some actual interval vectors in this system by way of example.

FIGURE 5.11

114

5.4.1 EXAMPLE: Figure 5.11 shows the mensural skeletons for motives (b), (c), and (d) from the Chopin sonata studied earlier (in section 4.3). The rhythmic motives are modeled by sets of time spans, and their interval vectors are tabulated on figure 5.12. Forming and reading these interval vectors becomes easy with practice. The forwards-oriented interval (1,1) labels the set-class of AODs D = (s,t) such that t begins right after s (1 s-length after s begins) and extends the same duration as s (1 times the length of s). Within set (b) we count three instances of such AODs. The AODs are formed by the first-and-second notes of the motive, its third-and-fourth notes, and its fourth-and-fifth notes. Thus the number 3 is entered on the table of figure 5.12, in the row of the table headed by the interval (1,1) and in the column of the table headed "vector of (b)." Set (c) includes only two AODs in the set-class (1,1): the first two notes of motive (c), and the last two notes of the motive. Remember: A pair of successive quarter notes in any tempo at any point in the piece (or any other piece any

Generalized Set Theory (1)

5.4.1

FIGURE 5.12

time) is canonically equivalent to such a pair of successive eighths or successive halves or successive quintuplet sixteenths. All the AODs just indicated belong to the same set-class, the set-class determined by the interval (1, 1) between the first and second members of each AOD. The two AODs of class (1,1) embedded in set (c) are counted on figure 5.12 by the number 2, entered to the right of the interval (1, 1) and in the column headed "vector of (c)." Set (d) also embeds two AODs of class (1, 1), namely the pair of half notes and the pair of whole notes at the end of the motive. Let us now consider the set-class corresponding to the forwards-oriented interval (3, 2). An AOD D = (s, t) belongs to this class if t begins 3 s-spans later than s begins, and lasts twice as long as s. The second quarter of (c) and the first half-note of (c) form such an AOD. So do the last quarter of (c) and the second half-note of (c). These two dyads are tabulated by the entry 2 in the (3, 2)-row and the (c)-vector-column of figure 5.12. The entry of 1 in the (11/2, !/2)-row arises from the AOD formed by the first and third notes of motive (d). In section 4.3 we noted a progressive "expansion" from motive (b), through motives (c) and (d). The progressive broadening of note values, from eighths to quarters to halves to whole notes, is obviously crucial. Our model does not address this aspect of the progression. But it does address and analyze well another aspect of interest, something we might call the "progres-

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5.4.2

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sive diversification" of the motives in their internal rhythmic structures. As one sees from the first column of figure 5.12, motive (b) concentrates on only a few intervals, most of which appear more than once. Motive (c) projects only two intervals that appear more than once (in the second column of the figure); no interval appears thrice (in that column). Motive (c) projects more intervals, and more diverse intervals, than (b). Motive (d) projects only one interval that appears more than once; that interval appears only twice. Motive (d) thus continues the process of diversification. The insensitivity of our interval vector to changes in tempo, a defect in some ways, is useful here: It enables us to compare motives (b), (c), and (d), each in its own intrinsic context. We touched on this idea earlier in connection with IFUNC(Y, Y). Motive (a), the motive of the opening Grave, does not appear on figure 5.11 or figure 5.12. If one ignores the anticipation of F(? = E natural in the music, then motive (a) is canonically equivalent to motive (d). 5.4.2 In connection with figure 3.3 (page 41), we earlier studied an "unrolling interval vector" for a set in a different GIS, a set pertinent to Webern's Piano Variations. The present GIS, like the earlier one, has an intrinsic chronology, so we can "unroll" its interval vectors too. The abstract method of doing so will involve a number of technical finesses. To begin the abstract study let us consider the imaginary string trio we discussed a short time ago, and let us imagine another of its passages, which we can symbolize as in figure 5.13.

FIGURE 5.13

116

We can model certain temporal aspects of this passage, as we did with the last one, by a set Y of time spans. The violin projects four time spans, (16,1), (17,1), (18,1), and (19,1); let us call these spans vnl, vn2, vn3, and vn4 respectively. The viola projects the two time spans (16,|) and (18f,f); let us call these spans val and va2. The cello projects the two time spans (16j, 2) and (18^, 1^); let us call these spans vcl and vc2. We can list the members of Y "in parts" as vnl, vn2, vn3, vn4, val, va2, vcl, vc2. Of course that is not their "order of appearance" in the music. But what isl

Generalized Set Theory (1)

5.4.2

We might try attack-ordering to list the members of Y "in order of appearance." Then we would list them as vnl, val, vcl, vn2, vn3, vc2, va2, vn4. For many purposes attack-ordering is natural, and we have seen how cogent it is in connection with dyads, intervals, and the canonical group. But we shall not want to use attack-ordering in connection with the way we perceive the members of Y "appearing." To see why not, imagine that we stop the music of figure 5.13 just after the attack of vn2, that is, just after time-point 17, and suppose that we ask just which time spans we have perceived up through that time. Obviously we have perceived vnl. But we have not yet perceived any other spans. True, we have heard both the other instruments attack other spans. But we do not yet know how long those spans are going to be, so we cannot claim to have perceived them as spans, using them e.g. to form intervals in an unrolling interval vector. For all we know as we listen at time-point 17, the viola may be intending to hold onto its note for 24 quarters. Now when we unroll the interval vector for Y in connection with figure 5.13, we are going to want precisely to "stop the music" of the figure at various stages, asking at each stage what intervals we have heard so far. As we have just seen, when we stop the music at time-point 17, we can be sure of having perceived only one span, namely vnl; hence we cannot say we have perceived any proper intervals at all so far. The attack-ordering for Y is deceptive in this connection. That ordering, beginning vnl, val, vcl, vn2,..., makes it seem as if the spans val and vcl have "already occurred" by the time vn2 occurs, attacking at time-point 17; hence it seems (wrongly) as if we ought to count the forwards-oriented intervals int(vnl, val) and int(vnl,vcl) and int(val,vcl) as "having already occurred" by the time we "get to" vn2. But this inference is wrong. As we have seen, no intervals have yet "occurred" by time-point 17, so far as our perceptions of spans and their interrelations are concerned. To reflect the true order of our span-perceptions, we shall want to use a different system of ordering, not attack-ordering but release-ordering. Given distinct spans s and t, s precedes t in the release-ordering if s ends before t ends, or if they end simultaneously and s is longer. If s and t correspond to musical events event 1 and event2, then s precedes t in the release-ordering when we perceive the time span during which event 1 has happened before we perceive the time span during which event2 has happened, or if we perceive both spans simultaneously and recall that event 1 began first. Release-ordering for the set Y thus enables us to articulate the music of figure 5.13 into stages that correspond to our evolving perceptions of time spans "having happened" as we listen. The members of Y in the releaseordering are vnl, vn2, vcl, val, vn3, vc2, va2, vn4. Furthermore, when we articulate the music into such stages, we shall want to demarcate the stages by the time points at which various spans are released, not at which they are attacked. Thus the finest possible articulation of the set Y = (vnl, vn2, vcl, val, vn3, vc2, va2, vn4) into stages for our purposes can be realized as follows.

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5.4.2

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Stage 1: We have heard Y t = (vnl, vn2) at time-point 18, the release of vn2. Stage 2: We have heard Y2 = (vnl,vn2,vcl) at time-point 18^, the releasepoint of vcl. Stage 3: We have heard Y3 = (vnl, vn2,vcl,val) at time-point 18f, the release-point of val. Stage 4: We have heard Y4 = all of Y at timepoint 20, the simultaneous release for vc2, va2, and vn4. By calculating how the interval vectors for Y 1} Y 2 , Y3, and Y4 develop, each expanding the counts of the last among the various intervals counted, we shall be able to model how our sense of intervallic structure evolves as we listen to the musical passage. We shall be able to use our formal model analytically, just as we used analogous machinery earlier in connection with the Webern passage and the expanding interval-counts of figure 3.3. Our work above can now be generalized. Given any set Y of time spans, first list Y as (s:, s 2 , . . . , SN) in the release-ordering. Next identify N or fewer "stages" associated with certain subsets of Y as follows. Stage 1 is articulated at the release of s2; it is associated with a certain subset Y x of Y. Y t = ( s t , s 2 ) unless s3 releases simultaneously with s2; in that case Y t = (s r , s2, s3) unless s4 also releases simultaneously at that time; in that case ... (etc. etc.). After Y! = ($!, s 2 , . . . , SM) has been found, Stage 2 is articulated by the release point of s M+1 . Stage 2 is associated with a certain subset Y2 of Y. Y2 = Y1 + (sM+1) unless s M+2 releases simultaneously with s M+1 (etc. etc.). And so on. Eventually one attains the release point of SN and exhausts the set Y. We can regard the stages as developing in a simple serial rhythm as stage 1, stage 2, stage 3, and so forth. Or we can regard them as developing in a "perceptual rhythm," the rhythm of the various release-points at which the stages articulate. (This is interesting but it oversimplifies the psychology of what is going on.) As the stages develop rhythmically, the evolving interval vectors of Y 1} Y 2 , etc. can be studied. Care must be taken here because the release-ordering of Y does not necessarily coincide with the attack-ordering. It is possible for sm to precede sn in the release-ordering, but to follow sn in the attack-ordering. (The different listings of Y in connection with figure 5.13 illustrate the possibility.) Should this happen, when we get to the stage that notices (the release of) sn in the release-ordering, we shall want to tabulate the forwards-oriented interval int(s n , sm) in our updated interval vector, not the backwards-oriented interval int(s m ,s n ). The reader who likes to fool with computer programming and who has a home computer with a color monitor will enjoy writing an "unrolling interval vector" program. The program will take a set Y of time spans, arrange it in release-ordering, determine the articulation-points of the various stages, and find the corresponding subsets Yj, Y 2 ,..., Y. The program will then compute the interval vector for Y^ and display it on the screen as follows. For each forwards-oriented interval (i, p) that is counted, a colored dot appears at the point (i, logp) on a half-plane grid, (i is always non-negative; log p is positive, zero, or negative.) If the interval appears only once in the set, the dot is violet;

Generalized Set Theory (1)

5.4.3

the more times the interval appears, the more the color of the dot moves toward the red end of the spectrum. (The background of the screen is either white or black.) After the program has computed the interval vector for Y x , it will update the count of various intervals so as to obtain the interval vector for Y 2 , changing the color of some dots on the screen as pertinent. Then it will update the count of various intervals to obtain the interval vector for Y3, and so on. The updating can be done quickly following the method of figure 3.3. (Remember that you may have to adjoin more than one releasing time-span at any new stage. Also remember to adjust for any new dyads that may be release-ordered but not attack-ordered.) The rhythmic updating of the screen can follow either the serial rhythm of the stages or their "perceptual rhythm" as discussed above, either in real time or suitably scaled for visual effect. 5.4.3 EXAMPLE: The technique of unrolling can be applied to EMB-related functions beyond the interval vector.

FIGURE 5.14

The "set" Y of figure 5.14, for example, is articulated into four stages. (We could articulate it farther, but we shall not do so here. Since neither the time unit nor the point zero is specified, Y is not strictly a numerical "set" within TMSPS, but I am assuming the reader will not mind a certain looseness in discourse at this point.) Figure 5.14 also displays "sets" X l 5 X 2 , X'l5 and X'2, all of which can be found embedded within Y. X\ is a canonical form of Xj and X'2 is a canonical form of X 2 . Figure 5.15 shows how the embedding numbers of the set-classes /XJ and /X 2 / within Y develop, as Y develops over the four stages. The values rise at Stage 4 because the dotted half releases there and the appearances of X\ and X'2, augmented (canonical) forms ofXtandX 2 ,can now be counted as "embedded" in Y. Figure 5.15 shows us how the set-class /X 2 / comes on late and strong, pulling ahead of (XJ at Stage 3 and then decisively ahead at Stage 4.

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5.4.4

Generalized Set Theory (1)

FIGURE 5.15

No doubt the reader has recognized Y as interpreting the opening of Brahms's G-Minor Rhapsody. The idea that "/X2/ comes on late and strong" is reinforced by the end of the closing group in the music, where the closing theme is liquidated rhythmically down to a succession of X 2 -forms alternating on the tonic and dominant of D minor. The rhythmic interpretation of figure 5.14 does not exclude other possible rhythmic readings of this music. E.g. one could read triplet eighths where there are triplet rests on figure 5.14; then /Xl/ and /X 2 / would come out in a tie on figure 5.15. But such other interpretations just as clearly do not exclude the reading of figure 5.14. The reader who consults the score will find ways enough in which the relation of quarter note to accompaniment changes after Stage 2 so as to support the reading of the figure. 5.4.4 NOTE: Through section 5.4 so far, we have focused upon the interval vector and more generally the EMB function, in connection with our non-commutative GIS of time spans. The technique of "unrolling in stages" which we applied to this study could also be applied in connection with IFUNC(X, Y), as we unroll either X or Y or both in stages. 5.5 NOTES: Let us return now to the most general abstract setting, that of a family S and a group CANON of operations on S. Following the suggestions of my writings elsewhere, we can explore numbers of interest beyond EMB(X, Y).7 We may define COV(X, Y), for example, the coverfng number of X in Y, as the number of forms of Y that include X. This is not necessarily the same number as EMB(X, Y), the number of forms of X that are embedded in Y. E.g. in atonal theory take X = (C, E) and Y = (C, E, G#); then EMB(X,Y) = 3 but COV(X,Y)=1. If S is finite then COV(X,Y) = EMB(Y,X), where Y and X are the complements of Y and X.

720

7. "Some New Constructions Involving Abstract Pcsets, and Probabilistic Applications," Perspectives of New Music vol. 18, nos. 1-2 (Fall-Winter 1979 and Spring-Summer 1980), 433-44.

Generalized Set Theory (1)

5.6

We may also consider SNDW(X,Y,Z), the sandwich number of Y between X and Z; this is the number of forms of Y that both include X and are included in Z. If we write 0 for the empty set, then SNDW(0,Y,Z) = EMB(Y, Z); if S is finite then SNDW(X, Y, S) = COV(X, Y). If Y' is a form of Y then SNDW(X,Y,Z) = SNDW(X, Y',Z); hence we can write SNDW(X,/Y/, Z) without ambiguity. But we cannot use /X/ or /Z/ for sandwich arguments in this way; SNDW(X,Y, Z) depends very much on the specific forms of X and Z being used as arguments. For example let Z be the C-major scale; let /Y/ be Forte-class 3-4, which we could write as /(B,C,E)/. Let Xj = (C, E). Then, allowing both transpositions and inversions as canonical, SNDW(X l5 /Y/,Z) = 2: There are 2 forms of (B,C,E) that can be sandwiched between (C, E) and the scale, namely (B, C, E) and (C, E, F). Now let X2 = (F, A). X2 and Xj belong to the same set-class, but SNDW(X 2 ,/Y/,Z) = 1, not 2: Only 1 form of (B,C,E) can be sandwiched between (F, A) and the scale, namely (E, F, A). Another interesting number is ADJOIN(X, Y, Z). This is the number of forms Y' of Y satisfying both (A) and (B) following. (A): Y' is disjoint from X. (B): There is some form of Z that includes both X and Y'. To illustrate what this number is inspecting, let X = (C, E), Y = (D, G), Z = the C-major scale. (C, E) + (D, G) lies within some major scale; so does (C, E) + (F, 6(7); so does (C, E) + (F#, B); so does (C, E) + (A, D). Exactly the 4 fourths (forms of Y) metioned in the preceding sentence have both the desired properties (A) and (B); other fourths (forms of Y) either contain C or contain E or do not add up together with (C, E) to lie within any major scale. So ADJOIN((C, E), (D, G), C-major scale) = 4. Inspecting properties (A) and (B), one sees that we can write ADJOIN(X,/Y/,/Z/); it follows that we can even write ADJOIN(/X/,/Y/,/Z/). 5.6 APPENDIX (optional): We prove here the crucial theorem stated earlier, on the relation of dyads and intervals in our non-commutative GIS for TMSPS, using the group of interval-preserving operations as CANONical. Here is that theorem stated again: Given attack-ordered dyads D x = (s^tj) and D2 = (s 2 ,t 2 ), then Dj and D2 are canonically equivalent if and only if imXs^tj) = int(s 2 ,t 2 ). The proof follows. Set Si = (a^xj, t t = (b^yO, s2 = (a 2 ,x 2 ), t2 = (b 2 ,y 2 ). Suppose first that D! and D2 are canonically equivalent; we shall show that int(s 1 ,t 1 ) = int(s2, t 2 ). Say that P = P (h%u) is the canonical operation mapping D x onto D 2 . P maps the members of the first dyad somehow onto the members of the second; conceivably the operation might transform s t into t2 and s2 into t t . But in fact that cannot happen in this situation: P transforms st into s2 and t x into t 2 . To see that, we use the fact that both dyads are attack-ordered. Since D! is attack-ordered, either ^ < bt or (at = b x and x t < y x ). If a1 < bl then ua1 < ubi and h + ua x < h + ub^ hence the P-transform of Sj begins before

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the P-transform of il. Since D2 is also attack-ordered, that means that the Ptransform of s t must be s2 and the P-transform of tl must be t 2 . (Otherwise we would have b2 < a 2 , contradicting the attack-ordering of D 2 .) So the case a x < bi leads to the desired result; let us investigate the case (a^ = b x and xi < vi)- In tnat case» h + uai — h + ubi and ux1again we infer that t < uy^ (P^), P(tj)) is the attack-ordering for D 2 , and so P(S!) = SjandP^) = t 2 as desired. Now that we have established the relations P^) = s2 and P(ti) = t 2 , the rest is easy: int(s 2 ,t 2 ) = int(P(s1),?(!!)) = int^,^) as claimed, because P is interval-preserving. Now we shall prove the converse. Supposing that int^, tj) = int(s 2 , t 2 ), we shall prove that D x and D2 are canonically equivalent. int(s 1 ,t 1 ) = ((bi — aj/x^yjxi) and int(s 2 ,t 2 ) = ((b2 — a 2 )/x 2 ,y 2 /x 2 ). So what we are assuming can be expressed by equations (A) below. (A): (bt - aj/xj = (b2 - a 2 )/x 2 ; y1/xl = y 2 /x 2 . Using some algebra applied to equation (A), we can derive (B); manipulation of (B) produces (C). (B): x ^ - x ^ = x 1 b 2 - x 1 a 2 ; yiX 2 = x t y 2 . (C): x^ - x 2aj = Xib2 - Xjb^ x^ = y2l Let g be the number such thatx t a 2 — x 2 aj = g = Xjb 2 — x 2 b 1 . Let u be the number such that x 2 /Xj = u = y 2 /y 1 . From the equation x x a 2 — x 2 a t = g we infer x^ = g + x 2 a x ; thence we infer a 2 = (g/x t ) -f (x 2 /x 1 )a 1 , or a2 = (g/Xj) + uaj. In similar fashion, we derive the other equations of (D) below.

(D): a 2 = (g/xO + ua f ; x 2 = ux x b2 = (g/x1) + ub1; y 2 = uy 1 . Seth = (g/x 1 ).Then(a 2 ,x 2 ) = (h + ua^uxj,while(b 2 ,y 2 ) = (h 4- ub 1? uyj. Thus, taking P = P (hiU) , we have s2 = P(s^ and ta = P(tj). Since Pis an interval-preserving operation and D2 = PCDJ, the dyads D! and D2 are canonically equivalent, q.e.d

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6

Generalized Set Theory (2): The Injection Function

This chapter is a pivot in a sort of modulation from the study of Generalized Intervals to the study of Generalized Transformations. The Injection Function can be defined, discussed, and applied to musical analysis without invoking the notion of interval or canonical operation at all. In that regard the Injection Function looks forward to the work lying beyond chapter 6. At the same time the Injection Function is strongly suggested by IFUNC and EMB, either of which it can be used to generalize when there is a GIS or a CANONical group at hand. In that regard the Injection Function logically continues the work of chapter 5; indeed it elucidates some problems that chapter 5 left hanging. It was hard for me to decide, as I pondered this dichotomy, how best to arrange my exposition of the material for this chapter. At first it seemed natural to emphasize aspects of continuity, first showing how the Injection Function grows out of IFUNC and EMB, and then moving gradually to higher and higher levels of abstraction. But when I drafted an exposition in that spirit, I did not like the effect. Looking back at IFUNC and EMB gave rise to so much interesting discussion that an endless time seemed to go by before I could get to discuss the more radical abstract features of the new construction. So I decided upon a different method of exposition. That is why I shall begin by discussing the Injection Function at a high level of abstraction, emphasizing its novelty in contrast to the material of chapter 5. Then, once the reader has become familiar with the new construct, I shall begin stitching in threads from here and there in chapter 5, gradually sewing together the seam I have created. At any rate, that is my plan. The reader who is not happy with it will I hope be able to follow along nevertheless without too much discomfort, once aware of it.

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Generalized Set Theory (2)

6.1 CONVENTIONS: We shall be concerned with a family S of objects, and with various transformations f that map S into itself. We do not assume that the transformations are necessarily operations (1-to-l and onto S). Operations are capable of entering into groups, e.g. canonical groups, groups of interval-preserving operations and/or transpositions in a GIS, and the like. A transformation that is not an operation can have no inverse transformation on S, and so cannot belong to any group of operations on S. As in chapter 5, we shall use the word "set" to denote a finite subset X of S; more exactly, we shall do so until explicit notice to the contrary. (Toward the end of chapter 6 we shall indicate how the work can be generalized to deal with infinite subsets of an infinite S.) 6.2.1 DEFINITION: Given sets X and Y, given a transformation f on S, then the injection number of X into Y for f, denoted LNJ(X, Y)(f), is the number of elements s in X such that f(s) is a member of Y. INJ(X, Y)(f) answers the question: "If I apply the transformation f to the set X, how many members of X will I thereby map into members of Y?" If f is 1-to-l, then those distinct members of X will map into distinct members of Y, so that INJ(X, Y)(f) will also be the cardinality of f(X) D Y, that is the number of elements that the sets f(X) and Y have in common. But when f is not 1-to-l, this is not necessarily the case. We might e.g. have 5,273,647 distinct members of X all mapping into one member of Y; in this case we might have INJ(X, Y)(f) = 5,273,647, while f(X) and Y might have only 1 common member. 6.2.2 EXAMPLE: Take S to be the twelve chromatic pitch classes. Define f as follows: f(s) is the pitch class C when s is any "white note"; f(s) is the pitch class F| when s is any "black note." Fix X = (C, C|, D, El», E). Take Y = (B, C#, D, E, F, F|). Then INJ(X, Y)(f) = 2. The 2 black notes of X map via f into the member F# of Y; the other notes of X, being white, do not map into any member of Y. They do not because the pitch class C, the image of all white notes under f, is not a member of Y. Now take Y = (F, F#). Then INJ(X, Y)(f) = 2 still and again, for the same reasons. Now take Y = X. Then INJ(X, Y)(f) = 3. The 3 white notes of X all map via f into the member C of Y = X; the other notes of X, being black, do not map into any member of this Y. (Ff, the image of all black notes under f, is not a member of this Y.)

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6.2.3 EXAMPLE: Take S to be the twelve chromatic pitch classes. We define a transformation called wedging-to-E, which we shall denote as WE. The transformation maps E into E and B\> into B\>. Every other pitch class when wedged

Generalized Set Theory (2)

6.2.3

advances one hour towards E along the clock of pitch classes, clockwise or counterclockwise by whichever route is shorter. The effect of W E on the clock of pitch classes is portrayed by figure 6.1.

FIGURE 6.1 W E is not an operation; it is neither 1-to-l nor onto. Nevertheless it is a useful transformation to have at hand for analyzing "Angst und Hoffen," the seventh song from Schoenberg's Book of the Hanging Gardens op. 15. Figure 6.2 shows the pitches in the two chords that form the opening "Angst und Hoffen" motto, chords X and Y.

FIGURE 6.2 Here INJ(X, Y)(wE) = 2: When the transformation W E is applied to the notes of X, 2 of those notes map into notes of Y. The D of chord X, specifically, maps into the E|? of chord Y and the B|? of chord X maps into the Bb of chord Y. Thus "two-thirds of X" is mapped into Y by the wedge. If only the G(? of X wedged into Y then all of X would wedge into Y. That would happen if the F|j of Y were an F natural. So we can consider the F|j of Y (in this context!) as a "wrong note" or a "blue note"; it substitutes for the F natural

725

6.2.3

726

Generalized Set Theory (2)

that "should" be there. Indeed, there is a lot of musical action later on in the song that involves the idea of "getting F|? (or E) to move to F." The F(? of Y is not just a blue note, though; it is also the point to which the wedge converges. In that connection it has a tonic character as a potential point of repose or arrival. We shall see later how this potentiality is realized in the music. The operation I = l|j£ = I| is also heavily involved in the effect of the music sketched by figure 6.2. Inversional symmetry about B|? is very strongly projected not only by sonic features but also by the visual layout of the chords, symmetrical about the middle line of the staff. In our new terminology, we can write INJ(X, X) (I) = 3: f or all of X maps into X under the transformation I. We can also observe that INJ(Y, Y)(I) = 2: f of Y maps into Y under that transformation. Once more we can point out that if only F-flat were Fnatural, and so on. (We can presume that F|?-for-F is the pertinent substitution here, rather than E|?-for-E in the upper register. Replacing E[? by E in the music here would result in the syntactically implausible chord (F|?, B[7, E).) Neither IFUNC nor EMB can adequately engage the ideas we have been considering in connection with INJ here. The injection function enables us to discuss several thematic functions for the F|? of chord Y: That note is at one and the same time a substitute for F natural, and the convergence-point of the wedge, and one center for the operation I. INJ also enables us to distinguish very different structural functions for the transformations I and W E . Specifically, I transforms each chord of figure 6.2 into something very like itself"; WE, in contrast, transforms an antecedent chord into something very like a consequent chord. We may think of I as an "internal" transformation and WE as a "progressive" transformation in this musical context. We shall pick up the theoretical implications of that notion later on. Figure 6.3 helps us hear how the transformational ideas under discussion persist, develop, and resolve over the last third of the piece. The figure comprises mainly the notes of the piano over this section, notes which carry the harmony in a homophonic texture. Figure 6.3(a) labels the chords involved. The last three notes of the vocal part are also included, beamed as an arpeggiated chord Z5 that belongs in this progression. The progression leads back to X and Y, "Angst" and "Hoffen," now as an outcome rather than a point of departure. Figure 6.3(b) indicates those notes of (a) which participate in E-wedging activity. Two notes of Z1 wedge into Z2 along the beams, two notes of Z2 wedge into Z3, and two notes (pitch classes) of Z3 wedge into Z4. (Bb of Z3 wedges as a pitch class into Bb of Z4; the registral symmetry of the high and low Bb pitches about the pitch E4 is nice.) Thus INJ(Z n ,Z n+1 )(w E ) is consistently = 2 for n = 1,2, and 3. All of Zj-as-2-note-pcset is projected into its successor set Z2 by the wedge; f of Z2 is projected into its successor, and |ofZ 3 .

Generalized Set Theory (2)

6.2.3

FIGURE 6.3

The F-with-a-question-mark, notated above the upper beam at Z4, indicates a "missing" F that breaks an otherwise consistent pattern of wedging and inversional balance. In this respect the missing F is exactly like the missing F on figure 6.2 earlier. It is like that F in breaking an E-wedge; it is like that F, too, in leaving the Eb of chord Z4 bereft of its I-partner, just as the E|? of chord Y on figure 6.2 was bereft of its I-partner. The note-against-note relations of figure 6.3(b) show that INJ(Z n ,Z n )(I) = 2 for n = 1 and 2; INJ(Z 3 ,Z 3 )(I) actually = 3. (Within the set Z3 of pitch classes, D and F# invert into each other, while B(? inverts into itself.) Hence the abrupt inversional imbalance at Z4 is all the more strongly felt, since INJ(Z 4 ,Z 4 )(I) = only 1. The metaphor of "imbalance" caused by "something missing" interacts well with the text: The singer is in a state of emotional discombobulation caused by the lover's absence. Continuing along figure 6.3(b), we hear how, when the progression Zt etc. recurs as the progression Z\ etc., the singer's final sign-off supplies the required F natural at the right moment, within the arpeggiated set Z5. The required Z5, which continues the patterns of wedging and inversion consistently, sets the word begehre = require. (The singer claims not to require the consolation of any friend.) Once the begehrte Z5 has appeared, the wedge can

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6.2.3

Generalized Set Theory (2)

and does converge all the way to the E of Z6, across the pickup chord Z'4 in the music. The level of progressive wedge-projection and internal I-projection is thus restored: INJ(Z'3,Z5)(wE) = 2, INJ(Z 5 ,Z 6 )(w E ) = 3; INJ(Z5, Z5)(I) = 2,INJ(Z 6 ,Z 6 )(I) = 3. The convergence of the wedge to its focal point E within Z6 coincides with and supports a big structural downbeat. The sonority of Z6 has earlier been associated with the word Seufzer, during the text, "meine Worte sich in Seufzer dehnen (my words trail off in sighs (or groans))." Figure 6.3.(b) portrays the sighing and trailing-off ideas very well. It also shows how the beamed sighing-progressions are framed by the elements (C, A[?) and (E) of the Seufzer-chord, which (for this reason and for others too) takes a big downbeat when it appears as Z6. Figure 6.3(b) shows how the wedging commences yet once more after Z6, continuing through X to Y, which is the end of the piece. Within X and Y, the pitches D5 and Et>5 of the music are brought down an octave, to be displayed as D4 and E|?4 noteheads on the figure; this shows clearly how the pitch classes D and E[? contribute to the final wedge. In particular, it brings out strongly how the final progression, Z6-Z7-X-Y, recapitulates the initial wedgestructure of the opening progression Z1-Z2-Z3-Z4 on figure 6.3(b). The "blue note" Fb of the final Y, shown on figure 6.3(a), takes on added significance in its "tonic" function, as it prolongs the downbeat E from the Seufzer-Z6(Dehnung). Figure 6.3(c) sketches in a format similar to (b) the influence of a subordinate wedge and inversion over this passage. That is wedging-to-F# and the inversion-operation J = Ipj| = l£. The symbols "A|?-G-F#//," at the beginning at bottom of the figure, show how the inner voice of the chords attains the goal and center F# of the wedge, getting there "from above" over chords Zi through Z3. Then from Z\ right on, all the way to X, the outer voices almost succeed in converging to F#(G[?), except that F is missing in the lower voice. Again we run into the thematic and structural "missing F!" The missing F now has a new structural function: It is missing as a semitone neighbor to Gj? in a wedge converging to G(?, the bass of the Angst-chord X. The earlier missing F was missing (inter alia) as a semitone neighbor to E in wedges converging to E, the bass of the Hoffen-chord Y. The T10-relation between the bass note of X and the bass note of Y is thus expanded into a larger T10-relation, a relation involving the respective wedges and inversions about those notes. The larger relation can be expressed by the

128

1. I discuss these ideas and others of the same sort elsewhere, exploring more systematically rhythm, meter, text setting, registers, doublings, and other features of the music. The reader who would like to go deeper into the piece itself will be interested by that article, which is less preoccupied than we must be here by theoretical constructions of various sorts. The article is "A Way Into Schoenberg's Opus 15, Number 7," In Theory Only vol. 6, no. 1 (November 1981), 3-24.

Generalized Set Theory (2)

6.2.3

transformational equations WE = T^w^T^1; I = T10JT101. Essentially, these equations "modulate by T10" the wedging and inversional transformations centering on F#, to obtain the analogous transformations centering on E. We shall explore the idea of such "modulation" more generally and more formally later on. The present remarks are meant to prepare that later exploration. The reader can hear in connection with the example at hand that the relation T10(Gb) = E has to do both with the bass line in the chord-succession X-Y, and also with the relation of WF* and J, in figure 6.3(c), to W E and I, in figure 6.3(b). The reader will then be able to summon that musical experience to mind during the abstract discussion of transformational "modulation" later on (in connection with Formula 6.7.2(c)). The formalities of INJ, applied to the transformations W E and I on S, have engaged figure 6.3(b) only partially. The visual layout of that figure conveys a good deal more than our formal machinery has so far described. For instance, the figure shows by its note-against-note layout that we do not simply have an Ab in Zt wedging to a G in Z2 and, independently, a C in Zl wedging to a C# in Z2; rather the "I-partnership" of (Ab, C) within Z^ wedges as a whole to the "I-partnership" of (G, C#) within Z2. In a similar sense, the missing F of Z4 is not just a missing note; it is a missing I-partner for Eb, without which the wedge cannot converge; as a missing partner it symbolizes the absent lover. And in a similar sense, the last line of text, "(that) I require the consolation of no friend," is symbolized exquisitely by the pitch class E as goal of the wedge and center of inversion: The pitch class has and needs no partner; it gets along by itself, perfectly self-centered in its Seufzer. The visual layout of the figure brings out such ideas; our transformational machinery has not as yet adequately engaged them. But it can be formally developed so as to do so. The operation I partitions the family S into distinct "transitivity sets" (Bb), (A,B), (Ab,C), (G,C#), (Gb,D), (F, Eb), and (E). I transforms the members of each transitivity set among themselves: I(Bb) = Bb; I(B) = A and I(A) = B; and so on. Such transitivity sets enable us to engage the notion of "I-partnerships" in our formal machinery. Many of the chords under consideration embed an entire transitivity set; some chords even embed two (e.g. Z3 = X = Angst and Z6 = Seufzer). Frequently a transitivity set embedded in one chord is transformed as a whole by W E into a transitivity set embedded in the next chord. Such ideas can be developed very abstractly in connection with the generalized INJ function. Here, it is formally important that the transformations I and W E commute. It is not remarkable that the visual aspects of figure 6.3 can be described by formal aspects of the INJ machinery when suitably extended. After all, one could hardly conceive the layout of the figure without some prior intuitions of W E and I as transformations; the formal extensions of the machinery, in that connection, amount only to making the relevant intuitions explicit.

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6.2.3

Generalized Set Theory (2)

Our investigations so far have focused on harmony and voice leading, as we explored the structural functions of various transformations in this song, using INJ. Now let us use INJ to explore some melodic functions of various transformations therein.

FIGURE 6.4

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Figure 6.4 will help us in this endeavor. The figure transcribes the pitches from the opening of the voice part, where they set the first line of text. Schoenberg's spelling projects throughout the phrase a strong visual inversional symmetry about the third line of the staff, where BW would appear as a center of I. Ordinal numbers 1, 2, ... , 10 appear under the first, second, ... tenth notes of the figure. The events of the melody are modeled here in a space S whose members are pairs (n, p), n being an ordinal number and p a pitch class. As a serial structure, the "melody" is modeled by an unordered set of ten such pairs; the elements of this set are the pairs (2, G!>), (1, D), (10, El>), (3, Et), and so on. "(2, G!>)" can be interpreted as saying "The second note is Gk" Arrows on the figure indicate transformational relations that will interest us here. Each arrow is labeled by a pair of symbols comprising a number (1, 2, 3, 5, or 6) and a letter (I or w). The number indicates how many ordinals later the transformed pitch class appears. The letter indicates a pitch-class transformation, w standing for WE. Thus the arrow labeled "6, w" which issues from the third note of the series, E\>, indicates that the note is transformed into a note appearing 6 order positions later, via the wedge transformation. The arrow from A to Cl>, labeled "2, I," indicates that the A is transformed into a note appearing 2 order positions later, via the / transformation. More formally, the transformation 6, w maps the element (3, E\>) of the melody into the element (3 + 6, w(EI>)) = (9, Fl>); the transformation 2, I maps the element (6, A) into the element (6 + 2, I(A)) = (8, Cl»). The transformations (6, w) and (2, I) are well defined by this formal method on the space S of pairs (n, p). (6, w) maps the pair (n, p) into the pair (n + 6, w(p)); (2, I) maps the pair (n, p) into the pair (n + 2, I(p)). The transformationsare not operations, w itself is not an operation on the twelve pitch

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6.2.3

classes; even beyond that, the ordinal aspect of the mappings prevents the transformations from mapping S onto itself. E.g. there is no (n, p) in S (with n a positive integer) such that (2,I)(n,p) = (1, Ab). The dotted arrows and question marks on figure 6.4 arise from the possibilities of considering the F|?s as substitutes for F naturals. If the flats were naturals, the dotted arrows would be solid and the question marks would disappear. We shall denote by XJJ, the set comprising the mth through nth events of the series (i.e. melody-set). Using the subsets XJJ,, we can apply familiar "unrolling" techniques to the situation, now using our injection numbers. For instance INJ(X*, X*) (f) = 2, where f is either (2, w) or (1,1), assuming we allow the dotted lines. The equation states: The set comprising (1, D), (2, Gb), (3, Eb), and (4, F(b)) contains "2" members that transform into the set under the transformation (2, w), and "2" members that transform into the set under the transformation (1,1), supposing that Fb is read "as if" F natural. This equation engages a significant "unrolling" when compared to the equation INJ(Xi,Xi)(f) = (only)l for the same transformations f. We can compare these internal transformations of X| with the internal transformations of Xf, the next 4-element subset of the melody. Xf comprises (5, C), (6, A), (7, Ab), and (8, Cb). INJ(Xf, Xf) (2,1) = 2: With respect to the new tetrad Xf, the transformation (2,1) plays the same internal role that (1,1) did in connection with the first tetrad X*. And, as the figure shows, (3,w) plays the same role with respect to Xf that (2,w) played with respect to X^. That is so even though INJ(Xf, Xf) (3, w) is only 1, not 2. Our arrow diagrams capture a certain picture of X* on the figure, as it appears bound together internally in a certain way by (1,1) and (2, w) arrows. The same kinds of arrow shapes capture a similar picture of Xf on the figure, as it appears bound together in a similar way by (2,1) and (3, w) arrows. Our model enables us to observe an interesting augmentation of ordinal distances, from the arrow transforms binding X| to the arrow transforms binding Xf. That is, within X* I-relations occur 1 note apart and w-relations occur 2 notes apart; within Xf these ordinal distances are expanded: Irelations occur 2 notes apart and a w-relation occurs 3 notes apart. This serial augmentation is particularly interesting because Xf takes only half the time to sing as did X?, in the clock time of the music. Our discussion of X*, Xf, I, and w is enhanced by the observation that no I-arrows and no w-arrows lead events of the first tetrad to events of the second, on figure 6.4. In our terminology, INJ(Xf, Xf) (f) = 0 when f is either (n, I) or (n, w), for any n. This observation specifically enhances our sense that the melody articulates into X* + Xf + Xg°, when heard in the context ofw and I relations. The italicized phrase is meant to recall our earlier discussion in connection with the various contexts of a melodic phrase within the Webern violin piece. Our sense of X* + Xf + X^0 in this context is further enhanced

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by the (6, w) arrow on figure 6.4, extending from inside X* over X| to Xg°. INJ(X}, X*°)(6, w) = 1, after INJ(Xf, X|)(n, w) had been zero for all n. The ordinal distance over which w functions continues to grow: The first w-arrow(s) had ordinal span 2 within Xf; the next w-arrow had ordinal span 3 within X|; now a w-arrow has ordinal span 6, between X* and Xg°. In this context the F flat of (9, F|?) in the melody is "correct"; it is in fact the goal of the wedge. One notes the care with which the high F[? is distinguished by the composer from the low F[?. When we hear the penultimate F|? as "correct" and ignore the last dotted arrow on figure 6.4, we get a sense of "ordinal expansion" over the phrase as regards not just w spans but also I spans. First I projects at ordinal distance 1 and w at distance 2; next I projects at distance 2 and w at distance 3; finally I and w both project at distance 5, and w projects at distance 6.

FIGURE 6.5

752

The injection function, like IFUNC earlier, enables us to discover and explore relationships our ears might not otherwise pick up quickly. Figure 6.5 (a), for instance, elaborates on the observation that there are 4 transformations f of form (n,T6) such that INJpC|,Xg)(f) is non-zero, allowing Ffr to represent F natural. So there are four T6 arrows on the figure, projecting al of X* progessively into all of Xf at a variety of ordinal distances. This contrasts sharply with the absence of any I or w arrows on figure 6.4 that led from anywhere within X* to anywhere within Xf. So far as the two tetrads in the vocal melody are concerned, we may put it that I and w are "internal" transformations, while T6 is "progressive." The structure of figure 6.5(a) is hard to pick up by ear alone because its predominant ordinal rhythm of "3 later" conflicts both with the motivic rhythm of the music, and with the 2-later rhythm established in figure 6.4 within X*.

Generalized Set Theory (2)

6.2.3

The structure of figure 6.5(b) is also easy to pick up by inspecting INJ^X^X*), hard to pick up at first by ear alone. It is reasonable to pay some attention to the hexad X* here because that set spans the transfer of the low F[? to the high Ffj in the melody, the first repeated pitch-class of the series. Inspecting the internal structure of the hexad with our machinery, one notes that there are relatively many transformations f of form (n, T8)-for-some-n such that INJ(X4,X4)(f) is high or positive. The arrows on the figure show how this works out analytically. For transformations f involving other T,, not as many analogous arrows would appear. The strong ordinal rhythm of the arrows on figure 6.5(b) is supported by the contour of the pitches—C, Ab, and both the F|?s are all turning points—and also by the rhythm and meter of the music. In this context, the melodic rise over the octave F[? takes place through an ornamented arpeggiation of an augmented triad F|?, C, A^, Ffr. One recognizes the triadic set of pitch classes from earlier discussion. It is Z 6 , the Seufzer triad into which the E-wedge will converge at the big downbeat near the end of the piece. And so on. One could combine the pitch classes or the pitches of the melody into formal pairs not only with the ordinal numbers 1 through 10, but also with the durations of the written notes, or with the time points at which the notes are attacked, or with the time spans of the written notes in our noncommutative GIS, and so on. One would get interesting results in each case. I have used ordinal numbers here because they furnish some new kinds of ideas about non-mensural rhythm, and because they guarantee that none of our transformations on the space of elements (n, p) can be operations. Even if OP is an operation on pitch classes, like I, the transformation (5, OP) cannot map our pair-space onto itself: There is no (n, p) such that (n + 5, OP(p)) = (2, q). So using the ordinal-number model gives INJ another opportunity to show how smoothly it handles transformations that are not operations. So far as the "new kinds of ideas about non-mensural rhythm" are concerned, we can take note that our pair-space is one useful way to model serial melody. Later we shall explore other interesting models for representing series of pitches or pitch classes (or anything else). As I mentioned earlier, the reader who is interested can find a more ample analysis of "Angst und Hoffen" for its own sake elsewhere in my writings. The interested reader might also wish to consult my analytic remarks elsewhere on "Die Kreuze," Number 14 from Pierrot Lunaire.2 Ideas of wedging and inversion are also engaged there. The pertinent wedge transformations are wedging-to-(C/C#) and wedging-to-(F#/G). wc/c# maps F#-to-F-to-E-to-Ebto-D-to-C#-to-C# and G-to-A|?-to-A-to-B|Ho-B-to-C-to-C. WF*/G transforms the pitch classes in analogous wise with respect to the focal goal-dyad 2. "Inversional Balance as an Organizing Force in Schoenberg's Music and Thought," Perspectives of New Music vol. 6, no. 2 (Spring-Summer 1968), 1-21. The discussion of "Die Kreuze" is on pages 4-8.

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F#/G. Figure 6.6 sketches a sense of how these transformations pertain to the opening of "Die Kreuze."

FIGURE 6.6 No pitch class p satisfies wc/c*(p) = F#. Likewise no p satisfies wF*/G(p) = D[?. This feature of the transformations is actually projected by the music, as the figure shows: The first F# and the last Db sit out their respective wedgegames. Iptf = Ic* naturally functions prominently in connection with the two wedges. l£ also figures in the music; it structures the second chord of figure 6.6 as an "internal" transformation. Let us stand back for a moment and think about the analytic uses to which we have put INJ so far. Nowhere in the discussion of the Schoenberg pieces have we used the word "interval" or even invoked the concept, except so far as it is implicit when we label certain operations as T 10 , T8, and so on. Nowhere, therefore, have we needed to use the fact that the family of pitch classes is a GIS. Nor did we need to suppose that our melodic space of elements (n, p) was a GIS, which in fact it was not. We have nowhere needed to suppose that the transformations we were inspecting were 1-to-l or onto; many in fact were not. From all this we get some idea of how generally the INJ construct can be applied in how great a variety of situations. We shall increase our sense of that variety now by studying another application of INJ to a situation not directly involving a GIS for the space S of elements. 6.2.4 EXAMPLE: Our space of elements for this study will be the family PROT of protocol pairs. A protocol pair is an ordered pair (p, q) of distinct (NB) chromatic pitch classes.3 There are thus 132 = 12 times 11 protocol pairs. A twelve-tone row can be regarded as a certain set within PROT: The pair (p, q)

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3. More generally, we could consider protocol pairs of distinct objects from any finite family, and the mechanics of our discussion coming up would obtain, so far as the theory was concerned.

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6.2.4

is in the set if and only if p precedes q in the row. Note that while the row imposes a certain ordering on the twelve pitch classes, the set under consideration is an w«ordered subset of PROT, i.e. an unordered collection of pitchclass pairs. Besides rows, we can consider other subsets of PROT that are consistent with our intuitions of "ordering pitch classes." To be consistent in this way, a set X must satisfy two conditions. First, we cannot intuit both p-preceding-q and q-preceding-p. Second, if we intuit p-preceding-q and q-preceding-r, then we intuit p-preceding-r. These two conditions translate into the two formal properties following, PO1 and PO2, which X must satisfy as a collection of pairs. (PO1): There is no (p, q) in PROT such that X contains both (p, q) and

(q,p).

(PO2): If (p, q) and (q, r) are members of X, then so is (p, r). Mathematically, a subset X of PROT that satisfies (PO1) and (PO2) is called a (strict) partial ordering of the pitch classes. The special partial orderings that correspond to rows are the "linear" or the "simple" orderings L; these subsets of PROT satisfy in addition the condition (SIMP) below. (SIMP): For any (p, q) in PROT, either (p, q) or (q, p) belongs to L. The set-theoretic condition matches our intuition that either p will precede q in the row, or q will precede p. Representing twelve-tone rows as linear orderings is attractive in many ways. For one thing it makes all rows conceptually equal. That is, it does not assign explicit or implicit priority to one row (e.g. the chromatic-scale row), from which other rows are explicitly or implicitly derived. The model assumes no a priori ordering of the pitch classes; any row orders them as well as any other row. This is very much in the spirit of the classical twelve-tone method. Other attractive features of the model will become apparent presently. In connection with the melody from Schoenberg's "Angst und Hoffen" a little while ago, we brought attention to the way in which series of pitches, pitch classes, and the like could be represented by pairs (n, p) consisting of ordinal numbers n and objects p. Now we have a different way of representing such series, provided they are non-repeating (NB). Our new representation allows us to apply set theory to a linear ordering L, together with its various transforms and other partial orderings X of interest as subsets of PROT. The old model represents the row of Schoenberg's Fourth Quartet by a family of pairs (1, D), (4, B|?), (3, A), (2, C#),... and so on: The first note of the row is D, the fourth note is 6)7, the third note is A, the second note is C#, and so on. The new model represents the same row by the family of pairs (A, B(?), (D, B[>), (C#, A), (D, C#),... and so on: A precedes B^, D precedes Bj?, C# precedes A, D precedes C#, and so on.

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No matter which formal model we use, it will still be convenient to use the notation D-C#-A-B[?-... for quick perusal. Partial orderings that are not rows can model many structures of interest in twelve-tone theory. The partial orderings X: and X 2 on figure 6.7 exemplify only two such types of structure from among many.

FIGURE 6.7 Xj models the small linear motive E-A-Bb. As a subset of PROT, X! contains the three protocol pairs (E, A), (E, 8(7), and (A, 6(7). The reader may check formally that this 3-element set satisfies conditions (PO1) and (PO2). X t models our being sure of the three cited precedence relations, and unsure of or indifferent to any other precedence relations. Ll is what I take to be "the row" of Schoenberg's Moses undAron; as a subset of PROT, Lj contains the pairs (A,Bb), (A,E),..., (A,C); (Bb,E), (Bb,D),..., (Bb,C), (E,D),..., (E,C);...;(B,C). X2 models an aggregate governing the soprano, alto, tenor, and bass voices of the four-part texture at the beginning of Variation 3 in Babbitt's Semi-Simple Variations* X2 contains the twelve pairs (B, D), (B, Eb), (D, Eb); (G, Bb), (G, F), (Bb, F); and so on. L2 models what I take as the row of the piece, that is, the succession of pitch classes formed by the first twelve notes in the soprano voice. We shall be examining various numbers INJ(L 1 ,X 1 )(f) and INJ(L2, X 2 )(f) in connection with a few analytic observations on the two pieces. The transformations f on PROT which will attract our interest are of types (l)-(4) following. (1) Transpositions of pairs, T;(p, q) = (Tj(p), Tj(q)); (2) in versions of pairs, I(p,q) = (I(p),I(q)); (3) retrogression of pairs, R(p,q) = (q, p); (4) combinations of types (1) through (3). These transformations are all well defined on PROT; they do not map any protocol pair into some pair (q,q) whose members are not distinct. The transformations are in fact oper-

136

4. Christopher Wintle provides a very useful analytic study of the piece in "Milton Babbitt's Semi-Simple Variations" Perspectives of New Music vol. 14, no. 2 and vol. 15, no. 1 (Spring-Summer/Fall-Winter 1976), 111 -54.

Generalized Set Theory (2)

6.2.4

aliens on PROT. They form a group isomorphic to the twelve-tone group of operations on rows. Furthermore, when we identify rows with subsets L of PROT, the sense of any row-operation coincides with the sense of the corresponding set-operation. E.g. if the set L corresponds to a certain row, then the set T3RI(L), as determined by the operations T3, R, and I on PROT, corresponds to the row formed by inverting, retrograding, and T3-ing the given row. Let J = 1^, the inversion operation that maps the pitch class A to the pitch class E. Then the small linear motive X, can be extracted from the row J(L,). We can see this by writing out the inverted row and italicizing the entries E, A, and Bl> of the motive Xt as they come along in X,-order: J(Lj) = E-E\>-A-B-B\>-C-F%-.... Another way of expressing the phenomenon is to point out that all three of the member pairs of Xj are member pairs of the inverted row: E precedes A in X} and also in J(Lj); A precedes Bl> both in Xl and in J(Lj); E precedes Bl> both in Xj and in J(Lj). That is, the three members of Xj, (E, A), (A, Bl>), and (E, Bl>), are also all members of J(Lr We reformulate this observation once more: There are three members of Lj whose J-transforms are members of Xr And finally, we can reformulate the observation into our present terminology most concisely: INJ(Lj, Xj)(J) = 3. Are there other inversion operations I such that INJ(Lj, Xj)(I) = 3? As it turns out here, there are not: J is the only one of the twelve inversion operations with that property. Working backwards through the semantic equivalencies of our observations in the paragraph above, we can interpret our most recent observation as telling us that J(Lj) is the only inverted form of Lj that serially embeds the small linear motive Xr We might say that Xj has a high "signature value" for J(Lj) among the twelve inverted forms of Lt: If we sense that an inverted form is at hand and we intuit the three protocol pairs of Xj clearly, that is enough information, abstractly, to identify J(Lj) as the specific form at hand. This property of Xj was noted by Michael Cherlin in connection with events near the opening of act 1, scene 2 in Moses und Aron.5 The scene portrays the brothers meeting in the desert; it begins with a lot of Aron music, light-textured, grazioso, piano and pianissimo, scored for solo flute accompanied by violins, harp and pianissimo horns. Then, just before Aron starts to sing, there is one measure of Moses music, scored for loud trombone and string bass. The trombone plays the short linear motive Xp extracted from the row-form J(Lj). The "signature motive" Xj is here attached to Moses as he steps forth on stage. Aron immediately thereafter begins to address Moses, singing the prime row-form combinatorial to J(Lj) and then J(L,) itself. Xl sounds particularly powerful here because it rearranges the opening trichord 5. The Formal and Dramatic Organization of Schoenberg's Moses und Aron (Ph.D. diss., Yale University, 1983).

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of L! , a trichord which had a strong tonic character as a harmony during the preceding (opening) scene of the drama. Babbitt has published a discussion of just such small linear signature motives in his own music.6 He gives the row of his compositionReflectionsas L = C-B-D-A-Db-Bb-E-F-G-Eb-Gb-Ab. Then he points out that the ordered trichord X = B-D-A is "uniquely characteristic [of the row] t within the transpositional sub-array, and of [one inverted form] to within the inversional sub-array." In our terminology, INJ(L, X)(Tj) is less than 3 unless i = 0; also INJ(L,X)(I) is less than 3 unless I is the one specific inversion J which Babbitt singles out. He then goes on to discuss the ordered tetrachor Y = B-D-A-Db. He notes that Y "is unique ... for the total array." In our terminology, INJ(L, Y) (f) is less than 6 for the forty-seven twelve-tone operations f other than f = T0; L itself is the only twelve-tone form of L that contains all of the six protocol pairs of Y. Y is a signature for L among its forty-eight forms; X is a signature for L among its twelve transposed forms; X is also a signature for J(L) among the twelve inverted forms of the row. Now let us turn out attention back to X 2 and L2 on figure 6.7. Th aggregate X 2 , considered as a subset of PROT, contains 12 member pairs. Hence INJ(L 2 , X 2 ) (f) can be at most 12, ifis an operation. (In that ca map N distinct members of L2 1-to-l into N distinct members of X 2 , so that N must be 12 or less.) The forty-eight specific transformations f that interest us here are in fact operations. As it turns out, none of our forty-eight operations f actually embed X 2 in some form of L 2 , satisfying INJ(L 2 ,X 2 )(f) = 12. However there are operations f that do {£ of the job, satisfying INJ(L 2 ,X 2 )(f) = 11. These operations are f = T! , f = RT7, f = J, and f = RT6 J, where J is the inversion l£ . Since the row is its own retrograde at the tritone, these four operations generate only two forms of L 2 , namely T t (L 2 ) = RT 7 (L 2 ) and J(L 2 ) = RT6J(L2). Figure 6.8 shows how well the ordering of the aggregate X 2 fits into each of these row-forms.

FIGURE 6.8 138

6. "Responses: A First Approximation," Perspectives of New Music vol. 14, no. 2 and vol. 15, no. 1 (Spring-Summer/Fall-Winter 1976), 3-23. The discussion is on page 10.

Generalized Set Theory (2)

6.2.4

The figure makes it visually clear how X2 fits '4i within" either row. Figure 6.8 (b) shows how only the pair (C#, F#) of X2 is not within the row T1(L2); the row contains instead the protocol pair (F#, C#)- Figure 6.8(c shows how only the pair (Bb, F) of X2 is not within the row J(L2); the row contains instead the protocol pair (F, B|?). If only the tenor voice of X2 went E-F#-C# instead of E-C#-F#, then the embedding of figure 6.8(b) would be perfect. Or, if only the alto voice of X2 went G-F-Bj? instead of G-B[?-F, then the embedding of figure 6.8(c) would be perfect. Or, yet again, if only the tenth and eleventh notes of all the rows involved were exchanged, then both embeddings would be perfect. This urge to "make small adjustments" with one set or another will be further discussed later. It is a typical feature of situations in which an INJ function almost attains a theoretical maximum possible value. X 2 , as mentioned before, is the aggregate governing soprano, alto, tenor, and bass voices at the opening of Variation 3 in the Semi-Simple Variations. Various (12-tone) forms of X2 govern SATB relations throughout Variation 3. SATB aggregates of similar format govern other variations, but none of those fit more than "{§ within" any form of the row L 2 . To put it in our terminology, if X is an aggregate governing SATB anywhere in the piece outside Variation 3, the INJ(L 2 ,X)(f) is at most 10, for each operation f we are considering. Thus we can say that the SATB-aggregates of Variation 3 are maximally compatible with forms of L 2 , compared to such aggregates from other variations. Statements of this sort are very useful to express structural differencesamong sections of a piece that sounds at first extremel geneous in texture throughout. That is particularly so when the statements can be backed up by precise measurements like -fj, y§, and the like. INJ helps us pinpoint and explore precisely other structural differences among sections of the composition. For example, let V and V be SATBaggregates from any one variation; then INJ(V, V) (T0) = either 0 or 4. That is, V and V will either have no common pairs or exactly 4 common pairs. Let V l 5 V 2 , ..., V5 be SATB-aggregates from the first, second, ..., fifth variations; then with two exceptions INJ(V m , V n )(T 0 ) is less than or equal to 2. That is, with two exceptions, aggregates Vn, and Vn from different variations will have only 2 or fewer common pairs. Since two distinct aggregates in this format could theoretically share as many as 11 common pairs, we can say that the level of "ordering cross-talk" between variations is very low, half as low as the level of cross-talk within each variation (INJ(V, V')(T0) = 4). Indeed that latter level (4 pairs out of a possible 11) is itself none too high. The two exceptions are these: INJ(V5, V 2 )(T 0 ) = 4 and INJ(V4, V 2 )(T 0 ) = 5. Wecan say that Variations 5 and 4 thus "talk with" Variation 2, so far as SATBaggregate ordering goes, at a level equalling or even surpassing the level of cross-talk within each individual variation. INJ(V4, V 2 )(T 0 ) = 5 is a maximum compared with other values of

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INJ(Vm, Vn)(T0). This observation suggests we devote special attention to aggregate-relations between V4 and V 2 . And when we do so, we shall notice a feature of the composition we might not quickly have come to notice otherwise. Whichever V4 and V2 we select to represent Variations 4 and 2 respectively, the two SATB-aggregates will share exactly one 3-note linear segment from among the four segments D#-B-E, Ab-C-G, C#-F#-D, and B[?-F-A. If Vm and Vn come from variations other than 4 and 2, the two SATB-aggregates will not share any 3-note linear segments. Let us call the family of four segments listed above the "pivot aggregate." Figure 6.9 shows how the pivot aggregate controls the tenor and bass voices of Variation 2, and the soprano and alto voices of Variation 4. The bar lines on the figure mark off SATB-aggregates within each variation.

FIGURE 6.9

INJ numbers bring quickly and effortlessly to our attention the fact that the relationship of figure 6.9 between Variation 4 and Variation 2 is a unique relationship between variations in the piece; it is not a ubiquitous feature of a large-scale design. As noted before, this sort of observation is very useful in bringing to our attention special discriminations within a composition that sounds at first extremely homogeneous.

140

6.3 In this section we shall explore further the notion of "if-only adjustments" in connection with INJ. The abstract notion can be formulated as follows. Suppose INJ(X, Y)(f) is near its theoretical possible maximum in a

Generalized Set Theory (2)

6.4

certain situation. The number might be close to the cardinality of X for instance, so that almost all of X is mapped into Y by f. Or f might be 1-to-l and the injection number might be very close to the cardinality of Y, so that the transformed set f(X) comes close to embedding Y. In such cases, a small adjustment in X or in Y might enable us to remove the "almost" component of the situation, bringing the injection number up to its theoretical maximum value. The parts of X or Y that do not quite fit may come under pressure to conform, giving rise in the music to urges for generating new material. We encountered an if-only situation in discussing the Semi-Simple Variations. Figure 6.8 earlier showed how the SATB-aggregate X2 was almost embedded in the rows Ti(L 2 ) and J(L2); more precisely "{i embedded." While discussing the figure, we mused about making if-only adjustments in either the aggregate or the row, to make the embedding work completely. So far as I can tell, the speculative adjustments do not correspond with musical pressures in Babbitt's piece. But in the Schoenberg song we examined earlier, similar speculative adjustments do correspond with strong musical pressures. In examining the progression of the 3-note chord X = Angst to the 3-note chord Y = Hoffen, we observed that INJ(X, Y)(wE) = 2 out of a maximum possible 3, and that INJ(X, Y)(I) also = 2 out of a maximum possible 3, "I" here meaning I| = ^Bb- "If only" the F|? of chord Y were adjusted to F natural, we noted, both the injection values of 2 above would rise to the maximum 3. In this connection the if-only speculation led to fruitful analytic ideas about the "missing F natural," the "missing I-partner," the missing lover, begehren, F[? as functional substitute for F natural, and so on. The interested reader will find an extensive treatment of if-only adjustment and its analytic implications elsewhere in my writings, in connection with the opening two chords of Schoenberg's piano piece op. 19, no. 6.7 I call the chords "rh" and "Ih" for right hand and left hand. TjL

6.4 The cited article goes on to discuss "progressive" and "internal" transformations in connection with the succession rh-lh. "Progressive" transformations make rh into something much like Ih; "internal" transformations make rh into something much like itself, or Ih into something much like itself, or both. The reader will recall that we used this nomenclature earlier, in connection with various transformations pertaining to harmony and melody in the song "Angst und Hoffen." We shall now extend the nomenclature and put it into a completely 7. "Transformational Techniques in Atonal and Other Music Theories," Perspectives of New Music, vol. 21, nos. 1-2 (Fall-Winter 1982/Spring-Summer 1983), 312-71, especially 33642.1 am indebted to Michael Bushnell for having observed the rewards that this sort of approach brings in analyzing the music. He worked with transposition operations only; I enlarge his aural field to include inversions as well.

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general theoretical setting, invoking only a family S of elements, certain transformations f on S, and the INJ function. Given sets X and Y, suppose we are inspecting the values of INJ(X, Y)(f), MT(X, X)(f), and INJ(Y, Y)(f) as f varies over a certain family INSPECT of transformations. For certain transformations f within the family INSPECT, the value of INJ(X, Y)(f) will be maximal, or at least relatively high subject to the constraints of the situation. We shall call these transformations progressive. They map a lot of X into Y. For certain transformations f within the family INSPECT, the value of MT(X, X)(f) or MT(Y, Y)(f) will be high. We shall call these transformations X-internal or Y-internal accordingly. A transformation which is both Xinternal and Y-internal can be called "internal (for the progression X-Y)." An X-internal transformation maps a lot of X into X. Intuitively, an X-internal transformation tends to extend/elaborate/ develop/prolong X in the music, while a progressive transformation tends to urge X onwards, to become something else (like Y). Progressive and internal transformations will tend to combine mathematically in certain interrelated ways, by their very natures. If I transform X to be much like itself, and then transform the result to be much like itself, it is likely that the composition of the two gestures will make X much like itself. That is, the composition of two X-internal transformations will tend to be X-internal. Similarly, the inverse of an X-internal operation will tend to be X-internal. Similarly, an X-internal transformation followed by an X-Yprogressive transformation will tend to be an X-Y-progressive transformation; and an X-Y-progressive transformation followed by a Y-internal transformation will tend to be X-Y-progressive. As a result, when we inspect the families of progressive and internal transformations pertinent to a given X-Y situation, we shall find those families tending to interrelate algebraically according to the considerations just surveyed. We can introduce other useful nomenclature. An f such that INJ(X, X)(f) is minimal or at least relatively small, given the constraints of X and INSPECT, can be called X-external. Such an f maps X largely outside itself. We can also define a dispersive transformation to be one that maps X largely outside Y, makeing the value of INJ(X, Y)(f) minimal or relatively small. These definitions avoid mentioning the complements of the sets X and Y in S, which may not be "sets" according to our definition if S is infinite. External and dispersive transformations tend to enter into typical algebraic relations with themselves, with each other, and with internal and progressive transformations. An X-internal transformation followed by an Xexternal one will tend to be X-external; a progressive transformation followed by a Y-external one will tend to be dispersive; and so on. A good example of dispersive transformations is furnished by measure 8

Generalized Set Theory (2)

6.4

of Schoenberg's op. 19, no. 6. This is the cryptic, very dense measure that precedes the final return of the chords rh and Ih in measure 9. Figure 6.10(a) reproduces measure 8.

FIGURE 6.10

Forte has noted that the music embeds many forms of the rh chord.8 For our purposes, we can observe that the music embeds four transposed forms of rh. Those are the four shown in figure 6.10(b): T2(rh), T5(rh), T7(rh), and T9(rh). Now if we let the interval i range from 0 through 11, we shall find that there are six values of i for which INJ(rh, Ih) (Tj) = 0; these values are i = 2,4, 5, 7, 9, and 0. That is, T2, T4, T5, T7, T9, and T0 are the six dispersive transposition operations, given the progression rh-to-lh. To put it another way, T2(rh), T4(rh), T5(rh), T7(rh), T9(rh), and T0(rh) = rh are the transposed forms of rh that have no common tones with Ih. As figure 6.10(b) shows us, four of these six forms are embedded within the music of figure 6.10(a). (The rest of that music does contain common tones with Ih.) And, as figure 6.10(c) shows us, a fifth dispersive form, T0(rh) = rh, ensues immediately thereafter, joining the parade of dispersive forms and thereby linking measure 8 to the downbeat of the final reprise at measure 9. The relevance of external transformations to traditional theory is illustrated by the "semi-combinatorial hexachord." If X is a 6-note set of pitch classes and I is an inversion operation that transforms X into its complement X, then I is an X-external transformation: INJ(X,X)(I) = 0. This is a good place to think about exploring how the injection function relates with set-conplementation when S is finite. We shall soon carry out that exploration, in section 6.6. Before that, it will be helpful to prove a theorem and a corollary about INJ(X, Y)(f) when f is an operation OP. 8. The Structure of Atonal Music, example 102 (p. 99).

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6.5.1

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6.5.1 THEOREM: If f is an operation OP, then INJ(X, Y)(OP) is the cardinality of OP(X) n Y, that is the number of common members shared by the sets OP(X) and Y. We took note of this theorem informally in the commentary following Definition 6.2.1 earlier. The fact is important enough to warrant formal verification. Proof: Let M = INJ(X, Y)(OP); let N = card(OP(X) n Y). Let x t , x 2 , . . . , X M be the distinct members of X that map into Y via OP. Since OP is 1-to-l, the elements OP(Xi), OP(x2), ..., OP(xM) are distinct members of OP(X) n Y. Thus OP(X) n Y contains at least M distinct members; N is greater than or equal to M. Now let y l 5 y 2 , ..., yN be the distinct members of OP(X)nY. Then OP^iX OP"1^)* ..., OP"1^) are N distinct members of X, each of which maps into Y under OP. So there are at least N distinct members of X that map into Y under OP; M is greater than or equal to N. q.e.d. The injection function applied to operations thus generalizes Regener's Common-Note Function, which was developed for the special case in which X and Y are sets of pitch classes and OP runs through the twelve transposition operations.9 6.5.2 COROLLARY: If f is an operation OP, then INJ(Y,X)(OP) = INJP^YXOP"1). Proof: An element z is a member of the set OP~ 1 (X)n Y if an only if OP(z) is in X and z is in Y; this is the case if and only if OP(z) is in X and OP(z) is in OP(Y), that is if and only if OP(z) is a member of the set X n OP(Y). So the transformation OP maps the set (OP~1(X)n Y) 1-to-l onto the set (XnOP(Y)). Therefore the two sets have the same cardinality: card(OP(Y)nX) = card(OP~ 1 (X)nY). Applying Theorem 6.5.1 to both sides of this equation, we infer the formula of the Corollary. Now we are ready to explore set-complementation in connection with INJ. It will simplify matters greatly to restrict our attention to transformations f that are operations OP in this context. We shall then be able to use the theorem and the corollary we have just proved. We mustjestrict the abstract setting by supposing S to be finite; then the complement X of a set X will be a formal "set" by our criterion, i.e. a finite subfamily of S. 6.6.1 THEOREM: Suppose S is finite. Given sets X and Y with complements X and Y; given any operation OP; then formulas (A) through (E) below obtain. 144

9. Eric Regener, "On Allen Forte's Theory of Chords," Perspectives of New Music vol. 13, no. 1 (Fall-Winter 1974), 191-212. The Common-Note Function is defined on page 202.

Generalized Set Theory (2)

6.6.2

(A): (B): (C): (D):

INJ(X, Y)(OP) = cardX - INJ(X, Y)(OP). INJ(X, Y) (OP) = cardY - INJ(X, Y) (OP). INJ(X,_Y)(OP) = cardY - cardX + INJ(X, Y)(OP). IfcardY_=cardX, then INJ(X, Y)(OP) = INJ(X,Y)(OP). (E): (Generalized Babbitt Hexachord Theorem) If cardX_= ^cardS, then INJ(X,X)(OP) = INJ(X,X)(OP).

Proofs:_ (A): The operation OP maps each member of X either into Y or into Y. So cardX = INJ(X, Y)(OP) (the number of X-members mapped into Y) plus INJ(X, Y)(OP) (the number of X-members mapped into Y). The formula follows. This argument works just as well for any transformation f. (B): Here it is essential that OP be an operation, so that we can apply Corollary 6.5.2. We write INJ(X, Y)(OP) = INJ(Y,X)(OP-1), via 6.5.2. This, via formula (A) just proved, = cardY - INJ(Y, X)(OP-1). And, applying 6.5.2 again, we infer that this number is indeed cardY — INJ(X, Y)(OP), as claimed. (C): INJ(X,Y)(OP) = cardY = INJ(X,Y)(OP), via (B); this is cardY - (cardX - INJ(X, Y) (OP)), via (A); this is cardY - cardX + INJ(X,Y) (OP), as desired. (D) is an obvious corollary of (C). And (E) is an obvious corollary of (D), setting Y = X. q.e.d. The methods of proof I have used are essentially Regener's. I have called formula (E) the Generalized Babbitt Hexachord Theorem because Babbitt's theorem, somewhat disguised, is a special case of this formula. For readers who may feel the disguise is perfect, I shall show the connection. Let X be a sixnote pitch-class set; let OP be a transposition operation Tj. Theorem 6.6.1 (E) above tells us that INJ(X,X)(TJ = INJ(X,X)(Ti). Theorem 6.5.1 then tells us that the cardinality of (Tj(X) n X) is the same as the cardinality of (T;(X) n X). Now the cardinality of (T;(X) n X) is the number of members of X that lie the interval i from some member of X. In other words, the number is IFUNC(X, X) (i). Similarly, the cardinalitypf (T}(X) n X) is IFUNC(X, X) (i). We have shown, then, that IFUNC(X,X)(i) = IFUNC(X,X)(i) for every i: the complementary hexachords contain the same number of i-dyads for each i. This is Babbitt's theorem. One sees that 6.6.1(E) is a very broad generalization. 6.6.2 EXAMPLE: Let us see how the Generalized Hexachord Theorem applies in another specific context. The reader will recall the space PROT of protocol pairs which we constructed earlier (6.2.4) in examining some topics from serial

145

6.6.2

146

Generalized Set Theory (2)

theory. We noted that the various twelve-tone rows can be regarded as those subsets L of PROT that are linear orderings on the pitch classes. In this model, the retrograde of the row L corresponds to the set-theoretic complement L of the set L in PROT. For a pair (p, q) is in the complement of the set if and only if p does not precede q in the row, which is the case if and only if p precedes q in the retrograde of the row, which is the case if and only if (p, q) lies in that portion of PROT corresponding to the retrograde row. The family PROT has 132 = 12 times 11 members. And any row, as a subset of PROT, contains exactly 66 protocol pairs. To see that, suppose the pitch classes of the row come in the serial order p l 9 p 2 , ..., p 12 . Then the subset L of PROT contains 11 pairs of form (Pi,p n ), and 10 pairs of form (p 2 ,p n ), ..., and 1 pair of form (p u , pn). The cardinality of the set L is thus 11 + 10 H + 1, which is 66. We are thus in a setting to which 6.6.1(E) pertains. card(S) = card(PROT) = 132; card(X) = card(L) = 66 = |card(S). In this setting, a row and its retrograde (complement) play a set-theoretic role formally analogous to that of a hexachord and its complement in traditional atonal theory. Here, 6.6.1 (E) tells us the following. Let OP be any operation on PROT. Given any row L with retrograde L, let N be the number of pairs (p, q) in L such that the pair OP(p, q) is also a precedence relation in L; let N' be the number of pairs (p', q') in L such that OP(p', q') is also a precedence relation in L. Then N = N'. In this connection, OP(L) need not itself be a row. Indeed L itself can be replaced by any set of cardinality 66 within PROT and the theorem remains ture, "L" now being simply the set-theoretic complement of L. But the application is of particular interest when we interpret the complementary sets L-and-L as row-and-retrograde. The twelve-tone operations Tj and I are induced on PROT by corresponding operations on individual pitch classes: Tj(p, q) = (Tj(p),Tj(q)); I(p,q) = (I(p), I(q)). But in general there need not be any operation op on pitch classes such that OP(p, q) = (op(p), op(q)). There is no such op, for instance, in the case of the retrograde operation R on PROT: R(p, q) = (q, p). Nor is there such an op for any of the operations to which R contributes, e.g. RTj and RI. One can also construct fancier operations on PROT not induced by pitch-class operations op. For example: If p and q are in the same wholetone scale, OP(p, q) = (p, q); if p and q are in opposite whole-tone scales, OP(p > q) = (T3(q),T9(p)). When L is a row and OP(L) is also a row, then INJ(L, L) (OP) measures the size of the largest partial ordering on the pitch classes which can be embedded in both OP(L) and L, i.e. whose protocol pairs are compatible with both those rows. That is because INJ(L, L)(OP) is the cardinality of OP(L) n L, as we know from 6.5.1.

Generalized Set Theory (2)

6.7.2

Now we shall show formally how INJ completely generalizes IFUNC when there is a GIS at hand. 6.7.1 THEOREM: Let (S, IVLS, int) be a GIS. Then for each interval i and for all sets X and Y, IFUNC(X, Y)(i) = INJ(X, Y)(T,). Proof: Let IFUNC(X, Y)(i) = M; let INJ(X, Y)(Tj) = N. We shall see that N must be at least as big as M, and that M must be at least as big as N. Since IFUNC(X,Y)(i) = M, there are M distinct pairs (Xj.yj), (x2, y2), ..., (XM, yM) such that xm lies in X, ym lies in Y, and int(xm, ym) = i. For each such pair, ym = Ti(xm). For m and n distinct, x m and x n are distinct members of X. (Otherwise we would have ym = Ti(xm) = Tj(xn) = yn, whence the pairs (xm, ym) and (xn, y n ) would not be distinct, contrary to supposition.) Thus X has at least M distinct members whose i-transposes lie in Y. That is, INJ(X, Y)(Tj) is at least as big as M. Or: N is at least as big as M. Now let Zi, z 2 , . . . , ZN be the N distinct members of X whose i-transposes lie within Y. (There are N such, since INJ(X, Y)(T;) = N.) For each such z n ,z n is in X, u = Tj(z n ) is in Y, and int(zn, u) = i. Therefore the pair (z n ,u) was counted as some (xm, ym) above. So every one of the N elements z 1 ,..., ZN is one of the M elements x l 5 . . . , X M . Hence M is at least as big as N. q.e.d. The logic of Theorem 6.7.1 can be visualized through the following aid. Imagine X and Y as two finite configurations of points in the Euclidean plane. Suppose i is the vector (directed distance) "to the right and up 30 degrees for a distance of 5 inches." We can ask: "From points of X to points of Y, how many distinct arrows can I draw that go to the right and up 30 degrees for a distance of 5 inches?" The answer to this question is IFUNC(X, Y)(i). We can also ask: "If I move the whole X-configuration to the right and up 30 degrees for a distance of 5 inches, how many points of the displaced configuration will then coincide with points of Y?" The answer to that question is INJ(X, Y)(Tj). One intuits easily that the two questions are logically equivalent. Next we shall explore what happens to INJ when the sets X and/or Y are transformed by some operation A. 6.7.2 THEOREM: Given a family S of objects, given sets X and Y, given a transformation f on S and an operation A on S, then formulas (A), (B), and (C) below obtain. (A): INJ(A(X),Y)(f) = INJ(X, Y)(fA). (B): INJ(X, A(Y))(f) = INJ(X, Y)(A~ 1 f). (C): INJ(A(X), A(Y)) (f) = INJ(X, Y) (A'1 f A).

147

5.7.2

Generalized Set Theory (2)

Proofs: (A): INJ(A(X), Y)(f) is the number of t within A(X) such that f(t) is a member of Y. Set t = A(s); then the family of such t is in 1-to-l correspondence, via A, with the family of s in X such that fA(s) is a member of Y. And the number of such s is exactly INJ(X, Y)(fA). (B): INJ(X, A(Y)) (f) is the number of s in X such that f (s) is a member of A(Y). Now f (s) is a member of A(Y) if and only if A"1 f (s) is a member of Y. So the number at issue is the number of s in X such that A~ 1 f(s) belongs to Y. And that is exactly INJ(X, YXA^f). (C): The formula follows at once from (A) and (B). q.e.d.

148

Formula 6.7.2(C) is of particular abstract interest. We can imagine that the shift from X-and-Y to A(X)-and-A(Y) reflects a "modulation" of the system by the operation A. For instance, if we are in a GIS and A is a transposition or an inversion, we are transposing or inverting (the sets of) the system accordingly. It is natural to ask: "If we modulate the system by A, what effect does that have on the INJ function?" At first one might suppose that INJ would remain unaffected by the modulation: INJ(A(X), A(Y)) = INJ(X, Y). But, as formula 6.7.2(C) shows, that is not in fact the case. Rather the INJ function is itself "modulated" according to the formula. We noted a specific instance of this general phenomenon earlier, during our analysis of "Angst und Hoffen." The wedge-structure of figure 6.3(b) converged on E; it was studied in connection with the transformations W E = wedging-to-E and I = inversion about E. The wedge-structure of figure 6.3(c) converged on F#; it was studied in connection with the transformations WF* = wedging-to-F# and J = inversion about F#. To get from the situation of (b) to the situation of (c), one "modulates the system by T2." The following equations obtain: F# = T2(E); wF* = T 2 w E T 2 " 1 ; J = T2IT2"1. The first equation relates the focal points of the two wedges, also the bass notes of the Angst and Hoffen chords. The second equation leads, via 6.7.2(C), to the relationship INJ(T 2 (X),T 2 (Y))(w F «) = INJ(X, YXT^w1^) = INJ(X, Y) (WE). The third equation leads via the same formula to the relationship INJ(T2(X),T2(Y))(J) = INJ(X, Y)(T 2 1 JT 2 ) = INJ(X, Y)(I). Thus, when we modulate the system from E-centricity to F#-centricity via the operation T 2 , then the wedge WF* plays the role, with respect to the modulated sets T2(X) and T2(Y), that the wedge W E originally played with respect to the sets X and Y. Similarly, the inversion J plays the role, with respect to the modulated sets, that the inversion I originally played with respect to the unmodulated sets. The reader will recall, perhaps, our earlier remarks on this subject by way of preparation (pp. 128-129). Here is another, more abstract, example of system-modulation. Let X be an atonal hexachord that inverts into its complement via the inversion I = IB. Then INJ(X,X)(I) = 0. Suppose some music projecting X-andits-complement "modulates" to a new section projecting T5(X)-and-//j-

Generalized Set Theory (2)

6.7.3

complement. Here I plays the role of "f" and T5 plays the role of "A" in Formula 6.7.2(C). We cannot suppose that INJ(T5(X), T5(X))(I) = 0: the new hexachord T5(X) will not invert into its complement via the inversion I, B/C inversion. Rather, T5(X), the modulated hexachord, inverts into its complement by the inversion J = T5IT~^. Formula 6.7.2(C) tells us this: INJ(T5(X), T5(X))(J) = INJ(X, X)(T-i JT5) = INJ(X, X)(I) = 0. Using formulas 3.5.6 (A) and (B), we can compute J = T 5 l£T 7 = I|T7 = Ij:. Thus T5(X) inverts into its complement about E and F (or about B!> and B). The system having modulated by T5, the transformation J = T5IT~* now plays the role that the transformation I originally played. Formula 6.7.2(C) tells us this sort of thing in great generality: When a system modulates by an operation A, the transformation f = AfA"1 plays the structural role in the modulated system that f played in the original system, in the sense that INJ(modulated X, modulated Y(f) = MT(X, Y)(f). 6.7.3 Theorems 6.7.1 and 6.7.2 enable us to generalize the abstract questions about IFUNC we asked earlier, toward the end of the section 5.1. We asked, for instance, under what circumstances in a GIS we would have IFUNCCXj, Xj) = IFUNC(X2, XJ. Via 6.7.1, we can rephrase the question, asking under what circumstances in a GIS we shall have INJ(Xj, Xj)(Tj) = INJ^Xj, XjXTj) for every transposition Tr And that question can easily be generalized: Given any family S of objects and any family INSPECT of transformation on S, under what circumstances shall we have INJ(Xj, Xj)(f) = IN](X2, XjXf) for every member f of INSPECT? We do not have to demand that S be in a GIS, or that INSPECT be a group; indeed, the question makes sense even if INSPECT is not a closed family (semigroup) of transformations. Likewise, we earlier asked under what conditions in a GIS we would have IFUNC(X,, XJ = IFUNC(Y,, Y2). We can generalize that question analogouly: Given any family INSPECT of transformations f on a family S of objects, under what conditions shall we have INJ(X,, X^f) = MT(Y,, Y2)(f) for every member f of INSPECT? The ideas of 6.7.2 enable us to expand that question even farther. Suppose we have a family S of objects, a family INSPECT of transformations f on S, and a group MDLT of "modulating operations" A on S. Under what conditions, given sets Xj, X^ Y,, and Y2, will there exist some modulation A such that when we modulate Yj and Y2 by A, obtaining Y', and Y'2, we shall have DSfJCX,, X^f) = INKY',, Y'2)(f) for every f in INSPECT? Via 6.7.2(C) this amounts to demanding that ESTJ(X,, X^f) and INJ(Yt, Y2(A~1fA) be equal numbers, for each member f of INSPECT. In the special case where we are in a GIS, where INSPECT is the family of transpositions, and where MDLT is the group of transpositions, the question asks under what conditions, given the four sets, there will exist some interval

149

6.7.4

Generalized Set Theory (2)

j such that for every interval i, IFUNCCX^XjXi) = IFUNC(Tj(Y1), Tj(Y2))(i) = IFUNCCY^Y^Gij- 1 ). (5.1.6 gives us the last equality. It is consistent with 6.7.2(C) because T^-i = T^TjTji The group of transpositions is aAm'-isomorphic to the group of intervals.) Besides using 6.7.1 and 6.7.2 to generalize earlier questions, we can also use them to help out with computations we earlier found difficult to execute. The formula following is a good example. 6.7.4 THEOREM: In any GIS let I = !„. Fix any referential element and set j = LABEL(u). Then for any sets X and Y, and for each interval i, IFUNC(I(X),I(Y))(i) = INJ(X, Y)(Pk) where Pk is the interval-preserving operation labeled by k = ji"1]"1. Proof (optional): IFUNC(I(X), I(Y)) (i) = INJ(I(X), I(Y)) (Tj), via 6.7.1. This, via 6.7.2, = INJ(X, Y)^"1^!). Set I^T,! = OP. It now suffices to prove that OP = P k , where k = ji"1.)""1. r1 = 1^(3.5.9). So OP = I^Tji;;. We can compute T^ by 3.5.6(A); the result is I*, where x = Tj(u). So OP = I"I*. And we can compute the composition of the two inversions by 3.5.8. It is PmTn, where m = LABEL(u)LABEL(x)-1 and n = LABELOO^LABELCv). Here n = e, so we have computed that OP = Pm, where m = LABEL(u)LABEL(x)~1. Since x = T,(u), LABEL(x) = LABEL(u)-i(3.4.3). Replacing LABEL(u) by j, we then have m = j(ji)"1 = ji-1j-1 = k. Thus OP = Pk as desired.

750

6.8 To demonstrate further the generalizing powers of INJ, we shall now use it to generalize Forte's K and Kh relations. For our generalization we suppose only a family S of objects and a group of operations on S which we shall call "canonical" for reasons known only to us. We shall denote the group by CANON. The complement of the set X will be denoted X. If S is infinite, X will not be finite, hence not a "set" in our terminology. Nevertheless we shall speak of its "cardinality," understanding the value infinity for the expression cardX. If S is finite, cardX will mean the finite cardinality of the set X. We shall restrict our attention to sets X and Y such that cardX < cardX and cardY < cardY. We can do this because Forte's K and Kh relations are not affected by the restriction. If cardX should be less than cardX, we can simply exchange the roles of the sets X and X as they do or do not enter into K or Kh relations with other sets. Having made that restriction, we may also suppose that cardX < cardY. Otherwise, we can simply exchange the roles of the sets X and Y in the arguments coming up. So our restrictions, in sum, are these: cardX < cardX, cardX < cardY < cardY. Given those restrictions, Forte's K and Kh relations hinge on the

Generalized Set Theory (2)

6.8

logical combination of two more primitive relations, which we can call K! and K2. (K x ): Some (canonical) form of X is embedded in Y. (K2): Some form of X is disjoint from Y (and hence embedded in Y). X and Y, subject to our restrictions, enjoy Forte's K relation if they enjoy either Kj or K 2 ; they enjoy his Kh relation if they enjoy both K^ and K 2 . Now the relations Kj and K 2 correspond to very natural aspects of INJ(X, Y), as that function takes arguments from the canonical group. Specifically, K t and K 2 are respectively equivalent to K\ and K 2 below. (K;): For some A in CANON, INJ(X, Y)(A) = cardX. (K'2): For some B in CANON, INJ(X, Y)(B) = 0. Now we can express Forte's relations, generalized, as follows: (K): INJ(X, Y) (A) takes on either its theoretical maximum value cardX (subject to our restrictions), or its theoretical minimum value 0, as A varies over CANON. (Kh): INJ(X, Y)(A) takes on both its theoretical maximum and its theoretical minimum values, as A varies over CANON. We can use the ideas of "progressive" and "dispersive" transformations (6.4), to rephrase K\ and K 2 yet once more, into the respective forms of K/{ and K 2 below. (K'i): CANON contains some maximally progressive transformation with regard to INJ(X, Y). (K2): CANON contains some maximally dispersive transformation with regard to INJ(X,Y). Let us return to relation K\ above. K't says there is some A in CANON such that INJ(X, Y)(A) = cardX, but it does not tell us how many such A there are. In case CANON is infinite, that may or may not be a meaningful question. When CANON is finite, the question is definitely interesting. By asking it, we can distinguish multiplicities of K! -ness about the K x relations of various sets: Some pairs of sets are "more K^-related" than others, Likewise, when CANON is finite, we can distinguish multiplicities of K2-ness about K2 relations, counting how many members B of CANON satisfy condition K 2 above. We can then attach pairs of multiplicity numbers to sets in the Kh relation, to indicate the multiplicities of K!-ness and K2-ness involved. For a specific example, take S to be the twelve chromatic pitch classes, and take CANON to comprise both transpositions and inversions. Take Y to be the black-note pentatonic scale. Take X to be the F#-major triad. Then we may say that X and Y are Kh-related with multiplicity (2,6): There are two members A of CANON embedding A(X) in Y, and there are six members B of CANON such that B(X) is disjoint from Y. Now, holding onto Y, take X to be the set (Ajj, B|?, Dfc>). The new X is Kh-related to Y with multiplicity (4,8). 6.9 Coming back from the specific example, let us return to the general

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6.10

Generalized Set Theory (2)

situation with CANON finite, still subject to our conditions on the cardinalities of X and Y. For each integer N between 0 and cardX inclusive, we can ask how many members A of CANON satisfy the equation INJ(X, Y)(A) = N. Our multiplicity-values aoove asked that question for N = cardX and for N = 0, but we can just as well ask it for every N in between. The answer we get, i.e. how many such A there are in CANON, is the value for the argument N ofRegener's Partition Function, here generalized.10 We shall denote that value byRGNPF(X,Y)(N). In this special case (CANON being finite), the Partition Function enables us to derive the EMB function by the formula below. RGNPF(X.Y)(cardX) EMB(X, Y) -

RGNpF(X)X)(cardx)-

That is, EMB(X, Y) is the multiplicity of Kj-ness about X-in-Y, divided by the multiplicity of Kj-ness about X-in-itself. The formula is given without proof. If X is symmetrical in some way, there may be several distinct members of CANON, other than the identity, that map X onto itself. If M members of CANON map X onto itself, then there will be M times as many distinct operations embedding X in Y, as there are forms of X embedded in Y. RGNPF counts operations; EMB counts forms. This accounts for the denominator of the fraction in the formula above. 6.10 OPTIONAL: In all our work so far with set theory, we have supposed the "sets" under scrutiny to be finite. This section will outline briefly the work needed to extend our results so that they can be applied to sets "of finite measure" in certain "measure-spaces." During this discussion we shall relax the use of the word "set" so as to conform to standard mathematical usage. Roughly speaking, that usage makes the word a synonym for "family" or "ensemble" of things. Given a family S of objects, a family FLD of subsets of S is called a fieldof subsets if it satisfies (1), (2), and (3) below. (1): 0 (the empty set) and S are members of FLD. (2): If X is a member of FLD, then so is its complement X. (3): If X and Y are members of FLD, then so is X u Y, their set-theoretic union. It follows deductively that whenever X and Y are in FLD, so are X n Y (which is the complement of X u Y), Y — X (defined as Y n X), and so on. A field of sets is called asigma-fieldif (4) below is also tr (4): Whenever Xj, X 2 ,... is a countable family of members of FLD, so is the countable union X = XiuX2u — X here is the set of s such that s is a member of at least one of the sets X n . "Countable" means "capable of being put into 1-to-l correspondence with the natural integers." 752

10. Ibid., p. 206ff.

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A measure on the field FLD is a function mes that assigns to each member X of FLD a value which is either a non-negative real number or infinity, in such fashion that mes(X u Y) = mesX + mesY whenever X and Y are disjoint. If FLD is a sigma-field, one supposes mes to be sigma-finite unless it is specified as not so. The condition for sigma-finiteness is that mes(UNION X n ) = SUM(mesXn) whenever X 1? X 2 ,... is a countable family of mutually disjoint members of FLD. The countably infinite sum is understood in the usual sense, as the limit of its partial sums; that limit may be infinity. We can deduce that if X is included in Y, then mesX < mesY. For mesX < mesX + mes(Y — X), mes being non-negative, and that number is mesY since X and (Y — X) are disjoint. We shall need one more concept. Given a field FLD of subsets of a family S, a transformation f on S is called measurable when, given any member Y of FLD, the set of s mapped by f into Y is also a member of FLD. Now we can generalize INJ. We suppose a family S, a field FLD of subsets, and a measure mes on FLD. By a "set of finite measure" we mean just that, a member X of FLD such that mesX is finite. If X and Y are sets of finite measure, and if f is a measurable transformation on S, then the family F of s such that f (s) belongs to Y is a member of FLD, so the family X n F is also a member of FLD. Furthermore, mes(X n F) < mesX, so X n F is a set of finit measure. We then define INJ(X, Y) (f) = mes(X n F). This number measures "how much of X" (according to mes) is mapped into Y by f. Using this definition for INJ and restricting X, Y, etc. to be sets of finite measure, we can generalize much of the machinery in chapter 6 as it stands. Theorem 6.5.1 and the formulas that follow from it must be restricted so that the operations at hand are "measure-preserving," satisfying mesOP(X) = mesX. We could derive more flexible and much more complicated formulas by allowing "measure-scaling" operations; OP is such if for some constant number "scale," mesOP(X) = scale • mesX. Our work on the Partition Func tion (6.9) may not generalize easily, since CANON is not likely to be finite if the sets X, Y, etc. are not finite. But often it will be possible to establish a "good" field of subsets within CANON, and a "good" measure on that field; then for any real numbers a and b with 0 < a < b, we can define RGNPF(X, Y)(a, b) to be the measure, within CANON, of the set of operations A satisfying a < INJ(X, Y)(A) < b. The mathematical ramifications of all this lie far beyond the scope of the present book. The work of chapter 6 up to the present is just one special case of the general system now being sketched. In that special case, FLD contains all subsets of S, so all transformations fare measurable; mesX is the cardinality of X, allowing infinity as a possible value. The sets of finite measure are exactly the finite sets, what we have been calling "sets" up until now. For another special case we can consider a non-musical setting. Let S be

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a Euclidean plane containing a Seurat painting. We take FLD to consist of all regions in the plane with a well-defined total area, and we use area as our measure. Let X be the region of the plane consisting of all animals in the painting; let Y be the region consisting of all plants. Let f be the transformation on S: move up and to the right at a 45-degree angle for 3 centimeters. Then INJ(X, Y)(f)/areaX answers the question: To what extent are the animals of the painting to be found 3-cm.-below-and-at-a-45-degree-angle-to-the-left-of plants? We now adopt a new measure, redmes. This measure assigns to each member of FLD the number of red dots it contains completely. (red)INJ(X, Y) (f )/redmesX now answers the question: To what extent do red dots within the animals of the painting lie 3-cm.-below-and-at-a-45-degreeangle-to-the-left-of plants? If we want to shift our attention to yellow dots, then (yellow)INJ(X, Y) (f )/yellowmesX answers the analogous question with respect to yellow dots. For our next example suppose we are analyzing a certain piece and we want to ask questions like this: Of the amount of time the violin is playing above high C during the piece, how much of that time will the clarinet be playing pianissimo or softer five seconds later? We can use a special case of the general model at hand. Fixing a referential unit of time and a referential timepoint zero, we take S = the real numbers, as modeling the continuum of timepoints within which our piece occurs. For each real number a, let Za be the set of numbers s satisfying s > a, and let Za be the set of numbers s satisfying s < a. Take FLD to be the smallest sigma-field that contains every Za and every Za. As it happens, there is essentially only one well-behaved measure on FLD that satisfies mes(Za n Zb) = (b — a) for every pair of numbers a < b. We shall use this measure; we can think of it as measuring the "amount of time" in a set. Take X to be the set of time-points in the piece during which the violin is playing above high C; take Y to be the set of time-points in the piece during which the clarinet is playing pianissimo or softer. Take f to be timepoint transposition by five seconds: The time-point f (s) occurs five seconds after s. Then the number we asked for above is INJ(X, Y)(f)/mesX. For a final example we shall explore the space S of time spans. We represent S by the Euclidean half-plane of number-pairs (a, x), where x is positive. FLD will be the smallest sigma-field containing all square-shaped regions within the half-plane. FLD contains all the sets we shall want to discuss here; e.g. it would contain the region comprising all animals in the Seurat painting. There is a unique measure on FLD that makes the measure of each square region its area; we shall call this measure "area." We may or may not want to use area as a measure for our musical purposes. There are other measures that have other desirable properties. For instance, in connection with our non-commutative GIS, there is an essentially unique measure that makes every interval-preserving operation measure-preserving: mesP(X) =

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mesX. We can call this measure "P-invariant measure." It is essentially unique in this sense: If mes' is any other P-invariant measure, then there is some positive number "scale" such that mes'(X) = scale • mesX for every member X of FLD. There is also adifferentmeasure on S, also essentially un makes every transposition-operation measure-preserving: mesT(X) = mesX. We can call this measure "T-invariant measure." If we are analyzing a specific piece, we may also want to use measures like the Seurat measure earlier that counted red dots. The following discussion will help develop a "Seurat model" for our analysis. Suppose the piece begins at the time point BEGIN and extends for the duration EXTENT. Then every time span that pertains to an event within the piece must satisfy the two constraints BEGIN < a and a + x < BEGIN + EXTENT. The set of time spans (a, x) satisfying those constraints forms a triangular region in the halfplane. This triangle plays a role in our model analogous to that of the rectangle on which Seurat painted, a bounded region within its plane. In applying time-span analysis to the piece, we are hearing events with well-articulated beginnings and durations, events whose temporal location and extent can be sensibly modeled by time spans. We may then fill in the triangle-of-the-piece by dots. A dot at the point (a, x) within the triangle models the location and extent of an event in the piece which begins at a and lasts x units of time. We can color certain dots red. Say these are the dots pertaining to stringed-instrument events. We can color certain dots yellow. Say these are the dots pertaining to events "above middle C." Orange dots will then pertain to events involving some stringed instrument (s) playing above middle C. Our model assumes the format of a triangular Seurat painting, and we can apply the ideas of "red measure," and so on. Given a time span (b, y), we can construct the family SHADOW(b, y) of all time spans (a, x) that "happen within" the time of (b, y). We constructed the shadow of (BEGIN, EXTENT) above; that was the triangle-of-the-piece. SHADOW (b, y) in general is the set of all spans (a, x) satisfying the two constraints b < a and a + x < b + y. This set forms a triangular region in the half-plane. Given two events in the piece, event 1 and event2, let (a, x) and (b, y) be the respectively pertinent time spans. Then (a, x) is in the SHADOW of (b,y) if and only if event 1 happens during the time of event2. Let us imagine a certain section of the piece which begins at the time point BEGSEC and lasts for a duration of DURSEC time units. Take X = SHADOW (BEGSEC, DURSEC). X is the set of all time spans pertaining to potential or actual events within the given section of the piece. Let BRASS be the family of time spans which articulate events played by brass instruments. (Those spans might be the green dots on our "painting.") Take Y = SHADOW(BRASS). A time span belongs to the set Y if and only if an actual or potential event to which it pertains happens or would happen completely during some brass event, e.g. a sustained brass note or chord. Take

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f to be transposition-by-the-interval-(4,2) in our non-commutative GIS. Thus the span f (a, x) begins 4 x-lengths after a, and lasts for a duration of 2x time units. We may then ask for the value of orange INJ(X, Y) (f). When we do so we are asking the following question. How many string events pitched above middle C are there within our given section of the piece such that if you start at the event, counting "ONE," and then beat three more event-durations, counting "-two-three-four," and then beat two more event-durations yet, counting "ONE-two," your new count of "ONE-two" will measure a span of time lying completely within the extent of some sustained brass note or chord?

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Networks (1): Intervals and Transpositions

We saw in Theorem 6.7.1 how INJ generalizes IFUNC when there is a GIS at hand: IFUNC(X, Y) (i) = INJ(X, YXT,). This relationship enables us not only to generalize IFUNC, but also to replace entirely the concept of intervalin-a-GIS by the concept of transposition-operation-on-a-space. Instead of thinking: "i is the intervallic distance from s to t," we can think: "Tj is the unique transposition operation on this space that maps s into t." We can even shift our attention, if we wish, from the atomic "points" s and t to the oneelement "Gestalts" X and Y, X being the set that contains the unique member s and Y being the set that contains the unique member t; then there is a unique member T; of the transposition-group satisfying INJ(X, Y) (Tj) > 0; the label i for this unique transformation is i = int(s, t). 7.1.1 By such thinking, we can replace the idea of GIS structure by the idea of a space S together with a special sort of operation-group on S. This special sort of group is what mathematicians call simply transitive on S. The group STRANS of operations on S is simply transitive when the following condition is satisfied: Given any elements s and t of S, then there exists a unique member OP of STRANS such that OP(s) = t. Given any s and any t in the space of a GIS, then there is a unique transposition-operation T satisfying T(s) = t, namely T = Tint(M). So the group of transpositions is simply transitive on the space of any GIS. Conversely, the following theorem is also true: Let S be a family of objects and let STRANS be a simply transitive group of operations on S; then there exists a GIS having S for its space and STRANS for its group of transpositions. We shall now prove that theorem. Given S and STRANS as described, let IVLS be an "index family," that is a family of elements i, j,... which can be put into 1-to-l correspondence with the family STRANS. We shall write "OP;"

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for that operation within STRANS that corresponds to the member i of the index family IVLS. Now we turn IVLS into a group by defining the binary combination ij = k in IVLS when (OPj)(OPj) = OPk in STRANS. The group IVLS, by this construction, is anti-isomorphic to the group STRANS. We have a space S and a group IVLS; next we define a function int from S x S into IVLS. Given r and s in the space S, we take int(r, s) to be that unique member i of IVLS such that OPj(r) = s. i is unique because STRANS is simply transitive. Holding onto r, s, and i above, suppose that int(s, t) = j in the same sense. That means OPj(s) = t. Then (OPj)(OPi)(r) = OPj(s) = t. By the group structure defined for IVLS, (OPj)(OP,) is OP(ij). Hence OP^r) = t. Then, by the construction of the function int, int(r, t) = ij. Thus int(r, t) = int(r, s) int(s, t); Condition (A) of 2.3.1 is satisfied. Now we shall show that Condition (B) of that definition is also satisfied, so that (S, IVLS, int) is a GIS. Given s and i, set t = OPj(s). By definition of int, int(s,t) = i. Iff is any member of S satisfying int(s, t') = i, then OPj(s) = t'. But OPj(s) is precisely t. So in this case t' = t. We have shown that, given s and i, there is a unique t satisfying int(s, t) = i. So we have shown that Condition 2.3.1(B) is satisfied. Thus (S, IVLS, int) is a GIS. And STRANS is the group of transpositions for this GIS; indeed Tj = OPi for every i in IVLS. To see this, we recall that T;(s), in any GIS, is the unique member of S that lies the interval i from the element s. Now in this particular GIS, the member of S that lies the interval i from s is OPj(s). Hence in this GIS Tj(s) = OP^s). That being the case for every sample s, T; = OPj as an operation on S. q.e.d.

158

7.1.2 By virtue of 7.1.1, all the work we have done with GIS structures since chapter 2 can be regarded as a special branch of transformational theory, namely that branch in which we study a space S and a simply transitive group STRANS of operations on S. From a strictly mathematical point of view, this would have been a more elegant way to develop the material of chapters 2 through 5. We could even have deferred the study of GIS structure until much later, after a more general exploration of transformations on musical spaces using CANON, INJ, and other such constructions. Yet there are also advantages to the order of presentation we have adopted in this book. By starting with a study of GIS structure, we have built a link between the historically central concept of "interval" and our present transformational machinery. To some extent for cultural-historical reasons, it is easier for us to hear "intervals" between individual objects than to hear transpositional relations between them; we are more used to conceiving transpositions as affecting Gestalts built up from individual objects. As this way of talking suggests, we are very much under the influence of Cartesian thinking in such matters. We tend to conceive the primary objects in our

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7.1.2

musical spaces as atomic individual "elements" rather than contextually articulated phenomena like sets, melodic series, and the like. And we tend to imagine ourselves in the position of observers when we theorize about musical space; the space is "out there," away from our dancing bodies or singing voices. "The interval from s to t" is thereby conceived as modeling a relation of extension, observed in that space external to ourselves; we "see" it out there just as we see distances between holes in a flute, or points along a stretched string. The reader may recall our touching on these matters in 2.1.5, where we pointed out how the historical development of harmonic theory has depended on such a projection of our intuitions into a geometric space outside our bodies, that is, the "line" of the stretched string, a space to which we can relate as detached observers. In contrast, the transformational attitude is much less Cartesian. Given locations s and t in our space, this attitude does not ask for some observed measure of extension between reified "points"; rather it asks: "If I am at s and wish to get to t, what characteristic gesture (e.g. member of STRANS) should I perform in order to arrive there?" The question generalizes in several important respects: "If I want to change Gestalt 1 into Gestalt 2 (as regards content, or location, or anything else), what sorts of admissible transformations in my space (members of STRANS or otherwise) will do the best job?" Perhaps none will work completely, but "if only ...," etc. This attitude is by and large the attitude of someone inside the music, as idealized dancer and/or singer. No external observer (analyst, listener) is needed. In 7.1.1 above we sketched a mathematical dichotomy between intervals in a GIS and transposition-operations on a space: Either can be generated formally from the characteristic properties of the other. More significant than this dichotomy, I believe, is the generalizing power of the transformational attitude: It enables us to subsume the theory of GIS structure, along with the theory of simply transitive groups, into a broader theory of transformations. This enables us to consider intervals-between-things and transpositionalrelations-between-Gestalts not as alternatives, but as the same phenomenon manifested in different ways. Consider figure 7.1 in this connection.

FIGURE 7.1

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The figure shows how the melodic motif of the "falling minor ninth (sixteenth)" is developed over Schoenberg's piano piece op. 19, no. 6, out from the intervallic structure of the opening chord in the right hand, a chord we have earlier called "rh." Figure 7.1 (a) displays rh, along with the three falling pitch-intervals that can be heard within it; they are notated on the figure as — 5, — 9, and —14 semitones. Figure 7.1(b) shows how the same network of intervals governs the scheme by which the falling-ninth motif is transposed over the course of the piece. This is interval-language. Alternatively, we could use transposition-language to put the matter as follows: The three transposition-operations T_ 5 , T_ 9 , and T_ 14 , which move the falling-ninth motif forwards in time over figure 7.1(b), are exactly those members of STRANS which move the individual pitches of rh downwards in space, as shown on figure 7.1 (a). "Forwards in time" and "downwards in space" are phenomena that work together in many ways over the course of the piece.1 But we do not have to choose either interval-language or transpositionlanguage; the generalizing power of transformational theory enables us to consider them as two aspects of one phenomenon, manifest in two different aspects of this musical composition: Intervals structure the referential sonority rh as an Unterklang; transpositions make the falling-ninth motif move forward through the piece. This, I think, is the sense in which we accept the symbol " — 5" on figure 7.1 (a) and the symbol " — 5" on figure 7.1(b) a legitimately the same, finding the usage suggestive rather than confusing. The two symbols are pointing at the same phenomenon, not at different phenomena. Before we leave figure 7.1, let us note that the chord of (a) and the various falling ninths of (b) have only one common pitch-class. This emphasizes that in comparing (a) to (b), we are talking about intervals and transpositions, not about pitches and pitch-elaboration (diminution). If figure 7.1 (b) were to have appeared five semitones lower, then one could argue that the basic phenomenon involved was that of the pitches B, F#, and A in the chord, elaborated to become BBt>, F#F, and AG# in the new figure 7.1(b). But this is not the case; the phenomenon under discussion involves intervals and transpositions, not pitches or pitch-classes and their structural ornamentation. The remainder of chapter 7 comprises further examples demonstrating various interrelationships of intervallic structures with transpositional progressions in the manner of figure 7.1(a)-(b), over a variety of musical styles. All these examples will involve intervals among pitches or pitch-classes. The distinction between pitch or pitch-class elaboration and intervallic/ transpositional interrelationship will not always be so clear, through the examples, as it was in figure 7.1. We shall concentrate mainly upon the

160

1. This aspect of the music is brought out very well in the analysis by Robert Cogan and Pozzi Escot, Sonic Design (Englewood Cliffs, N. J.: Prentice-Hall, 1976), 50-59.

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7.2

intervallic/transpositional interrelationships, for those are our current focus of study. But we shall also discuss from time to time how those interrelationships interact with pitch or pitch-class elaborations.

FIGURE 7.2 7.2 EXAMPLE: Figure 7.2 shows the beginning of the Zauber motive from Wagner's Parsifal, so far as pitch classes are concerned. The motive appears with a variety of rhythms in the music drama. I shall call the serial network of figure 7.2 "Zauber" or "Z" for present purposes. Forms of the motive do not appear in the foreground of the music until Kundry's first-act entrance (ride); figure 7.2 shows Z as its pitch classes appear during the kiss in the middle of act 2. Figure 7.3 sketches melody and harmony for the phrase that introduces the Motive of Faith in the Prelude to act 1. This is long before Z has appeared in the foreground. Yet the intervallic structure of the Z motive governs the plan of modulations for the phrase, as the arrows on the figure show us.

FIGURE 7.3

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In figure 7.3 one hears not only the intervals of modulation but also the specific pitch classes Ab-Cb-Ebb~Eb being tonicized; these are the pitch classes for Z which were displayed in figure 7.2 above. Of course we hear the music of figure 7.3 long before the music of figure 7.2 (second-act kiss). On the other hand, we do hear the local keys of figure 7.3 elaborating a pitch-class variation on a structure related to Z, namely the Liebesmahl motive that opened the opera: Ab-C-Eb-F-(etc). In this context, hearing the successive tonics Ab-Cb-Ebb-Eb of figure 7.3 as a variation on the Liebesmahl helps us hear Zauber itself, when it appears in the foreground later on, as a variant of the Liebesmahl.

762

Figure 7.4 shows the succession of local tonics during the transformation music of the first act. The principal local tonics are represented by open noteheads; measure numbers above them indicate where the cadential tonicizations discharge. Usually these discharges coincide with entries of important motives; the names of the motives appear on the figure preceding the measure numbers: BELL, GRAIL, AG = Agony, and LM = Liebesmahl. Some filledin noteheads also appear on the figure; these represent local tonics which are subordinated to the principal tonics in various ways. The B tonic at measure 1084 is a structural dominant preparation for the E tonic at measure 1092. The B tonic takes the Grail Motive, not the Bell Motive like other tonics from measure 1074 to measure 1106. The C tonic at measure 1115 is supported by no special motivic entry; it is a structural neighbor to the Db = C# tonics that surround it by open noteheads on the figure. After the neighboring event and the enharmonic shift, the predominant motive on the figure changes, from BELL to AG (measure 1123 and following). Then, after measure 1140, AG disappears and BELL returns. The cadence at measure 1140 is deceptive. The

Transformation Graphs and Networks (1)

7.2

implied tonic is A|>, which appears as an open notehead on the figure; the substitute root that actually carries the Liebesmahl entry is F(?. The situation at measure 1148 is similar, in fact sequential. Up to measure 1140 the open noteheads are organized by serial forms of the Zauber motive. These forms are beamed on figure 7.4 and labeled as Z t , Z 2 , Z3, and Z4, meaning the first Z-form, the second Z-form, and so on.2 Each Z-form is a retrograde inversion of the Z-form that precedes it, specifically that RI form which uses for its opening two notes the final two notes of the preceding form. Thus Z2 begins E(?-E-, taking its point of departure from the end of Z^-Eb-E. Likewise Z3 begins G-B|?-, taking its point of departure from the end of Z2,-G-B|?. It does not much matter whether we call figure 7.4 up to measure 1140 a pattern of "intervals" among tonics, or of "transpositions" among tonics, or even of "modulations" among keys: We are talking about the motivic exfoliation of one phenomenon in various ways. "Modulations" is only awkward if we try to associate the term with Schenkerian notions of pitch or pitch-class prolongation. That is clearly not happening here (up to measure 1140), and since Schenker himself rejected the word for his discourse, I feel free to use it in this non-Schenkerian connection. Indeed I prefer it (for that reason) to the word "tonicizations." Continuing along figure 7.4, we note that the open noteheads followin measure 1140 build up to one beamed form of the BELL motive. It is very significant that this structural Bell Motive appears "in A|j." The deceptive cadence at measure 1140, featuring an entry of the Liebesmahl in the trombones, forcefully recalls the same event right at the curtain-rise of act 1. In this connection, pitch-class relations clearly are important. We are to understand the Ab of measure 1140 as the tonic of the opera; we are to understand the structural Bell Motive of measures 1140-50, beamed on figure 7.4, as a prolongation of that tonic; and we are thereby to understand the mammoth local tonic C of measures 1150-62ff. as the third degree of that Ab. It is just this section of act 1 that enables us (with the help of the Parsifal bells) to be sure that the C tonic which ends the act is indeed a functional III of A(? (inter alia). To support this hearing, we can note that the four pitch-classes of the Bell Motive in Ab, portrayed by the beamed noteheads of measures 1140-50 on figure 7.4, are in fact a serial permutation of the four pitch-classes that begin the Liebesmahl (and the opera): Ab-C-Eb-F. The pitch-class relation is very striking, yet Wagner apparently goes to great lengths to conceal it. The deceptive cadences at measure 1140 and measure 1148 help to do so; so does the absence of the Bell Motive in Ab from the foreground of the music here (and so far as I can recall anywhere else in the music). Whatever private games 2. I am grateful to Thomas Christensen for calling Zj and Z3 to my attention.

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Wagner may have been playing, it is safer for us as analysts to treat the permutational relation between the Bell Motive and the Liebesmahl as intervallic, rather than pitch-class prolongational. Figure 7.4 enables us to explore another motivic relationship involving BELL. This motive has the same organizing function after measure 1140 that Zauber had before measure 1140, that is the function symbolized by grouping open noteheads on the figure with beams. The compositional relationship invites us to explore the intervals to which those beams give rise in each case, and indeed we can hear an interesting intervallic relationship now that we are primed to listen for it.

FIGURE 7.5 Figure 7.5 shows how we can hear BELL as an overall progression of - 3, elaborated by two subprogressions of 7. Compare this network of intervals to that of figure 7.2, which displayed Zauber as an overall progesssion of 7, elaborated by two subprogressions of 3 (inter alia). Let us now return our attention to the network of Z-forms displayed by figure 7.4 up to measure 1140. We shall have much more to say later about the serial technique of Rl-chaining that links successive Z-forms. When this technique is applied to any given serial motive over and over, alternate forms of the motive-chain will be transposed forms of each other, the interval of transposition depending upon the serial structure of the motive. In this specific case, Z3 = T 10 (Z 1 )andZ 4 = T10(Z2). The repeated Rl-chaining thus gives rise to "structural sequencing" on figure 7.4: Measures 1096-1140 (with A[?) "sequence" measures 1074-1100 structurally. (The musical foregrounds of the two passages are not related sequentially.) The sequencing-interval of 10 in this particular case is a dispersive interval for Z as an unordered set: INJ(Z, Z)(T10) = 0; T10(Z) has no common notes with Z. The dispersive function is clear on figure 7.4, where the sequencing forms of Z fill up chromatic pitch-class space very diligently. Indeed, the open noteheads of the figure up through the A|? of measure 1140 constitute a non-repeating ten-note series. (F# and B are missing. It is amusing, if far-fetched, to imagine them as representing the absent Klingsor.) 164

7.3 EXAMPLE: Figure 7.6 graphs an intervallic/transpositional structure which we shall call "the nuclear gesture with pickup." The nuclear gesture

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7.3

FIGURE 7.6

comprises the pitch-class interval (transposition) 4, subarticulated into two intervals (transpositions) of 2. The pickup gesture consists in approaching something by pitch-class interval/transposition 5. Figure 7.7 shows through a variety of examples how the nuclear gesture dominates intervallic/transpositional configurations over the last movement of Brahms's Horn Trio op. 40. It may seem awkward to be using pitch-class intervals in this context, since the piece is so highly structured registrally and is so diatonic. Our reasons will become clear by the end of this discussion. For other purposes it would make more sense to use pitch intervals in semitones up, or diatonic intervals in scale steps up, or degree intervals in scale steps up modulo 7. (a) of the figure displays the basic motive of the movement. The nuclear gesture with pickup governs the first half of the motive; the second half is governed by what we shall call the "complementary gesture," which articulates 8 into 10 + 10. (b) of the figure sketches the bass line for the opening phrase. We hear how it is governed as a whole by the nuclear gesture. Now each stage of that gesture is inflected by its own pickup. The octave leap on B|?, which appears in parentheses at the end of (b), is an important motive of the piece. Here it interrelates with the octave B[? that delimits the ambitus of motive (a). (c) of the figure shows the melody beginning to descend from its first climax. The complementary gesture governs the melodic structure as a whole; each stage of that gesture is inflected by a pickup. The pickup interval of 5 does not get complemented; it always remains 5, never becoming 7. (d) shows how the network of (c) gets tightened rhythmically at the approach to the last recurrence of motive (a) within the first group. Over (a) through (d), the intervallic/transpositional interrelationships can be analyzed as fallout from the pitch and pitch-class motives. The nuclear and complementary gestures are always applied to Ejj-F-G and G-F-Efr. functioning as degrees 1-2-3 and 3-2-1 in E(? major. The pitch and pitchclass motives, with their degree functions, can claim priority here: In (a)-(b)(c)-(d) we can deduce the intervallic relations of interest from stronger pitch and pitch-class relations, but we cannot run such deductions in the other direction. So a traditional Schenkerian approach to (a)-(b)-(c)-(d) reveals more than does our intervallic/transpositional approach. The same can still be said of (e), which shows the melody at the opening of

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FIGURE 7.7 166

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FIGURE 7.7 (continued)

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an episode that begins toward the end of the second group and eventually leads to the closing theme. The principal pitch-classes that carry the structure of our (e)-network are still G and F, even though they are now degrees 6 and 5 in Bk (The whole passage takes place over a dominant pedal in that key.) Intervallically and transpositionally, (e) shows how transposition-by-10, which governs the sequence, exfoliates from the interval-of-10 between G and F; this interval of 10 develops the "subinterval of 10" from the complementary gesture, blowing it up rhythmically. Each stage of the 10-gesture is inflected by a pickup. In passages (f) through (h), the intervallic/transpositional networks take on a life of their own. They become autonomous structures; no longer subordinated to concomitant local pitch or pitch-class events, they rather interact with such events or perhaps even determine them, (f) shows an extended sequential passage from the development section. The model for the sequence is sketched on (f) over measures' 103-05 "etc." Over measures 104-05, the pedal note B of the ostinato figure combines with the opening two notes of the legato theme that follows, projecting the complementary gesture. (At "etc." the legato theme stops moving stepwise.) The pedal pitch-class B is inflected by a pickup. To hear the gestures here, one must listen to pitch classes, not pitches, (f) continues on, showing how the sequence from measure 103 etc, through measure 113 etc., to measure 123 etc., projects the nuclear gesture. The mobile harmony of (f) achieves as its goal a reattained dominant of El> major. Thereupon the horn launches an unusually extended solo passage, (g) shows first how the opening eight measures of the solo, measures 137-44, are structured by the complementary gesture with pickup. By projecting that gesture at this pitch-class level, Brahms finally gets his "octave-Bt" idea to interact with the system of nuclear and complementary gestures. (The octaveBl> idea stood apart from those gestures in passages (a) and (b) above.) As a result of this interaction Brahms generates a Gl> at the end of the complementary gesture, a G!> standing in an 8-relation to the initial Bl> of that gesture. We are aware of the 8-relation from B\> to Gl> as it carries over into the music of measures 145-47. And that gives us a clue as to what is going on over measures 145-49 more broadly. Another clue is furnished by rhythmic augmentation: The pickup motive of measure 137 etc. is augmented rhythmically to measure 145 etc., and the complementary-gesture motive of measure 141 is blown up rhythmically to measure 149 etc. These clues explain the new intervallic/transpositional gesture graphed in (g) over measures 145-49 (with pickup): An overall interval of 4 is subarticulated into 8 + 8. The subinterval 8 of the new gesture identifies with the overall interval 8 of the complementary gesture—both span Bl>-to-GI> during passage (g). And, just as rhythmic value are multiplied by 2 in passing from measures 137-44 to measures 145-56, so intervallic values are also multiplied by 2. Instead of the complementary gesture 8 = 10 + 10, we now have an expanded gesture involving twice those

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7.4

numbers: 2 - 8 = 2-10 + 2-10or, modulo 12,4 = 8 + 8. The new gesture will therefore be called "the complementary gesture times 2." Our ears follow the intervallic idea "times 2" here because of the rhythmic augmentation. Each stage of the new gesture takes its own pickup. (h) shows how the new gesture was already presented in the first movement, derived in a musical manner that was almost the equivalent of an algebraic demonstration. While the horn melody in the treble clef chains 10intervals into suspension-patterns sequencing through 8-intervals, the concomitant bass line indicates how those 8-intervals in turn chain up to form the complementary-gesture-times-2. Each stage of the times-2 gesture, in the bass of (h), takes its own pickup; we must use the fundamental bass in this connection for the first stage. I have chosen for example (h) the reprise-form of this passage rather than its form in the exposition. That is because the pitch classes in the bass line of (h) as given make it easy for the reader to hear the relation of (h) to measures 145-49 in (g).

FIGURE 7.8 7.4 EXAMPLE: Figure 7.8(a) sketches the opening of the Minuet from Beethoven's First Symphony. The C pedal bass is omitted from the first complete measure, signifying that we are to understand an F root and a G root for the two harmonies of that measure in the present context. This is an oldfashioned way of hearing, especially at the usual tempo for the piece, but we shall find that the old-fashioned hearing is of interest here. Those who have trouble hearing an F root, given the C bass, will be helped to do so by summoning up the memory of the slow movement. That movement, the last music we have heard before figure 7.8(a) begins, has just ended in F major. Figure 7.8(b) aligns beneath (a) a sketch for the opening of the entire

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symphony. In the deceptive cadence of measure 2, a C harmony is understood as interpolated between the G7 and the A-minor harmonies. This too is an oldfashioned but consequential hearing in the context. The progression of G to (interpolated) C is bracketed. On (a) above, the repeated progression of G to C harmony is also bracketed. Ignoring the repetition of the bracketed harmonic progression in (a), counting the theoretically interpolated C harmony in (b), and considering the first complete measure of (a) to contain F and G harmonies, we can then refer both passages to a common network governing the progression of roots, a network suggested by figure 7.9.

FIGURE 7.9

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The intervals on the arrows are familiar ratios from fundamental bass theory. They are expressed here as fractions between 1/2 and 1, to capture the "falling" sense of the root progressions. The representation of chord-changes by "intervals" between roots is not quite adequate. We shall improve the model in chapter 8; meanwhile the somewhat inadequate representation will serve. The root-intervals have an impressive tradition behind them, exemplifying the desire of earlier theorists to subsume intervals and transformations (here harmonic changes) into one general phenomenon (the fundamental bass). Traditional criticism would describe the relation between the Adagio and the Minuet by saying that the latter parodies the former, as a passage in a satyr-play may parody a dramatic theme treated majestically in an earlier drama of a tetralogy. Our "common network" of figure 7.9 enables us to propose a different relation between the passages, one that does not infer so much structural priority for the Adagio from its temporal priority in the composition. Namely, we can conceive both (a) and (b) of figure 7.8 as different realizations, in different environments, of one underlying abstract gesture, a gesture symbolized by figure 7.9. The mere possibility of this shift in

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7.4

critical stance is important, no matter which stance one prefers in viewing the particular art-work at hand. Consider this analogy. When one first encounters Mr. X, one sees him in formal attire, discharging an important and solemn public professional duty. When one next encounters him, he is surrounded by screaming children and friends in a park, his mouth filled with potato chips, rushing down a hill carrying a football in one hand and an open can of beer in the other. Here we would surely be reluctant to adopt a view analogous to that of the traditional criticism, claiming that the first X is a norm for X-ness and the second X is a parody of the first. Rather, our attitude would be analogous to the alternative view. We might notice in X's body, features, movements, voice, and the like certain things common to both occasions; from that we would infer a certain abstract "X-ness," and we would say that this X-ness was being manifested on both occasions, albeit differently in different environments. A dramatist or novelist might first introduce us to X either at the public occasion or in the park. One could also imagine an open-form play or novel, in which the author allowed either scene to precede the other at the choice of the performers or the reader. Traditional criticism would attack this idea on the grounds that we cannot separate our concept of X-ness from the particular way in which we have built up that concept through time. And so on; having suggested the philosophical and aesthetic issues our investigations engage, I shall not pursue the matter farther here. I do, nevertheless, want to work out some specific critical consequences that emerge when we take figure 7.9 as a norm for both figure 7.8(a) and figure 7.8(b). First, let us analyze the "tail" on our normative figure 7.9, that goes from the C harmony to the A-minor harmony, as a secondary feature of the structure. (The minor harmony is a "secondary triad.") Imagine the node containing the A-minor harmony, then, as erased from figure 7.9, along with the arrow labeled "5/6." It is then possible to analyze the rest of the figure as a succession of three V-I cadences moving along the circle of fifths. This interpretation of the graph is projected by the music of figure 7.8(b), the opening Adagio. There we hear a motive, the motive repeated with harmonic variation, and the motive repeated again with rhythmic variation. The local tonics for the motive-forms move along nodes of figure 7.9 located at successive arrowheads, first F, then C, then G. (We have dropped the tail from the figure.) This music, in sum, projects a "progressive" reading of figure 7.9-as-norm. The opening of the Minuet, in contrast, projects a different way of reading figure 7.9-as-norm. Here the figure is interpreted as manifesting a sense of balance in its cadence structure, not a progressive chaining of tonics. To explore the cadential sense of balance, let us construct an intervallic graph called CADENCE, shown in figure 7.10. Now let us apply CADENCE to the analysis of figure 7.9-as-norm,

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FIGURE 7.10 always ignoring the tail. The leftmost side of figure 7.9 exhibits CADENCE formatting the subnetwork C-F; G-C. The rightmost side of figure 7.9 exhibits CADENCE formatting the subnetwork G-C; D-G. This suggests hearing the entire normative network of figure 7.9 (without tail) as comprising two CADENCES, the right side of the first CADENCE being elided into the left side of the second CADENCE. According to this reading, figure 7.9 without tail is to be understood as a contraction of figure 7.11.

FIGURE7.il

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This interpretation of figure 7.9 illuminates the repeated G-C progression within the Minuet theme, marked by the brackets on figure 7.8(a): That projects the normative repeated G-C progression which we see on figure 7.11. The pitch classes that fill the nodes of figure 7.11 are the roots of various harmonies. Some harmonies are (local) tonics; others are dominants. The tonic pitch classes fill nodes that have arrowheads pointing to them. Figure 7.12 isolates the tonics of figure 7.11 and organizes them in a network of their own. This network in fact fills the nodes of the CADENCE graph (figure 7.10) yet again. The relation is hard to see because the diagonal arrows of figure 7.10 go to the right while those of figure 7.12 go to the left. But the CADENCE

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7.4

FIGURE 7.12

graph, as a configuration of nodes and labeled arrows, knows no "right" and "left." We are using those visual distinctions here to indicate musical chronology, not graph-structure as such. Musical chronology is naturally crucial. We may hear a dominant precede its tonic, symbolized e.g. by the rightpointing diagonal arrows on figure 7.10; we may hear a dominant follow and inflect its tonic, symbolized e.g. by the left-pointing arrows on figure 7.12. But in this particular system of root-relations, the normative graph itself displays only a dominant node and a tonic node connected by a labeled arrow; chronological distinctions in this system function not as norms but as different interpretations of the normative graphs. That is a feature of classical fundamental-bass theory, not of our graphic structures per se. We have many systems of rhythmic "intervals" at hand within which we could construct analogous graphs that would enforce musical chronology. The opening of the Minuet interprets the basic norm of figure 7.9 in the manner of figures 7.11 and 7.12. The music specifically reads the normative network as a backwards-laid-out CADENCE of local tonics (figure 7.12), diminuted into two balanced fowards-laid-out CADENCEs of roots (figure 7.11). We have already noted how the repeated G-C progression within the Minuet theme helps to bring out this balanced structure. The idea of balance is also projected melodically to some extent: The first CADENCE of roots harmonizes the tetrachord G-A-B-C in the principal melodic line; the second CADENCE harmonizes the answering similar tetrachord D-E-F#-G. Of course the irrepressible thrust of the rising scale works against this feeling of balance, along with other features of the passage. It is curious that figure 7.8(a), so ebullient in its texture, should project a reading of our underlying harmonic norm as balanced, while figure 7.8(b), so stately and measured in its texture, should project a reading of the same norm as progressive. Figure 7.13 continues from the end of figure 7.8, asserting a structural correspondence between the next four measures of the Minuet, arriving at a structural dominant, and the continuation of the first movement all the way up to the dominant which prepares the first big tonic tutti in the Allegro. The correspondences displayed here are mostly melodic, serial, and motivic, involving particularly the thematic leading-tone-to-tonic idea. Fundamental-bass

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FIGURE 7.13

structure is not involved to the extent it was in connection with figure 7.8. Figure 7.13 is really not our business here, but I think the reader will be glad to hear it anyway. One would not easily conceive listening for its correspondences without having first noted the correspondences of figure 7.8. Then too, to the extent one assents to the analytic implications of figure 7.13, that confirms the propriety of figure 7.8 and our work that issued therefrom. And of course figure 7.13 is interesting in its own right. It owes an enormous debt to the theoretical ideas of Schenker. 7.5 The interested reader will find in an article by John Peel a sensitive and analytically revealing use of small transposition-graphs and networks for discussing a passage from Schoenberg's String Trio.3 John Rahn has published a discussion of the theme from the second movement of Webern's Symphony op. 21 that bears very suggestively on ideas we have been considering throughout this chapter.4 Casting his discussion in the form of an ear-training exercise, Rahn directs students' attention to networks of tritones and networks of semitones; he also directs their attention to concomitant rhythmic structures. One could perhaps construct an appropriate GIS for his discourse, a GIS involving pitch and rhythm together in a direct-product system. Aspects of his analysis might then be recast and extended, to involve networks of direct-product intervals. 3. John Peel, "On Some Celebrated Measures of the Schoenberg String Trio," Perspectives of New Music vol. 14, no. 2 and vol. 15, no. 1 (Spring-Summer/Fall-Winter 1976), 260-79. 4. John Rahn, Basic Atonal Theory (New York: Longman, 1980), 4-17.

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Transformations

Figure 7.9, the normative fundamental-bass network we constructed for the Beethoven symphony, exhibited a feature we discussed early in chapter 7. On the one hand, we could conceive of its arrows as signifying intervals between individual elements, in this case roots. On the other hand, we could also conceive of the arrows as signifying transformational relations between Gestalts, in this case chords or harmonies, or even potential keys. Resolving the ambiguity, we could also conceive of the arrows as denoting some tertium quid, a phenomenon whose manifestations include both harmonic intervals between pitch classes and transformational relations between chords. To conceive such a tertium quid, in the form of the fundamental bass and its progression, was Rameau's supreme inspiration. And some of the problems to which that conception gives rise are neatly pinpointed by figure 7.9 as well. That is particularly the case as regards its A-minor "tail," the tail we conveniently docked before starting our earlier discussion. The interval of 5/6, which labeled that tail, does inform us correctly that the root A, as a pitch class in just intonation, is 5/6 of the root C (modulo the pertinent congruence relation). But the numerical ratio does not tell us that the A harmony is minor rather than major. If we transpose a C-major chord by the interval 5/6 we obtain an A-major chord, not an A-minor chord. Thus, when we pass from the C-node to the A-node on figure 7.9, we are really applying some transformation other than harmonic-transposition-by-(5/6), some transformation which is more than a synonym or isomorphic image for that interval. 8.1.1 We can solve this problem very elegantly by adopting and modifying some ideas from the function theories of Hugo Riemann. Our space will consist not of pitch classes but of objects we shall call "Klangs" Each Klang

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is an ordered pair (p, sign), where p is a pitch class and sign takes on the values + and — for major and minor respectively. The Klang models a harmonic object with p as root or tonic, an object whose modality is determined by the sign. We can transpose a Klang by transposing its pitch class while preserving its sign; thus (C, +) transposed by 5/6 is (A, +). Rather than writing T(C, +) = (A, +) here, we shall write the symbol for the transformation to the right of its argument: (C, +)T = (A, +). The reader will recall our having discussed such "right orthography" a long time ago, in section 1.2.4. Right orthography will conform much better than left orthography to our intuitions in the contexts we shall be exploring just here. The one special thing we have to watch is that the order of composing transformations is reversed under right orthography: If f and g are transformations, then (Klang)fg = ((Klang)f)g denotes Klang-transformed-by-f, all transformed by g; this was denoted by "gf(Klang)" in left orthography. We define the operation DOM on Klangs: DOM is transposition by the inverse of the dominant interval. Thus (p,sign)DOM = (q, sign), where q is that pitch class of which p is the dominant. We can read this equation as telling us that (p, sign) becomes the dominant 0/(q, sign). On a graph we could have a (C, +) node, an (F, +) node, and an arrow labeled DOM from the first to the second. Right orthography conforms nicely to the visual layout of the graph here: Being at (C, +) and following an arrow labeled DOM we arrive at (F, +); that is, (C, +)DOM = (F, +). In a similar spirit we define the operation MED: (p, sign) becomes the mediant of its MED-transform. For example, (C, + )MED = (A, —), and (C, —)MED = (Ab, +). If we are at (C, + ) on a graph and follow an arrow labeled MED, we arrive at (C, + )MED = (A, -). Now we can rewrite the network of figure 7.9 as a network of Klangtransformations, rather than fundamental-bass intervals. Figure 8.1 is the result. The transformations DOM and MED drive the network of figure 8.1 in a

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FIGURE 8.1

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8.1.1

natural musical way. In general, DOM and MED will always drive similar Klang-networks in the same way. That would happen even where an instance of (F, +) occurred in some music followed by an instance of a dominantrelated (C, +). We could use the visual layout of an analytic network to reflect the musical chronology, having a DOM arrow pointing to the left from a (C, 4-) node to an (F, +) node. In that case we would be saying: "(C, +) refers (back), as dominant, to (the preceding) (F, +)." The DOM arrow, here as before, makes a dependent Klang point at the local tonic Klang to which it refers. For the configuration of nodes and arrows as a configuration, it is immaterial whether the DOM arrows point right or left (or up or down). Our unusual definition of DOM is what makes the graphs move naturally in this way. The orientation of all the DOM arrows would be reversed if we had chosen the more usual alternative idea, in defining a "dominant" Klang-transformation. The usual idea is represented by the transformation DOM', which takes a Klang and transforms it into its own dominant. E.g. (F, +)DOM' = (C, +): "Being at (F, + ), take its dominant and obtain (C, 4-)." Observe how poorly figure 8.1 would fit our kinetic intuitions about the music under study if we reversed all the DOM and MED arrows, using DOM' and the analogous MED' instead. This elucidates one problem with the conceptual structure of Riemann's function theories. His dominants, other than secondary dominants, do not point to their tonics via implicit DOM arrows. Rather the tonics point to their dominants, generating them by implicit DOM' arrows. Then the dominants just sit around, not going anywhere. This conceptual flaw in Riemann's approach makes his musical analyses subject to inertia and lifelessness, seldom doing justice to the power and originality of his theoretical ideas. An even more basic problem for Riemann was that he never quite worked through in his own mind the transformational character of his theories. He did not quite ever realize that he was conceiving "dominant" (whether DOM or DOM') as something one does to a Klang, to obtain another Klang. Here, I conjecture, he was unduly influenced by a desire to promote his notation as a substitute for Roman-Numeral notation; I think it was this desire that led him to conceive "dominant" and the like as labels for Klangs in a key, rather than as labels for transformations that generate Klangs from a local tonic (along the lines of DOM'), or that urge the Klangs of a key towards their tonics (along the lines of DOM). We may continue to explore other transformations on the family of Klangs, following or modifying Riemann. We can define SUBD, the formal inverse of DOM. "(F, +)SUBD = (C, +)" means that F major becomes the subdominant of C major. Even though SUBD = DOM"1 in a group of operations, the arrows on the graphic format enable us to distinguish a SUBD arrow read forwards from a DOM arrow read backwards. We also have left and right directionality at our disposal in this connection, to represent musical chronology. Thus, analyzing a plagal cadence in C major, we would draw a

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SUBD arrow pointing from (F, +) on the left to (C, +) on the right. Analyzing a tonic-dominant progression in F major, say an opening phrase terminating with a half-cadence, we would still put (F, + ) on the left and (C, -1-) on the right, but now we would draw a DOM arrow pointing leftwards, from (C, +) to(F, +). In similar vein, we can define and explore SUBM, the formal inverse of MED. We can consider other sorts of transformations too. For example, we can define REL, the operation that takes any Klang into its relative minor/ major. (C, +)REL = (A, -); (C, -)REL = (Eb, +). REL is not the same operation as MED or SUBM: (C, -)REL = (Eb, +) but (C, -)MED = (Ab, +); (C, +)REL = (A, -) but (C, +)SUBM = (E, -). We can also define PAR, the operation that takes any Klang into its parallel minor/major, (p, sign)PAR = (p, —sign). We can define Riemann's "leading-tone exchange" as an operation LT: (C, +)LT = (E, -); (E, -)LT = (C, +). We can also define more exotic operations on Klangs. For instance we can define an operation SLIDE that preserves the third of a triad while changing its mode: (F, +)SLIDE = (F#, -);(F#, -)SLIDE = (F, +). The SLIDE relations between (F, +) and (F#, —) can be heard in the last movement of Beethoven's Eighth Symphony, where the F-major theme that begins on the note A, the third of the triad, is transformed at measures 379-91 into F# minor, where it begins on the same A; the theme slides back into F major at measure 392. A SLIDE relation between (C, +) and (C#, —) can be heard over measures 103-10 in the slow movement of Schubert's posthumous Bb-Major Piano Sonata. Over those measures, thematic material which we expect to hear in C# minor is presented in C major instead. Using such transformations to label arrows, we can construct networks that could not be conceived using only intervals-and-transpositions. For example, figure 8.2 displays interesting relations between a "Tarnhelm network" (a) and a "Valhalla network" (b). The Tarnhelm network of (a) takes (B, +) as a tonic for the motive in its own context; it asserts structural relaxation on that Klang. This seems legitimate; besides, Wagner not infrequently interprets the motive in a larger context of B minor or B major, e.g. at the end of G otterdammerung I, or at the end of Tristan. (E, + ) on network (a) is bracketed to indicate that this Klang functions by implication only; (E, —) substitutes for it in the music. The Valhalla network, figure 8.2(b), asserts the indicated relations among the principal Klangs over the first presentation of the Valhalla theme in Das Rheingold, during measures 1-20 of scene 2. We shall be concerned with the "modulating" part of the network, the part that extends from measure 7 onwards. That is why the events of measures 1 -6 are in parentheses. Graphs (a) and (b) make visually clear a strong functional relationship between the Tarnhelm progression and the modulating portion of the Valhalla theme, a relationship which it is difficult to express in words.

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8.1.2

FIGURE 8.2

8.1.2 Obviously, we could not construct the graphs of figure 8.2 using intervals-cum-transpositions. The fact is obvious if by "intervals" we mean intervals between the notes of a fundamental bass. However, might we not be able to think of "intervals" here in some more extended formal sense? The matter bears theoretical examination, if only for the sake of review. The reader will recall how we showed in section 7.1.1 that the entire notion of a GIS can be developed formally from a given family S of objects and a given simply transitive group STRANS of operations on S. When a GIS is developed therefrom, STRANS becomes the group of transposition operations for that GIS. Let us take S to be a family comprising a given Klang and all other Klangs that can be derived from it via any chains of DOM, MED, SUBD, and SUBM transformations. Let STRANS be the group generated by the four cited operations. Since SUBD = DOM"1 and SUBM = MED"1, STRANS is generated by DOM and MED. The reader can now verify that DOM itself is generable by MED: DOM = (MED)(MED). That is, given any Klang, if it becomes the mediant Klang of a second Klang which in turn becomes the mediant Klang of a third Klang, then the first Klang is the dominant of the third Klang. E.g. ((C, +)MED)MED = (A, -)MED = (F, +) =

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(C, +)DOM;((C, -)MED)MED = (Ab, + )MED = (F, -) = (C, -)DOM. Since DOM = MED2, the group STRANS here is generated by MED alone: The operations of STRANS are precisely the powers of MED, including MED° = IDENT and MED~ n = (MED"1)" = SUBMn. Once we have this insight into the structure of our group STRANS here, it is not difficult to show that the group is in fact simply transitive on the defined family of Klangs, whatever the system of intonation we are using. Then we can regard the powers of MED as formal "intervals" on that family, in the sense of 7.1.1. We may then regard figure 8.1 as a formal "intervallic" network among Klangs, since it involves among its transformations only powers of MED. In this system, the Klang (C, —) is derived from (C, +) by 7 applications of MED: (C, +)MED7 = (A, -)MED6 = (F, +)MED5 = (Bb, +)MED3 = (Eb, +)MED = (C, —). Thus MED7, applied to a major Klang in some systems of intonation, will yield the parallel minor Klang. However, MED7 applied to a minor Klang will not yield the parallel major Klang, no matter what the system of intonation. E.g. (C, -)MED7 = (F, -)MED5 = (Bb, -)MED 3 = (Eb, -)MED = (Cb, +). Thus MED7 is not the same operation as PAR on the family of Klangs at hand. Now the graphs of figure 8.2(a) and (b) reference both the operations SUBM and PAR. It follows that we shall not be able to find a simply transitive group on a suitable family of Klangs that enables us to consider figure 8.2(a) and (b) as formally "intervallic" graphs. Our group would have to include both SUBM and PAR; containing SUBM, it would contain MED = SUBM'1; then the group would contain both MED7 and PAR. Then the group could not be simply transitive. E.g. given elements (C, +) and (C, —) in our family of elements, there would not be one unique member of the group transforming the former Klang into the latter; both MED7 and PAR would do the job. We could only salvage the formalities of this situation by cleaving so firmly to just intonation that we were willing to admit an infinite number of distinct Klangs (C0, +), (Clf +), (C2, +),..., (C0, -), (C,, -), (C2, -),... and so on, whose roots lay syntonic commas apart. Then (C0, +)MED 7 and (C0, +)PAR would have different formal values: The former would be (C_ 1? —) and the latter would be (C0, —). We would end up with a game board like that of figure 2.2 earlier, only with yet a third dimension reversing the modality of each Klang on the board. Clearly, this model would be at least awkward for analyzing Wagner's music. (It does engage some of the performance problems, especially for the singers.)

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8.2.1 We have seen that non-intervallic transformation-networks can be revealing in connection with Klangs. They can also be revealing in connection with serial transformations. Let us consider a series s of pitches or pitchclasses s 1? s 2 , ..., SN. We can apply to s the Rl-chaining operation RICH. RICH(s) is that retrograde-inverted form of s whose first two elements are SN-! and S N , in that order. Thus if s = A-C-Eb-E (a form of Zauber), then

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8.2.2

RICH(s) is Eb-E-G-Bb, and RICH(RICH(s)) is G-Bb-Db-D. The RICH transform of RICH(s), being a retrograde-inverted form of a retrogradeinverted form of s, must always be some transposed form of s. In the Zauber example above, the interval of transposition is 10: G-Bb-Db-D is the 10transpose of A-C-Eb-E. For another example, let us examine s = Eb-B-Bb~D-C#-C-F#-EG-F-A-G#. This is the prime row of Webern's Piano Variations op. 27. RICH(s) is A-G#-C-Bb-Qf-B-F-E-Eb-G-F#-D, and RICH(RICH(s)) is then F#-D-Db~F- etc. As before, the RICH transform of RICH(s) is a transposed form of s, but now the specific interval of transposition is different. F#-D-Db-F-etc. is the 3-transpose, not the 10-transpose, of Webern's row. In general, when we RICH the RICH-transform of an abstract pitch or pitch-class series s = s t , s 2 ,..., SN, the transposed form of s that results will be TJ(S), where the interval of transposition is i = int(s l5 s N ) + int(s 2 ,s N _ 1 ). (We are supposing that N > 2. The formula is given without proof.) Thus the interval of transposition for the Zauber series A-C-Eb-E was 10 = 7 + 3 = int(A, E) + int(C, Eb), while the interval of transposition for Webern's row Eb-BA-G# was 3 = 5 + 10 = int(Eb,G#) + int(B, A). It follows: When we define the operation TCH as (RICH) (RICH), then TCH(s) is always some transposed form of s, but just which transposed form depends on the internal structure of any given argument s upon which TCH is operating. Specifically, if i = int(s l5 s N ) + int(s 2 ,s N _ 1 ), then TCH(s) = Tj(s). We shall call i here "the TCH-interval for s." The TCH interval for a retrograde or an inverted form of s will be the negative (group inverse) of this i; the TCH interval for a retrograde-inverted form of s will be the same as the TCH interval for s itself. (These facts follow from the formula defining i above.) 8.2.2 EXAMPLE: Let us turn our attention once more to figure 7.4, the earlier example from Parsifal, inspecting the first, second, third, and fourth forms of Zauber that are beamed thereon and labeled as Z t , Z 2 , Z3, and Z4. Considering Z as a serial motive, we can graph Wagner's transformational technique here by figure 8.3.

FIGURE 8.3

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We have already discussed how the TCH operation makes Z3 the 10transpose of Z x and Z4 the 10-transpose of Z 2 . During our earlier discussion of the passage, we pointed out how the chaining technique creates structural sequencing as a result. We can "hear" the structural sequence in the note heads of figure 7.4, even though the foreground events of the music over measures 1074-1100 are quite different from those of the music over measures 1096-1140. 8.2.3 EXAMPLE: Webern is fond of using RICH and TCH, especially as rowtransformations in his serial works. Figure 8.4 graphs two instances from the Piano Variations, (a) of the figure coincides with the thematic middle section of the first movement; (b) coincides with the frantically syncopated variation in the last movement. The nodes on figure 8.4 are understood to contain various forms of the row. On (a) of the figure, T5 is not TCH. The row-form that fills both the leftmost and the rightmost nodes of (b) is the prime row-form cited earlier, the form which opens the third movement. The same form also fills the rightmost node of (a). This is the first occurrence of the prime form in the piece; after

182

FIGURE 8.4

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8.2.5

measure 37, the music continues to project the prime form of the row as it launches the big thematic reprise of the movement. The prime form that fills the rightmost node of (b) launches the coda of the third movement and the entire piece. The total rhythm of this coda matches that of the first-movement theme reprised at 1,37: In both cases we hear a rhythmic ostinato whose repeated units project four attacks and a rest in steady note-values. 8.2.4 If we excise any four consecutive nodes from figure 8.4(b), along with the arrows that connect them, we shall have essentially the same graph as that of figure 8.3. The same transformations are arranged and combined in the same structure of nodes and arrows, even though the contents of the nodes are Wagnerian in one case and Webernian in the other. We shall say that the two networks-of-series are isographic. The isography would not obtain if we wrote "T10" for TCH on figure 8.3 and "T3" for TCH on figure 8.4(b): T10 and T3 are not the same transformation. 8.2.5 EXAMPLE: Let us define another operation on series, an operation called MUCH. MUCH(s) is that retrograde-inverted form of s whose beginning overlaps the ending of s to the maximum possible extent. Figure 8.5(a) shows how Bach chains MUCH and RICH in the first Two-Part Invention. Figure 8.5(b) shows Bach's transformational technique in a graph which is very similar to the earlier graphs involving RICH. OP is the transformation RICH-after-MUCH; OP' is the transformation MUCH-after-RICH. MUCH and RICH do not commute on the family of all series, but the OP-interval-oftransposition for any given s is always the same as the OP'-interval for the

FIGURE 8.5

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8.2.6

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retrograde-inversion of s. In the specific example at hand, that interval is 2diatonic-steps-down. Bach's foreground sequence is constructed by a method very similar to that used by Wagner and Webern, in building their structural sequences. Figure 8.5 shows how artfully Bach's transformational technique uses the characteristics of his motive to fit his meter. 8.2.6 EXAMPLE: Wagner also uses RICH in the foreground. Figure 8.6 sketches the opening of the "Todesverkiindigung," Die Walkure, act 2, scene 4, starting from three measures before the scene begins.

FIGURE 8.6

184

The scene is dominated by the Fate motive, whose melodic component FATE is bracketed over measures 1-2 on the figure. FATE functions both as a 3-note series and as an intervallic network, 2 = (— 1) + 3 semitones. The interval 2 of that network is originally spanned from A to B, and the thematic pitch-idea of "A to B" thereby goes along with the prime form of the FATE motive/series/network. Locally, "A to B" helps us hear FATE as related to the

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8.2.6

(head of the) LOVE motive, which has just been repeated over and over in the melody preceding measure 1. The repeated LOVE fragment also projects an overall sense of "A to B"; as it repeats, it inflects A by a pickup F# and thereby defines a total ambitus of F#-to-B. These features enable us to extract the intervallic network of figure 8.7(a).

FIGURE 8.7

Figure 8.7(b) displays the FATE network for purposes of comparison. A 3-arrow and a 2-arrow are common to both networks. Another way of relating the motivic fragments is to regard LOVE as an essential A-to-B elaborated by an F# pickup 3 below A, while regarding FATE as an essential A-to-B elaborated by a G# pickup 3 below B. This view attributes emphatic priority to pitch relations and even pitch hierarchies, at the expense of intervallic relationships per se. We shall see that the intervallic structures of the "Todesverkiindigung," like those of the Brahms Horn Trio studied earlier, take on a more autonomous role as the music develops. We can hear another strong relation of LOVE and FATE in the music through the harmonization of LOVE that immediately precedes FATE: A-toB within LOVE is supported by C-to-B in the bass; the total matrix thereby projects a pitch-class form of FATE. Figure 8.6 indicates the pitch-class intervals of the FATE network on the LOVE matrix accordingly. A 2-arrow connects A to B within the melody of LOVE; that 2-arrow is subarticulated into a "3"-arrow, from A to the C below, and a (— l)-arrow, from C in the bass to B in the bass. The bass B supports and doubles the B of the LOVE melody as a pitch class. Supposing a fundamental bass for the A minor harmony of LOVE, an A that goes under and hence conceptually "before" the C of the basso continue, we can even identify a serial form of FATE embedded in this LOVE music, namely A-C-B. This is the unique form of the FATE series, other than the prime form, that embeds the essential A-to-B gesture. Starting from that A-C-B, we can hear a RICH chain of FATE forms proceeding along the bass line of figure 8.6. A-C-B before measure 1 chains

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#.2.6

186

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into C-B-D across measure 1; C-B-D in turn chains into B-D-C#, which carries the harmony through the bass of the FATE motive proper. The chaining continues on in the bass line, through D-C#-E, C#-E-D#, and even farther (across the DEATH music). We recognize the sequencing of the FATE bass as a typical TCH sequence, created by Rl-chainmg. The interval of the sequence, that is the TCH interval for the FATE bass, is 2. Computing the interval specifically from the series A-C-B, our point of departure in the bass, we have i = int(A, B) + int(C, C) = int(A, B), which is 2. So we see that 2-as-TCH-interval, governing the sequence, is the "same" 2 as the 2 that spans A-to-B, that is int(A, B). We do not need such heavy transformational machinery, just to hear that the 2-sequences of the music are related to the A-B gesture of the motive. But the transformational machinery clarifies just how the relation is worked out, and it attributes thereby a special and characteristic formal function to the compositionally prominent interval in this serial context. The FATE series is Rl-chained in the melody, too. Just as the bass chain began with A-C-B, one form of FATE that embeds A-to-B, so the melodic chain begins with A-G#-B (measures 1-2), the other form of FATE that embeds A-to-B. Again the TCH interval is 2 = int(A, B) + int(G#, G#) = int(A, B). Again the musical sequence from measures 1-2 to measures 5-6 is carried by a TCH sequence, now in the melody. The melodic chain of measures 1-8, covered by a slur on figure 8.6, is recalled in summary at the end of the DEATH motive, covered by another slur over measures 10-12 on the figure. The last stage of FATE under the slur, B-A#-C#, recurs yet again at measure 13, to launch the transposed return of all the FATE music, which starts the chaining underway all over again. Via the relationships just mentioned, we can hear that the melodic B-A#-C# of measures 13-14 is in the TCH-relation to the A-G#-B of measures 1-2. Hence the large sequence, the sequence that carries all of measures 1-12 into all of measures 13-24, is itself a TCHsequence, a product of Rl-chaining. It would be possible and even legitimate to draw many 2-arrows upon figure 8.6, e.g. transposing measures 1-4 into measures 5-8, transposing measures 10-llj of the melody into measures 11-12 of the melody, and transposing all of measures 1-12 into all of measures 13-24. We are certainly strongly aware of the interval 2 in these connections. The resulting graph would be inadequate, though, so far as it suggested that the relations involved were only transpositional, the results of expanding the interval 2 = int(A, B) into T2-relations between larger Gestalts. We would miss the function of 2 as a TCH interval in two independent FATE chains, one in the bass and one in the melody, each chain launched by one of the two FATE-forms that embed A-to-B. We would miss hearing how the sequences in the music hang on the pertinent stages of RI chaining in the outer voices. Figure 8.8 shows how a motive I shall call FATE' is Rl-chained over

Transformation Graphs and Networks (2)

8.2.6

FIGURE 8.8

measures 56 and following. This is where Siegmund "looks into the eyes of the Valkyrie," as Brunnhilde puts it later on. The subinterval 3 of the FATE network 2 = (—1) + 3 now becomes the overall interval of the FATE' network 3 = (— 1) + 4; (—1) remains a subinterval of FATE'. The pitch class A of FATE' is bereft of its FATE-partner B, just as Siegmund will be lonely in Valhalla, bereft of his sister/wife. The comparison is suggestive because the F# and A of FATE' recall the F# and A of LOVE, displayed earlier in figure 8.7(a). Interval 3, the TCH-interval for FATE', is precisely int(F#, A)(= int(F#, A) + int(E#, E#)). On figure 8.8, the FATE' series goes through four TCH-sequences. Its course is therefore isographic to the bassline chain of figure 8.6, which also went through four TCH-sequences. RICH and TCH thus enable us to hear a way in which the bass line of figure 8.6 and all of figure 8.8 project the same overall transformational gesture. Obviously we could not hear such a relation using T2 in one case and T3 in the other; those are different transformations. After Siegmund has looked into Brimnhilde's eyes he goes into an extended Wagnerian stichomythy, asking a series of questions about Valhalla which Brunnhilde answers in turn. What really concerns Siegmund, we discover, is the idea that Sieglinde may not be able to accompany him to Valhalla. His concern is well founded. As he asks each question, he sings or sings along with an entry of the DEATH motive. Figure 8.9 tabulates his questions in brief, along with the keys in which the DEATH melody enters therewith, during measures 70-133.

FIGURE 8.9

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We are intensely aware of Siegmund's concern for Sieglinde even before his final questions, because the keys of figure 8.9 and the intervals among their tonics project the pitch classes and intervals of the LOVE motive, as diagrammed in figure 8.7 (a) earlier. During the repeated LOVE music just before the scene change, Siegmund had calmed Sieglinde, and she is lying asleep in his arms during the whole exchange with Brunnhilde. 8.3.1 EXAMPLE: RICH and TCH have provided good examples of serial transformations that are not intervallic/transpositional but are nonetheless very suggestive in connection with node/arrow analytic graphs. The retrograde operation on series is another such example. So are the various operations RT and RI, T being some transposition and I some inversion. (For a given u and a given v, RI* and RICH are not the same operation, as they operate upon a variety of series.) Webern's piece for string quartet, op. 5, no. 4, demonstrates other suggestive serial operations. The operation TLAST transposes a series by its last interval; the operation TFIRST transposes a series by its first interval. Hence TFIRST"1 transposes a series by the complement of its first interval. TLAST makes the last note of a given series the next-to-last note of the transformed series; TFIRST"1 has a sort of "dual" effect, in that it makes the first note of a given series the second note of the transformed series. Figure 8.10 shows three series of pitch classes, arranged in a network that involves TFIRST'1 and TLAST in this "dual" relationship.

FIGURE 8,10

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The three series cited by the figure are projected by the three appearances of a prominent unaccompanied rising motive FLYAWAY in Webern's piece. Each of these unaccompanied appearances ends a major section of the work. The order in which figure 8.10 cites the three series is not the chronological order in which the music presents the forms of FLYAWAY. In the music, the form that begins on A(7 occurs last; indeed it is the last event of the piece. The visual "centrality" of this form on figure 8.10 portrays a cadential function for the event, transformationally "balanced" as it is between the other two forms. Just so, the formal "centrality" of a tonic Klang, balanced between its

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8.4

dominant and subdominant, is often projected temporally by a final cadence in which the tonic is the last event. While it would be perfectly possible to label the arrows of figure 8.10 as "T8" and "T3" rather than TFIRST'1 and TLAST, the transpositional labels would conceal, not reveal, the balancing centrality of the form beginning on A|?, the form that ends the piece. The intervals of transposition are of course very important in other connections. 8.3.2 EXAMPLE: We define operations FLIPEND and FLIPSTART on series of three pitches or three pitch-classes. FLIPEND transforms the series S1-s2-s3 into the series s^Sj-a, where a is the inversion-about-s3 of s2. (a = Is*(s2); int(s3,a) = int(s2, s3).) The inverse operation FLIPEND"1 then transforms the series ti-t2-t3 into the series ti-b-t^ where b is the inversionabout-t2 of t3. (To verify this, given t l s t 2 , and t3, set Si = t l 5 s2 = b-as-b-isdefined, and s3 = t 2 . Then observe that FLIPEND transforms the s series into the given t series.) Dually, FLIPSTART transforms S1-s2-s3 into a-s1-s3, where a is the inversion-about-Si of s2; then FLIPSTART"1 transforms ti~t2-^-3 into t2-b-t3, where b is the inversion-about-t2oftl. Figure 8.11 shows what happens when FLIPEND and FLIPSTART"1 are chained in alternation, starting with one series of three pitches in (a) and another in (b). The arrows above the staff indicate applications of FLIPEND; the arrows below the staff indicate applications of FLIPSTART"1. FLIPEND and FLIPSTART are my own names for transformations used by Jonathan W. Bernard in studying how Varese's music expands, contracts, and displaces registral space.1 8.4 The serial transformations just studied, i.e. RICH, TCH, MUCH, TLAST, TFIRST, FLIPEND, and FLIPSTART, are all easily generalized to operate on series whose elements are members of an abstract commutative GIS. In the non-commutative case, it is not clear just how some of the operations are to be defined; different possibilities are equally plausible. For instance, given the abstract series s = s l5 s2, ..., a, b within some abstract GIS, let us try to define RICH(s) abstractly. We can consider three possibly different series, t, u, and v, as plausible candidates for "RICH(s)." t is the retrograde of I(s), where I is (a/b)-inversion l£. u is the retrograde of J(s), where J is (b/a)-inversion I£ = (I*)"1, v is a series constructed as follows. Let the serial intervals of s be i t = int(s1, s2), i2 = int(s2, s 3 ),..., i N _j = int(a, b). The series v starts with the element a, and then proceeds according to the succession of intervals i N _ 1} i N _2 ,... ,i2,i!. Elements a and b will be the first and second elements of all three series t, u, and v. t and u are both retrograde1. Jonathan W. Bernard, The Music of Edgard Varese (New Haven and London: Yale University Press, forthcoming).

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8.4

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FIGURE 8.11 inverted forms of s. The serial intervals of v are the same as the intervals of s in reverse order. If the GIS is commutative, t, u, and v will all be the same series. If the GIS is not commutative, t, u, and v may be three distinct series.

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8.5 In section 8.1 we studied Klang-transformations as potential labels for arrows on graphs, arrows that could not be analyzed adequately by intervallic/ transpositional ideas alone. In sections 8.2 and 8.3 we studied a number of serial transformations in the same connection. Inversional relations between elements in a GIS, even a well-behaved commutative GIS, may also give rise to non-transpositional arrows on graphs, that is I-arrows. Consider figures 8.12 (a) and (b). The pitch classes on the networks stand for the pitches presented at the opening of the second movement in Webern's Piano Variations. There, the pitches come in pairs articulated by a rhythmic motive that remains essentially constant over the events portrayed in the figure. Graph (a) analyzes the pitches

Transformation Graphs and Networks (2)

8.4

FIGURE 8.12

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8.4

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of the phrase in an intervallic/transpositional network. One could add more interval-arrows connecting certain pitches by criteria other than immediate temporal succession. Graph (b) analyzes the same pitches using not only some intervallic/transpositional arrows (shown as curved) but also some inversional arrows (shown as straight). "I" is inversion about the pitch A4. Graph (b) reflects the rhythmic motif of the music by a visual motif, the vertical I-arrow. It represents by a visual symmetry the mirror symmetry of the pitch registration. It makes manifest the row-structure of the passage, and the way in which that structure interacts with the dux/comes structure of the musical canon. Graph (b) thus reflects more clearly than graph (a) not only the compositional structure of the passage but also our foreground perception of its shapes. Let us denote by IPAIR the graph consisting of two nodes connected by a two-way I-arrow. Now "I" will mean pitch-class inversion about A. As a configuration of pitch classes, network 8.12(b) has four subnetworks manifesting the graph IPAIR. Those are the subnetworks relating Bt?-and-G#, A-and-A, F-and-C#, and D-and-E; the subnetworks are all isographic. The network of row-forms being used is also isographic: Two row-forms are being used, each of which is the I-inversion of the other. Also manifesting IPAIR isographically is a network interrelating the "antecedent" and "consequent" row forms that respectively control the first twelve and the second twelve notes of the third movement. Antecedent and consequent forms there are I-inversions, each of the other. The consequent is presented in the music isorhythmically to the antecedent. IPAIR, in short, is a thematic graph.

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9

Transformation Graphs and

Networks (3): Formalities

It is time now to become formal about what we are doing when we draw nodes, arrows connecting some pairs of nodes, and names of transformations labeling those arrows, sometimes also putting pitches, Klangs, series, row-forms, or other objects "into" the nodes. I have already worked out most of these formalities elsewhere.1 Their treatment here differs somewhat, mainly in that the present exposition is more general. 9.1.1 DEFINITION: By a node I arrow system we shall mean an ordered pair (NODES, ARROW), where NODES is a family (i.e. set in the mathematical sense), and ARROW is a subfamily of NODES x NODES, i.e. a collection containing some ordered pairs (N l9 N 2 ) of NODES. We say that nodes N t and N2 are "in the arrow relation" if the pair (N l 5 N 2 ) is a member of the collection ARROW. For present purposes, we shall stipulate that every node is in the arrow relation with itself. That is, we assume that (N, N) is a member of ARROW for every node N. On figure 9.1, the nodes Mt and M 2 are not in the arrow relation; neither are M 2 and Mj. Nodes M4 and M 5 are in the arrow relation; so are M 5 and M4. Nodes M 5 and M6 are in the arrow relation; M6 and M 5 are not. Arrows from M! to M! , from M 2 to M 2 , and so on, are all understood on the figure. 9.1.2 DEFINITION: Nodes N and N' in a node/arrow system communicate if 1. David Lewin, "Transformational Techniques in Atonal and Other Music Theories," Perspectives of New Music vol. 21, nos. 1-2 (Fall-Winter 1982/Spring-Summer 1983), 312-71. The formalism there is confined to pages 360-66. The article consists mostly of analyses, together with exemplary graphs and networks, that will interest the reader who has enjoyed chapters 7 and 8 of the present book. The music studied there overlaps that studied here only slightly.

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9.1.3

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FIGURE 9.1

there exist nodes N 0 , N t , ..., Nj which satisfy criteria (A), (B), and (C) following. (A): N0 = N. (B): For each j between 1 and J inclusive, either (Nj.^Nj) is in the ARROW relation, or else (NJ5 Nj.J is. (C): Nj = N'. The criteria demand a finite unbroken path of forwards-or-backwards arrows which starts at N and ends at N'. The nodes M t and M 2 of figure 9.1 communicate. But neither Mj nor M 2 communicates with M4. The communication relation among nodes is easily proved to be reflexive, symmetric, and transitive. Hence "communication" is an equivalence relation on the NODES of a node/arrow system. As figure 9.1 suggests, the nodes within any equivalence class all communicate, each with any other, while nodes in different equivalence classes do not communicate. 9.1.3 DEFINITION: A node/arrow system is connected if any two nodes communicate. The system displayed in figure 9.1 is not connected. It can be analyzed into two component subsystems, each of which is connected; there is no communication between any node of one subsystem and any node of the other. This sort of structure is typical of disconnected node/arrow systems. Any disconnected system can be analyzed into component connected subsystems that do not communicate with one another. Each such subsystem is a pair (NODESa, ARROWJ, where NODESa is an equivalence-class of NODES under the communication relation, and ARROWa is that subcollection of ARROW comprising pairs of nodes from NODEa. As we proceed, we shall find the following construct necessary and useful.

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9.1.4 DEFINITION: An arrow chain from node N to node N' in a node/arrow system is a finite series of nodes N0, N x , . . . , N, satisfying criteria (A), (B), and (C) below.

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9.2.2

(A): NO = N. (B): For each j between 1 and J inclusive, (N^, Nj) is in the ARROW relation. (C): Nj = N'. The criteria demand a finite unbroken path of forwards-orientedarrows, starting at N and ending at N'. We are now ready to "label" our arrows formally with symbolic transformations. At this stage of our work, the nodes do not yet "contain" any objects to be transformed, so we shall represent the eventual transformations abstractly, as members of some abstract semigroup. 9.2.1 DEFINITION: A transformation graph is an ordered quadruple (NODES, ARROW, SGP, TRANSIT) satisfying criteria (A), (B), (C), and (D) below. (A): (NODES, ARROW) is a node/arrow system. (B): SGP is a semigroup. (C): TRANSIT is a function mapping ARROW into SGP. (D): Given nodes N and N', suppose that N0, N1? ..., Nj is an arrow chain from N to N'. Suppose that M 0 , M 1 } ..., MK is also an arrow chain from N to N'. For each j between 1 and J inclusive, let Xj = TRANSIT(Nj.j^Nj). For each k between 1 and K inclusive, let yk = TRANSIT(Mk_1, Mk). Then the semigroup product X j . . . x2xi is equal to the semigroup product yK ... y 2 yi-

FIGURE 9.2 The setting in which criterion (D) holds sway is illustrated by figure 9.2. Symbolically, the eventual "contents" of N' will be the Xj-transform of the... of the x2-transform of the x^transform of the eventual "contents" of N. That is, the contents of N' will be the (Xj... x 2 x ^-transform of the contents of N, in the sense of our left-orthographic convention. Likewise, the contents of N' must also be the (yK ... yjy^-transform of the contents of N. To ensure that no contradiction can possibly arise, we must ensure that the semigroup products are equal. And that is what Criterion (D) of the definition does. 9.2.2 OPTIONAL: Criterion (D) also enables us to prove that for each node N,

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TRANSIT(N, N) must be some idempotent member of SGP. Tp prove this, consider the following two formal arrow-chains from N to N: N0 = N t = Nj = N; M0 = M j = M 2 = MK = N. Set Xj = TRANSITCN^NO, y t = TRANSITCM^Mi), y 2 = TRANSIT^, M 2 ). Then, via the criterion, x =V v i 2 i- But x1? y1} and y2 are all the same member of SGP, namely TRANSIT (N, N). So that element is idempotent, as claimed. Criterion (D) then ensures that whenever M0, M l 5 . . . , MK is an arrow chain from N back to N, the product yK ... y^ (as in 9.2.1) is equal to the idempotent TRANSIT(N, N). The only idempotent member of a group is the identity element. (If zz = z, then zzz"1 = zz"1 and z = e.) Hence, when SGP is a group, TRANSIT (N, N) must be the identity. 9.2.3 DEFINITION: An operation graph is a transformation graph in which SGP is a group. Now we shall render formal the idea of "putting objects into" the nodes of a transformation graph, and transforming the objects about as indicated by the graph and by some appropriate semigroup of transformations. That is the idea of a transformation network, which we shall now define, distinguishing it formally from the transformation graph whose nodes its objects fill. 9.3.1 DEFINITION: A transformation network is an ordered sextuple (S, NODES, ARROW, SGP, TRANSIT, CONTENTS) having the features (A) through (D) below. (A): S is a family of objects (that are to be transformed in various ways). (B): (NODES, ARROW, SGP, TRANSIT) is a transformation graph such that SGP is a semigroup of transformations on S. (C): CONTENTS is a function mapping NODES into S. "CONTENTS(N)" can be read: "the contents of node N." (D): Given nodes N t and N2 in the ARROW relation; if f = TRANSIT(N1,N2), then f (CONTENTS^)) = CONTENTS(N2). To see how feature (D) of the construction reflects our intuition, we can inspect figure 9.3. The figure depicts nodes N t and N2 in the arrow relation. On the graph, the member f of SGP labels the arrow from N! to N2; that is, f =

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FIGURE 9.3

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9.3.3

TRANSIT(N1,N2). f is some transformation on S, according to feature (B) of 9.3.1. Inspecting figure 9.3 further, we see that $! is the CONTENTS of N! and s2 is the CONTENTS of N 2 . In this situation, (D) of the definition assures us that we shall have f^) = s 2 . 9.3.2 DEFINITION: An operation network is a transformation network for which SGP is a group of operations on S. 9.3.3 THEOREM: Let S be a family of objects. Let GP be a group of operations on S. Let (NODES, ARROW, GP, TRANSIT) be an operation graph whose node/arrow system is connected. Let N0 be a node; let s0 be a member of S. Then there exists a unique operation-network having S for its objects and (NODES, ARROW, GP, TRANSIT) for its graph such that s0 is the CONTENTS of N0. The theorem is given without proof. It says that all the contents of all the nodes in a connected operation-network are uniquely determined, once we know the contents of any one node. Intuitively, we can follow some path of forwards-or-backwards arrows from the given N0 to any other node, since the system is connected. As we go along that path, we can fill in the CONTENTS of each node by applying the indicated TRANSIT operation when we traverse an arrow forwards, or the inverse (NB) of that operation when we traverse an arrow backwards. Criterion (D) of 9.2.1 enables us to infer that it does not matter which path we might follow from N0 to a specific other node in this connection; the end result will be the same as regards the necessary CONTENTS of that other node. To get some sense of why this works, let us look at the graph displayed in figure 9.4.

FIGURE 9.4 Here, we are supposing A, B, C, and D to be operations upon some family S. Fixing some member s0 of S, let us construct an operation-network having S for its objects and figure 9.4 for its graph, such that s0 is the CONTENTS of node N0 on the figure. Since s0 is the CONTENTS of N0 and A is the

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TRANSIT-operation from N0 to N3, the CONTENTS of N3 on the figure will have to be A(s0). (We know this by 9.3.1(D).) Let us write "s3" for "the necessary CONTENTS of node N3." Then we shall have to have s3 = A(s0). Now what about s l5 the necessary CONTENTS of node Nj? Since B is the TRANSIT-operation from N x to N3, we shall have to have s3 = 6(5!). Since we have already determined what s3 must be, and we are searching for s1, we can infer that the Si we are looking for is derived as s t = B"1 (s3). Note how we have leaned on the fact that B is an operation at this point in our construction. When we write "Sj = B"1^)," we are using the idea that a unique s t is well defined by the relation s3 = B^); that idea in turn rests on the supposition that B is 1-to-l and onto. So, given our initial s0, we have derived the necessary CONTENTS s3 and Sj for the nodes N3 and N t in the operation network we are constructing: s3 = A(s0); Si = B'^SS). What about s2, the necessary CONTENTS of N2? Here, it seems at first that we have two different choices for s2. Since an arrow labeled D points from N x to N 2 , we must have s2 = D^). But also, since an arrow labeled C points from N2 to N3, we must have s2 = C"1^). It is just here that 9.2.1 (D) comes to our rescue. The criterion tells us in this situation that, in fact, D(SJ) = C~1(s3). Therefore it only seems that we have "two" choices for S 2 ; in fact the value of s2 is well determined. Let us see just how we can infer from Criterion 9.2.1(D) that D(SJ) = C~1(s3). The Criterion notices an arrow-chain going from N x directly to N3; it also notices an arrow-chain going from N x through N2 to N3. It informs us that a certain algebraic relation must therefore obtain among the TRANSIToperations B, D, and C that link the nodes along those arrow-chains. The relation must obtain, that is, for us to have spoken at all of a well-formed "operation graph." The relation given by the Criterion here is B = CD. Now Si was chosen to satisfy the relation B(Si) = s3. So we can infer that CD(S!) = s3. And thence we can infer that D^J = C"1^), which is just what we wanted to show. Obviously, the sort of situation we have just examined in connection with the arrow-chains of figure 9.4 can get very complicated on a general operation-graph. The way in which Criterion 9.2.1 (D) helps us out is basically the same, though.

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During the musical discussions of chapters 7 and 8 we invoked from time to time the concept of "isography." It would seem that we can now define that concept rigorously: Two transformation networks are isographic if they "have the same graph." But that definition is not yet formal enough. For suppose we want to assert an isography between the networks (S, NODES, ARROW, SGP, TRANSIT, CONTENTS) and (S', NODES, ARROW, SGP', TRANSIT', CONTENTS'), where S and S' are different families of objects. Those two networks have the same node/arrow system, but they

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9.4.3

cannot have the same graph. The graph of one is (NODES, ARROW, SGP, TRANSIT); the graph of the other is (NODES, ARROW, SGF, TRANSIT'). SGP', a semigroup of transformations on S', cannot be "the same as" SGP, a semigroup of transformations on S. And therefore TRANSIT', a function taking on values in SGP', cannot be "the same as" TRANSIT, a function taking on values in SGP. Moreover, the concept of isography is particularly suggestive exactly when S' and S are different families of objects, as above. S for example might be the twelve chromatic pitch classes, while S' might be a family of set-forms, motive-forms, or row-forms; SGP could comprise transpositions and inversions on pitch classes, while SGP' comprised transpositions and inversions of set/motive/row forms. To make our intuitions about isography work out formally here, we need the concept of isomorphism between one transformation graph and another. We can then define two networks to be isographic if their graphs are isomorphic. To define the isomorphism of graphs, we shall in turn have to define the isomorphism of node/arrow systems. 9.4.1 DEFINITION: Two node/arrow systems, (NODES, ARROW) and (NODES', ARROW), are isomorphic if there exists a 1-to-l map NODEMAP of NODES onto NODES' such that for every pair (Ni,N 2 ) of NODES, (N 1} N 2 ) is in the ARROW relation if and only if (NODEMAP(N1), NODEMAP(N2)) is in the ARROW' relation. A function NODEMAP having the indicated property will be called an isomorphism of (NODES, ARROW) with (NODES', ARROW'). 9.4.2 DEFINITION: Given two transformation graphs (NODES, ARROW, SGP, TRANSIT) and (NODES', ARROW, SGP', TRANSIT'), the two will be called isomorphic if there exists a pair (NODEMAP, SGMAP) having properties (A), (B), and (C) following. (A): NODEMAP is an isomorphism of (NODES, ARROW) with (NODES', ARROW'). (B) SGMAP is an isomorphism of SGP with SGP'. (C): For every pair (N x , N2) in ARROW, TRANSIT'(NODEMAP(N1), NODEMAP(N2)) = SGMAP(TRANSIT(Nt, N2)). The pair (NODEMAP, SGMAP) will be called an isomorphism of the first graph with the second. 9.4.3 DEFINITION: The transformation networks (S, NODES, ARROW, SGP, TRANSIT, CONTENTS) and (S', NODES', ARROW, SGP', TRANSIT', CONTENTS') are isographic if the transformation graphs (NODES, ARROW, SGP, TRANSIT) and (NODES', ARROW, SGP',

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TRANSIT') are isomorphic. If (NODEMAP, SGMAP) is an isomorphism of the first graph with the second, then that pair is an isography of the first network with the second. 9.4.4 EXAMPLE: Figure 9.5 shows some transformation networks, all of which are isographic. (a), (b), and (c) will be familiar from our recent discussion (in section 8.5) of Webern's Piano Variations.

FIGURE 9.5

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Let us work out the isographies of the figure very formally. We can fix the node/arrow system that underlies all the graphs and networks: NODES is a two-element family; every pair of NODES is in the ARROW relation. Graph (a) has as its SGP the group of operations on pitch classes that contains the identity E and the operation I = l£. (This is a group, since II = E.) Graph (a) has as its TRANSIT function the function TRANSIT^, N x ) = TRANSIT(N2, N 2 ) = E; TRANSIT^, N2) = TRANSIT(N2, Nj) = I. Graph (b) has the same SGP and the same TRANSIT function as graph (a). So in this special case, graph (b) is in fact literally "the same as" graph (a). Graph (c) however is "different" (though isomorphic). Its semigroup comprises two operations on twelve-tone rows, not two operations on pitch classes. The row-operations are E (which leaves any row alone) and I (which inverts any row about the pitch class A). The TRANSIT function for graph (c) maps ARROW into this new semigroup of row-operations. The semigroup of row-operations, while "new," is isomorphic with the old semigroup of pitchclass operations, under the correspondence of pitch-class-operation E with row-operation E and pitch-class-operation I with row-operation I. We can take this map of the old semigroup into the new one as our formal SGMAP. And we can take as NODEMAP the identity map of our fixed NODES onto itself. The (NODEMAP, SGMAP) is a formal isomorphism of graph (a) (or graph (b)) with graph (c). Therefore (NODEMAP, SGMAP) is a formal isography of network (a) (or network (b)) with network (c).

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9.5.2

Networks (a) and (b) are derived from the opening of the second movement in Webern's piece. Suppose we played the music a semitone higher; then we could derive networks (d) and (e) of the figure instead. Here J is inversion-about-B)?. For graphs (d) and (e) the semigroup consists of the identity operation E on pitch classes and the inversion-operation J on pitch classes. The new semigroup is isomorphic with semigroup (a) under the map SGMAP(E) = E; SGMAP(I) = J. Using this SGMAP and the identity map on NODES as NODEM AP, we establish a formal isomorphism of graph (a) (or graph (b)) with graph (d) (or graph (e)). Networks (a) (or (b)) and (d) (or (e)) are thereby isographic. The notions of isomorphism we have just been exploring can be extended suggestively to more general notions of "homomorphism." 9.5.1 DEFINITION: Given node/arrow systems (NODES, ARROW) and (NODES', ARROW), a mapping NODEMAP of NODES into NODES' is a homomorphism of the first system into the second if (NODEMAP^j), NODEMAP(N2)) is in the ARROW' relation whenever (N l 5 N 2 ) is in the ARROW relation. NODEMAP is a homomorphism onto if it maps NODES onto NODES' in a special way: Whenever N't and N'2 are in the ARROW relation, there exist N t and N 2 in the ARROW relation such that N\ = NODEMAP(Ni) and N'2 = NODEMAP(N2). A homomorphism NODEMAP is 1-to-l as a homomorphism between systems if it is 1-to-l as a map of NODES into NODES'. Under these definitions, a 1-to-l homomorphism of one system onto another is an isomorphism in the sense of 9.4.1, and an isomorphism in that sense is a 1-to-l homomorphism of the first system onto the second. Here, the special definition of "homomorphism onto" in 9.5.1 above is crucial. It is possible for NODEMAP to be a homomorphism of (NODES, ARROW) into (NODES', ARROW) and also a 1-to-l map of the family NODES onto the family NODES', without being an isomorphism of the two systems. That is so because the second system may "have more arrows." For example we could simply take NODES' = NODES and add more arrows to ARROW for ARROW'. Then the identity map of NODES into NODES' is a homomorphism of (NODES, ARROW) into (NODES', ARROW') and also a 1-to-l map of NODES onto NODES', but it is clearly not an isomorphism of the two systems. It is not a homomorphism of the first system onto the second, in the full sense of 9.5.1. 9.5.2 DEFINITION: Given transformation graphs (NODES, ARROW, SGP, TRANSIT) and (NODES', ARROW', SGP', TRANSIT'), a homomorphism of the first graph into/onto the second is a pair (NODEMAP, SGMAP) having features (A), (B), and (C) below.

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(A): NODEMAP is a homomorphism of the node/arrow system (NODES, ARROW) into/onto the system (NODES', ARROW). (B): SGMAP is a homomorphism of the semigroup SGP into/onto the semigroup SGP'. (C): For every pair of nodes (N l 5 N 2 ) in the ARROW relation, TRANSIT'(NODEMAP(N1), NODEMAP(N2)) = SGMAP(TRANSIT(N15 N2)). The graph homomorphism (NODEMAP, SGMAP) is defined to be 1-to1 if both NODEMAP and SGMAP are 1-to-l maps, of NODES into NODES' and SGP into SGP' respectively. According to these definitions, an isomorphism of the first graph with the second (in the sense of 9.4.2 earlier) is precisely a 1-to-l homomorphism of the first graph onto the second.

FIGURE 9.6

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9.5.3 EXAMPLE: Earlier (section 7.3), we studied graphs (a) and (b) of figure 9.6 in connection with Brahms's Horn Trio. We called graph (a) the "complementary gesture," and graph (b) the "complementary gesture times 2." The intervallic augmentation that transforms graph (a) into graph (b) is in fact a formal homomorphism. To verify this, let us begin by attaching the name (NODES, ARROW) to the three-node node/arrow system common for both graphs. Take NODEMAP to be the identity map on NODES. NODEMAP is then, trivially, an isomorphism of the node/arrow systems involved for the two graphs. Take SGPa, the semigroup for graph (a), to be the group of the twelve chromatic pitch-class intervals. Take SGPb, the semigroup for graph (b), to be (provisionally) the same group. The values of TRANSITa and TRANSIT,, are as indicated on graphs (a) and (b) of the figure. Take SGMAP to be the mapping of the interval i into the interval 2i, a map that transforms SGPa into SGPb. That is, take SGMAP(i) = 2i for each interval i. Then SGMAP is a homomorphism of SGPa into SGPb: SGMAP (i + j) = SGMAP(i) + SGMAP(j) (mod 12). As defined, SGMAP is neither 1-to-l nor onto. Requirements (A) and (B) for Definition 9.5.2 are now verified, as regards a potential homomorphism of graph (a) into graph (b). It remains to verify requirement (C) of the definition. This is easily done by inspecting the

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9.5.4

numbers labeling the three arrows on each graph: If interval i labels any arrow on graph (a), then interval 2i labels the corresponding arrow on graph (b). That is, if i = TRANSIT^, N2), then 2i = TRANSIT'(N^Nj). Or, yet more formally, TRANSIT'(NODEMAPCNO, NODEMAP(N2)) = TRANSIT'(N15 N2) = 2i = SGMAP(i) = SGMAP(TRANSIT(Nl5 N2)), as demanded by requirement (C) of the definition. The requirement is also satisfied in case N2 = N t = N: TRANSIT'(NODEMAP(N), NODEMAP(N)) = TRANSIT'(N, N) = 0 = 2 - 0 = SGMAP(O) = SGM AP(TRANSIT(N, N)). The case N! = N2 = N is trivial here, as it always will be when the semigroups involved are groups; when the semigroups are not groups, TRANSIT'(N', N') and TRANSIT(N, N) will be idempotents within semigroups that may have many idempotents, and the requirement of the definition is not trivially satisfied. To make our graph homomorphism here a "homomorphism onto," we need only redefine SGPb as the group of all even intervals. Then SGMAP takes SGPa onto SGPb, and the graph homomorphism thereby becomes "onto" in the sense of 9.5.2. It is not, of course, 1-to-l. (SGMAP is still not 1-to-l.) 9.5.4 EXAMPLE: For this example we shall use the word "tritone" to mean a collection of two pitch-classes spanning that interval. There are six tritones in that sense: (C, F#), (C#, G), (D, Ab), (E[>, A), (E, B[?), and (F, B). Transposing a tritone by a pitch-class interval i has the same effect on the set as transposing it by interval i + 6. So for instance transposing (C,F#) by 5 yields (F, B); transposing (C, F#) by 11 also yields the unordered set (F, B). Accordingly, we can define six formal "transposition operations" on the family of tritones. We shall call the six operations O-or-6, l-or-7,2-or-8,3-or-9,4-or-10, and 5-or-l 1. The six operations form a simply transitive group on the family of tritones. We can therefore construct a GIS having the tritones for its objects and the six operations for its formal intervals. We write "int((C, F#), (F, B)) = 5-or-l 1," and so on. Consider the map SGMAP that takes pitch-class interval i into tritoneinterval i-or-(i + 6). This map is a homomorphism from the group of pitchclass intervals, onto the group of tritone intervals. That is, if we transpose a given tritone by i-or-(i + 6), and then transpose the resulting tritone by j-or(j + 6), we shall end up having transposed the original tritone by (i + j)-or(i + j + 6), all mod 12 of course. SGMAP is part of a homomorphism that transforms the graph of network (a), in figure 9.7, onto the graph of network (b). We imagine arrows labeled "0" from each node of (a) to itself, and arrows labeled "O-or-6" from each node of (b) to itself. The NODEMAP for the graph homomorphism takes the two top nodes of (a) into the top node of (b), and the two bottom nodes of (a) into the bottom node of (b). The CONTENTS of various nodes on networks(a) and (b) in the figure help us see why

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9.5.5

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FIGURE 9.7

NODEMAP is a musically plausible function here. But the homomorphism of the graphs does not depend formally upon the CONTENTS with which the nodes are filled in the networks. The graphs as such know nothing of these contents; they know only the node/arrow systems, the semigroups involved in labeling the arrows, and the TRANSIT functions that provide those labels. 9.5.5 EXAMPLE: Figure 9.8(a) transcribes the opening phrase from the first example in the Scholica Enchiriadis that shows the Symphony of the Diatesseron.2 (b) of the figure graphs the melody "Nos qui vivimus"; the numbers measure steps up or down in the mode, (c) of the figure is a network whose node/arrow system is disconnected; the network exhibits the vocal lines of Principalis and Organalis separately. Graph (b) is a homomorphic image of graph (c). The homomorphism works as follows: NODEMAP takes the first Principalis node of (c) and the first Organalis node of (c) both into the first node of (b); NODEMAP takes the second Principalis node and the second Organalis node of (c) both into the second node of (b); and so on; SGMAP is the identity map of SGP onto itself, where SGP is the pertinent group of intervals, that is the group of distances in steps up a scale. Graph (b) is however not a homomorphic image of graph (d) under the analogous NODEMAP. For there is no possible SGMAP, mapping our intervals homomorphically into themselves, that satisfies both SGMAP(l) = 1 and SGMAP(3) = 0. Any homomorphism SGMAP that satisfies SGMAP(l) = 1 must satisfy SGMAP(3) = SGMAP(1 + 1 + 1) = SGMAP(l) + SGMAP(l) 4- SGMAP(l) = 1 + 1 + 1 = 3 . The graph of (d) can be constructed as a formal "product" of graph (b) with graph (e). But that is quite another matter. Network (d) must be distin204

2. The transcription is taken from Oliver Strunk, Source Readings in Music History (New York: W. W. Norton, 1950), p. 130.

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FIGURE 9.8

9.5.5

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guished not only from (c) but also from (f) and (g). (f) is a network whose graph is (b); each node of (f) contains a network whose graph is (e). (f) is thus a network-of-networks; the arrows on (f) labeled 1, — 1, and 0 transpose entire (e)-networks. (f) models the thought, "We are singing (the graph of) 'Nos qui vivimus,' singing diatessera ((e)-networks) as we go." (g) is a network whose graph is (e); each node of (g) contains a network whose graph is (b). (g) is thus, like (f), a network-of-networks. It reflects a way in which Organalis might think: "Principalis is singing 'Nos qui vivimus'; I too am singing 'Nos qui vivimus,' and my relation to Principalis is governed by the Symphony of the Diatesseron (the symbol '3' on the graph)." Our ability to form the "product network" of (d), and the networks-ofnetworks displayed in (f) and (g), is heavily dependent on the following aspects of the situation: The transformations involved in (b), that is T15 T_ l 5 and T0, are all operations; also the transformation T3 involved in (e) is an operation; also T3 commutes with T l 9 with T_ 15 and with T0. We shall not pursue the abstract theory of such matters any farther here.3 Earlier, in connection with the vocal line of the song "Angst und Hoffen" (figure 6.4), and again in connection with our study of twelve-tone rows as families of "protocol pairs," (section 6.2.4), we discussed various techniques for modeling series of pitches, pitch classes, or other objects. Figure 9.8(b) suggests yet another technique: We can regard a series as a certain type of transformation-network. Just what type is a matter we shall clarify and make formal later on (in section 9.7.7). The terminology we shall develop there will tell us that a network can model a series if the node/arrow system is "precedence-ordered and linearly ordered under that ordering." The three examples we have just studied, 9.5.3, 9.5.4, and 9.5.5, show us that there is a lot of variety in the forms that graph-homomorphisms can assume. In 9.5.3, NODEMAP was an isomorphism of the node/arrow system and SGMAP was a proper homomorphism of the semigroup, i.e. not an isomorphism. In 9.5.4, NODEMAP was onto but not 1-to-l, while SGMAP was again a proper homomorphism. In 9.5.5, NODEMAP was not 1-to-l, while SGMAP was an isomorphism. We now turn away from isographies, isomorphisms, and homomorphisms, to explore some different matters. When we use a transformation-

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3. T3, a gesture which we can call "climb three rungs up on the modal ladder," corresponds to the concept of dia + tesseron, given the difference in the manner of counting rungs. T3 is not the same gesture as RISE(4/3), meaning "rise so as to get higher in the harmonic pitch-ratio of 4-to3." RISE(4/3) does not commute with the transformations TI and T_j of the example. That is of course a salient problem of the style. And that is how our machinery views the problem. Our machinery also provides us with the formally different models of (d), (f), and (g), which give us interestingly different ways of thinking about what is going on in the Symphony.

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9.6.3

network to discuss events in a musical passage, the configuration of its node/arrow system may allow us to isolate and discuss formal properties of certain nodes that implicitly assert corresponding formal functions for the CONTENTS of those nodes. In this connection, it is interesting to study "input nodes" and "output nodes" for node/arrow systems, and hence for graphs and networks that involve those systems. 9.6.1 DEFINITION: An input node for a node/arrow system is a node IN to which no proper arrows point. That is, if (N, IN) is in the ARROW relation, we must have N = IN. Analogously, an output node is a node OUT from which no proper arrows issue. That is, if (OUT, N) is in the ARROW relation, we must have N = OUT. 9.6.2 EXAMPLE: In the network of figure 9.9, the node on the left is an input node and the node on the right is an output node.

FIGURE 9.9

The reader will recognize the graph as the "complementary gesture" from the Brahms Horn Trio. In discussing figure 7.7(g) earlier, we noted how the motif of the Bb octave-leap, when led into the complementary gesture, generates the pitch class G[?. And we explored some consequences of that generation. Here we can note that our intuitions about "putting in" Bj? and "getting out" G[? correspond nicely to the formal input and output functions of the left hand and right hand nodes on figure 9.9. 9.6.3 EXAMPLE: Figure 9.10 shows a network of Klangs whose graph we

FIGURE 9.10

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9.6.4

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earlier called CADENCE, when we were studying passages from Beethoven's First Symphony (in section 7.4). The Klang (C, +) that fills the left node of the figure is the same as the Klang that fills the right node. But the function of the Klang as CONTENTS(left node) is different from '^function as CONTENTS (right node). In the former capacity, the Klang is an input; in the latter capacity it is an output. The input and output functions for (C, +) in the network reflect very well in this setting two of the three principal ideas about tonicity that have governed most theories of tonality since the eighteenth century. The input function reflects the idea of tonic-as-generator, a tonic that asserts itself in the very act of sounding a tone, setting a musical process in action, a tonic which generates other tones through that action. The output function reflects the idea of tonic-as-goal, a tonic that appears at the end of a completed gesture as a point of repose towards which events have been moving. The third principal idea asserts as tonic a center of balance in a well-balanced structure. That idea, too, is manifest in the visual aspect of figure 9.10: The figure balances about the two nodes containing (C, +). We cannot completely translate out input/output formalities into ideas about tonicity. For instance we do not want to assert (G, +) or (F, +) on figure 9.10 as "tonics," even though the former Klang fills an input node and the latter Klang an output node. Nevertheless the input/output formalities are suggestive in connection with tonal theory. 9.6.4 EXAMPLE: Figure 9.11 shows a network whose nodes are filled by forms of the FATE motive from the opening of Die Walkiire, act 2, scene 4, the "Todesverkiindigung." We discussed this network in connection with figure 8.6 earlier. The horizontal arrows on the figure are labeled by the TRANSIToperation RICH; the curved arrows are labeled by TCH. The transformation

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FIGURE9.il

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9.7.1

BIND which labels the diagonal arrows takes a series of pitch classes and transforms it into that one of its retrograde-inverted forms which has the same first and last notes. BIND commutes with TCH. Figure 9.11 has precisely one input node, the node indicated at the lower left. The input node is filled by the FATE-form A-C-B, which thereby acquires a special generative function for the network. This formal status of A-C-B as network-generator corresponds very well with the musical priority we assigned the motive-form in our earlier analysis. There we noted how the entire opening of the "Todesverkiindigung" grows out of the measures immediately preceding the scene, measures in which the A-B gesture of the LOVE motive is harmonized with C-B in the bass, A functioning also as fundamental bass below C and hence implicitly asserting a relation of Abefore-C. The three protocol pairs (A, B), (C, B), and (A, C) determine the motive A-C-B as a partial ordering. Dramatically, the (A, B), (C, B) and (A, C) protocols in the music just before scene 4 represent Siegmund's relation to the sleeping Sieglinde, a relation which metaphorically lies under the entire Siegmund/Brunnhilde scene just as Sieglinde lies under it literally. The input and output functions we have been exploring are essentially rhythmic aspects of node/arrow systems, in a certain structural sense: Input nodes "happen before" other nodes with which they communicate; output nodes "happen after" others with which they communicate. More generally, one observes that the arrows of any node/arrow system have a formal rhythmic structure of their own, a structure which can engage musical rhythm in varied and sometimes complicated ways. Our practice of laying out graphs visually so that most arrows go from left to right on the page has made it easy for us to put off investigating the issues that arise when we try to match the internal arrow-flows of a network with the temporal flow of the music upon which the network comments. Here, now, we shall attempt to explore some of those issues, though we can hardly do them justice in one section of one chapter. We recall the definition of an "arrow chain" from node N to node N' in a node/arrow system; it is a finite series of nodes N0, N t , . . . , Nj such that N0 = N, Nj = N', and (N^, N3) is in the ARROW relation for each j between 1 and J inclusive (9.1.4). 9.7.1 DEFINITION: An arrow chain (as above) is proper if there is at least one j between 1 and J inclusive such that (Njs N^) is not in the ARROW relation. Intuitively, the definition demands at least one "one-way arrow" along the chain. (The implicit arrow between any node and itself counts as "twoway" in this connection.) Another way of intuiting the definition is to think of a proper arrow-chain as one that cannot be "walked backwards."

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9.7.2 DEFINITION: In a node/arrow system, node N precedes node N', and N' follows N, if there exists some proper arrow-chain from N to N'. One must be careful to distinguish the relation "N precedes N'" from the relation "N is in the ARROW relation to N'."

FIGURE 9.12 On figure 9.12, for instance, M t precedes M3 because M1-M2-M3 is an arrow-chain from M^ to M 3 that involves a one-way arrow (from M 2 to M3). But M! is not in the ARROW relation to M 3 . M t is, on the other hand, in the ARROW relation to M 2 . But Mt does not precede M2: There is no arrowchain from M! to M 2 which involves any one-way arrow. 9.7.3 DEFINITION: A node/arrow system is precedence-ordered if there is no pair of nodes (N, N') such that N both precedes and follows N'. The reason why we speak of a precedence-ordered system as "ordered" is implicit in the following theorem.

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9.7.4 THEOREM: Let (NODES, ARROW) be a precedence-ordered node/ arrow system. Let PRECEDENCE be the family of node-pairs (Ni, N 2 ) such that N: precedes N 2 . Then PRECEDENCE is a (strict) partial ordering on NODES. That is, PRECEDENCE satisfies conditions (PO1) and (PO2) below. (PO1): There is no pair (N l 9 N 2 ) such that both (N 1 ,N 2 ) and (N^NJ are members of PRECEDENCE. (PO2): If (N! , N 2 ) and (N 2 , N3) are both members of PRECEDENCE, then so is (N 1? N 3 ). Proof: (PO1) for PRECEDENCE is equivalent to the condition of 9.7.3, which is true in a precedence-ordered system. (PO2) is obvious: If there is a "good" arrow-chain from N x to N2 and a "good" arrow-chain from N2 to N 3 , then there will be a "good" arrow-chain from N! to N3. q.e.d. The reader will recall (PO1) and (PO2) from section 6.2.4 earlier, where we invoked them to characterize collections of protocol-pairs that were (strict) partial orderings on the twelve pitch-classes. Here we use the same mathematical conditions to characterize (strict) partial orderings on NODES. A precedence-ordered system is at least potentially compatible with our naive sense of chronology. When used for analytic purposes, that system will

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9.7.5

not have to assert that one musical event both "precedes" and "follows" another, in the strictly formal sense of 9.7.2. There is nothing intrinsically correct or good about avoiding such assertions, but it is useful to have at hand a formal criterion that characterizes those particular node/arrow systems which enable us to avoid them.4 9.7.5 OPTIONAL: We can be quite precise mathematically about what we mean in saying that a precedence-ordered system is "potentially compatible with our naive sense of chronology." Readers who do not care can move on from here to section 9.7.6. Given the family of NODES, let us review what we mean when we speak of a (strict) partial ordering on NODES. We mean a collection P of nodepairs, a collection that satisfies conditions (PO1) and (PO2) following. (PO1): There is no pair (N t , N 2 ) such that both (N t , N2) and (N2, NJ are members of P. (PO2): If (N l 5 N 2 ) and (N 2 ,N 3 ) are both members or P, then so is (Ni, N3). Theorem 9.7.4 told us that P = PRECEDENCE is a (strict) partial ordering on NODES in a precedence-ordered node/arrow system. A partial ordering L on NODES is "linear" or "simple" when, given any distinct nodes N and N', either (N, N') or (N', N) is a member of L. The following theorem can be proved: If NODES is finite, containing J nodes, and if L is a linear ordering on NODES, then the members of NODES can be arranged in a series Nj,, N 2 , . . . , N, such that (NJ5 N k ) is a member of L if and only if j is less than k. The partial ordering P is "weaker than" the partial ordering Q, and Q is "stronger than" P, when every node-pair that is a member of P is a member of Q, and Q contains some pair that P does not contain. That is, this relation between P and Q obtains when P is strictly included in Q as a set of node-pairs. A partial ordering P is "maximally strong" when there is no partial ordering Q stronger than P. The following theorem can be proved: Every maximally strong partial ordering is linear, and every linear ordering is maximally strong. When NODES is finite, the following theorem can also be proved: Given any partial ordering P, there exists some maximally strong (i.e. linear) ordering L which is either equal to P or stronger than P. When NODES is not finite, the same theorem can be proved if one makes an additional logical assumption which need not concern us here. We can apply all these theorems as follows: Given a precedence-ordered node/arrow system, there exists a linear ordering L of NODES which is either 4. Jonathan Kramer develops a very interesting sense in which he claims that the first movement of Beethoven's F-major String Quartet op. 135 begins with its ending. Kramer does this in the article, "Multiple and Non-Linear Time in Beethoven's Opus 135," Perspectives of New Music vol. 11, no. 2 (Spring-Summer 1973), 122-45. The last movement of Haydn's D-major String Quartet op. 76, no. 5, might serve as another example.

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stronger than PRECEDENCE or equal to PRECEDENCE. Assuming that NODES is finite (which we shall assume from now on), we can then L-order the notes of this precedence-ordered system as a series Nj, N2, ... , N;, in such wise that (N, Nk) is a pair within L if and only if j is less than k. Since PRECEDENCE is weaker than or equal to L, it follows that j must be less than k whenever N. precedes Nk in the system. So if we imagine a chronology in which Nj "happens first," N2 "happens second, ". . . , and N; "happens last," this chronology cannot be violated by the precedence relation of the system. That is, whenever N. formally precedes Nk, N. will "happen before" Nk in the L-chronology. If PRECEDENCE is not itself linear, there will be more than one linear ordering stronger than PRECEDENCE; accordingly there will be more than one "linear chronology" of the above sort with which the precedence relation is compatible.

FIGURE 9.13 For example, in the precedence-ordered system of figure 9.13 we can take either of the two left-hand nodes as "N," and the other one as "N2," in imposing a linear chronology; we can similarly take either of the two righthand nodes as "N3" and the other as "N4." This reflects the structure of PRECEDENCE here, which makes each left-hand node precede each righthand node, while neither left-hand node precedes the other and neither righthand node precedes the other. The groupings of segments within the various possible linear chronologies are typical.

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9.7.6 The gist of section 9.7.5 may be summarized as follows: When a finite node/arrow system is precedence-ordered, its J nodes can be labeled by the numbers 1 through J in such fashion that when j is less than k, it is possible for the j* node N. to precede the k* node Nk, but impossible for Nk to precede N. This means that we can always display the system visually on a page (in theory) using a format in which all one-way arrows go from left to right. We must be very careful to recognize that the words "precedence" and "precede" in the paragraph above refer to formal aspects of the node/arrow configuration, and not necessarily to the musical chronology of any passage upon which a network using that node/arrow system may be commenting. Even when the node/arrow system is precedence-ordered, it is perfectly pos-

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9.7.6

sible for a node N to precede a node N' in a network, while the contents of N are heard after the contents of N' in the pertinent music. Figure 9.14 will help us explore the possibility.

FIGURE 9.14 (a) of the figure shows a network of Klangs. All the Klangs are understood to be major, except for the lower-case e|?-Klang, which is minor. The network models the harmonic progression at the opening of the slow movement in Beethoven's Appassionato Sonata. The Ej?-major Klang is bracketed to indicate that the Klang is not actually sounded but is theoretically understood. The fourth sonority heard in the music is modeled by two Klangs. It is first understood as a G(?-major Klang (with added sixth); then it is understood as an eb-minor Klang (with minor seventh, inverted). This is Rameau's double emploi. The arrow goes only one way, from G[? to e(? but not back. The operation REL takes a Klang into its relative minor/major. The left-to-right format of (a) arranges the nodes in an order that corresponds to the order of events in the music. This order is not compatible with the precedence relation of the node/arrow system, even though that

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system is precedence-ordered. We see this incompatibility on (a) in the form of some one-way arrows that point from right to left, (b) of the figure rearranges the nodes on the page, in a new visual format compatible with the precedenceordering; on (b), all one-way arrows point from left to right. We shall discuss later the box containing the word START and the arrow issuing from that box; the reader should ignore them for the time being. The left-to-right format of (b), while respecting the one-way arrows, violates the musical chronology of the passage. A special aspect of this violation is the clarity with which it accents the input function of the two Gb nodes. The input function violates musical chronology: Db, not Gb, is what gets this music under way. The input function also violates our sense of structural "priority": Db, not Gb, is the Klang structurally prior to all others here. The Gb Klangs should be manifest as inflecting such a Db "point of departure" for the tonal structure; the Gb Klangs should not themselves appear as structural "points of departure," which figure 9.14(b) seems to make them. These concerns are very well addressed by Schenkerian theory, which provides us with an apparatus of hierarchical levels, voice-leading events, and subordinate Klangs harmonizing voice-leading events. With such a model in mind, we can easily see that the configuration of Klangs displayed in figure 9.14((a) or (b)) is not an adequate representation for the way tonality controls this passage. The representation fails to model the middleground progression Db-Ab-Db supporting a sustained melodic fifth degree; it fails to model the inflection of that fifth degree by its upper neighbor twice, once within the opening Db Klang of the middleground and once on the way from Db to Ab Klangs within the middleground; it fails to show how the Gb Klangs support that neighboring inflection of the principal tone 5. Later on, we shall construct a "Schenkerian network" that can address these matters within our network format. Meanwhile, we can observe that figure 9.14, incomplete as it is for analytic purposes, still does represent a foreground configuration of Klangs that engages a valid part of our musical experience. It would impoverish, not refine, that experience to explain away the peripatetics of the configuration as "merely" incidental, as "naught but" the harmonization of voice leading, as "only" preparing the eventual dominant, and the like. More specifically, even though the formal Gb inputs on figure 9.14(b) must yield somehow to overall D(? priority in a complete analysis, the formal Gb inputs still reflect an interesting feature of the music, a feature it would be easy to overlook if one were to plow through the foreground to a middleground level prematurely. One hears this feature upon singing over the music mentally while looking at network-format (b). The input Gb nodes are then sensed strongly as "carriage returns," especially the second one. This feeling of "carriage return" on the Gb harmonies, sensed as one reads figure 9.14(b) in the musical chronology, interacts very effectively with the phrasing of the passage.

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9.7.6

It would be hard to express the carriage-return function so precisely in any other theoretical vocabulary. The function marks those precise moments in the listening experience at which we shoot back from right to left on the figure, violating the sense of the one-way arrows. More formally: The carriage-return moments are precisely those moments in the listening chronology at which that chronology violates precedence-ordering. At all other moments, listening chronology is compatible with precedence-ordering. By using the expression "precedence-ordering," we are implicitly supposing that the node/arrow system to which we are applying this concept is precedenceordered, with all that this entails about left-to-right, and so on. Our theoretical machinery enables us to pinpoint carriage-return moments, define them precisely, and attribute a special theoretical function to them. All this duly noted, we have still not resolved the theoretical problems raised by the formal priority our model assigns to Gj? nodes over D[? nodes as "input" to the flow of events, and to the tonal structure. One line of attack on the problems is suggested by the box on figure 9.14(b) containing the word START, and by the arrow from that box to the indicated D[? node. We may formally adjoin the box and the arrow to the node/arrow system, and the contents START to the network, so as to help ourselves along. The START node is an input node, and we can declare a formal convention that it supersedes all other input nodes in function. When we start at the START node, we cannot reach the G|? nodes without traversing some arrows backwards. We might use just that feature of the system as a formal criterion for assigning a special sort of subordinate status to those nodes. There is no problem walking arrows backwards here because our transformations are all operations. Our analytic criterion for pointing the START arrow at the indicated D[? node could be diachronic (because the music starts there) ar synchronic (because that node begins the foreground elaboration of a higherlevel tonic function). Another formal device for modeling a discrepancy between precedenceordering and musical chronology is indicated by figure 9.15. The figure

FIGURE 9.15

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attaches to each node of figure 9.14(b) a certain time span; the musical event corresponding to the contents of that node occurs over that time span. Formally, we are constructing a new apparatus which we might call a "time-spanning network." That is a transformation network together with a function TIMESPAN that maps each node into a certain time span. In the case of a network whose contents are already time spans, we could take TIMESPAN(N) = CONTENTS(N). A time-spanning network could model via TIMESPAN the exact time spans over which its events occur; that is the case with figure 9.15. Or TIMESPAN(N) could model a certain range of time during which CONTENTS (N) might occur. Instead of attaching time spans to the nodes of a network in this way, we could also attach time spans to the contents of those nodes. On figure 9.15, for example, instead of a node N with CONTENTS(N) = Gb and TIMESPAN(N) = (3.5, .5), we could have a node N whose CONTENTS are the ordered pair (Gb,(3.5, .5)). On the revised figure 9.15, the family of transformations would have to be more complicated. Each graphtransformation would be an ordered pair comprising both a Klangtransformation and a time-span-transformation. Yet another formal device at our disposal is to incorporate Schenkerian transformations into a network format. Figure 9.16 indicates one way in which this might be done. The contents of the nodes in this network are ordered triples (Klang, degree, level). Thus (Ab, 5, 2) denotes an Ab-major Klang supporting a fifth degree in the structural melodic voice at level 2. The operation PROJ + incre-

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FIGURE 9.16

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9.7.6

ments the level of its operand, transforming (Kng, deg, lev) into (Kng, deg, lev + 1). Klang and degree are thereby PROJected one level closer to the foreground. The operation PROJ— is the inverse of PROJ + ; it decrements the level of (Kng, deg, lev), transforming it into (Kng, deg, lev — 1). Since the objects (Kng, deg, lev) are purely formal, we can always reference levels "lev + 1" and "lev — 1" formally, even when they have no analytic pertinence to a given situation. That is desirable here, in order to make PROJ-f and PROJ — well-behaved context-free operations. To save space on figure 9.16, the PROJ arrows have all been drawn as two-way, signifying PROJ + or PROJ — as appropriate. Within each level on the figure, each transformation is specified by a pair (Klangtrans, degtrans). Klangtrans is the pertinent Klang transformation, and degtrans is the pertinent degree transformation. Thus an arrow labeled (DOM, SUST) from (Kng, deg, lev) to (Kng', deg', lev) indicates that the Klang Kng is the dominant of the Klang Kng', while degree deg sustains to become degree deg' = deg. An arrow labeled (SUED, N +) from (Kng, deg, lev) to (Kng', deg', lev) indicates that Kng is the subdominant of Kng', while degree deg is the upper neighbor to degree deg'. Distinguishing levels in the manner of figure 9.16 enables us to make input terminology conform better to our intuitions. The Gb nodes of figure 9.16 are indeed still input nodes, but we can now say that they are "input at level 3. In the same sense, we can say that the node containing (Ab, 5,2) is "input at level 2," distinguishing it in this capacity from the node containing (Ab, 5,3). The Db nodes of level 2 are both output nodes at level 2; the Db nodes of level 3 are all output nodes at level 3. The Db node of level 1 is both input and output at that level, according to our definition (9.6.1). Figure 9.16 may be made to engage rhythmic mensuration by attaching time spans to its nodes along one of the lines suggested earlier. Determining where to end the time span for (Db, 5,2) and where to begin the time span for (A)?, 5,2) is an interesting methodological and phenomenological problem.5 a number of assertions seem plausible. My own preference is to carry the Db time span right up to the Ab Klang, and also to begin the At? time span right after the Db Klang stops sounding. The two time spans would then overlap on level 2, and the Gb-eb-Eb part of level 3 would all occur during the time span of the overlap. This satisfies my hearing, and it is also an elegant way to elaborate the theoretical idea behind the double emploi. Figure 9.16 as it stands is not equivalent to a Schenkerian reading, which would devote less attention to Klangs that do not project Stufen, more attention to the bass line and to the essential counterpoint between the outer 5. Fred Lerdahl and Ray Jackendoff discuss pertinent matters at length in their important book, A Generative Theory of Tonal Music (Cambridge, Mass, and London: MIT Press, 1983). The interested reader can explore various ways in which their tonal tree-structures resemble and differ from transformation-networks of the sort displayed by figure 9.16.

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FIGURE 9.17

voices.6 I think it likely, though I am not certain, that actual Schenkerian graphs could be represented in network formats of the sort under present consideration, when suitably extended. 9.7.7 A short time ago, in connection with our study of "Nos qui vivimus," we observed that a series of objects could be modeled by a certain type of transformation network. Now we are in a position to specify formally just

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6. A good Schenkerian analysis of the theme as a whole is presented by Allen Forte and Steven E. Gilbert in their Introduction to Schenkerian Analysis (New York: W. W. Norton, 1982), 154-56.

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9.7.7

what type. Specifically, a formal "melody" can be defined as a transformation network whose node/arrow system is precedence-ordered, and linearly ordered under that ordering. According to our work so far, that means there will be one and only one way of labeling the J nodes with the numbers 1 through J so as to be compatible with the one-way arrows of the system. This concept of "melody" is very elaborate, for it carries within it the idea of transforming earlier events to later ones, along the arrows of the network, by transformations from a specified semigroup. That idea was not implicit in our earlier models for series. Different configurations of arrows can give rise to one and the same precedence-ordering on a given family of NODES. When that ordering is linear, corresponding networks whose nodes have the same contents will nevertheless be formally different "melodies" by our definition. We should find a better word than "melody" if we want to continue to work along these lines. For instance, the two node/arrow systems of figure 9.17(a) and (b) give rise to one and the same precedence-ordering. Networks (a) and (b) are formally different "melodies."7

7. "Melody," however, is exactly the proper term if we want to follow and extend the usage of Ernst Kurth, when he claims that "the basis of melody is, in the psychological sense, not a succession of tones... but rather the impetus of transition between the tones." Figures 9.17 (a) and (b) depict different transition-structures. The cited text appears in Grundlagen des linearen Kontrapunkts (Bern: Drechsel, 1917), p. 2. ("Der Grundinhalt des Melodischen ist im psychologischen Sinne nicht eine Folge von Tonen... sondern das Moment des tfbergangs zwischen den Tonen.")

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10

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FIGURE 10.1

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10.1 EXAMPLE: Figure 10.1 sketches some motivic work from the passage opening the development section in the last movement of Mozart's G-Minor Symphony, K.550. Up to measure 133, the entire orchestra except horns is playing the indicated line, allegro assai and essentially staccato. There is one exception: The quarter note B which appears on the figure at the end of measure 128 is actually an eighth rest and a sixteenth triplet, filling in the

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diminished fourth scalewise under a slur. I am supposing that it is legitimate for us to consider that gesture a variation on the motivic model of the figure, inter alia. From the pickup of measure 127 until measure 133, the passage projects forms of a pitch motive PM, comprising a diminished fourth up followed by a diminished seventh down. This motive is Rl-chained in the manner familiar to us by now; the transformation RICH takes us along the chain of PMforms (E-Ab-B), (Ab-B-Eb), (B-Eb-Ffl), (Eb-Ffr-Bb), (FJ-Bb-Ctf), (Bb-Ctf-F), and (C#-F-G#). The TCH interval is a falling fourth. On figure 10.1, the four prime forms of PM are indicated by the brackets numbered 1,2, 3, and 4a/b. The form (C#-F-G#) is bracketed twice, by brackets 4a and 4b. This reflects an interesting ambiguity about its rhythmic location, an ambiguity we shall soon investigate. Beneath the staff on the figure a series of numerical durations appears. The numbers label the distances in quarter-note beats between the time points at which successive notes are attacked. I am supposing that we hear an ictus at the barline of measure 128. (Later on, we shall hear how the music of measures 125-27 prepares this ictus.) The barline of measure 128 thereby articulates the duration of five quarters, between the two B naturals, into (2 + 3) quarters. The three attacks within PM, together with the ictus, define the durational series 1 + 2 + 2 (quarters) as a rhythmic setting for the pitch idea. We shall call 1 + 2 + 2 the "durational motive" DM. On the figure, bracket 1 is placed around PM so as to articulate DM; the corresponding durationnumbers 1, 2, and 2 below the staff are also bracketed. Bracket 2 also articulates DM, now as the rhythmic setting for the next TCH-form of PM. Bracket 3 is placed around the next TCH-form of the pitch motive, that is around F#-Bb-C#. Here the rhythmic setting is no longer DM itself, but rather an augmented (rhythmically transposed) form of DM: 2 + 4 + 4. The rhythmic transposition will be denoted as T2, multiplying all durations by 2. The next TCH stage of the pitch motive is C#-F-G#. Bracket 4a gives this stage the durational setting 4 + 4 + 2; bracket 4b gives it the setting 4 + 2 + 2. Both these durational settings are serial transformations of DM. The series of 4a retrogrades the elements of series 3: 4 + 4 + 2 retrogrades 2 + 4 + 4. The series of 4b inverts the elements of series 3: 4 + 2 + 2 inverts 2 + 4 + 4. We can regard the inversion as multiplicative, about the numerical product 8: 8 divided by 2,4, and 4 (series 3) yields 4,2, and 2 (series 4b). Or we can regard the inversion as additive, about the numerical sum 6:6 take away 2, 4, and 4 (series 3) yields 4,2, and 2 (series 4b). Whichever way we think of the inversion, durational series 4b is a retrograde-inversion of durational series 4a. In fact, series 4b is precisely RICH(series 4a). Thus the transformational motif of Rl-chaining, very audible in the pitch structure of the passage, is also projected in the durational structure. The

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musical effect is bewildering on first hearing, because both the durational series involved in the RICH relation, that is both 4a and 4b, are heard as alternative rhythmic settings for one and the same pitch motive. And this effect strikes our ears just as we are beginning to adapt to a heavy reliance on motivic listening, having temporarily lost the local tonic. Bracket 5 shows how the play of durational motives continues on through the change of texture at measure 133. The durational series for bracket 5, 2 + 2 + 4, is a retrograde-inversion (multiplicative or additive) of series 3, just as series 4b was a retrograde-inversion of series 4a. Since series 3 was a multiplicative transposition of DM, durational series 5 is also a multiplicative retrograde-inversion of DM itself. In fact series 5 is precisely the RICH-transform of DM, using multiplicative inversion: (multiplicative) RICH(1 +2 + 2 ) - 2 + 2 +4.

FIGURE 10.2

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Figure 10.2 summarizes in a network the transformational interrelations of the durational motive-forms so far surveyed. On the figure, T is multiplicative T 2 ,1 is the pertinent multiplicative inversion, and RICH is considered to be defined multiplicatively also. The RICH-arrow between the node marked "DM = 1; 2" and the node marked "5" suggests some aural explorations. Going back to figure 10.1, we can hear the rhythmic RICH-relation between bracket 1 and bracket 5 by focusing on a linking rhythmic element, that is, the rhythmic identity of measure 127 (at the end of bracket 1) with measure 132 (at the beginning of bracket 5). This is exactly the linking aspect of the Rl-chain involved. We hear that the high Ab of measure 127 and the low G# of measure 132 are the registral boundaries for figure 10.1. The enharmonically equivalent A(? and G# are tied together, too, by the diminished-seventh harmony implied over measure 127; this harmony implicitly recurs under bracket 5. We can hear the rhythmic RICH-relation between bracket 2 and bracket 5 by focusing on measures 129-30 as an intermediating stage. That pair of measures spans the same durational series as series 5: 2 + 2 + 4. And in

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10.1

measures 129-30 the series 2 + 2 + 4 is explicitly linked to the end of bracket 2, as the RICH-transform of series 2. This rhythmic RICH-transformation is easy to pick up aurally because it coincides with a RICH-transformation of the pitch motives that correspond: B-E|?-F#, with rhythm 1 + 2 + 2, is transformed into RICH(B-Eb-F#) = Eb~F#-Bb, with rhythm RICH(1 + 2 + 2) = 2 + 2 + 4. It remains then only to hear that measures 132-33, under bracket 5, reproduce the attack rhythm of measures 129-30, the intermediating stage. And this is quite possible if one hears measures 127-29 and measures 130-32 as a pair of three-measure groups, or if one hears measures 129-31 and 132-34 as such a pair. Let us return now to brackets 6 and 7 on figure 10.1. Bracket 6 applies to the winds only, whose durational series is 4 + 2 -I- 2 over this span. The series recapitulates the series of 4b. Despite the radical change of texture, we are aided in hearing this relation by the boundary tones of measures 133-35, specifically by the opening high F of measure 133, the final C# at measure 135, and the low G# at the end of measure 134. Those three notes keep alive in permuted order the pitch classes C#, F, and G# from the PM-form of measures 131-32, spanned by bracket 4b. Bracket 7 shows how the rhythm of the winds during bracket 6 becomes diminuted approaching the barline of measure 135, into the diminuted series 2 + 1 + 1. This diminution (multiplicative transposition by ^) undoes the effect of the earlier augmentation (transposition by 2), the augmentation we underwent in passing from bracket 2 to bracket 3. Accordingly, durational series 7 bears to series 1 = 2 = DM the same relation that series 6 = 4b, the augmentation of series 7, bore to series 3, the augmentation of series 1 = 2 = DM. Thus series 7 inverts series 1 either multiplicatively or additively: 2 divided by 1,2, and 2 (series 1) yields 2,1, and 1 (series 7); alternatively, 3 take away 1,2, and 2 also yields 2, 1, and 1. Figure 10.3 extends figure 10.2 to display some of the new relationships involving series 6 and series 7. J is the pertinent multiplicative inversion \\. Our study has shown us how the straightforward Rl-chaining in one dimension of this music both suggests and conceals a very elaborate transformational network, also involving RICH-relations, in another dimension. The whole discussion is grossly oversimplified as regards both dimensions. We have not, for instance, considered the rhythmic implications of the PM-form that goes from the G# of measure 132 to the F of measure 133 in the bassoon (an octave below figure 10.1), and thence through the stepwise filler down to the C# of measure 135 (again an octave below the figure). This form retrogrades in register the PM-form of measures 131-32. The stepwise filling-in of the diminished fourth F-to-C# is a huge rhythmic expansion, in retrograde, of the tiny sixteenth-note triplet figure that filled in from B to Eb in measures 128-29. We have also not considered the crucial way in which the music gets to figure 10.1. The Rl-chaining of PM actually begins earlier; it involves the

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FIGURE 10.3 all-important high D|? of the rocket theme, the main theme of the movement and the source of the diminished-seventh leaps. Figure 10.4 helps us explore this a bit.

FIGURE 10.4

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The beamed arrows on the figure pick out a PM-form that extends the RI chain of pitch motives one TCH stage backwards from our earlier analysis. The rhythmic setting of the PM-form A-D[?-E is not related to DM by any standard serial transformation. It does, however, interact with the ictus at the barline of measure 128. Figure 10.4 allows us to hear the source of the 3 quarters' duration between the ictus and the following B. That duration is in rhythmic sequence with the 3 quarters' duration between the high D[? and the E natural of measure 126. The 3 quarters' sequence causes the rhythmic identity of bracket 1 with bracket 2 to extend backwards. Not only do EAJ7-B-ictus and B-Eb-Fft-Bb demarcate the same durational series DM = 1 + 2 + 2, but also Dj?-E-A[7-B-ictus and ictus-B-Eb-F#-Bb demarcate

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70.2

the same series 3 + 1 + 2 + 2. In fact, the rhythmic identity extends back even farther: A-Db-E-Ab-B-ictus and B-ictus-B-Eb-F#-Bb demarcate the same series 2 + 3 + 1+2 + 2. Only after the Bb of measure 130 does the rhythmic parallelism break off: After that Bb the first duration of 4 appears, launching us into the transformational complexities of figure 10.2.

FIGURE 10.5

10.2 EXAMPLE: Figure 10.5 sketches some structural aspects of Bartok's "Syncopation," no. 133 from Mikrokosmos, vol. 5. Our ability to hear prolongation in this sort of context is the subject of an important study by Roy Travis.1 Travis tries to hear (GBD) + (Eb F#) as a tonic chord for the piece. It works better to hear (G) + (A#C#D#F#) in that capacity. This is the final sonority of the piece and the only complete harmony attacked by both hands simultaneously, an event which occurs at the final barline. That is, it is the only harmony which does not involve a "Syncopation." The syncopations of the music bring out a broad variety of structural contrasts between the two hands. The left hand plays white notes; the right hand plays black notes. The left hand plays "down" both metrically and sotto; the right hand plays "up" both metrically and sopra (with its black notes). The white notes in the left hand arrange themselves into triadic formations within a G Mixolydian mode; the black notes in the right hand arrange themselves within a pentatonic F# mode. The left hand's chord of reference is a G-major |, a chord with G on the bottom; the right hand's chord of reference is an F# added-sixth harmony in 3 inversion, a chord with F# on the top. Over measures 1-10 of figure 10.5, one sees how these elements contend during the opening section of the piece. The black notes win out. The opening (D#F#) of the figure is spelled (EbF#) by the composer, to emphasize its local dependency on the left-hand G tonic. But as the section progresses, (EbF#) becomes (EbGb) and shows that it can move down an octave to displace the 1. "Toward a New Concept of Tonality?" Journal of Music Theory, vol. 3, no. 2 (November 1959), 257-84.

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left-hand G at measure 9. The black dyad celebrates the success of its incursion by bobbing triumphantly back up to its original register, with a crescendo to sf; now the composer spells it (D#F#), reflecting its forthcoming orientation towards the right-hand tonic F#. The next section of the piece, measures 11-25, shows (D#F#) continuing to exercise its registral mobility: It moves up an octave over that span of the piece, along with C# and eventually A#. The octave transfer, represented by a slur on figure 10.5, plows through the climactic D of measure 18. This D, as fifth of G, shows that left-hand, G-oriented material can also become registrally mobile. Ds and Gs continue to move downwards in register from measure 25 to the end, while the right-hand chord of measure 25 moves up another octave during that time. Bartok puts (D#F#) back into its original spelling, as (E|?F#), from measure 25 on. In the right-hand scheme of things, as displayed on figure 10.5, the climactic D5 of measure 18 is apparently "only" a chromatic passing tone. But D5 is also the pitch about which the sonority A#4, C#5, D#5, F#5 is inversionally symmetrical. That sonority, as it appears with a structural downbeat at measure 25, projects the chord-of-reference for the right hand. Also, the pitch class D is a center of inversion for both the blacknote collection and the white-note collection. In both the left hand and the right hand over measures 11-25, RIchaining provides the means of harmonic progression. The left hand produces the Rl-chain of registrally ordered trichords A-C-E, C-E-G, E-G-B, G-B-D. There is even a hint that this process continues on in register, as the right hand takes over the climactic D5 at measure 18 and leads it on up to F#5. But the black tonic F# is too clearly foreign to the white-note chain, for this hint to develop further. Indeed the left hand emphasizes just that point, at measure 25, by providing the white F natural (not the black F#) as a chordal third above D, within the sonority G-B-D-F. The F natural breaks the RIchain of the left hand, which cannot pass the color barrier. Just as the Rl-chain of white trichords in the left hand appears beamed over measures 11-25 on figure 10.5, so do two analogous Rl-chains of black trichords in the right hand. The first chain breaks at F#-G#-B; rather than continuing on to the next RICH-stage G#-B-C#, the right hand substitutes a different form of the same trichord, namely G#-A$-C#, and then starts a new Rl-chain therefrom. The color barrier is again involved: Once the white B has been generated by the Rl-chain in the right hand, it must be replaced by the black A# = Bj?. So, just when the Rl-chaining is about to carry F#-G#-B on to G#-B-C#, A# substitutes for B and the next trichord is G#-A#-C# instead. The substitution preserves the outer voices of the trichord involved, and also preserves its set-class. The replacement of B by A# recapitulates in a setting of F# tonicity the replacement of B by B[? that we heard earlier, at measure 3, in a setting of G tonicity. The double Rl-chaining of the right hand over measures 11 -25 is also

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10.3

FIGURE 10.6

involved with a cadential proportioning (balance) of the trichord forms. Figure 10.6 displays this proportioning by a pertinent transformationnetwork. The straight diagonal arrows are RICH transformations. Among its input and output chords, figure 10.6 manifests a classical "triple proportion": C#-D#-F# (input) is to Ftf-G#-B (output) as G#-A#-C# (input) is to C#-D#-F# (output). The triple proportion also involves an isography of the left side, on figure 10.6, with the right side, on the same figure. That is, the relation of the material grouped by the two right-hand beams, over measures 11-25 on figure 10.5, is an isography. Intervals of 5 span the trichords and (therefore) measure the TCH-transpositions. They make the triple proportion here sound almost like I-IV-V-I, especially since the pitch class F#, appearing at the bottom of the first "I" and at the top of the final "I" in the I-IV-V-I, has a certain tonicity about it as regards the pitch structure of the right hand. Perhaps this sound led Travis to hear a functional I-IV-V-I governing the tonal structure of the piece as a whole. 10.3 EXAMPLE: Figure 10.7 sketches aspects of the opening seven measures from the first of Prokofieff's Melodies op. 35.1 am indebted to Neil Minturn for bringing this passage to my attention. The four-fiat signature on the figure is mine; Prokofieff writes no signature. The harmony in the music is far from traditional. Yet it is diatonic enough, and the outer voices are diatonic enough, so that some harmonic events stand out as "strange." Foremost among these are the cadence harmonies, E minor over the last half of measure 2 and Eb minor over the last half of measure 5. At both these cadences we expect Eb-major harmony. We can analyze the cadential substitutions by using the terminology of Klangs and Klang-transformations. The network of figure 10.8 (a) does so, and also brings the D-major Klang of measure 6 into the picture: D major, the SLIDE transform of Eb minor, bears to the latter Klang the same relation

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FIGURE 10.8

228

which E|j major, a Klang heard in the upper voices of measure 1 and expected at measure 2^, bears to E minor, the Klang actually heard at measure 2j. Figure 10.8 (a) does not attempt to engage the C-major and F-minor triads of measure 6 in its Klang-network. The proportion among the four

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10.3

triads of measure 6 (with pickup) is clear enough aurally, supported as it is by a rhythm-and-contour motive: E[? minor is to C major as F minor is to D major. But the proportion seems hard to portray by Klang-relations that mesh with our aural intuitions. Formally, we can of course note that F minor is the minor subdominant of C major, and that C major is the major dominant of F minor: (F, -) (PAR) (SUED) = (C,+) and (C, +) (DOM) (PAR) = (F, -). But how functional are these relations in a context that emphasizes the proportion (E(? minor)-is-to-(C major) as (F minor)-is-to-(D major), all after a cadence in Et>? And what sort of Klang relation might we hear in this tempo and rhythm governing the terms of that proportion? Can we hear, for instance, (Eb, -)(PAR)(REL)(PAR) = (C, +) here? Our attempt to hear any functional Klang-relation is bound to be hindered by the parallel voice-leading of the chords involved here. It seems more fruitful to analyze the triadic formations of measure 6 using the discourse of pitch-class sets and pitch-class transformations. Indeed we can fruitfully study the triadic formations of the entire passage using that discourse. Figure 10.8(b) shows the result. To save space, major and minor triads are denoted there by upper- and lower-case letter names. The new analysis describes the triadic structure of the passage as follows, ignoring sevenths and added sixths. First, E minor substitutes for an expected Efc> major in the cadence at measure 2^; the transformation involved is inversion about G, as depicted by the leftmost vertical arrow on figure 10.8(b). (Since we are now talking about pitch-class sets and not Klangs, IQ is a pertinent transformation while SLIDE, which inverts a Klang about its mediant, is no longer pertinent.) The pitch class G is a plausible center of inversion; it is highly accented by its boundary functions within the opening melodic phrase of measures 1-3. In particular, the substitution of E-minor harmony for E[? major occurs exactly when the melody reaches its climax G, and the E-minor harmony remains around while the melody drops back to the low G, where it sits until its next phrase begins in measure 4 (on that G). The next vertical arrow on figure 10.8(b) shows inversion-about-F#-andG relating the E|?-major harmony we expect to the Ej?-minor harmony we hear, when the next cadence arrives at measure 5j. This inversion, which replaces the pitch class G of the E(?-major harmony by the pitch class F# = G[? of Eb minor, thereby develops the chromatic relation of the pitch classes G and F#, the relation which governed the first chromatic event of the melody (and of the music as a whole), when F# appeared on the second beat of measure 1. The structuring power of that F# is then developed as shown by the next vertical arrows on figure 10.8(b). F# becomes a new center of inversion: Inversion-about-F# relates E^-minor harmony to D-major harmony, and also (NB) C-major harmony to F-minor harmony, all during the new thematic material of measure 6 and following. E[? -minor and D-major harmonies mark

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the temporal boundaries of the little phrase, while C-major and F-minor harmonies mark its registral boundaries. The repeated rhythm-and-contour motif within the phrase (of measure 6 with pickup) groups E[j minor with C major, and F minor with D major. Since all the minor/major relations of figure 10.8(b) so far have been analyzed as pitch-class inversions, the figure also explores hearing E[? minor become C major by inversion (about F), and hearing F minor become D major by inversion (about G). Those inversions are depicted by horizontal arrows on the figure. Ip and I % thereby inflect the central inversion, Ip"j|, for the little phrase of measure 6. The structural centrality of F# between F and G in this arrangement recalls the melodic position of F# in measure 1, where it mediated between a preceding melodic G and a subsequent melodic F. In connection with the vertical arrows of figure 10.8(b), we have noted how the progressing centers of inversion develop the pitch classes G and F# which are so characteristic of the incipit motive in the melody. We can also adopt a purely intervallic stance toward the progression of inversional centers, and toward the structure of the incipit motive. Figure 10.9 elaborates that idea.

FIGURE 10.9

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(a) of the figure is a network whose nodes contain the inversion operations associated with the vertical arrows of figure 10.8(b). The T M arrows mean T n Ig = Ip^ and T n Ip# = Ip*. The family S of operands here is the family of inversion-operations, and SGP is the group of left-multiplicationsby-transposition-operations, as that group operates on the stipulated S. Figure 10.9(b) is essentially extracted from the accompaniment of measure 1, where it supports the thematic G-F#-F gesture of the melody. The network implied by figure 10.9(b) is isographic to the network of figure 10.9(a). The networks discussed in connection with figures 10.8 and 10.9 are musically compelling at least to the extent that one would want to show how

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10.4

they relate to other aspects of the music, as one goes on to explore those other aspects. These topics, for instance, would repay such a study: the "minor-third root relations" within measure 6 (with pickup); the chromatic line, rising in parallel minor thirds, that starts with Eb-and-Gb at the pickup to measure 6 and ends with F#-and-A in the middle of measure 6; the tonal structure of the bass line, either in Ab major or in a Mixolydian Eb. 10.4 EXAMPLE: The opening section of Debussy's Reflets dans I'eau, up to the reprise at measure 35, is rich in interrelations between transformational networks and other sorts of musical structures. A sketch for the passage appears as figure 10.10. The opening motive X plays a strong generative role. Particularly full of import are the transpositions T and T' that respectively take Db to Eb and Ab to Eb within the motive. Figure 10.11 (a) isolates this structure for study. Depending on various contexts to come, we sometimes hear the interval associated with T as one diatonic step up, sometimes as a major second up, sometimes as two semitones up, and sometimes as the pitch-class interval 2. The interval associated with T' varies similarly depending on the GIS supplied by the context. Figure 10.11 (b) shows how motive Y, which follows X in the music, can be derived from X. Ab and Eb, connected by the T'-arrow, remain within Y. The T-relation of 10.11 (a), between Db and Eb within X, is "folded in" both temporally (serially) and in registral space, to become the T"1 relation displayed in 10.1 l(b), between F and Eb within Y.2 The mathematical logic of the X-to-Y transformation has the following implication: If X should become T-transposed into T(X), then the new Trelation between the pitches in order positions 1 and 3 of T(X) will specifically engage the relation Eb-to-F, thereby retrograding the last two pitches of Y. And in fact X is T-transposed into T(X) right after Y has sounded. The reader can follow these events along on figure 10.10. The new pitch Bb4 of T(X) is thereby generated in the principal melodic line. Within T(X), Bb-to-F is in the T'-relation. And Bb, as the high point of T(X), is the T-transpose of Ab, the high point of X. Measures 5-8 repeat measures 1-4. As a result of the repeat, we hear T(X) return to X via T"1; Bb also returns to Ab via T"1; and so on. Thus the change from T to T"1, a prominent characteristic of the change from the internal structure of X to the internal structure of Y, now characterizes on a larger rhythmic level the change from the progression X-T(X) (measures 1-3), to the progression T(X)-X (measures 3-5). 2. The reader will recall our discussing earlier just such kinds of "folding" transformations in connection with serial trichords. That was in section 8.3.2, where we examined the transformations FLIPEND and FLIPSTART. The transformation taking X to Y here is neither of those, but it is of the same genre.

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FIGURE 10.10

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10,4

FIGURE 10.11 At measure 9 the new motive Zj appears in the melody. The way in which Zj pulls together the pitches and intervals of X, Y, T(X), and the repeat of those cells is aurally clear. The transformational situation is already so complex that it would require an inordinate amount of discussion to describe this synthesis adequately in words. Let us just explore a few of its features, as displayed by figure 10.12.

FIGURE 10.12 The T"1 relation from F to E|>, which we heard within Y, is recapitulated at the opening of Z t . The T"1 relation from Bfc> to At?, which we heard when T(X) returned to X at measure 5, is recapitulated by the last two notes of Z t . And the relation between those two T"1 relations, within Zj, is an inverse Trelation. Figure 10.12 shows how this recalls the T'-features of X and T(X). Beneath Zj in the accompaniment, a new idea makes its appearance. This idea involves a continuous chromatic rise of a certain object (here a certain 3-note chord), over a total span of three semitones. The graph of this idea will be called CHR; it is depicted in figure 10.13.

FIGURE 10.13

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On Figure 10.10 we see how a CHR network fits homophonically beneath Z^ The first, second, and third of the resulting 4-note chords in measure 9 all have different structures. The fourth and last 4-note chord has the same structure as the first. That is because the CHR network in the accompaniment has risen 3 semitones overall, while Zl in the melody has also risen 3 semitones overall, from its opening F to its closing Ab- The homophony thus identifies the 3-arrow on the CHR graph with the interval that spans F-to-Ab across all of Z x . Because of the ways in which Z t synthesizes earlier motives, the F-to-Ab is heard in turn as part of a permuted Y embedded within Zj. So the 3-arrow of the CHR graph can ultimately be traced back to the Ab-F dyad within Y. The homophony between Zj^ and its CHR accompaniment is marked by a crescendo. The crescendo is to become a significant thematic element. In measure 10 the music of measure 9 is repeated and extended. The crescendo recurs. In the melody the repetition gives rise to a rotated form of Z l 5 marked "rot Z:" on figure 10.10. Rot Zl is Bb-Ab~F-Eb; it embeds serially the original form of Y, Ab-F-Eb, and precedes this Y by its overlapping inverse-RI-chained form Bb-Ab-F. (Bb-Ab-F is RICH'^Ab-F-Eb).) This relationship is more or less inherent in the derivations of X, Y, T(X), their repetitions, and Z:. When the music of measure 9 is extended during measure 10, a new motive Z2 arises as shown on figure 10.10. Z2 is articulated, and associated with Z l s by its contour and its rhythm. Figure 10.12 earlier analyzed Z x as a pair of T"1-related dyads in an inverse-T' relation. Z2 can be similarly analyzed as a pair of T"1-related dyads in a T-relation. Figure 10.14 displays that analysis.

FIGURE 10.14

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The accompaniment below Z2 projects a new network whose graph is CHR, a new network isographic to the network of measure 9 in the accompaniment. The new network is the T-transpose of the old, as indicated on figure 10.10 by the T arrow leading from under measure 9 to under measure 10. The extended accompaniment within measure 10 rises 5 semitones from its point of departure, in contrast to the accompaniment of measure 9, which rose

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10.4

FIGURE 10.15

only 3 semitones from the same point of departure. Figure 10.15(a) is a graph conveying this idea. Graph (b) of Figure 10.15 is a homomorphic image of graph (a). The two lower nodes of (a) each map into the one low node of (b), and the semigroups (groups) of intervals for the graphs are isomorphic. Figure 10.15(c) is a network whose graph is (b). We recognize that the pitches of (c) build a serial form of Y in the precedence ordering, namely the series F4-AJ?4-8^4. This is Y inverted about F4-and-A|?4. The retrograde of the form was Rl-chained into Y within the rotated Zl motive we recently examined. We earlier noted that F4 and A[?4 within Y were identified with the 3-semitone rise of the chromatic accompaniment during measure 9. During measure 10, the F4 and B[?4 of inverted-Y (that is, of figure 10.15(c)) are similarly identified with the 5-semitone rise of the extended chromatic accompaniment. The inverted Yform F4-A(74-B|?4, and the network of figure 10.15(c), are brought out by the homophony in the same way as was the F-A[> dyad during measure 9: F4, Aj?4, and Bfc>4 are the three notes of the melody in measure 10 that are supported by "dominant-thirteenth" harmonies; no other harmony appears more than twice during measure 10. Figure 10.16 is a "product network." It adjoins beneath a copy of figure 10.15(c) the bass notes of the dominant-thirteenth chords in an isographic network. These are exactly the bass notes of measures 9-10 which are in pitchclass relation 3 to the melody above them. Beyond participating in that relation, the pitches A|?2 and D|?3 on figure 10.16 also function as temporal and registral boundaries for the bass line over measures 9-10. The tonality of the piece further reinforces their structural significance.

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FIGURE 10.16

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Figure 10.16 shows how the pitch and pitch-class interval 3 is proliferating as a constructive element of the composition. The interval began as a secondary phenomenon; its complement spanned Ab-to-F within Y, which appeared as the difference between the T'-related Ab-to-Eb and the T"1related F-to-Eb- F-to-Ab then assumed greater prominence in the melody as the temporal boundary for Z x , supported by the concomitant 3-semitone rise of CHR in the accompaniment. Figure 10.16 shows how the melodic F-to-Ab is verticalized in the harmony, and how the interplay of horizontal and vertical 3-intervals next generates a structural Cb in the bass. That Cb is the first "middleground" chromaticism of the piece; it will take on formidable proportions hereafter, especially after measure 18. In that connection, Cb will often be heard in conjunction with F and Ab, following its prototypic generation on figure 10.16. Returning again to figure 10.10, let us now examine the large structure of the principal melodic line over measures 1-17. The pitch C5 of measure 10 is the climax of this line, which has been using the rising T and T'-inverse transformations to ascend up to that point from the initial Db4. The line is completely diatonic, so the leading tone C5 makes a strong effect as a provisional climax. The effect is somewhat concealed by the chromatic harmonization and by the possibility of hearing C5 as a neighbor to Bb4 on a subordinate level, even though the harmony does not support the neighboring function. Still, as one listens to the top staff of figure 10.10 by itself through measure 17, it is clear that there is unfinished business for the principal melodic voice in its upper register. That business will not be fully discharged until well beyond measure 35, where our present analysis will stop. Nevertheless, we shall hear before measure 35 further important developments engaging C5 and Db5 in the principal melodic voice. After the provisional climax on C5 in measure 10, the principal melody

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10.4

sits for a while on B|?4 and then returns over measure 14 back down to Dj?4, its original point of departure. The melodic descent embeds rotated Z l f Bb-Ab-F-Eb, on the strong sixteenths of measure 14. Figure 10.17(a) shows this and also shows how a rotated form of T(X) within the figuration proceeds back down via T"1 to the correspondingly rotated form of X.

FIGURE 10.17

The sixteenth-note appoggiaturas, all by descending whole tones, give the T"1 idea a heavy workout in the forefront of the melodic texture. Within measure 14, the second half of the melody, F-Eb-Eb-Db, is in T'-relation to the first half of the melody, Bb-Ab~Ab~Gb. The T' relations of Bb to F, and of Ab to Eb, are very familiar by now. Figure 10.17(b) shows a further T'relation, one that involves the retrograde CHR gestures in the alto voice of the same measure. Indeed, the entire second half of measure 14 is in T'-relation to the entire first half of the measure. Figure 10.10 brackets a pentachord called MAGIC which it asserts as controlling the first half of measure 14; T' (MAGIC) then controls the second half. Tonally, the music has progressed from the tonic pedal of measures 1-8, through the dominant that opens measure 9, to the dominant-of-the-subdominant that ends measure 10 under the melodic Bb4. After the peripatetic harmonies that prolong the melodic Bb4, the harmony discharges its subdominant obligation with T' (MAGIC) in the second half of measure 14, and the tonal idea that carries measures 15-17 is a plagal cadence supporting a prolonged Db4 in the melody. Figure 10.10 shows how T' (MAGIC) leads into that cadence. I forego with great reluctance analyzing the music hereabouts in greater detail, particularly the ingenious chaining of Y, rotated X, and rotated Zl series in the outer voices of measures 16-17, and the set-theoretic relations of those formations to the vertical sonorities there. At measure 17^ and following in the music (represented as measure 18 on figure 10.10), the cadential F of the plagal 4-3 gesture is confirmed by its own verticalized Y-form F-D(natural)-C. The first interval of Y thereby expands in structural power, transposing Y = Ab4-F4-Eb4 into T_3(Y) = F4-D4-C4. Earlier, figure 10.16 showed us how the original Ab-F dyad

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FIGURE 10.18

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within Y generated Cb, the minor third above Ab; just so, the same Ab-F now generates D natural, the minor third below F. Figure 10.18 shows how this idea is expressed very precisely by the three verticalized Y-forms within measure 18. The music ties T_3(Y) and its D natural together with Y and with J(Y) = A\?-C\?-D\>, the inverted form which originally generated Cb in the bass of figure 10.16. J(Y) appeared there as T3I(Y), where I is inversion about F-and-Ab: J(Y) appeared in the bass of figure 10.16 coupled at the 3-interval beneath a structural melodic I(Y) = F-Ab-Bb. Within the soprano line of measure 18 there are also references to permuted prime forms of melodic Y and melodic X, as indicated on figure 10.10. These recollections help get the new large section of the piece underway, by recalling material associated with the opening. Figure 10.10 also draws attention to the chromatic voice-leadings D-Eb and C-Cb during measure 18; when the material repeats the voice leadings are reversed and then repeated. The chromatic notes D natural and Cb involved in the voice-leading gestures arise as already discussed in connection with figure 10.18. I shall call the characteristic rhythm and contour that govern the second half of measure 18 the "ruffling motive"; here the wind first ruffles the surface of the pond. The ruffling motive is bound together with the Cb events we have just explored, including the C-Cb voice leading, the vertical Y of figure 10.18, and the vertical J(Y) of the same figure. The motive arpeggiates a Tristan chord upwards and then partially arpeggiates a £5 harmony downwards, all within the registral confines of the bass F3 and the upper note Ab4. The Tristan chord is in the correct spacing at the right pitch-level, once the doubling Ab3 of the ascending ruffle is removed. Over measures 20-21, the Tristan-harmony-cum-ruffle-motive moves up quasi cadenza in literal sequences, 3 semitones per stage, until it gets essentially two octaves higher. The Tristan chord no longer contains any doublings. Figure 10.19 graphs the beginning of the cadenza sequence. Now that D natural, as well as Cb, is on the scene along with F and Ab, the minor third or 3-semitone interval can be fully unleashed; the sequence at hand unleashes it. Figure 10.19 has striking features in common with figure 10.16 earlier, where the 3-interval first began to flex its muscles in a chromatic

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10.4

FIGURE 10.19

context. Both the figures start with their outer voices on the basic 3-dyad Fand-Ab. In both, the outer voices then move by the melodic 3-interval, to project Ab-and-Cb (G#-and-B) in the homophony. Both figures project forms of the Y motive: In figure 10.161(Y) = F-Ab-Bb above is coupled to J(Y) = T3I(Y) = Ab-Cb-Db below; figure 10.19 embeds Y and J(Y) subnetworks in the ruffling as shown, and then sequences all Y-forms indefinitely. I have no idea what private commentary on Wagner Debussy may have intended by his use of the Tristan chord and his continued sequencing of it by 3 semitones, a sequence which Wagner artfully only suggests in the first-act Prelude. Debussy's sequential cadenza, unleashing the hitherto restrained 3interval, makes perfect sense in his own composition, of course. The first transposition of the Tristan chord, which Debussy spells as G#-D-F#-B, is a subset of the MAGIC pentachord D-Cb~Gb-(Bb)-Ab. Ruffle-cum-Tristan sequences eight times in the music, arriving at measure 22 essentially two octaves above its point of departure. Figure 10.10 shows the Tristan chord at measure 22 only one octave higher, with its upper voices rearranged. That is to accommodate the subsequent voice-leading into measure 24, and thence into measure 27 and measure 30; those events are pretty clearly in the correct structural registers where figure 10.10 portrays them.

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For the music between measure 22 and measure 24, where the cadenza again becomes mesure and a new theme appears, I hear two possible readings. One of them, marked "CHR and ret CHR?" on figure 10.10, is reflected in other notational aspects of the figure hereabouts. This reading takes the descending minor third from F to D, first introduced on the figure at measure 18, and fills it in by a retrograde CHR line; in counterpoint to this, an ascending (prime) CHR line connects the Ej? of Tristan to the G|? of MAGIC; MAGIC then saturates the music from measure 23^ to measure 27. The ascending CHR line of this reading, from E|? to G^-within-MAGIC, retrogrades the opening gesture of figure 10.17(b) earlier, which showed a retrograde CHR line proceeding from Gb-within-MAGIC to E[?. On figure 10.10, brackets demarcate the CHR groups of four events into which the voice leading is articulated by this reading. Since the earlier CHR networks of measures 9 and 10 were associated with thematic crescendi, and since the retrograde CHR events of measure 14 (figure 10.17(b)) could easily take a diminuendo, it would be useful to explore in connection with this reading how best to structure the "poco a poco cresc. e stringendo" that begins at measure 20. Does it stop at measure 22? Does it intensify there and continue on right up to the arrival of the D natural at measure 23|? Does it continue even past that, right up to the (subito?) ppp mesure at measure 24, where the low Ab pedal comes in? I have put parentheses and a question mark on figure 10.10, together with the crescendo sign under measure 22 there, to suggest and emphasize these questions. The other possible reading I hear is indicated on the figure by the annotation "Voice-exchange Eb/F?" According to this reading, the basic grouping is not of four CHR events, followed by ppp mesure; rather the basic grouping is of three events within the Tristan harmony, followed by the arrival of MAGIC harmony at the D natural that launches the uwmeasured section of the cadenza, at measure 23^. This is the point where the progressively more agitated ripples of the water turn into turbulence; at the (subito) ppp mesure of measure 24 we presume that the burst of wind abruptly stops. The idea of putting the structural downbeat for the MAGIC arrival not at the obvious measure 24, but rather just at the moment where periodic wave motion turns to "aperiodic" turbulence, in measure 23^, is extraordinarily poetic. I would enjoy trying to play the passage this way for a small group of close friends; I am not sure how well I could project it to a large public audience. Again, it would be very helpful to have more indications for tempo and dynamics between "pp, poco a poco cresc. e stringendo" at measure 20 and "(subito?) ppp, mesure" at measure 24. As it is, the pianist's decisions about tempo and dynamics will very much affect the sense of our alternate readings, and viceversa. I write "(subito?) ppp" because I can conceive that Debussy heard a diminuendo from a MAGIC arrival at measure 23^ to ppp at measure 24, even though he did not write such a diminuendo. In that case, one imagines the

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10.4

wind dying down more gradually, making the turbulence subside gradually into regular ripples again. The rippling figuration over measures 24-29 ornaments a basic succession of two harmonies: The MAGIC harmony just under discussion moves to a dominant-ninth structure at measure 27, as shown on figure 10.10. The figure also shows, by notations under the staff, how the progression involves precisely the two chromatic voice-leading gestures already exposed during measures 18-19, that is D-to-Eb and Ct>-to-C. Within the overall progression, each of the two harmonies is prolonged by (or accompanies) a form of a new motive I call V, marked in parentheses on figure 10.10. V stands for "variable"; the motive always proceeds by step up, third-leap up, step down, step down, and step down, ending on the same note with which it started, but the sizes of the "steps" and the "third" vary considerably, sometimes even within one V-statement, and the rhythm of the motive is also extremely plastic. These features of V presumably represent the perpetual mutations of things seen through reflets dans I'eau. (One thinks of Monet's pond at Giverny.) The descending part of the second V-statement here hooks up with the rippling figuration just before measure 30, to project the descending hexachordal line BJ7-A[7-G(?; F-Et?-D[?. We have heard this descending hexachord before, namely in the melody of measure 14; there the B|?-A[7-Gb segment was also launched by MAGIC harmony. At measure 30 the V motive carries the final Dfr of the hexachord on down to C5. We noted earlier that C5 was an interesting climax for the principal melodic line over measures 1-17, and that the melody seemed to have unfinished business in its upper register. Debussy is now, at measure 30, turning his attention to some of that business. All at once, we hear bursting forth from the ripples a Z-form in a principal melodic line, a CHR-related network of trichords in parallel motion, and a crescendo; these were all aspects of the music during its earlier rise over measures 9-10. The Z-form Z 3 , articulated like Z2 earlier by rhythm and contour, circles around the critical pitch C5. It does touch D[?5, but there is some question as to how essential D[?5 sounds in the melodic line. Hitherto, all motivic Trelations have been both by one diatonic scale degree and by two semitones (or by major second). Now we have to respond motivically to a relation that is and is not the same, between C5 and D|?5. This relation is quintessential^ by one diatonic scale degree, but it is just as quintessentially by only one semitone (or by minor second). Is this, or is this not, a bona-fide "T-relation?" The answer, of course, is "Yes and no." We shall not get far arguing the question in that form; but it will be very much worth our while to explore the ways in which the answer is ambivalent. More generally, let us explore further ways (beyond the issue of T-relation) in which we feel ambivalent about the idea that the line might rise from C5 to D|?5 here. The sense of structural rise seems strongly contradicted by the retrograde

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forms for Z3 and CHR. Since we associated prime Z-forms with rising, over measures 9-10, we naturally feel that a retrograde Z-form is falling. And of course the retrograde CHR network is falling very decisively. But the crescendo is motivically important too. No matter how much the retrograde motives sound falling, the music is still rising in dynamic, like the music of measures 9-10 that first led the melody up to C5. Debussy would surely have written a diminuendo from mf, not a crescendo, had he heard a completely subsiding effect here. It is difficult but crucial for the pianist to husband the dynamics scrupulously, so the crescendo can proceed past mf and on to f without actually attaining ff. (Think of narrow-bore brass!) To support further the idea that the line might be rising to D[?5, we can hear that D[?5 gets strong, if fleeting, harmonic support within the Z3-form. D[?5 is specifically supported by T' (MAGIC) harmony, continuing to follow the precedent of measure 14. Figure 10.20 shows what I mean.

FIGURE 10.20

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Now the Dj? at the end of measure 14, the model for the D|?5 of figure 10.20 in the relation just pointed out, is (or becomes) a very strong cadence tone; its function as an essential tone is not in doubt. By analogy, the D^S of measure 30 sounds that much more essential. But now a new consideration arises. The very strength of the fit between figure 10.20 and measure 14 reminds us that everything shown on figure 10.10 from measure 22 through measure 30 has in effect been transferred up an octave from the Tristan chord of measure 20, which served as a launching pad. Figure 10.20, in particular, is still "an octave up" from measure 14. According to this reading, the retrograde Z3 at measure 30 is an octave transfer of a form that actually belongs an octave lower in the large melodic structure. C5 and Dt? 5 within ret Z3 are to that extent not in the structural climax register at all; they are octave transfers of C4 and D[?4, from the cadence register. That view is afforded support by the melodic doublings hereabouts: Both the second Vstatement and the retrograde Z3 motive are doubled in the music an octave below where they appear on figure 10.10.

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10.4

Still, no matter what the pianist does with the balance of these doublings, there has been no written dynamic heretofore as loud as mf, let alone mf with a crescendo to f, and that aspect of measure 30 by itself makes it hard to deny that we should hear C5 and D[?5 as climactic events to some extent, and not just as doublings or transfers of C4 and D|?4. In sum, the C5 and D(?5 of the retrograde Z3 at measure 30 are ambivalent in many ways: It is unclear to what extent they address the issue of unfinished business in the climax register of the principal line, and to what extent they represent C4 and D[?4 transferred temporarily up an octave; it is also unclear to what extent ret Z3 is "rising," and to what extent it is "falling." The music plays with these ambiguities in the immediate sequel, via the little descending arpeggios that alternate C with Db first in register 6, then in register 5, and then in register 4. (The arpeggios are not shown on figure 10.10.) Eventually the ret Z3 figure returns down the octave, as shown on figure 10.10. Or one could put it that the doubling in the upper octave drops away. In any case, the registral issue is temporarily resolved in favor of the lower register. We become particularly convinced of this at measure 35, where the reprise begins. There we recognize the notes of Z3 in the lower register as the notes of an alto voice that fits underneath the X motive of the soprano. We might say that Z3 belonged in the lower octave all the time; it was only tossed up an octave higher at measure 30 by the agitated activity of the waters, so to speak, as a rare and curious submarine object that flashed momentarily into view and then sank again. (One supposes that D|?4 is the surface of the pond.) As Z3 subsides under the surface at the reprise, the unfinished business in the climax register remains unfinished. But the discussion of figure 10.10 has ended. I should discuss, if only briefly, significant later events in the climactic register of C5 and above. The key to this analysis is the passage sketched in figure 10.21.

FIGURE 10.21 The tune is supported by a cadential progression in Efr major over a tonic pedal. The dynamic, ff and crescendo, is uniquely climactic for the composition. (The unexpected bursting forth of Eb-major harmony a measure before figure 10.21 is only f, with a hairpin. Again the pianist must husband the dynamics very meticulously.) Here B[?4 and C5, that could not rise farther at

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measure 10 and rose to Db5 only problematically at measure 30, finally make their breakthrough. Indeed, they get as far as £[7 5, with no question about it. Db 5, significantly, is not on the scene now that we are in the key of Eb. And C5 is not a leading tone in the new key. The "breakthrough" form of V at measures 57-58 directly follows two Vessays of a very different character. Debussy tells us that the Bb4 and C5 of the breakthrough are to be compared with the same pitches in measure 9-10, by his use of the retrograde T(Z X ) motive. Whereas in measure 10 only the Bb-Ab of Z x moved up to T(Bb-Ab) = C-Bb within Z2, now all of Zj = F-Eb-Bb-Ab has moved up to T(Z1) = G-F-C-Bb, which is then retrograded in measures 58-59 so as to descend from the climactic V-statement. Further, the whole tonality of the piece has temporarily moved up from Db to Eb = T(Db), to accommodate and support the T-transformation of Zl. This is indeed an extraordinary expansion of the original T-relation which, the reader will recall, obtained precisely between the notes Db and Eb = T(Db), the notes which started and ended the initial motive X. The potential for the change of tonality was perhaps already latent in the transformation of X to T(X) during measure 3. To the extent that T means "two semitones," T(X) would have had to be supported by Eb harmony in measure 3 to have the same "meaning" as X in measure 1. We suggested in discussing measure 30 that the rise from C5 to Db5 within a Z-form could not be completely convincing because the minor second (or distance of one semitone) is not completely T-ish in character. Now, over measures 56 and following, the change of tonality from Db to Eb provides an appropriate and enormous T-ish boost, enabling the melody to rise definitively beyond C5.3

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3. For a sensitive appreciation of dynamics and contours rising and falling over this piece on a large scale, and also for a convincing view of the E|? major breakthrough as it articulates a large rhythmic design, the reader is referred to Roy Howat, Debussy in Proportion (Cambridge, England: Cambridge University Press, 1983), 23-29.

Appendix A:

Melodic and harmonic GIS Structures; Some Notes on the History of Tonal Theory

Chapter 2 surveyed a variety of musical spaces pertinent to theories of Western tonality. In some of these spaces, our intuitions of directed distance or motion from one position to another were measured by steps along some melodic scale, diatonic or chromatic, linear or modular. In other spaces, our intuitions were measured by numbers reflecting harmonic relationships of various kinds, or by moves on a game board derived from harmonic relationships. The richness of tonal music, and of music in related idioms, is much enhanced by the ways in which a variety of such intuitions come into play. We can review in this connection the Protean meanings of "the interval from F to At?" in Reflets dans I'eau. We sense a harmonic interval within the Dt?-major triad and the Tristan Chord; we sense also a melodic interval moving two steps along a diatonic scale in D(? major, or along a diatonic hexachord (the cadential descending hexachord B^AbGbFEbDj?); to some extent we can even hear F-to-At? as one melodic step along the pentatonic scale DbE^FA^Bt?, as we listen to the melody at the beginning of the piece; finally we also hear F-to-Aj? as spanning three semitones along a chromatic scale, once the CHROM figure comes onto the scene. A transformational approach enabled us to sidestep these ambiguities in chapter 10, referring there to transformations T and T' that mapped D[> to E|? and Aj? to £[7 respectively; we could conceive T and T' as transpositions by "intervals" in any-or-all of the conceptual spaces involved; then we could compute a corresponding transformation T-1T' which mapped A(? to F in any-or-all of the spaces, a transformation worked out musically in the change from motive X to motive Y. Such transformational discourse is particularly useful to discuss architectural features of Reflets that obtain no matter what sorts of intervallic

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Appendix A

intuitions one considers. For instance, the tonic of the climactic fortissimo E|? major is in a T-relation to the tonic of D[? major no matter what kind of interval, in what kind of melodic or harmonic space, we consider T to be transposing D(? by. On the other hand, transformational discourse is correspondingly impoverished when it conies to exploring the varieties of spatial and intervallic intuitions at hand, and the ways in which the music brings those intuitions into play, each with the others. In connection with the big climax of Reflets, for instance, the major mode of the fortissimo music in E[? favors certain kinds of intervallic intuitions over others, when we hear Ej? major, not E|? minor, in relation to D[? major. The reader may also recall our discussion of the high C-Db in the principal melodic line, and our question: "Is this, or is this not, a T-relation?" That question implicitly involves very broad questions about the premises of the composition: To what extent is the piece diatonic-melodic, so that one scale-step is one scale-step, regardless of its acoustical size? To what extent is the melos chromatic, so that one semitone is something necessarily very different from two semitones, even if both are spanning one diatonic step? To what extent is "the interval" attached to T heard in a harmonic context that gives it a size somewhere between 10/9 and 9/8, but not as small as 16/15? To what extent can techniques of melodic motivic transformation, involving the rhythm and contour of the Z motive in particular, alter our impressions in any or all of these respects? and so on. Exactly these ambiguities must be appreciated, if we are adequately to appreciate the conceptual tensions of the local climax involving C5 and D|?5, beyond its high register and relatively high dynamic level. We return, then, to the variety of intervallic intuitions surveyed in chapter 2. Pertinent syntheses of these intuitions are essential not only for many occasions in critical listening and analysis, but also for many abstract theoretical purposes. Indeed, such syntheses are among the greatest triumphs in the history of Western music theory, and their neglect or failure has led to some of the more embarrassing moments in that history. Among the latter, we may cite Rameau's argument that the harmonic intervals of 5/4 and 6/5 may be exchanged in relative register within the harmonic triad, so as to derive the minor triad from the major. This may be allowed, he says, since 5/4 and 6/5 are both "thirds."* But at the time he says this, he has not as yet presented us with any scale along which we can measure distances of "three" degrees, and he has assured us very strongly that melody is in any case thoroughly subordinate to harmony.2

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1. Jean-Philippe Rameau, Traite de I'harmonie reduite a ses principes naturels (Paris: Ballard, 1722). "... la difference du majeur au mineur qui s'y rencontre n'en cause aucune dans le genre de 1'intervale qui est toujours une Tierce de part & d'autre;..." (p. 13). 2. Ibid. "On divise ordinairement la Musique en Harmonic & en Melodic, quoique celle-cy ne soil qu'une partie de 1'autre, & qu'il suffise de connoitre I'Harmonie, pour etre parfaitement instruit de toutes les proprietez de la Musique,..." (p. 1).

Appendix A

One might try to replace Rameau's implicitly melodic argument about the "thirds" by a suitable harmonic argument: The intervals 5/4 and 6/5 are adjacent within the senario; they divide the interval 3/2 = 6/4 harmonically, and so arithmetically when reversed; therefore such a reversal is logical. But one could argue in exactly the same fashion for the intervals 4/3 and 5/4, as they divide 5/3. Would Rameau, then, have accepted the argument that we ought to consider the harmony G4-C5-E5 as functionally equivalent to the harmony G4-B4-E5, by the analogous reasoning? Obviously, he would not have; such reasoning there would violate the principle of the Fundamental Bass. Helmholtz, however, might have been willing to entertain the argument; indeed he actually asserts harmonic equivalence of a certain sort between the six-four position of the major triad and the six-three position of the minor triad. Both those positions comprise the highly consonant verticalities of a fourth, a major third, and a major sixth; having the same vertical-interval content, they are thereby the "most consonant" close positions for their respective pitch-class sets.3 Among the triumphal syntheses mentioned earlier a high position must be reserved for Zarlino's Istitutioni harmoniche.* Book 1 discusses intervals as phenomena in a harmonic space. Book 3 discusses intervals all over again as phenomena in melodic space, and synthesizes that approach with the mathematical ideas of book K Abstract harmonic ratios are accessible to our perception (as well as our intellect) because they can be filled in by notes of a diatonic series in melodic space; conversely, articulated segments of a unidirectional diatonic series make sense to our understanding (as well as our perception) because of the harmonic relations obtaining between the boundaries of the segments. This way of interrelating harmonic and melodic space has much in common with central aspects of Schenker's theories, in particular with Schenker's understanding of the Zug, and even specifically of the Urlinie. Schenker's mature theory contains another triumphal synthesis of harmonic and melodic space, understood now in the context of functional tonality.5 Even Zarlino has an embarrassing moment, confusing melodic with harmonic space, when he comes to discuss the minor sixth. He wants to analyze the major and the minor sixths as analogous structures. Specifically, he says that they "are composed ... from the fourth plus the major third, or 3. Hermann Helmholtz, Die Lehre von den Tonempflndungen als physiologische Grundlage fur die Theorie der Musik, 2d ed. (Brunswick: Friedrich Vieweg und Sohn, 1865). "..., so folgt hieraus, dass die Quartsextenlage des Disaccords wohllautender ist als die fundamental, und diese besser als die Sextenlage. Umgekehrt ist die Sextenlage beim Mollaccord besser als die fundamentale, und diese besser als die Quartsextenlage." (p. 325). 4. Gioseffo Zarlino, Istitutioni harmoniche, 2d ed. (Venice: Senese, 1573). Facsimile republication (Ridgewood, N. J.: Gregg Press, 1966). 5. Heinrich Schenker, Neue musikalische Theorien und Phantasien, vol. 3, Derfreie Satz (Vienna: Universal Edition, 1935).

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the minor third." 6 He might have continued: So, in a major mode such as Ionian, the fourth G3-C4 plus the modal major third C4-E4 yields the major sixth G3-E4, while in a minor mode such as Dorian, the fourth A3-D4 plus the modal minor third D4-F4 yields the minor sixth A3-F4. The modal idea is perfectly clear. Elsewhere, too, Zarlino makes a great point of the modal relation between major and minor thirds in harmonic contexts; indeed he even points to this as a specific resource for harmonic variety, beyond the resources of the senario itself: Some chords have a major third or tenth over the bass, others a minor third or tenth.7 One wishes, then, that he would have produced G3-(C4)-E4 and A3-(D4)-F4, to illustrate a modal analogy between the two sixths as being "composed ... from the fourth plus the major third, or the minor third." Unfortunately he does not do so. Probably he was not as sensitive as we are to the thirds above the modal tonics C4 and D4 in the harmonic structures above; those thirds are not over the bass notes of the structures. Whatever his motivation, he attempts to realize the analogy of the sixths as a feature of his harmonic space rather than his modal theory, and that leads him into confusion. He has to adjoin the number 8 to the senario in order to get the harmonic ratio 8:5 at hand for the minor sixth, and then he has to argue that the proportion 8:6:5 is somehow analogous to the proportion 5:4:3 in his harmonic world. He even claims that 6 is a "harmonic mean term" between 8 and 5; this is simply false if "harmonic" is to mean anything at all in the context.8 We may fairly put his argument into modern dress by regarding it as an attempt to draw a direct analogy between the major sixth G3-E4, as divided by C4, and the minor sixth E4-C5, as divided by G5. Of course this does not work. In particular, the conjuction of the minor third E4-G4 below with the fourth G4-C5 above is not at all the same thing as the conjunction of some fourth below with some minor third above, as in the Dorian modal sixth A3-(D4)-F4. Zarlino could also, of course, have analyzed the sixths as arising by inversion from the thirds. But this approach would have been foreign to his purpose, for then the sixths would not have been primary features of his harmonic space, somehow embedded within the senario. Besides, the sixths that arise from inverting thirds have a very different modal character from the sixths that interest (or should interest) Zarlino. That is exactly the problem

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6. Zarlino, 1st. harm., book 3, chapter 21. "... sono composte ... della Diatesseron et del Ditono, over del Semiditono;..." (p. 193). 7. Ibid.,book3,chapter31. "... lavarietadell'Harmonia...nonconsistesolamentenella varieta delle Consonanze, che si trova tra due parti; ma nella varieta anco delle Harmonic, la quale consiste nella positione della chorda, che fa la Terza, over la Decima sopra la parte grave ..., overo che sono minori ...; overo sono maggiori..." (p. 210). 8. Ibid., book 1, chapter 16. "... tal proportione tra 8 & 5 termini son capaci di unmezzano termine harmonico, che e il 6;..." (p. 33).

Aooendix A

with his "minor" sixth 8:5 (E4-C5). To the extent that we hear it in a "Cmajor" modal context as third-degree-to-octave, inverting tonic-to-thirddegree = C4-E4 = 5:4, a major third, the sixth itself has a "major" modal character about it, despite its small size. Contrast that with the modal character of A3-F4 in a D-Dorian context, as fifth-degree-(through-tonic-)to-thirddegree: This sixth, in its context, has a "minor" modal character as well as a minor absolute size. Similarly, F4-D5 in D-Dorian, inverting D4-F4, has a "minor" modal character despite its large size, while G3-E4 in C-Ionian has both a "major" modal character and a large size. Zarlino had no Stufen theory that could enable him to make such discriminations. And yet it is quite possible that, even if one had been available to him, he might have rejected it. He would have been uncomfortable making the meaning of his harmonic intervals so dependent on the contextual assignment of a modal tonic. For him this would have weakened the context-free universality of his harmonic theory. Schenker, quite willing to assign structural priority to contextual modal tonics inter alia, uses his Stufen theory to powerful effect in related connections. On the other hand, he finesses certain problems about the universality of minor harmonic structures which Zarlino attempts to confront, and succeeds in confronting to a remarkable extent. Hindemith makes an interesting synthesis of melodic and harmonic spaces.9 He tries to show that a chromatic scale from C2 to C3 is filled by those pitches, and only those pitches, which lie in "closest" harmonic relation to C2 within a certain harmonic space. We ignore the overtones of C2; then G2, within the desired scale-segment, is harmonically "close" to C2 because the second partial of G2 is the third partial of C2. F2, within the desired scalesegment, is "close" to C2 since the third partial of F2 is the fourth partial of C2. And so on, casting away harmonic octave-replicates of pitches already generated (which, happily, do not lie within the desired scale-segment). For the most part, this works quite well, though a bit of strain is perceptible in the construction of certain secondary relationships. The essence of Hindemith's achievement was not just to find pitch classes that can be represented by pitches within a chromatic scale. After all, Zarlino and his forerunners could do that well enough and better. Rather, the achievement was to have shown how pitches within a melodically well-packed region, a chromatic scale from C2 to C3, could be regarded as pitches also within a harmonically well-packed region around the tonic, harmonically well-packed according to Hindemith's special algorithms for generating harmonic pitch-space. The one pitch with which Hindemith has trouble is A[?2, the minor sixth above the tonic C2. Curiously enough, his troubles resemble Zarlino's troubles with the minor sixth. Hindemith can generate E2, E[?2, and A2 without using partials of those 9. Paul Hindemith, Unterweisung im Tonsatz: Theoretischer Teil (Mainz: B. Schott's Sonne, 1937), 47-61.

249

Appendix A

pitches, or of C2, that involve numbers greater than 6. E.g.: The fifth partial of Eb2 is the sixth partial of C2; the third partial of A2 is the fifth partial of C2. But in order to find A[?2 by this method, he would have to have used an eighthpartial relationship: The fifth partial of A|?2 is the eighth partial of C2. This relation would presumably have made A(?2 too "remote" in harmonic space; besides, it might have given rise to awkward questions about seventh-partial relationships. (Zarlino has to deal with the analogs of such questions, when he adjoins the number 8, but not the number 7, to his senario.) Presumably for reasons of these sorts, Hindemith produces not A[?2 but Afc> 1 by his algorithm; A)?! is a unique pitch which he generates in this way outside the octave C2-C3. Then, without much explanation, he brings A|? 1 up an octave, so that it will lie within his desired scale-segment.10 The foregoing discussion of ways in which some theorists have attempted to integrate harmonic and melodic tonal spaces, or have failed to integrate them, is not meant to be exhaustive or even representative. It is rather intended to show that we do not really have one intuition of something called "musical space." Instead, we intuit several or many musical spaces at once. GIS structures and transformational systems can help us to explore each one of these intuitions, and to investigate the ways in which they interact, both logically and inside specific musical compositions.

250

10. Ibid. "The frequency ... of the fourth overtone [of C2] is now divided by 5 ... and so generates ... Ab 1 ..., whose second overtone ... is inserted in our store of pitches. (Die Schwingungzahl ... des vierten Obertones c1 wird nunmehr noch durch 5 ... geteilt ... und erzeugt so ... das lAs ..., dessen zweiter Oberton ... in unseren Tonvorrat eingereiht wird.)" (p. 54).

Appendix B:

Non-Commutative Octatonic GIS Structures; More on

Simply Transitive Groups

Let S be the octatonic family of pitch classes comprising C, Cft, Dft, E, Fft, G, A, and Aft. Eight of the standard "atonal" operations on the twelve pitchclasses transform S into itself; these operations are T0, T3, T6, T9, l£*, IG, 1°» and l£*. The eight operations form a group on the twelve pitch-classes and therefore, mapping S into itself, induce a group of corresponding operations on S; we shall call those corresponding operations RO, R3, R6, R9, K, L, M, and N respectively. It is not hard to verify that the latter group is simply transitive on S: Given members s and t of S, there is a unique OP, among the eight cited operations on S, satisfying OP(s) = t. (If t is in the same diminished-seventh chord as s, OP will be RO, R3, R6, or R9; if t is in the opposite diminished-seventh chord from s, OP will be K, L, M, or N.) We shall call this simply transitive group of operations STRANS1. The operations RO, R3, R6, and R9 may be thought of as "rotations," to justify the use of the letter R in their names. We can define another group of operations on S, STRANS2, as follows. RO and R6 (as above) are members of the group; so are two "queer" operations Q3 and Q9. Q3 rotates each of the diminished-seventh chords within S, but in opposite directions; it maps C to Dft, Dft to Fft, Fft to A, A to C, and also Cft to Aft (not to E), Aft to G, G to E, and E to Cft. Q9 is the inverse operation to Q3; it maps C to A , . . . , Dft to C, and also Cft to E,..., and Aft to Cft. Besides RO, Q3, R6, and Q9, STRANS2 also contains four "exchanging" operations XI, X2, X4, and X5. XI exchanges pitch classes within S that lie one semitone apart; it thus maps C to Cft, Cft to C, Dft to E, E to Dft, Fft to G, G to Fft, A to A#, and Aft to A. X2 exchanges pitch classes that lie two semitones apart; it maps C to Aft, Aft to C, Cft to Dft, Fft to E, and so on. X4

257

Appendix B

252

exchanges pitch classes that lie four semitones apart; it maps F# to A#, E to C, G to D#, and so on. X5 exchanges pitch classes that lie five semitones apart; it maps A to E, D# to A#, F# to C#, and so on. It can be verified that STRANS2 is a group of operations on S, and that it is simply transitive. Using the method discussed in 7.1.1, we can develop a GIS structure for S in which the members of STRANS1 are exactly the formal transposition operations. We can call this structure GIS1 = (S, IVLS1, intl). In GIS1, then, applying any one of the operations RO, R3, R6, R9, K, L, M, or N to a member s of S amounts formally precisely to "transposing" the given s by a suitable corresponding interval of IVLS1. We must be careful to distinguish the operations K, L, M, and N, which are "GIS 1-transpositions" under this formalism, from the operations l£* etc. that gave rise to them; !£* etc. are inversion-operations in a different GIS, a GIS involving a different family of (twelve not eight) objects, a different group of (twelve not eight) formal intervals, and a different function int. Likewise, and more subtly, we must distinguish the octatonic GIS 1-transpositions RO, R3, R6, and R9 from the dodecaphonic atonal-GIS-transpositions T0, T3, T6, and T9. As it turns out, the members of STRANS2 are exactly the intervalpreserving operations for GIS1. Every member of STRANS2 commutes with every member of STRANS1. In fact, the members of STRANS2 are precisely those transformations on S that commute with every member of STRANS1. Using the method of 7.1.1, we can develop another GIS involving the family S, a GIS for which the members of STRANS2 are exactly the formal transposition operations. We can call this structure GIS2 = (S, IVLS2, int2). In this GIS, applying any of the operations RO, Q3, R6, Q9, XI, X2, X4, or X5 to a member s of S amounts precisely to transposing s, formally, by a suitable corresponding interval of GIS2. The interval-preserving operations for GIS2 are exactly the members of STRANS1; those are in fact precisely the transformations on S that commute with every member of STRANS2. Either GIS1, or GIS2, or both, might lead to results of interest in analyzing a variety of octatonic musics. STRANS2 and STRANS1, which figure as groups of interval-preserving operations in those respective GIS structures, are thereby also likely candidates for CANONical groups of operations in a variety of set-theoretical studies. The STRANS1-forms of a set within S are exactly the dodecaphonically transposed and inverted forms of the set that lie within S. The STRANS2-forms of a set within S are in general a more novel sort of family. Taking (C, E, G), for instance, we apply to it in turn the operations RO, Q3, R6, Q9, XI, X2, X4, and X5; its STRANS2forms are thereby computed as RO(C, E, G) = (C, E, G), Q3(C, E, G) = (D#, Qf, E), R6(C, E, G) = (F#, A*, C|), Q9(C, E, G) = (A, G, AJ), XI (C, E, G) = (Cfl,DJF,FJF) f X2(C,E,G) = (A#, Ffl, A), X4(C,E,G) = (E,C,D#), and X5(C, E, G) = (G, A, C). If Y is any one of those eight sets, and Y' is any other one, and f is any one of the eight operations in STR ANSI, then the number of

Appendix B

members of Y whose f-transforms lie within Y is the same as the number of members of Y' whose f-transforms lie within Y': JNJ(Y, Y)(f) = INJ(Y', Y')(f). More generally, if f is any one of the eight operations in STRANS 1, and A is any one of the eight operations in STRANS2, and Y and Z are any sets whatsoever within S, then INJ(Y, Z)(f) = INJ(A(Y), A(Z))(f): the number of members of Y whose f-transforms lie within Z is the same as the number of members of A(Y) whose f-transforms lie within A(Z). As an exercise, the reader may consider a new family of pitch classes, S = (C, C|, E, F, Gf, A), and develop on the new S two analogous simply transitive groups of operations. More generally, suppose now that S is any family of objects and that STRANS is any simply transitive group of operations on S. Consider the family STRANS' of transformations f on S such that f commutes with every member of the given group STRANS. It can be proved that STRANS' is itself a simply transitive group of operations on S, and that every transformation A which commutes with every member of STRANS' is (already) a member of the given group STRANS. When S is considered as a GIS whose formal transpositions are the members of STRANS, then the members of STRANS' will be the interval-preserving operations. Dually, when S is considered as a GIS whose formal transpositions are the members of STRANS', then the members of STRANS will be the interval-preserving operations. If STRANS is commutative, then STRANS' will be precisely STRANS itself.

253

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Index

ADJOIN, 121 Anti-homomorphism, 14 Anti-isomorphism: 14; i-to-Tj, 46, 47, 150 AOD: 113-15; and interval-preserving operations, 121-22 Argument, of function, 1 ARROW. See Node/arrow system Arrow chain: 194-95; proper, 209 Associative: law, 5; binary composition, 5 Attack-ordered dyad, 113, 121-22. See also AOD Attack ordering, 117 Babbitt, Milton: and beat classes, 23; SemiSimple Variations, 136, 138-40, 141, figs. 6.7-6.9; Reflections, 138; hexachord theorem, 145 Bach, Johann Sebastian, Two-Part Invention #1, 183-84, fig. 8.5 Bart6k, B61a, "Syncopation," 225-27, figs. 10.5-10.6 Beat class, 23 Beethoven, Ludwig van: First Symphony, 169-74, figs. 7.8-7.13, 8.1; Quartet op. 135, 2lln; Sonata Appassionata, 213-17, figs. 9.14-9.16 Bernard, Jonathan W, 62/», 189 Binary composition, 5 BIND. See Serial transformations Boundary tones, 94, 95, 98, 102-03, 235 Brahms, Johannes: G-Minor Rhapsody, 119-21, figs. 5.14-5.15; Horn Trio, 16569, 202-03, 207, figs. 7.6-7.7, 9.6 Bresnick, Martin, 67n5 Bushnell, Michael, 90, 141n Cann, Richard, 85n7 Canonical: group, 104, 106, 111, 113, 15052; equivalence, 104, 113, 121-22; forms, 105, 119 Carriage return, 214-15 Carter, Elliott, First String Quartet, 23n3, 62, 67-74, figs. 4.2-4.3 Cartesian: product, 1; thinking, 158

Central: defined, 7; interval, 49, 52-55 passim Cherlin, Michael, 137 Chopin, Fr£d6ric, B-Flat-Minor Sonata, 8587, 114, figs. 4.6, 5.11 Christensen, Thomas, 163n Chronology: 173, 177; and precedence, 210-18. See also Precedence Closed, family of transformations, 4 Cogan, Robert, and Escot, Pozzi, 160n Collection, of objects, defined, 1 Coltrane, John, 62 Common-note function, 144 Commutative: defined, 6; GIS, 50, 53, 56, 58 Complement, of set, 144-45 Composition: of functions, 2; binary, 5 Cone, Edward X, 39n Congruence, defined, 10 CONTENTS. See Network, transformation COV, 122 Debussy, Claude, Reflets dans I'eau, 231-44, 245-46, figs. 10.10-10.21 Diatesseron, symphony of, 204-06, fig. 9.8 Direct product: of semigroups, 15; of groups, 15, 61; of intervals, 93-94; of GISs, 37-46, 174 Dispersive. See Transformations DOM, 176-78, 229 Double emploi. See Rameau Duration classes, 24, 25 EMB: defined for sets, 105; for set classes, 106; as probability, 107, 111; properties, 107-08; topological model, 108-11, figs. 5.9-5.10; and Partition Function, 152 Embedding number. See EMB Environment, of a set, 97-98 Equivalence classes: defined, 8; canonical, 104 Equivalence relation: defined, 7; arising from function, 7; and partitioning, 8;

255

Index Equivalence relation (continued) examples, 9; in quotient GIS, 33-35; canonical, 104 External. See Transformations

256

If-only adjustments, 140-41 IFUNC: defined, 90; examples, 89-99, figs. 5.1-5.7; maximum values, 94, 95; properties, 99-101; as probability, 101; and Zsets, 103; questions about, 103-04; unFamily: of objects, denned, 1; finite, 144rolling, 120; generalized by INJ, 147; ef45 fect of inversion on, 150 FLIPEND. See Serial transformations Induced map, of quotient family, 10 FLIPSTART. See Serial transformations INJ: defined, 124; free of GIS structure, Forms, of a set, 105 134; and operations, 144-45; and setForte, Allen: 9n, 103, 104, 143, 150; and complements, 144-45; as generalized Gilbert, Steven E., 218/1 IFUNC, 147; for transformed X and Y, Function: defined, 1; onto, 3; one-to-one, 3; 147-49; questions about, 149-50; generalinverse of, 3; and equivalence relation, 7 izing K and Kh, 150-51; for infinite sets, Functional: orthography, 2; equality, 2; 153, 154, 156 equations, 2-3 Injection function, 95, 99. See also INJ Fundamental bass, 169, 170, 173, 175, 185, Injection number. See INJ 209, 247 Input: node, 207, fig. 9.9; Klang, 208, 215, fig. 9.10; motive, 209; trichords, 227 Int, 26 Generalized Interval System (GIS): defined and discussed, 26-28; review of prelimiInternal. See Transformations nary examples, 28-30; and intuitions, Interval: intuitions about, xi-xii, 16, 17-20, 250. See also Commutative; Direct prod25-26, 74-75, 245-46, figs. 0.1, 4.4; reuct; IFUNC; Interval-preserving operaplaced conceptually by transposition, xiii, tions; Interval-reversing transformations; 157-60, 245-46; scarce, 102-03; forwards Inversions; LABEL; Non-commutative (backwards) oriented in time-span GIS, GIS; Octatonic; Quotient; Simply transi113-14; between roots, 170; for TCH, tive group; Timbral GIS; Time spans; 181; varying in Debussy, 231, 245-46; in Transpositions Rameau, 246-47; in Zarlino, 247-49. See also Central; GIS; IFUNC; Vector Graph, 171, 190-92; transformation, deInterval-preserving operations: and fined, 195; operation, defined, 196; isoLABEL, 47; defined, 47-48; form a morphism, 199; homomorphism, 201-06; product, 204, fig. 9.8(d) group, 48; characteristic behavior, 48-49; are sometimes transpositions, 49; comGreer, Taylor, 95n mute with transpositions, 50; combined Group: of operations, 4; abstract, 6; direct product, 15; homomorphism, 14-15; quowith inversions, 55; effect on IFUNC, 99; in time-span GIS, 111-13; and AODs, tient, 15; of intervals, 25-26; of transpositions, 46-47; of interval-preserving 121-24; octatonic, 252 operations, 48; locally compact, 103; can- Interval-reversing transformations, 58-59 onical, defined, 104. See also Canonical; Inverse: function, 3; element of semigroup, 6; operation, 56 Simply transitive group Inversions: defined in a GIS, 50-51, fig. 3.7; and LABEL, 51; when functionally Harvey, Jonathan, 24n equal, 52, 53; combined with transposiHasty, Christopher, 43n tions, 54; combined with interval-preservHaydn, Joseph, Quartet op. 76, no. 5, 211/j ing operations, 55; combined one with Helmholtz, Hermann, 247 another, 56; inverses of, 56; effect on Hexachord: semi-combinatorial, 143, 148IFUNC, 101, 150; as graph transforma49; Babbitt Theorem, 145 tions, 190-92, fig. 8.12 Hindemith, Paul, 75, 249-50 Homomorphism: of semigroup, 13; and nat- Isography: informally mentioned, 183, 187, 192; defined, 198-200; further examples, ural map, 14; of group, 14-15; of node/ 227, 230, 234 arrow system, 201; of graph, 201-06, 235 Isomorphism: of semigroups, 13; i-to-Pj, 48; Howat, Roy, 244n of node/arrow systems and graphs, 199 IVLS, 26 Ictus, 42-44, figs. 3.4-3.6 Identity: operation, 4; element for semiJones, James Rives, 39« group, 5

Index K, and Kh, 150-51 Klang: 175-81, 213-14, 216-17, 227-29; input and output, 207-08, fig. 9.10 Kramer, Jonathan, 211n Kurth, Ernst, 219/t LABEL: defined and discussed, 31-32; for T,(s), 47; for Pi(s), 47; for Itfs), 51 Lerdahl, Fred, and Jackendoff, Ray, 217« Lewin, David, 32n, 42«, 60n, 88n, 89n, 120n, 128n, 141n, 193n Ligeti, Gyorgy, Poeme symphonique, 23/i3, 67 Linear ordering: 135,146; and precedence, 211-12, 219 LT, 178 Mapping: defined, 1; of a set, 88 Maximum values: of IFUNC, 94; of INJ, 140-41, 142 Measurable transformations, 153 Measure: abstract, 153; on time-span GIS, 154-55 MED, 176-77 Melody, models for, 133, 219, fig. 9.17 Minturn, Neil, 227 Modular musical spaces, 17, 20-25 passim, 36. See also Quotient, GIS Modulation of transformational system, 129, 148-49 Moorer, James A., and Grey, John, 84n Morris, Robert D., 104n, 105/j Mozart, Wolfgang Amadeus, G-Minor Symphony, 220-25, figs. 10.1-10.4 MUCH. See Serial transformations Nancarrow, Conlon, Studies for Player Piano, 23n3, 66-67 Natural map: onto quotient family, 8; onto quotient semigroup, 12 Network, 164, 165, 168, 170, 172, 176, 177, 178, 192; transformation, defined, 196; connected operation, 197; operation defined, 197; isography, 198-200; product, 206, 235, figs. 9.8(d), 10.16; as model for series, 206, 218; Schenkerian, 214, 21618, fig. 9.16; and time spans, 215-16, 217, fig. 9.15; and melody, 219, fig. 9.17 Node/arrow system: defined, 193; communication in, 193-94; connected, 194; arrow chain in, 194-95; isomorphism of, 199; homomorphism of, 201. See also Precedence NODEMAP. See Node/arrow system, isomorphism of, homomorphism of NODES. See Node/arrow system

Non-commutative GIS, 50, 58. See also Octatonic; Time spans Octatonic: scale, 17; non-commutative GISs, 251-53 One-to-one. See Function Onto. See Function; Graph, homomorphism Operations: defined, 3; group of, 4; on a GIS, 46-59; and INJ, 144-45. See also DOM; Graph; LT; MED; PAR; REL; Serial transformations; SLIDE; SUED; SUBM Ordering. See Attack ordering; Linear ordering; Partial ordering; Precedence; Release ordering Orthography, left and right, 2, 176 Output: node, 207, fig. 9.9; Klang, 208, fig. 9.10; trichords, 227

PAR, 178, 229 Partial ordering: of pitch classes, 135-40 passim, figs. 6.7-6.9; of nodes, 209, 211 Partition Function, 152 Peel, John, 174 Pitch notation, xiii Precedence: relation, 210; ordering, 210, 219; and chronology, 210-18 Probability: and IFUNC, 101; and EMB, 107, 111 Product: of graphs, 204, fig. 9.8(d); network, 206, 235, figs. 9.8(d), 10.16. See also Cartesian; Direct product Progressive. See Transformations Prokofieff, Serge, Melodies op. 35, 227, figs. 10.7-10.9 PROT: defined, 134; and rows, 134-35, 146 Protocol pairs, 134, 211. See also PROT Quotient: family, 8; semigroup, 10-12; group, 15, 29; GIS, 29, 32-37 Rahn,John, 174 Rameau, Jean-Philippe: 73, 175, 246-47; double emploi, 213, 217 Reflexive property, 7 Regener, Eric, 103n, 144, 145, 152 REL, 178, 213, 229 Release ordering, 117 RICH. See Serial transformations Rl-chaining: 164, 235; in both pitch and rhythm, 221-22; in 1. and r. hands, 22627. See also Serial transformations, RICH and TCH Riemann, Hugo, 22n, 73, 175, 177 Row, as set in PROT, 134-35

257

Index Schenker, Heinrich: 165, 174, 214, 247, 249; and networks, 216-18, fig. 9.16 Schoenberg, Arnold: Violin Fantasy, 10103, fig. 5.8; "Angst und Hoffen," 125-33, 148, figs. 6.1-6.5; "Die Kreuze," 133-34, fig. 6.6; Moses und Aron, 136, 137-38, fig. 6.7; Piano Piece op. 19, no. 6, 141, 143, 160, figs. 6.10, 7.1 Scholica Enchiriadis, 204 Semigroup: of transformations, 4; abstract, 5; multiplicative notation, 5; congruences and quotients, 10-12; homomorphism and isomorphism, 13-15; direct product, 15 Serial transformations: RICH and TCH, 180-88 passim, 221-24, 226-27; MUCH, 183, fig. 8.5; RT and RI, 188; TLAST and TFIRST, 188-89, fig. 8.10; FLIPEND and FLIPSTART, 189, fig. 8.11; and commutative GIS, 189-90; I, 192; BIND, 208-09 Set: mathematical, 1; in a GIS, 88; environments of, 97-98; class, 105; in a family S, 124; transitivity, 129; theory in infinite case, 152-56 SGMAP. See Graph, isomorphism, homomorphism SGP. See Graph, transformation Signature motive, 137-38 Simple ordering. See Linear ordering Simply transitive group, 157; as transposition group, 157-58; on Klangs, 179-80; on octatonic family, 251-53; and its commuting group, 253 Slawson, Wayne, 85 SLIDE, 178, 227 SNDW, 121 Stockhausen, Karlheinz: rhythmic theory, 24n; Gruppen, 24n; Aus den sieben Tagen, 62; Klavierstuck XI, 66 Strunk, Oliver, 2Q4n SUED, 177-78, 229 SUBM, 178 Symmetric property, 7 TCH. See Serial transformations TFIRST. See Serial transformations Timbral GIS, 82-84, 84-85, fig. 4.5 Time spans: defined, 60; commutative GIS for, 61; non-commutative GIS for, 74-81,

258

112, 154-56, fig. 4.4; and networks, 21516, 217, fig. 9.15 Time unit: discussed, 61; contextual, 67 TLAST. See Serial transformations TMSPS, 60 Transformations: and intuition of musical space, xiii; defined, 3; internal and progressive, 126, 132, 134, 141, fig. 6.6; external and dispersive, 142-43, 164; modulated, 148-49; measurable, 153. See also Graph; Network; Operations; Serial transformations Transitive property, 7 Transpositions: can replace intervals conceptually, xiii, 157-60, 245-46; defined in GIS, 46; form a group, 46-47; and LABEL, 47; sometimes preserve intervals, 49; commute with interval-preserving operations, 50; combined with inversions, 54; effect on IFUNC, 100; form simply transitive group, 157; in octatonic GISs, 252 Travis, Roy, 225, 227 Tristan chord, 238-40, 242 Unfolding. See Unrolling; Vector Unrolling: interval vector, 116-19; EMB, 119-20; IFUNC, 120; INJ, 131 Value, of function, 1 Varese, Edgard, 189 Vector: unfolding interval, 44; interval, 98, 104, 114, fig. 5.12; M-class, 106-07 Wagner, Richard: Parsifal, 161-64, 181-82, figs. 7.2-7.5, 8.3; Tarnhelm and Valhalla networks, 179, fig. 8.2; Todesverkiindigung, 183-88, 208-09, figs. 8.6-8.9, 9.11. See also Tristan chord Webern, Anton: Piano Variations, 38-44, 181, 182-83, 190-92, 200, figs. 3.1-3.6, 8.4, 8.12, 9.5; Pieces for Violin and Piano, no. 3, 90-99, figs. 5.2-5.7; Pieces for String Quartet, op. 5, no. 4, 188-89, fig. 8.10 Wedging, 124-32 passim, figs. 6.1-6.4, 6.6 Wintle, Christopher, 136n Zarlino, Gioseffo, 73, 247-49, 250 Zero time point, 63 Z-sets: generalized by IFUNC, 103-04; generalized further by INJ, 149-50

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