Explaining Tonality: Schenkerian Theory and Beyond

Explaining Tonality

Eastman Studies in Music Ralph P. Locke, Senior Editor Eastman School of Music (ISSN 1071–9989) Additional Titles in Music Theory, Analysis, and Aesthetics Analyzing Wagner’s Operas: Alfred Lorenz and German Nationalist Ideology Stephen McClatchie Berlioz’s Semi-Operas: Roméo et Juliette and La damnation de Faust Daniel Albright The Chansons of Orlando di Lasso and Their Protestant Listeners: Music, Piety, and Print in Sixteenth-Century France Richard Freedman Concert Music, Rock, and Jazz since 1945: Essays and Analytical Studies Edited by Elizabeth West Marvin and Richard Hermann Elliott Carter: Collected Essays and Lectures, 1937–1995 Edited by Jonathan W. Bernard Historical Musicology: Sources, Methods, Interpretations Edited by Stephen A. Crist and Roberta Montemorra Marvin Music and Musicians in the Escorial Liturgy under the Habsburgs, 1563–1700 Michael Noone

Music and the Occult: French Musical Philosophies, 1750–1950 Joscelyn Godwin “The Music of American Folk Song” and Selected Other Writings on American Folk Music Ruth Crawford Seeger, edited by Larry Polansky and Judith Tick The Music of Luigi Dallapiccola Raymond Fearn Music Theory in Concept and Practice Edited by James M. Baker, David W. Beach, and Jonathan W. Bernard The Pleasure of Modernist Music: Listening, Meaning, Intention, Ideology Edited by Arved Ashby Schumann’s Piano Cycles and the Novels of Jean Paul Erika Reiman The Sea on Fire: Jean Barraqué Paul Griffiths Theories of Fugue from the Age of Josquin to the Age of Bach Paul Mark Walker

A complete list of titles in the Eastman Studies in Music Series, in order of publication, may be found at the end of this book.

Explaining Tonality Schenkerian Theory and Beyond

Matthew Brown

University of Rochester Press

Copyright © 2005 Matthew Brown All Rights Reserved. Except as permitted under current legislation, no part of this work may be photocopied, stored in a retrieval system, published, performed in public, adapted, broadcast, transmitted, recorded, or reproduced in any form or by any means, without the prior permission of the copyright owner. First published 2005 University of Rochester Press 668 Mt. Hope Avenue, Rochester, NY 14620, USA www.urpress.com and Boydell & Brewer Limited PO Box 9, Woodbridge, Suffolk IP12 3DF, UK www.boydellandbrewer.com ISBN: 1–58046–160–3 Library of Congress Cataloging-in-Publication Data Brown, Matthew, 1957Explaining tonality : Schenkerian theory and beyond / Matthew Brown. p. cm. Includes bibliographical references and index. ISBN 1-58046-160-3 (hardcover : alk. paper) 1. Schenkerian analysis. 2. Tonality. I. Title. MT6.B87E9 2005 781.2⬘58–dc22 2005007617 A catalogue record for this title is available from the British Library. This publication is printed on acid-free paper. Printed in the United States of America.

Contents Figures

vii

Preface

xiii

Introduction. Theoretical and Meta-Theoretical Issues Basic Goals and Assumptions Building and Testing Theories Six Criteria for Evaluating Theories 1. Schenker and the Quest for Accuracy Fux and Strict Counterpoint “The Heinrich Maneuver” “The Complementarity Principle” 2. Semper idem sed non eodem modo Conceptual Origins Prototypes Transformations Levels Fallout 3. What Price Consistency? Sequences Reconsidered Sequences and Counterpoint Analytical Implications 4. Schenker and “The Myth of Scales” Modes and Scales in Traditional Theory Schenkerian Theory and Scales Schenkerian Theory and Modal Inflections Schenkerian Theory and Exotic Inflections Schenkerian Theory and the Emergence of Functional Tonality 5. “Pleasure is the Law” The Limits of Schenkerian Theory Debussy, “C’est l’extase langoureuse” Debussy, “La mort des amants” Schenkerian Theory and Twentieth-Century Music

1 2 12 18 25 27 41 56 66 67 72 76 83 91 99 103 117 126 140 142 146 151 158 162 171 172 186 192 202

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6. Renaturalizing Schenkerian Theory Naturalizing Schenkerian Theory Schenkerian Theory as a Model of Expert Functional Monotonal Composition Conclusion

209 211

Notes

239

Bibliography

267

Index

281

222 234

Figures I.1. I.2. I.3. I.4. I.5. I.6. I.7. I.8. I.9. I.10.

Explaining tonality A procedure for composing typical tonal melodies Five forms of passing tone Counterfactual conditionals ‘The Covering-Law Model’ Explaining suspensions A procedure for generating 7–6 suspensions ‘The Hypothetico-Deductive Method’ The logic of falsification Six criteria for evaluating theories

3 5 6 7 8 9 11 13 16 19

1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9.

From strict counterpoint to functional tonality Fuxian cantus firmi First Species counterpoint Prototypical counterpoints in Fifth Species Dissonances in florid counterpoint Triads in three- and four-voice textures First Species in three voices Cadence patterns in two, three, and four voices Parallel and direct perfect octaves and fifths in three and four voices Differences in the behavior of triads and Stufen The major-minor system Laws of melodic motion and closure Polyphonic melodies Parallels by doubling and figuration Beethoven, Piano Sonata, Op. 2, no. 3, 1st movement, mm. 47–51 Parallels by combinations of harmonic and non-harmonic tones Laws of relative motion and closure Laws of vertical alignment Consonant non-harmonic tones and dissonant harmonic tones

26 30 32 33 34 36 37 39

1.10. 1.11. 1.12. 1.13. 1.14.

1.15. 1.16. 1.17. 1.18.

40 42 44 45 47 48

49 50 51 52

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1.19. 1.20. 1.21. 1.22. 1.23. 1.24. 1.25. 1.26. 1.27. 1.28.

Figures

Neighbor tones and suspensions as passing motions Implied tones and the nota cambiata Successive seventh chords Displacement and accented dissonances Beethoven, Piano Sonata, Op. 81a, 1st movement, mm. 230–42 Laws of harmonic classification Chord function vs. chord derivation Laws of harmonic progression Laws of chromatic generation Rectification of Phrygian
II

2.1. Schenker’s concept of prototypes, transformation, and levels 2.2. The non-recursive nature of Fuxian species counterpoint 2.3. Schenkerian Ursätze in C Major 2.4. Horizontalizing transformations 2.5. Filling in transformations 2.6. Harmonizing transformations 2.7. Reordering transformations (non-recursive) 2.8. Composing out 2.9. Schenker’s deep-middleground paradigms 2.10. Divided Urlinien 2.11. Derivation of divided Urlinien 2.12. The explanatory scope of Schenkerian theory 2.13. Schenker’s sketch of “The Representation of Chaos” from Haydn’s Creation 2.14. Alternative sketch of “The Representation of Chaos” 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8.

Sequences A typical ascending-fifth sequence Deriving ascending-fifth sequences Restacking ascending-fifth sequences Deriving ascending-third sequences Deriving descending-fifth sequences Deriving descending 5–6 sequences Deriving alternative descending-fifth sequences

52 54 55 55 56 59 60 61 62 63

69 71 73 78 80 81 82 85 86 88 90 92 94 97 102 104 105 107 108 109 111 112

Figures

3.9. 3.10. 3.11. 3.12. 3.13. 3.14. 3.15. 3.16. 3.17.

ix

Deriving ascending 5–6 sequences Simple mixture Double mixture Fux’s prototypical cantus firmi Typical two-voice counterpoint in First Species Typical two-voice counterpoints in Fourth Species Fux’s three-voice prototypes Typical three-voice counterpoints in Fourth Species Parallel motion in mixed species with three and four voices Parallel linear progressions within a single Stufe Parallel linear progressions between different Stufen Schenker’s analysis of Bach’s Little Prelude in C Major, BWV 924 Alternative analysis of Bach’s Little Prelude in C Major, BWV 924 Schenker’s analysis of Bach’s Prelude in C Minor, WTC I, BWV 847, mm. 1–18 Alternative analysis of Bach’s Prelude in C Minor, WTC I, BWV 847, mm. 1–18 Two analyses of the Prelude from Bach’s Partita No. 3 for Solo Violin Analysis of Bach, French Suite in D Minor, BWV 812, Minuet II

113 114 115 118 118 119 120 120

4.1. Scale membership and tonality 4.2. Schenker’s account of mixture 4.3. Beethoven, “Heiliger Dankegesang,” String Quartet, Op. 132 4.4. Graph of Beethoven, “Heiliger Dankegesang,” String Quartet, Op. 132 4.5. Brahms, “Vergangen ist mir Gluck und Heil,” Op. 14, no. 6 4.6. Graph of Brahms, “Vergangen ist mir Gluck und Heil,” Op. 14, no. 6 4.7. Chopin, Etude, Op. 10, no. 5 4.8. Graph of Debussy, Prélude à “L’Après-midi d’un faune,” mm. 30–37

145 148

3.18. 3.19. 3.20. 3.21. 3.22. 3.23. 3.24. 3.25.

122 123 125 127 129 131 132 133 135

152 153 155 157 159 160

x

Figures

4.9. Van den Toorn’s analysis of the opening of Stravinsky, Petrouchka 4.10. Cadences in fifteenth-century music 4.11. The Artusi-Monteverdi debate 4.12. Renaissance modal polyphony and functional tonality 5.1. Parallel dominant seventh chords 5.2. Free dissonances 5.3. Non-functional successions. Beethoven, Appassionata Sonata, Op. 57, 1st movement, mm. 62–87 5.4. Extreme chromaticism. Graph of Reger, Piano Quintet, Op. 64, mm. 1–8 5.5. Incomplete transferences of the Ursatz 5.6. Interpolations in Debussy, “La sérénade interrompue” (Préludes, Book 1, no. 9) 5.7. Graph of Debussy, “C’est l’extase langoureuse,” mm. 1–18 5.8. Graph of Debussy, “C’est l’extase langoureuse,” mm. 18–35 5.9. Graph of Debussy, “C’est l’extase langoureuse,” m. 36ff 5.10. Evolution of ‘The Sigh Figure’ in Debussy, “C’est l’extase langoureuse” 5.11. Global view of Debussy, “C’est l’extase langoureuse” 5.12. Debussy, “La mort des amants,” mm. 1–12 5.13. Debussy, “La mort des amants,” mm. 12–18 5.14. Debussy, “La mort des amants,” mm. 19–29 5.15. Debussy, “La mort des amants,” mm. 30–45 5.16. Global view of Debussy, “La mort des amants” 5.17. Prolonged dominant-seventh chords 5.18. Schubert, “Die Stadt,” Schwanengesang, no. 11 5.19. Functional tonality and twentieth-century tonal practices 6.1. Naturalizing music theory 6.2. Schenker’s derivation of the major system from ‘The Chord of Nature’

162 164 168 169 174 176 180 181 183 184 188 189 190 191 192 194 196 198 200 201 203 206 208 210 212

Figures

6.3. Lerdahl/Jackendoff’s derivation of Bach, Prelude in C, WTC I 6.4. Learning curve for expert monotonal composition 6.5. Sloboda’s “Diagram of typical compositional resources and processes” 6.6. Schenker’s account of expert monotonal composition 6.7. The scope of music theory

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218 221 228 231 232

Preface Few terms in music theory are more profound and more enigmatic than ‘tonality.’ First coined by Alexandre-Étienne Choron in his “Sommaire de l’histoire de la musique” (1810), it was popularized by François-Joseph Fétis in the 1830s and 1840s and has subsequently remained an essential part of theoretical discourse. Choron originally used the term to denote music in which notes are related functionally to a particular tonic, the tonic triad. This particular brand of tonality is often known as ‘functional tonality’ and is characteristic of works written by composers such as Handel, J. S. Bach, Scarlatti, C. P. E. Bach, Haydn, Mozart, Beethoven, Schubert, Schumann, Mendelssohn, Chopin, and Brahms. But, as Choron’s term has gained currency, so it has expanded its meaning considerably. Nowadays, the term is often used in a very general sense to denote any music that focuses melodically and/or harmonically on some stable pitch or tonic. This definition covers a broad range of music from many cultures and many time periods, from Medieval plain chant to various twentieth-century idioms. Of the many attempts to explain the nature of functional tonality, perhaps the most comprehensive was undertaken by Heinrich Schenker (1868–1935).1 In his monumental triptych, Neue musikalischen Theorien und Phantasien, he systematically investigated the ways in which lines and chords behave in functional tonal contexts. In the first volume, Harmonielehre (1906), he explained how functional harmonies (or Stufen) are organized into progressions (or Stufengang).2 In the second volume, Kontrapunkt (1910, 1922), he explained the basic properties of tonal voice leading (or Stimmführung).3 And in the final volume, Der freie Satz (1935), Schenker showed how the principles outlined in the Harmonielehre and Kontrapunkt operate recursively across entire monotonal compositions.4 But what sorts of relationships did Schenker count as tonal or, to be more precise, functionally monotonal? Why do these relationships work in some ways and not others? Why should we prefer Schenker’s theory of functional monotonality to its competitors?

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The purpose of this book is to try to answer these questions. The Introduction explores some of the general methodological issues that arise when we try to build, test, and evaluate a plausible theory of tonality. It begins by outlining the main ingredients of such a theory, namely concepts, laws, and procedures, and describes some of the problems that they raise. The Introduction goes on to discuss six criteria that theorists typically use to evaluate the success of their models. These criteria include accuracy, scope, fruitfulness, consistency, simplicity, and coherence. With this broad framework in place, the central portion of the book uses these six criteria to illuminate the foundations of Schenkerian theory. The conclusion describes some of the ways in which Schenkerian theory might develop in the future. There are several reasons why I have decided to address these issues at the present time. Like many music theorists, I am attracted to Schenker’s work because it offers us not only a powerful model for explaining the tonal system, but also a flexible tool for analyzing the practices of functional monotonal composition. The benefits of such an approach seem clear enough; instead of laboriously labeling each successive harmony with its own Roman numeral, the analyst can study the contrapuntal behavior of those harmonies, both within the local context of an individual phrase and within the global context of the piece as a whole. Although Schenkerian theory deals primarily with matters of functional harmony and voice leading, it often leads to important insights about a work’s motivic, rhythmic, and formal structure. This fact is amply demonstrated by Schenker’s best graphic analyses, such as those published in Der Tonwille (1921–24), Das Meisterwerk in der Musik (1925–30), and the Fünf Urlinie-Tafeln (1932).5 Furthermore, since the process of graphing particular pieces often teaches us new ways to hear music and understand the processes of musical composition, Schenkerian analysis can be of great help to performers and composers alike. At the same time, however, I am also intrigued by the formal properties of Schenker’s model. This fascination is something that I share with many other music theorists. Some, such as Milton Babbitt, Allan Keiler, Fred Lerdahl and Ray Jackendoff, have noted certain parallels between Schenker’s account of tonal relationships and Naom Chomsky’s account of grammatical relationships in

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natural language.6 Others, such as Michael Kassler, James Snell, and Stephen Smoliar, have even tried to model Schenkerian theory on the computer.7 I see my work as part of these particular traditions. That being said, Schenkerian theory is still the subject of considerable debate. For starters, although Schenker’s ideas are widely discussed in the music theory community, they are still shrouded in mystery. For example, Jonathan Dunsby has recently accused aficionados of promoting “a cabalistic image of Schenker” that treats his work as “a secret body of knowledge that can be applied but never fully exposed.”8 To make matters worse, when Schenker’s arguments are scrutinized, they often appear to be disjointed, cryptic, and even illogical. William Benjamin has described these shortcomings as follows: “While it is true in many instances that the problem lies with Schenker’s way of putting things and not with the formal relationships he has in mind, there are other cases where contradictory lines of reasoning go to the heart of his level-relating style. In these cases the problem results in a conflict between his artistic-compositional side and his formal-theoretic side.”9 Such conflicts are not, however, easy to resolve; as William Rothstein has observed, the ways in which theorists respond to these issues depends as much on their individual interests, temperament, psychological makeup, and their broader sense of the prevailing Zeitgeist, as on anything Schenker may or may not have written.10 Opinions vary enormously. Some regard Schenker’s work as a theory of musical structure, some as a theory of organic coherence, some as a theory of structural levels, some as theory of voice leading, and some as a theory of tonality.11 I prefer, however, to treat it as a theory of functional monotonal composition. It is this view that I will defend in this book with arguments drawn from analytical philosophy and cognitive science. Another point of contention is the widely held belief that Schenkerian theory is a closed system, incapable of adaptation. Edward Laufer expresses this view most strongly in his review of Ernst Oster’s English translation of Der freie Satz.12 According to him, “Schenker’s concepts as such are complete: they call for no extensions or modifications.” Elsewhere, he defends the narrow scope of Schenker’s project by noting: “It is ridiculous to demand as a criterion of validity that an approach be applicable to all times and

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musics; for it is sufficient that Schenker revealed, in ever new ways, the masterpieces of the classic era: altogether more great music than anyone could hope to come to terms with in several lifetimes.”13 While I am certainly sympathetic to Laufer’s point of view, I would add two important provisos. First, no one has ever offered a convincing argument to show why we should accept Schenker’s concepts as necessary and sufficient for explaining functional tonality. As I see it, such an argument requires us to reconstruct the theory systematically from the ground up. This is precisely what I hope to accomplish in the present volume. Second, although I have no problem accepting that Schenkerian theory is designed to explain functional monotonal pieces of the Common-Practice Period, I acknowledge that such music is part of a broader historical continuum. Eventually, Schenkerians must respond to Joseph Kerman’s complaint that they treat tonal practice as “an absolutely flat plateau flanked by bottomless chasms.”14 I am not suggesting, however, that we should simply revive the approach adopted by Felix Salzer in Structural Hearing; much as I admire Salzer’s broad outlook on music history, I do not think his work provides a satisfactory response to this issue.15 What we need is a way to redirect our thinking about tonality so that we can explain not only why Schenker’s theory works so well for functional monotonal compositions, but also how this theory enhances our understanding of tonality in general. Another important area of debate concerns the ways in which Schenkerian analysis interfaces with the processes of hearing and composing. Although I have no doubt that the way in which we graph a piece depends on how we hear that piece, I do not accept the common view that producing a Schenkerian analysis simply means learning to hear music more effectively. For one thing, such a view leaves the theory open to the charge of circularity levelled by Eugene Narmour in his book Beyond Schenkerism.16 According to Narmour, when Schenker tried to demonstrate that a piece could be derived from a given prototype, he always knew in advance what the prototype should be. By bending the piece to fit his preconception, Schenker was guilty of circular reasoning. From a methodological standpoint, this is a very serious charge and, so far as I can tell, is one that has never been successfully dismissed by the Schenkerian community.

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For another, there is strong evidence from Schenker’s own writings that he was not trying to explain how ordinary people hear a particular piece, but rather to explain how expert composers conceive of their music. For example, in his essay “Forsetzung der Urlinie-Betrachtung: I” from Das Meisterwerk I (1925), Schenker insisted that it is the composer’s business to compose out chords and that of the listener and the performer “to retrace” this path from foreground to background.17 Given this claim, I remain unpersuaded by the notion that Schenkerian analyses give us “artistic statements, in music, about music.”18 I firmly believe that, though they may be expressed as tones on a staff, Schenkerian analyses should be regarded not as pieces of music, but rather as models of music.19 Like any model, these graphs capture some aspects of the music and not others. I would argue that they model an expert composer’s internalized knowledge of functional monotonality.20 My goal, then, is to respond to these various challenges. Since I am mainly motivated by methodological concerns, I have adopted something of a compromise when referring to Schenker’s works. For one thing, Schenker’s views clearly changed over time; his goals in writing the Harmonielehre and Kontrapunkt I were not exactly the same as those of the Fünf Urlinie-Tafeln and Der freie Satz. Having said that, I still believe that there is an underlying continuity to Schenker’s thought. This continuity stems from two basic claims: 1) the laws of tonal voice leading are transformations of the laws of strict counterpoint and are intimately related to certain laws of functional harmony; and 2) complex tonal progressions can be explained as transformations of simple tonal prototypes. Given my interest in these claims and their theoretical implications, I will focus more on the unity of his thought, than on its evolution. Similarly, although one must always be sensitive to nuances in the meaning of particular Schenkerian terms, I have generally quoted from standard English translations. My rationale is simple; while I am aware of the extraordinary problems that arise whenever we try to render Schenker’s often convoluted prose into English, I do not want to become sidetracked with the daunting task of retranslating every passage. Anyone interested in Schenkerian theory must eventually compare the English translation with the original; I have

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tried to give citations in such a way that this can be accomplished as easily as possible. Although this book has been largely written from scratch, it nonetheless draws on material from several published and unpublished papers. The Introduction develops ideas originally outlined in two published papers written with Douglas Dempster, “The Scientific Image of Music Theory,” Journal of Music Theory 33/1 (1989), pp. 65–106, and “Evaluating Music Analyses and Theories: Five Perspectives,” Journal of Music Theory 34/2 (1990), pp. 247–79. This work was subsequently updated in two unpublished lectures, “The Scientific Image of Music Theory: Ten Years On,” given at Eastman School of Music in the fall of 1996, and “Choosing between Music Theories: Six Things to Think About” delivered at SUNY Buffalo in the fall of 1998. Chapters 1 and 2 expand arguments first presented in my Ph.D. dissertation, “A Rational Reconstruction of Schenkerian Theory” (Cornell, 1989), and in my paper “Rothstein’s Paradox and Neumeyer’s Fallacies,” Intégral 12 (1998), pp. 95–132. Whereas chapter 3 is entirely new, chapter 4 draws on another unpublished paper, “Schenker and ‘The Myth of Scales,’ ” presented at the Oxford Music Analysis Conference and the annual meeting of the Society for Music Theory in Baltimore in 1988. Chapter 5 borrows from lectures delivered at the University of Texas at Austin in the fall of 2001 and at Oxford University in the spring of 2003. Chapter 6 is mostly new but draws on material that I have been developing with Panayotis Mavromatis. I would like to take this opportunity to thank several people for helping me along the way. Much of the blame for this book must go to Arnold Whittall for introducing me to Schenkerian theory in my undergraduate years at King’s College, London. Since then, he has been a pillar of support and very kindly read a draft of this manuscript. This particular project started life in 1983–86, when I was a Junior Fellow at the Society of Fellows, Harvard University. I would like to thank the Society for its unqualified support during those years. Since that time, my thinking about Schenkerian theory has also been shaped by discussions with four other people—Doug Dempster, Dave Headlam, Panayotis Mavromatis, and Bryce Rytting. The fact that I have collaborated with most of them should indicate just how much they have taught me. I must also thank all my

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graduate students at Eastman for listening to me and for reading chunks of text—Don Traut and Bill Marvin deserve special mention—and John Brackett for inviting me to be involved with his dissertation. Special thanks must go to Frank Samarotto for carefully reading the manuscript. Most recently, I must thank Wayne Alpern and the Mannes Institute for Advanced Studies in Music Theory for inviting me to discuss my work in three workshops in the summer of 2002. I am extremely grateful to the members of my workshop—Kofi Agawu, Richmond Browne, Allan Cadwallader, William Drabkin, Yayoi Everett, Dora Hanninen, Dan Harrison, Peter Kaminsky, Richard Kaplan, Steve Larson, Nicolas Meeus, Margus Partlas, Giogio Sanguinetti, Carl Schachter, and Joe Straus. In terms of the production of the book, I would like to thank Dariusz Terefenko and Ciro Scotto for help in preparing the examples, though I must admit that I was never able to reproduce Schenker’s interlocking slurs to my satisfaction. And I must thank my editors Louise Goldberg and Ralph Locke for their patience and encouragement. Finally, I owe a special thank you to Milton Babbitt for encouraging me to publish my ideas in book form. I would like to express my gratitude to the following people and publishers for permission to quote from their works and publications: Cambridge University Press, for permission to use materials from the English translation of Schenker’s Das Meisterwerk in der Musik: (The Masterwork in Music, ed. William Drabkin, trans. I. Bent, R. Kramer, J. Rothgeb, and H. Siegel), vols. 1 and 2, © 1994, 1996. MIT Press, for permission to use the figure showing the derivation of J. S. Bach’s Prelude in C Major, WTC 1, from Fred Lerdahl and Ray Jackendoff, A Generative Theory of Tonal Music, © 1983. John Rothgeb, for permission to use materials from his edition/translation of Schenker’s Kontrapunkt I & II (Counterpoint I and II, ed. John Rothgeb, trans. John Rothgeb and Jürgen Thym, 2 vols., [New York, Schirmer, 1987]). Universal Edition, for permission to use materials from Schenker’s Harmonielehre (1906) and from Schenker’s Der freie Satz (1935), copyright © Universal Edition A.G. Vienna.

Introduction

Theoretical and Meta-Theoretical Issues What should we expect from a successful theory of tonality? Why should we prefer one theory of tonality over another? To what extent do theories of tonality pose the same methodological problems as theories in other domains? Although these are surely basic questions for any music theorist to ask, they are by no means easy ones to answer. In part, the difficulties stem from the fact that the term ‘tonality’ has come to mean different things to different people; as mentioned in the preface, some theorists use it very generally to denote music that centers on a stable pitch or tonic, whereas others use the term more restrictively to denote music that centers functionally on a particular tonic triad. But difficulties also arise because theorists often disagree about what they take to be the goals of their work. Once again, opinions differ widely. Some believe that theory building is an explanatory pursuit akin to the natural and social sciences, whereas others believe that it is a critical activity, analogous to art criticism or literary theory. As a result, some theorists deal exclusively with the internal properties of tonal music, whereas others insist that these properties cannot be studied apart from their cognitive, aesthetic, historical, and ideological context. The purpose of this Introduction is not to address these issues in a systematic manner, but rather to pinpoint some of the methodological concerns that shape my own particular views about tonal theory. I will proceed from the assumption that, before we can assess the cognitive, aesthetic, historical, and ideological implications of a particular theory, we must first see how that theory explains why tonal music behaves in some ways and not others. Since I believe that, at some level, we process our knowledge of music separately from our knowledge of other domains, I find it useful to treat music

2

Explaining Tonality

theory autonomously from other disciplines. Furthermore, since I also believe that tonality is basically a general property of voice leading and harmony, I will focus my attention on explaining these phenomena. This does not mean, however, that I am uninterested in thematic, rhythmic, or formal relationships. Rather, I do so because I am concerned with explaining the tonality of music rather than explaining tonal music per se. In fact, I readily accept that there is much more to understanding the latter than simply explaining its tonality. My discussion has three main parts. Part 1 outlines what I take to be the main goals of any theory of tonality: 1) to develop a vocabulary of concepts for describing what relationships count as tonal; 2) to discover a set of covering laws for explaining why these relationships work in some ways in ways and not others, and 3) to devise procedures for explaining how to produce specific tonal relationships. Next, part 2 discusses some of the issues that arise in testing a particular theory of tonality. Finally, part 3 discusses six criteria that theorists frequently use to pick one theory of tonality over another: accuracy, scope, consistency, simplicity, fruitfulness, and coherence.

Basic Goals and Assumptions Although music theorists formulate theories of tonality for a variety of reasons, three seem to be especially important. The first is to develop a vocabulary of concepts for describing what relationships count as tonal or, more specifically, functionally tonal. Concepts are terms we use to categorize our observations into broad types. According to ‘The Classical Theory of Concepts,’ defining a concept involves establishing a set of necessary and sufficient conditions that something must satisfy if it is to fall under that concept.1 Music theorists have traditionally expended considerable effort on developing concepts to describe a wide range of tonal relationships. Many of these concepts allow us to describe how notes behave linearly. For example, when describing the tonal properties of the music shown in figure I.1a (Beethoven, Six Variations, WoO 70), we might observe that the thirty-second note B in the treble clef,

Theoretical and Meta-Theoretical Issues

3

Figure I.1. Explaining tonality. a. Beethoven, Six Variations, WoO 70, mm. 13–16.

b. Beethoven, Piano Sonata, Op. 22, 4th movement, mm. 19–21.

m. 13, passes between A and C. This account presupposes that the concept ‘passing tone’ can be defined as an unaccented dissonance that moves by step between two consonances a third apart. Such a definition conveys the necessary condition that passing tones move by step and is sufficiently precise to differentiate passing tones from other dissonances, such as neighbor tones, cambiatas, and suspensions. Alternatively, we might use quite different concepts to describe how notes behave harmonically. In the case of figure I.1a, we might observe that the passage begins and ends on a G triad with the root in the bass. This description presumes that the concept ‘triad’ can be defined as a chord with a root and two other members a third and a fifth above. The latter definition conveys the necessary condition that triads are built from thirds and fifths, and yet is sufficiently broad to encompass major, minor, augmented, and diminished triads. It goes without saying that both of these descriptions tell us significant things about the tonal properties of figure I.1a. And yet,

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Explaining Tonality

neither one actually explains why Beethoven’s music is ‘in’ G major. The problem is that to explain why the passage is tonal, it is not enough to describe what melodic tones and triads are present; we must also say why they are related to each other in some ways and not others. To do so, music theorists invoke various laws of harmony and voice leading. These laws are general claims about the ways in which melodic tones and triads usually behave in tonal contexts.2 They are often, but not always, expressed in a conditional form: if X occurs in context Y, then Z will happen. For example, to explain why the passage in figure I.1a establishes the key of G major, we might invoke the following law: “If the leading tone appears in tonal contexts, then it normally ascends by half step onto the tonic.” This law explains why the melody rises from F
to G in mm. 13–14, why the alto part follows suit in mm. 14–15, and even why the same theme is in B-flat major when Beethoven transposes it up a minor third in his piano sonata, Op. 22 (see figure I.1b). Having said this, it is important to note that the law governing leading tones is not true all of the time. In figure I.1a, for example, the alto F
in mm. 15–16 moves to B3 and not G4, presumably to avoid doubling the soprano part. Since most laws of tonality are generally but not universally true, they are best classified as law-like generalizations. Besides introducing concepts to describe what relationships are tonal and invoking law-like generalizations to explain why tonal music behaves in some ways and not others, theorists often have yet another important goal: to explain how specific tonal relationships are produced. This task requires them to develop a set of procedures. Procedures consist of strings of commands that are usually expressed in conditional form: to produce X, do Y, then Z, and so on. Over the centuries, tonal theorists have developed procedures for accomplishing a variety of tasks from harmonizing a scale to composing a prelude from a given figured bass. Take, for example, figure I.2 (A procedure for composing typical tonal melodies). This procedure has six basic steps. First, pick a final tonic for the melody as a whole (figure I.2a). Second, begin the melody on a member of the tonic triad and end with a stepwise descent onto the tonic (figure I.2b). Third, pick a climax note midway through the melody and not more than an octave above the tonic (figure I.2c). Fourth, reinforce the tonic at the opening (figure I.2d). Fifth, join the opening to the climax and

Theoretical and Meta-Theoretical Issues

5

Figure I.2. A procedure for composing typical tonal melodies. a. Pick a final tonic for the melody as a whole. 1 b. Begin the melody on 8, 5,  3, or  1 and end with a stepwise descent onto the tonic. Cadence 3  2 1 c. Pick a climax about two thirds through the melody and not more than an octave above the tonic. Climax Cadence 4 3 2 1 d. Reinforce the tonic at the opening.

3

4

7

1

Climax 4

Cadence  2 1

e. Join the opening to the climax and the climax to the cadence. Climax Cadence 4 4 7 3 1 2 3  2 1 f. Fill in any details and check to see that the melody has a good overall shape and that it satisfies any general laws of melodic motion. Climax Cadence 7 4  3  2  3 4 6 1 2 3 2 1

the climax to the cadence (figure I.2e). Sixth, fill in any details and check to see that the melody has a good overall shape and satisfies any general laws of tonal voice leading, for example, that leading tones normally ascend by half step onto the tonic (figure I.2f). The preceding discussion has highlighted the central role concepts, laws, and procedures have traditionally played in tonal theory, but it is important to realize that these components are a lot more difficult to deal with than we might initially suppose. Take, for example, concepts. While it is certainly possible to find necessary and sufficient conditions for many concepts, cognitive scientists have found that certain concepts cannot be defined in this manner. Instead, they tend to define such concepts by appealing to the notion of prototypes.3 As Alvin Goldman explains: Concepts are represented in terms of properties that need not be strictly necessary but are frequently present in instances of the concept. These

6

Explaining Tonality properties are weighted by their frequency or by their perceptual salience. A collection of such properties is called a prototype.4

He adds: “Under the prototype view, an object is categorized as an instance of a concept if it is sufficiently similar to the prototype, similarity being determined (in part) by the number of properties in the prototype possessed by the instance and by the sum of their weights.”5 Although Goldman does not explicitly say so, the ‘perfect’ prototype may not actually ‘exist’ in the world at all; it may be an idealization that combines features from many different individuals. We can illustrate these points by reconsidering our definition of passing tones (see figure I.3, Five forms of passing tone). Although we defined passing tones as unaccented dissonances that move by step between two consonances a third apart, some passing tones do not satisfy this definition. In figure I.3a, for example, the pitch B in m. 1 seems to behave as a passing tone, even though it is consonant, and in figure I.3b the notes F and E in m. 1 are both dissonant and seem to connect two consonances a fourth, not a third, apart. More remarkably, figure I.3c contains an accented passing tone, figure I.3d includes a chromatic passing tone, and, if you believe Schenker, figure I.3e contains a leaping passing tone (or springender Durchgang)!6 In other words, it is much easier to think about passing tones in terms of prototypes and variants, than it is to provide a necessary and sufficient definition that works in all cases.7 Laws, too, pose their own problems.8 To begin with, not all generalizations are law-like.9 Take, for example, the claim that all pieces in G major have a key signature of one sharp. Even if true, which it is not, this generalization does not stand up as a law because it does not explain why there is any connection between having a signature of one sharp and establishing the key of G. To ensure that particular Figure I.3. Five forms of passing tone.

Theoretical and Meta-Theoretical Issues

7

generalizations are law-like, Nelson Goodman and others have suggested they should support so-called counterfactual conditionals.10 Counterfactual conditionals are hypothetical statements that suggest what would have been the case had things occurred differently (see figure I.4, Counterfactual conditionals). For example, when explaining why the cadence in figure I.4a establishes the key of G major, we might invoke our law governing leading tones. In this case, the F
in the first chord ascends by half step to the G in the final sonority. We might support this law-like generalization by noting that if the phrase had been in F
major, then the F
in the first chord would have descended to E
, before moving back onto F
for the final chord (see figure I.4b). Since we know that the piece in question is in G major, our remarks about what might have happened if the piece were in F
major are known as counterfactual conditionals. While counterfactual conditionals have proved very useful in helping us determine whether a particular generalization is indeed law-like, they nonetheless raise their own sets of questions; it

Figure I.4. Counterfactual conditionals.

8

Explaining Tonality

Figure I.5. ‘The Covering-Law Model’. ‘The Covering-Law Model’ Statement of initial conditions Statement of covering laws Statement about phenomena to be explained.

is unclear not only how to ensure that they are relevant in any given context, but also that they can be used to support all law-like generalizations. It is also debatable whether law-like generalizations are always necessary and sufficient for explanations. Certainly, many experts believe that scientific research is fundamentally law seeking or nomothetic.11 This prompted Carl Hempel and Paul Oppenheim to advance ‘The Covering-Law Model’ of explanation.12 According to them, explanations are arguments in which the premises are sets of covering laws and initial conditions, and the conclusion is some statement about the phenomena to be explained (see figure I.5, ‘The Covering-Law Model’). If the laws are universal and the arguments are deductively valid, then the result fits ‘The DeductiveNomological Model,’ and if the laws are not universal and the arguments are only inductively valid, then they conform to ‘The Inductive-Statistical Model.’ Figure I.6 (Explaining suspensions) illustrates what Hempel and Oppenheim had in mind. Suppose, for example, that we want to explain why a particular suspension C resolves by step to B (see figure I.6a). We might do so by invoking a simple law of tonal voice leading: namely, that suspensions normally resolve down by step onto consonances (see figure I.6b). Given the initial conditions that the seventh C–D on the down beat of m. 2 is dissonant and that the dissonance is a suspension, this law-like generalization allows us to deduce that the dissonant tone C on the down beat of m. 2 will resolve down by step onto the consonant tone B in m. 2. This is a perfectly acceptable explanation. Although ‘The Covering-Law Model’ certainly produces acceptable explanations, it is unclear whether covering laws are absolutely necessary for all plausible explanations. In particular,

Theoretical and Meta-Theoretical Issues

9

Figure I.6. Explaining suspensions.

b.

c.

The seventh C-D on the down beat of m. 2 is dissonant

The sixth B-D on the weak beat of m. 2 is consonant

This dissonance is a suspension

This consonance is a resolution.

Covering Laws

Suspensions generally resolve down by step consonances

Suspensions generally resolve down by step onto onto consonances

Explanation

Resolution on weak beat is a consonant sixth

Suspension on down beat is a dissonant seventh

Initial Conditions

critics have suggested that some types of explanation, such as so-called functional explanations used in biology, or narrative explanations found in history, do not necessarily involve covering laws, at least not in any explicit way.13 Functional explanations explain how particular parts of a complex system help to reinforce the system as a whole. For example, when explaining the tonal motion of Beethoven’s “Waldstein” Sonata, we might note that, in the first movement, the function of the recapitulation is to recompose the tonal motion of the exposition, that the function of the exposition is to modulate from the first key (C major) to the second key (E major), and that the function of this modulation is to create a pattern of tonal tension, and so on. Though this explanation seems plausible enough, it is unclear what covering laws it uses. Historical narratives often proceed on similar lines. For example, when explaining why the climax of a given aria appears on a high ‘C’ we might note that this aria was written for a particular tenor to sing at La Scala, and that high ‘C’ was his top note. Although this explanation seems to make historical sense, we would never suggest that, as a general rule, composers always make sure that the climax of an aria necessarily corresponds to the highest note in the singer’s register.14

10

Explaining Tonality

By the same token, Sylvain Bromberger and others have raised doubts about whether covering laws are sufficient for all explanations.15 We can paraphrase their point by comparing the explanation given in figure I.6b with the one shown in figure I.6c. In figure I.6b, we explained why the dissonant note C on the down beat of m. 2 resolved down by step to a consonant note B by invoking the law-like generalization that suspensions normally resolve down by step. However, we can also use the same covering law in a quite different way. This time we might start with the initial conditions that the sixth B–D on the weak beat of m. 2 is consonant and that it resolves a suspension on the preceding down beat. Since our covering law states that suspensions generally resolve down by step, we can deduce that the consonant note B is preceded by a dissonant note C. Although this argument is logically consistent, it does not carry the same weight as the argument given in figure I.6b. The problem is that the argument in figure I.6b explains the causal connections between the suspension and the resolution, whereas the one in figure I.6c does not. This, in turn, suggests that it is not enough to invoke covering laws in our explanations; our explanations must also be able to explain how one event causes another. One way to guarantee such casual connections is by reformatting our covering laws in procedural form.16 As mentioned earlier, procedures are strings of commands that we express in conditional form: to produce X, do Y, then Z, and so on. This point is illustrated in figure I.7, (A procedure for generating 7–6 suspensions). This procedure involves three distinct steps: 1) take an upper voice that descends by step C to B (figure I.7a); 2) add a lower voice that moves in parallel sixths below, E to D (figure I.7b); and 3) displace the first note of the upper voice over the second note of the lower voice (figure I.7c). This step produces the 7–6 suspension. Significantly, this procedure implies all of the same knowledge as the explanation given in figure I.6b. In particular, it implies that suspensions generally resolve down by step. And yet, the procedure adds something extra: it also indicates that suspensions are caused by displacing the upper voice over the lower voice.

Theoretical and Meta-Theoretical Issues

11

Figure I.7. A procedure for generating 7–6 suspensions. a. Take an upper voice that descends by step from C to B.

b. Add a lower voice that moves in parallel sixths below, E to D.

c. Displace the first note of the upper voice over the second note of the lower voice to create a 7–6 suspension.

Whatever advantages procedural explanations may give us, they can, however, be slippery things to deal with. Reconsider, for a moment, the procedure given in figure I.2. This strategy identified six basic steps for composing a typical tonal melody. In order for us to determine whether the procedure is successful or not, we must decide whether or not our new melody resembles the melody in figure I.1. But what does it mean to say that two melodies resemble one another? To answer this question we must invoke some notion of similarity, but it is by no means obvious what this step involves. As the philosophers Quine and Ullian explain: “Everything is similar to everything in some respect. Any two things share as many traits as any other two, if we are undiscriminating about what to call a trait; things can be grouped in no end of arbitrary ways.”17 In other words, we can judge similarity in different ways, depending upon what examples we take as prototypical and on how we decide to measure similarity. It seems, then, that in describing what relationships are tonal, explaining why these relationships create music that behaves in some ways and not others, and explaining how to produce specific tonal relationships, music theorists draw on a rich assortment of

12

Explaining Tonality

concepts, laws, and procedures. Although each component raises its own methodological issues, a given theory of tonality takes a specific cluster of concepts, laws, and procedures, and structures it in a particular way. If these theories are successful, then we normally expect that these knowledge structures will give us explanations and predictions that are coherent, reliable, and capable of being tested empirically by other theorists. And they should work for all and only all tonal music, or at least all and only all functional pieces. But how do we go about building such a theory? How, in fact, do we check to see that it actually covers all and only all music that we classify as tonal? Let us see how we might answer these new questions.

Building and Testing Theories According to conventional wisdom, theorists use a very simple strategy for coming up with their theories: they make guesses and then they test them.18 This approach is summarized in figure I.8 (‘The Hypothetico-Deductive Method’).19 In fact we can subdivide ‘The Hypothetico-Deductive Method’ (or H-D) into four basic steps: First, observe some distinct phenomenon in a well-defined test sample (figure I.8a). Second, guess some laws that seem to explain these observations (figure I.8b). Third, deduce some predictable consequences that are implied if these laws are correct (figure I.8c). Fourth, see if these predictions are confirmed by further observations (figure I.8d). If the predictions are indeed confirmed, then theorists carry on using their laws; if they aren’t, then they must either modify them, or they must replace them with new laws and start the procedure all over again. This final step is crucial to the entire process; ideally, it implies not only that the predictions are testable, but also that they are testable by someone else and under different conditions. Now, if we want to make a theory to explain the behavior of tonal voice leading and harmony, then we might adopt the following plan. We might begin by using certain familiar concepts to study a specific corpus of pieces that a given community regards as

Theoretical and Meta-Theoretical Issues

13

Figure I.8. ‘The Hypothetico-Deductive Method’. a. Observe phenomenon in some well-defined test sample. b. Guess laws to explain these observations. c. Deduce some consequences that are implied if the laws are correct. d. See if these predictions are confirmed by further observations. If they are, then keep using the new laws, if they aren’t, then modify them or replace with some new laws and start procedure over.

quintessentially tonal. Chances are we would pick pieces by such composers as Handel, J. S. Bach, Scarlatti, C. P. E. Bach, Haydn, Mozart, Beethoven, Schubert, Schumann, Mendelssohn, Chopin, and Brahms.20 Next, we might develop general laws that cover the behavior of these concepts. For example, we might generalize about how lines and chords behave in specific contexts. We could then predict how the lines and chords might behave in other contexts, preferably those that are slightly different from the ones used to make the original theory. If our predictions are confirmed, then we will keep on using our theory to explain the behavior of similar works from the same corpus; if, however, they are disconfirmed, then we must either modify the theory or find an alternative one that does work. So much for conventional wisdom; we find it perpetuated in many introductions to science. But when we look at how theorists actually work, we soon find that the process of building and testing theories is a good deal more complex than figure I.8 suggests. To begin with, it is very unlikely that any music theorists would actually build a new theory of tonality from scratch. Instead, they are more likely to start by taking some preexisting model and seeing if individual laws stand up to close scrutiny. Once they encounter a problem, then they will propose new covering laws. But disconfirming existing laws and confirming new laws is no easy task; on the contrary, these activities are riddled with problems and inconsistencies.21 These difficulties stem from the fact that, as David Hume famously remarked, “all inferences from experience, therefore, are effects of

14

Explaining Tonality

custom, not of [logical] reasoning.”22 Hume’s claim does not mean that inferences from experience are necessarily untrue; rather it suggests that they always fall short of certainty. As a result, our theories will always be fallible, though we may not always know where they will fail. Besides recognizing the general problems noted by Hume, philosophers have discussed several other difficulties. One of the most famous is ‘The Raven Paradox.’23 This paradox arises because law statements of the form “for all x, if x is F, then x is G” are logically equivalent to those of the form “for all x, if x is not G, then x is not F.” If x stands for piece of music, F stands for Beethoven and G stands for tonal, then the first law-statement “all pieces of music by Beethoven are tonal” is equivalent to the observation that a particular non-tonal piece, say Babbitt’s Philomel, is indeed not by Beethoven. What is paradoxical is that an atonal piece by Babbitt should count as evidence that confirms the generalization that all pieces by Beethoven are tonal; after all, Babbitt’s music appears to have little in common with Beethoven’s and it hardly seems relevant to any claims about whether the latter is tonal or not. Relevance also features prominently in another paradox known as ‘The Grue Paradox.’ This paradox was first discussed by Nelson Goodman.24 According to him, the issue of whether a generalization is supported by its instances depends on the nature of the properties that appear in that generalization. Paraphrasing Goodman, let us imagine a new property, ‘gronality’ which we define as follows: a piece is ‘gronal’ if it is classified as tonal before the year 2010 and atonal after that point. Now consider the following generalizations: 1) all works by Burt Bacharach are tonal and 2) all pieces by Burt Bacharach are ‘gronal.’ All works examined before the year 2010 will support not only the first generalization, but also the second one. This result is problematic because we want to use our generalizations to predict what will happen at some later date; as it stands, we have no basis for knowing whether the piece will be tonal or ‘gronal.’ To resolve this paradox, Goodman proposed that the law-like status of a generalization is a matter of entrenchment and projectibility. According to him, a predicate is entrenched if it is true as a matter of historical fact and has been used to formulate

Theoretical and Meta-Theoretical Issues

15

true predictions.25 He suggests that this property is the only one that allows us to project what will happen in the future. Tonality is just such a predicate; it is a trait that we naturally project from past observation to future expectation. ‘Gronality’ is not, however, because we have no reason to suppose that Burt Bacharach wrote music that can been classified as tonal at one point in time and atonal at some later date.26 ‘The Grue Paradox’ leads to a more general problem in confirmation; even if we agree on the same body of evidence, there is no reason to suppose that this data can be explained by only one theory; as we have seen, we can always invent new predicates, such as ‘grue-ness’ or ‘gronality,’ that capture some aspect of the piece. This means that, in principle at least, the evidence always underdetermines theories; there are always a variety of theories that will accommodate any given set of data. Pierre Duhem and W. V. Quine have gone even further to claim that, taken on its own, a particular piece of experimental evidence is seldom used to falsify an entire theory, because each element of the theory is somehow related to another element in the theory. As Quine puts it, “our statements about the external world face the tribunal of sense experience not individually but only as a corporate body.”27 In other words, “any seemingly disconfirming observational evidence can always be accommodated to any theory.”28 This claim is usually known as ‘The Duhem-Quine Thesis.’ Although extremely controversial, ‘The Duhem-Quine Thesis’ is significant because it threatens to undermine the most famous alternative to H-D. Given the many paradoxes of confirmation, Karl Popper and others have suggested that, instead of defending their theories by finding more and more supporting evidence, scientists should actually spend their time trying to show that some hypotheses are false.29 In this sense, the guiding principle of testability is not confirmation but falsification. The rationale behind Popper’s thinking is simple enough and is apparent from the arguments given in figure I.9 (The logic of falsification). According to Popper, H-D seems to follow the plan given in figure I.9a. Let us assume that, if a particular explanation E is valid, then it will make a given prediction P. When researchers test this prediction and find that it is indeed accurate, they regard this as confirmation of their

16

Explaining Tonality

Figure I.9. The logic of falsification. a.

c.

If Explanation E is valid, then Prediction P is true.

b.

If X → Y

Prediction P is true

Y

∴ Explanation E is valid

∴ X (invalid)

If Explanation E is valid, then Prediction P is true.

d.

modus tollens If X → Y

Prediction P is false

⫺Y

∴ Explanation E is false

∴ ⫺X (valid)

explanation. But, according to Popper, this line of reasoning is invalid. Since explanation E was used to come up with prediction P, prediction P cannot then be used to confirm explanation E. That would commit the fallacy of affirming the antecedent (see figure I.9b). To avoid this problem, Popper insists that H-D should be used, not to confirm, but to falsify a theory. This means that, given explanation E and prediction P, knowing that P is false allows us to deduce that E is false as well (see figure I.9c). Such an argument follows the principle of modus tollens given in figure I.9d. Popper used the notion that H-D can be used to falsify an explanation to reach two important conclusions. First, he proposed that even our best knowledge is fallible or conjectural. To quote him, “we cannot reach certainty . . . all we can do is to criticize [our theories], and to test them, as severely as our ingenuity permits.”30 Second, Popper decided that “the criterion of the scientific status of a theory is its falsifiability, or refutability, or testability.”31 He even claimed that falsifiability serves as a criterion for making demarcations between science and non-science; whereas scientific theories must always have discrete boundaries and can be falsified, non-science need not have such boundaries and cannot be falsified. Critics, however, have responded that strict falsification is hard to uphold in light of ‘The Duhem-Quine Thesis.’ If, as Duhem and Quine insist, “any seemingly disconfirming observational evidence can always be accommodated to any theory,” then it is hard to see how we can decisively falsify any theory. Each time we come up with a counter example, we can simply adjust our theory to make it

Theoretical and Meta-Theoretical Issues

17

fit. If we cannot use specific observations to falsify a given theory, then we cannot pick one theory over another on purely evidential grounds. Popper’s claim that falsifiability provides us with a definitive means of discriminating science from non-science seems, therefore, too strong.32 As it happens, Popper’s views have also been challenged by recent findings in cognitive science. Research by Tweney, Doherty, Mynatt, and others has suggested that scientists do not usually set out simply to falsify existing theories; on the contrary, they normally start out by seeking confirmatory data; only when this data has been obtained does it make sense to engage in rigorous falsification.33 Thus, while it might be true that successful theories are initially conjectural, the accumulation of supporting evidence will eventually move them beyond that status.34 Most people do, in fact, believe that theories become more strongly confirmed the more supporting evidence has been amassed. This point suggests that our understanding of what makes a successful music theory must eventually take account of the ways in which music theorists actually work, rather than simply relying on their logical or empirical content. All in all, just as ‘The Covering Law Model’ provides an idealized picture of explanation, so ‘The Hypothetico-Deductive Method’ presents an idealized account of how music theorists confirm or refute a particular theory. While the latter conveys many aspects of how music theorists work, the process of building and testing theories involves a far more complex interplay between confirmation and falsification. As we have seen, this process is always open ended; music theorists do not begin with a blank slate, they do not have foolproof methods, and they do not reach definitive solutions. Instead, they plunge in medias res. They start working within the context of an existing music theory, even if they know some portions of that theory are surely wrong. They then try to overcome certain specific problems, using the rest of the theory to support their work. To borrow an image from Neurath and Quine, this situation is like that facing sailors at sea on a leaking boat.35 Unable to rebuild their vessel from the keel up in a dry dock, the crew is forced to fix the leaks while adrift on the open water. As they work on leaks in one area of the boat, the

18

Explaining Tonality

sailors rely on the remaining timbers to keep the craft afloat. But as one leak is patched so another appears; bit-by-bit the boat becomes transformed into something new. In fixing the leaks, music theorists typically try to balance what Quine has described as “the drive for evidence and the drive for system.”36 According to him, the former demands that “theoretical terms should be subject to observable criteria, the more the better, the more directly the better, other things being equal” while the latter insists that these terms “should lend themselves to systematic laws, the simpler the better, other things being equal.” Quine adds, “If either of these drives were unchecked by the other, it would issue in something unworthy of the name scientific theory: in the one case a mere record of observations, and on the other a myth without foundation.”37

Six Criteria for Evaluating Theories It should be clear by now that the task of building and testing music theories is not only a lot messier than we might suppose, but it is also plagued by many of the same methodological problems as theories in other disciplines, especially the natural and social sciences. In very general terms, we have seen that these problems often involve finding effective ways to balance the drive for evidence with the drive for system. Although this all sounds reasonable enough, we can spell out more clearly how such a balance might be achieved. Following, Kuhn, Quine and others, we can invoke several concrete criteria for doing so: figure I.10 (Six criteria for evaluating theories) includes the notions of accuracy, scope, fruitfulness, consistency, simplicity, and coherence.38 The list is by no means exhaustive; other criteria, such as completeness, elegance, or even ‘coolness,’ could easily be added. Figure I.10, however, gives us a good place to start our inquiry. For convenience, the six criteria are divided into two types—those that relate to the evidential basis of theories and those which relate to the systematic aspects of a model. The horizontal arrow at the top of the figure suggests that there may be inherent conflicts between the evidential concerns (accuracy, scope, and fruitfulness) and the systematic concerns

Theoretical and Meta-Theoretical Issues

19

Figure I.10. Six criteria for evaluating theories. Evidence

System

Accuracy

Consistency

Scope

Simplicity

Fruitfulness

Coherence

(consistency, simplicity, and coherence). The vertical arrows along the sides suggest that there may be a similar tension between members of the same type. The first criterion on our list is accuracy. Since a successful theory should explain why certain phenomena behave the way they do and predict what will happen in new situations, we will surely want the most exact explanations and predictions possible. Although our theories can never be completely accurate, we generally see increased precision as a virtue, so that, given two theories, we usually prefer the one that is more accurate, other things being equal. And yet, it is by no means obvious how to measure the accuracy of competing models. Norwood Hanson, Thomas Kuhn, Paul Feyerabend, and others have insisted that since our observations about the world may be theory-laden, decisions about what constitutes evidence will be determined by our theoretical prejudices.39 Indeed, to paraphrase Quine, “Our judgments about what there is are always embedded in some sort of theory; we can substitute one theory for another but we cannot detach ourselves from theory altogether and see the world unclouded by any preconception of it.”40 If competing theories reflect widely different values, there may be no neutral grounds for comparing their accuracy. In such cases the two theories are said to be incommensurate. Critics, however, have countered that the problems of theory-ladenness and incommensurability are greatly exaggerated. While it may be true that observations tend to be theory laden, this doesn’t mean that we can never distinguish observational terms from theoretical terms. Indeed, as Quine points out, “theoretical sentences grade off to observation sentences”; some observations come with

20

Explaining Tonality

negligible theoretical baggage, while others come with a lot.41 Compare, for example, the observation that the first movement of the “Eroica” Symphony begins with an E triad with the claim that the movement is “in” E. Whereas the former involves few theoretical assumptions and is readily apparent to most listeners, the latter is highly theory laden and presupposes an elaborate theory of key relations.42 But even though many of our observations are biased, theorists working with different frameworks are still able to reach some degree of consensus in specific cases. In this respect, the main issue is that of intersubjective testability rather than of objectivity per se. This notion of intersubjective testability through bias is most remarkable when the various biases are not only different, but also contradictory. I refer to this as ‘The Hostile Witness Principle.’ Very simply, this principle suggests that a particular claim gains force when it is confirmed by theories that are directly opposed to one another. Such situations often arise because, as Richard Boyd points out, “A particular experiment can be conducted on the basis of a methodology that—however theorydependent—is not committed to either of the two contesting theories.”43 The second criterion in figure I.10 is scope. Just as we want our theories to be as accurate as possible, so we also put a premium on their breadth of coverage. This means that, given two theories, we normally prefer the one that covers the larger array of pieces or wider range of properties, other things being equal. Perhaps the most common way to expand the scope of our theories is by subsuming hitherto separate theories under a single scheme. This is known as ‘Theory Reduction.’44 For example, if we proposed a theory of functional monotonality that subsumes the theory of tonal voice leading with the theory of functional monotonal harmony, then we should prefer it to a rival theory that explains only the behavior of tonal voice leading or that explains only the behavior of functional monotonal harmony. This does not mean, however, that generality is always a good thing; on the contrary, some theories are so general that they lose their explanatory force. That’s the snag with Anne Elk’s theory of the brontosaurus.45 While it may be true that the only thing common to all brontosauruses is that they “are

Theoretical and Meta-Theoretical Issues

21

thin at one end, much, much thicker in the middle and then thin again at the far end,” this account is so general that it is trivial. The notion of ‘Theory Reduction’ has likewise been questioned. While there are certainly situations in which the model seems to apply, it does not explain every option. Kuhn, for example, has suggested that explanatory scope can expand through conceptual innovations or paradigm shifts, rather than the addition of new laws or the reduction of one theory into another.46 To overcome these difficulties Philip Kitcher and others have advocated the notion of ‘Theoretical Unification.’ According to Kitcher, the success of theories depends on “minimizing the number or patterns of derivation employed and maximizing the number of conclusions generated.”47 When evaluating the success of our theories, we do not simply want to keep duplicating results in familiar pieces; we also want to use our concepts, laws, and procedures to predict how things will behave in other, perhaps novel, works and disclose new phenomena or previously unnoted relationships among those already known.48 To do this, we must be able to predict every consequence and not merely a smattering of special cases.49 This idea represents the third criterion in figure I.10, namely fruitfulness. Very simply, given two theories of functional monotonality, we prefer the one that makes the more fruitful predictions, other things being equal. According to Kuhn, the criterion of fruitfulness “deserves more emphasis than it has yet received.”50 Just as it is hard to measure the accuracy of rival theories, it is also difficult to assess their fruitfulness, especially if the theories draw on widely different bodies of empirical data. This issue is troubling because successful theories often evolve considerably over time; it may take a long while for theorists to appreciate just how fruitful a theory may be and even longer to consider all of its ramifications. As a result, fruitfulness may not play a significant role when a theory is originally presented to the world but will become more significant as that theory matures. Whereas our first three criteria concern the drive for evidence, our fourth criterion concerns the drive for system. When formulating a music theory, we will want it to be as internally consistent as possible, other things being equal. Inconsistencies are bad because they prevent us from making concrete predictions; if we cannot make concrete predictions, then we cannot subject our work to

22

Explaining Tonality

rigorous testing, especially by other people. It is important to stress, however, that even internal consistency is a matter of degree; the more comprehensive our theories become, the less likely they are to be internally consistent. For example, although Schenker insisted that the process of composing out is bound to the laws of tonal voice leading and harmony, his graphs often violate the law prohibiting parallel perfect octaves and fifths. As we will see in chapters 2 and 3, this inconsistency has enormous consequences for understanding various tonal phenomena, especially sequences. Assuming one has satisfied the preceding constraints, the drive for system often prompts us to evaluate rival theories according to their complexity; as a rule we prefer a simple theory to a complicated one, other things being equal. This fifth criterion from figure I.10 is commonly known as Ockham’s Razor or ‘The Principle of Parsimony’ and is almost as old as theorizing itself.51 The rationale is obvious enough: the simpler the theory, the easier it is to apply and the less prone it is to error.52 Nevertheless, Ockham’s Razor can leave some scars. Simplicity is, to some extent, in the eye of the beholder; if taken to extremes it can end up being a downright liability.53 As we will see in chapter 1, the theory of functional equivalence seems to run afoul of this very issue. By restricting tonal harmonies to just three basic functions (tonic, dominant, and subdominant), the theory oversimplifies the richness of many tonal progressions. In such cases simplicity comes into conflict with other values, especially accuracy and scope. Finally, we come to the sixth criterion from figure I.10. Suppose that we are faced with two theories that are equivalent empirically and systematically, that is, both are equally accurate, both have a similar scope, both are fruitful, both are consistent to the same degree, and both are comparable in their level of simplicity. What grounds, then, do we have for picking one theory over the other? One answer is to see if they are coherent with theories in related disciplines, other things being equal.54 If, for example, one theory of functional monotonality is coherent with current theories of music cognition, whereas the other is not, then we have good reason for preferring the former to the latter. Conversely, if our theories cannot be embedded or made compatible with known principles of music cognition, then we have good reason to be suspicious of them.

Theoretical and Meta-Theoretical Issues

23

Of course, coherence between theories in different domains is an extremely difficult thing to achieve, especially since the two spheres of inquiry may have such different methodological bases; it is no easy task to find appropriate bridge laws that bind one discipline to another. Nevertheless, coherence is still a goal worth striving for. So far, we have sketched some general reasons for invoking the six criteria given in figure I.10. We have also seen that these criteria are often at odds and that theorists are often forced to trade one off against another. These last ideas are significant because they help us to explain how different theorists can reach a consensus on certain methodological issues without agreeing on the particulars of any given theory. The reason for this is simple: music theorists might evaluate theories according to the same criteria, but weight each one differently in any given context. Take, for example, the differences between Schenker’s original theory and that of his student, Felix Salzer.55 As we will see in chapter 1, Schenker put a premium on the accuracy of his model. He went to great lengths to make sure that he could explain the minutest details of tonal voice leading. But, to achieve such a high degree of precision, Schenker confined himself to functional monotonal music of the CommonPractice Period. Felix Salzer, meanwhile, focused his attention on explanatory scope; he wanted to explain the tonal properties of a broad spectrum of music from the Middle Ages to the Twentieth Century. To do this, Salzer had to modify many of Schenker’s ideas, thereby sacrificing much of their accuracy and predictive power. History has shown that Salzer’s trade was not worth the price. As time has gone by, music theorists have generally found Schenker’s approach more robust than Salzer’s. This seems in keeping with the general notion that theorists value accuracy most highly among the evidential values; once accuracy is gained, theorists are unwilling to relinquish it without a fight. Just as theorists seem to value accuracy over scope and fruitfulness, so they also seem to value consistency over simplicity and coherence. Again the rationale is clear. Consistency guarantees that claims can be tested inter subjectively; and inter subjective testability is one of the hallmarks of rational discourse. Simplicity, meanwhile, is an advantage in application, but it is ultimately much better to be complex and consistent than simple and inconsistent.

24

Explaining Tonality

Besides providing us with a mechanism for evaluating rival theories, we can also use the preceding model to explain how music theory might progress as a discipline. Now, there can be little doubt that “progress” has become something of a dirty word in musicological circles these days. Indeed, as Richard Taruskin notes, “Few historians today subscribe to overtly teleological or deterministic models.”56 The reasons for this are not hard to find. Recent scholarship has tended to focus on those properties of music whose significance depends on historical or social context. Once music is treated as a social construct, the notion of progress simply smacks of anachronism and cultural imperialism. Change, yes; progress, no. Yet, even if we balk at the idea of progress in musical composition, we may still accept the notion of progress in music theory. The two things are, in fact, quite different. Whereas the former is concerned with producing aesthetic experiences, the latter is concerned with producing knowledge and understanding. Knowledge and understanding are two things that can clearly improve. They progress when a community of theorists acknowledges that a new cluster of concepts, laws, and procedures is more accurate, more expansive, more fruitful, more consistent, more parsimonious, and more coherent than its predecessors. It is for these reasons that Schenker’s theory of functional monotonality is superior to its precursors. And we have every reason to suppose that even more successful theories of tonality will be developed in the future.

1

Schenker and the Quest for Accuracy Of all epistemic values, none is more important to the music theorist than the quest for accuracy. Whether formulating concepts, developing explanatory laws, or devising effective procedures, music theorists always try to provide accurate accounts of the music they are analyzing. If their methods fall short, then they will try to devise new concepts, laws, and procedures that fulfill these expectations. And so it was for Heinrich Schenker. He began from the simple observation that the laws of strict counterpoint (or Der strenge Satz) are not accurate enough to explain the richness of functional tonality (or Der freie Satz). To account for these anomalies, he transformed the laws of strict counterpoint through the addition of functional harmonies, or Stufen. To quote from the Harmonielehre: “[Tonal] composition, then, appears as an extension of strict [counterpoint]: an extension with regard to both the quantity of [tonal] material and the principle of its motion. What is responsible for all these extensions is the concept of the Stufe.”1 Later, in Kontrapunkt I–II, he reiterated this view, claiming that functional relationships must be understood only as transformations or “prolongations” of strict counterpoint.2 As it stands, Schenker’s claim can be interpreted in two quite different ways. It could mean that the laws of strict counterpoint still operate in tonal contexts but, through the intervention of Stufen, they operate more freely. This interpretation is the one endorsed by most music theorists.3 The disadvantage with this view, however, is that functional tonality extends or transforms strict counterpoint is some ways, but not in others. If we generalize about when or why these extensions occur, then we inevitably end up proposing a new set of covering laws. This latter option is precisely the one shown in figure 1.1 (From strict counterpoint to functional

26

Explaining Tonality

Figure 1.1. From strict counterpoint to functional tonality.

tonality). Quite simply, it takes certain basic principles of voice leading and interprets them in three different contexts; interpreting them within a world of intervals allows us to explain the behavior of strict two-voice counterpoint; interpreting them within a world of simple triads allows us to explain the behavior of strict three- and four-voice counterpoint; and interpreting them within a world of Stufen allows us to explain the behavior of functional tonality. To defend this second interpretation, this chapter begins by taking another look at the laws of strict counterpoint as presented

Schenker and the Quest for Accuracy

27

by Fux in Gradus ad Parnassum. Part 1 shows that, even in Fuxian Species counterpoint, the laws of strict counterpoint actually change as the individual lines become more elaborate and as their context changes from the intervallic world of two voices to the triadic world of three or more voices. Continuing Fux’s line of argument, part 2 shows how Schenker changed the laws still further when they operate in the context of functional tonality. We will refer to these changes as ‘The Heinrich Maneuver.’ Finally, part 3 demonstrates that Schenker’s laws of functional voice leading are intimately related to various laws of tonal harmony. We will refer to this interconnection between harmony and counterpoint as ‘The Complementarity Principle.’ The conclusion explores some ramifications of this argument; it suggests that, by eliminating some familiar anomalies from conventional theories of voice leading and harmony, Schenker upheld an utterly traditional epistemic value— the quest for accuracy.

Fux and Strict Counterpoint Few music theory treatises have been more influential than Johann Joseph Fux’s Gradus ad Parnassum.4 First published in 1725, this volume became an instant success, circulating throughout Europe both in its original Latin and in German, Italian, French, and English paraphrases and translations. Indeed, Fux’s discussion of species counterpoint not only shaped the thinking of composers such as Haydn, Mozart, and Beethoven, but it also served as model for later treatises, such as Schenker’s Kontrapunkt.5 Yet, as Joel Lester and others have stressed, the Gradus stands out more for its pedagogical success than for the originality of its ideas. Fux is famous for his Five Species, but as Lester and others have shown, the notion of the species can be traced all the way back through Berardi’s Documenti armonici (1687) and Miscellanea musicale (1689, Book 1) to Bononcini’s Musico prattico (1673, 1688), Zacconi’s Prattica di musica seconda parte (1622), Diruta’s Il transilvano Part 2 (1609) and even to Lanfranco’s Scintille di musica (1533).6 Furthermore, these treatises endorse many of the same laws as texts written by Zarlino, Gaffurius, Guilielmus Monachus, Tinctoris, and Prosdocimus, to name but a few.

28

Explaining Tonality

When thinking about Fuxian species counterpoint, it is important to underscore a few basic points. First, like many of his predecessors, Fux distinguished between the artificial world of counterpoint and the real world of musical composition; in the former, notes are considered as abstract entities, independent of any motivic, rhythmic, or formal implications, but in the latter, they are understood in terms of their motivic, rhythmic, and formal function. Such distinctions are not, of course, unique to music theory. For example, physicists invariably begin by studying particular phenomena under idealized conditions with perfect vacuums and frictionless pulleys. Only when these simplified situations can be explained do they extend their arguments to cover real world situations where vacuums leak and pulleys move more erratically. Even when physicists model the real world, they do not mistake the model for the real thing. After all, when simulating nuclear reactions, researchers don’t expect to blow up their laboratories. Second, to explain the behavior of contrapuntal lines, Fux tried to formulate an explicit set of covering laws. These laws cover three main areas: 1) how individual lines move and reach closure; 2) how polyphonic lines move in relation to one another; and 3) how unstable (or dissonant) tones behave in relation to stable (or consonant) tones. Nevertheless, Fux’s generalizations are not as precise as we might like them to be: sometimes they are inconsistent and sometimes they are demonstrably incomplete. For example, Lester has noted that Fux’s treatments of raised leading tones and melodic tritones are contradictory.7 Meanwhile, David Lewin has proposed a new ‘Global Rule’ which states that “for every note X of the counterpoint line lying above (below) the cadence tone, some note lying one step lower (higher) than X must appear in the line at some point subsequent to X.”8 Lewin also claims that the counterpoint should not be “overarticulated by a strong closure before the final cadence is reached.”9 Third, even within the realm of strict counterpoint, Fux assumed that the behavior of contrapuntal lines changes as the individual lines become more elaborate and as the number of voices increases from two to three or more. To explain changes of the first type, Fux used the notion of the species. He identified five distinct species, with First Species corresponding to simple note-against-note

Schenker and the Quest for Accuracy

29

counterpoint and Fifth Species to florid counterpoint. Second, Third, and Fourth Species are intermediate stages that introduce progressively more complex dissonances: Second Species introduces the passing tone, Third Species, the nota cambiata, and Fourth Species, the suspension. To deal with changes of the second type, Fux systematically discussed all five species, first in two voices, then in three, and then in four. Let us now look at these changes in more detail, starting with First Species in two voices. When we think about First Species counterpoint, it is important to begin by considering the structure of the cantus firmus. For convenience, these are presented in figure 1.2 (Fuxian cantus firmi). According to Fux, the six tunes shown in figure 1.2a epitomize the principles of good melodic writing. Globally, each one constitutes a single phrase of music; each has a clearly defined beginning, a single climax, and an emphatic cadence at the end. They are also clearly modal; each one begins and ends on a given modal final and each one is strictly diatonic. Locally, Fux’s cantus firmi primarily move by half or whole step, they avoid successive unisons, and they avoid leaps of a seventh or diminished/ augmented intervals, as well as consecutive leaps in the same direction (see figure 1.2b). These simple observations are expressed as general laws in figure 1.2c. These laws set out to explain why particular notes appear in a given cantus firmus and why they behave in specific ways and not others. For convenience these laws are classified in several ways. On the one hand, figure 1.2c distinguishes between main laws, which explain what normally happens in a melody, and subordinate laws, which explain significant exceptions to the norm. On the other hand, it distinguishes between local laws, which operate from one note to the next, and global laws, which operate across the melody as a whole. Turning to figure 1.3 (First Species counterpoint), we see that First Species counterpoints resemble cantus firmi in some ways but not others. Like a cantus firmus, each counterpoint in figure 1.3a forms a single coherent phrase with a clear beginning, a welldefined climax, and a stepwise motion onto the final cadence. Locally, it primarily moves by whole- or half-step and uses leaps only sparingly (see figure 1.3b). Unlike a cantus firmus, however, the counterpoint contains repeated notes (e.g., B in mm. 8–9) and

30

Explaining Tonality

Figure 1.2. Fuxian cantus firmi. a. Fux’s prototypical cantus firmi.

b. Interval content. Melody 1 Melody 2 Melody 3 Melody 4 Melody 5 Melody 6

U m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8 – 2 5 1 1 1 – – – – – – – – 2 3 2 1 – – – – – – – 1 – 1 5 2 2 – – 1 – – – – – – 1 4 2 3 3 – – – – – – – – 4 4 2 1 – – – – – – – – – 3 4 2 1 1 – – – – – – –

Total 10 9 11 13 11 11

Schenker and the Quest for Accuracy

31

Figure 1.2. (continued). c. Laws of melodic motion/closure for a cantus firmus. If a cantus firmus is perfectly closed, then it begins on
1 and ends
2–
1.

GM

If a cantus firmus moves from one note to another, then successive notes are usually a whole- or a half-step apart and never repeat the same note.

LM

If leaps do occur, then they are never larger than an octave and never encompass diminished/augmented intervals or the interval of a seventh.

LS

(G⫽global, L⫽local, M⫽main, S⫽subordinate)

ends by ascending chromatically G–A at the final cadence.10 Since Fux promoted the independence of lines, he put a premium on contrary motion between the cantus firmus and the counterpoint (see figure 1.3c). At cadences, for example, the final perfect consonance is always approached in contrary motion from the nearest imperfect consonance, either with a major sixth expanding to an octave or a minor third contracting to a unison. Furthermore, Fux insisted that when the cantus firmus and counterpoint do move in the same direction, they can never produce parallel or direct perfect octaves and fifths between successive notes. Figure 1.3d also shows that First Species counterpoints are always consonant with the cantus firmus; we will refer to this idea as ‘The Consonance Constraint.’ According to Fux, unisons, octaves, fifths, and their compounds are classified as perfect consonances, whereas thirds, sixths, and their compounds are classified as imperfect consonances. All other intervals, seconds, fourths, sevenths, and their compounds plus all diminished/augmented intervals, are counted as dissonant. Although figure 1.3d suggests that First Species textures often include more imperfect than perfect consonances, Fux insisted that the cantus firmus and the counterpoint always begin and end on perfect consonances. These generalizations are presented as covering laws in figure 1.3e. Once again, these laws are classified as main or subordinate and as local or global.

32

Explaining Tonality

Figure 1.3. First Species counterpoint. a. Prototypical counterpoint in First Species.

b. Interval content of the counterpoint. U m2 M2 m3 M3 P4 A4 1 3 3 2 – 1 –

P5 –

m6 1

M6 –

m7 –

c. Relative motion between cantus firmus and counterpoint. Contrary Oblique Similar Parallel 3 6 1 1 1 d. Vertical intervals between cantus firmus and counterpoint. U 2 3 4 A4 5 6 – – 4 – – 1 5 e. Laws of melodic motion/closure for a counterpoint. If a counterpoint is perfectly closed, then it begins on
8 or
5 and ends
7–
1.

GM

If a counterpoint moves from one note to another, then successive notes are usually a whole- or a half-step apart, though they can occasionally repeat the same note.

LM

If leaps occur, then they are never larger than an octave and never encompass diminished/augmented intervals or the interval of a seventh.

LS

If leaps occur, then they seldom appear consecutively in the same direction and are normally approached/departed by step in the opposite direction.

LS

(G⫽global, L⫽local, M⫽main, S⫽subordinate)

M7 –

P8 –

Parallel 6 2

7 –

8 2

Total 11

Total 11

Total 12

Schenker and the Quest for Accuracy

33

Figure 1.4. Prototypical counterpoints in Fifth Species. From Fux, The Study of Counterpoint, Figs. 82 and 87.

Significant as they may be, the differences in behavior between cantus firmi and simple counterpoints are less pronounced than those between simple and florid counterpoints. We can see this very clearly by comparing the examples in figure 1.3 with those in figures 1.4a–b (Prototypical counterpoints in Fifth Species). For example, whereas the counterpoint in figure 1.3 has only one note for every member of the cantus firmus, those in figures 1.4a and 1.4b have up to four. And, unlike the counterpoint in figure 1.3, the one in figure 1.4a crosses the cantus firmus (e.g., m. 2) and includes the chromatic tone B (e.g., m. 5). But the most striking difference between the counterpoints in figure 1.3 and those in figures 1.4a and 1.4b is that the latter are no longer strictly consonant with the cantus firmus. In fact, we find three specific forms of dissonance (see figure 1.5, Dissonances in florid counterpoint). First there is the passing tone. These dissonances move by step in a single direction between two consonances a third apart. For example, in figure 1.4a, the dissonant E in m. 2 passes by step from D to F. Although passing tones usually move between two adjacent consonances, they can appear consecutively: if, for example, the cantus firmus holds the note B, the counterpoint can descend by step from the sixth G, through the diminished fifth F and perfect fourth E onto a third D.11 Fux’s second type of dissonance is the nota cambiata.12 Whereas passing

34

Explaining Tonality

Figure 1.5. Dissonances in florid counterpoint. Passing tone

Nota cambiata

Appears on beat 2 and moves by step between two consonances a third apart.

Second Species

Two/three notes in the counterpoint against one in cantus firmus.

Appears on beats 2, 3, and 4. Consecutive passing tones of a diminished fifth and a perfect fourth can appear on beats 2 and 3.

Third Species

Four notes in the counterpoint against one in the cantus firmus.

Appears on beat 2, a step below consonance on beat 1 and leaps to a consonance a third below on beat 3.

Third Species

Four notes in the counterpoint against one in the cantus firmus.

Fourth Species

Two notes in the counterpoint against one in the cantus firmus.

Lower neighbor can sometimes appear on beat 2. Suspension

Appears on beat 1. Sounds as a consonance on previous weak beat and resolves onto a consonance a step below.

tones always move by step, the cambiata is approached by step and departed by leap. Consider figure 1.4b, m. 2. Here, the dissonant G on beat 2 is preceded by a consonant A and followed by a consonant E. The third type of dissonance is the suspension. Whereas passing tones and cambiatas fill in the space between two consonances, suspensions displace one strand of counterpoint against another. This is clear in the final cadence from figure 1.4a. Here the two voices basically move in parallel sixths F/D–E/C before arriving on the octave D/D. The suspension is formed when the consonant D of the counterpoint in m. 10 is displaced over E in the cantus firmus on the down beat of m. 11. Having become dissonant, D finally descends onto the consonant C at the end of the bar. Since suspensions stem from displacements rather than diminutions, Fux classified them as essential dissonances and passing tones/cambiata

Schenker and the Quest for Accuracy

35

as non-essential dissonances.13 That being said, figure 1.4a contains several cases where the resolution of a suspension can be ornamented by step (e.g., mm. 7–8) or by leap (e.g., mm. 8–9). So far, we have noted that the behavior of intervals can be specified precisely in two-voice counterpoint: consonances (unisons, octaves, fifths, thirds, sixths, and their compounds) are stable and control the contrapuntal motion, whereas dissonances (seconds, fourths, sevenths, and their compounds, as well as all augmented and diminished intervals) are unstable and play a subordinate role. But Fux realized that when two, three, or more counterpoints are added to a cantus firmus, the behavior of these additional lines changes even more: “as the number of voices increases, [so] the rules are to be less rigorously observed.”14 According to him, these changes occur because three-voice textures are no longer controlled by intervals per se, but rather by specific amalgamations of intervals known as triads: “[T]he harmonic triad should be employed in every measure if there is no special reason against it.”15 We will refer to this idea as ‘The Triadic Constraint.’ According to Fux, harmonic triads are three-note collections that include unisons, thirds, fifths, or octaves above a given bass note. According to this definition, diminished triads are not harmonic triads, nor are so-called first- and second-inversion triads. There are, in fact, only twelve harmonic triads within the modal system: these are listed at the top of figure 1.6 (Triads in three- and four-voice textures). Significantly, Triads 1–8 contain members of the Gamut; some include only the white notes, while the others include B. Meanwhile, Triads 9–12 contain the chromatic notes C, F, and G; they normally appear at cadences in Dorian, Mixolydian, and Aeolian modes. Fux sometimes replaced the diminished triad B–D–F with the minor triad B–D–F, especially in Mixolydian mode. Although Triads 13–17 are not harmonic triads per se, they appear in figure 1.6 because they do occur at cadences; the diminished triads D–F–B and G–B–E contain members of the Gamut, whereas the others use the chromatic tones C, F, and G. The shift from the intervallic world of two-voice counterpoint to the triadic world of three- and four-voice counterpoint has several important implications. For one thing, it erodes the distinction

36

Explaining Tonality

Figure 1.6. Triads in three- and four-voice textures. 1. 2. 3. 4. 5. 6. 7. 8.

C D E F G G A B

E F G A B B C D

G* A* B* C* D* D* E* F*

Major Minor Minor Major Major Minor Minor Major

5/3 or 6/3 5/3 or 6/3 5/3 or 6/3 5/3 or 6/3 5/3 or 6/3 5/3 or 6/3 5/3 or 6/3 5/3 or 6/3

9. 10. 11. 12.

D E A B

F G C D

A* B* E* F*

Major Major Major Minor

5/3 or 6/3 5/3 or 6/3 5/3 or 6/3 5/3 or 6/3

[Mixolydian] [Aeolian] [Dorian] [113–114]

13. 14. 15. 16. 17.

D G E A B

F B G C D

B E C F G

Diminished Diminished Diminished Diminished Diminished

6/3 6/3 6/3 6/3 6/3

[Dorian] [Mixolydian] [Aeolian]

* ⫽ Fux’s harmonic triads

between perfect and imperfect consonances. We have already seen that two-voice textures always begin and end on perfect consonances. This, however, is no longer the case in three- and four-voice textures. As shown in figure 1.7 (First Species in three voices), threevoice textures often begin and end on imperfect consonances: figures 1.7a and 1.7b both open with sonorities that contain the third, but not the fifth and figure 1.7c begins and ends in the same way. The only time when Fux still favors perfect over imperfect consonances is in the final cadence; figure 1.7a ends on bare octaves and figure 1.7b ends on an octave plus a fifth. Besides weakening the distinction between perfect and imperfect consonances, ‘The Triadic Constraint’ also blurs the distinction between consonant and dissonant intervals. Since the behavior of intervals is determined from the bass, the upper voices can be consonant with the bass and dissonant among themselves. The cadence in figure 1.7b offers a good case in point. Although perfect and augmented fourths are always dissonant in two-voice counterpoint, they behave as consonances when they appear

37 Figure 1.7. First Species in three voices. From Fux, The Study of Counterpoint, Figs. 104, 105, 106.

38

Explaining Tonality

between the upper voices in three- and four-voice contexts. In this case, the final measures contain the perfect fourth A–D followed by an augmented fourth G–C. Since the behavior of the fourth depends upon its harmonic context, the distinction between harmonic and non-harmonic tones starts to be more significant in threeand four-voice textures than the distinction between consonance and dissonance. The significance of ‘The Triadic Constraint’ is easy to see in figure 1.8 (Cadence patterns in two, three, and four voices). In two-voice textures, cadences are normally marked by a stepwise descent
3–
2–
1 in the cantus firmus and a stepwise motion
1–
7–
1 in the counterpoint (see figures 1.8a–b). Since it does not matter whether the counterpoint is above or below the cantus firmus, these lines are completely invertible at the octave. But in threevoice textures, things are more complicated. Using our two-voice cadence as a prototype, figures 1.8c–e show what happens when the third voice is added to them. If the cantus firmus is in the bass, then the third voice usually contains the notes
5–
4–
5 or
5–
4–
3 (see figures 1.8c and 1.8d). Significantly, this new line is only partially invertible at the octave: it can appear above or below the counterpoint
1–
7–
1, but never below the cantus firmus. This is because
5 creates a dissonant fourth with
1. If, however, the third voice is added below the two-voice prototype (see figure 1.8e), then it typically leaps
1–
5–
1. Unlike the other voices, this one is never invertible at the octave; on the contrary, it is the quintessential bass part and as such is quite different in character from the cantus firmus and the other counterpoint. Figure 1.8f then shows how these prototypical voice-leading patterns normally appear in four-voice cadences. ‘The Triadic Constraint’ has other effects on the behavior of contrapuntal lines. In two-voice textures, counterpoints rarely contain consecutive leaps in the same direction, but in three- and four-voice textures, such phenomena are actually quite common. Figure 1.7c offers us some good examples: in mm. 5–9 the middle counterpoint leaps from F via C to E and back through C to A, and in mm. 2–5, the lowest counterpoint leaps from F to D to A to F. We can find similar adjustments to the laws of relative motion (see figure 1.9, Parallel and direct perfect octaves and fifths in three and

Schenker and the Quest for Accuracy

39

Figure 1.8. Cadence patterns in two, three, and four voices. Two-voice cadences.

Both voices are completely invertible at the octave. Three-voice cadences.


3–
2–
1 and
1–
7–
1 are completely invertible at the octave,
5–
4–
3 is partially invertible at the octave (never in bass), and
1–
5–
1 is never invertible at the octave (only in bass). Four-voice cadences.


3–
2–
1 and
1–
7–
1 are completely invertible at the octave,
5– 4
–
3 is partially invertible at the octave (never in bass), and
1–
5–
1 is never invertible at the octave (only in bass).

four voices). In two-voice contexts, for example, parallel perfect octaves and fifths do not occur between successive down beats when they are separated by a falling third (see figures 1.9a and 1.9b). Yet they can occur in three- or four-voice textures, “for the sake of the harmonic triad” (see figures 1.9c and 1.9d). Notice, too, that repeated tones and consecutive leaps appear in both of the counterpoints. The preceding discussion has shown that, even according to Fux, the laws of strict counterpoint change as the individual lines become more elaborate and as their context changes from a texture

40

Explaining Tonality

Figure 1.9. Parallel and direct perfect octaves and fifths in two, three and four voices. From Fux, The Study of Counterpoint, Figs. 29, 27, 28, 120, 173.

with two-voices to one with three or more voices. This is because two-voice textures are controlled by ‘The Consonance Constraint,’ whereas three- and four-voice textures are controlled by ‘The Triadic Constraint.’ As these contexts change, so the laws of voice leading change as well. But, as Fux was fully aware, species counterpoint is still several steps removed from actual tonal practice. We are left to wonder how the laws of strict counterpoint must be altered to account for the idiosyncrasies of functional tonality.

Schenker and the Quest for Accuracy

41

Figure 1.9 (continued).

“The Heinrich Maneuver” One writer who specifically addressed this issue was Heinrich Schenker. In his Harmonielehre (1906) and Kontrapunkt I–II (1910, 1922), he demonstrated that, in functional contexts, the behavior of lines is influenced by a new condition, which we will refer to as ‘The Stufe Constraint.’ As he explained in the Harmonielehre: “[Tonal] composition differs from strict counterpoint, in so far as the former possesses Stufen, which articulate its content, and in so far as [they] allow for a much wider range of freedom in voice-leading.”16 He added: Where in strict [counterpoint], we have notes consonant to those of the cantus firmus, we have, in [tonal] composition, the Stufe. Where, in strict [counterpoint], we have a dissonant passing tone, we have, in [tonal] composition, free voice-leading, a series of intermediate chords, unfolding in free motion.17

According to him, “Stufen then resemble powerful projector lights: in their illuminated sphere the parts go through their evolution in a higher and freer contrapuntal sense, uniting in [discrete harmonies], which, however, never become [an] end in themselves but

42

Explaining Tonality

Figure 1.10. Differences in the behavior of triads and Stufen. a, b. Influence of Stufen on harmonic progressions. From Fux, The Study of Counterpoint, Figs. 118, 117.

Reordering of diminutions. c. First Species.

d. Third Species.

e. Tonal Counterpoint.

always result from the free movement.”18 For convenience we will classify these extensions of strict counterpoint under the rubric of ‘The Heinrich Maneuver.’ This point is clarified in figure 1.10 (Differences in behavior of triads and Stufen). Take, for example, the passages shown in figures 1.10a and 1.10b. Although both satisfy the laws of strict counterpoint and are triadic, the progression I–II–I6 in figure 1.10a is extremely rare in tonal contexts, whereas the progression I6–VII6–I

Schenker and the Quest for Accuracy

43

in figure 1.10b is very common. This suggests that there may be constraints on what strings of triads can appear in tonal contexts. These constraints are connected with chord function. Figures 1.10c to 1.10e show another important difference. Figure 1.10c gives a simple cadence in First Species. Although there are many ways in which this pattern can be elaborated in Third Species, Fux insisted that the final
1 must always be preceded by
7 (see figure 1.10d). In functional tonal composition, however, this cadence pattern can be transformed in other ways (see figure 1.10e). Assuming that this cadence articulates the progression I6–VII6–I, then the upper voice actually implies two strands of counterpoint: a soprano voice that moves
1–
7–
1 and an alto voice that descends
5–4
–
3. In other words, it is a so-called polyphonic or compound melody. The descending line
7–6
–
5–4
actually connects the soprano voice C with the alto voice G. Since the measure expands a single Stufe, the soprano C need not sound on the final beat; according to Schenker, this pitch is “mentally retained” throughout the entire measure before it resolves onto D. But what exactly are Stufen and how do they influence tonal voice leading? Very simply, whereas triads are merely collections of stacked thirds and fifths, Stufen are triads that function within a tonality, this function being indicated by a Roman numeral. For us to assign a Roman numeral to a given triad, we must be able to show that triad is related to the tonic. The notion of functionality helps to explain why some chord strings are possible in tonal contexts and others are not. As shown in figure 1.11(The major-minor system), Schenker acknowledged that there are seven possible Stufen. In major keys, I, IV, and V Stufen are represented by major triads; II, III, and VI Stufen by minor triads; and the VII Stufe by a diminished triad (see figure 1.11a). Meanwhile, in minor keys I, IV, and V Stufen are represented by minor triads; II, III, VI, and VII Stufen by major triads; and the II Stufe by a diminished triad (see figure 1.11b). Schenker invoked the concept of simple mixture to swap triads between parallel keys and the concept of secondary mixture to change the qualities of II, III, VI, and VII in major, and II, III, VI, and VII in minor.19 These are shown in figures 1.11c and 1.11d. According to Schenkerian theory, double mixture allows the composer to borrow triads from the parallel key and change their

44

Explaining Tonality

Figure 1.11. The major-minor system. Mixture is a transformation that allows us to move from one quadrant to another: simple mixture allows us to move from one diatonic quadrant to another; secondary mixture allows us to move from a given diatonic quadrant to its secondary; double mixture allows us to move from a given diatonic quadrant via the other diatonic quadrant to the other secondary quadrant. Major

Minor o

o

a. Diatonic

|I, ii, iii, IV, V, vi, vii

|

i, II, ii , III, iv, v, VI, VII| b. Diatonic

c. Secondary

|I, II, III, IV, V, VI, vii, VII

|

i, ii, iio, iii, iv, v, vi, vii | d. Secondary

Tonicization is a transformation that derives the missing triads by shifting locally to another major-minor system. Diminished triads serve as position finders in the new system: iº iiº iiiº iiiº ivº ivº/vº iv/v IV/V vº viº viº viiº

viiº/II, iiº/vii viiº/II, iio/vii viiº/III, iiº/ii viiº/IV, iiº/ii viiº/IV(V), iiº/iii viiº/V, iiº/iii iv/II, iii/II, iii/III, ii/III, ii/IV, vii/V, vii/VI, vi/VI, vi/VII, v/VII IV/II, III/II, III/III, II/III, II/IV, VII/V, VII/VI, VI/VI, VI/VII, V/VII viiº/VI, iiº/iv viiº/VI, iiº/iv(v) viiº/VII, iiº/v viiº/VII, iiº/vi

quality. Since diminished triads appear only as VII in major keys and II in minor, they serve as position finders; for example, instead of treating the diminished triad C–E–G as io in C major-minor, we can regard it either as VII of D major or as II of B minor.20 Although figure 1.11 covers a wide range of Stufen, it does not include those on IV/V. This omission suggests that these Stufen cannot be derived directly from the tonic by mixtures. Instead, Schenkerian theory explains them by another process, tonicization. Tonicization allows an individual note or Stufe to function temporarily as a tonic and, in so doing, implies that the piece temporarily shifts to a new major-minor system. For example, to generate the major triads F–A–C and G–B–D in C major, we must temporarily shift to a new key; these triads can then arise as the subdominant of D major (IV/II), the dominant of B major (V/VII), the Phrygian II of F minor (II/IV), the mediant of E minor (III/III), the submediant

Schenker and the Quest for Accuracy

45

Figure 1.12. Laws of melodic motion and closure. a. Fux’s Laws of Strict Counterpoint

b. Revised Laws of Tonality

If the cantus firmus is perfectly closed, then it begins on
1 and ends
2–
1.

GM

If a melody is perfectly closed, then it begins on 8
, 5
, or 3
, and ends 2
–
1.

If a cantus firmus moves from one note to another, then successive notes are usually a whole- or a half-step apart and never repeat the same note.

LM

If a melody moves from one note to another, then successive notes are usually a step apart.

If leaps do occur, then they are never larger than an octave and never encompass diminished/augmented intervals or the interval of a seventh.

LS

If leaps occur, then they do so when the melody shifts from one harmonic tone to another or from one contrapuntal voice to another.

If leaps occur, then they seldom appear successively in the same direction and are normally approached/departed by step in the opposite direction.

LS

(G⫽global, L⫽local, M⫽main, S⫽subordinate)

of B minor (VI/VII), or as the sub-tonic of A minor (VII/VI). In other words, although it is impossible to generate Stufen on IV/ V directly from the tonic, they can be generated indirectly from I as II/IV, II/III, III/III, III/II, IV/II, V/VII, VI/VII, VI/VI, or VII/VI. Elsewhere, Douglas Dempster, Dave Headlam, and I have referred to this idea as ‘The IV/V Hypothesis.’21 We believe that ‘The IV/V Hypothesis’ is sufficiently well confirmed to stand as a subordinate law of functional harmony. Having introduced Stufen into the mix, we can now assess their impact on tonal voice leading. Figures 1.12a–b (Laws of melodic motion and closure) compare Fux’s laws of strict counterpoint with Schenker’s laws of tonal voice leading. Starting with the structure of melodic lines, it is clear that although Schenker certainly accepted that functional tonal melodies mostly move by whole- or half-steps and normally end by descending
2 to
1 (this is part of what he meant by the term melodic fluency or Der fliessende Gesang), he also recognized that this motion is controlled by ‘The Stufe Constraint.’22 Indeed, Schenker went so far as to claim that “all melody is the composing-out of sonorities.”23 Since Stufen can only be expressed

46

Explaining Tonality

by major, minor, and diminished triads, it follows that melodic motion must always occur between triadic intervals, such as thirds, fourths, fifths, sixths, and octaves. Now the condition “between tones of the tonic triad” in the law of melodic motion forces us to change both of our main laws: globally, the essential melody need not begin on
1; it can now start on
5 and
3; locally, since
2 and
1 cannot be members of the same Stufe, the melody must end with the step wise descent
3–
2–
1. Of course, if melodies ultimately descend from
3, then they can immediately be classified as major or minor. By dropping the phrase “whole- or half-steps” the new law also allows for steps of an augmented second, provided that they appear between harmonic tones. Among other things, this means that lines can skip from 6
to
7 in progressions from II6 to V.24 Once the preceding adjustments have been made, we can explain how melodic leaps arise in functional triadic contexts. First of all, they can occur when the essential melody moves from one harmonic tone to another. The opening theme from the first movement of Beethoven’s “Eroica” Symphony is just such a melody: it arpeggiates the tones of the tonic triad E–G–E–B–E–G–B–E.25 Notice how this theme contains several consecutive leaps in the same direction. But melodic leaps can also occur when the essential melody moves to another voice in the essential counterpoint. This process gives rise to polyphonic or compound melodies. We can illustrate this idea in figure 1.13 (Polyphonic melodies) by comparing the Sarabande and Double from Bach’s Partita No. 1 for Solo Violin, BWV 1002. As shown in figure 1.13a, the Sarabande is a polyphonic composition built from at least four lines of counterpoint. Sometimes the four voices are given explicitly, such as at the start of the piece, but more often they are implied by the laws of tonal counterpoint. For example, since we know that tonal harmonies must always contain the third of the chord we know the dominant triad in m. 8 must have an A in the tenor voice. And, since it seems likely that the final B in m. 7 belongs to the alto voice, we can infer that the tonic chord at the start of m. 7 has B as its alto. Meanwhile, figure 1.13b shows how the Double horizontalizes this four-voice framework to create a seamless polyphonic melody. In other words, the prominent leaps of the Double are created by moving between different contrapuntal voices of the Sarabande.

Schenker and the Quest for Accuracy

47

Figure 1.13. Polyphonic melodies. a. Bach, Partita No. 1 for Solo Violin, BWV 1002, Sarabande, mm. 1–8.

b. Bach, Partita No. 1 for Solo Violin, BWV 1002, Sarabande: Double, mm. 1–8.

Just as Schenker used ‘The Stufe Constraint’ to explain why melodies behave differently in strict counterpoint than in functional tonality, so he also used it to explain anomalies in the relative motion between lines. Schenker was quite adamant that tonal counterpoint mostly moves by contrary or oblique motion and that

48

Explaining Tonality

Figure 1.14. Parallels by doubling and figuration. Beethoven, Piano Sonata, Op. 2, no. 3, 1st movement, mm. 47–51. From Schenker ed., Brahms Octaven und Quinten, Ex. 5a.

the foreground “fundamentally prohibits parallel octaves and fifths.”26 But he also recognized that parallel perfect octaves or fifths sometimes occur in actual tonal pieces; in fact, he reconsidered almost 100 of the 140 or so examples cataloged in Brahms’s famous study Oktaven und Quinten.27 According to him, these anomalous parallels arise in two distinct ways. First, they stem from unessential voices. Figure 1.14 (Parallels by doubling and figuration) gives a simple illustration. This short extract from the first movement of Beethoven’s Piano Sonata, Op. 2, no. 3, contains two sets of parallel perfect octaves between the tenor and the bass, B/B–C/C in mm. 48–49 and A/A–B/B in mm. 50–51. These are marked by the parallel lines in the musical example. But, according to Schenker, these parallel octaves actually arise from doubling.28 The melody in the right hand is actually polyphonic: the soprano voice descends from D (m. 48) through C (m. 50) to an implied B (m. 51); the alto voice descends from B (mm. 47–48) through A (mm. 49–50) to G (m. 51); and the tenor voice articulates the neighbor motion G (mm. 47–48) F (mm. 49–50) G (m. 51). The imitation of the melody in the bass clef (m. 48ff.) simply doubles this counterpoint. Second, parallel perfect octaves and fifths can arise from unessential tones. In Der freie Satz, Schenker listed the following possibilities: a principal note with an accented or unaccented passing tone or with a neighboring note; a passing tone with an anticipation, with an accented passing tone, or with a neighboring note; a neighboring note with another neighboring note, with the concluding turn of a trill, or with a suspension; the resolution of a suspension with a passing tone, with another suspension, and so forth.29

Schenker and the Quest for Accuracy

49

Figure 1.15. Parallels by combinations of harmonic and non-harmonic tones. a. Passing tone with anticipation. Bach, St. Matthew Passion, Part 2, No. 16, m. 8. From Schenker ed., Brahms Octaven und Quinten, Ex. 29.

b. Accented and unaccented passing tones. Cherubini, Missa Solemnis in D Minor, Kyrie 2, mm. 70–73. From Schenker ed., Brahms Octaven und Quinten, Ex. 55.

c. Simultaneous neighbor tones. Mozart, Cosi fan tutti, Act 2, No. 19, m. 22. From Schenker ed., Brahms Octaven und Quinten, Ex. 53.

Figure 1.15 (Parallels by combinations of harmonic and nonharmonic tones) illustrates some of these configurations.30 In figure 1.15a, for example, the parallel perfect fifths B/E–A/D arise from a passing motion E–D–C in the tenor voice combined with an anticipation A in the soprano. Similarly, the parallel perfect fifths C/F–B/E in figure 1.15b occur because the soprano passes from C

50

Explaining Tonality

Figure 1.16. Laws of relative motion and closure. If a counterpoint is perfectly closed, then it begins on
8 or
5 and ends
7–
1.

GM

If a texture is perfectly closed, then the melody begins on
8,
5, or
3 and ends
2–1
, the alto ends 7
–1
, the tenor ends
5–
4–
3, and the bass leaps
5–
1.

If a counterpoint moves from one note to another, then it mainly moves in contrary motion with the cantus firmus.

LM

If the contrapuntal lines move from one note to another, then they mainly move in contrary motion or in parallel thirds or sixths.

If a counterpoint and the cantus firmus move in the same direction, then parallel perfect octaves and fifths do not occur between successive notes.

LS

If two essential lines move in the same direction, then parallel perfect octaves and fifths do not occur between successive harmonic tones.

LS

If parallel perfect octaves and fifths occur, then they arise from doubling/figuration or from combinations of harmonic and non-harmonic tones.

(G⫽global, L⫽local, M⫽main, S⫽subordinate)

to A, whereas the alto passes from G to C. Finally, figure 1.15c shows how parallel perfect fifths arise when an accented neighbor tone C sounds against the chord tone F. Taking these various cases into account, we can revise our subordinate laws (see figure 1.16, Laws of relative motion and closure). These new laws state: “[I]f two essential lines move in the same direction, then parallel perfect octaves and fifths do not occur between successive harmonic tones”; and “[I]f parallel perfect octaves and fifths occur, then they arise from doubling/figuration or from complex combinations of harmonic and non-harmonic tones.” In other words, permissible parallels arise from complexities in the voice leading: “It is as if two people who have no contact with one another simply pass in the street without an exchange of greetings.”31 With regard to the behavior of unstable/stable tones, Fux’s laws again differ from Schenker’s (see figure 1.17,). Like Fux before him, Schenker certainly believed that Laws of vertical alignment consonances are more fundamental than dissonances. As he put it,

Schenker and the Quest for Accuracy

51

Figure 1.17. Laws of vertical alignment. If a counterpoint is added above or below a cantus firmus, then it always begins/ends on a perfect consonance.

GM

If contrapuntal lines are added to a melody, then they normally begin and end on members of the tonic triad.

If the counterpoint moves from one note to another, then each note is normally consonant with the cantus firmus.

LM

If the contrapuntal lines move from one note to another, then each verticality is basically triadic.

If dissonances occur, then they move by step to and/or from consonances.

LS

If non-harmonic tones occur, then they move by step between harmonic tones or by leap between contrapuntal lines.

(G⫽global, L⫽local, M⫽main, S⫽subordinate)

“[C]onsonance manifests an absolute character, dissonance, on the contrary, a merely relative and derivative one: in the beginning is consonance! The consonance is primary, the dissonance is secondary!”32 Nevertheless, one of the most important consequences of ‘The Stufe Constraint’ is that it erodes the distinction between consonance and dissonance. Indeed, figure 1.18 (Consonant non-harmonic tones and dissonant harmonic tones) shows that functional tonality not only contains dissonant harmonic tones, but also consonant nonharmonic tones. Figures 1.18b and 1.18c give two examples of the former; according to Schenker, the VII Stufe in figure 1.18b and the II Stufe in figure 1.18c may be diminished, but are treated as if they are consonant.33 Meanwhile, figure 1.18a gives a good example of a consonant non-harmonic tone: the pitch A in m. 1 serves as a passing tone, even though it is consonant with the bass. Schenker, however, tried to go one stage further by suggesting that all non-harmonic tones stem from passing motion. In Kontrapunkt I, for example, he noted that “the apparently free dissonance must be understood as the clearly established internal element of a latent passing motion.”34 This is shown in figure 1.19 (Neighbor tones and suspensions as passing motions). As shown in figure 1.19a, he suggested that the neighbor motion C–D–C is a transformation of the passing motion C–D–E, while in figure 1.19b he proposed that

52

Explaining Tonality

Figure 1.18. Consonant non-harmonic tones and dissonant harmonic tones.

Figure 1.19. Neighbor tones and suspensions as passing motions. a. Schenker, Kontrapunkt II, Ex. 123.

b. Schenker, Kontrapunkt I, Ex. 400.

the suspension C–B is a transformation of the passing motion D–C–B.35 Later, in volume 2 of Der Tonwille (1921), he reiterated his claim that non-harmonic tones arise from passing motion: Consonance lives in the triad, dissonance in [the] passing [tone]. From the triad and from passing [tone] stem all the phenomena of tonal life: the triad can become a Stufe, the passing tone can be modified to become a neighbor

Schenker and the Quest for Accuracy

53

note or accented passing tone, anticipation, a dissonant syncopation, and the seventh of a chord.36

But by the time Schenker completed Der freie Satz, he had softened his position; in par. 66 he claimed that “the dissonance appears only as a passing tone or as a syncopation,” and in par. 106–12 he included neighbor motion as an independent transformation.37 Now he simply claimed that “the traversal of the Urlinie is the most basic of all passing motions.”38 Given that non-harmonic tones arise from step motion between harmonic tones, how do we explain cambiatas, appoggiaturas, changing notes, and other leaping dissonances? Once again, Schenkerian theory offers two types of explanation. The first relies on implied tones. In figure 1.20a (Chopin’s Mazurka, Op. 30, no. 4, mm. 129–30), Schenker explained in his graph the string of parallel seventh chords by invoking implied suspensions.39 The second explanation treats leaping dissonances as byproducts of motion between polyphonic voices. We can see how this might work in figure 1.20b. Here the nota cambiata is explained in terms of an implied motion to an inner voice: the alto voice passes B–C–D, while the soprano temporarily moves down through D to hit the alto C. Schenkerian theory uses much the same strategy to explain the behavior of consecutive non-harmonic tones. Although such things rarely occur in strict counterpoint, they are a dime a dozen in tonal composition. As John Rothgeb has pointed out, “The linear progression is but an extension of the basic passing-tone concept of second species counterpoint in that it allows for passing motions within larger intervals than a third.”40 Sometimes, however, consecutive non-harmonic tones arise from motion between different polyphonic voices. For example, a double neighbor tone C–B–D–C might be derived from flipping between the soprano and alto voices: in this case, the B might belong to the alto line, whereas the D might belong to the soprano. Rothgeb cites a more extreme example from Schenker’s unpublished Generalbasslehre that seems to contain adjacent seventh chords (see figure 1.21, Consecutive seventh chords). According to Schenker, this passage, “basically reduces to an 8–7 motion [above a stationary bass]; but the passing tone [A] in the bass disguises this

54

Explaining Tonality

Figure 1.20. Implied tones and the nota cambiata. a. Implied tones. Chopin, Mazurka, Op. 30, no. 4, m. 129ff.

From Schenker, Der freie Satz, Fig. 54.6.

b. Nota cambiata. From Schenker, Kontrapunkt I, Ex. 347.

fact, and thereby causes insurmountable difficulty for the theory teacher and forces him to speak of a succession of seventh chords.”41 In other words, the consecutive non-harmonic tones arise from two separate, but simultaneous, levels of contrapuntal motion: the bass A is simply a passing tone and does not support an independent Stufe, whereas the soprano F is a passing tone between G and E in the underlying progression V–I in C major. This phenomenon is common in mixed species and will be discussed in chapter 3. Finally, whereas suspensions are the only possible accented non-harmonic tones in strict counterpoint, many other types of accented non-harmonic tones occur in tonal contexts through the concept of displaced intervals. Figure 1.22 (Displacement and

Schenker and the Quest for Accuracy

55

Figure 1.21. Consecutive seventh chords. From Schenker, Generalbasslehre, p. 47. From John Rothgeb, “Schenkerian Theory: Its Implications for the Undergraduate Curriculum,” Music Theory Spectrum 3 (1981): 146.

Figure 1.22. Displacement and accented dissonances.

accented dissonances) shows how displacements can used to generate accented passing tones (figure 1.22a), accented neighbor tones (figure 1.22b), and appoggiaturas (figure 1.22c). Displacement can even account for more radical deviations from strict counterpoint, such as the ones found in figure 1.23 (Beethoven’s Piano Sonata, Op. 81a).42 By displacing the right and left hands, Beethoven superimposed the tonic and dominant chords, thereby creating an effect of extraordinary beauty. So far, we have considered Schenker’s explanation of the relationship between strict counterpoint and functional tonality. Schenker saw many connections between the two; in both cases, melodic lines mostly move by step, converge on the tonic at final cadences, and move between stable and unstable verticalities. But he also saw subtle differences; these arise because strict counterpoint is bound either by ‘The Consonance Constraint’ or ‘The Triadic Constraint,’ whereas tonal voice leading is controlled by ‘The Stufe Constraint.’ We have seen that, through ‘The Heinrich Maneuver,’ Schenker not only

56

Explaining Tonality

Figure 1.23. Beethoven, Piano Sonata, Op. 81a, 1st movement, mm. 230–42. From Schenker, Harmonielehre, Ex. 132.

reinterpreted the basic principles of melodic motion, relative motion, and the behavior of unstable/stable tones so that they applied to functional tonality, but he also supported these basic principles with an alternative network of subordinate laws. Schenker was well aware of what he had done: in Kontrapunkt I, for example, he promised to “justify each prescription and restriction and elaborate how the application of [these rules] more or less changes in the context of [tonal] composition.”43 He believed that “by this procedure,” he could “contribute best toward eliminating that unfortunate confusion of counterpoint and composition as well as its sad consequences.”44

“The Complementarity Principle” From the preceding discussion, it should be clear that ‘The Heinrich Maneuver’ ties the principles of tonal voice leading to the behavior of tonal harmonies. This observation reflects Schenker’s general belief that it is impossible to understand functional tonality adequately from a purely contrapuntal or a purely harmonic perspective. Indeed, as he put it in Harmonielehre: “[O]ne note, or even more, may be heard merely horizontally, while the vertical is to be totally disregarded; for other notes, on the contrary, the vertical concept is far more important.”45 For convenience, we will refer to this interrelation of line and chord as ‘The Complementarity Principle.’

Schenker and the Quest for Accuracy

57

This principle can, in fact, be stated even more strongly: although every tonal event can be interpreted contrapuntally or harmonically, any account that is purely contrapuntal or purely harmonic will necessarily be incomplete.46 ‘The Stufe Constraint’ does not simply list what essential harmonies are possible, it also explains how they conform to “their own secret law of progression.”47 To understand what ‘The Stufe Constraint’ involves, let us briefly compare traditional explanations of functional harmony with those offered by Schenker. In general terms, tonal theorists usually address three main issues. First, they are concerned with showing how stable harmonies are distinguished from unstable harmonies. Most theorists believe that functional tonal music is basically built from triads and seventh chords, which they classify in terms of their quality (major, minor, diminished, and augmented) and their inversion (root, first, second, and third). We can use the term ‘inversional equivalence’ to refer to the notion that the identity of a triad is independent of its vertical ordering. Most theorists also classify triads into seven types identified by seven Roman numerals. Second, tonal theorists are interested in determining how successive harmonies are arranged to create typical functional progressions. To do this, many theorists invoke the idea of ‘functional equivalence’: they propose that the seven essential harmonies fulfill three basic functions—tonic (I, VI, and III), dominant (V and VII), and subdominant (IV, II, and VI). They then suggest that prototypical tonal progressions follow the scheme tonic (T)–subdominant (S)–dominant (D)–tonic (T). Third, tonal theorists are concerned with understanding how chromatic harmonies arise in functional tonal contexts. Since most theorists assume that functional tonal music is fundamentally diatonic, they usually regard chromaticisms as surface deviations from this basic system. Schenker’s outlook on these three topics was, however, anything but conventional; by including voice-leading elements in his theory of harmony, he was able to simplify the principles of harmonic classification and harmonic progression considerably, thereby increasing their flexibility and accuracy. In the first case, he used the laws of tonal voice leading to limit the number of essential harmonies to major, minor, and diminished triads and he was very suspicious of inversional equivalence.48 To be specific, since the

58

Explaining Tonality

interval of the perfect fourth is dissonant when it occurs above the bass, Schenker seldom treated 6/4 sonorities as functional harmonies, and since augmented triads do not appear within the major or minor systems, he did not count them as essential either. Finally, and perhaps most importantly, Schenker rejected the idea that seventh chords normally behave as functional harmonies. To quote from the start of Kontrapunkt I: The Stufe exists in our perception only as [a] triad; that is, as soon as we expect a Stufe, we expect it first of all only as [a] triad, not as a seventh chord. In this sense, the seventh is absolutely not an a priori element of our perception comparable to the fifth and the third; it is rather an event a posteriori, which we understand best of all with reference to the function associated with it; that is, we understand it in retrospect as a passing tone, or as a means of chromaticization, or the like.49

Obviously, if seventh chords are ultimately created by contrapuntal motion, then so must more abstruse sonorities, such as ninths, thirteenths, and augmented-sixths. These are shown in figure 1.24 (Laws of harmonic classification). Schenker also used the laws of tonal voice leading to shed light on the behavior of harmonic progressions (or Stufengang). He was quite clear that cadential closure was not simply a contrapuntal phenomenon, it also depends on the distinctive motion from dominant to tonic: In order to gain insight into cadences in [tonal] composition it is important to recognize that there the closure is no longer based on the horizontal line alone but rather (and to a larger degree) on the harmony of the vertical [dimension], or, more precisely, on the succession from the V Stufe to I.50

But while he certainly accepted the functional priority of the tonic and dominant Stufen, Schenker rejected the notion of functional equivalence: he denied that the seven Stufen necessarily fulfill just three basic functions—tonic, subdominant, and dominant. To quote from Kontrapunkt I, “How can one claim to have understood the [tonal] ‘system’ if its individual Stufen, except I, IV, and V, are deprived of their independence and thus of their attractive capability of assuming various functions?”51 Schenker added, “[I]t is the functional versatility of the Stufe that is the basis of [tonal] practice,

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59

Figure 1.24. Laws of harmonic classification. a. Traditional Laws of Harmony

b. Revised Laws of Tonal Harmony

If a melody is harmonized, then it is mainly supported by major, minor, diminished, or augmented triads, and seventh chords on seven degrees.

LM

If a melody is harmonized, then it is mainly supported by major, minor, diminished triads on seven degrees.

If these triads appear in succession, then these seven degrees serve one of three functions—tonic (T), subdominant (S), or dominant (D) (functional equivalence).

LM

If a triad appears, then it always has the root and the third, with any member in the bass (inversional equivalence).

LS

If a triad appears, then it has the root and the third, with only these members in the bass.

If the triad doubles notes, then it normally doubles the root, then the fifth, then the third, but not
7.

S

If the triad doubles notes, then it normally doubles the root, then the fifth, then the third, but not 7
.

If non-harmonic tones appear, then they arise from seventh chords or motion between triads.

LS

If non-harmonic tones appear, then they arise from motion between harmonic tones or contrapuntal voices.

(G⫽global, L⫽local, M⫽main, S⫽subordinate)

and this, of course, at least presupposes its independence!”52 Although there will certainly be times when II chords behave the same way as IV chords, there will be other times when they do not; this means that the notion of functional equivalence cannot be generalized across the entire range of tonal progressions.53 Taking Schenker’s argument further, even if two Stufen do behave in the same way, they need not arise for the same reasons (see figure 1.25, Chord function vs. chord derivation). Figure 1.25a gives a simple progression I–V–I with a descent
3–
2–
1 in the soprano. Next, figure 1.25b shows how we can compose out this progression by a leaping passing tone in the bass, thereby generating an incomplete upper neighbor motion in the soprano. We can assume that the II6 in figure 1.25c is generated in much the same

60 Figure 1.25. Chord function vs. chord derivation.

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61

Figure 1.26. Laws of harmonic progression. a. Traditional Laws of Harmony If triads appear in succession, then they are normally arranged as T-S-D-T.

b. Revised Laws of Tonal Harmony GM

If a tonal progression is maximally closed, then it ends by moving from V to I.

LS

If another essential harmony occurs, then it does so from motion between I and V.

way. But what about the II Stufe in figure 1.25d? This one seems to be derived from the upcoming dominant rather than from the preceding tonic.54 The same can also be said of the D-major Stufe given in figure 1.25e. In other words, figure 1.25 indicates that different predominant chords may serve the same function, even though they may be generated in quite different ways. Schenkerian derivations are simply more accurate than functional explanations.55 Assuming that Schenker’s seven Stufen cannot be reduced to three functional categories, how can we explain the behavior of harmonic progressions? The answer is, in fact, surprisingly easy; according to Schenker, they arise from the process of composing out: As a consequence of voice-leading constraint[s], all those individual harmonies that arise from the progression of the various voices are forced to move forward. All the transient harmonies which appear in the course of a work have their source in the necessities of voice-leading [par. 178, 180].56

Since “Stufen are inextricably bound up with counterpoint,” we can reformulate some new laws, as shown in figure 1.26 (Laws of harmonic progression).57 Lastly, Schenker recognized that tonal voice leading has an important influence on the behavior of tonal chromaticisms. Obviously, one of the biggest differences between strict counterpoint and functional tonal composition lies in the area of chromaticism. As Schenker himself pointed out, whereas strict counterpoint is primarily diatonic and avoids direct chromatic successions, functional tonal composition uses mixture and tonicization to create the entire spectrum of chromaticisms.58 In some cases, these chromatic

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Explaining Tonality

Figure 1.27. Laws of chromatic generation. a. Traditional Laws of Harmony

b. Revised Laws of Tonal Harmony

If a melody is harmonized by triads, then these triads are mainly diatonic.

LM

If a melody is harmonized by triads, then these triads are mainly diatonic.

If chromaticisms occur, then they substitute for or elaborate diatonic triads.

LS

If chromaticisms occur, then they arise from mixture or tonicization.

LS

If harmonies appear on IV/V, then they are always indirectly related to I.

successions will be direct: “In contrast to strict counterpoint (see Kpt. I, II/2, par. 28, and Kpt. II, III/1, par. 25), [tonal] composition permits a succession of chromatic tones.”59 But, since tonal composers still try to avoid juxtapositions of this latter sort, Schenker believed that “the prohibition is in a certain sense reestablished.”60 In particular, he suggested that they can be avoided by techniques such as motion from an inner voice, neighbor motions, and linear progressions.61 This point is summarized in figure 1.27 (Laws of chromatic generation). To illustrate what Schenker had in mind, we need only consider his discussion of the Phrygian II. As we all know, the supertonic Stufen are often used to reinforce the dominant Stufe in perfect authentic cadences (see figure 1.28a, Rectification of Phrygian II). In such contexts, the soprano voice will normally descend from
2 to
1 as the bass moves from V to I. But when the Phrygian II is added before the dominant, the voice leading becomes more complicated (see figure 1.28b). Since the Phrygian II typically has 
2 in the soprano and since changing this scale degree to
2 for the dominant Stufe would create the direct chromatic succession 
2–2, Schenker insisted that the melody should descend from 
2 to
7 in the alto voice. He referred to the modification of 
2 as rectification (or Die Richtigstellung).62 Occasionally, Schenker even used this same principle in reverse. For example, to explain the special ‘Phrygian’ effects at the end of Chopin’s Mazurka, Op. 41, no. 2, in E Minor, Schenker

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63

Figure 1.28. Rectification of Phrygian II. Adapted from Schenker, Five graphic Analyses, No. 5, Chopin, Étude in C Minor, Op. 10, no. 12.

suggested that
2 or F is rectified in reverse to create 
2 or F, though once again the music avoids a direct chromatic succession.63 In a paper cowritten with Douglas Dempster and Dave Headlam, I have described another more remarkable situation in which contrapuntal devices are used to eliminate one specific type of direct chromatic succession.64 We began by showing that, according to Schenkerian theory, direct connections between I and IV/V cannot occur in tonal contexts. We then considered examples in which Stufen on IV/V arise from contrapuntal motion. Among the most remarkable examples of this appears in the Scherzo to Beethoven’s String Quartet, Op. 59, no. 1. Although this movement is in B major, the coda contains a brief excursion to E minor. We argue that this outburst actually arises from a contrapuntal expansion of the dominant F, which is reached in m. 445. Instead of resolving onto a tonic B, it shifts to a diminished seventh on F (m. 446). This latter sonority tonicizes E (m. 450) before returning to F (m. 458) to set up the final arrival on B (m. 460). The F and E therefore serve as double neighbors to the dominant F, and, as if to draw attention to the significance of this pattern F–F–(G)–E–F, Beethoven presents it locally within the final cadence (mm. 469–76). In other words, the

64

Explaining Tonality

same procedures govern the behavior of surface melodies and larger harmonic progressions. All in all, ‘The Heinrich Maneuver’ and ‘The Complementarity Principle’ demonstrate the intimate connections between voice leading and harmony in functional tonality. These two basic principles allowed Schenker to show not only how the traditional laws of strict counterpoint are transformed in functional contexts by the influence of Stufen, but also how the traditional laws of functional harmony can be modified by grounding them in the laws of tonal voice leading. By classifying these laws along the lines suggested above, he had good reason for supposing that these new laws are both necessary and sufficient for explaining functional tonality. Schenker was able to make these connections between line and chord because he recognized that Stufen play a crucial role in both domains: in his words, they are “the essential generator of all [musical] content.”65 He was prompted to take these steps precisely because traditional theories of counterpoint and harmony proved to be inaccurate. Such observations are significant for several reasons. From a pedagogical perspective, we have good reason to update current textbooks on tonal theory. On the one hand, we can provide students with a more persuasive account of why Fuxian species counterpoint helps us understand functional tonality. We can tell them that Fux teaches us about the behavior of contrapuntal lines as they exist in the simplified world of intervals, whereas functional tonality exists in the messy world of functional triads, or Stufen. On the other hand, we can offer students an explicit list of laws that cover the behavior of functional voice leading. These new laws allow us to abandon the rather dubious notion that in free composition, great composers sometimes break the rules of strict counterpoint simply because they are great composers. From a methodological perspective, our observations also underscore the importance of accuracy to the development of music theories. Schenker was obviously very concerned about whether the laws of strict counterpoint were adequate for explaining the behavior of tonal voice leading. Instead of dismissing abnormalities by appealing to the liberties of genius or to the extra-musical allusions of specific

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65

pieces, he tried to build a theory that provided a more accurate fit with the music he was studying. In this sense Schenker’s work falls within a quite normal pattern of theoretical inquiry. Indeed, as Quine puts it: The tension between law and anomaly is vital to the progress of science. The scientist goes out of his way to induce it. Sir Karl Popper well depicts him as inventing hypotheses and then making every effort to falsify them by cunningly devised experiments.66

Quine adds that “it is the tension between the scientist’s laws and his own breaches of them that powers the engines of science and makes it forge ahead.”67 Given that Schenker left us with an empirically testable theory of functional monotonality, our next job is to find the anomalies that it surely contains; if we are able to fix them up, then we can keep the engines of music theory firing on all cylinders.

2

Semper idem sed non eodem modo In the previous chapter, we saw how Schenker’s concern for accuracy motivated him to refine the traditional laws of strict counterpoint and functional harmony; he did so by constraining the laws of counterpoint harmonically and by grounding the laws of harmony contrapuntally. If Schenker’s only contribution to music theory had been to devise more accurate laws of tonal voice leading and harmony, then his place in music history would have been assured. After all, these new laws overcome technical problems that had perplexed theorists for several centuries. But Schenker took another crucial step: he reformulated his new laws in a procedural form as a system of prototypes (Ursätze), transformations (Verwandlungen), and levels (Verwandlungs-Schichten, Stimmführungs-Schichten, or Schichten). This system allowed him to reach two important conclusions: 1) all functional monotonal pieces can be derived from a single prototype; and 2) there are only three possible prototypes for all functional monotonal compositions. There are several reasons why these conclusions are so important. On the one hand, they allowed Schenker to achieve the sort of theoretic unification described in the Introduction. Indeed, whereas music theorists had traditionally treated counterpoint and harmony as largely separate phenomena, Schenkerian theory insists that they are irrevocably intertwined. This synthesis is undoubtedly a major step forward in our understanding of tonal relationships. On the other hand, these results allowed Schenker to widen the explanatory scope of tonal theory; instead of simply explaining tonal motion across an individual phrase, he could now explain tonal motion across an entire monotonal composition. Schenker achieved this goal by showing that the same laws of functional voice leading and harmony operate both in the small and in

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67

the large. This radical insight is encapsulated in his famous motto semper idem sed non eodem modo or “always the same but not in the same way.”1 Remarkable though it may be, Schenker’s system of prototypes, transformations, and levels does raise a number of interesting questions. How, in fact, are the individual components of the system related to the general laws of functional voice leading and harmony outlined in chapter 1? Why did Schenker insist that his prototypes, transformations, and levels have some forms but not others? Why was he so confident that this system is capable of generating all and only all complete, continuous monotonal pieces? Does his motto semper idem sed non eodem modo really stand up to critical scrutiny? This chapter will answer these questions. Part 1 starts by looking at the conceptual origins of Schenker’s work. Since Schenker’s system of prototypes, transformations, and levels has some obvious connections with Fux’s concept of species counterpoint, we will compare the one with the other. After part 1 has shown certain important differences between the goals of Schenker and those of Fux, part 2 examines Schenkerian prototypes in more detail. Among other things, it suggests that they summarize in an optimally compact way the main laws of tonal motion mentioned in chapter 1. Next, part 3 describes the various ways in which these prototypes can be transformed. In particular, it explains why these transformations are finite in number. Part 4 then looks at the various ways in which Schenker ordered his transformations into discrete levels. The main focus will be on showing how, in principle at least, the process of generation preserves the local laws of tonality outlined in the previous chapter. Finally, Part 5 considers some implications of the preceding discussion. In particular, it looks at Schenker’s preference for analyses in which the same patterns of transformation occur within and between levels.

Conceptual Origins As mentioned above, Schenker’s thinking about functional tonality is dominated by the basic idea that complex tonal progressions can be explained as transformations of simple tonal prototypes. For him,

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Explaining Tonality

the processes of transformation have several implications. First, they imply that whenever a prototype is transformed, it will be elaborated with new harmonic and melodic material. Schenker drew attention to this idea by invoking a number of colorful terms, such as Auskomponierung (composing out), Mehrung (increase of content), and Auswicklung (unwinding).2 Second, these processes imply that when a prototype is elaborated, the new material may behave somewhat differently from the original prototype. For example, although Schenker’s prototypes are fundamentally consonant and diatonic, they can be transformed to create progressions that are highly dissonant and chromatic. Such changes are what Schenker had in mind when he used terms like Umwandlung (reshaping) and Umbildung (recasting).3 Schenker was not, of course, the first person to think about music in terms of prototypes, transformations, and levels; on the contrary, the same ideas lie at the heart of many theoretical projects, including Fuxian species counterpoint.4 Fux’s interest in the process of elaboration is obvious enough; in codifying the Five Species he set out to show how cantus firmi can be elaborated with simple or florid counterpoints. But as we saw in chapter 1, Fux also realized that the behavior of the contrapuntal lines alters from one species to the next; whereas counterpoints are always strictly consonant with the cantus firmus in First Species, they can be dissonant in Fifth Species. Furthermore, as counterpoints become more elaborate, so they can include progressively more repeated tones and even chromaticisms. But while Fuxian species counterpoint has some obvious connections with Schenker’s system of prototypes, transformations, and levels, it nonetheless differs in crucial respects. For one thing, the theoretical status of a cantus firmus is very different from that of a Schenkerian prototype. Fux’s cantus firmi are essentially examples of effective modal melodies. Each one begins and ends on the final of a given mode and each one moves onto the reciting tone at points of structural significance. Meanwhile, Schenker’s prototypes are far more general in scope. Instead of representing a specific melody per se, they encapsulate certain underlying principles of melodic construction. These principles guide both the local behavior of individual melodies and the global behavior of entire pieces. Schenker

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69

Figure 2.1. Schenker’s concept of prototypes, transformation, and levels. From Schenker, Der freie Satz, Fig. 1.

conveyed such levels of abstraction very beautifully at the start of Der freie Satz, in a chart given here as figure 2.1 (Schenker’s concept of prototypes, transformation, and levels). This chart shows that the tonality of a given foreground (Vordergrund) can be generated from the diatony of the given background (Hintergrund) through various levels of the middleground (Mittelgrund). Each prototype (Ursatz) contains an upper line (Urlinie) in the upper register and a bass arpeggiation (or Bassbrechung) that articulates the upper fifth, i.e., it moves from I to V and back to I. To reinforce the global nature of his prototypes, Schenker even included a paragraph in Der freie Satz that differentiates prototypes from cadences.5 For convenience, we will refer to the notion that functional monotonal pieces derive from a single prototype as ‘The Global Paradigm.’ As it happens, ‘The Global Paradigm’ did not come to Schenker overnight; it actually developed in his mind over a period of at least twenty years. In fact, he started to toy with the concept of a prototype as early as the Harmonielehre and Kontrapunkt I and even criticized C. P. E. Bach for underestimating their significance. But at this point in time his prototypes were purely local phenomena and certainly did not control an entire piece.6 Over the next decade, however, they became wider in scope.7 In his edition of Beethoven’s Piano Sonata, Op. 110 (1914), for example, Schenker explained how the development section of the second movement

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Explaining Tonality

could be derived from a single 32-bar progression. Six years later, in his comments to Beethoven’s Piano Sonata, Op. 101 (1920), he even coined the term Urlinie. Nevertheless, it was not until the early issues of Der Tonwille (1921) that he proposed a single Urlinie for an entire piece and the fifth issue (1923) that he derived a whole work from a single Ursatz.8 Another important difference between Fux’s conception of species counterpoint and Schenker’s system of prototypes, transformations, and levels is that the latter is recursive and rule preserving, whereas the former is not. Recursion is a term borrowed from mathematics. In very general terms, a musical system is recursive if it posits certain starting states, such as a prototypical harmonic progression, and derives more complex states, or progressions, by repeatedly applying a given set of transformations. This system is also rule preserving if every derived state or progression conforms to the same underlying laws of voice leading and harmony as the prototype. If the musical system is indeed recursive and rule preserving, then the processes of generation and reduction will be the reverse of each other. We will refer to these ideas as ‘The Recursive Model.’ There are good reasons to suppose that Fuxian species counterpoint is neither recursive nor rule preserving. To begin with, Fux never claimed that Fifth Species counterpoints can be generated by repeatedly elaborating First Species prototypes, nor did he suggest that Second, Third, and Fourth Species are intermediate stages of generation. Furthermore, if Fuxian species counterpoint were indeed recursive and rule preserving, then we would be able to reduce all of the species to First Species prototypes. But this is not necessarily the case. Figure 2.2 (The non-recursive nature of Fuxian species counterpoint) gives settings of a single cantus firmus in First, Second, Third, and Fourth Species. Although these settings all satisfy the laws of strict counterpoint, they cannot be reduced to a plausible First Species prototype, like the one in figure 2.2a. Take, for example, the counterpoint given in figure 2.2b. When we try to reduce the counterpoint in mm. 3–5 from two notes per measure to one, we soon run into difficulties. If we regard the G in m. 4 as ornamental, then parallel octaves occur between the downbeats of mm. 4 and 5. Conversely, if we regard the C in m. 4 as ornamental, then we create parallel fifths with the main note A in m. 3. Many

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71

Figure 2.2. The non-recursive nature of Fuxian species counterpoint. From Fux, The Study of Counterpoint, Figs. 11, 36, 57, 76.

of the same issues arise in mm. 4–5 of figure 2.2c. But the problems are even more acute in figure 2.2d. Here, the string of suspensions is created by displacing the counterpoint over the cantus firmus. If we try to normalize this displacement, then the resulting First Species prototype consistently violates the law prohibiting parallel perfect fifths (c.f., mm. 3–7). Whatever their similarities, it seems that Fux’s concept of species counterpoint and Schenker’s system of prototypes, transformations, and levels have quite different goals. Whereas Fux used his cantus firmi to illustrate a well-composed melody in each mode, Schenker used his prototypes to explain the general principles of

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Explaining Tonality

melodic construction as they operate locally within a phrase and globally across an entire piece. And, whereas Fux’s species teach students certain general techniques of elaboration, Schenker’s system of prototypes, transformations, and levels explains how the extraordinary diversity of functional tonality stems from the working out of certain underlying principles of counterpoint and harmony. To show how this system is, in principle at least, both recursive and rule preserving, let us now consider its main components in detail, starting with Schenker’s prototypes.

Prototypes For anyone concerned with the explanatory scope of Schenkerian theory, it is important to reconsider the various covering laws mentioned in chapter 1. In very general terms, these laws cover six areas: 1) how individual lines move and reach closure; 2) how polyphonic lines move in relation to one another; 3) how unstable tones behave in relation to stable tones; 4) how stable harmonies are distinguished from unstable harmonies; 5) how successive harmonies are arranged to create typical functional progressions; and 6) how chromatic harmonies arise in functional tonal contexts. As such, these laws seem to be necessary and sufficient for explaining tonal relations. Within each domain, we classified these laws in several ways. On the one hand, we distinguished main laws from subordinate laws: the former explain how melodies normatively behave, whereas the latter explain significant exceptions to that norm. On the other hand, we distinguished local laws from global laws; the former explain how one note moves to the next, whereas the latter explain how the melody moves as a whole. The great advantage of classifying the covering laws in this way is that it allows us to structure our knowledge about tonal music; this structure becomes very important when we reformulate our laws as prototypes, transformations, and levels. As we saw in chapter 1, Schenker believed that, locally, melodies mainly move by step and, globally, they are maximally closed if they begin on
8, 5
, or
3 and end
2–
1. Similarly, he acknowledged that contrapuntal lines tend to move in contrary motion or in parallel thirds, sixths, and tenths and

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73

never produce parallel perfect octaves or fifths between successive Stufen. Schenker also accepted that the only stable harmonies are functional triads in root or first inversion. According to him, chord successions are controlled contrapuntally, with closure most strongly articulated by the harmonic progression V–I. And, he recognized that functional progressions are primarily diatonic in nature. If we now look at Schenker’s prototypes, we soon see that they summarize these main laws in an optimally compact way. For convenience, the three parts of figure 2.3 (Schenkerian Ursätze in C Major) show the three basic prototypes for C major. Each one consists of a stepwise descent from a headtone (Kopfton), either
3, 5
, or
8, to
1. This simple passing motion is supported by a bass arpeggiation I–V–I. Whereas Schenker notated the upper line and bass arpeggiation in whole notes, he included a prototypical inner voice in black note heads. Elsewhere, he referred to the precise location of the prototype as the obligatory register (obligate Lage).9

Figure 2.3. Schenkerian Ursätze in C Major. Adapted from Schenker, Der freie Satz, Figs. 9, 10, 11.

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Explaining Tonality

With respect to the laws of melodic motion and closure, it is clear that the upper line follows the local law of moving by step and the global law of beginning on
8,
5, or
3 and ending
2–
1. The upper line and bass arpeggiation likewise obey the main laws of relative motion: the three essential lines close
2–
1,
7–
1, and
5–
1, whereas the outer voices essentially move in contrary or oblique motion with the upper line descending from the headtone to
1 and the bass arpeggiation ascending from I to V. Similarly, each prototype follows the main laws of vertical alignment by beginning and ending on members of the tonic Stufe. Schenker’s prototypes also conform to the main laws of functional harmony: each one contains three Stufen arranged to form the quintessential functional progression I–V–I. This progression is not only diatonic, but it also defines the tonic C in the most unambiguous manner possible. Having shown how Schenker’s prototypes summarize the main laws of functional tonality in an optimally compact way, we can now respond to several alternative accounts of their structure. For starters, although we have seen certain connections between Schenkerian theory and Fuxian species counterpoint, one should not read too much into Carl Schachter’s claim that Schenker’s prototypes are basically Second-Species constructs.10 While Schachter is certainly right to suggest that their upper lines derive some of their logic from the principle of passing motion, it is important to remember that these lines do not belong to the purely intervallic world of strict counterpoint; on the contrary, they clearly belong to the world of Stufen, something that is underscored by the Roman numerals marked under the bass arpeggiation in figures 2.3a–c.11 We can use similar arguments to rebuff Peter Westergaard’s critique of
5 and
8 lines.12 Westergaard has suggested that, since these upper lines contain unsupported stretches (Leerlaufen), they are conceptually inferior to
3 lines. This notion seems to be confirmed by the fact that the Fünf Urlinie-Tafeln and Der freie Satz contain a deluge of
3-line graphs and a dearth of
8-line readings. The chief problem with Westergaard’s position is that Schenker himself did not insist the upper line should be completely supported; on the contrary, in par. 69 of Der freie Satz, he specifically acknowledged that they can contain passing tones.13 Since
5 and
8 lines do indeed conform to the general laws of tonal voice leading

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75

outlined in chapter 1, there are no grounds for discriminating against them. This does not mean that 5
and
8 lines will necessarily be as common as
3 lines; on the contrary, there are good reasons why they may be rare. For example,
8-lines present the composer with a fairly narrow set of options; it is perhaps small wonder that they tend to be found in specific types of music, such as Baroque preludes.14 By the same token, we can respond to David Neumeyer’s revised list of prototypes. In a series of thought-provoking papers, Neumeyer has proposed that: 1) there are other forms of upper line, including some that rise, for example,
5–
6–
7–
8; 2)
8 lines properly belong to the middleground not the background; and 3) some pieces are controlled by three-voice prototypes consisting of “a structural soprano and a structural alto—above a bass.”15 These alternative prototypes stem in part from the work of Schenker’s pupil, Felix-Eberhard von Cube.16 The chief drawback with Neumeyer’s additions is that they no longer conform to the laws of tonal motion mentioned above. Most obviously, his rising lines contradict the law that melodies reach maximum closure when they descend
3–
2–
1. Since 8
lines satisfy this and our other laws, it is hard to see how they can be rejected as Neumeyer suggests.17 Having said that, Neumeyer’s case for three-part prototypes is utterly convincing. For one thing, it is certainly borne out by the examples shown in figures 2.3a and b. For another, Joseph Lubben has noted that in Der Tonwille, Schenker often used the term Aussensatz to denote “a structure of two upper voices above the Stufen.”18 Furthermore, since Schenker clearly used so-called polyphonic transformations (for example, unfolding and motion from an inner voice) at the deep middleground, it suggests certain inner voices must be present at some prior level or derivation. As we will see in chapter 3, these inner voices also help us resolve certain inconsistencies in Schenker’s generation of sequences. Finally, we can address Eugene Narmour’s charge that Schenkerian theory commits the fallacy of affirming the consequent.19 Narmour claims that Schenkerian theory is circular because it sets out to show that all tonal compositions can be generated from various prototypes. According to Narmour, however, the analyst must know the nature of these prototypes in advance in

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Explaining Tonality

order to make a reduction. Since Schenkerian analyses seem to bend the notes to fit the theory, they are, in Narmour’s opinion, specious. Given the ways in which Schenkerian analyses are normally presented, it is hard to disagree with Narmour; at times the arguments do indeed seem viciously circular. But the foregoing account minimizes these problems by suggesting that the process of confirming Schenkerian theory is a lot more complex than Narmour suggests. We have seen that the explanatory laws underpinning Schenkerian theory were actually discovered empirically in the Harmonielehre and Kontrapunkt I, long before Schenker formulated his concept of a single tonal prototype. These laws are, in fact, extensions or transformations of well-established laws of counterpoint. By classifying them along the lines suggested in chapter 1, we have good reason to suppose that they are both necessary and sufficient for all functional tonal music. After spending the next decade studying a broad range of functional monotonal compositions, Schenker discovered empirically that he could reformulate this set of explanatory laws in terms of prototypes, transformations, and levels. There is a sense, then, in which the principles governing Schenker’s prototypes can be confirmed independently without the need for graphing; whatever circularity remains stems from the kind of bootstrapping process by which Schenker came up with his results.

Transformations So far, we have seen that Schenker’s prototypes summarize the main laws of tonal voice leading and harmony in an optimally compact way. They are the simplest possible expressions of a given key. But we also know from chapter 1 that these particular laws do not explain every aspect of functional tonality; on the contrary, we also introduced a number of subordinate laws to cover deviations from these norms. Among other things, these exceptions allow us to explain why leaps can occur in melodic lines, why melodic lines can contain a whole host of dissonances and not just the simple passing tone, why functional progressions can include harmonies other than I and V, and why these progressions can contain a variety of

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chromaticisms. To explain these diverse phenomena and satisfy the various subordinate laws, Schenker invoked a set of transformations or prolongations. Although Schenker did not define his arsenal of transformations as precisely as we might wish, we can divide them into four main groups. The first group horizontalizes a given Stufe by presenting the constituent harmonic tones successively rather than simultaneously (see figure 2.4a), the first of Horizontalizing transformations.20 Of these, the simplest is repetition (Wiederholung). In a nutshell, repetition expands a particular Stufe by taking a note in the soprano or bass voice and duplicating it exactly. In figure 2.4b, the tonic chord in C major is composed out by repeating the soprano pitch G. Significantly, repetition can generate new material before or after the original Stufe: this new material is said to be “front-related” if it appears before and “back-related” if it appears after. Whereas repetition duplicates a particular tone in a given register, the other members of this group create leaps. To begin with, register transfer creates octave leaps by projecting a tone from one register to another. Schenker referred to this idea in general by the term Lagenwechsel and introduced specific transformations to denote ascending register transfers (Höherlegung), descending transfers (Tieferlegung), and alternations between a given pair of registers (Koppelung). An ascending register transfer is given in figure 2.4c. Like repetition, register transfer can occur in the soprano or the bass. Meanwhile, arpeggiation (Brechung) creates leaps by taking the soprano or bass note and moving to another member of the same Stufe. For example, in figure 2.4d the soprano voice arpeggiates the tonic Stufe by leaping from the third to the root. Significantly, arpeggiations can appear successively and even in the same direction. Just like repetition, they can be applied before or after the original Stufe. Just as register transfer and arpeggiation create leaps by moving from one harmonic tone to another, so unfolding, voice exchange, and reaching over create leaps by moving from one polyphonic voice to another. The simplest of these is unfolding (Ausfaltung). As shown in figure 2.4e, this transformation creates a single line by moving from the soprano E to the alto C. Following Schenker, it’s marked by a diagonal beam. Voice exchange (Stimmentausch), however, is more complicated; it involves swapping notes between two

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Figure 2.4. Horizontalizing transformations. a. Transformation

Domain

Schenker’s Discussion

Repetition (Wiederholung)

Single line Single harmony

None, but implied, DfS, Fig. 21

Register transfer (Hohelegung, Tieferlegung, Koppelung)

Single line Single harmony

DfS, par. 147–54, 238–41 Fig. 47–49, 106–8

Arpeggiation (Brechung)

Single line Single harmony

DfS, par. 125–28, 230 Fig. 40, 100

Unfolding (Ausfaltung)

Multiple lines Single harmony

DfS, par. 140–44, 234 Fig. 43–45, 103

Voice exchange (Stimmtausch)

Multiple lines Single harmony

DfS, par. 236–37

Reaching over (Ubergreifen)

Multiple lines Multiple harmonies

DfS, par. 129–34, 231–32 Fig. 41, 101

voices (see figure 2.4f). Here the soprano E in the first Stufe becomes the alto E in the second Stufe, while the alto C in the first Stufe becomes the soprano C in the second. The final transformation, reaching over (Übergreifen), seems to combine unfolding and voice exchange; as shown in figure 2.4g, the soprano and alto E/C of the first tonic Stufe are horizontalized, and then the soprano E connects with the alto D of the second Stufe and the alto C connects with the soprano F of the second Stufe.

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Although the first group of transformations can create a wide range of material, they have an enormous limitation; since they cannot create step motions, they cannot generate non-harmonic tones. To overcome this deficiency, Schenker introduced four more transformations (see figure 2.5a, Filling in transformations). The simplest of these is neighbor motion (Nebennote).21 As shown in figure 2.5b, neighbor motion produces a step motion between two repeated tones. In this case, the neighbor tone F elaborates the repeated tone E in the soprano. The next transformation—linear progression (or Zug)—fills in the leap produced by a register transfer or an arpeggiation. Figure 2.5c shows how the third-progression E–F–G connects the harmonic tones E and G by a passing tone F. The next two transformations produce step motion between different voices of adjacent Stufen or repetitions of the same Stufe. Figure 2.5d gives a simple example of motion from an inner voice (Untergreifen). Here a single melodic line is created by a step motion between the alto C of the opening tonic Stufe and the soprano E of the adjacent tonic chord. Figure 2.5e shows an analogous example of motion to an inner voice. This time the melody is created by a step motion from the soprano E of the first Stufe to the alto E of the second. The fact that the soprano moves to an inner voice suggests that this transformation is closely connected with reaching over, hence the parallel nature of Schenker’s original terms.22 Besides accounting for an almost limitless array of melodic patterns, Schenker also needed to generate an array of new Stufen from the tonic and dominant triads of the prototype. This meant producing not only every diatonic Stufe, but all of their chromatic counterparts as well. He did this through a third group of transformations (see figure 2.6a, Harmonizing transformations). The term “harmonize” is easy enough to explain (see figure 2.6b). Once a given Stufe has been horizontalized, the derived tone can be supported harmonically by a root or first inversion triad, typically ()III, IV, V, and ()VI. In such cases, Schenker insisted that the generating Stufe is always conceptually present; he referred to this idea by the concept of a mentally retained primary tone (der festgehaltene Kopfton).23 Once the resulting span is filled in, the various non-harmonic tones can also be harmonized. Although these non-harmonic tones are usually harmonized by root or first inversion triads, they are sometimes supported

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Figure 2.5. Filling in transformations. a. Transformation

Domain

Schenker’s Discussion

Neighbor motion (Nebennote)

Single line

DfS, par. 106–12,196–202 Fig. 32, 76–80

Linear progression (Zug)

Single line

DfS, par. 113–24, 203–29 Fig. 33–39, 81–99

Motion from inner voice (Untergreifen)

Multiple lines Multiple harmonies

DfS, par. 135–39, 233 Fig. 42, 102

Motion to an inner voice

Multiple lines Multiple harmonies

DfS, par. 203

by a dominant-seventh chord. This latter option arises when the melody moves by step through 4
(for example, by passing motions 5
–4
–3
or 3
–4
–5
and neighbor motion
3–4
–
3). In such cases, we can support 4
with a dominant-seventh sonority.24 Instead of harmonizing derived tones by a complete triad, Schenkerian theory also allows another option. Suppose that a given Stufe has been horizontalized and that the space between its members has been filled in by step by a linear progression. This linear progression can be supported by adding a linear progression that proceeds in parallel thirds, sixths, or tenths. These thirds can then be harmonized as before (see figure 2.6c). We will discuss this strategy in more detail in chapter 3. Under normal conditions, harmonize and addition produce only diatonic Stufen—that is, major, minor, or diminished triads from the appropriate major or minor system. To create chromatic Stufen, Schenkerian theory introduces two transformations—mixture (or Mischung) and tonicization (or Tonikalisierung). As mentioned in

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Figure 2.6. Harmonizing transformations. Transformation

Domain

Schenker’s Discussion

Harmonize

Single harmony Single/Multiple lines

None, but implied, c.f. Retained tones DfS, par. 93, 115

Addition

Single tone Single/Multiple lines

DfS, par. 221–29 Fig. 95–99

Mixture (Mischung, Phrygische II)

Single harmony Single/Multiple lines

DfS, par. 102–5, 193–95 Har., par. 26–30, 38–52 Fig. 28–31, 73–75

Tonicization (Tonikalisierung)

Single tone / harmony Single/ Multiple lines

Har., par. 132–62

chapter 1, there are three basic types of mixture: simple mixture allows for the interchange of triads between parallel keys (see figure 2.6d); secondary mixture allows for changes in the quality of triads in the major and minor systems (see figure 2.6e); and double mixture borrows triads from a parallel key and then changes their quality (see figure 2.6f). Meanwhile, tonicization creates chromaticisms by changing, albeit temporarily, from one tonal system to another. In the case of figure 2.6g, the chromatic tone F is created by tonicizing the dominant Stufe with an applied dominant harmony. Whereas the first three groups of transformations generate new tones and, as such can be used recursively, the final group modifies

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Figure 2.7. Reordering transformations (non-recursive). Transformation Delete (Vertretung) Displacement (Der uneigentliche Intervalle)

Domain

Schenker’s Discussion

Single line

DfS, par. 145–46, 235, 244–46, Fig. 46, 104.

Single/Multiple lines

DfS, par. 158, 261

or reorders tones that have already been generated (see figure 2.7a, Reordering transformations). These new transformations cannot normally be used recursively, though they can be used at early stages in the generative process. The first of these is known as deletion. Schenker clearly believed that the effect of a particular tone can sometimes be felt, even though this tone is not actually present in the score (see figure 2.7b). He described these virtual or deleted tones under the rubric of substitution (Vertretung); this more general idea suggests that particular notes do not behave exactly as they appear and can therefore by replaced by other notes. Schenker normally placed such implied notes in parentheses in his graphs.25 The second transformation is displacement and it shifts tones from one point to another (see figure 2.7c).26 Schenker referred to these tones as displaced or inauthentic intervals (Die uneigentliche Intervalle) and notated them with a diagonal line. As we noted in chapter 1, displacements can be applied to non-harmonic, as well as harmonic tones. Now that we have surveyed Schenker’s list of transformations, we are still left with a couple of nagging questions: why, in fact, should we suppose that this list is complete and why should we suppose that this list of transformations is powerful enough to generate all and only all tonal pieces? In answering these questions, it is important to remember that Schenker’s transformations are intimately related to the subordinate laws of tonal voice leading and harmony outlined in chapter 1; the list of transformations is

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simply as comprehensive as this body of covering laws. For example, register transfer, arpeggiation, unfolding, voice exchange, and reaching over satisfy the subordinate law that melodic leaps arise when the melody shifts from one harmonic tone to another or from one polyphonic voice to another. These transformations account for every possible way in which a single line can be created from two polyphonic voices.27 Similarly, neighbor motion, linear progression, motion to and from inner voice, represent the only ways to fill in the space between horizontalized harmonic tones. When combined with displacement, these transformations can generate the full range of non-harmonic tones, ranging from suspensions and anticipations to appoggiaturas, cambiatas, and other more exotic phenomena. Finally, Schenker’s arsenal of harmonizing transformations conform nicely with the general laws covering the behavior of diatonic and chromatic harmonies; they allow us to generate not only the entire range of diatonic chord progressions, but also the full array of chromaticisms.

Levels Besides proposing that any complex tonal surface can be explained as a composing out of some simple progression, ‘The Recursive Model’ also presumes that whenever a given progression is expanded by the recursive application of a given transformation, the resulting progression conforms to the same laws of voice leading and harmony as the starting progression. To quote from Der freie Satz, “The principles of voice-leading, organically anchored, remain the same in background, middleground, and foreground, even when they undergo transformations. In them the motto of my work is embodied, semper idem sed non eodem modo.”28 According to our classification of laws given in chapter 1, it is only the local laws that are preserved from one level to the next: these local laws guarantee that melodic motion will mostly move by step, that contrapuntal lines will not include parallel perfect octaves or fifths between successive harmonic tones, that harmonic progressions will mostly be triadic, diatonic, and follow the basic law of harmonic closure. It is important to mention, however, that Schenker

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wasn’t always able to achieve this goal; as we will see in chapter 3, he was sometimes inconsistent in his treatment of the laws prohibiting parallel perfect octaves and fifths. We can illustrate these ideas with a simple example (see figure 2.8, Composing out). Figure 2.8a gives a progression C–G–C with a step wise descent in the soprano E–D–C. The first thing we must do is to specify its function within a particular key. This step is especially important if we want to transform the triad chromatically, because we can specify whether it serves as a mixture or even as a member of some secondary key area. In this particular case, figure 2.8b places the progression within the context of C major, hence the Roman numerals I–V–I. Once we have interpreted the individual harmonies as Stufen within a key, we can also compose them out using the horizontalizing transformations from figure 2.4. In this particular case, we can arpeggiate the opening tonic Stufe to create the bass motion C–E. We can then harmonize the derived tone to create our first derived triad in figure 2.8d this creates a I6 sonority (see figure 2.8c). Although there may, in principle, be many options, ‘The Recursive Model’ prohibits those that violate the local laws of tonal voice leading and harmony. This means, for example, that the derived tone cannot produce parallel perfect octaves or fifths with members of the original progression. If we want to continue generating material, then we have two options: we can either repeat the same moves on our first derived triad, or we can fill in the tone space between this derived triad and our initial triad. The second option requires that we use a transformation from figure 2.5. In this case linear progression to produce the bass motion C–D–E (see figure 2.8d). Having created the passing tone D, it is then harmonized in figure 2.8e. Once again, the specific harmony will be constrained by the local laws of relative motion and vertical alignment; our derived Stufe cannot produce parallel perfect octaves or fifths with its neighbors and it can initially take the form of only a major, minor or diminished triad. Notice how the resulting progression I–VII6–I6 is slightly different from the underlying progression I–V–I, but it still conforms to our basic laws of tonal voice leading and harmony. In particular, the subordinate progression mostly moves by step, it does not include

85 Figure 2.8. Composing out.

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Explaining Tonality

Figure 2.9. Schenker’s deep-middleground paradigms. From Schenker, Der freie Satz, Figs. 15, 16, 18.

any parallel perfect octaves and fifths, and it contains lines that converge on the tonic 7
–1
and 2
–1
. Although figure 2.8 illustrates several distinct stages of transformation, it marks a significant point in the generative process. Schenker referred to this stage as the deep, or first-level, middleground. In part 2 of Der freie Satz, Schenker cataloged a broad range of deep-middleground paradigms.29 These are listed in figure 2.9 (Schenker’s deep-middleground paradigms). Looking through these paradigms, it soon becomes clear that the main feature of this level is to fill in the tone space created by the first two notes of the bass arpeggiation. For Schenker, the most basic motion involves horizontalizing the opening tonic Stufe to produce the progression I–I6–V–I or I–III–V–I, the latter giving rise to what he referred to as a Terzteiler (third divider).30 But Schenker also supposed that the

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87

bass arpeggiation is filled out by passing tones.31 He then explained the deep-middleground progressions I–IV–V–I and I–II–V–I, as incomplete passing motions in the bass; he even notated such progressions with a pair of interlocking slurs to convey the idea that they articulate “the contrapuntal-melodic step of a second.”32 Besides focusing on the paradigms given in figure 2.9, Schenker restricted the use of transformations in several other ways. First, to avoid violating the local law prohibiting parallel perfect octaves and fifths, he excluded the progression I–VI–V–I at the deep middleground because this progression creates parallel perfect fifths between the soprano and the bass
3/VI–
2/V. Second, since he believed that each piece lies in a specific register, Schenker limited the use of transformations so that any derived tones remain as close to this register as possible. For example, he excluded ascending arpeggiations, linear progression, voice exchanges, or reaching over from the headtone, upper neighbors from
8, and so forth. Third, since prototypes are bound by the laws of melodic and harmonic closure, Schenker discouraged transformations between
2/V and
1/I. According to him, “at the first level,
2/V–
1/I provides33 no opportunity for prolongation in contrapuntalmelodic terms.” At later levels, however, he relaxed his position; as he explained in par. 189, “the arpeggiation of the descending fifth can proceed through the third only.”34 Fourth, since Schenker wanted to preserve the distinction between different forms of prototype, he shied away from adding transformations that immediately convert one prototype into another. For example, he preferred not to compose out a
3-line with a preliminary descent from
5–
3 since that transformation would create a
5-line descent at the deep middleground. Schenker also discussed one other important strategy for transforming prototypes at the deep middleground, namely, divisions of the upper line (der Gliederung des Urlinie-Züges). As shown in figure 2.10 (Divided Urlinien), these divisions have very distinct forms. In
3- and
5 lines, they involve an interrupted version of the descent (Unterbrechungen) followed by a complete descent. In the case of
3-lines, the result is an upper-line pattern
3–
2//
3–
2–
1; in the case of
5 lines,
5–
4–
3–
2//
5–
4–
3–
2–
1. Schenker used a divider symbol (Teiler) to mark the two segments. In
8-lines, however, he

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Figure 2.10. Divided Urlinien.

subdivided the upper line at
5, thereby producing the motion
8–
7–
6–
5//
5–
4–
3–
2–
1. He made this adjustment because the descending seventh from
8 to
2 can be reinterpreted as an ascending second from
1 to
2. This latter motion contradicts the notion that tonal melodies typically descend by step onto
1 and therefore conflicts with Schenker’s conception of the upper line. As it happens, Schenker had a very good reason for dividing the upper line at the deep middleground: this strategy provided him with a means for explaining the behavior of certain formal types, a point that he made perfectly clear at several moments in Der freie Satz. In par. 94, for example, he declared: “Interruption has the quality of heightening the tension toward
1; particularly, it opens the way to two- or three-part forms, a1–a2 or a1–b–a2.”35 He added

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89

that “it is even the basis of the extended form of the sonata, with exposition, development, and recapitulation.”36 Provocative as they may be, divisions of the upper line do, however, raise an interesting question: how, in fact, do divided lines derive from prototypes? Unfortunately, Schenker gave inconsistent answers to this question.37 On certain occasions, he seems to suggest that the first
2/V belongs to the prototype and that the subsequent I–V progression is front-related to the final I. This interpretation seems to conform with graphs published in Der freie Satz, especially figures 21, 23–28, 32.7, and 33–35. At other times, however, he intimated that the final
2/V belongs to the prototype. This alternative implies that the dividing dominant is backrelated to the opening I. Such a view seems more consistent with the analyses presented in Fünf Urlinie-Tafeln, such as his sketch of Bach’s setting of the chorale “Ich bin’s, ich sollte büssen” from the St. Matthew Passion. If we accept the claim that prototypes summarize the main laws of tonal voice leading and harmony, then the latter response seems preferable to the former. In particular, we know that one of the main goals of the prototype is to explain why a given piece closes in the tonic. It is for this reason that the upper line descends
2–
1, the alto line ascends
7–
1, and the bass descends V–I. Unfortunately, the front-related prolongation of the final I advocated by Schenker in Der freie Satz seems to obscure the connection between the background dominant and the final tonic. Figure 2.11 (Derivation of divided Urlinien) shows how we might derive a divided upper line in the manner implied by his sketch of the Bach Chorale “Ich bin’s, ich sollte büssen.” First, the headtone is repeated; second, the opening repeated tone is harmonized with a tonic Stufe; third, this derived tonic is arpeggiated in the bass; and, fourth, the new bass tone G is harmonized with a new dominant Stufe. Besides adding intermediate Stufen and divisions of the upper line, the deep middleground is also the source of other transformations, though they appear at a slightly later stage of generation. For example, Schenker allowed the headtone to be composed out by preceding or front-related material, provided this material ascends onto the headnote. He referred to this as a preliminary ascent (Anstieg). Such ascents can derive from an ascending register

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Explaining Tonality

Figure 2.11. Derivation of divided Urlinien.

transfer, an ascending arpeggiation, an unfolding, or an ascending linear progression. Schenker also allowed passing motions between the intermediate Stufe and its companions. These extra motions can be clearly seen in the deep-middleground paradigms given as figures 15–18 in Der freie Satz.38

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91

Although it is fairly easy to describe the characteristics of Schenker’s prototypes and his deep middleground paradigms, the situation becomes a lot harder as we pass through the middleground and foreground levels. The reason for this is that as we approach the surface, our graphs will capture the individual characteristics of each piece with greater and greater specificity: above all, thematic and rhythmic features will emerge as the process reaches the surface. In this respect, background levels capture what is common among the entire class of tonal composition, whereas foreground levels convey what is idiosyncratic to a particular piece. Having said that, several specific possibilities should be mentioned. For one thing, Schenker showed how VI can be used as an intermediate Stufe, provided that another intermediate Stufe is added to eliminate the parallels mentioned earlier.39 For another, the progression from V to I can be filled out by a motion through III.40 Schenker also allowed transformations to the final tonic chord. In fact, these prolongations will often correspond to the coda.41

Fallout The preceding sections have shown that, according to Schenkerian theory, any complete, continuous, functional monotonal piece can be generated from a single prototype by the recursive application of certain transformations. As shown in figure 2.12 (The explanatory scope of Schenkerian theory), this idea expands the scope of traditional tonal theory not only by showing how line and chord interact with one another, but also how they do so both locally and globally. However, given the complexity of most functional monotonal pieces, we have every reason to suppose that there may be more than one way to derive a particular surface from a given prototype, provided that each derivational scheme follows the prescribed laws. In practice, however, it is clear that Schenker endorsed some derivations and not others. Our next job is to find suitable criteria for making such a choice. Why, in fact, are some readings deemed preferable to others? Unfortunately, this question is not an easy one to answer. Certainly, there is no magic formula. When pressed, Schenker

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Figure 2.12. The explanatory scope of Schenkerian theory.

and his disciples generally throw their hands in the air and insist that analysis is a ‘creative’ not a ‘scientific’ activity; they vehemently deny that it can be reduced to any sort of algorithm.42 Yet, in actuality, Schenker did leave us with a few tantalizing clues. In particular, his sketches show that he put a premium on analyses in which the same patterns of derivation appear not only at the same level of transformation, but also between different levels. This point was clearly made by Milton Babbitt some forty years ago: Schenkerian theory of tonality, in its structure of nested transformations so strikingly similar to transformational grammars in linguistics, provides rules of transformation in proceeding synthetically through levels of composition. Since many of the transformational rules are level invariant, parallelism of transformation often plays an explanatory role in the context of the theory (and, apparently, an implicitly normative one in Schenker’s own writing).43

In fact, Schenker took special delight when these patterns coincided with surface motivic shapes. He referred to them as hidden repetitions (Der verborgene Wiederholungen) and stressed that “repetitions

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of this kind have nothing to do with ‘motive’ repetitions” in the normal sense of the term.44 We can see the pros and cons of such parallelisms when we consider figure 2.13 (Summary of Schenker’s sketch of “The Representation of Chaos” from Haydn’s Creation).45 Perhaps the most striking feature of Schenker’s reading is that it includes a recurring fourth progression C–B–A–G in the bass; this pattern appears five times at the foreground, each one being marked by square brackets. According to him, this progression is made up of two descending seconds—C–B and A–G. These two-note pairs are motivic in function and derive from the prominent descent A–G at the opening of the movement. Schenker also found the fourthprogression C–B–A–G projected across mm. 1–39 of the middleground: as shown in figure 2.13a, he linked the opening C to the B4^ in m. 31 to the A in m. 38 to the G in m. 39. While the sketch in figure 2.13 reveals many important insights about Haydn’s remarkable score, Schenker was sometimes a little overzealous in his search for descending fourth-progressions. Particularly dubious is the one he found in mm. 18–27. It is hard to see how the C and B in mm. 18–19 initiate a span; on the contrary, they articulate a clear half cadence in the key of E, with the C supporting an augmented-sixth chord. Similarly, by linking the B in m. 19 to the A and G in m. 27, Schenker glossed over the arrival onto D in m. 21. This is perplexing because m. 21 not only articulates the first, albeit weak, authentic cadence in the piece, but it also marks the introduction of what Tovey regards as the second group material.46 This material eventually returns over a dominant pedal in the Recapitulation in mm. 44–49. Other readings are possible, however; see figure 2.14 (Alternate sketch of “The Representation of Chaos”). The underlying prototype for the piece is shown in figure 2.14a. Next, figure 2.14b shows how this progression is composed out contrapuntally at the deep middleground by the descending fourth progression C–B–A–G in the bass and how upper voices are displaced over the structural dominant to create the progression I–V4^–3%–I. Notice how the augmented-sixth chord serves to tonicize the dominant Stufe. In figure 2.14c, this progression is transformed by an initial ascent from C to E in mm. 1–8. This motion is

94 Figure 2.13. Schenker’s sketch of “The Representation of Chaos” from Haydn’s Creation. From Schenker, The Masterwork in Music 2, pp. 102–3.

Figure 2.13. (continued).

95

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Explaining Tonality

supported by a descending fourth-progression C–B–A–G in the bass. Figure 2.14d then fills out mm. 13–30 by two nested motions from an inner voice, the larger one of which extends from the alto A in m. 13 through B, C, and D, to the soprano E in m. 31, and the more local one extends from the alto A in m. 13 through B, C, and C, to D in m. 21. These rising spans are then harmonized in figure 2.18e. The former leads to the passing 6/4 sonority in m. 30 while the latter moves to D. This complex motion completely obscures Schenker’s fourth-progression C–B–A–G in m. 18ff. and suggests that the fourth span in mm. 13–19 is generated after the analogous spans in mm. 1–8 and 40–44. Besides preferring derivations in which the same patterns occur within and between levels, Schenkerians also prefer derivations that show connections with other related pieces. Such decisions suggest that composers actually learn by reworking tonal models from one piece to another. A nice case in point has been discussed by David Beach and others. When analyzing several piano sonatas by Mozart, Beach noticed an interesting pattern; he found that in certain cases the return to the tonic for the Recapitulation was accomplished not by a simple motion to the dominant, but by a large-scale harmonic progression V–III–I.47 Beach showed that this pattern appears in several sonatas: the first and third movements of the Piano Sonata in F, K. 280, the first movement of the Piano Sonata in F, K. 332, and the first movement of the Piano Sonata in B, K. 333. A similar pattern can be found in the first movement of Beethoven’s Sonata for violin and piano, Op. 24.48 Although other composers certainly used this strategy, it seems extremely likely that Beethoven learned the pattern from Mozart; many of Beethoven’s works are clearly modeled on those of his illustrious forebear.49 On one leaf from the “Kafka” papers, Beethoven admitted as much by noting that a passage in C minor was “stolen from the Mozart symphony in C minor [sic].”50 This chapter has shown just how important ‘The Recursive Model’ and ‘The Global Paradigm’ were to Schenker’s thinking. According to ‘The Recursive Model,’ Schenkerian transformations can generate an almost limitless range of tonal surfaces when they are recursively applied to their own output. Since the prototypes and

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Figure 2.14. Alternative sketch of “The Representation of Chaos.”

transformations satisfy the various laws of tonal voice leading and harmony, recursive use of the transformations will necessarily produce new pieces that also satisfy them. If we insist that all complete continuous monotonal pieces can be derived according to ‘The Global Paradigm,’ then Schenkerian transformations provide us with a recursive definition of the infinite set of tonal pieces. In the words of Schenker’s motto, ‘The Recursive Model’ and ‘The

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Global Paradigm’ do indeed show us how functional monotonal pieces are “always the same, but not in the same way.” Seen in broader context, however, Schenker’s motto also gives us an insight into his overall goals as a theorist. Schenker is often criticized for reducing functional monotonal music to some Procrustean formula, akin to “Three Blind Mice.”51 In a trivial sense, this characterization is right on target. Schenker was clearly preoccupied with showing how tonal music conforms to certain very simple principles. But this observation disguises a more profound point: by showing that “all of the [tonal] master works manifest identical laws of coherence,” Schenker demonstrated “a diversity in essential nature among the masters.”52 According to him, we can appreciate the significance of specific works only if we understand the general mechanisms that shape them. Or, to put it another way, “It is an inevitable principle that all complexity and diversity arise from a single simple element rooted in the consciousness or the intuition.”53 This notion is not really so different from those of theories found in other empirical disciplines, such as biology: indeed, just as Darwin tried to explain the diversity of life through a few fundamental processes, so Schenker tried to explain the extraordinary richness of functional tonal music via a limited number of tonal prototypes, transformations, and transformational levels. Furthermore, Schenker’s motto also underscores his concern for the scope of his theory. ‘The Recursive Model’ shows that Schenker was indeed concerned with trying to explain the broadest range of surfaces possible. To do this, he introduced an array of transformations that was large enough to account for all conceivable melodic configurations, all conceivable non-harmonic tones, and all conceivable chromaticisms. In this way, ‘The Recursive Model’ is intimately connected with ‘The Heinrich Maneuver’ and ‘The Complementarity Principle.’ Meanwhile, ‘The Global Paradigm’ allowed him to account for the behavior of particular lines and particular Stufen not only within the local context of an individual phrase or period, but also in the global context of an entire composition. This move expanded the scope of tonal theory considerably.

3

What Price Consistency? There can be little doubt that music theorists value consistency as much as any epistemic value. The reasons for this are clear enough. Claiming that something and its opposite are both true creates difficulties in making predictions; though prediction may not be the sole purpose of scientific inquiry, it is always the bottom line. To quote Quine: “[Prediction] is what gives science its empirical content, its link with nature. It is what makes the difference between science, however high flown and imaginative, and sheer fancy.”1 When we confirm a theory, we do so through “the verification of its predictions.”2 The more predictions we verify, the more confident we will be about using our theory. The same can be said about building and testing music theories, as Schenker made perfectly clear near the start of Kontrapunkt I. According to him: In this study, the beginning artist learns that tones, organized in such and such a way, produce one particular effect and none other, whether he wishes it or not. One can predict this effect: it must follow. Thus tones cannot produce any desired effect just because of the wish of the individual who sets them, for nobody has the power over tones in the sense that he is able to demand from them something contrary to their nature. Even tones must do what they do.3

Several pages later, Schenker reinforced the point by noting that he was primarily interested in describing the abstract effects that a particular tone might have on the motion of a voice and not the psychological effects it might have on a listener: “Tones mean nothing but themselves; they are as living beings with their own social laws.”4 Although Schenker went to great lengths to ensure the consistency of his theory, he repeatedly ran into problems in one particular

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area: the treatment of parallel perfect octaves and fifths. To understand the source of these contradictions, it is important to remember that tonal voice leading is founded on the notion that contrapuntal lines tend to move in contrary motion or in parallel thirds and sixths. In strict counterpoint, if a given pair of lines moves in the same direction, then they can never produce parallel perfect octaves and fifths between successive notes. But in functional monotonality, a given pair of voices can contain parallel perfect octaves and fifths provided that those lines involve doublings and figuration or combinations of harmonic and non-harmonic tones. This subtle change from strict counterpoint to functional tonality is an essential component of ‘The Heinrich Maneuver.’ We can also infer from Schenker’s motto, semper idem sed non eodem modo, that this principle applies consistently at all structural levels. At least, that is what we are led to believe. Unfortunately, Schenker was not always able to preserve the laws of tonal voice leading from one level to the next; he ran into particular difficulties maintaining the laws prohibiting parallel perfect intervals. William Benjamin has described the problem as follows: [O]n the one hand, [Schenker] accounts for certain foreground events in terms of the need to get rid of parallel fifths or octaves at a middleground level; on the other, he justifies certain parallel fifths and octaves in the foreground by noting that they are no longer present once a reduction to the next-higher level (a middleground) is accomplished.5

For his part, Schenker was perfectly aware of these inconsistencies; in par. 161 of Der freie Satz, he claimed that the foreground “fundamentally prohibits parallel octaves and fifths,” but was forced to concede that “the middleground frequently displays forbidden successions, it is then the task of the foreground to eliminate them.”6 We can illustrate the problem with a simple example. Figure 3.1 shows mm. 1–16 of Minuet II from Bach’s French Suite I in D Minor, BWV 812. As shown in figure 3.1a, the passage has two eight-bar phrases, each of which ends with a cadence in D minor. Each phrase is sequential in nature. These sequences are normally explained either in terms of the texture’s outer-voice counterpoint, in this case the repeating intervallic pattern 5–10–5–10, or in terms of local bass motion—in this case the repeating pattern of descending

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fifths transposed by descending seconds. Such explanations are the ones we normally encounter in undergraduate theory classes or harmony textbooks.7 Nevertheless, difficulties arise when we try to reconcile such explanations with a Schenkerian derivation of the music. From a global perspective, the most striking feature of the sequences is that they both support a stepwise descent in the upper voices from A to D, with A ornamented by B
, G by A, and F by G. If we remove the ornamental notes and their supporting harmonies, then we are left with a chord succession I–VII–VI–V–I that violates the law prohibiting parallel perfect octaves and fifths (see figure 3.1b, ‘The Parallel Problem’). These inconsistencies suggest that our common-sense distinction between structural and ornamental tones conflicts with the general laws of tonal counterpoint. This conflict, which we will refer to as ‘The Parallel Problem,’ is particularly vexing for those who believe that Schenkerian analysis give us “artistic statements, in music, about music.”8 After all, how can we support such a view if our analyses include relationships that do not typically occur in tonal surfaces? This is not, however, the only problem with sequences. If we take the distinction between structural and ornamental notes a step further, then we might suppose that the first chord in each portion of the sequence has priority over the second (see figure 3.1c, ‘The Top-Down/Bottom-Up Problem.’). According to this scheme, the A chord in m. 7 should have priority over the D chord in m. 8. But this reading is unconvincing; as the goal of the entire phrase, the final tonic D in m. 8 surely has priority over the dominant in m. 7. Thus, describing sequences in “bottom-up” terms as some sort of repeating melodic, contrapuntal, or harmonic pattern conflicts with treating the phrase in “top-down” fashion as a prolongation of the tonic D. This topic, which we will refer to as ‘The Top-Down/ Bottom-Up Problem,’ is significant because it suggests that there may be inherent differences between the conventional ways in which we describe musical surfaces and the more radical ways in which Schenkerian theory derives them. In short, sequences present us with some thorny questions. Can we, in fact, generate sequences like the one in figure 3.1 without violating the basic laws of tonal voice leading? Can we resolve

102 Figure 3.1. Sequences. a. Bach, French Suite in D Minor, BWV 812, Minuet II, mm. 1–16

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Figure 3.1 (continued).

‘The Parallel Problem’ and ‘The Top-Down/Bottom-Up Problem’ using Schenkerian theory? This chapter sets out to answer these questions. Part 1 takes another look at sequential patterns like the one in figure 3.1 and suggests some concrete ways to overcome the hazards mentioned above. Next, Part 2 shows how these solutions can be grounded in familiar principles of counterpoint and in Schenker’s own concept of combined linear progressions; among other things, it shows that there are interesting connections between sequences and pedals. Part 3 then considers the analytical consequences of these ideas and presents detailed readings of several pieces, including the Minuet shown in figure 3.1.

Sequences Reconsidered To shed light on the nature of sequences, we will begin by looking at some familiar patterns, starting with the one in figure 3.2 (A typical ascending-fifth sequence). This particular phrase has two main components: a sequence and a cadence. The former consists of pairs of fifth-related harmonies that are transposed down a third: C–G,

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Figure 3.2. A typical ascending-fifth sequence.

A–E, F–C, D–A. The latter, meanwhile, consists of a typical progression II5^ –V–I. Innocuous as these observations may seem, they actually suggest that each component encapsulates a quite different principle of voice leading. These differences are most obvious in the soprano and alto voices. Indeed, figure 3.2 clearly shows that, while the two upper voices move by parallel thirds during the sequence, they ultimately converge on the tonic at the cadence. This point has important ramifications; among other things, it confirms the intuition that parallel motion creates a sense of openness, whereas convergence creates a sense of closure. We might also note that the soprano and alto lines make contrapuntal sense on their own; the other voices could be omitted and the phrase would still define the tonic C. This point is supported by the fact the two upper voices can potentially be supported by a long tonic pedal in the bass. We will return to this connection between sequences and pedals later in the chapter. But for the time being, we can simply conclude that sequences are ultimately contrapuntal, rather than harmonic, in origin and that they might be generated from parallel voice leading, rather than from the outer voice counterpoint. This observation allows us to generate the pattern in the manner shown in figure 3.3 (Deriving ascending-fifth sequences). Figure 3.3a begins by presuming that the sequence occurs within the context of a single phrase, in this case one that descends by step 8–7  3– 2– 1 and ends with the cadential progression II5^ –V –I. Since this derivation is context-dependent, we can avoid ‘The Top-Down/ Bottom-Up Problem.’ Next, the soprano tone  3 is transferred up an octave and the intervening space is filled by a stepwise descent

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Figure 3.3. Deriving ascending-fifth sequences.

(see figure 3.3b). In figure 3.3c, this descent is harmonized by a string of parallel thirds. Finally, figure 3.3d harmonizes this string of thirds according to two principles: each third must be supported by a root triad and each triad should not produce parallel perfect octaves and fifths with its neighbors. Significantly, the laws of tonal

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voice leading dictate that only one such harmonization is possible from a given starting chord. For example, although the third D/B can belong to triads on G or B, the latter creates parallel perfect octaves and fifths with the initial tonic chord. This leaves G as the only option. Similarly, when the third D/B descends to A/C, the latter can be harmonized by triads whose roots are F and A. But since the triad on F creates parallel perfect octaves and fifths with the preceding G chord, the triad on A must be picked. In other words, we can resolve ‘The Parallel Problem’ by generating the sequence from the upper voices and not the bass. The same strategy can be used with suitable adjustments to derive other sequences. Perhaps the easiest way to change the pattern in figure 3.3 is to invert the voices contrapuntally, thereby transforming the chain of parallel thirds into a chain of parallel sixths. This process is shown in figure 3.4 (Restacking ascendingfifth sequences). In figure 3.4a, the soprano spans an octave  8 to  1. The final stepwise descent  2– 1 is harmonized by a cadential progression II5^ –V–I. Next, figure 3.4b supports the stepwise descent of the soprano with a string of parallel sixths from E to E in the tenor voice. Finally, these parallel sixths are harmonized by root triads (see figure 3.4c). As with figure 3.3, the starting chord and laws of counterpoint dictate that only one such harmonization is possible. With further adjustments, we can generate still more sequences (see figure 3.5, Deriving ascending-third sequences). Figure 3.5a starts again with a phrase model consisting of a descent  3– 2– 1 in the soprano supported by a simple cadential progression I–II5^ – V8–7–I. As before, the soprano tone  3 is transferred up an octave. The intervening octave space is then filled by step (see figure 3.5b) and harmonized by a string of parallel thirds in the alto voice (figure 3.5c). But in figure 3.5d, the soprano E from the first third E/C is displaced over the second alto note B to create the pattern 3–4–3. Similar displacements occur when C/A descends to B/D, when A/F descends to G/E, and when F/D descends to E/C. Instead of creating a chain of parallel thirds, this process produces the succession 3–4–3–3–4–3–3–4–3–3–4–3. Although figures 3.5a–d set the soprano and alto voices against a single tonic pedal, figure 3.5e harmonizes the descent 4–3 as a passing motion; E/B–D/B are supported by a triad on E, C/G–B/G by a triad on C, A/E–G/E by

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Figure 3.4. Restacking ascending-fifth sequences.

a triad on A, and F/C–E/C by a triad on F. For variety, several chromaticisms have been added; these introduce applied chords that tonicize the triads on VI, IV, and II. Whereas the sequence in figure 3.5 elaborates the chain of parallel thirds by displacing the lines to create the pattern 3–4–3, other sequences can be created by inserting intermediary tones (see figure 3.6, Deriving descending-fifth sequences). This sequence is, of course, analogous to the one in figure 3.1. Figure 3.6a begins with a stepwise descent 5 –4 –3 –2 –1 in the soprano. This motion is supported by a

108 Figure 3.5. Deriving ascending-third sequences.

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Figure 3.6. Deriving descending-fifth sequences.

simple cadential progression I–II5^ –V8–7–I. Next, the soprano part is elaborated with escape tones (see figure 3.6b) and the new line is harmonized by a string of parallel thirds (see figure 3.6c). Figure 3.6d then harmonizes the soprano and alto lines exclusively with root triads. Once again, there is only one possible harmonization. Although the

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third A/F belongs to triads on D and F, the former is impossible because it produces parallel perfect octaves and fifths with the opening tonic chord. Similarly, when the third A/F leaps to F/D, the latter can be harmonized only by a triad on B because a triad on D creates parallel perfect octaves and fifths with the preceding F harmony. Figure 3.4 modifies the sequence in figure 3.3 by inverting the lines contrapuntally; the soprano line in figure 3.4 was originally the alto line in figure 3.3, and the tenor line in figure 3.4 was originally the soprano line in figure 3.3. But, if the alto line in figure 3.3 is placed in the bass, below the tenor voice, then this no longer the case; the resulting pattern is now reclassified as a descending 5–6 sequence. Figure 3.7 (Deriving descending 5–6 sequences) shows how to generate such a configuration. Figure 3.7a starts with a descent  3– 2– 1 in the soprano, harmonized by the cadential progression I–II5^ –V8–7–I. In figure 3.7b, 3 in the soprano line is transferred up an octave and the intervening space is filled by a stepwise descent. Figure 3.7c then supports the octave descent with a string of parallel tenths in the bass. Finally, figure 3.7d adds inner voices to this parallel string. Although these tenths can be harmonized in other ways—perhaps by a string of parallel 6/3 sonorities—figure 3.7d follows the same scheme as figure 3.3, thereby underscoring the close connections between these two forms of sequence. In the same vein, we can develop further transformations of our model. One such transformation is shown in figure 3.8 (Deriving alternative descending-fifth sequences). Figure 3.8a starts with a descent  3– 2– 1 in the soprano, harmonized by the cadential progression I–II5^ –V8–7–I. In figure 3.8b, the alto voice G moves up by step through A and B to C, with each of these notes repeated. This motion from an inner voice is then harmonized in parallel thirds (see figure 3.8c). Finally, figure 3.8d harmonizes the chain of thirds with root chords. Given the starting chord and the laws of counterpoint, only one harmonization is possible. Notice how the tenor part rises by step from C to G before descending back to E. A similar strategy is shown in figure 3.9 (Deriving ascending 5–6 sequences). Once again, figure 3.9a starts with a descent  3– 2– 1 in the soprano, harmonized by the cadential progression I–II5^ –V8–7–I. In figure 3.9b, the alto moves up by step from G to C, and in figure 3.9c it is supported by parallel thirds with the tenor.

111 Figure 3.7. Deriving descending 5–6 sequences.

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Figure 3.8. Deriving alternative descending-fifth sequences.

Next, figure 3.9d repeats each new member of the alto and tenor voice; it also displaces them so as to create the pattern 3–4–3–4–3–4–3. Finally, in figure 3.9e, this pattern is then harmonized by alternating root and first inversion triads. As in figure 3.5, a few chromaticisms have been added; these allow us to treat the 6/3 sonorities as applied dominants.

113 Figure 3.9. Deriving ascending 5–6 sequences.

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Figure 3.10. Simple mixture.

We have already shown in figures 3.5 and 3.9 that sequences often allow composers to introduce chromaticisms; in both cases, applied chords were added to tonicize particular harmonies. But there is another important source of chromaticism in tonal contexts, namely the principle of mixture; figure 3.10 (Simple mixture)

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Figure 3.11. Double mixture.

and figure 3.11 (Double mixture) derive two chromatic sequences. Figure 3.10 presents a variant of the descending-fifth sequence that we derived in figure 3.6. Figure 3.10a begins with a stepwise descent in the soprano from  5 to  1, though the third is mixed  5– 4 –
 3– 2– 1. This motion is supported by a simple cadential progression V8–7–I.

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Next, the soprano part is elaborated with escape tones (see figure 3.10b) and the resulting line is harmonized by a string of parallel thirds (see figure 3.10c). Figure 3.10d then harmonizes the soprano and alto lines exclusively with root chords. Again, this harmonization is the only one possible. Although the third A/F in m. 2 belongs to triads on D and F, the former is impossible because it produces parallel perfect octaves and fifths with the opening tonic chord. Similarly, when the third A/F leaps to F/D, the latter can be harmonized only by a triad on B
because a triad on D creates parallel perfect octaves and fifths with the preceding F harmony. Notice how the D
harmony
II serves as a pre-dominant at the cadence. Figure 3.11 follows the same strategy, though the double mixture is created by changing the quality of the triads on B
, E
, and A
from major to minor. To sum up, figures 3.3–3.11 resolve the problems of generating sequences in two quite different ways. On the one hand, they overcome the ‘The Top-Down/Bottom-Up Problem’ by deriving sequences within the context of a phrase. This move guarantees that the goal of the sequence is always specified before its surface features are completely worked out. On the other hand, figures 3.3–3.11 sidestep ‘The Parallel Problem’ by deriving the sequence from parallel motion in the upper voices and not necessarily from the counterpoint between the outer voices. Not only does this approach contrast with most conventional accounts of sequences, but it also confirms the notion that sequences are basically contrapuntal, rather than harmonic in nature. Indeed, by showing that the bass motion is ultimately controlled by the upper-voice counterpoint, figures 3.3–3.11 also imply that harmonic function in a Riemannian sense emerges from contrapuntal motion. This point is evident both inside the sequence, where functionally related harmonies derive from parallel step motion, and at the cadence, where the penultimate dominant chord converges on the final tonic, with the soprano descending  2– 1, the alto ascending  7– 1, and the predominant chord converges on the dominant chord, with 4 and 6 both moving to  5. These derivations even suggest that interesting connections can be found between sequences and pedals. But this contrapuntal view of the sequence begs its own set of questions: to what extent can we find precursors of sequences in traditional

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contrapuntal theory, especially Fuxian species counterpoint, and how do the derivations in figures 3.3–3.11 fit with Schenkerian theory? In order to answer these questions, we must take a closer a look at the principles of strict counterpoint as formulated by Fux and assess their impact on Schenker’s thinking.

Sequences and Counterpoint At first sight, it seems utterly perverse to search for the origins of sequences in the pages of Gradus ad Parnassum. After all, Fux called for variety in melodic writing and discouraged the use of recurring melodic patterns (or monotonia) in strict counterpoint.9 And yet, his examples often contain an important component of the sequence, namely, strings of notes that move in a single direction by some fixed interval. This point is clear from figure 3.12 (Fux’s prototypical cantus firmi): although these melodies avoid exact repetitions of given melodic shapes, many of them end with a string of descending steps. In fact, Melody 6 ends with a string of three, Melody 1 with a string of four, and Melody 5 with a string of five! The significance of these stepwise descents becomes more apparent when we add a simple counterpoint to the texture (see figure 3.13, Typical two-voice counterpoint in First Species). Indeed, although both voices primarily move in contrary motion, they often move by parallel thirds or sixths when the cantus firmus descends by step at the cadence. These parallel strings actually create a strong feeling of motion that makes the closure at the cadence all the more compelling. In Fourth Species, this sense of propulsion is heightened when the parallel intervals are displaced to create chains of suspensions (see figure 3.14 , Typical two-voice counterpoint in Fourth Species). Textures with three or more voices provide even more opportunities for parallel motion by step. The reasons are clear: if the new counterpoint moves in contrary motion with the cantus firmus, then it will inevitably move in similar motion to the existing counterpoint; and if it moves in contrary motion with the counterpoint, then it will inevitably move in similar motion with the cantus

118 Figure 3.12. Fux’s prototypical cantus firmi.

Figure 3.13. Typical two-voice counterpoint in First Species. From Fux, The Study of Counterpoint, Fig. 22.

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Figure 3.14. Typical two-voice counterpoints in Fourth Species. From Fux, The Study of Counterpoint, Figs. 73, 74.

firmus. Fux himself offered some intriguing examples in his discussion of First Species in three voices: instead of recycling one of the cantus firmi from figure 3.12, he introduced some new prototypes in which the lowest voice simply ascends by step from C through D, E, and F to G, before ending back on C (see figure 3.15, Fux’s threevoice prototypes). The first prototype (figure 3.15a) has three descending thirds between the upper parts, whereas the second (figure 3.15b) has three ascending thirds between the lowest voices. In both of these cases, the soprano essentially moves in contrary motion with the bass. It seems very likely that Schenker had these prototypes in mind when he showed how Ursätze are usually transformed at the deep middleground.10 Similar examples are shown in figure 3.16 (Typical three-voice counterpoints in Fourth Species). Figure 3.16a is interesting for a couple reasons. In the first place, Fux harmonized the chain or 7–6 suspensions in the upper-voice parallels at the end of the passage with a sequential bass line F–C–D–A–D. For another, he included a pair of 7–6 suspensions against the repeated tone G in the bass in mm. 3–4. In his text, Fux singled out this passage, noting that the bass can be extended to create a pedal

120 Figure 3.15. Fux’s three-voice prototypes. From Fux, The Study of Counterpoint, Figs. 91, 92.

Figure 3.16. Typical three-voice counterpoints in Fourth Species. From Fux, The Study of Counterpoint, Figs. 141, 142.

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Figure 3.16 (continued).

point as shown in figure 3.16b. According to him, the results are “not only correct but even very beautiful.”11 As it happens, figures 3.16a–b open the door for mixed species. Although Fux did not address this topic systematically in Gradus ad Parnassum, he did include a few tantalizing examples, given here in figure 3.17 (Parallel motion in mixed species with three and four voices). In figure 3.17a, he put the cantus firmus in the bass, the soprano in Second Species, the alto in Third, and the tenor in Fourth. Notice how the stepwise descent in the bass near the end of the example supports quasi-sequential motion in the upper voices. Similarly, in figure 3.17b, he showed a three-voice texture in which the melody moves in First Species, the alto in Fourth, and the bass in Second. In this case, the parallel motion of the upper voices starting in m. 4 is supported by a sequential bass part, F–G, E–F, D–E, and C–D. These examples suggest that pedals and sequences often spring from the same source: pedals can sometimes be created when chains of stepwise parallel lines move against a static voice, whereas sequences can arise when such strings are supported by a recurring bass pattern. This ties in nicely with our discussion of figures 3.3–3.11. Whereas Fux avoided sequences, Schenker was openly hostile to them. His response was simply to reject them altogether. For example, when discussing the passage in figure 3.1a, he scoffed at the idea that the progression I–IV–VII–III–VI–II–I in mm. 1–8 is an “idle sequence” and insisted that it simply defines the tonic D

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Figure 3.17. Parallel motion in mixed species with three and four voices. a. From Fux, The Study of Counterpoint, Fig. 204.

b. From Fux, The Study of Fugue, Ex. 61.

minor.12 Later, in the essay “Das Organische der Fuge” (1926), Schenker claimed that the word sequence “has no validity” and that “the mere fact of its existence as a theoretical term does not lend it any credibility as a concept.”13 And, in Der freie Satz (1935), he rejected the term on the grounds that it described local details without explaining global processes.14 Yet, like Fux, Schenker certainly recognized the significance of parallel stepwise counterpoint, especially over a pedal. This is already apparent in his discussion of mixed species in the latter portions of Kontrapunkt II (1922). Here, Schenker specifically referred to examples from Fux like those shown in figures 3.16–3.17. According to him, it is irrelevant whether the florid counterpoints are consonant or dissonant with the bass; all that matters is that they make contrapuntal sense with

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Figure 3.18. Parallel linear progressions within a single Stufe. From Schenker, Der freie Satz, Fig. 97.2.

each other.15 This crucial idea fits with our observation that the soprano and alto parts in figure 3.3 make contrapuntal sense in their own right. Matters become more complex when Schenker shifted his attention from the purely intervallic world of strict counterpoint to the triadic world of functional monotonality. Now the stepwise parallel strings are constrained by the underlying harmonic motion of the music. Schenker took up this particular issue with a vengeance in his discussion of parallel linear progression in par. 224–26 of Der freie Satz.16 In fact, these paragraphs appear in a general discussion of the various ways in which two or more linear progressions can be combined (par. 221–29); they mention sequences only in passing.17 The main thrust of Schenker’s argument is clear: when two or more linear progressions are combined, one is primary (or leading) and the others are secondary. To prioritize one linear progression over another, Schenker insisted that each progression must be evaluated according to the order in which it is generated from the background.18 This is readily apparent from figure 3.18 (Parallel linear progressions within a single Stufe). Here Schenker sketched a short passage from the third movement of Mozart’s Piano Sonata in A Minor, K. 310. He suggested that the soprano voice leads and the alto follows, and that both voices move over a pedal A. Given the essential role that Urlinien play in generating the melodic profile of a piece, leading progressions are frequently found in the soprano voice. But, as Schenker was quick to point out, they need not be confined to this register: “once one has decided whether the leading linear progression is in the lower or in the upper voice, one must understand the counterpointing progressions as upper or

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lower thirds, tenths, or sixths.”19 These points are implied in figures 3.3–3.11. Since these derivations proceed from a single stepwise descent that we subsequently harmonized in thirds or sixths, they necessarily treated one linear progression as the leader. Furthermore, we have already noted that the various parallel voices can be redistributed in the texture and even inverted contrapuntally. We used these strategies to derive the sequences in figure 3.4 from those in figure 3.3. Besides evaluating the status of linear progressions, Schenker also classified parallel linear progressions in two ways.20 The first way is according to the length of the leading progression. Since he believed that genuine linear progressions can span only the intervals contained within triads, this means that the leader will normally span a third, fourth, fifth, sixth, or octave; in the case of figure 3.18, for example, it articulates an octave span. That being said, it is important to note that the follower need not be the same length as the leader. Thus, whereas the leader in figure 3.18 projects an octave line, the follower is far more ad hoc in nature and barely a linear progression at all. Indeed, as William Rothstein has recently pointed out: “[T]he one instance in which Schenker admits non-harmonic Züge is in the case of two Züge moving in parallel thirds, sixths, or tenths.”21 In this respect, the “leading” Zug arpeggiates the harmony; the “follower” goes along passively for the contrapuntal ride. The second way in which Schenker classified parallel linear progressions is according to whether they horizontalize a single Stufe or whether they fill in the space between two Stufen. We have already seen the first possibility in figure 3.18; in this case the parallel linear progressions compose out a single A-major chord. The second possibility is shown in figure 3.19 (Parallel linear progressions between different Stufen). This figure derives an ascending 5–6 sequence along the lines shown in figure 3.9. Now, however, the phrase as a whole is controlled by a stepwise descent  5–4– 3 in the soprano and a cadential progression I–II5^ –V7–I (see figure 3.19a). Before reaching the cadence, the alto line ascends G–A– B–C–D–E (see figure 3.19b). In figure 3.19c, every note of the alto voice is repeated, and in figure 3.19d this string is harmonized with alternating thirds and fourths. Finally, in figure 3.19e the pedal is replaced by a string of alternating root and first-inversion chords.

125 Figure 3.19. Parallel linear progressions between different Stufen.

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As a result, the alto voice appears to span a minor seventh from G to F. But this span is not, in Schenker’s terms, a true linear progression; instead of composing out a single line of counterpoint, the span actually connects the alto voice of the opening tonic Stufe with the soprano voice of the II5^ . The span is therefore an example of what Schenker termed “motion from an inner voice.” Although he devoted a paragraph of Der freie Satz to so-called seventh progressions (Die Septzüge), Schenker conceded that they usually arise from changes of register between voices, rather than from genuine linear progressions within a single voice.22

Analytical Implications Having solved ‘The Parallel Problem’ and ‘The Top-Down/BottomUp Problem’ and grounded these solutions in contrapuntal principles and combined linear progressions, let us now consider some analytical implications. We will try to show how particular sequential patterns can be derived within the context of specific pieces. A good place to start is with Bach’s Little Prelude in C Major, BWV 924. This short through-composed prelude contains two sequences and a dominant pedal. The first sequence appears in mm. 1–3 and consists of the pattern C–G, D–A, E in the bass. This pattern leads to a second sequence in mm. 3–6 that descends A–B–C; E–F– G–A; C–D–E–F. The long dominant pedal in mm. 7–17 eventually resolves onto a tonic chord at the final cadence in m. 18. Schenker himself analyzed this prelude on at least two occasions: in Der Tonwille 4 (1923) and again in Der freie Satz.23 His feadings are conflated in figure 3.20 (Schenker’s analysis of Bach’s Little Prelude in C Major, BWV 924). In Der freie Satz, Schenker’s main insight was to suggest that the Urlinie is composed out at the deep middleground by a pair of unfoldings E–C and B–D (see figure 3.20a–b).24 He suggested that the unfolding E–C is inverted contrapuntally to create an upper sixth and that this interval is filled to create a sixth progression E–F–G–A–B–C. This sixth progression is then split into two segments, E–F–G and A–B–C, with the latter segment transferred down an octave (see figure 3.20c).25 In Der freie Satz and Der Tonwille, Schenker derived the first sequence from the

127 Figure 3.20. Schenker’s analysis of Bach’s Little Prelude in C Major, BWV 924. Adapted from Schenker, Der Tonwille, vol. 4, and Der freie Satz, Fig. 43.b.

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first segment by harmonizing it in parallel tenths with the bass E/C–F/D–G/E (see figures 3.20c–d). He followed much the same strategy in Der Tonwille to derive the second sequence (mm. 3–6). This pattern stems from a 5–6 motion between the outer voices that is subsequently elaborated by motion in the inner voice. As it stands, figure 3.20 offers a fascinating analysis of Bach’s prelude, one that captures both the broad sweep and subtle details of the musical fabric. And yet, this reading is not without its problems. Indeed, although Schenker deliberately avoided any mention of the term sequence and overcame ‘The Top-Down/Bottom-Up Problem’ by deriving both sequences from a global prototype, he was unable to avoid ‘The Parallel Problem’ because he still derived both sequences quite conventionally from the outer-voice counterpoint. In the first sequence, he sidestepped the parallels by omitting the inner voices altogether; he simply claimed that any middleground parallels are avoided by inserting intervening harmonies to create the pattern 10–5–10–5–10.26 In the second sequence, he finessed ‘The Parallel Problem’ by deriving all four voices simultaneously. Once again, he insisted that the purpose of the foreground was to cover up the implied parallel perfect octaves and fifths of the middleground, in this case by a string of 5–6 motions. Figure 3.21 (Alternative analysis of Bach’s Little Prelude in C Major, BWV 924) tries to overcome ‘The Parallel Problem’ in figure 3.20. Figures 3.21a–b correspond to the progressions shown in figures 3.20a–b. Unlike figure 3.20, which derives the two sequences from linear interval patterns in the outer voices, figure 3.21 derives them from chains of parallel thirds in the upper register. Figure 3.21c includes a chain of ascending thirds E/C–F/D–G/E– A/F. These thirds are elaborated in two ways: the ascent E/C–F/D– G/E is elaborated by lower thirds E/C–D/B–F/D–E/C–G/E, and the progression from G/E to A/F is transferred down an octave and filled in G/E–F/D–E/C–D/B–C/A–B
/G–A/F (see figure 3.21d). Once the string of thirds in the upper voices has been generated, it can be harmonized in the manner shown in figure 3.21e. Significantly, there is only one way to harmonize this string; this harmonization inevitably produces the two sequences found at the surface of the music. Besides circumventing ‘The Parallel Problem,’ figure 3.21 also reveals some interesting connections between the

129 Figure 3.21. Alternative analysis of Bach’s Little Prelude in C Major, BWV 924.

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sequences in mm. 1–6 and the pedal in mm. 7–17; in particular, the ascending sixth progression E–C is balanced by a descending sixth progression B–D. The main difference between the sequence and the pedal is that the former harmonizes each successive parallel third with a new chord. We encounter many of the same analytical issues when we try to analyze Bach’s Prelude in C Minor, WTC I, BWV 847. Once again, Schenker discussed the piece on several occasions. In Harmonielehre (1906), he cited mm. 1–4 as examples of a subdominant fifth and a pedal point.27 Almost twenty years later, he published an extended analysis of the entire prelude in Die Musik (1923); this analysis was subsequently revised for the final portion of his essay “Das Organische der Fuge,” from Das Meisterwerk in der Musik 2 (1926).28 And, finally, in Der freie Satz, Schenker included a reduction of mm. 1–18 in his discussion of combined linear progressions; this reduction refines his analyses from the 1920s.29 Broadly speaking, Bach’s C-minor prelude follows the same basic format as the Little Prelude in C, though it is somewhat larger in scope.30 Like its diminutive cousin, the C-minor prelude begins with a long expansion of the tonic (mm. 1–18) that includes prominent sequential motion. The sequence eventually leads to an elaborate dominant pedal (mm. 21–33). This pedal finally moves onto a tonic harmony in mm. 34–38. With regard to the initial tonic expansion (mm. 1–18), Schenker rightly observed that it has four distinct components. As he noted in the Harmonielehre, mm. 1–4 use a simple pedal to establish the tonic C minor. Next, mm. 5–11 consist of a passing motion from E
to B
in the Urlinie. Although mm. 11–14 continue the downward trajectory of the Urlinie, they change the accompanying lines in order to avoid a premature arrival on C. Finally, mm. 14–18 use a 5–6 exchange to complete the octave descent E
–E
in the Urlinie and return to the tonic chord. Schenker’s derivation of mm. 1–18 is summarized in figure 3.22 (Schenker’s analysis of Bach’s Prelude in C Minor, WTC 1, BWV 847, mm. 1–18). Figure 3.22a gives the foreground graph that originally appeared in “Das Organische der Fuge.” Next, figure 3.22b gives the middleground reduction that he presented in Der freie Satz. When we compare these two readings, it is clear that they both rely heavily on the string of parallel tenths that occur between

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Figure 3.22. Schenker’s analysis of Bach’s Prelude in C Minor, WTC I, BWV 847, mm. 1–18. From Schenker, The Masterwork in Music 2, pp. 48–49.

the soprano and bass. According to Schenker, the main body of the sequence, mm. 5–11, is generated from a string of descending 7–6 sequences, akin to those found in Fourth Species textures like the one cited in figure 3.14a. His view of the prototypical voice leading is given here as figure 3.22c. Notice how Schenker suggested how the line might continue in mm. 12–18; he added the hypothetical bass tones F–E
–D–C in parentheses. Although this interpretation initially seems plausible, it seems less satisfactory when we compare figure 3.22 with figure 3.23 (Alternative analysis of Bach’s Prelude in C Minor, WTC 1, BWV 847,

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Figure 3.23. Alternative analysis of Bach’s Prelude in C Minor, WTC I, BWV 847, mm. 1–18.

mm. 1–18). In figure 3.22a, Schenker proposed that mm. 5–11 project a string of parallel first-inversion chords, separated by applied chords. For example, the motion from A
6 in m. 5 to G6 in m. 6 is elaborated by an applied harmony that tonicizes G6. But in figure 3.22b these applied chords are reduced out, leaving a succession of parallel perfect fifths E
/A
–D/G–C/F–B
/E
. Figure 3.23 tries to overcome these problems: it circumvents ‘The Parallel Problem’ that mars figure 3.22 and it maintains Schenker’s position that dissonances are ultimately passing in nature. Like figure 3.7 this derivation places the string of parallel tenths in the soprano and bass

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Figure 3.24. Two analyses of the Prelude from Bach’s Partita No. 3 for Solo Violin. BWU 1006 From Schenker, The Masterwork in Music 1, pp. 40–41.

(see figure 3.23a) and, like figure 3.9, it repeats each successive soprano note (see figure 3.23b). Finally, instead of harmonizing the repeated tones with root chords, figures 3.23c and 3.23d support them with inversions; this strategy preserves the prominent parallel tenths between the outer voices. One of the most interesting features of Bach’s C-minor prelude is the fact that it uses apparent sequential motion to support an octave descent in the upper voice. The descent occurs at the middleground in figure 3.22. But it does raise the possibility that such motions might occur at even deeper levels. This possibility is clearly shown in Figure 3.24 (Two analyses of the Prelude from Bach’s Partita No. 3 for Solo Violin, BWV 1006).31 Significantly, Schenker supported the descent  8– 1 in the Urlinie with a sequential harmonic progression I–V/VI–VI–V/IV– IV–V/II–II–V–I (see figure 3.24a). However, figure 3.24b presents an alternative derivation analogous to the one presented in figure 3.5; in particular, it adapts Schenker’s reading to

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highlight the chain of descending parallel sixths. This being said, it is important to stress that sequential settings of  8-line Ursätze are not common, and it is quite possible that Schenker himself would have regraphed the Prelude later in his career. One reason for this is that  8-line prototypes tend to divide  8–5 – 1 with an intermediate motion onto the dominant, rather than  8–4 – 1 as in this particular prelude.32 Nevertheless, the graph in figure 3.24 certainly captures many special features of the work, and it confirms David Smyth’s claim that 8 -line Ursätze often have important formal implications.33 Given that the preceding analyses have focused on the ways in which Bach used sequences and pedals, it seems fitting to round things off by reconsidering Minuet II from his French Suite No. 1. As we saw in figure 3.1a, the first part of this piece has two eight-bar phrases. These phrases both begin and end in the tonic D minor, and both are built from the same sequential progression I–IV–VII– III–VI–II–V–I. The second section also opens with another eightmeasure phrase (mm. 17–25), though it ends with a clear half cadence in D minor. Motivically, this phrase develops the main material from m. 1. Bach presents this gesture three times in the bass and tenor registers on D (m. 17), G (m. 18), and A (m. 19), and twice more in the soprano on C (m. 21) and D (m. 22). Just like mm. 9–16, these motivic statements are accompanied by continuous eighth notes. Having reached the half cadence in m. 25, Bach brings back verbatim the entire opening section (mm. 25–40). Unlike the first section, however, mm. 17–40 are not repeated. Although Schenker did not publish a graph of this particular movement, we can fill this gap in the manner shown in figure 3.25 (Analysis of Bach’s Minuet II, French Suite in D Minor, BWV 812). Figure 3.25a suggests that the piece can be derived from a  5line prototype in D minor. Figure 3.25b then shows how the half cadence in m. 24 is generated by a division of the Urlinie at the deep middleground. Next, figure 3.25c derives mm. 1–16 as a descent from  5 to  1 just like mm. 33–40. Figure 3.25d then shows how the descending fifth sequences in mm. 8–16 and 33–40 are generated from third chains in the upper voices; this derivation follows the general scheme outlined in figure 3.6. Finally, figure 3.25e derives the same sequence in mm. 1–8 and 25–32 from analogous chains of thirds. Unlike mm. 9–16 and 33–40, however, the

135 Figure 3.25. Analysis of Bach, French Suite in D Minor, BWV 812, Minuet II.

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chain of parallel thirds is submerged beneath a soprano line that rises from A (m. 1) through B
(m. 2) and C (m. 3) to D (m. 4). Once m. 4 is reached, the soprano line takes over the lower third of the chain, thereby demonstrating Bach’s fondness for invertible counterpoint. Yet again, the derivation shown in figure 3.25 avoids both ‘The Parallel Problem’ and ‘The Top-Down/BottomUp Problem.’ In this chapter, we have used our discussion of sequences as a pretext for considering the consistency of Schenkerian theory. We have seen that Schenker was ultimately inconsistent in the way he treated parallel perfect octaves and fifths. Although he insisted that such phenomena do not occur when a given pair of voices move between successive harmonic tones, his analyses are littered with parallels, especially when the music is sequential. When these anomalies appear at the foreground, Schenker claimed that they can be eliminated by appealing to the behavior of the middleground, and when they appear at the middleground, he proposed that they can be eliminated at the foreground. To resolve these inconsistencies, we found a new way to generate sequences, one that focused less on the outer voice counterpoint and more on the stepwise motion of the upper voices. This strategy conformed very nicely with Schenker’s own discussion of combined linear progressions in par. 224–26 of Der freie Satz. Besides eliminating an important inconsistency in Schenkerian theory, this solution has several important consequences. For one thing, it suggests that, although certain aspects of tonal motion are controlled by the outer voice counterpoint, others can be understood only in terms of the inner voices. When graphing a particular piece, the analyst should not simply trace the motion of the soprano and bass voices; he or she should also monitor the behavior of the tenor and alto voices. This point ties in with our discussion of Ursätze in chapter 2. For another, our discussion has shown that harmonic function is intimately connected to voice leading. In particular, we found that bass motion by fifth inevitably arises when the upper voices move by step in a single direction by parallel thirds or sixths. This observation fits in nicely with our discussion of ‘The Complementarity Principle’ in chapter 1. In the same vein, we

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have also noted that sequences and pedals are actually related phenomena. This point is interesting for a couple of reasons. On the one hand, it provides further justification for  8- and  5-line Ursätze; instead of thinking of them as containing “unsupported stretches,” we can think of them as descending across a pedal. On the other hand, by showing connections between pedals and sequences we can explain why both often appear in the same piece. As we saw in our analysis of Bach’s Little Prelude in C, BWV 924, the underlying counterpoint of both can, in fact, be very close indeed. Another important consequence of this solution is that it underscores a fundamental methodological difference between Schenker’s concerns and those of many other music theorists. As mentioned in the Introduction, there are important differences between describing what happens in a piece of music and explaining why these things happen or how to make them happen. While many music theorists are concerned with describing music in “bottom-up” terms as a string of surface events, Schenker was intent on explaining how these surfaces are generated “top-down” from tonal prototypes. This dramatic shift in perspective does not mean that conventional descriptions of sequences are necessarily wrong or that Schenker himself never took time to describe surface events. Nothing could be further from the truth. All empiric inquiries must start from careful descriptions of phenomena and, like any good empiricist, Schenker often provided the reader with extremely vivid descriptions of how a piece sounds. For example, in his analysis of Bach’s Little Prelude No. 1 in C Major, BWV 924, he described the music in narrative terms, suggesting that Bach wanted “to spin a tale” and create suspense by “exquisite tensions and convolutions . . . an insatiable desire for first-rate suspense and intricacy.”34 Yet, Schenker’s emphasis on “top-down” processes underscores that there is more to understanding music than describing its local effects; describing pieces and deriving them are simply not the same thing. Derivation requires something more. But why should “top-down” derivations mean more to Schenker than “bottom-up” descriptions? Who, in fact, needs to understand the significance of global prototypes? To answer these questions, it is helpful to take Schenker’s narrative metaphor a bit further.

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Imagine, for a moment, that we have just seen a “whodunnit” at the local theater. The production involves at least three different groups of people: the author of the play, the performers, and the audience. Of these, it is clear that the author must have some “topdown” sense of the play; if not, then, it is unclear how he/she could reveal the various clues in the right order so that the final revelation is convincing. Equally, the audience must not know who dunnit, at least on first viewing. Such knowledge would deprive them of the thrill of deducing that Professor Plum bludgeoned Miss Scarlet with the candlestick in the library. Their response will initially be governed by “bottom-up” concerns. The actors, however, seem to stand somewhere in between. To understand a character’s motivations and personality, the actors must know who perpetrated the crime and yet they must not give the game away to the audience. The ability to know something without betraying this knowledge would appear to be the essence of acting. By the same token, it is composers rather than listeners who must know in advance the global structure of pieces, and it is the task of the performer to mediate between these two groups. Schenker made his point quite clear on several occasions. In his essay “Forsetzung der Urlinie-Betrachtung: I,” from Das Meisterwerk in der Musik 1 (1925), he noted that “the composer’s business is the composingout of a chord; this task leads him from a background Ursatz through prolongations and diminutions to a foreground setting.” Meanwhile, “It is up to the reader or player, conversely, to retrace the path from foreground to the background.”35 Schenker reinforced his point a few lines later. When analyzing the opening to Mozart’s Sonata in A, K. 331, he uncharacteristically placed his foreground sketch above those of the middleground: “The voiceleading strata are deliberately viewed from the perspective of the observer not from that of the composer—that is, they are presented, as an exception, from the foreground to the background.”36 The same idea was apparently behind Schenker’s claim in Der freie Satz that “the ability in which all creativity begins—the ability to compose extempore, to improvise fantasies and preludes—lies only in a feeling for the background, middleground, and foreground.”37 In other words, the tension between traditional accounts of the sequence and Schenkerian derivations stems from the fact that the

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goals of traditional music theory are ultimately quite different from those of Schenkerian theory. Whereas traditional music theory is often motivated by a desire to describe how we, as informed listeners, experience a piece as it flows from beginning to end, Schenkerian theory is more concerned with explaining how expert tonal composers tacitly understand how their music fits together locally and globally. This does not mean, of course, that listening and composing are completely unrelated activities or that informed listeners understand music in fundamentally different ways to expert composers. If there were no points of intersection, then it is hard to understand why informed listeners are able to recognize and value exceptional feats of compositional prowess. But it is important to recognize that they may be different and that such differences have enormous consequences for the music theorist.

4

Schenker and “The Myth of Scales” “My [theory] shows that the art of music is much simpler than present-day [theories] would have it appear.”1 For anyone reading Der freie Satz, this statement sticks out like a sore thumb. Schenker’s intricate analyses often make pieces look anything but simple; at times they seem even more complex than the scores themselves. And yet, there can be no doubt that Schenker meant what he said; he expressed the same sentiment in various ways on other occasions. When discussing the structure of prototypical cantus firmi in Kontrapunkt I, for example, Schenker defended the quest for simplicity on ethical grounds, claiming that “it be may be artistically ethical to match a simple situation only with a simple beginning—that is, a beginning with a tonic—instead of contradicting its simplicity with a complicated beginning.”2 Similarly, at the end of his essay “Eläuterungen,” he announced: The art of geniuses is as simple as the simplest passing-tone progression; but for precisely that reason it is for ever inimitable and unattainable to nongeniuses. The latter lack a bond that is for them unfathomably God-derived, the bond which connects back to ultimate simplicity; consequently they go in a thousand different directions, always anxious because they have no sense of origin.3

In the same vein, he noted, “[W]here art is concerned, only geniuses pertain, for they bring to bear the utmost economy in feeling and creative activity.”4 Just as Schenker saw simplicity as a hallmark of musical genius, so philosophers also see it as a guiding principle of theory construction. Indeed, it was regarded as a basic epistemic value, long before William of Ockham supposedly authored the famous dictum Entia non multiplicanda sunt praeter necessitatem.5 There are times, however,

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when simplicity is no longer a virtue: as the philosopher Paul Churchland notes, “simply to hold fewer beliefs from a given set is . . . to be less adventurous, but it is not necessarily to be applauded.”6 Churchland adds that the desire for simplicity may even end up being counterproductive, in which case the result is “perversity not parsimony!”7 Nevertheless, before we dismiss Schenker’s claims as mere wishful thinking, it is worth considering one specific situation in which the desire for parsimony was uppermost in Schenker’s mind: this is his treatment of modes and scales. Modes and scales have, of course, played a pivotal role in shaping our notions of music theory. Over the centuries, music theorists have expended considerable energy discussing a broad spectrum of modes and scales, ranging from the so-called Church modes to major, minor, and chromatic scales, and even a plethora of exotic scales. They have primarily been guided by the simple belief that the behavior of a particular piece is determined by intervallic properties of some source scale. In some cases, they have assumed that harmonic systems derive from scales. We will refer to this particular assumption as ‘The Myth of Scales.’ Seen from Schenker’s perspective, however, the musical soundscape looked very different. Instead of viewing the behavior of functional monotonal music in terms of diatonic scales, Schenker envisioned it in terms of prototypes and recursive transformations. His picture of the tonal universe was revolutionary because it proposes a single system of prototypes, transformations, and transformational rules, rather than myriad interacting scale-based systems. Through the concepts of mixture and tonicization, he was able to explain not only how tonal surfaces can be highly chromatic, but also how they can contain a wide variety of modal and exotic inflections. Conceptually, this new system represents a dramatic step forward in theoretical simplification. For convenience, we will address the issue of modality and exoticism in five stages. To begin with, we will discuss the role modes and scales play in traditional theory; we will see just how strongly entrenched ‘The Myth of Scales’ is in the theoretical community. Next, we will consider Schenker’s general discussion of scales and his case against ‘The Myth of Scales.’ We will then use Schenkerian theory to help us understand the behavior of

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Common-Practice works that have a strong Lydian or Dorian flavor. Next, we will analyze certain exotic passages from a Schenkerian perspective—these passages include pentatonic, whole-tone, and octatonic material. Finally, we will see how Schenkerian theory might shed light on the emergence of functional tonality in the fifteenth and sixteenth centuries. Once again, we will see that Schenker’s work provides a simplified solution to some familiar problems in the history of music theory.

Modes and Scales in Traditional Theory At least since Classical Antiquity, music theorists have tried to explain the behavior of melodies by appealing to some concept of mode or scale. Ancient Greek music theory developed a system of eight modes, a corrupt version of which was transmitted into Western music theory during the Middle Ages in order to classify plainchants. Medieval theorists adopted this system for several reasons.8 For one thing, performers needed an effective way to proceed from one chant to another; this was of paramount importance in singing antiphons and psalms. For another, performers needed a way to learn and teach the enormous repertory of chants that appear throughout the liturgical year; grouping them into families provided one way to do this. For some theorists, the impetus may have been purely academic—to widen the frontiers of knowledge. Whatever their motivation, Medieval theorists generally used four criteria to explain such classification schemes: final; range or ambitus; species of fourth, fifth, and octave; and reciting tone. Although these explanations worked for many chants, they were not always completely accurate; in some cases, chants were even rewritten to fit with theoretical norms; in others, recalcitrant chants were ignored completely. Theorists sometimes compensated for other anomalies by discussing irregular or imperfect modes. By the turn of the sixteenth century, however, theorists ran into problems when they tried to extend their system of modal classification to polyphonic music. At first, they classified pieces according to the mode of the tenor; but as they tried to be more specific, the difficulties soon multiplied. Some of them stemmed from the fact that individual strands of the polyphony had different finals

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and different ranges, whereas others stemmed from the fact that each line was constrained by a growing sense of triadicity. Even today, the task of explaining mode in polyphony remains one of the thorniest problems in contemporary music scholarship. The problems of applying melodic categories to harmonic systems multiply when we try to use major and minor scales to explain tonal music written from the Common-Practice Period. Certainly, there has been no shortage of attempts to do so. Many music theory textbooks claim that the properties of functional triadic tonality derive from those of diatonic scales. According to William J. Mitchell, for example, “[T]he major and minor scales, which form the basis of this study of harmony, are diatonic.”9 Similarly, Edward Aldwell and Carl Schachter claim that “[f]rom the time of the ancient Greeks through the nineteenth century, most Western art music was based on diatonic scales.”10 Scholarly publications have likewise promoted this point of view. For example, Pieter van den Toorn has remarked: Tonality is viewed here in its more restricted sense as a hierarchic system of pitch relations based on the diatonic major scale (the C scale) . . . the historical development of which can be traced from the beginning of the seventeenth century to the end of the nineteenth.11

Or, to quote Richard Taruskin: Just as we get our sense of Mozart’s C major not only from his use of the “C scale on C” but also from the way the “black keys” are related hierarchically to the tones of the scale, so, if we are able to conceive of the octatonic collection as a tonality, we must be able to account for the use of the “other” four tones in relation to it.12

Taruskin adds that, in Mozart’s case, “even the simplest minuet or sonatina movement will contain tones foreign to the C scale that defines its key.”13 But what does Taruskin really mean when he says that Mozart’s C major is referable to the “C scale on C”? How does this scale define the key of C major? How are the chromatic notes in Mozart’s simplest minuet or sonatina related hierarchically to the members of this scale? Are these “wrong” notes really intrusions of some other scale type? Do the properties of the tonal system really depend on those of major and minor scales?

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As it happens, there are good reasons to be cautious about answering these questions. Even on an intuitive level, we know that scale membership is neither necessary nor sufficient for determining the tonality of a passage or piece (see figure 4.1, Scale membership and tonality). It is easy, for example, to imagine how a key might be defined by progressions that do not contain every note of the relevant scale. Take, for example, the progression I–V–I in figure 4.1a; this passage clearly lacks two members of the C-major scale—4
and 6
. Similarly, the mere presence of a given scale need not guarantee that a passage is “in” the corresponding key. As shown in figure 4.1b, a passage built exclusively from the notes of a C-major scale might actually be in A Aeolian, D Dorian, E Phrygian, and so forth. To complicate matters further, we can also establish a tonality using progressions built from pitches outside the diatonic collection. Figures 4.1c and 4.1d give some simple cases in point. Figure 4.1c shows a short progression in D minor. Here, the opening chords move from I to VI6 and back via VIIo7 to I5, while the final cadence tonicizes D by the familiar succession VII6–I. The progression contains several interesting chromaticisms. The B (m. 1) is a simple mixture. The A (m. 3) seems to imply a local tonicization of C (VII), though the leading tone is immediately raised to tonicize D. Meanwhile, in figure 4.1d, we find a progression from the tonic of E to the dominant of F. This motion is achieved by a common-tone progression onto a diminished-seventh sonority in m. 2. The chromatic tones D and F in this sonority are both mixtures. The B(⫽A) and D in m. 2 then appear as tonicizations; they inflect the subsequent V$5 of F in m. 3. These last two examples show very clearly that tonality does not simply depend on the presence of the “right” notes, but rather on the fact that particular notes appear in the “right” order according to some general laws of tonal voice leading and harmony. Figures 4.1c and d are particularly telling in this regard; although both passages are tonal, neither is referable to the appropriate diatonic scale. Figure 4.1c is not built exclusively from the notes of a D-minor scale and figure 4.1d is not derivable from an E-major scale. In both cases, the passages are built exclusively from the notes of the octatonic scale shown in figure 4.1e.

145 Figure 4.1. Scale membership and tonality.

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Schenkerian Theory and Scales One indication that Schenker was suspicious about ‘The Myth of Scales’ is that he seldom, if ever, discussed scales explicitly. Yet, through a number of passing comments, he left us in no doubt about his views. Among the most colorful of these appears at the start of Kontrapunkt 1, in a general discussion of prototypical cantus firmi. Although Schenker’s immediate goal was to discuss the essential behavior of tonal melodies, he took the opportunity to explain why a simple appeal to scale type provides an inadequate explanation.14 If we ignore his condescending attitudes to music written before Bach and to music theorists before him, Schenker’s objection to ‘The Myth of Scales’ is clear enough: since scales can at best describe only purely linear relationships, they are incapable of explaining how voice leading and harmony interact in functional monotonal contexts.15 In other words, scales may describe what pitches are present in a given context, they do not explain why these pitches are related in some ways and not others. Whatever value they may have as “descriptive tools” for classifying melodic patterns, scales have little power to explain the behavior of specific notes or chords.16 To support these bold claims, Schenker began his onslaught against scales with a brief synopsis of modal theory. According to him, the church modes were “modest efforts to categorize horizontally conceived melodies” whose purpose was “simply to capture theoretically the beginning and end of a given melody as well as other relationships in the course of the horizontal line.”17 He added, “Earlier periods further divided, for similar descriptive purposes, all modes into authentic and plagal or so-called perfect, imperfect, and mixed modes just to catalog and categorize the various melodic phenomena.”18 Schenker insisted, however, that once theorists began to recognize that melodic lines behave differently in polyphonic contexts, then it became necessary for them to consider the vertical as well as the horizontal dimension of music. As he put it: Consequently they need no longer limit themselves to providing a highly detailed horizontal description; rather, by the application of harmonic criteria (even to the horizontal line—compare Harmony, par. 76), therefore precisely by virtue of their deeper penetration, they are able to reveal all the more accurately the true inner core of the melody.19

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Schenker made similar arguments about attempts to explain exotic harmonies by appealing to exotic scales: “Even though the intentions in all these cases seem to be so diverse, the error is one and the same, since neither the so-called church modes nor exotic scales should be considered real systems.”20 To be considered a ‘real system’ in any explanatory sense, Schenker insisted that the system must not only list what pitches appear, it must also explain how they behave. This inevitably requires formulating explicit laws of voice leading and harmony, and rejecting ‘The Myth of Scales.’ If Schenker thought that scales do not represent ‘real systems’ in any explanatory sense, how did he explain the diatonic nature of the tonal system? How did he get rid of any primitive notion of scale membership? Quite simply, Schenker believed that the diatonic properties of functional monotonal music arise from composing out triads. To quote from Der freie Satz: The series of tones thus created in the Urlinie, represents diatony (Diatonie). In the narrowest sense, Diatony belongs only to the Urlinie. But, in accord with its origin, it simultaneously governs the whole contrapuntal setting [of the Ursatz], including the bass arpeggiation and the passing tones . . .. Diatony therefore does not stem from the so-called Greek or Gregorian modes, but rather from the composing-out process, which is governed by the principle of the fifth.21

In other words, diatonic scales are not the basis of functional monotonality; rather they result from composing out triadic prototypes. But Schenker went further. Since he believed that tonal surfaces could be fully chromatic, Schenker proposed that modal and exotic effects could be produced by mixture and tonicization. For example, in the Harmonielehre, he included two charts that demonstrate how various modal inflections can arise from varying degrees of mixture (see figure 4.2, Schenker’s account of mixture).22 Figure 4.2a shows the major system, with its natural third, sixth, and seventh degrees, on the top staff, whereas the minor system, with its lowered third, sixth, and seventh degrees, appears on the bottom staff. Between these two extremes, there are six rows, each one corresponding to the six possible combinations of natural and lowered degrees. In Rows 1, 2 and 3, the lowered third, sixth, and seventh degrees appear individually: Row 1 illustrates the ascending melodic

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Figure 4.2. Schenker’s account of mixture. a. Schenker, Harmonielehre, par. 41, p. 110.

minor scale and Row 3 Mixolydian mode. In Rows 4, 5, and 6, the various pairs of lowered degrees occur: Row 4 illustrates the harmonic minor scale and Row 5 Dorian mode. The arrows along the right hand side of figure 4.2a indicate that mixture can occur in varying degrees.23 Figure 4.2b lists the tonic, subdominant, and dominant triads for each row.24 Since simple mixture cannot account for all modal inflections, Schenker was forced to invoke tonicization as well. For example, to explain so-called Lydian mode, Schenker suggested that 
4 arises to tonicize
5; similarly, he suggested that Phrygian  II arises from a mixture within the minor system. Although he did not discuss Locrian mode, he presumably derived it from mixtures of 
2, 
3, 
5, 
6, and 
7. Once Schenker realized that he could explain a wide range of different modal inflections using mixture and tonicization, he was

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Figure 4.2 (continued). b. Schenker, Harmonielehre, par. 48, p. 117.

able to reject the notion that the modes were systems equivalent to tonality. To quote him: In dropping the Dorian, Phrygian, Lydian, and Mixolydian “rows” we have apparently reduced the number of possible relationships into which each tone could enter, to the detriment of its vitality and egoism. However, this

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Explaining Tonality loss is merely apparent only. The tone itself bravely stood its ground, and it seems it was the tone itself that forced the artist to leave the door ajar for relationships of a Mixolydian, Dorian, etc. character, even when the artist no longer believed in the validity of those systems.25

As he explained, “If the so-called Dorian and Mixolydian qualities are established by the major and minor systems alone (these understood correctly, of course), then why should we burden ourselves with still more independent systems?”26 Schenker made similar claims about the origins of exotic inflections in functional monotonal contexts. At the start of Kontrapunkt 1, he outlined the traditional view of exotic scales: Finally, a parallel exists in that the Orientals, exactly like our ancestors— and this is proof enough!—submit to the puerile preoccupation with scale systems they commit to paper simply by following the horizontal direction of the melodies. They assume, for example, a pentatonic system consisting of five degrees: C D E. G A.C or C D E. G A.C; or a heptatonic system consisting of seven degrees: F G A B C D E F (Chinese), D E F G A B C D (Japanese), C D E F G A B C (Gypsy), F G A B C D F (Chinese wholetone scale), or C D E F G A B C (Indian), and so forth. In addition, all these so-called systems, like our old church modes, can begin with any of its tones, whereby the number of systems is increased to monstrous proportions.27

But once again, he suggested that these phenomena stem not from alternative systems, but from transformations within the tonal system: Skillful artists, still, have always successfully limited the problem of musical exoticism in practice. They solved it by attempting to make the original melodies of foreign peoples (often original only because of their imperfections and awkwardness) accessible to us through the refinements of our two tonal systems. They expressed the foreign character in our major and minor— such superiority in our art, such flexibility in our systems!28

In short, Schenker did not deny that Common-Practice composers wrote music that sometimes sounds modal or exotic; rather, he denied that these inflections can be explained merely by invoking various independent scale systems. In place of this plethora of scale systems, Schenker drew on just two processes: mixture and tonicization. This

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step represents a remarkable simplification of theoretical concepts. To see the consequences of Schenker’s arguments, let us now see how he analyzed some specific pieces with clear modal inflections.

Schenkerian Theory and Modal Inflections Schenker offered his most detailed account of modal inflections at the opening of the Harmonielehre in two chapters entitled “Die übrigen Systeme (Kirchentonarten)” and “Mischungen.”29 Here he used the notions of mixtures and tonicization to analyze various modal passages from works of the Common-Practice Period. One extract that he discussed in particular detail is given in figure 4.3 (Beethoven, “Heiliger Dankgesang,” String Quartet in A Minor, Op. 132).30 Beethoven apparently completed this piece after recovering from a serious illness in 1825. He celebrated his recovery by inscribing the score with the phrase “Holy Song of Thanksgiving to the Divinity by a Convalescent, in the Lydian Mode.”31 As if to underscore the highly personal nature of the movement, Beethoven even marked the contrasting D-major sections “Feeling new strength” and the final section “With most intimate feeling.” Although much has been written about this remarkable movement, most commentators take for granted that the opening section is indeed in Lydian mode. The signs are clear enough. As Schenker himself pointed out, there is the chorale-like feel of the half notes, the strong tendency to use triads in root position, the clear avoidance of any extreme chromaticisms, and perhaps most significantly of all, the consistent preference for B rather than B.32 Given these factors and Beethoven’s own title, the case for a Lydian interpretation seems almost overwhelming. Yet, while Schenker certainly acknowledged the strong modal tendencies of the “Heiliger Dankgesang,” he insisted that these traits stem not from the Lydian system per se, but from chromaticisms within the overall key of F major: It could be objected that the two B’s in measures 5 and 23 (first and fourth part) are incompatible with F major and can be explained only if we presuppose the Lydian system as [the] basic key. This objection can be countered: The two B’s, as they appear here, are in no way incompatible with our F

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Figure 4.3. Beethoven, “Heiliger Dankegesang,” String Quartet, Op. 132. From Schenker, Harmonielehre, Ex. 47.

major. They result from a trivial chromatic trick, which we use everyday and on only slight occasion to emphasize the cadence and to underline the F major character of the composition.33

In Schenker’s opinion, “[I]t is true that the composer’s intention to avoid the B-flat is particularly noticeable—an intention which, in art, unfailingly entails punishment; it is not true, however, that, in accordance with that intention, the Lydian mode is presented convincingly.”34 Schenker’s own analysis divides the opening section into five parts. The first ends with a deceptive cadence in F major (mm. 5–6);

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Figure 4.4. Graph of Beethoven, “Heiliger Dankegesang,” String Quartet, Op. 132.

the second, modulates to C major and ends with an imperfect authentic cadence (mm. 11–12); the third is in C major and closes with a half cadence (mm. 17–18); the fourth leads back from C major to a perfect authentic cadence in F major (mm. 23–24); and the fifth modulates to D major and ends with a half cadence (mm. 29–30).35 The ramifications of this analysis are shown in figure 4.4 (Graph of Beethoven, “Heiliger Dankgesang,” String Quartet, Op. 132). It suggests that the passage is built from a distinctive prototype I–I6–V/V–V–I or VI. As shown in figure 4.4a, this prototype controls the opening phrase of the movement. Notice how the progression from I–I6 is mirrored by a descent from A to F in the upper voice. Figure 4.4b then shows how this same pattern governs the opening section as a whole. The local motion to V in mm. 4–5 is now projected as the larger tonicization of C in mm. 11–12. Beethoven eventually returns to F via the I6 in m. 20. This sonority leads to the

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prominent cadential progression V/V–V–I in mm. 23–24. Beethoven now presents the rising inner line C–D–E–F in the alto rather than the tenor voice. Despite the obvious modal tendencies of the surface, there are several clues that the music does not conform to the principles of sixteenth-century polyphony. For example, the descending fifths A–D–G in m. 7 of the second violin part are unlikely in Palestrina’s style, as are the successions C–F–G–B–C in mm. 11–12 and C–A–C–A–G–F–B–C in mm. 25–28. As it happens, the reading in figure 4.4 stands in direct contrast to the one presented in a paper by Kevin Korsyn. Besides many differences in detail, the most striking divergence between the two analyses is that the former is in F major, whereas the latter is interpreted in F Lydian. Although Korsyn’s sketch purports to be Schenkerian in nature, it is very hard to reconcile with Schenker’s own interpretation.36 For Schenker, the power of the tonal system was ultimately much greater than that of the Lydian system: “[Beethoven] had no idea that behind his back there stood that higher force of Nature [that] led his pen, forcing his composition into F major while he himself was sure he was composing in the Lydian mode, merely because that was his conscious will and intention.”37 He added, “Is that not marvelous? And yet it is so?” Before leaving Schenker’s explanation of Lydian inflections, it is worth noting that he used much the same arguments in his analysis of Chopin’s Mazurka Op. 24, no. 2 in Kontrapunkt 1. According to him: With this passage, however, Chopin by no means intends to establish the old [Lydian] system as equivalent (to major and minor) and as independent; this is sufficiently clear from the refined artistry he uses in the introduction as well as the harmonization in general to provide the listener with the absolute certainty of only C major and F major.38

Schenker concluded, “Thus, the passage in question simply contains a few features of artist archaism, a highly ingenious trick, such as could befall Chopin occasionally in the midst of his fantastic improvisations.”39 Whereas Schenker drew on the concept of tonicization to explain the Lydian features of Beethoven’s “Heiliger Dankgesang,” he invoked the notion of mixture to explain the Dorian qualities of the music shown in figure 4.5 (Brahms’s song “Vergangen ist mir

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Figure 4.5. Brahms, “Vergangen ist mir Gluck und Heil,” Op. 14, no. 6. From Schenker, Harmonielehre, Ex. 50.

Glück und Heil,” Op. 48, no. 6). These qualities are not hard to spot. The song is written in four-part chorale style and is clearly centered on the tonic D. Except for a single B triad in m. 23, Brahms consistently favors the pitch B to its diatonic counterpart. To emphasize the modal qualities even further, Brahms frequently uses the lowered leading tone, C. Nonetheless, Schenker insisted that these modal inflections stem not from the Dorian system, but from mixtures within the key of D minor. His analysis is worth quoting at length: The artist here clearly aims at writing in Dorian mode on D. This results from the mere fact that he omitted the key signature B in a composition really written in D minor. Brahms, too, guided by his desire to compose in

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Explaining Tonality Dorian mode (just like Beethoven, in the previous example, aiming at the Lydian) strictly avoids any B with one single exception in the second-to-last measure.40

He continued: And yet I insist: None of the B’s occurring in this beautiful chorale is to be derived, as Brahms believed, from the Dorian scale as such; we must substitute, rather the following explanations. The first bars constitute, basically, the A minor scale; hence the B is justified merely in consideration of that key. It is true that, with the C of m. 2, the composition changes to D minor. If in this D minor the IV Stufe is presented with the third B natural rather than with the diatonic third B, the idea of D minor remains nevertheless alive in the listener. More than that, we recognize here the very B natural which we employ in our daily practice in D major/minor (cf. par. 38ff.) and, to boot, in this same sequence IV3–V3, without sacrificing in any way the identity of the D minor! That Brahms abstains from using the B in the subsequent development (mm. 10–13) is simply explained by the motion that the composition is taking toward C major.41

Schenker concluded, “This example, too, demonstrates how music itself holds on to the minor mode even where the artist’s intention aims at the Dorian system.”42 As it stands, Schenker’s analysis leaves a lot to be desired. For one thing, the opening phrase does not really center on A minor; on the contrary, it clearly moves to a half cadence in D minor. For another, mm. 10–13 may indeed move towards C major, but this motion is temporary to say the least. In fact, the goal of the progression is the cadence in D at m. 16. An alternative interpretation is given in figure 4.6 (Graph of Brahms, “Vergangen ist mir Glück und Heil,” Op. 48, no. 6). The most obvious feature of this piece is that it repeats two basic chunks of material: mm. 1–6 are repeated as mm. 6–11, and mm. 11–16 are repeated as mm. 17–22. The piece ends with a single three-bar phrase. The tonal structure of mm. 1–6 (6–11) is particularly intriguing and provides clues to the layout of the piece as a whole. The passage consists of two distinct phrases: mm. 1–3 move from the tonic chord D (m. 1) to a half cadence on A (m. 3); mm. 4–6 then move back from the dominant (m. 4) to a perfect authentic cadence on the tonic D (m. 6). From a contrapuntal perspective, the motion to the dominant in mm. 2–3 (7–8) is quite distinctive; it is marked by a clear melodic motion

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Figure 4.6. Graph of Brahms, “Vergangen ist mir Glück und Heil,” Op. 14, no. 6.

A–C–B–A–G–F–E over a bass motion A–E–F–C–D–A. The descending trajectory of this figure culminates in the motion from E to D at the cadence in mm. 5–6 (10–11). Seen in this context, the B’s in mm. 2 (7), 4 (9), and 5 (10) all arise as simple mixtures, while the C’s in mm. 3 (8), 4 (9), and 5 (10) serve to tonicize D. If we compare the counterpoint of the first chunk with that of the second, we see that the former is a variant of the latter. The second chunk begins with a sequential motion from a triad on A to a triad on C in mm. 11–14 (17–20). This motion is answered by a progression E–F–C–D–A–D that recalls the progression in mm. 2–3 (7–8). This time, however, the soprano motion B–A–G–F–E is buried in the inner voices: B–A appear in the alto, G–F–E occur in the tenor. The new soprano follows the alto line of mm. 2–3 (7–8). Once again, the B’s in mm. 12 (17), 13 (18), 15 (20) arise as mixtures and the C in m. 16 (21) serves to tonicize D. The closing three-bar phrase (mm. 22–24) then mirrors the end of the first chunk: the final cadential descent is almost identical in both cases, A–G–E–D. In both cases the Dorian traits arise from the process of simple mixture. It should be clear from the preceding analyses of the “Heiliger Dankgesang” and “Vergangen ist mir Glück und Heil” that in Schenker’s mind, Beethoven and Brahms were able to simulate modal music using the resources of the tonal system. Instead of invoking separate Lydian or Dorian systems, he explained these inflections through the concepts of mixture and tonicization. Schenker drew on these same ideas in his discussions of other modal pieces. In Kontrapunkt 1, for example, he claimed that,

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although the chorale melody “Gelobet seist du Jesu Christ” might appear to have a Mixolydian quality, Bach’s settings are “obviously grounded in G major.”43 And, in his discussion of Hassler’s melody “O Sacred Head Sore Wounded” in Der freie Satz, Schenker mentioned that Bach’s settings of the tune paid “only an outward tribute to the Phrygian System that people of the time believed in.” Instead he regarded the melody as firmly rooted in the major mode.44

Schenkerian Theory and Exotic Inflections So far we have seen how Schenker used the concepts of mixture and tonicization to explain modal inflections in functional monotonal pieces from the Common-Practice Period. But what about exotic inflections? How did he explain their appearance in CommonPractice composition? As mentioned earlier, it seems that Schenker conceived of exotic inflections in much the same way that he treated modal inflections. This suggests that they, too, can be explained by mixtures and tonicizations. Once again, Schenker provided his most suggestive account of the matter in Kontrapunkt I: Think, for example, of Haydn’s and Beethoven’s Schottische Lieder, Schubert’s unique Divertissement à l’hongroise, the Hungarian Dances by Brahms, the Slavonic Dances by Dvorák and the Norwegian Dances by Grieg, as well as Scheherazade by Rimsky-Korsakov, among others. The point in all these cases was not to loosen our system in order to incorporate a foreign one, but, on the contrary, to use our major and minor systems to express the foreign element, which does justice in a certain sense to a primeval state of music but needs to be adjusted in some way to suit the needs of a more advanced art.45 ˆ

Unfortunately, Schenker did not support his assertion with extensive analyses of these pieces; of the works listed above, he offered only a brief sketch of mm. 1–15 of Schubert’s Divertissement à l’hongroise, Op. 54, in Der freie Satz (Fig. 89.2). But elsewhere Schenker left us with more concrete clues about how we might derive specific exotic effects. For example, in a brief discussion of Chopin’s “Black-Key” Etude in G, Op. 10, no. 5, he gave a hint at how he might explain the work’s apparently pentatonic surface (see figure 4.7, Chopin, Etude, Op. 10, no. 5). Having taken exception to Leichentritt’s scalar explanation, Schenker

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Figure 4.7. Chopin, Etude, Op. 10, no. 5. From Schenker, The Masterwork in Music 1, p. 92.

claimed that the melody derives from orthodox tonal transformations: The right-hand figuration is no ‘jolly tune,’ still less has it anything to do with a ‘pentatonic scale.’ The omission of C and F from the figuration is explicable solely in terms of the association of the third progression B–A–G with the neighbour-tone motive D–E–D, i.e., the association of

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Figure 4.8. Graph of Debussy, Prélude à “L’Après-midi d’un faune?” mm. 30–37.

two perfectly good diatonic motives; as soon as the other voices contribute the C and F to these, the major diatonic mode [Dur-Diatonie] is secured.46

The third progression and neighbor-tone motive are marked on the score in figure 4.7. Taking the last example a bit further, we can begin to see how Schenkerian theory might explain intrusions of whole-tone material. The notion that whole-tone material can be explained within the tonal system is not, of course, without precedent. On the contrary, writers such as Schoenberg and Tovey have suggested that they often stem from altered dominant harmonies.47 But from a Schenkerian perspective, such phenomena can be explained with much greater precision. Consider, for a moment, mm. 30–37 from Debussy’s Prélude à “L’Après-midi d’un faune.”48 This passage divides into two parts: the first contains a diminution of the famous flute theme set against the whole-tone collection B–C–D–F–G–A, whereas the second contains another statement against the other whole-tone collection C–D–E–F–G–A. Meanwhile, figure 4.8 (Graph of Debussy, Prélude à “L’Après-midi d’un faune,” mm. 30–39) suggests that these harmonies arise from a complex prolongation of a dominant Stufe.49 The upper line is created by a motion from an inner voice: F in m. 30 ascends by step through G, A, B, and C, to C in m. 37. The inner parts mirror this succession: the viola part moves up from B through C (mm. 31–33) to D and E

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(mm. 34–36); the second violin part moves up from F through G (mm. 31–33), G and A (mm. 34–36); and the cello, bass, and bassoon parts move up from C through D and D (mm. 31–33) to E, F and F (mm. 34–36). The final sonority in m. 36 (F–A–E–C) functions as an altered secondary dominant (V7/5 of V) that resolves onto V9 of E in m. 37. If we become more adventurous still, we can use Schenkerian theory to explain even more radical pieces, such as the opening of Stravinsky’s Petrouchka. This famous passage has been analyzed from many standpoints. One of the most influential is summarized here in figure 4.9 (Van den Toorn’s analysis of the opening of Stravinsky’s Petrouchka). Although he concedes that the first Tableau does not contain any blocks of explicit octatonic material, van den Toorn insists that “ ‘the chromatic’ (non-diatonic) pitch elements and intervals . . . may be heard and interpreted as referentially octatonic, as prompting a form of diatonic-octatonic interpenetration.”50 In particular, he regards the notes in the passage (see figure 4.9a) as an interaction between a D or Dorian scale transposed onto E (top line of figure 4.9b) and an octatonic scale beginning on E (bottom line of figure 4.9c). Van den Toorn apparently invokes both scales because the D scale on E lacks the crucial F and the octatonic scale has an extra B and A. According to him, this diatonic-octatonic interaction is “a matter of consequence” because it anticipates “the (more) fully committed Collection III framework of ‘The Petrouchka Chord’ in the second tableau.”51 Instead of treating this material as a by-product of interacting diatonic and octatonic systems, it can be regarded as a Schenkerian transformation of a D triad. Here, the upper line descends by step D–C–B–A and the lower parts move by neighbor progressions— A–G–A and D–E–D. Notice how Stravinsky avoids parallel fifths for the final tonic by suspending the B to create a 6–5 succession. Within this pattern, the C serves to tonicize D and the B arises as a mixture, thereby averting a diminished supertonic Stufe, and creating the all-important motivic tetrachord. What makes this alternative reading so interesting is that it is not so different from the one we gave for Brahms’s song “Vergangen ist mir Glück und Heil” in figure 4.6. In fact, the two explanations are similar because the

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Figure 4.9. Van den Toorn’s analysis of the opening of Stravinsky, Petrouchka. From Van den Toorn, The Music of Igor Stravinsky, Exx. 20 and 21.

pitch content of both passages is exactly the same, even though the two pieces sound quite different stylistically. This point is all the more ironic if we recall the two progressions given in figures 4.1c and 4.1d: not only do these two progressions have exactly the same pitchclass content, but these pitches belong to the same octatonic scale, D–E–F–G–A–B–B–C–D. Whether or not we regard these passages as genuinely octatonic is a matter for debate, but the two examples should certainly erode our blind faith in ‘The Myth of Scales.’

Schenkerian Theory and the Emergence of Functional Tonality So far, we have seen how Schenkerian theory explains modal and exotic inflections as they appear in functional monotonal contexts.

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Through mixture and tonicization, the theory can account for a wide array of such effects. Surprisingly perhaps, this claim fits in nicely with Carl Dahlhaus’s views about nineteenth-century harmonic practice: Regardless of the milieu being depicted, exoticism and folklorism almost invariably make do with the same technical devices: pentatonicism, the Dorian sixth and Mixolydian seventh, the raised second and augmented fourth, non-functional chromatic coloration, and finally bass drones, ostinatos, and pedal points as central axes.52

In other words, nineteenth-century composers created generically exotic and modal music; they did not compose music that authentically recreated music from another specific culture. Now this is an important point because it suggests that, as it stands, Schenkerian theory cannot explain the behavior of fifteenth- and sixteenth-century music adequately. Indeed, just as the laws of functional tonal voice leading and harmony must be modified in order to explain the behavior of much twentiethcentury tonal music, so they must also be altered to explain these earlier repertories. Let us now see if we can use the simple processes of adding voices and changing harmonic environments to explain the emergence of triadic composition in the fifteenth century and the shift from Prima to Seconda Prattica in the seventeenth century. Although there have been many attempts to explain the rise of tonality, one of the most provocative has been offered by Don Randel in his paper, “Emerging Triadic Tonality in the Fifteenth Century.”53 Focusing on the origins of perfect authentic cadences, he claims that “triadic” tonality came to the fore when contrapuntal textures thickened from three to four voices (see figure 4.10, Cadences in fifteenth-century music). Randel notes that there are several ways to cadence in three-voice textures—some involve step motion in the bass (figures 4.10a–b), whereas others involve the leap of a fifth in the bass (figures 4.10c–e). But when a fourth voice is added, only one cadence satisfies the laws of counterpoint and that has a leap of a fifth in the bass (figure 4.10f).54 Randel suggests that it is not anachronistic to label these last two harmonies with the Roman numerals V and I.55

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Figure 4.10. Cadences in fifteenth-century music. From Randel, “Emerging Triadic Tonality in the Fifteenth Century,” Exx. 1a–e and Ex. 2.

Obviously, Randel’s general observation about the addition of a voice is perfectly in sync with the arguments presented in chapter 1. Indeed, his paper was an important stimulus for this discussion. But it is important to stress that “triadic tonality” is not quite the same thing as “functional tonality”; on the contrary, the latter not only requires that triads are the norm, but it also requires that these triads are related functionally according to the system of Stufen

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established for the major-minor system in figure 1.11. This amounts to shifting from the ‘The Triadic Constraint’ to ‘The Stufe Constraint.’ Now, this change in harmonic environment has several implications for Randel’s argument. For one thing, it prevents us from classifying the progression in figure 4.12f as tonal in a functional sense; in such contexts, the alto line normally lands on
3, not
5 in the final chord. For another, ‘The Stufe Constraint’ excludes progressions like I–II–I6 from occurring in tonal contexts; such chord successions violate the functionality of Common-Practice music. Since ‘The Stufe Constraint’ is more restrictive than ‘The Triadic Constraint,’ it seems that there is more to functionality than the simple addition of voices; other constraints are required, and these require subtle changes in the laws of counterpoint. To cite an obvious case in point, since we know that sequences are far more common in tonal music than in modal music and since we saw in chapter 3 that they can be derived from strings of parallel thirds or sixths that ascend or descend by step in a single direction, we can surmise that functional tonality emerged as composers began to experiment with these new contrapunttal configurations. In other words, although Randel may be right that some aspects of functional tonality began to emerge even before 1500, it is not clear that all aspects were present. Certainly we can frequently find cadential progressions like those shown in figure 4.10f, but it does not follow that they operate within the larger array given in figure 1.11. Until we are sure that they do, we should be cautious about using Roman numeral analysis for this repertory. Using the same arguments, we can also clarify the meaning of Tinctoris’s term res facta. Although this term has been the subject of considerable controversy, Bonnie Blackburn has given a particularly careful reading of Tinctoris’s texts.56 She suggests that Tinctoris specifically associated the term “counterpoint” with twovoice textures either in note-against-note or florid style.57 According to her: [R]es facta differs from counterpoint in that it consists of “three, four, or more parts” . . . and these parts are mutually bound to each other according to the “law and order of concords,” that is each part must follow the rules of counterpoint with respect to each other part, which is not true of counterpoint, in which the added voice or voices need only be consonant with the tenor.58

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Blackburn concludes that the differences are matters of compositional procedure; “counterpoint is successive” whereas “res facta may be composed simultaneously or successively.”59 Nevertheless, when Tinctoris claimed that res facta has “three, four, or more parts” that are “mutually bound to each other,” he may have implied that res facta is controlled, not by ‘The Consonance Constraint,’ but by ‘The Triadic Constraint.’ As we saw in chapter 1, this condition allows us to explain why the interval of a fourth can behave as a consonance or as a dissonance. Tinctoris made this point perfectly clear in Liber de arte contrapuncti.60 For him, the distinction between res facta and counterpoint seems to hinge on changing from one harmonic environment to another, rather than on changing from successive to simultaneous composition. This suggests that Tinctoris was already aware that the behavior of contrapuntal lines changes according to the number of lines and the harmonic environment in which they appear. Adding voices and changing harmonic environments have an impact on the transition from Prima to Seconda Prattica around 1600. When we think about this topic, it is important to remember that Fuxian species counterpoint is an abstraction from actual musical practice; this is as true of sixteenth-century modal polyphony as it is of eighteenth-century functional tonality. One important difference is that whereas Fuxian counterpoint confines the preexistent material to a single voice, the cantus firmus, sixteenth-century modal polyphony usually spreads the preexistent material imitatively throughout the entire texture. But other differences arise because such music seems to conform to some general principles of modal harmony. One writer who was fully aware of this point was Knud Jeppesen; when he adapted Fuxian principles to cover Palestrina’s style he included brief discussions of modal harmony.61 Other scholars agree: for example, as Andrew Haigh has shown, “each mode had its characteristic pattern of distribution of harmonies and cadences, which differed from all the rest.”62 Although Haigh does not present a picture of modal harmony analogous to the one given for tonal harmony in figure 1.11, we know that the two pictures would differ in some important respects. According to ‘The Stufe Constraint,’ for example, the behavior of each Stufe is the same in each key; the only difference between one key and another is that

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the individual Stufen are transposed. But, in modal contexts the behavior of Stufen does change from one mode to another. For example, whereas diminished triads appear only on VII in major keys and II in minor keys, they appear on VI in Dorian, V in Phrygian, IV in Lydian, and III in Mixolydian. Furthermore, it is unclear that modal harmony requires the complex system of chromatic operations described in chapters 1 and 2. John Clough has even suggested that there are subtle changes in the behavior of chromaticisms from the Renaissance to the Baroque.63 Once we distinguish modal and tonal harmony, we can clarify the celebrated debate between Artusi and Monteverdi. In his infamous diatribe against modern music, Artusi complained about nine extracts from Monteverdi’s madrigals “Cruda Amarilli” (Book 5, 1605) and “Anima mia perdona” (Book 4, 1603). These passages apparently violate established laws of counterpoint, thereby bringing “confusion and imperfection of no little consequence.”64 Monteverdi, however, defended his actions on the grounds that he followed a new way thinking, so-called Seconda Prattica, rather than the established laws of Prima Prattica.65 Such deviations were motivated by a desire to make “the words the mistress of the harmony.”66 However, we can offer a slightly different explanation: Monteverdi did not follow the precepts of Prima Prattica because he was interested in exploring an environment that was richer in its harmonic functions. In fact, many of the irregular dissonances in Monteverdi’s music can be explained using the same procedures that Schenker used in chapter 1. For example, figures 4.11a–c (The Artusi-Monteverdi debate) list the first three passages cited by Artusi. Figures 4.11d–f then show how these dissonances arise either from implied register transfers or from motion between different contrapuntal voices. Furthermore, the passages show a marked tendency towards the progression I6–VII6–I and away from the string I–II–I6. In short, Artusi’s criticisms miss the mark because they interpret Monteverdi’s music within the wrong harmonic environment; as we have seen, changing from a modal to a tonal environment has an enormous effect on voice leading. The preceding discussion has touched on some of the issues that arise in formulating an appropriate set of laws of voice leading and harmony for modal music. We have seen not only that the laws

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Figure 4.11. The Artusi-Monteverdi debate. From Strunk, Source Readings in Music History, p. 395.

governing modal voice leading and harmony are different from those governing functional monotonality, but also that they are different from those governing Fuxian strict counterpoint. This suggests that strict counterpoint, modal counterpoint, and functional tonality can be connected along the lines shown in figure 4.12 (Renaissance modal polyphony and functional tonality). Figure 4.12 makes no claims about whether or not we can express the laws of modal counterpoint recursively as prototypes/transformations and whether pieces in a particular mode can be generated from specific prototypes. Recent work by Panayotis Mavromatis gives us good reason to be optimistic, but until we have pinned down the precise laws of voice leading and harmony for modal music, we

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Figure 4.12. Renaissance modal polyphony and functional tonality.

cannot hope to deal with the issue adequately.67 In the meantime, we can be content to know that Schenkerian theory has at least given us a theoretical framework within which to tackle the problem. To sum up, Schenker rejected ‘The Myth of Scales’ not because scales and modes are irrelevant to music theory, but rather because they have only limited explanatory value. Although they provide us with useful categories for classifying melodic lines, scales and modes are much less effective at explaining how melodic lines behave in functional triadic contexts. Schenker’s response to this shortcoming was simple; instead of deriving music from scales or modes, he believed that these scales arise from composing out essential harmonies. They are products, rather than primitives within the system. This does not mean that major and minor scales have no part in Schenkerian theory; among other things, their effect can still be felt in his decision to classify his essential harmonies into seven Stufen. But it does mean that they have little

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explanatory power. Although some will find this concern with parsimony quite liberating, others will find it utterly perverse. After all, scales and modes are among our most cherished theoretical concepts. And yet, recent research by music psychologists has also challenged ‘The Myth of Scales.’ Mary Louise Serafine, for example, has suggested that “scales . . . have figured disproportionately in music research, chiefly through their influence on the design and conception of studies.”68 David Huron has reached similar conclusions: according to him, “In comparison to most of the world’s music, Western music tends to be highly harmonically oriented. Where scales provide the basis for predominantly melodic music, examining the harmonic properties of these scales may be inappropriate.”69 It is remarkable that we are only just beginning to realize the full implications of Schenker’s most audacious ideas.

5

“Pleasure is the Law” Although music theorists aim to make their theories as accurate, broad, consistent, and simple as possible, they are also keen to apply them to situations or phenomena for which those theories were not originally intended. By casting the empirical net ever wider, theorists can not only test the limits of their work, but they can also open up new avenues of research. Such extensions are signs of the theory’s fruitfulness. And so it is for Schenkerians. There would seem to be two main ways in which Schenkerians can achieve these goals. First, they can apply their methods to music that lies outside Schenker’s original core sample. This is not a difficult step to take because Schenker focused his attention on pieces by a fairly narrow range of composers, from Bach to Brahms; it is quite easy to think of other composers whose music has many of the same tonal properties. Second, they can use their explanations of a work’s harmony and voice leading to illuminate other aspects of its composition. In fact, Schenker’s own analyses often provide crucial insights about a work’s thematic, rhythmic, and formal structure. This chapter tackles both of these issues by offering Schenkerian analyses of two early songs by Claude Debussy: “C’est l’extase langoureuse” from the Ariettes oubliées (1887, 1903) and “La mort des amants” from the Cinq poèmes de Charles Baudelaire (1887–89). I have chosen these pieces for two reasons. For one thing, although Schenker dismissed Debussy for delighting in “the mediocrity of French taste,” both songs have highly cultivated tonal structures.1 They clearly demonstrate Debussy’s intimate knowledge of functional monotonality. For another, by focusing on a pair of songs, we can show how Schenkerian analysis sheds light not only on Debussy’s tonal practices, but also on other aspects of his compositional technique. For convenience, the chapter has four main parts.

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Part 1 considers some of the methodological problems that arise when Debussy’s music is analyzed from a Schenkerian perspective. We will see how, in principle at least, the theory deals with parallel chords, dissonant prolongations, non-functional successions, extreme chromaticism, modal/exotic inflections, incomplete structures, and parenthetical passages/interpolations. Next, Parts 2 and 3 use these ideas to present Schenkerian analyses of Debussy’s two songs “C’est l’extase langoureuse” and “La mort des amants.” In particular, we will see how Schenkerian analysis can be fruitfully used not only to explain many anomalous features of these works, but also to provide fresh insights about their formal structure and meaning. Finally, Part 4 suggests how Schenker’s ideas might be expanded to explain some types of twentieth-century music.

The Limits of Schenkerian Theory For anyone interested in testing the limits of Schenkerian theory, Debussy’s music is an ideal subject. Although most of his works can be classified as tonal in a general sense, they often extend or contradict the specific laws of functional monotonality. As Debussy’s contemporaries were quick to point out, his works challenge almost every aspect of functional voice leading and harmony.2 In terms of its voice leading, Debussy’s music often violates traditional laws prohibiting parallel chords and free dissonances; not only does it contain strings of parallel triads, sevenths, and ninths, but it frequently treats dissonances without preparation or resolution. In terms of its harmony, his music is infused with non-functional progressions, extreme chromaticism, and modal/exotic inflections. To complicate matters further, these works often disrupt the principles of tonal closure and continuity through their use of incomplete structures and interpolations. As a result, they may behave functionally at a local level, but quite differently at a global level. Debussy was, of course, well aware of these issues; throughout his letters and journal articles, he launched a bitter campaign against the value of conventional music theory and what he regarded as its “silly obsession with overprecise ‘forms’ and ‘tonality.’ ”3 For example, in a letter to Pierre Louÿs (22 January 1895), Debussy denounced

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accepted notions of chord function, claiming that “tonic and dominant had become empty shadows of use only to stupid children.”4 Similarly, he rejected traditional distinctions between consonance and dissonance: “Nothing is more mysterious than a consonant chord! Despite all theories, both old and new, we are still not sure, first, why it is consonant, and second, why the other chords have to bear the stigma of being dissonant.”5 Elsewhere he scoffed at any prohibition against parallel sonorities and even proposed that tonality should be fully chromatic and “enriched by other scales.”6 According to him, “There is no theory. You simply have to listen. Pleasure is the law.”7 When viewed against the backdrop of nineteenth-century tonal theory, then, Debussy’s music seems to defy explanation; it is hardly surprising that contemporary theorists were able only to catalog and classify each anomaly. But for Schenkerians, the situation is rather different. Since Schenker explained similar anomalies in the music by composers of the Common-Practice Period, the main issue is to decide whether the deviant aspects of Debussy’s music are matters of degree or of kind.8 And since we cannot be sure when Schenkerian theory ceases to be applicable, we must consider each piece, case by case. With those thoughts in mind, let us now see how Schenkerian theory explains many of the anomalies found in Debussy’s music. Few aspects of Debussy’s music are more striking and more problematic than his fondness for parallel chords. Widely discussed by his contemporaries, parallel triads and sevenths appear throughout Debussy’s music, and may even extend for considerable periods of time. Debussy was not, however, the first composer to indulge in such practices; on the contrary, we can find examples of the same phenomenon in various works from the Common-Practice Period. A particularly good example comes from Chopin’s Mazurka Op. 30, no. 4 (see figure 5.1a, Parallel dominant seventh chords). From a conventional standpoint, this passage is extremely problematic because it projects a string a parallel dominant seventh chords; such strings do not normally occur in functional contexts. Yet, Schenker thought that the passage could be explained as a string of implied suspensions. His sketch from Der freie Satz, given here as figure 5.1b, derives the chain of parallel sevenths from an underlying

174 Figure 5.1. Parallel dominant seventh chords. a. Chopin, Mazurka, Op. 30 no. 4

b. Schenker, Der freie Satz, Fig. 54.6.

c. Schenker, Der freie Satz, Fig. 53.3.

e. Alternative sketch of Chopin, Mazurka, Op. 30, no. 4.

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succession 5, 4–5, 4–5.9 But this derivation does require clarification. As Schenker showed in another graph, the passage in question actually occurs within a larger progression I–V–I in C
minor (see figure 5.1c).10 The various unfolding symbols suggest that the passage arises from a motion between different polyphonic voices, especially the soprano D# and the alto F#. This idea is clarified in figure 5.1d. Here, the unfolding of the sixth D
–F
is filled out chromatically and supported by a string of parallel thirds and sixths in the manner described in chapter 3. Since these parallel sonorities are created contrapuntally by combinations of non-harmonic tones, they do not violate Schenker’s revised laws of tonal voice leading that we discussed in chapter 1. Another hallmark of Debussy’s style is his free handling of dissonances. In tonal contexts, dissonances always behave in narrowly circumscribed ways. They are generally thought to derive from seventh chords with the seventh invariably resolving down by step. Although the seventh is normally prepared in secondary sevenths, such as II7 or IV7, this is not always the case for dominant and leadingtone sevenths. Debussy, however, often treated dissonances in bold new ways: as René Lenormand and others have noted, he not only introduced many new forms of dissonance, but he also used them without preparation or resolution and even prolonged them for long periods of time. Once again, we can find precedents for these licenses in earlier repertories. Take, for example, the opening nine measures of Bach’s Prelude in A, BWV 942. As shown in figure 5.2a (Free dissonances), this remarkable passage is filled with pungent dissonances. For example, on the second beat of m. 1, the right hand leaps from the consonant tone A to the dissonant D. More remarkably, perhaps, the final beat of m. 2 contains the triplets D
–C–B in the right hand against A in the left hand. While it is hard to explain the dissonances in figure 5.2a in conventional terms, it is much easier to explain them from a Schenkerian perspective. This is because Schenker proposed that all non-harmonic tones ultimately stem from linear motion between Stufen. Through displacements, ellipses, and motion between voices, his theory can explain a much wider range of non-harmonic tones than its rivals.11 Schenker even coined the term Tonklötze to describe sonorities in which non-harmonic tones are fused with

176 Figure 5.2. Free dissonances. a. Bach, Prelude in A Minor, BWV 942.

177 Figure 5.2 (continued). a. Bach, Prelude in A Minor, BWV 942 (continued).

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Figure 5.2 (continued). b–d. Graphs of Bach, Prelude in A Minor, BWV 942.

harmonic tones.12 Added-sixth chords are a good case in point: such sonorities arise when the fifth of a triad is elaborated with its upper neighbor. Since Schenker believed that non-harmonic tones arise contrapuntally, he accepted that tonal surfaces can be almost continually dissonant, provided that all non-harmonic tones move between triadic intervals. Such motion need not, however, be

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confined to the surface of the piece; on the contrary, it can be generated at deep structural levels. Figures 5.2b to 5.2d present a slightly modified version of Schenker’s sketch of it shown in figure 5.2a.13 According to this reading, it can be derived from a simple progression in I–V/IV–IV–V7–I in A minor (see figure 5.2b). The opening tonic is then connected contrapuntally to the V/IV sonority in m. 7 by a string of passing chords (see figure 5.2c). Finally, figure 5.2d shows how the striking dissonances mentioned above are created both by moving from one contrapuntal voice to another and by displacing, eliding, and subverting the voices in various ways. Schenker had no doubt whatsoever that the surface of this prelude could be completely explained by the principles of tonal voice leading: on the contrary, at the end of his essay he claimed that his reading “demonstrates how the necessities of voice-leading prompt the imagination of a genius to find solutions that not only meet such needs but transcend them, radiating excellence and beauty in all directions.”14 Along with his use of parallel chords and free dissonances, Debussy is also famous for his fascination with non-functional successions. According to conventional wisdom, harmonic progressions are classified as non-functional if they do not follow the scheme: tonic–pre-dominant–dominant–tonic. Schenker, however, was far less rigid in his prognosis; although he accepted the functional priority of I and V, he believed that the behavior of other Stufen depends primarily on the necessities of voice leading. This approach proves especially useful for explaining the behavior of non-functional successions. Take, for example, mm. 65–87 from the first movement of Beethoven’s “Appassionata” Sonata (see figure 5.3, Non-functional successions).15 Although the bass projects as a non-functional string of major thirds (A–E–C–A), Schenker suggested that this string is generated from the upper voice counterpoint and not from some third cycle. According to him, the upper voice inserts enharmonic material so as to avoid the direct chromatic succession C–C. Elsewhere, he also noted that the bass line does not arpeggiate a single Stufe; it simply transfers the primary note A from one register to another.16 In other words, the string of major thirds is produced by the voice leading; it is an effect and not a cause.

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Figure 5.3. Non-functional successions. Beethoven, “Appassionata” Sonata, Op. 57, 1st movement, mm. 62–87. From Schenker, Der freie Satz, Fig. 114.8.

Debussy’s music is no less remarkable for its pervasive use of chromaticism. Again, this is something that Schenkerian theory is particularly well equipped to explain.17 As we saw in previous chapters, Schenker treated chromaticism as an essential feature of the tonal system and even claimed that composers can never write too chromatically in tonal contexts.18 Schenker explained these chromaticisms in two ways: as mixtures or tonicizations. Through these processes, he was able to generate almost the entire range of chromatic Stufen directly from I. The only exceptions, in fact, are Stufen on
IV/V and, according to ‘The
IV/V Hypothesis,’ such sonorities can be generated only indirectly from I. Besides claiming that tonal surfaces can be almost continuously chromatic, Schenker also recognized that the full range of chromatic tones can be generated as far back as the deep middleground, provided these tones conform with the basic principles of tonality. Given this constraint, William Mitchell was quite right to suggest: “The more intense the chromaticism, the greater the need to relate individual sonorities to a broad context.”19 We will refer to this as ‘Mitchell’s Axiom.’ ‘Mitchell’s Axiom’ is particularly useful when we are confronted with bold harmonic experiments, such as those found in a work like “The Representation of Chaos” from Haydn’s Creation. As we saw in chapter 2, Schenker explained the extraordinary surface of this work as fallout from a massive fourth-progression in the bass that extends from m. 1 to m. 40. Yet, although Schenker was extremely flexible in his handling of highly chromatic music, he frequently criticized many latenineteenth-century composers not for writing too chromatically, but rather for failing to treat these chromatic elements with sufficient care and attention. He felt that these composers often failed to

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Figure 5.4. Extreme chromaticism. Graph of Reger, Piano Quintet, Op. 64, mm. 1–8.

coordinate the harmony and voice leading: they either obscured the sense of harmony altogether, or they stretched it “too thin to support or sustain with any security the complexities of the voice leading.”20 To illustrate his point, Schenker cited the opening of Reger’s Piano Quintet, Op. 64. According to him: The opening is by far the most flexible part of the first movement. What follows is in many respects more confused. But, I ask: do we really hear C minor or is it rather in E major? What is the significance, in particular, of mm. 6–8 in themselves and in relation to the coherence of the whole? Not by chance would the harmonic movement be difficult to perceive (obviously, considered in the context of E major-minor, it is perhaps as follows: I3–II [Phryg.]–IV and finally II3 as if it wants to move to the dominant) the only question remains: what does the motion of the Stufen tell us,—from where has it come, to where does it go? In what ways do these Stufen want to serve directly the principal key of C minor and how should the alleged E majorminor finally present itself in relation to C minor? Where is the solution to this problem? Nowhere in the work are there particulars concerning the principal key, (and) only with effort do subsequent events refer to preceding ones, and if one such connection occurs it is too paltry, too trivial, too short.21

Schenker concludes: “There is no plan to the keys; no plan in the tonicized Stufen—everything is merely a large, homogeneous, irrational mass.” Figure 5.4 gives a possible reduction of mm. 1–8. Another important consequence of Schenker’s work is that it offers a radically new way of explaining how modal and exotic inflections can arise in tonal contexts; whereas conventional theory explains them as products of some alternative scale system, Schenkerian theory generates them from mixtures and tonicizations.

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As suggested in chapter 4, Schenker believed that composers experimented with modal and exotic materials, not to undermine the tonal system, but rather to reveal its flexibility and scope. We adopted this strategy in chapter 4 to explain a wide variety of modal/ exotic pieces from Beethoven to Debussy. The great advantage of this approach is that it allows us to show how composers can allude to different modes and scale types within a single composition. So far, we have considered some of the ways in which Debussy challenged the local principles of tonal voice leading and harmony. But these are by no means the only problems posed by his music. On the contrary, Debussy also challenges the ways in which tonal relationships are projected at a global level. This issue is important because we have presumed that Schenkerian theory explains the behavior of complete, continuous monotonal compositions. By invoking the terms “complete” and “continuous,” we imply that it may be possible for tonal pieces to be in some sense incomplete and/or discontinuous. Significantly, Schenker considered both possibilities in his writings. To begin with, Schenker discussed two ways in which pieces might be classified as incomplete: those that do not present the tonic at the opening and those that do not return to the tonic at the end. He considered pieces of the first type near the end of Der freie Satz in a section entitled “Incomplete transferences of the Ursatz.”22 According to him, tonal compositions sometimes omit the first tone of the bass arpeggiation. The resulting progression “is ‘closed-off’ from what precedes it” and “points only to the forthcoming I.”23 Schenker referred to these progressions as “auxiliary cadence progressions” and cited the example of Chopin’s Prelude, Op. 28, no. 2. As shown in figure 5.5a (Incomplete transference of the Ursatz), he interpreted the piece as an auxiliary cadence progression in A minor. Schenker also acknowledged that pieces sometimes omit the final tonic. In the case of Bach’s short prelude, BWV 999 (see Figure 5.5b), he suggested that the “composed-out I–V can only be understood as a prelude.”24 Schenker even cited one piece, Chopin’s Mazurka, Op. 30, no. 2, in which it is unclear whether the piece omits the opening or the final tonic. As shown in figure 5.5c, he could not decide whether the Mazurka is in B or F
.25 Another possibility, however, is to conclude that the Mazurka actually

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Figure 5.5. Incomplete transferences of the Ursatz. a. Chopin, Prelude, Op. 28, no. 2. From Schenker, Der freie Satz, Fig. 110.3.

b. Bach, Prelude in C Minor, BWV 999. From Schenker, Der freie Satz , Fig. 152.6.

c. Chopin, Mazurka, Op. 30, no. 2. From Schenker, Der freie Satz, Fig. 152.7.

progresses from B to F
; many theorists refer to this option as “directional tonality” or “progressive tonality.”26 Claiming that a work moves from one key to another does not mean that the individual keys necessarily violate the principles of functional tonality; on the contrary, both keys can still satisfy every local law of tonal voice leading and harmony. But it does mean that the piece as a whole cannot be derived from a single tonal progression. In this respect, ‘directional tonality’ contradicts ‘The Global Paradigm’ and challenges the preeminence of monotonality. As for the matter of discontinuities, Schenker was fully aware that the global motion of a piece can be sidetracked in various ways. He made this point perfectly clear near the start of Der freie Satz: In the art of music, as in life, motion toward the goal encounters obstacles, reverses, disappointments, and involves great distances, detours, expansions,

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Figure 5.6. Interpolations in Debussy, “La sérénade interrompue” (Préludes, Book 1, no. 9).

interpolations, and, in short, retardations of all kinds. Therein lies the source of all artistic delaying, from which the creative mind can derive content that is ever new.27

Figure 5.6 (Interpolations in Debussy, “La sérénade interrompue,” Préludes, Book I, no. 9) gives a good example of these phenomena. Although the piece is basically in B minor, Debussy disrupts the tonic with two interpolations in D major, mm. 80–83 and 86–88: these interpolations not only create violent discontinuities in the surface of the music, but they actually quote the main theme from “Le matin d’un jour de fête” (third movement of Ibéria).28 Since these intrusions really “belong” to another piece, they do not derive from the prelude’s tonal prototype. Instead, they illustrate what Graham George has termed “interlocking tonality.”29 Since each local key may still conform to the local laws of tonal voice leading and harmony, the practice of interlocking tonality need not distort our sense of functionality; it simply undermines our sense of monotonality.

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It should be clear from the preceding discussion that, in principle at least, Schenkerian theory has built-in mechanisms for explaining many anomalies that we typically find in Debussy’s music. This is not to say, however, that the theory can necessarily explain the tonality of every piece by Debussy; but it does mean that we can determine, case by case, why some of his pieces sometimes sound tonal and sometimes do not. We have also seen that the explanatory scope of Schenkerian theory is much wider than many suppose. This point is especially important because tonal theorists are still debating whether eighteenth- and nineteenthcentury music can actually be explained by a single theory. Schenker clearly believed that it can. He proposed a comprehensive theory that purports to explain functional monotonal works from Bach to Brahms. But Gregory Proctor and others think otherwise; they distinguish between two overlapping practices, ‘classical diatonic tonality,’ in which chromaticisms arise from the interaction between diatonic scales, and ‘nineteenth-century chromatic tonality,’ in which chromaticisms derive from a single chromatic scale.30 In responding to Proctor, et al., it is important to note that we have already established a framework for understanding changes in tonal practice. According to this scheme, tonality is best regarded as a family of musical languages. These languages conform to the basic principles that melodies primarily move by step, that they converge on the tonic at cadences, and that they differentiate between stable and unstable sonorities. Each language, however, interprets these principles within a different harmonic environment: strict counterpoint interprets them within an environment built from consonant and dissonant intervals, modal counterpoint within an environment built from a limited range of triads, and functional tonality within an environment built from the full range of diatonic and chromatic Stufen. Although the practices of tonal composition clearly changed between the eighteenth and nineteenth centuries, it is unclear that they exist within yet another harmonic environment. Certainly, nineteenth-century composers experimented with the principles of directional and interlocking tonality, but these practices challenge the notion of monotonality and not principles of functionality, at least as Schenker explained

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them. The beauty of Schenkerian theory is that it is powerful enough to explain surfaces that are almost continuously dissonant and chromatic.31

Debussy, “C’est l’extase langoureuse” Now that we have shown how Schenkerian theory has built-in mechanisms for explaining many anomalous features of Debussy’s style, let us now see how it helps explain the tonality of his two pieces: “C’est l’extase langoureuse” and “La mort des amants.” These songs are an appropriate choice for two reasons. First, they pose the specific technical problems discussed earlier. In fact, we will encounter examples of parallel chords, free dissonances, non-functional successions, extreme chromaticism, modal/exotic inflections, incomplete structures, and parenthetical passages/ interpolations. We will also see that Schenkerian theory provides us with ways to explain these phenomena. Second, the two songs are significant because their texts display many of the essential ingredients of symbolist poetry. By showing how Debussy used tonal relationships to highlight the structure and meaning of each poem, we can see the composer’s strong ties to symbolist/decadent aesthetics. We will begin by looking at “C’est l’extase langoureuse.” Composed in 1887 and published in the set Ariettes oubliées (1903), this song is not only one of Debussy’s best early compositions, but also one of his most sophisticated experiments in tonal composition. The text for this song comes from Verlaine’s collection Romances sans paroles (1874).32 Le vent dans la plaine Suspend son haleine. (Favart) C’est l’extase langoureuse, C’est la fatigue amoureuse, C’est tous les frissons des bois Parmi l’étreinte des brises,

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C’est, vers les ramures grises, Le choeur des petites voix. O le frêle et frais murmure! Cela gazouille et susurre, Cela ressemble au cri doux Que l’herbe agitée expire . . . Tu dirais, sous l’eau qui vire, Le roulis sourd des cailloux. Cette âme qui se lamente En cette plainte dormante, C’est la nôtre, n’est-ce pas? La mienne, dis, et la tienne, Dont s’exhale l’humble antienne Par ce tiède soir, tout bas? Each of the three verses has six lines that rhyme a, a, b, c, c, b, and each one ends with a complete syntactic unit. Reading through the poem, one is immediately struck by Verlaine’s preference for words containing the sounds “s” and “t.” In verse 1, for example, we find the phrases “C’est la fatigue,” and “C’est tous les frissons”; in verse 2, the phrases “Cela gazouille” and “cela ressemble”; and in verse 3, the phrases “Cette âme” and “C’est la nôtre.” These sounds give Verlaine’s text an aspirate quality that captures the main subject of the poem, namely the fatigue of love. Three words stand out in this regard: “extase” in verse 1, “expire” in verse 2, and “exhale” in verse 3. Their ordering seems to match a more general progression in the poem’s meaning from declarative statements about the lovers’ present feelings to more personal speculations about their future. In verses 1 and 2, for example, Verlaine compares the lovers’ shivers to particular sounds—the faint rustle of trees (“les frissons des bois”), the swish of grass (“[le] cri doux Que l’herbe agitée expire”), and the rumble of pebbles rolling under water (“sous l’eau qui vire, Le roulis sourd des cailloux”). But in verse 3, the mood changes, as the lovers realize that they will probably never share the same experience again.

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Figure 5.7. Graph of Debussy, “C’est l’extase langoureuse,” mm. 1–18.

When composing “C’est l’extase langoureuse,” Debussy was clearly intent on projecting the form and meaning of Verlaine’s poem. Take, for example, Debussy’s setting of verse 1 (mm. 1–18). Here, Debussy captured the sultry atmosphere of the poem in several different ways. Most obviously, he took as his main motive, the descending second G
–F
. This gesture, which pervades the piano and the vocal parts in mm. 1–10, is the quintessential sigh figure; it conveys not only the breathlessness of the lovers, but also the hopelessness of their relationship. Debussy reinforced this idea by starting the song with an auxiliary cadence progression in E major: as shown in figure 5.7 (“C’est l’extase langoureuse,” mm. 1–18), the opening V9 sonority hangs in mid air for six bars, before resolving, via a passing seventh on G
(mm. 7–8), onto I in m. 9–10. Having set the tone of the song, Debussy develops his material in mm. 11–18. Starting in m. 11, the accompaniment includes a syncopated rhythm—sixteenth-note, eighth-note, dotted sixteenth-note; this pattern helps to give the music a gentle lilt that carries through much of the song. Measures 11–18 then move back from the tonic to the dominant. Figure 5.7 shows that this progression is controlled contrapuntally by a stepwise ascent from G
to D
in the upper voice. The motion from D$2–B in mm. 17–18 is especially striking because it anticipates a similar motion in verse 3. Whereas verse 1 establishes the song’s main material, verse 2 (mm. 18–35) seems to move off in other directions. Motivically, it

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Figure 5.8. Graph of Debussy, “C’est l’extase langoureuse,” mm. 18–35.

focuses on a new chromatic motive. Originally introduced in the voice on G
in m. 24, this gesture returns four times on C
in mm. 28–33. These statements appear within a more general modulation from E major to D major. As shown in figure 5.8 (“C’est l’extase langoureuse,” mm. 18–35), the verse starts with a compressed reprise of the opening but, instead of resolving down to B, C
becomes part of an added-sixth chord on E. Debussy continues to highlight C
in the next few bars by unfolding the fourth C
–G
in the voice and piano (mm. 20–21) and by passing from C
through B to A
in the piano (mm. 22–24). This passing motion is harmonized by the simple progression C
9 to F
9, or ii9–V9 in B major. But the song suddenly changes trajectory in m. 24. Indeed, over the next few bars, there is a massive voice exchange between the voice part and the right hand of the piano, leading to a repetition of the chromatic motive on C
in m. 28. From that moment on, D asserts itself as tonic. Although there is no authentic cadence, the sonority in m. 28 clearly serves as a dominant to the tonic chord in m. 29 with the seventh G (4 ) resolving down by step onto F
( 3). Debussy reinforces the arrival on D with a plagal cadence IV3–3–I in mm. 32–36. Verse 3 (m. 36ff) then weaves its way gently back to E. At first, the texture is saturated with the sigh figure and syncopated rhythm, but in mm. 44–46, the voice brings back its line from mm. 3–4. This reminiscence leads to two statements of the chromatic motive in mm. 46 and 47 and to further reminiscences of the sigh figure and the syncopated motive in m. 48ff. In the meantime, Debussy

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Figure 5.9. Graph of Debussy, “C’est l’extase langoureuse,” m. 36ff.

reestablishes E by means of a common-tone progression built on the pitch A. Figure 5.9 (“C’est l’extase langoureuse,” m. 36f.) shows that the harmonies shift effortlessly from D7 (mm. 36 and 38) via B7 (mm. 37 and 39), and G9 (mm. 40 and 42), to C
13 (m. 41), before arriving back on B7 in mm. 44–45. At this point, however, Debussy transfers A into the upper register for the return of the opening gesture. He even recalls the chromatic passing motion C
–B
–B in the inner voice as B–B
–C
. Although there is no doubt that the tonic returns in m. 46, its effect is undercut by the lingering presence of the added-sixth C
and the return of the chromatic motive. It is only after the motive is expanded in the bass as G–F–E (mm. 48–52) that the song lands on a pure tonic triad. Perhaps because Verlaine’s final question is rhetorical, Debussy replaces the final question mark with a full stop. It should be clear from the preceding discussion that Debussy went to elaborate lengths to articulate the form of Verlaine’s poem both motivically and tonally. But we are still left to wonder how his setting conveys the poem’s overall progression from ecstasy to ennui. To address this issue, let us now put our observations into a wider perspective. As we saw earlier, the song has two main motives: the sigh figure, which dominates verses 1 and 2, and the chromatic motive, which comes to the fore in verse 2. Despite their obvious differences, these gestures are, in fact, closely related; we

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Figure 5.10. Evolution of ‘The Sigh Figure’ in Debussy, “C’est l’extase langoureuse”.

Figure 5.11. Global view of Debussy, “C’est l’extase langoureuse”.

can see these connections at the start of the song. After Debussy has introduced the sigh figure in mm. 1–6, he ends the voice part with the descent C
–B
–B (mm. 7–9). This descent not only implies that mm. 1–10 are a giant version of the sigh figure, but it also anticipates the downward spiral of the chromatic motive. This possibility becomes clearer in mm. 22–23, when the piano part descends from C
through B to A
. And, when the chromatic motive eventually appears on C
, it descends first to A
(mm. 28, 30, and 32) and then to A (mm. 33–34). The arrival onto A is important because it gives rise to the common-tone progression in mm. 36–45. It subsequently resolves down by step through G
to F
in m. 45, eventually reaching E in m. 48. In other words, the gradual transformation of the sigh figure into the chromatic motive inscribes an overall melodic descent from C
, through B and A
to A and from A through G
and F
to E. This process is summarized in figure 5.10 (Evolution of the sigh figure in “C’est l’extase langoureuse”). The significance of this point becomes even clearer when we compare figure 5.10 with the Schenkerian graph given in figure 5.11

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(Global view of “C’est l’extase langoureuse”). Significantly, this sketch suggests not only that “C’est l’extase langoureuse” can be derived from a 5 -line prototype, but also that the main events of the middleground are marked by the emergence of the chromatic motive. For example, the chromatic descent C
–B
–B in mm. 7–9 serves as an upper neighbor to the headtone B. Starting in m. 20, C
is restored as an incomplete upper neighbor to B; it is then prolonged by the passing motion C
–B–A
in mm. 22–23, before descending chromatically through C, B and B to A in mm. 34–35. Once A is reached in the upper line, it is subsequently prolonged by common-tone harmonies until mm. 44–45. A eventually resolves down by step through G
to F
, reaching E in bar 48. Significantly, the song reaches cadences on E in the piano at m. 46, two bars before it reaches E in the voice. This disjunction is remarkable because it undermines our sense of closure at the end of the song; just as the long descent conveys the lovers’ desire for satisfaction, so the conflict between harmonic and melodic closure suggests that reality never lives up to expectation.

Debussy, “La mort des amants” Debussy faced many of the same compositional issues when setting Baudelaire’s poem “La mort des amants.” He apparently completed this song between 1887 and 1889, just after “C’est l’extase,” and published it in the set Cinq poèmes de Baudelaire in February 1890. The text for this song comes from Baudelaire’s incomparable collection Les fleurs du mal (1861).33 Nous aurons des lits pleins d’odeurs légères, Des divans profonds comme des tombeaux, Et d’étranges fleurs sur des étagères, Écloses pour nous sous des cieux plus beaux. Usant à l’envi leurs chaleurs dernières, Nos deux coeurs seront deux vastes flambeaux, Qui réfléchiront leurs doubles lumières Dans nos deux esprits, ces miroirs jumeaux.

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Un soir fait de rose et de bleu mystique, Nous échangerons un éclair unique, Comme un long sanglot, tout chargé d’adieux; Et plus tard un Ange, entr’ouvrant les portes, Viendra ranimer, fidèle et joyeux, Les miroirs ternis et les flammes mortes. This wonderful sonnet contains two quatrains that rhyme a, b, a, b, followed by a sestet that rhymes c, c, d, e, d, e. Though it describes the union of two lovers, the poem is one of several texts concerned with the nature of death. In typical decadent fashion, Baudelaire implies that the inevitable consequence of sexual pleasure is that it ends; to underscore this fact, he wrote the poem entirely in the future tense. Baudelaire compares the human soul to a mirror (“nos deux esprits, ces miroirs jumeaux”) and the heart to a flame (“Nos deux coeurs seront deux vastes flambeaux”); as the poem unfolds, he reminds us that mirrors tarnish and that flames can be extinguished. According to him, it is only by the intervention of a new lover, in the guise of an Angel, that the protagonists can rekindle their passions. This rather pessimistic message is not dissimilar to that of “C’est l’extase langoureuse”—it suggests that the lovers will achieve true satisfaction only in the future, never in the present. In setting “La mort des amants,” Debussy was again very careful to highlight the work’s intricate poetic structure. The first quatrain introduces the song’s main thematic and tonal material. The song has two main motives, both of which appear in the opening measures: an arpeggiated figure in the upper register of the piano (see figure 5.12a, motive X) and a chromatic gesture in the lower register D–E–E–F–F (mm. 1–2) and B/D–C/E–D/F (m. 3) in the lower register (see figure 5.12b, motive Y). The arpeggiated figure is especially interesting because it bears a striking similarity to the main theme of the piano piece “Clair de lune” from Debussy’s Suite bergamasque (see figure 5.12c).34 The first quatrain also establishes the song’s principal tonality G. As shown in figure 5.12d, the piece actually opens with an auxiliary cadence progression in G (mm. 1–5). This progression leads contrapuntally to an authentic cadence in G (mm. 7–8). This cadence is interesting on several

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Explaining Tonality

Figure 5.12. Debussy, “La mort des amants,” mm. 1–12. a. Motive X.

b. Motive Y.

c. Debussy, “Clair de lune,” Suite bergamasque.

counts. For one thing, the piano includes a variant of the chromatic line that rises from B, through C, D and D, before resolving down from E to D. This gesture becomes important later in the song. For another, Debussy captures the image of a tomb in mm. 7–8 by plumbing the depths of the vocal register. By the end of the quatrain, however, the music moves from G to the dominant D; this

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Figure 5.12 (continued). d. Graph of Debussy, “La mort des amants,” mm. 1–12.

modulation is confirmed by a perfect authentic cadence in D at m. 12. This cadence is beautifully prepared by a whole-tone chord in m. 11; besides intensifying the upcoming cadence, this striking sonority allows Debussy to bring back the chromatic line from mm. 1–2, this time starting on E in m. 11. The second quatrain of “La mort des amants” (mm. 12/13–18) begins very much like the first; the arpeggiated figure appears in the upper register of the piano with the chromatic line (D–E–E–F) in counterpoint underneath. As the music unfolds, the chromatic line becomes increasingly prominent and it leads inexorably up from D (m. 12) through E, F, F, G, G, A, A, and B to B/C (m. 18). At the same time, the vocal line stands out for its ingenious word painting (see figure 5.13a, Mirroring in the melody): when the text describes how lovers’ hearts will be reflected in each other’s souls, the melodic line mirrors itself, descending first from D through B to F, E, and D (m. 16), and then ascending from D through E, F, and A back to D (m. 17). For the most part, the second quatrain prolongs the dominant; as shown in figure 5.13b (Graph of “La mort des amants,” mm. 12–18), it is reaffirmed as a local tonic in mm. 14–15 by a simple V–I progression. In m. 18, however, the music seems to point back to G; as it stands, the minor seventh A–C– E–G would seem to serve as II7 of G. But, instead of cadencing on G at the start of the final sestet, Debussy respells A/C enharmonically as G
/B and immediately thrusts the first half of the sestet (mm. 19–29) in the direction of E

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Explaining Tonality

Figure 5.13. Debussy, “La mort des amants,” mm. 12–18. a. Mirroring in the melody.

b. Graph of Debussy, “La mort des amants,” mm. 12–18.

major, with a perfect authentic cadence emphatically reached in mm. 22–23 (see figure 5.14a, Graph of “La mort des amants,” mm. 19–29). Significantly, the use of the rising chromatic motive C
–D
–E–E
–F
at the cadence in mm. 22–23 not only recalls the previous arrival on G in mm. 7–8, but it also anticipates the cadence in mm. 103–6 of the Prélude à ‘l’Après-midi d’un faune’ (see figures 5.14b and 5.14c). Once E major has been firmly established, Debussy develops the cadence motive against a rising chromatic line B–B
–C
–D–D
–E–E
–F
–G
–A–B–C
in the bass (mm. 22–26). Motivically, Debussy places the sigh figure on the downbeat, to convey the charged meaning of Baudelaire’s text “comme un long sanglot, tout chargé d’adieux.” Katherine Bergeron has noted that this passage seems to recall the “Good Friday” music from Act 3 of Wagner’s Parsifal (see figure 5.14d).35 Given the complex erotic/religious nature of Wagner’s opera, this allusion seems to resonate with Baudelaire’s subsequent reference to the angel. In any case, the passage culminates with a return to A7 in m. 29, spelled this time as G
–C–D
–F
.

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This chord prepares for the second half of the final sestet (mm. 30–45). Starting in m. 30, Debussy shifts the direction of the harmony yet again by recalling the opening arpeggiated figure this time transposed into the context of C major (see figure 5.15a, Graph of “La mort des amants”). Unlike the opening, however, he omits the chromatic gesture entirely and immediately transposes the arpeggiated figure into the context of E major. Nevertheless, in m. 38, he returns to A7 for the word “ranimer” and again in m. 41 for the last line of text. This last recollection finally leads to an emphatic cadence in the tonic. To reinforce the sense of closure, Debussy even accompanies the final line of the sonnet—“Les miroirs ternis et les flammes mortes”—with a descending chromatic line. This line mirrors the rising lines in mm. 1–2 and 13–18 (see figures 5.15b-d). Debussy rounds off the song with a short coda for the piano; we hear our last statements of the arpeggiated figure and chromatic gesture over a tonic pedal. So far, we have seen how Debussy managed to articulate the subtle subdivisions in Baudelaire’s text, but we are still left to see how he managed to convey its pessimistic message that true satisfaction is something that cannot be achieved in the present. Whereas Debussy created the sense of ennui in “C’est l’extase langoureuse” by initiating the long chromatic descent in the upper line but by failing to coordinate melodic and harmonic closure at the end, he produced similar effects in “La mort des amants” by the use of incomplete progressions and parenthetical passages/interpolations. These points are readily apparent in figure 5.16 (Global view of “La mort des amants”), a Schenkerian reading of the entire song. This sketch highlights the fact that the song opens with an auxiliary cadence progression V^4–3%–I in G and that the first quatrain subsequently modulates to the dominant. The long chromatic ascent in the lower register from D to B/C in the second quatrain creates a sense of sexual tension that seems to require resolution through a cadence in G in m. 19. But, though the progression arrives on the expected predominant ii7 in m. 18, the final sestet systematically delays this cadence with one digression after another: in mm. 19–29, Debussy interpolates a passage in E major that conjures up images of Parsifal and anticipates the Prélude à “l’Après-midi d’un faune”; in mm. 30–40 he inserts sequential statements of the

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Explaining Tonality

Figure 5.14. Debussy, “La mort des amants,” mm. 19–29. a. Graph of Debussy, “La mort des amants,” mm. 19–29.

b. Motive Y at cadence in “La mort des amants,” mm. 20–23.

c. Motive Y in cadence in Prélude a “l’Après-midi d’un faune,” mm. 103–6.

opening material. Each digression is framed by seventh chords on A/G
: the first is framed by the sonorities in mm. 18 and 29 and the second by those in mm. 29, 38, and 41. It is only after this last version that Debussy lets the sonority lead to a conclusive cadence in G. Once we reach G, Debussy finally gives us the main theme against a stable tonic chord. The preceding analyses of “C’est l’extase langoureuse” and “La mort des amants” have suggested just how sensitive Debussy

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Figure 5.14 (continued). d. Wagner, Parsifal, “Good Friday” cadence, Act 3.

was to the pessimistic overtones of both texts. By insisting that the reality of the present never lives up to our expectations, both poems express a central theme in symbolist/decadent aesthetics. But of all the texts dealing with this topic, why did Debussy set these particular poems? We can, I think, offer a couple of answers. On the one hand, both poems are mentioned explicitly or implicitly in one of the most famous symbolist texts of the period—J. K. Huysmans’ sensational book À rebours (1884). This lurid novel, which scandalized Parisian society, describes the decadent lifestyle of Huysmans’ alter ego, Duc Jean Floressas des Esseintes. He is “the modern man par excellence, . . . painfully conscious that his pleasures are finite, his needs infinite.”36 Though fixated with sex, Des Esseintes was concerned less with celebrating sexual pleasure per se and more with confessing his most perverse thoughts. Such confessions were ultimately motivated by a sense of Catholic guilt and misogyny. As Jean Pierrot points out, “If the decadents discovered sexuality, however, it was only for the most part to reject it, or at least to reject its normal forms.”37 Given these facts, it is hardly surprising that Des Esseintes adored the works of Baudelaire; after all, Baudalaire’s

200 Figure 5.15. Debussy, “La mort des amants,” mm. 30–45. a. Graph of Debussy, “La mort des amants,” mm. 30–42.

b. Debussy, “La mort des amants,” mm. 39–43.

c. Debussy, “La mort des amants,” mm. 1–5.

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Figure 5.15 (continued). d. Debussy, “La mort des amants,” mm. 16–18.

Figure 5.16. Global view of Debussy, “La mort des amants”.

works are full of misogynist overtones. More to the point, however, Des Esseintes was so enamored of “La mort des amants” that he had it copied on vellum and hung over his mantle piece. He also adored Verlaine and, though he did not mention “C’est l’extase langoureuse” by name, his description of the poet’s work bears a striking similarity to that text.38 And Verlaine, of course, had an infamous homosexual liaison with Rimbaud. It is surely not by chance, then, that Debussy’s interest in the poetry of Verlaine and Baudelaire blossomed in 1884–85 after reading À rebours.39 Another reason why these texts may have spoken so directly to Debussy around 1887 is that he was recovering from his notorious affair with Mme Blanche de Vasnier. Debussy first met the aspiring singer and her husband in the early 1880s; he became her accompanist and she became his inspiration. Before leaving for Rome in

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1885, Debussy even gave her a collection of thirteen songs, commonly known as “The Vasnier Songbook.” But by the time he returned to Paris for good in 1887, the affair was apparently over; Marcel Dietschy has even suggested that in 1888, when Debussy inscribed the copies of the Ariettes oubliées “to Mme Vasnier, in grateful homage,” he did so as a last farewell to a past love.40

Schenkerian Theory and Twentieth-Century Music Up until now, our discussion has tried to demonstrate the fruitfulness of Schenker’s ideas by using them to explain the tonal structure of Debussy’s music. Although the two songs, “C’est l’extase langoureuse” and “La mort des amants,” certainly challenge the limits of Schenkerian theory, they still lie within its outer reaches; they mark, if you will, a boundary point at which the theory starts to break down. We are left to wonder, however, what happens when we try to analyze music that is even more remote tonally. Can we use Schenkerian methods to account for the behavior of music that is no longer built from functional triads? Can Schenkerian theory be adapted to provide accurate explanations of music composed in the twentieth century? If so, how? As we begin to answer these questions, it is important to note that they are still matters of fierce debate. Robert P. Morgan has suggested, in fact, that this conflict stems in part from tensions within Schenker’s own writings.41 According to him, there is no doubt that Schenker despised twentieth-century music, especially for the way it breaks down the distinction between consonance and dissonance. For Schenker, this distinction was absolute; in Der freie Satz, he insisted that, since dissonances are always derivative, they can never serve as goals of motion and can never be prolonged in their dissonant state.42 And yet, Morgan also claims that Schenker contradicted himself in two ways. For one thing, Schenker violated his own prohibition against “dissonant prolongations.” Morgan points out that Schenker not only included composed-out seventh chords in his graphs, but he also inserted a paragraph on seventhprogressions in Der freie Satz.43 Some of these graphs are shown in figures 5.17a-e (Prolonged dominant-seventh chords). For another,

203 Figure 5.17. Prolonged dominant-seventh chords. a. Haydn, Piano Sonata in E-flat Major (Hob. XVI:49), 1st movement, development section. From Schenker, Der freie Satz, Fig. 62.1.

b. Beethoven, Leonore Overture No. 3, Adagio. From Schenker, Der freie Satz, Fig. 62.2.

c. Beethoven, Piano Sonata, Op. 81a, 1st movement, development section. From Schenker, Der freie Satz, Fig. 62.4.

d. Bach, Prelude in C Major, WTC I, m. 24ff. From Schenker, Der freie Satz, Fig. 62.5.

e. Beethoven, Symphony No. 3, 1st movement, development section. From Schenker, Der freie Satz, Fig. 62.10.

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Explaining Tonality

Schenker may have designed his methods to explain music from Bach to Brahms, but he also used them to analyze music from Stravinsky’s Concerto for Piano and Wind Instruments.44 According to Morgan, this analysis “unwittingly provided the foundation for a theory of twentieth-century tonal structure based on ‘dissonant tonics.’ ”45 Meanwhile, other experts, such as Edward Laufer, have warned against reading too much into the Stravinsky analysis; they insist that the task of building a theory of prolongation for twentiethcentury music is fraught with technical difficulties.46 For example, although Laufer admits that twentieth-century music behaves “linearly” and may even have “a background of some sort,” he concedes: There is no triad to be prolonged: some contextually derived associative sonority must take its place. The concepts of consonance and dissonance, as technically defined, therefore cannot exist, nor can, strictly speaking, the notions of passing and neighbor notes where these were dissonant events. Their attendant constraints, which provided motion and delays, must be compensated for by other kinds of embellishing and traversing motions. There is probably no generalized fundamental line: it could not now be diatonic.47

Laufer adds, rather pessimistically, “If there is no technically consistent, non-speculative basis, then anything goes, and likewise nothing.”48 We can respond to these arguments in several ways. To begin with, Morgan exaggerates the contradictory nature of Schenker’s work. His first contradiction is, in fact, more apparent than real. In chapter 1 we saw that, through ‘The Stufe Constraint,’ Schenkerian theory actually relies less on distinguishing consonances from dissonances and more on distinguishing harmonic tones from nonharmonic tones. This allows for dissonant harmonic tones (for example, VII Stufen in major keys and II Stufen in minor keys) as well as consonant non-harmonic tones (e.g, 5–6 motions). More to the point, however, it is also clear from the sketches in figure 5.17 that dominant-seventh chords can be prolonged in some ways, but not in others. If we classify Schenkerian transformations along the lines shown in chapter 2, then it seems that dominant-seventh

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chords can indeed be horizontalized and that the horizontalized tones can be filled with passing or neighbor tones. Most of the graphs in figure 5.17 also suggest that so-called seventh-spans actually arise, not from composing out the members of a single Stufe, but rather from filling in the space between an inner voice and an upper voice that descends by step (for example, figure 5.17b). But these graphs do not show cases in which prolonged dominantseventh chords can be harmonized to create new Stufen. This suggests that composed-out dissonances cannot produce new levels. In other words, Morgan exaggerates the inconsistencies in Schenker’s work because he treats prolongation as a single process; as we have seen, prolongations come in several distinct flavors. Morgan also claims that Schenker was contradictory in his views about twentieth-century music and that his analysis of Stravinsky’s Concerto for Piano and Winds sets a precedent for composing out dissonant tonics. To see what this means, Figure 5.18 (Schubert, “Die Stadt,” Schwanengesang, No. 11) compares Morgan’s analysis (figure 5.18a) with Schenker’s (figure 5.18b). According to Morgan, the most striking feature of this song is the diminished-seventh chord C–E–F
–A; this sonority controls the introduction, verse 2, and the coda. Although Morgan admits that this referential sonority is subordinate to the C-minor harmony governing verses 1 and 3, he treats the diminished-seventh underlying verse 2 as a dissonant tonic and claims it seems almost completely stable. Morgan believes that such dissonant tonics foreshadow the future; he cites various other examples by Schubert, Liszt, Wagner, and Scriabin, in which augmented triads, diminished sevenths, and other dissonances seem to act as stable goals of motion. With regards to Morgan’s analysis, there are significant differences between his concept of prolongation and the one advocated by Schenker (figure 5.18b). As we saw in chapter 2, ‘The Recursive Model’ is based on two crucial ideas: 1) all complex tonal progressions are transformations of simple tonal prototypes; and 2) whenever a prototype is transformed the resulting progression conforms to the same laws of tonal voice leading and harmony as the prototype itself. Morgan clearly endorses the first idea, but not the second; although he treats complex surfaces as

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Explaining Tonality

Figure 5.18. Schubert, “Die Stadt,” Schwanengesang, no. 11. a. Morgan’s analysis. From Morgan, “The dissonant prolongation.”

b. Schenker’s analysis. From Der freie Satz, Fig. 103.4.

prolongations of simple progressions, he does not try to connect the process of generation with any general laws of voice leading. In other words, his analysis shows how a given referential sonority is projected from one level to another, but not how these sonorities are transformed contrapuntally to create new sonorities. Furthermore, Morgan is unable to generalize beyond the specific piece in question; he cannot predict what sonorities will be referentially significant in any given piece or how those sonorities generate any new material. In other words, Morgan’s approach may help us describe the main features of a given piece, but it cannot explain why that piece behaves like other pieces of the same class or kind.

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These are the very problems that Laufer touches on in the passage cited earlier. In particular, he notes that, whereas functional music prolongs triads, twentieth-century pieces may derive from other contextually derived harmonies. He also insists that without a clear distinction between consonance and dissonance, it is hard to define any general principles governing contrapuntal motion. And Laufer doubts that pieces can be generated from any generalizable prototypes. But whereas Laufer does not indicate how we might overcome these problems, chapters 1 and 2 offer some basic guidelines. Our first step might be to isolate certain specific repertories of Post-Tonal music. These repertories should be aurally distinct from one another. Our next step might be to look for general laws that cover the local and global behavior of the constituent lines and chords. Ideally, these laws will cover six areas: 1) how individual lines move and reach closure; 2) how polyphonic lines move in relation to one another, 3) how unstable tones behave in relation to stable tones; 4) how stable harmonies are distinguished from unstable harmonies; 5) how successive harmonies are arranged to create idiomatic progressions; and 6) how stable harmonies are inflected coloristically. Just as the precise laws of step motion, melodic convergence, and vertical alignment must be modified when we shift from the intervallic world of strict counterpoint to the triadic world of functional monotonality, so they must be different when we shift to some new non-triadic context. These new harmonic environments might be based on seventh chords, quartal harmonies, or even more complex pitch-class sets. This point is shown in figure 5.19 (Functional tonality and twentiethcentury tonal practices). Once we have discovered appropriate laws of voice leading and harmony for each repertory, we can try to represent them as a system of prototypes, transformations, and levels. In some cases, the prototypes and transformations will look a lot like tonal transformations, but there is no reason to suppose that they will always be analogous. Nor should we expect that we can necessarily assume these laws can be reformatted as a recursive and rule preserving system; as we saw in chapter 1, even Fux was unable to formulate the laws of strict counterpoint in such a fashion. But, while it is premature to speculate about what these new prototypes, transformations,

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Figure 5.19. Functional tonality and twentieth-century tonal practices.

and levels will look like, there is no a priori reason why such new theories cannot be found for many types of twentieth-century music. To paraphrase Laufer, the resulting theories will not be Schenker’s, but they will still owe much to him.49

6

Renaturalizing Schenkerian Theory Ever since the birth of their discipline, music theorists have endeavored not only to find specific concepts, laws, and procedures to explain musical phenomena, but also to connect them with concepts, laws, and procedures in other domains. This process is known as naturalizing music theory.1 The reasons for undertaking such a task are clear enough; naturalizing music theory allows us to see how our knowledge of music coheres with our knowledge in other domains. We want our theories to be coherent because we know that our understanding of music is shaped by many external factors. Furthermore, as we saw in the Introduction, coherence provides us with a concrete criterion for choosing between theories that are otherwise equivalent on evidential and systemic grounds. If we are confronted with two theories that are equally accurate, equivalent in explanatory scope and predictive power, and equally consistent and parsimonious, then we will prefer the one that is most coherent with related disciplines. Attempts to ground music theory conceptually have traditionally taken one of two main tacks (see figure 6.1, Naturalizing music theory). Many theorists have tried to connect their explanations of music with the acoustic properties of notes. Up until the seventeenth century, they grounded their work in string divisions, but in the eighteenth, nineteenth, and early twentieth ceturies, they have turned their attention to the nature of the overtone series. More recently, however, an increasing number of theorists have moved away from acoustics and have tried to support their work with appeals to the principles of music psychology. This strategy presumes that our theoretical concepts and analytical methods should reflect the ways in which human beings actually listen to and think about music.

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Figure 6.1. Naturalizing music theory.

As it happens, Schenker and his followers have set about naturalizing their theory in both ways. Schenker himself picked the acoustic route; throughout his career, from the Harmonielehre to Der freie Satz, he connected the principles of tonality to the physical properties of the overtone series. As he explained, “[T]herefore art manifests the principle of the harmonic series in a special way, one which lets ‘The Chord of Nature’ shine through.”2 In fact, we find the same basic arguments at the start of Der freie Satz as we do in the opening of the Harmonielehre, some thirty years earlier. Fred Lerdahl and Ray Jackendoff, meanwhile, have developed a theory of pitch prolongation which, though similar to Schenker’s, is purportedly based on listener psychology. According to them, a successful theory of music provides “a formal description of the musical intuitions of a listener who is experienced in a musical idiom.”3 Influential as the writings of Schenker and Lerdahl/Jackendoff may have been, this chapter proposes a rather different way to naturalize Schenkerian theory. Like Lerdahl and Jackendoff, this alternative model takes the psychological tack; unlike them, however, it places greater emphasis on the relationship between listening and composing, and on the ways in which we acquire our knowledge of tonal relationships. This change of focus has important consequences for the testability of the theory; by shifting our attention to the connections between composition and listening and to the ways in which musicians gain their expertise, we will inevitably draw on quite different types of evidence than those offered by Lerdahl and Jackendoff.

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Naturalizing Schenkerian Theory There can be little doubt that Schenker took the task of naturalizing tonal theory very seriously. This is readily apparent from his desire to show that “all of the masterworks manifest identical laws of [tonal] coherence.”4 Besides unearthing these fundamental laws of tonality, Schenker wanted to link them to the physical properties of sound. On the one hand, he tried to relate specific transformations to various “natural” processes. In fact, he frequently claimed not only that individual tones expressed a natural urge to be tonicized, but also that tonalities have an inherent tendency to change from major to minor or vice versa. Later, in Der freie Satz, Schenker maintained that each prototype composes out the socalled Chord of Nature (Der Naturklang): according to him, ‘The Chord of Nature’ was the fundamental unifying element in functional monotonal music. We will refer to this as ‘The Chord of Nature Argument.’ To see just how important natural processes were to Schenker’s thinking, we need only consider his attempts to generate the major system from the overtone series.5 Schenker discussed the origins of the major system most extensively at the start of his Harmonielehre. Figure 6.2 (Schenker’s derivation of the major system from “The Chord of Nature”) summarizes the main components of his argument. To begin with, since Schenker believed that only the first five overtones of any given note are audible, he proposed that the major triad must be the fundamental element of the tonal system (figure 6.2a). He referred to this fundamental triad as ‘The Chord of Nature.’ Since the fifth is the strongest interval after the octave, Schenker then used this interval to create a cycle of fifths above the tonic pitch (figure 6.2b). Next, he claimed that since, if sounded, each member of this cycle will produce its own overtone series, the cycle can be rewritten as a chain of major triads (figure 6.2c). To counteract the outward motion of the “natural” cycles in figures 6.2b and 6.2c, Schenker proposed that the “Artist” construes another cycle of fifths that descends back to the fundamental (figure 6.2d). This descending cycle likewise implies triads on each of its members (figure 6.2e). After compressing the outward and inward cycles into an octave (figure 6.2f), the subdominant triad is

212 Figure 6.2. Schenker’s derivation of the major system from ‘The Chord of Nature’. a. Schenker, Harmonielehre, Ex. 18.

b. Schenker, Harmonielehre, Ex. 19.

c. Schenker, Harmonielehre, Ex. 20.

d. Schenker, Harmonielehre, Ex. 29.

e. Schenker, Harmonielehre, Ex. 29.

f. Schenker, Harmonielehre, Ex. 30.

g. Schenker, Harmonielehre, Ex. 34.

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Figure 6.2 (continued). h. Schenker, Harmonielehre, Ex. 35.

produced by a fifth under the tonic (figure 6.2g). Schenker then asserted that “the content of the more distant fifths, starting with the second fifth, is tempered and modified by the content of the fundamental and by the fifths immediately above and below.”6 This process eliminates the chromatic pitches shown in figures 6.2c–6.2g and produces the diatonic major system (figure 6.2h). Although Schenker placed great stock on ‘The Chord of Nature Argument,’ his derivation of the major system hardly stands up to close scrutiny. As Suzannah Clark rightly points out, this derivation hinges on what Schenker regarded as ‘The Mysterious Five.’7 According to him, whereas “Nature . . . knows only the fifth that appears in the overtone series, and no other kind of fifth,” Man . . . “is led to the fifth (as well as other intervals) in various applied ways.”8 The snag is that magic numbers have very dubious explanatory status. Why should we pick one magic number over another? Why not three or seven? Once magic numbers are removed, Schenker was unable to offer any compelling reasons why the ascending cycle in figure 6.2c stops at VII, or why the descending cycle in figure 6.2g should extend beyond the tonic to the subdominant. And his explanation of why the chromatic triads are eliminated from figures 6.2g and 6.2h seems strained to say the least: why should the content of more distant fifths be tempered by more immediate ones? Stepping back from figure 6.2, Schenker’s naturalism leads to several other difficulties. For example, Schenker was surely on the right track when he noted that “tones have lives of their own, more independent of the artist’s pen . . . than one would dare to believe.”9 After all, a perfect authentic cadence sounds conclusive quite independently of who wrote it. And yet, the distinction between the products of “Nature” and those of the “Artist” seems arbitrary to say the least. There is something particularly odd about claiming that the natural minor system is artificial: what does it

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mean to say that Beethoven’s “Eroica” Symphony is more natural than his fifth symphony? Unfortunately, Schenker did not answer this question; he simply denied that it is possible to derive the minor system from any natural process. Besides forcing him to regard the minor system as inferior to the major system, Schenker’s naturalism also led him to the equally dubious notion that non-triadic music is necessarily inferior to functional triadic music. By denying the value of music outside the domain of the so-called Common-Practice Period, Schenker blithely cast aside large repertories of World music, as well as Western art music from earlier or later periods. It is small wonder, then, that Joseph Kerman has accused Schenker of viewing music history “as an absolutely flat plateau flanked by bottomless chasms.”10 Schenker’s blindness on this issue is utterly entwined with his impoverished form of naturalism. One obvious way to avoid these problems is to stop grounding our theory of functional monotonality in some culturally independent phenomenon, such as the overtone series or ‘The Chord of Nature,’ and acknowledge that the theory is intended to explain and predict the properties of one particular type of music. By linking the theory to specific pieces produced by particular communities, we can regard the laws of tonal voice leading and harmony not as absolute truths, fixed for all times and all people, but rather as empirical generalizations that apply to some historically and culturally defined corpus for analysis. Having taken this step, we can then explain the significance of the major and minor systems in very different ways. For example, instead of deriving the major system from the overtone series and the minor system by analogy, we could show empirically that in functional monotonal music of the Common-Practice Period the most common triads are those that are diatonic to a particular key: in C major, the most common chords will be major triads on C, F, and G, minor triads on D, E, and A and diminished triads on B; and in C minor, they will include minor triads on C, F, and G, major triads on E
, A
, B
, G (at cadences), and diminished triads on D. In other words, we can reject figures 6.2a–g and still keep figure 6.2h. In fact, such a count has already been performed by H. Budge.11 If we treat tonality as a property of some specific culture and time period, then we have taken the first

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steps down the psychological route mentioned earlier and we can begin to engage Lerdahl and Jackendoff’s brand of naturalism. When we discuss the work of Lerdahl and Jackendoff, it is important to stress that they do not set out to defend Schenkerian theory per se. Rather, they attempted to produce a psychologically grounded theory of tonality that happens to resemble Schenkerian theory in many ways. These similarities range from sharing a common concern with hierarchy to adopting some of the same transformations and prototypes. Yet, Lerdahl and Jackendoff are quick to spot the differences between their work and Schenker’s. As they explain: Schenker can be construed (especially in Der freie Satz) as having developed a proto-generative theory of tonal music—that is, having postulated a limited set of principles capable of recursively generating a potentially infinite set of tonal pieces. But, remarkable and precursory though his achievement was, he did not develop a formal grammar in the sense one would expect nowadays of a generative theory. Moreover, his orientation was not psychological (as that of generative linguistics is), but artistic; the chief purpose of his theory was to illuminate structure in musical masterpieces.12

Though they aspire to fulfill the same tasks, Lerdahl and Jackendoff concede that their focus is primarily on musical cognition. So what aspects of human cognition do Lerdahl and Jackendoff use to support their theory of tonality? In fact, they mostly borrow from two quite separate traditions of cognitive psychology. On the one hand, Lerdahl and Jackendoff base many of their hypotheses on the principles of Gestalt psychology. They do so for several reasons. Like Gestalt psychologists, Lerdahl and Jackendoff are keen to treat perception as a dynamic process; they believe that it relies on the active, though often unconscious, participation of the person, as well as on the recognition that our local perceptions are guided by global concerns.13 Gestalt psychologists also implied that our mental representations of music consist not only of lists of pitches, but also of abstract relations among them.14 These ideas are clearly crucial to Lerdahl and Jackendoff’s work, especially their discussion of archetypes. On a more specific level, Lerdahl and Jackendoff also connect many of their preference rules to particular Gestalt principles. For example, they specifically invoke the Law of Prägnanz

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along with the principles of continuity, proximity, similarity, inclusiveness, and closure.15 Principles of Gestalt Psychology Law of Prägnanz:

Psychological organization will be as good as conditions allow. All cognitive experiences will tend to be as organized, symmetrical, meaningful, simple, and regular as they can be, given the pattern of brain behavior at any given moment. Principle of proximity: When stimuli are close together they tend to be grouped together as a perceptual unit. Principle of inclusiveness: When there is more than one figure, we are most likely to see the figure that contains the greatest number of stimuli. Principle of similarity: Objects that are similar in some way tend to form perceptual units. Principle of continuity: Once patterns in stimuli are established, we expect them to continue. Principle of closure: Incomplete figures in the physical world are perceived as complete figures.

They support their case with more recent research by Shiman, Shepard, Marr, and others.16 On the other hand, Lerdahl and Jackendoff draw extensively on the field of generative linguistics. As they make perfectly clear, they set out to build a theory that is analogous to the grammatical models proposed by Noam Chomsky and others.17 Such models attempt to characterize what human beings know, often unconsciously, when they speak a language. They want to explain how humans are able to understand and create an indefinitely large number of sentences, most of which they have never heard before. Chomsky and others model this knowledge by a formal system of principles or rules called a grammar; each grammar describes or generates the possible sentences in the language.18 Lerdahl and Jackendoff do recognize that there are important differences in methodology between music theory and linguistic theory; however they adopt much of the same formalisms as their linguistic models. Despite its extraordinary impact on contemporary music theory, Lerdahl and Jackendoff’s work is not immune from criticism. To begin with, they certainly show important connections between

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formalism and mental representation, but they seem to tailor their formalism to fit Chomskyan models, rather than adjust their formalism to fit the peculiarities of tonal music. As a result, they fail to capture certain essential features of the listening experience. Take, for example, their insistence that functional tonality can be represented by a strict context-free grammar. Although many aspects of tonal music can indeed be represented in this way, it is clear that others cannot. Imagine, for example, the simple pattern in C major with the stepwise descent  3– 2– 1 in the soprano and the progression I–V–I in the other voices. If tonal grammar is, indeed, context free, then it should be possible to compose out the initial tonic Stufe with a motion to  3/VI. The problem is that, when considered within the context of the progression from I to V, the  3/VI Stufe creates parallel octaves and fifths with the adjacent  2/V Stufe. In other words, how we compose out a given Stufe depends in part on the context in which that Stufe appears. As we saw in chapter 3, this issue becomes very important in the generation of sequences; it is striking to see that Lerdahl and Jackendoff run into consistent problems when they grapple with sequential passages. As shown in figure 6.3 (Lerdahl and Jackendorff’s derivation of Bach’s Prelude in C, WTC I), their prolongational reductions often contain parallel perfect octaves and fifths between successive Stufen. Since such parallels are not directly audible at the surface of the music, their reduction does not seem to conform with the way people actually hear the passage. If Lerdahl and Jackendoff are unable to capture this basic intuition about tonal music, it is hardly surprising that Schenkerians often find their analyses of more complex passages arid from an interpretative perspective.19 Other difficulties stem from Lerdahl and Jackendoff’s rather narrow conception of what a theory of tonality should do; they go too far in distinguishing between the activities of listening and composing, and they do not go far enough in discriminating between different levels of expertise. In the first case, we have already seen that, whereas Schenker saw his goal in compositional terms as creating a production system for generating an infinite number of tonal pieces, they see their goal in psychological terms as a means for describing how people represent tonal relationships when they listen. There are good reasons, however, why we should

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Figure 6.3. Lerdahl/Jackendoff’s derivation of Bach, Prelude in C, WTC I. From Lerdahl and Jackendoff, A Generative Theory of Tonal Music, p. 262.

not separate these activities too sharply. For one thing, we have absolutely no reason to suppose that when expert composers listen to music, they process their knowledge in different ways than when they compose. On the contrary, the evidence suggests that expert listening requires similar mental representations to expert composition.20 For another, although ordinary listeners are unable to comprehend immediately every aspect of an expert composition, such as Beethoven’s “Eroica” symphony, they can still appreciate some of

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its features; if not, then it is hard to understand why they are able to recognize and value exceptional feats of musicianship. In the second case, Lerdahl and Jackendoff focus their thoughts exclusively on the “experienced listener”; they pay little or no attention to explaining how experienced listeners actually acquire and even refine their knowledge. This omission is most serious with respect to their claims about the audibility of long-range tonal motion. Although they seem to attribute the ability to hear prototypes to all experienced listeners, there is good reason to suppose that this is not actually the case. Nicholas Cook, for example, has conducted experiments to show that music students do not hear tonal closure on a large scale.21 Such a result is hardly surprising given the fact that expert Schenkerians often disagree when picking the prototype for a given piece. As Robert West, Peter Howell, and Ian Cross stress, “Lerdahl and Jackendoff’s model may overestimate the perceptual and cognitive propensities as well as powers of even musically educated listeners. In fact, the features used to pattern a piece of music on a single hearing may well differ from those used if the listener is given time to study a piece.”22 To make matters worse, Lerdahl and Jackendoff make no effort to interface their theory with traditional approaches to analysis and theory; they completely ignore the idea that their theory should be coherent with what we already know from these areas. To negotiate these problems, it is useful to reconsider the topics discussed in chapters 1 and 2. As we saw earlier, Schenkerian theory can be boiled down to four basic claims. First, the laws of strict counterpoint must be transformed to explain the behavior of tonal voice leading. This is ‘The Heinrich Maneuver.’ Second, the laws of tonal voice leading and harmony are interdependent. We referred to this as ‘The Complementarity Principle.’ Third, any complete, continuous tonal progression can be derived recursively from a simple string of essential lines and essential harmonies using a finite set of transformations. We referred to this as ‘The Recursive Model.’ Fourth, any complete, continuous, monotonal composition can be derived recursively from a single prototypical string of essential lines and essential harmonies, using a finite set of transformations. This is ‘The Global Paradigm.’

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If we think of these four claims as forming a single schema, then they provide us with a mechanism both for connecting listening with composition and for explaining how ordinary people acquire the necessary skills to become experts (see figure 6.4, Learning curve for expertise at functional monotonal composition). This figure suggests that, whereas ordinary listening might simply involve understanding ‘The Heinrich Maneuver’ and ‘The Complementarity Principle,’ the process of composition requires that a person can internalize the relevant laws of tonal voice leading harmony as a set of prototypes, recursive transformations, and levels. In this sense, ‘The Recursive Model’ and ‘The Global Paradigm’ are especially important to composers. Furthermore, progressing from the first claim to the last would seem to require greater and greater levels of expertise. For example, the laws of voice leading implied at the top of figure 6.4 reflect certain basic intuitions about melodic motion and closure, the relative motion of lines, and the relative stability of vertical intervals, whereas the ability to perceive lines and chords together might require a certain level of training. Similarly, recognition of prototypes and transformations would seem to demand intimate knowledge of specific repertories, but the capacity to control an entire tonal piece would seem to involve a profound understanding of large-scale organization. This again seems to be a sign of expert behavior. In other words, figure 6.4 provides us with a simple mechanism for explaining how people acquire expertise at understanding and composing functional monotonal music. And there are other bonuses too. Significantly, figure 6.4 reflects the order in which Schenker actually developed his theory. In fact, he outlined the basic elements of ‘The Heinrich Maneuver’ and ‘The Complementarity Principle’ before 1914 in his Harmonielehre and Kontrapunkt, he developed ‘The Recursive Model’ in his editions of the late Beethoven sonatas from 1913 to 1920, and he explained ‘The Global Paradigm’ in his writings of the 1920s and 1930s—Der Tonwille, Das Meisterwerk in der Musik, Fünf Urlinie-Tafeln, and Der freie Satz. Figure 6.4 also fits with the broad outline of traditional theory instruction. For example, counterpoint is ideally taught first, and large-scale form last. But we are still left with some basic questions: To what extent does this model map onto the things we know

Renaturalizing Schenkerian Theory Figure 6.4. Learning curve for expert monotonal composition.

221

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about music cognition? How might we go about testing the model’s cognitive implications? In the remainder of this chapter, we will begin to offer answers to these two provocative questions.

Schenkerian Theory as a Model of Expert Functional Monotonal Composition Before we consider some of the specific ways in which figure 6.4 maps onto recent research in music psychology, it is important to stress that there are important methodological differences between the goals of music theory and those of experimental/cognitive psychology. Whereas the former is concerned with explaining the behavior of specific pieces or repertories of music that exist in the external world, the latter is concerned with explaining the internal mental processes that allow human beings to create or understand such music. Although these two activities may be related, the connections may not always be direct, and the manner of testing the respective theories may not always be identical. It is also important to remember that just as we can explain musical structures at different levels, so we can also explain psychological processes at different levels.23 These levels range from the psycho-acoustic level of explaining how people actually hear individual sounds to the higher cognitive levels of explaining how people understand complex musical structures, such as themes, harmonies, and forms. Again, we cannot assume that these different levels of explanation map onto one another in any simple way. Although there has been no systematic attempt to examine Schenkerian theory from a psychological perspective, many specific laws of tonal voice leading and harmony have already been confirmed by psychological experiments.24 To begin with, consider the laws of melodic motion. In chapter 1, we saw that tonal melodies usually begin on 8, 5, or  3 and reach maximum closure on  2– 1; that they primarily move by step; and that, if leaps occur, then they do so when the melody shifts from one harmonic tone to another or from one contrapuntal voice to another. Research by Carol Krumhansl, J. J. Bharucha, and others have shown that, in tonal contexts, listeners do indeed perceive pitches of the tonic

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triad as more stable than others, and that unstable tones generally move by step onto stable or “anchored tones.”25 Leonard Meyer, Eugene Narmour, and others have also shown that listeners expect leaps to be followed by step motions, though their explanations of gap filling is not immune from criticism.26 As Diana Deutsch has shown, pitch proximity may be an advantage in processing efficiency; this suggests that it might be tied to the Gestalt Law of proximity.27 The issue of melodic closure is less clear cut. Burton Rosner and Eugene Narmour have shown that  2– 1 does indeed produce strong melodic closure, but they also note that “listeners showed no closure preference for a V–I progression with scale steps 28  2– 1 over those with 7– 1 or  2– 3.” This result essentially confirms the closure rules for other polyphonic voices, but it does not give a preference to  2– 1 in the soprano. To account for the latter, we must remember that, globally, well-formed melodies tend to rise before falling to a cadence. Although this is an important aspect of what Schenker referred to as melodic fluency, there does not seem to be any experimental evidence to support this view. Similarly, we can find some support for the laws of relative motion. In chapter 1, we noted that at points of maximum closure, the “soprano” voice normally ends on  2– 1, the “alto” ends 7– 1, the “bass” 5– 1, and the “tenor” 5– 3; that the soprano and bass essentially move in contrary motion or parallel thirds and sixths; that parallel perfect octaves and fifths do not occur when two essential lines move in the same direction; and that, if parallel perfect octaves and fifths do occur, then they arise from doubling and figuration or from combinations of non-harmonic tones. As mentioned above, Rosner and Narmour have effectively confirmed the first law, even though they do not give preference for  2– 1 as a soprano. Unfortunately, less attention has been paid to the predominance of contrary motion and the lack of parallel perfect octaves and fifths. There does not appear to be a systematic study of whether people hear permissible parallel octaves and fifths as our laws suggest. In chapter 1, we also suggested that melodies generally move between triadic tones and that if non-harmonic tones occur, then they move by step between harmonic tones or by leap between contrapuntal lines. The discussion of unstable tones by Krumhansl,

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Bharucha, et al., is not confined to harmonic tones; it also covers non-harmonic tones. Again, they show that unstable tones generally move by step onto the “anchoring tones,” that stable tones are perceived to be more closely related to each other than unstable tones, and that unstable tones are perceived to be more closely related when the stable tone follows the unstable tone than vice versa.29 Unfortunately, there seems to have been little attempt to investigate the relative stability of different types of non-harmonic tones. As for the laws of tonal harmony, we identified several specific laws of harmonic classification. These laws stated that tonal music is essentially built from seven diatonic triads and that these triads must contain the third and normally double the root, then the fifth, and then the third. In a series of experimental studies, Krumhansl, Bharucha, and others have shown that, in tonal contexts, composers and listeners favor the diatonic over chromatic triads, and that they regard the tonic, dominant, and subdominant as the most stable of these diatonic triads. Rosner and Narmour, however, have raised doubts about the functional claim that the seven triads fulfill just three harmonic functions; they suggest that VII triads do not normally substitute for V.30 No reference to any psychological studies of the doubling laws have been found. When we turn to the laws of harmonic progression, we find strong confirmation for our basic law, namely that maximum closure occurs when the dominant moves to the tonic. To quote Rosner and Narmour: [L]isteners never equated the harmonic closure of perfect authentic cadences with that of plagal ones. In comparing isolated chord pairs, subjects always judged V–I (in whatever guise) significantly more closed than IV–I. In terms of closure, listeners even preferred III–I progressions with scale steps 7–8 in the soprano over IV–I, albeit weakly.31

They add that, “in formulating a generative structural constant like the Ursatz, Schenker appears to have been quite correct to isolate the V–I progression over all others.” The situation is much the same for our laws of chromatic generation. In fact, there is considerable experimental support for the notion that tonal music is fundamentally diatonic. Krumhansl and Bharucha have stressed that, although diatonic tones are generally

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considered more stable than nondiatonic tones, “a nonchord tone that is nondiatonic but anchored is more stable that one that is diatonic but not anchored.”32 Unfortunately, we do not have similar support for the notion that chromaticisms arise from mixtures and tonicization; there appear to be no published systematic studies of this point. This is not the case for ‘The IV/
V Hypothesis’: according to Krumhansl, listeners do indeed hear a disjunction when two triads a tritone apart are presented simultaneously. In the case of triads on C and F, she observes: The tendencies of unstable tones in these keys would often work in opposition. The tone F, for example, would be highly unstable in the key of C major (with a strong tendency to resolve to G), but would be the most stable tone in F major. Similarly, the tone C would be highly unstable in the key of F major (with a strong tendency to resolve to C), but would be the most stable tone in C major.33

In other words, it is hard to understand either triad in the context of the other. Stepping back from the individual laws, there is some evidence to support ‘The Heinrich Maneuver’ per se. Although music psychologists do not seem to be interested in testing the similarities and differences between strict counterpoint and tonal composition, David Huron has provided a wealth of experimental support for the psychological bases of traditional laws of counterpoint.34 In particular, he suggests that the laws of voice leading are, indeed, different in monophonic, homophonic, and polyphonic repertories. And with regard to ‘The Complementarity Principle,’ Bharucha claims that listeners tend to hear tonal melodies in relation to some underlying chord progression. As he puts it: The melodies used in the first two experiments were thought to imply chords but were never explicitly accompanied by them. Subjects abstracted an underlying chord from the melody, in line with Schenker’s theory. The explicit sounding of a chord was avoided to test the view that harmonic considerations are decisive even for melodies heard alone. This has been borne out, even for the musically untrained.35

Such statements fit in well with Huron’s observation, quoted at the end of chapter 4, that tonal music is more harmonically driven than other types of music.

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Just as we can be optimistic about the psychological basis of our specific laws of tonal voice leading and harmony, as well as the precise dynamics of ‘The Heinrich Maneuver’ and ‘The Complementarity Principle,’ so we also have good reason to be optimistic about the cognitive significance of ‘The Recursive Model’ and ‘The Global Paradigm.’ There is good reason to suppose that ordinary people are able to abstract prototypes from complex surfaces and that as they gain expertise, so they are able to abstract them across longer and longer spans. And there is good reason to suppose that expert composers are indeed able to abstract a single prototype across a complete, continuous monotonal piece. The notion of prototypes has become a staple in psychological research at least since the early 1970s.36 It has likewise attracted considerable attention from music psychologists, such as Diana Deutsch and John Feroe, John Sloboda, Mary Louise Serafine, and others.37 For the present purposes, two studies are of particular interest. John Sloboda and David Parker have shown that when people memorize well-formed tonal melodies they build a simplified mental model of the underlying structure and then fill in structurally marked slots according to general constraints about what is appropriate to the piece or genre. They also suggest that different levels of structure are available to people with different amounts of expertise: “musicians code harmonic relationships that seem less accessible to non-musicians.”38 Mary Louise Serafine, meanwhile, has supported the notion that the capacity to perceive prototypes increases with experience.39 After a series of experiments involving short, unaccompanied melodies, she concluded that “simple underlying structures were accessible to subjects at age 8 and above, but examples of the more complicated structures involved in harmony and compound melody yielded equivocal findings.”40 This having been said, it is one thing to claim that people understand tonal music using prototypes and quite another to claim that they are doing so using the specific prototypes and transformations posited by Schenkerian theory; no extensive research appears to have been conducted to confirm the latter. Researchers in human cognition have also argued that, as people gain expertise in specific domains, so they rely more and more on

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prototypes. Starting in the 1960s, psychologists discovered that, whereas novice chess players simply learn rules for moving individual pieces, grand masters conceive of games according to large meaningful patterns. For example, Adrian De Groot, and later William Chase and Herbert Simon, have shown that grand masters can reproduce complex patterns of chess pieces in short periods of time when these patterns belong to games, but not when they are randomly placed on the board.41 Grand masters apparently do so by comparing the pieces to a library of prototypical games or portions of games that they have stored in a highly organized way in their long-term memories. This library of successful games and portions of games allows them to consider strings of moves in one go, thereby streamlining their play considerably. More recently, Michelene Chi and Robert Glaser have noted that the results of De Groot, Chase, and Simon have been replicated in other domains, from reading circuit diagrams and architectural plans, to recalling computer programs. They also stress that this ability reflects more sophisticated ways of organizing knowledge rather than superior perceptual capacity.42 For his part, John Sloboda has drawn attention to the role global prototypes play in composing music. As he explains, “[T]he art of composition lies, in part, in choosing extensions of initial thematic ideas that honour superordinate constraints.”43 Without such overall mental images, it is hard to explain how expert composers are able to produce large quantities of music so fast, how they are able to recall several pieces at once, and how they can recall pieces out of order. Sloboda gives such global prototypes a prominent place in his overview of the processes of musical composition. This is shown in figure 6.5 (Sloboda’s diagram of typical compositional resources and processes). It is clear from the arrows in figure 6.5 that Sloboda does not regard these plans as absolutely fixed: on the contrary, he concedes that “they can . . . be changed in light of the way a particular passage ‘turns out.’ ” 44 Yet he nonetheless regards them as fundamental to the way expert composers work. Sloboda supports his claims with four main types of evidence. First, he turns to composers’ own workings. To quote him: [S]ketches and notebooks, if datable, can show how a composition grew and changed over the time during which it exercised the composer’s

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Figure 6.5. Sloboda’s “Diagram of typical compositional resources and processes.” Adapted from Sloboda, The Musical Mind, p. 118.

mind. . . . Even where sketches do not exist we can sometimes discover something about the compositional sequence from the final manuscript.45

Sloboda finds evidence of global plans in the workings of specific pieces by Beethoven, Stravinsky, and Mozart. Although he does not actually say so, composers sometimes leave explicit outlines of entire movements or pieces. For example, Lewis Lockwood has noted that, in Beethoven’s sketchbooks, “incipits of movements often appear, sometimes connected by words showing briefly what the intended order of sections or movements is to be.”46 Among the best examples are those for the String Quartet in C Minor, Op. 131.47 Of course, it is important to stress that composers’ workings do present a number of technical problems for the researcher; these are problems that often cut across the boundaries between cognitive psychology, music theory, and historical musicology. But the potential value of such documents is still enormous.48 Second, Sloboda supports his case with evidence from composers’ personal testimonies, citing four letters by Richard Strauss (1949),

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Beethoven (1822), Mozart (1789?), and Roger Sessions (1941). Of these, perhaps the most interesting is the one attributed to Mozart, which explicitly mentions the idea that composers have an overall mental image of the entire work when they write. The letter states: “[A]ll this fires my soul, and provided I am not disturbed, my subject enlarges itself, becomes methodized and defined, and the whole, though it be long, stands almost complete and finished in my mind, so that I can survey it, like a fine picture or a beautiful statue, at a glance.”49 What is most interesting is that Schenker himself cited this famous letter in par. 301 of Der freie Satz, along with other comments by C. P. E. Bach, Haydn, Beethoven, and Brahms, that also confirm the notion of some sort of global paradigm. Just as there are problems dealing with composers’ workings, so there are problems dealing with their personal testimonies. For example, the authenticity of the Mozart letter quoted by Sloboda and Schenker has been disputed; most musicologists now accept that it is a forgery. Ironically, although the letter may be phony, Oswald Jonas and Otto Erich Deutsch still claim that it sounds as if it had been written by Mozart!50 Third, Sloboda appeals to direct observations of individual composers working in real time. In particular, he relies on a verbal protocol obtained from an unnamed composer who was given the problem of writing a piano fugue, as well as on one Sloboda himself used in composing a choral work.51 Verbal protocols are statements by an expert of what they are doing, while they are doing it; though they are by no means easy things to deal with, such statements have, in fact, become essential tools in research concerning problem solving. Although Sloboda provides few details of the fugue protocol, he notes that his own protocol emphasizes an interplay between local and global concerns: All the way through the protocol I find myself using some feature of melody or harmony in previously written material as the starting point for a continuation. On the other hand, there are many cases in which super-ordinate considerations of harmony and structure prescribe changes which modify or overwrite the essentially imitative strategies which generate the first continuations.52

Sloboda ends by suggesting some of the ways in which protocols can be developed to generate more fruitful scientific research into composition.

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Fourth, Sloboda cites the results of research into the process of improvisation. This time, however, his remarks about global plans are somewhat equivocal: when comparing improvisation to the task of telling stories, Sloboda again stresses the importance of some overall scheme, but when discussing the nature of jazz improvisation, he suggests that in improvisation “the crucial factor is the speed at which the stream of invention can be sustained, and the availability of things to do which do not overtax the available resources.”53 As it happens, there is considerable evidence that classical composers traditionally learned their craft by improvising pieces from global prototypes. A recent paper on the origins of the keyboard prelude by Panayotis Mavromatis has shown that fifteenthcentury organists were taught to compose and improvise by learning to elaborate simple polyphonic prototypes or fundamenta.54 Sixteenthcentury Italian preludes draw on similar prototypes based on so-called intonatione. Mavromatis demonstrates how these same prototypes can be found in preludes written by seventeenth- and eighteenth-century composers such as Buxtehude and Bach. Similar observations have been made by Richard Hudson in his research on mode in guitar music from the early seventeenth century.55 As it happens, the work of Mavromatis and Hudson confirms Schenker’s own claims about the role global prototypes play in improvisation: The ability in which all creativity begins—the ability to compose extempore, to improvise fantasies and preludes—lies only in a feeling for the background, middleground, and foreground. Formerly such an ability was regarded as the hallmark of one truly gifted in composition, that which distinguished him from the amateur or the ungifted.56

Schenker went on to stress that studying the fantasies, preludes, cadenzas, and embellishments of the expert composers should be a “high priority” for all music instruction.57 He even claimed that improvisation provides a crucial means of developing a composer’s musical memory.58 We can even go further to suggest that improvisation plays a central role in the process of internalization; by creating new pieces on the fly, budding composers are able to internalize the declarative laws we outlined in chapter 1 into the procedural form of an Ursatz, recursive transformations and levels.

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Figure 6.6. Schenker’s account of expert monotonal composition. From Schenker, Der freie Satz, Fig. 13.

As it stands, then, we have good reason to believe that the schema in figure 6.4 gives us a plausible account of how, through increased expertise, ordinary listeners can become expert composers of functional monotonal music. It suggests that, as children, human beings develop the basic notions of step motion, convergence on the tonic, and motion between stable tones. These notions may be linked to even more basic cognitive capacities, such as the Gestalt principles of continuity, proximity, similarity, inclusiveness, and closure. The ability to perceive lines and chords together and to recognize prototypes seems to demand more extensive exposure to specific repertories. The capacity to comprehend tonalities across entire compositions would seem to require even higher levels of expertise. For the record, the schema presented in figure 6.4 stands in direct contrast to Schenker’s own account of expert composition. According to him, genius is born, not made: “The master composer enters the scene in isolated instances—the man of moderate ability is always there. Never can there be a connection between them!”59 To underscore this point, Schenker included his own illustration, given here as figure 6.6 (Schenker’s account of expert monotonal composition). Here he insisted that there is no way to bridge the gap between the capacities of the average person (Durchschnitt) and those of the genius (Genie). Unfortunately, Schenker’s example was cut from Oswald Jonas’s second edition of Der freie Satz and hence from Ernst Oster’s subsequent English translation. This chapter has considered the role coherence plays in building and testing music theories. We have taken the cognitive path to naturalization because knowledge of music does not exist in the environment; it exists in our heads and is shaped by the basic processes of human cognition. Of course, our knowledge of music theory is also shaped by knowledge drawn from other areas, such as

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Figure 6.7. The scope of music theory.

acoustics, cultural and biographical information, aesthetics, and so forth. We can convey this in the simple chart given in figure 6.7 (The scope of music theory). Following the traditional division of music theory into musica practica and musica speculativa, the chart lists four practical areas that inform and are informed by such knowledge of music theory—composition, performance, music history, and analysis—and below it suggests four speculative areas that do the same—acoustics, cognitive studies, social studies, and aesthetics. Notice how each arrow points in two directions; this suggests there is a constant flow of knowledge from one discipline to the other. In a very general way, the chart shown in figure 6.7 conforms to the image of research in music theory outlined at the start of this book. In the Introduction, we suggested that music theory is always open-ended; music theorists do not begin with a blank slate, they do not have foolproof methods, and they do not reach definitive solutions. Instead, they plunge in medias res. They always start working within the context of an existing music theory, even if they know some portions of that theory are surely wrong. They then try to overcome certain specific problems, using the rest of the theory to support their work. As they fix one problem, so another appears; bit-by-bit the theory becomes transformed into something ever new. And so it is on a higher level with the chart shown in figure 6.7. Music scholarship is always open ended; composers, performers, historians, and theorists all rely on received knowledge from other areas, even though they know that some this information is probably false. As they try to solve problems within their

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own area of interest, so they take for granted any received knowledge they might have from these other disciplines. As problems are solved in one area, others will appear; gradually music scholarship as a whole becomes transformed into something new. In this respect, our knowledge of music theory is part of a more general fabric of knowledge that Quine and Ullian refer to as ‘The Web of Belief.’60 As we propose the chart given in figure 6.7, it is important to stress that, although music theory is placed at the center, this does not mean that it necessarily has any epistemic priority over the other areas of inquiry. On the contrary, music theory appears at the center simply because it happens to be that area of inquiry that I have chosen to discuss in this particular book; scholars working in other areas will place their own discipline center stage and the other disciplines on the peripheries. This is as it should be. After all, figure 6.7 does not present a hierarchy of knowledge about music; it simply shows a network in which ideas from one domain flow to another. This last point is important at the present time because scholars are constantly trying to prioritize one area of research over another. On the one hand, many historians criticize music theorists for ignoring the cultural context of the works they analyze; they insist that unless such knowledge is taken into account, music cannot be understood adequately. On the other hand, many music theorists criticize historical musicologists for offering superficial analyses of musical compositions; they insist that there is more to understanding a piece than understanding the cultural circumstances from which it came.61 By presenting figure 6.7 as a network rather than a hierarchy, I claim that there is no a priori reason for prioritizing one domain over another. Each discipline can offer valuable insights about the nature of music; we should all learn to respect these differences and look for constructive ways for allowing scholars working in one domain to communicate with those working in another.

Conclusion In his well-known essay “The Americanization of Heinrich Schenker,” William Rothstein claims that, however they choose to regard their work, Schenkerians must eventually meet “the challenge posed by the ‘scientific’ theorists.”1 During the course of this book, we have seen one way in which this challenge might be met. We have seen that Schenkerian theory is explanatory insofar as it explains why certain notes appear in particular tonal contexts, why these notes behave in some ways and not in others, and how we can actually generate specific tonal relationships. It does so by invoking an appropriate set of concepts and covering laws, which it represents in procedural form as a system of prototypes, transformations, and levels. When picking one analysis over another, Schenkerians are guided by some of the same criteria as scholars working in the natural and social sciences, namely, they are motivated by a desire for accuracy, scope, consistency, simplicity, fruitfulness, and coherence. Besides accepting Rothstein’s challenge, this book has responded to some of the other issues outlined in the Preface. We have avoided the “cabalistic image of Schenker” abhorred by Dunsby by showing how Schenkerian theory is based on explicit laws of tonal voice leading and harmony. Since these laws are testable intersubjectively, we can not only reject Schenker’s appeal to “magic numbers” or other mystical forces, but we can also counter the charge of circularity as leveled by Narmour. As mentioned earlier, the explanatory laws underpinning Schenkerian theory were actually discovered empirically in the Harmonielehre and Kontrapunkt I, long before Schenker formulated his concept of a single tonal prototype. By the same token, we have eliminated some of the inconsistencies discussed by Benjamin. What is most remarkable is that we did so by using some of Schenker’s own ideas. This does not mean that we have necessarily resolved every anomaly in Schenker’s work, but it does mean that we have overcome one particularly pressing problem. In addition to shoring up the methodological foundations of Schenkerian theory, we have also clarified several other areas of

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debate. For one thing, we have responded to Laufer’s claim about the completeness of Schenker’s concepts. By grounding Schenkerian theory in a set of covering laws, we have been able to show why it posits just three forms of prototype, why it proposes a finite number of transformations, and why it orders the generative levels in some ways but not in others. We have also placed some definable limits on what kinds of music this theory can explain. In chapters 4 and 5 we specified what some of these limits might be and have suggested how, in principal, other repertories conform to related sets of covering laws. This move allows us to counter Kerman’s charge that Schenkerian theory necessarily sets functional tonality apart from the more general history of Western music. Finally, we have indicated that the interrelationship between hearing, performing, and composing is a lot more complex than many experts seem to suppose. Schenkerian theory does not simply model how ordinary people “hear” a piece of music; rather it tries to identify what types of internalized knowledge expert composers use when they try to create successful monotonal music. When reconstructed in the preceding manner, Schenker’s intellectual achievements do indeed seem formidable and go well beyond those of any other twentieth-century music theorist. And yet, there is no reason to suppose that the theory discussed in the Neue musikalischen Theorien und Phantasien is beyond refinement or extension. Indeed, if we take the image of Neurath’s boat seriously, then we can safely assume that Schenker’s work has its own share of leaks and that it is our job to find and patch them. By way of conclusion, it is worth mentioning a few obvious areas in need of patching. To begin with, chapter 1 showed how Schenker refined the traditional laws of strict counterpoint and functional harmony in order to provide more accurate explanations of how lines and chords behave in functional tonal music. These new laws may indeed be more powerful than their predecessors, but they should not be taken as the final word. Take, for example, the case of so-called direct chromatic successions. Although Schenker admitted that such successions can sometimes occur in functional contexts, he nonetheless conceded that they are often avoided. What he did not do, however, was to propose a general law to explain why direct chromatic successions are possible in some situations, but not in others. If we are able

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to discover such a law, then we might be able to offer fresh insights about music not only from the Common-Practice Period, but also from the late nineteenth and twentieth centuries. Similarly, chapter 2 demonstrated how Schenker reformulated his revised laws of tonal voice leading and harmony as a system of prototypes, transformations, and levels. By subsuming the theories of tonal voice leading and harmony within a single framework, Schenker was able to expand the scope of tonal theory considerably. But this is by no means the only way to achieve such a goal. One could, for example, unify the theory of functional monotonality with general theories of tonal rhythm and tonal form. These endeavors are well underway: on the one hand, Carl Schachter, William Rothstein, and others have been busy trying to link Schenkerian theory to a theory of tonal rhythm; on the other hand, Joel Galand, Charles Smith, Janet Schmalfeld, and others have been connecting Schenkerian theory to a theory of tonal form.2 Both of these projects hold great promise for the future. In chapter 3 we addressed the issue of consistency. In particular, we overcame certain inconsistencies in Schenker’s treatment of parallel perfect octaves and fifths by invoking his own discussion of combined linear progressions. This strategy proved extremely useful in generating sequences. But this is surely one of many contradictions in Schenker’s work. To eliminate the others, we must have a clear idea of what these other contradictions are and why they occur. One obvious way to do so is by formalizing the theory on the computer. Building on the work of Kassler, Snell, Smoliar, and others, Panayotis Mavromatis and I have set about this very task by implementing Schenkerian theory as a definite clause grammar in Prolog. This implementation was possible only by thinking about Schenkerian theory as a set of declarative rules, like those described in chapter 1, and by reformatting them as a production system, in the manner described in chapter 2. Of course, other implementations can surely be devised, depending on how the theory is reconstructed. In the same vein, chapter 4 illustrated one way in which Schenkerian theory simplifies our understanding of tonal relationships: by showing how modal and exotic phenomena can be generated by the processes of tonal transformation, it is able to reduce

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the number of theoretic systems enormously. Once again, other examples of simplification can surely be found. For example, we might find ways to simplify Schenker’s notion of composing out and reduce the total number of transformations. In this respect, the concept of “reaching over,” which we discussed in chapter 2, seems to be a prime target; Schenker’s account of the term in Der freie Satz is particularly obscure. Yet again, the computer promises to be a powerful tool in helping us streamline the theory even further. As for the matter of fruitfulness, chapter 5 tackled this issue in two different ways: on the one hand, it explained the tonality of music that lies outside Schenker’s original corpus, and on the other, it used our explanations of a work’s tonal structure to shed light on other aspects of its composition. In the first case, we used Schenkerian theory to analyze two works by Debussy, a composer whose music Schenker despised intensely. We saw that the theory has built-in mechanisms for explaining many anomalous features of Debussy’s style and that these mechanisms can be extended to explain the tonality of music composed after Debussy’s death. In the second case, by analyzing two songs from a Schenkerian perspective, we were able to shed new light on Debussy’s approach to text setting. These insights tied in nicely with what we know about Debussy’s interest in symbolist aesthetics. Finally, chapter 6 described how Schenkerian theory is indeed coherent with recent research in music cognition. This discussion implied that there is much to be gained from grounding our music theories cognitively. The chapter closed by suggesting that our image of Schenkerian theory offers a plausible account of how ordinary listeners can become expert composers of functional monotonal music; it required both the learning of explicit laws and the internalization of these laws in some procedural form. This model is, in effect, a model of how people learn to master functional monotonal composition. Since this model has extraordinary implications both for music psychology and music pedagogy, it clearly needs to be tested in a systematic manner. Although we cannot predict how Schenkerian theory will develop over time, there is every reason to suppose that it will continue to have an enormous impact on the ways in which we think about tonal music. Obviously, historians must continue to

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investigate the origins and development of Schenker’s thought. It is, after all, vital that we have an accurate picture of where his ideas came from and why they evolved as they did. Similarly, analysts must continue to produce Schenkerian readings of individual pieces. Strange as it may seem, we still lack extensive analyses of many central works in the tonal canon. In this way, we can enhance our understanding of the tonal system with a more complete picture of tonal practices. And theorists must continue to prod and poke at Schenker’s ideas. In this respect, our role in history is no different from his; where Schenker patched the leaks in traditional theories of tonal voice leading and harmony, so we must patch the leaks in Schenker’s theory of functional monotonality. As we replace each plank, so we will expose new and perhaps more profound problems. It is testimony to Schenker’s brilliance as a thinker that he was able to steer the discipline of music theory in a new direction.

Notes Preface 1. For detailed lists of Schenker’s works and works about Schenker, see David Beach, “A Schenker Bibliography,” Journal of Music Theory 13, no. 1 (1969): 2–37; David Beach, “A Schenker Bibliography 1969–1979,” Journal of Music Theory 23, no. 2 (1979): 275–86; David Beach, “The Current State of Schenkerian Research,” Acta Musicologica 57 (1985): 275–387; Larry Laskowski, Heinrich Schenker: An Annotated Index to His Analyses of Musical Works (New York: Pendragon Press, 1978); Nicholas Rast, “A Checklist of Essays and Reviews by Heinrich Schenker,” Music Analysis 7, no. 2 (1988): 121–32; Benjamin McKay Ayotte, Heinrich Schenker: A Guide to Research (New York: Routledge, 2004); and David Carson Berry, A Topical Guide to Schenkerian Literature: An Annotated Bibliography with Indices (New York: Pendragon, 2004). 2. Heinrich Schenker, Neue musicalische Theorien und Phantasien, vol. 1: Harmonielehre (Stuttgart and Berlin: Cotta, 1906); in English as Harmony, ed. Oswald Jonas and trans. Elisabeth Mann Borgese (Chicago: University of Chicago Press, 1954). 3. Heinrich Schenker, Neue musicalische Theorien und Phantasien, vol. 2: Kontrapunkt I (Stuttgart and Berlin: Cotta, 1910); Kontrapunkt II (Vienna: Universal, 1922); in English as Counterpoint I and II, ed. John Rothgeb and trans. John Rothgeb and Jürgen Thym (New York: Schirmer, 1987), corrected ed. (Ann Arbor, MI: Musicalia Press, 2001). 4. Heinrich Schenker, Neue musicalische Theorien und Phantasien, vol. 3: Der freie Satz (Vienna: Universal, 1935), 2nd ed. (Vienna: Universal, 1956); in English as Free Composition, ed. and trans. Ernst Oster (New York: Longman, 1979). 5. Heinrich Schenker, Der Tonwille, vols. 1–10 (Vienna: A. Guttman, 1921–24); in English as Der Tonwille Pamphlets in Witness of the Immutable Laws of Music, 5 vols., vol. 1 ed. William Drabkin (Oxford: Oxford University Press, 2004); Das Meisterwerk in der Musik, vols. 1–3 (Munich: Drei Masken Verlag, 1925, 1926, 1930); in English as The Masterwork in Music 1–3, ed. William Drabkin (Cambridge: Cambridge University Press, 1994, 1996, 1999); and Fünf Urlinie-Tafeln (Vienna: Universal, 1932), in English as Five Graphic Analyses, ed. Felix Salzer (New York: Dover, 1969). 6. Milton Babbitt, “The Structure and Function of Music Theory,” College Music Symposium 5 (1965): 49–60; Allan Keiler, “The Syntax of Prolongation I,” In Theory Only 3, no. 5 (1977): 3–27; Allan Keiler, “The Empiricist Illusion: Narmour’s Beyond Schenkerism,” Perspectves of New Music (1978): 161–95; Allan Keiler, “On Some Properties of Schenker’s Pitch Derivations,” Music Perception 1,

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no. 2 (1983–84): 200–228; Fred Lerdahl and Ray Jackendoff, A Generative Theory of Tonal Music (Cambridge, MA: MIT Press, 1983). 7. For example, see Michael Kassler, “Explication of the Middleground of Schenker’s Theory of Tonality,” Miscellanea Musicologica 9 (1977): 72–81; James Snell, “Design for a Formal System for Deriving Tonal Music” (Master’s thesis, SUNY Binghamton, 1979); and Stephen Smoliar, “A Computer Aid for Schenkerian Analysis,” Computer Music Journal 4 (1980): 41–59. 8. Jonathan Dunsby, “Recent Schenker: The Poetic Power of Intelligent Calculation (or the Emperor’s Second Set of New Clothes),” Music Analysis 18, no. 2 (1999): 263. 9. William Benjamin, “Schenker’s Theory and the Future of Music,” Journal of Music Theory, 25, no. 1 (1981): 163. 10. William Rothstein, “Review: Articles on Schenker and Schenkerian Theory in The New Grove Dictionary of Music and Musicians, 2nd ed.,” Journal of Music Theory 45, no. 1 (2001): 206. 11. See, for example, Matthew Brown, “Review: Hedi Siegel ed., Schenker Studies and Allen Cadwallader ed., Trends in Schenkerian Research,” Music Theory Spectrum 13, no. 2 (1991): 265–73. 12. Edward Laufer, “Review: Heinrich Schenker, Free Composition (Der freie Satz), trans. Ernst Oster,” Music Theory Spectrum 3 (1981): 161. 13. Laufer, “Review: Heinrich Schenker, Free Composition,” 159–61. 14. Joseph Kerman, Contemplating Music (Cambridge, MA: Harvard University Press, 1985), 85. 15. Felix Salzer, Structural Hearing (New York: Boni, 1952). 16. Eugene Narmour, Beyond Schenkerism (Chicago: University of Chicago Press, 1977). 17. Schenker, “Further Consideration of the Urlinie: I,” The Masterwork in Music I (1925), trans. John Rothgeb, 105. 18. Benjamin, “Schenker’s Theory and the Future of Music,” 160. Benjamin is not alone is claiming that Schenker graphs can be regarded as music. For example, to quote Arthur Maisel, “Schenker was a superior music theorist because he grew more and more to think of music as music: the graphs are music—not words, not pictures, not anything else.” Arthur Maisel, “Talent and Technique: George Gershwin’s Rhapsody in Blue,” in Trends in Schenkerian Research, ed. Allan Cadwallader (New York: Schirmer, 1990), 69. 19. I am not the only person to hold this view. For example, William Rothstein distinguishes between notes, which he regards as musical entities represented in a score, and tones, which he treats as analytical abstractions inferred from the piece. According to him, “The tradition of Schenkerian graphing has been that only tones, not notes, are shown in graphs, even at foreground levels.” See William Rothstein, “On Implied Tones,” Music Analysis 10, no. 3 (1991): 295. 20. For an extensive discussion of this point, see Matthew Brown, Debussy’s ‘Ibéria’: Studies in Genesis and Structure (Oxford: Oxford University Press, 2003).

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Introduction 1. For a summary of ‘The Classical Theory of Concepts,’ see two articles by Edward E. Smith, “Concepts and Thought,” in The Psychology of Human Understanding, ed. Robert J. Sternberg and Edward E. Smith, 19–49 (Cambridge: Cambridge University Press, 1988); and “Concepts and Induction,” in Foundations of Cognitive Science, ed. Michael L. Posner, 502–5 (Cambridge, MA: MIT Press, 1989); see also Alvin I. Goldman, Philosophical Applications of Cognitive Science (Boulder, CO: Westview Press, 1993), 126ff. 2. See Matthew Brown and Douglas J. Dempster, “The Scientific Image of Music Theory,” Journal of Music Theory 33 (1989): 65–106; and Brown and Dempster, “Evaluating Music Analyses and Theories: Five Perspectives,” Journal of Music Theory 34 (1990): 247–79. 3. See Smith, “Concepts and Induction,” 505ff. 4. Goldman, Philosophical Applications of Cognitive Science, 128. 5. Ibid., 128. 6. See Schenker, Counterpoint II, part 6, chap. 3, par. 17, pp. 235–42; and Free Composition, par. 172, p. 62. 7. For a long discussion of the role concepts play in music theory, see Mark DeBellis, Music and Conceptualization (Cambridge: Cambridge University Press, 1995). 8. For general accounts of these problems, see John Losee, A Historical Introduction to the Philosophy of Science, 3rd ed. (Oxford: Oxford University Press, 1993); William Bechtel, Philosophy of Science: An Overview for Cognitive Science (Hillsdale, NJ: Erlbaum, 1988); Robert Klee, Introduction to the Philosophy of Science (Oxford: Oxford University Press, 1997); Philip Kitcher, “Explanatory Unification and the Causal Structure of the World,” in Scientific Explanation, ed. Philip Kitcher and Wesley C. Salmon, 410–505, Minnesota Studies in the Philosophy of Science 13 (Minneapolis: University of Minnesota Press, 1989); Philip Kitcher, The Advancement of Science (Oxford: Oxford University Press, 1993); David Papineau, “Philosophy of Science,” in The Blackwell Companion to Philosophy, ed. Nicholas Bunnin and E. P. Tsui-James, 290–324 (Oxford: Blackwell, 1996); Wesley Salmon, “Four Decades of Scientific Explanation,” in Scientific Explanation, ed. Philip Kitcher and Wesley C. Salmon, 3–219, Minnesota Studies in the Philosophy of Science 13 (Minneapolis: University of Minnesota Press, 1989); Martin Curd and J. A. Cover, Philosophy of Science: The Central Issues (New York: Norton, 1998). 9. The classic presentation of this problem has been offered by Henry Kyburg. According to him, we could explain why a sample of salt dissolves by claiming that it does so because a magician in a funny hat has cast a spell on the salt. If we assume that all samples of salt hexed by the particular spell dissolve in water, then this explanation conforms to Hempel and Oppenheim’s version of ‘The Covering-Law Model.’ The snag, of course, is that the magician’s spell has nothing to do with the solubility of salt in water; there must be some way of

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excluding accounts like this one. For details, see Henry Kyburg, “Comment,” Philosophy of Science 35 (1965): 147–51. 10. Nelson Goodman, “The Problem of Counterfactual Conditionals,” Journal of Philosophy 44 (1947): 113–28. 11. To quote Peter Railton, “Where the orthodox covering-law account of explanation propounded by Hempel and others was right has been in claiming that explanatory practice in the sciences is in a central way law-seeking or nomothetic. Where it went wrong was in interpreting this fact as grounds for saying that any successful explanation must succeed either in virtue of explicitly invoking covering laws or by implicitly asserting the existence of such laws”; Peter Railton, “Probability, Explanation, and Information,” Synthese 48 (1981): 248–49, cited in Salmon, “Four Decades of Scientific Explanation,” 162. 12. Carl Hempel and Paul Oppenheim, “Studies in the Logic of Explanation,” Philosophy of Science 15 (1948): 135–75. 13. For an extensive discussion of the relevance of functional explanations to music theory, see John Brackett, “Philosophy of Science as Philosophy of Music Theory” (Ph.D. diss., University of North Carolina, Chapel Hill, 2003). The classic defense of historical narratives can be found in Michael Scriven, “Definitions, Explanations, and Theories,” in Concepts, Theories, and the MindBody Problem, ed. H. Feigl, M. Scriven, and G. Maxwell, 99–195, Minnesota Studies in the Philosophy of Science 2 (Minneapolis: University of Minnesota Press, 1958); and Wesley Salmon, Scientific Explanation and the Causal Structure of the World (Princeton, NJ: Princeton University Press, 1984). 14. See Carl Hempel, “The Logic of Functional Explanation,” in Aspects of Scientific Explanation and Other Essays in the Philosophy of Science, 297–330 (New York: Free Press, 1965). 15. Sylvain Bromberger, “An Approach to Explanation,” in Analytic Philosophy, ed. R. J. Butler, 72–105 (Oxford: Blackwell, 1968). Bromberger’s original argument concerns a flagpole. Imagine a flagpole of height h casts a shadow of length l. Knowing the length of the shadow, the angle of elevation of the sun (initial conditions), the laws of light propagation and elementary geometry, we can deduce the height of the flagpole. Although such an argument is deductively valid and follows ‘The Covering-Law Model,’ it does not constitute an explanation because the flagpole causes the shadow and not vice versa. See also Hempel, Aspects of Scientific Explanation, 429–30. 16. For an extensive discussion of the role causality plays in scientific theories, see Judea Pearl, Causality: Models, Reasoning, and Inference (Cambridge: Cambridge University Press, 2000), especially the epilogue entitled “The Art and Science of Cause and Effect,” 331–58. Stathis Psillos gives a nice survey of the philosophical problems posed by the concept of causation in his recent book, Causation and Explanation (Chesham, Bucks.: Acumen, 2002). 17. W. V. Quine and J. S. Ullian, The Web of Belief (New York: McGraw-Hill, 1978), 87. 18. Richard Feynman, The Character of Physical Law (New York: Modern Library, 1994), 150.

Notes, pp.12–17

243

19. See Losee, A Historical Introduction, 158ff.; Bechtel, Philosophy of Science, 22ff. 20. The claim that Handel, J. S. Bach, Scarlatti, C. P. E. Bach, Haydn, Mozart, Beethoven, Schubert, Mendelssohn, Schumann, Chopin, and Brahms wrote music that is quintessentially tonal does not mean that their music is typical tonal music. Indeed, one might argue that it is precisely because their music is in many respects untypical that it has been canonized. Having said this, their music did serve as a benchmark to which other composers continually aspire. As we will see in chapter 6, this fact suggests that what Schenkerian theory models is not tonal music in general, but an expert composer’s internalized knowledge of functional monotonality. It is also worth mentioning that within the repertory of music written by Handel, J. S. Bach, Scarlatti, C. P. E. Bach, Haydn, Mozart, Beethoven, Schubert, Mendelssohn, Schumann, Chopin, and Brahms, Schenker’s sample of pieces seems to have been more or less random. 21. See R. G. Swinburn, “The Paradoxes of Confirmation—a Survey,” American Philosophical Quarterly 8 (1971): 318–30. 22. David Hume, An Enquiry Concerning Human Understanding, 2nd ed., reprint (La Salle IL: Open Court Press, 1966), 46. 23. This original version of this paradox presumes that the claim “If X is a raven, then it is black” is logically equivalent to the claim that “If X is not black, then it is not a raven.” This equivalence suggests that the claim “ravens are black” is confirmed by appealing to any other object that is not black, such as a red shoe. 24. See Nelson Goodman, “The New Riddle of Induction,” in Fact, Fiction, and Forecast, 2nd ed. (Indianapolis, IN: Bobbs-Merrill, 1965). For an extensive discussion of the issue, see Douglas Stalker, ed., Grue: The New Riddle of Induction (Chicago: Open Court Press, 1994). According to Goodman’s classic formulation of the problem, an object is grue if it is green before time t and blue after t. If t is the year 2005 and prior to that date we observe an object that is green, then we cannot assume conclusively that it is indeed green and not grue. 25. Simon Blackburn, “Entrenchment,” Oxford Dictionary of Philosophy (Oxford: Oxford University Press, 1994), 121. 26. Losee, A Historical Introduction, 211–13; Quine and Ullian, The Web of Belief, 87–89. 27. W. V. Quine, “Two Dogmas of Empiricism,” in From a Logical Point of View (Cambridge, MA: Harvard University Press, 1953), 41. 28. Klee, Introduction to the Philosophy of Science, 62. 29. Karl Popper, The Logic of Scientific Discovery, rev. ed. (New York: Harper and Row, 1968). 30. Karl Popper, “The Place of Mind in Nature,” in Mind in Nature, ed. Richard Q. Elvee (New York: Harper & Row, 1982), 32. 31. Karl Popper, Conjectures and Refutations: The Growth of Scientific Knowledge, 3rd ed. (London: Routledge, 2002), 48. 32. For a lively account of the problems of demarcation between science and non-science, see Patrick Grim, ed., Philosophy of Science and the Occult (Albany, NY: SUNY Press, 1982).

244

Notes, pp. 17–20

33. R. D. Tweney, M. E. Doherty, and C. R. Mynatt, On Scientific Thinking (New York: Columbia University Press, 1981). See also Bechtel, Philosophy of Science, 37; and K. I. Manktelow and D. E. Over, Inference and Understanding: A Philosophical and Psychological Perspective (London: Routledge, 1990), 133. 34. Papineau, “Philosophy of Science,” 294. 35. Otto Neurath originally referred to this image in his article “Protokollsätze,” Erkenntnis 3 (1932): 204–8. This essay is translated by George Schick as “Protocol Sentences,” in Logical Positivism, ed. A. J. Ayer, 199–208 (New York: Free Press, 1959). Quine mentions Neurath’s account in his “Identity, Ostension, and Hypostasis,” in From a Logical Point of View, 65–79 (Cambridge, MA: Harvard University Press, 1961; 2nd ed. 1980). It became a mainstay of Quine’s later writings. For details see Matthew Brown, “Adrift on Neurath’s Boat: The Case for Naturalized Music Theory,” Journal of Musicology 15, no. 3 (1997): 330–42, esp. 330 and 337–38. 36. W. V. Quine, “What Price Bivalence?” in Theories and Things (Cambridge, MA: Harvard University Press, 1981), 31. 37. Ibid. 38. My six criteria follow those listed by Thomas Kuhn in his essay, “Objectivity, Value Judgment, and Theory Choice,” in The Essential Tension (Chicago: University of Chicago Press, 1977), 320–39. I have simply refined his notion of consistency. According to him, “a theory should be consistent, not only internally or with itself, but also with other currently accepted theories applicable to related aspects of nature.” (321–22). I use the term ‘consistency’ to refer to Kuhn’s internal consistency and ‘coherence’ to refer the idea of external consistency with other theories. Although Quine and Ullian’s list is slightly different from the one presented by Kuhn, it captures many of the same ideas; see The Web of Belief, 64–82. For a handy discussion of Kuhn’s views and the role of epistemic values in general, see Curd and Cover, Philosophy of Science, 83–253. See also Hilary Putnam, “The Philosophers of Science’s Evasion of Values,” in The Collapse of the Fact/Value Dichomoty and Other Essays (Cambridge, MA: Harvard University Press, 2002), 135–45. 39. See Norwood Hanson, Patterns of Discovery (Cambridge: Cambridge University Press, 1958); Thomas Kuhn, The Structure of Scientific Revolutions, 3rd ed. (Chicago: University of Chicago Press, 1996); P. K. Feyerabend, Against Method (London: New Left Books, 1975). 40. This paraphrase is cited by Alfred Ayer, Philosophy in the Twentieth Century (New York: Random House, 1982), 157. Although Quine could not remember where this quote came from, he agreed with its message (personal communication). 41. W. V. Quine, “Five Milestones of Empiricism,” in Theories and Things, 71. 42. For a summary of the debate between Paul Churchland and Jerry Fodor on theory-laden observation, see DeBellis, Music and Conceptualization, 80–116.

Notes, pp. 20–25

245

43. See Richard Boyd, “The Current Status of Scientific Realism,” in Scientific Realism, ed. Jarrett Leplin (Berkeley: University of California Press, 1984), 53. 44. For a classic account of ‘Theory Reduction,’ see Ernst Nagel, The Structure of Science: Problems in the Logic of Scientific Explanation, 2nd ed. (Indianapolis, IN: Hackett, 1979), 336–97. 45. Graham Chapman et al., The Complete Monty Python’s Flying Circus: All the Words (New York: Pantheon, 1989), 2:118–20. 46. Kuhn, The Structure of Scientific Revolutions. 47. Kitcher, “Explanatory Unification and the Causal Structure of the World,” 477. 48. Kuhn, “Objectivity, Value Judgment, and Theory Choice,” 322. 49. According to Feynman, “Every theory that you make has to be analyzed against all possible consequences, to see if it predicts anything else as well.” Feynman, The Character of Physical Law, 32. 50. Kuhn, “Objectivity, Value Judgment, and Theory Choice,” 322. 51. Although this notion does not appear in this form in the writings of William of Ockham (ca. 1285–1347), it can be traced back as far as Aristotle, see Robert Audi, ed., The Cambridge Dictionary of Philosophy (Cambridge: Cambridge University Press, 1995), 545. 52. See W. V. Quine, “Atoms,” in Quiddities: An Intermittently Philosophical Dictionary (Cambridge, MA: Harvard University Press, 1987), 12. 53. Quine and Ullian, The Web of Belief, 71. 54. See Robert L. Causey, Unity of Science (Dordrecht: D. Reidel, 1977). 55. Salzer, Structural Hearing. 56. Richard Taruskin, “Review: Kevin Korsyn, “Towards a Poetics of Musical Influence,” and Joseph N. Straus, Remaking the Past,” Journal of the American Musicological Society 46 (1993): 125.

Chapter 1 1. Schenker, Harmony, par. 88, p. 159. Used by permission of the University of Chicago Press. 2. For example, in Counterpoint I, he noted, “The phenomena of [tonal] composition, then, are invariably to be understood only as [transformations] the prolongations of those principles.” Schenker, Counterpoint I, introduction, p. 13. Similarly, at the end of Counterpoint II, he claimed that tonal relationships can be treated “as prolongations of the fundamental laws.” Schenker, Counterpoint II, part 6, introduction, p. 176. For rather different interpretations of this passage, see Joseph Dubiel, “When You Are a Beethoven: Kinds of Rules in Schenker’s Counterpoint,” Journal of Music Theory 34 (1990): 291–340, esp. 292–93; and Robert Snarrenberg, Schenker’s Interpretative Practice (Cambridge: Cambridge University Press, 1997), 9–53.

246

Notes, pp. 25–43

3. For example, according to Kirnberger, “What is forbidden in strict style is not only permissible in freer style but often sounds very good because the expression is often assisted by such deviations from the rules. This is particularly true in situations where disagreeable passions are to be expressed.” Kirnberger, The Art of Strict Musical Composition, trans. David Beach and Jürgen Thym (New Haven, CT: Yale University Press, 1982), 152. 4. Johann Joseph Fux, Gradus ad Parnassum (Vienna: Johann Peter van Ghelen, 1725); ed. and trans. Alfred Mann as The Study of Counterpoint, rev. ed. (New York: Norton, 1973). 5. See Alfred Mann, Theory and Practice: Great Composers as Teachers and Students (New York: Norton, 1987). 6. Joel Lester, Compositional Theory in the Eighteenth Century (Cambridge, MA: Harvard University Press, 1992), 26–31. 7. Ibid., 33–34. 8. David Lewin, “An Interesting Global Rule for Species Counterpoint,” In Theory Only 6 (1981): 19–44. Lewin states his law on pp. 22 and 25. 9. Ibid., 24. 10. According to Mann, “The repetition of a tone—the only way of using oblique motion in the first species—may occur occasionally in the counterpoint; however, the same tone should not be repeated more than once.” See Fux, The Study of Counterpoint, 29n. 11. See ibid., 50n. 12. Theorists disagree about whether or not neighbor notes should be included in third species. Fux generally did not include them, though he included a lower neighbor in at least one example; see the penultimate measure of Fig. 132 in Fux, The Study of Counterpoint, 92. Mozart seems to have followed this practice, including lower neighbor tones in third species, especially at cadences; see H. Hertzmann, B. B. Oldman, D. Heartz, and A. Mann, eds., Thomas Attwood: Theorie- und Kompositionsstudien bei Mozart, in Wolfgang Amadeus Mozart. Neue Ausgabe sämtlicher Werke X: 30/i (Kassel: Bärenreiter, 1965), 53–61. Schenker, however, argued against their use. 13. Fux, The Study of Counterpoint, 55–56. Fux’s distinction is quite different, of course, from Kirnberger’s distinction between essential and unessential dissonances. 14. Ibid., 139. 15. Ibid., 71. 16. Schenker, Harmony, par. 88, p. 159. Used by permission of the University of Chicago Press. 17. Ibid. 18. Ibid., par. 85, p. 155. Used by permission of the University of Chicago Press, © 1954. 19. For details, see Matthew Brown, “The Diatonic and the Chromatic in Schenker’s Theory of Harmonic Relations,” Journal of Music Theory 30 (1986): 1–34.

Notes, pp. 44–56

247

20. For an extensive discussion of position finders, see Richmond Browne, “Tonal Implications of the Diatonic Set,” In Theory Only 5 (1981): 3–21. 21. Matthew Brown, Douglas J. Dempster, and Dave Headlam, “The
IV(V) Hypothesis: Testing the Limits of Schenker’s Theory of Tonality,” Music Theory Spectrum 19 (1997): 155–83. 22. Schenker, Counterpoint I, part 1, chap. 1, par. 20, pp. 94–100. 23. Schenker, “The Prelude of Bach’s Partita No. 3 for Solo Violin [BWV 1006],” trans. John Rothgeb, The Masterwork in Music 1 (1994), 47. 24. For example, see Schenker’s graph of Beethoven’s Rondo a capriccio, Op. 129, esp. mm. 302–316, in Free Composition, Fig. 134.6. 25. See Schenker’s discussion of the theme in The Masterwork in Music 3 (1999), 11ff. 26. Schenker, Free Composition, par. 161, p. 56. 27. Schenker, “Brahms’s Study, Octaven und Quinten,” 166. 28. Ibid., 30–31 and 156–58. 29. Schenker, Free Composition, par. 164, p. 59. 30. Schenker, “Brahms’s Study, Octaven und Quinten,” 154–56. 31. Schenker, Free Composition, par. 161, p. 56. 32. Schenker, Counterpoint I, part 2, chap. 1, par. 2, p. 111. 33. Schenker, Counterpoint II, part 3, chap. 1, par. 3, p. 3. 34. Schenker, Counterpoint I, part 2, chap. 4, par. 10, p. 283. 35. Schenker discussed the derivation of neighbor tones in Counterpoint II, part 3, chap. 3, par. 2, pp. 76–77. He dealt with suspensions in Counterpoint I, part 2, chap. 4, par. 4–5, pp. 266–67. 36. Schenker, “Laws of the Art of Music,” trans. Robert Snarrenberg, Der Tonwille 1 (1922): 51. 37. Schenker, Free Composition, par. 66 and 106–12, pp. 32 and 42–43. 38. Ibid., par. 5, p. 12. 39. Ibid., par. 164, p. 59. 40. John Rothgeb, “Strict Counterpoint and Tonal Theory,” Journal of Music Theory 19 (1975): 278. 41. Schenker, Generalbasslehre (ms.), 47, trans. John Rothgeb in “Schenkerian Theory: Its Implications for the Undergraduate Curriculum,” Music Theory Spectrum 3 (1981): 146. 42. For details, see Matthew Brown, “A Rational Reconstruction of Schenkerian Theory,” (Ph.D. diss., Cornell University, 1989), 212ff.; Donald G. Traut, “Counterpoint, Form, and Tonality in the First Movement of Stravinsky’s Concerto for Piano and Wind Instruments,” Master’s thesis (L.S.U., 1995), 48–51; and Donald G. Traut, “Revisiting Stravinsky’s Concerto,” Theory and Practice 25 (2000): 65–86. 43. Schenker, Counterpoint I, introduction, part V, p. 16. 44. Ibid. 45. Schenker, Harmony, par. 54, p. 121. Used by permission of the University of Chicago Press.

248

Notes, pp. 57–65

46. See Brown, “A Rational Reconstruction,” 86. I would like to thank Andrew Cohen for helping me formulate ‘The Complementarity Principle.’ 47. Schenker, Counterpoint I, part 2, chap. 1, par. 2, p. 112. Indeed, as he put it in the preface, at the start of “all musical technique is derived from two basic ingredients: voice leading and the progression of Stufen.” Counterpoint I, preface, p. xxv. 48. Schenker, Counterpoint II, part 3, chap. 1, par. 14, p. 10. 49. Schenker, Counterpoint I, preface, p. xxxi. 50. Ibid., part 1, chap. 2, par. 23, p. 105. 51. Ibid., part 1, chap. 1, par. 5, p. 23. 52. Ibid., part 1, chap. 1, par. 5, p. 24. 53. For general discussions of Schenker’s rejection of functional equivalence, see Robert Wason, Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg (Ann Arbor, MI: UMI, 1985), 126–27; Brown, “A Rational Reconstruction,” 192–99; and Matthew Brown and Robert Wason, “Review of Heinrich Schenker, Counterpoint, translated by John Rothgeb and Jürgen Thym,” Music Theory Spectrum 11 (1989): 232–39, esp. 237–38. Another area of concern is that of substitution. Obviously, the notion of functional equivalence presupposes the notion that some chords can be substituted for another. But it is not clear that adequate guidelines are given about when a substitution can take place. For example, the following chords are treated as equivalent— I/VI, IV/II, and V/VII. Now imagine a simple progression I–II6–V–I. Suppose we substitute the opening I with a VI, the penultimate V with a VII6 and the final I with another VI. Instead of having an unambiguous progression in C major, we would produce a modal progression in A minor! 54. Schenker’s notation in Free Composition of Figs. 15.2c–d, 15.3c, 15.6, 16.2c, 16.3c, 16.5, and 18.3 seems to confirm this point. 55. For a different interpretation, see Eytan Agmon, “Functional Harmony Revisited: A Prototype-Theoretic Approach,” Music Theory Spectrum 17 (1995): 196–214. 56. Schenker, Free Composition, par. 84, p. 35. 57. Ibid., par. 79, p. 35. 58. See, for example, Schenker’s discussion of direct chromatic successions in Counterpoint I, part 1, chap. 2, par. 4, p. 46ff.; and Free Composition, par. 249, pp. 91–92. 59. Schenker, Free Composition, par. 250, p. 92. 60. Ibid., par. 249, p. 91. 61. Ibid., par. 233 and 249, pp. 83 and 91–92. 62. Ibid., par. 194, p. 71. 63. Ibid. 64. Brown, Dempster, and Headlam, “The
IV/V Hypothesis,” 155–83. 65. Schenker, Counterpoint I, preface, p. xxxi. Emphasis in original. 66. Quine, Quiddities, 8. 67. Ibid.

Notes, pp. 67–74

249

Chapter 2 1. Besides Free Composition, Schenker also used this motto starting with vol. 1 of Der Tonwille (1921) and in each part of Counterpoint II. 2. See Schenker, Free Composition, par. 45, p. 25. 3. See ibid. 4. For example, Schenker claimed that his principles of reduction are analogous to those traditionally under the rubric of diminution theory, see Free Composition, par. 49, p. 26. 5. See ibid., par. 28, p. 17. 6. In Harmony, Schenker mainly reduced passages to fairly local prototypes. Yet, there are times when he was also thinking in wider terms. For example, in par. 131, he claimed: “In the form of established keys we have the same progression of Stufen, albeit at a superior level. For the sake of the construction of content in a larger sense, the natural element of Stufengang is elevated correspondingly.” Harmony, par. 131, p. 246 (used by permission of the University of Chicago Press). For a general discussion of Schenker’s views, see Matthew Brown, “The Diatonic and the Chromatic in Schenker’s Theory of Harmonic Relations,” Journal of Music Theory 30 (1986): 14–16; Carl Schachter “Analysis by Key: Another Look at Modulation,” Music Analysis 6, no, 3 (1987): 289–318; and Rothstein, “Review: Articles on Schenker and Schenkerian Theory in The New Grove Dictionary of Music and Musicians, 2nd ed.,” Journal of Music Theory 45, no. 1 (2001): 208–9. 7. See also Schenker, Beethoven neunte Sinfonie (Vienna: Universal, 1912), ed. and trans. John Rothgeb as Beethoven’s Ninth Symphony (New Haven, CT: Yale University Press, 1992); and Schenker, ed., Beethoven Piano Sonata, Op. 109 (Vienna: Universal, 1913); Beethoven Piano Sonata, Op. 110 (Vienna: Universal, 1914); Beethoven Piano Sonata, Op. 111 (Vienna: Universal, 1915); Beethoven Piano Sonata, Op. 101 (Vienna: Universal, 1920). For an excellent general history, see William Pastille, “The Development of the Ursatz in Schenker’s Published Works,” in Trends in Schenkerian Research, ed. Allan Cadwallader, 71–85 (New York: Schirmer, 1990); and William Pastille, “Heinrich Schenker, Anti-Organicist,” 19th-Century Music 8, no. 1 (1984): 34–35. 8. Pastille, “The Development of the Ursatz,” 74–76. 9. See Schenker, Free Composition, par. 8, p. 12, and par. 268–70, pp. 107–8. 10. For example, according to Schachter, “Schenker conceives of the fundamental structure as a kind of second-species counterpoint with dissonant passing tones, rather than as a first-species counterpoint restricted to consonances.” Schachter, “A Commentary on Schenker’s Free Composition,” Journal of Music Theory 25, no. 1 (1981): 126. More recently, Robert Snarrenberg refers to Schenker’s theory as a “second-species model of tonality”; see Snarrenberg, “Competing Myths: The American Abandonment of Schenker’s Organicism,” in Theory, Analysis and Meaning in Music, ed. Anthony Pople (Cambridge: Cambridge University Press, 1994), 39, 54.

250

Notes, pp. 74–79

11. See, for example, par. 84 (p. 154) of Schenker’s Harmony, entitled “The Lack of Stufen in Strict Counterpoint.” 12. Peter Westergaard, An Introduction to Tonal Theory (New York: Norton, 1975), 426n. For a very helpful survey of Westergaard’s work, see Stephen Peles, “An Introduction to Westergaard’s Tonal Theory,” In Theory Only 13, no. 1–4 (1997): 73–93. 13. Schenker, Free Composition, par. 69, p. 32. 14. For a recent survey of the relevant literature, see David Smyth, “Schenker’s Octave Lines Reconsidered,” Journal of Music Theory 43, no. 1 (1999): 101–33. 15. David Neumeyer, “The Urlinie from 8 as a Middleground Phenomenon,” In Theory Only 9, no. 5–6 (1987): 3–25; David Neumeyer, “The Ascending Urlinie,” Journal of Music Theory 31, no. 2 (1988): 271–303; and David Neumeyer, “The Three-Part Ursatz,” In Theory Only 10 (1987): 3–29, esp. 3. 16. See Susan Tepping, “An Interview with Felix-Eberhard von Cube,” Indiana Theory Review 6 (1982–83): 77–103. 17. The same point can be used to counter Peter Westergaard’s charge that 3-lines are conceptually superior to 5 - and 8-lines, see An Introduction to Tonal Theory, 426n. 18. Robert Joseph Lubben, “Schenker the Progressive: Analytic Practice in Der Tonwille,” Music Theory Spectrum 15 (1993): 65ff. 19. Eugene Narmour, Beyond Schenkerism (Chicago: University of Chicago Press, 1977). 20. Schenker explained this point very nicely at the opening of Counterpoint II: “Now the triad reaches us by both routes, but with only this difference of effect: in their vertical dimension, it sounds in its complete, palpable, physical totality, so to speak, while the horizontal dimension unrolls step by step, through the detour of melodic evolution.” Schenker, Counterpoint II, part 3, chap. 1, par. 2, p. 2. Significantly, he anticipated this idea in the Harmony by claiming 1) that the concept of an interval “is bound to and limited by the concept of its harmonizability” and 2) that the harmonic element “has to be pursued in both directions: the horizontal as well as the vertical.” Schenker, Harmony, par. 76, p. 134. 21. Schenker confirms the idea that neighbor motion implies a prior repetition in Free Composition, par. 108, p. 42. 22. According to Schenker, “The descending linear progression always signifies a motion from the upper to the inner voice; ascending linear progression denotes a motion from the inner voice to the upper voice.” See Schenker, Free Composition, par. 203, p. 73. 23. For discussions of mentally retained tones, see Schenker, Counterpoint II, part 3, chap. 2, par. 2, esp. pp. 57ff.; Schenker, “Further Considerations of the Urlinie: II,” trans. John Rothgeb, The Masterwork in Music 2 (1996), 3ff.; and Schenker, Free Composition, par. 93 and 204, pp. 38–39 and 73, etc. William Rothstein considers the term in detail in his thesis, “Rhythm and the Theory of Levels” (Ph.D. diss., Yale University, 1981), 91ff.

Notes, pp. 80–93

251

24. For clarification of Schenker’s views, see “Further considerations of the Urlinie: II,” trans. John Rothgeb, The Masterwork in Music 2 (1996), 1–22. 25. For an extensive discussion of implied tones, see William Rothstein, “On Implied Tones,” Music Analysis 10, no. 3 (1991): 289–328. 26. Although Schenker did not include this transformation in Parts II and III of Free Composition, he used it freely in his graphs. For an extensive discussion of displacement, see Rothstein, “Rhythm and the Theory of Structural Levels.” 27. Matthew Brown, “A Rational Reconstruction of Schenkerian Theory” (Ph.D. diss., Cornell University, 1989), 128–29. The argument runs as follows. Imagine that a soprano voice moves from the note S1 to the note S2 and an alto moves from the note A1 to the note A2. The notes of these voices can be connected in many ways. If S1 is linked to A1 (or A1 to S1) or S2 to A2 (or A2 to S2), then the result is an unfolding . If S1 is joined to A2 and A1 to S2, then the result is a voice crossing or a voice exchange. If S1 is separate from A2 but A1 is bound to S2, then the result is motion from an inner voice. Finally, if A1 is separate from S2 but S1 is linked to A2, then the result is motion to an inner voice or reaching over. 28. Schenker, Free Composition, chap. 1, section 3, pp. 5–6. 29. These are adapted from Ibid., Figures 14–18, par. 53–76, pp. 29–34. 30. Ibid., par. 55, pp. 29–30. 31. Ibid., par. 56, p. 30. 32. Ibid., par. 56, p. 30. Emphasis in original. 33. Ibid., par. 65, p. 31. 34. Ibid., par, 189, p. 69. Emphasis in original. 35. Ibid., par. 94, p. 39. 36. Ibid. 37. See Bryce Rytting, “Structure vs. Organicism in Schenkerian Analysis,” (Ph.D. diss., Princeton University, 1996). 38. Oster adds one in Free Composition. 39. Schenker, Free Composition, par. 186–87. 40. Ibid., par. 189–90. 41. See ibid., par. 304. 42. See ibid., pp. xxiii and xxiv. 43. Babbitt, “The Structure and Function of Music Theory,” College Music Symposium 5 (1965): 59–60. 44. See Schenker, Free Composition, par. 254, p. 100. See also ibid., par. 30, and the section on diminution, par. 251–66. For a general discussion of the concept, see Charles Burkhart, “Schenker’s ‘Motivic Parallelisms,’ ” Journal of Music Theory 22, no. 2 (1978): 145–75. 45. Schenker published his graph of this remarkable piece in his essay “The Representation of Chaos from Haydn’s Creation,” trans. William Drabkin, The Masterwork in Music 2 (1996), 97–105. 46. Donald Francis Tovey, “The Creation,” in Essays on Musical Analysis, vol. 5 (Oxford: Oxford University Press, 1937), 114–46.

252

Notes, pp. 96–121

47. David Beach, “A Recurring Pattern in Mozart’s Music,” Journal of Music Theory 27, no. 1 (1983): 1–29. 48. Charles Burkhart, Anthology for Musical Analysis, 4th ed. (New York: Harcourt Brace, 1986), 284. See also Schachter, “Analysis by Key,” 298; and Carl Schachter, “The Sketches for Beethoven’s Sonata for Piano and Violin, Op. 24,” Beethoven Forum 3 (1994): 107–25. It must be stressed, however, that Mozart was by no means the only composer to use this strategy; not only does Heinrich Koch hint at this procedure, but Joel Galand gives a good example from Haydn’s Symphony 85 in his excellent article, “Form, Genre, and Style in the Eighteenth-Century Rondo,” Music Theory Spectrum 17, no. 1 (1995): 37–39. 49. For an excellent overview of Beethoven’s knowledge of Mozart’s music and possible models, see Lewis Lockwood, “Beethoven before 1800: The Mozart Legacy,” Beethoven Forum 3 (1994): 39–52. 50. Joseph Kerman, ed., Ludwig Beethoven: Autograph Miscellany from circa 1786 to 1799: British Museum Additional Manuscript 29801, ff. 39–162 (London: British Museum, 1967), vol. 1 (facs.), fol. 88r, vol. 2 (transcription), p. 228, and comment on p. 293. 51. Schachter, “Analysis by Key,” 289. 52. They also demonstrate Schenker’s claim that, although all tonal masterworks are based on the same laws of tonal voice leading and harmony, they do so in extremely diverse ways. See Schenker, Free Composition, appendix H, p. 160; originally part 1, chap. 1, section 4, p. 6. 53. Ibid., par. 29, p. 18.

Chapter 3 1. Quine, Quiddities, 162. 2. Quine and Ullian, The Web of Belief, 100. 3. Schenker, Counterpoint I, part IV, p. 14. 4. Ibid., part IV, p. 16. 5. Benjamin, “Schenker’s Theory and the Future of Music,” 163. 6. Schenker, Free Composition, par. 161, p. 56. 7. For a comprehensive account of sequences in general, see Adam Ricci, “A Theory of the Harmonic Sequence” (Ph.D. diss., University of Rochester, 2004). 8. Benjamin, “Schenker’s Theory and the Future of Music,” 160. 9. As Alfred Mann observes, “The forming of sequences (the so-called monotonia) ought to be avoided as far as possible.” See Fux, The Study of Counterpoint, ed. and trans. Alfred Mann (New York: Norton, 1973), 54. 10. Whereas Figures 91 and 92 of Gradus ad Parnassum conform to 5-line paradigms, Figure 98 composes out a 3-line, 3– 4 –3–2–1. 11. Fux, The Study of Counterpoint, 98.

Notes, pp. 122–128

253

12. According to Schenker, “the Stufen I–IV–VII–III–VI–II–V–I simply define D minor without any tendency to sequence or modulate.” Schenker, Counterpoint I, part 1, chap. 1, par. 5, p. 26. 13. Schenker, “The Organic Nature of Fugue,” trans. Hedi Siegel, The Masterwork in Music 2 (1996), 48n. 14. To quote him: “One cannot speak of ‘melody’ and ‘idea’ in the work of the masters; it makes even less sense to speak of ‘passage,’ ‘sequence,’ ‘padding,’ or cement’ as if they were terms that one could possibly apply to art. Drawing a comparison to language, what is there in a logically constructed sentence that one could call ‘cement’? How does one distinguish an ‘idea’ from ‘cement’?” Schenker, Free Composition, par. 50, p. 27. 15. Schenker, Counterpoint II, part 6, chap. 1, par. 2, p. 180. 16. Schenker, Free Composition, par. 221–29, pp. 78–82. 17. According to Schenker, “double counterpoint therefore takes its place in the ranks of such fallacious concepts as the ecclesiastical modes, sequences, and the usual explanation of consecutive fifths and octaves.” Schenker, Free Composition, par. 222, p. 78. 18. Schenker, Free Composition, par. 221, p. 78. 19. Ibid. 20. Ibid., par. 224, p. 79. 21. Rothstein, “Review: Articles on Schenker and Schenkerian Theory” in The New Grove Dictionary of Music and Musicians, 2nd ed.,” Journal of Music Theory 45, no. 1 (2001): 209–10. 22. See Schenker, Free Composition, par. 215. 23. Schenker, “Bach’s Little Prelude No. 1 in C Major, BWV 924,” trans. Joseph Dubiel, in Der Tonwille ed. William Drabkin, 1:141–44 (Oxford: Oxford University Press, 2004). For a catalog of Schenker’s analyses of this Prelude, see Larry Laskowski, Heinrich Schenker: An Annotated Index of His Analyses of Musical Works (New York: Pendragon Press), 22–23. Perhaps the most significant difference between the reading in Der Tonwille and the reading in Free Composition is that in the former, he treated mm. 1–7 as an ascending fourth span G–A–B–C, but in the latter, he regarded them as an ascending sixth span. This change reflects Schenker’s growing commitment to the three basic forms of the Ursatz and the codification of his voice-leading transformations, especially that of unfolding. 24. Schenker included his sketch of the prelude in his general discussion of unfolding. See Free Composition, par. 140–42. He noted Bach’s decision to support the bass arpeggiation with unfoldings in the upper voices in Free Composition, par. 243, p. 87ff. Elsewhere Schenker mentioned that the Prelude is a perfect example of a one-part form. See Free Composition, par. 307, p. 131. 25. See Schenker, Free Composition, par. 224. 26. To quote Schenker, “At the same time, interpolation of the fifthprogressions serves to remove consecutive fifths.” Schenker, “Bach’s Little Prelude,” 141. See also Schenker, Free Composition, par. 283, pp. 116–17.

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Notes, pp. 130–143

27. Schenker, Harmonielehre, ex. 36 (subdominant fifth), and p. 416 (pedal point). 28. Schenker, “The Organic Nature of Fugue,” 48ff. Schenker’s analysis was printed earlier as “Joh. Seb. Bach: Wohltermperiertes Klavier, Band I, Praeludium C-Moll,” Die Musik 15, no. 9 (June 1923): 641–51; on the differences between the two essays, see Siegel’s note on p. 48 of “The Organic Nature of Fugue.” 29. Schenker, Free Composition, par. 221 and 224, fig. 95e4. 30. Schenker specifically draws parallels between the two preludes; see “The Organic Nature of Fugue,” 50n40. 31. Schenker, “The Prelude of Bach’s Partita no. 3 for Solo Violin [BWV 1006],” trans. John Rothgeb, The Masterwork in Music 1 (1994), 40–41. 32. See Schenker, Free Composition, par. 43, 77 (especially Fig. 19b), and 100–101 (especially Fig. 27a–b). 33. David Smyth, “Schenker’s Octave Lines Reconsidered,” Journal of Music Theory 43, no. 1 (1999): 101–33. 34. See Schenker, “Bach’s Little Prelude No. 1 in C Major,” 142. 35. Schenker, “Further Consideration of the Urlinie: I,” trans. John Rothgeb, The Masterwork in Music 1 (1994), 104–5. 36. Schenker, “Further Consideration of the Urlinie: I,” 105. 37. Schenker, Free Composition, chap. 1, sec. 4, p. 6.

Chapter 4 1. Schenker, Free Composition, introduction, xxiii. I have changed Oster’s wording slightly. 2. Schenker, Counterpoint I, part 1, chap. 2, par. 1, 33–34. Emphasis in original. 3. See Schenker, “Elucidations,” in The Masterwork in Music 1 (1994), 113–14. 4. See ibid., 113. 5. This maxim is usually translated as “entities should not be multiplied beyond necessity.” 6. Paul M. Churchland, “Ontological Status of Observables,” in Images of Science: Essays on Realism and Empiricism, ed. Paul M. Churchland and Clifford A. Hooker, 40–41 (Chicago: University of Chicago Press, 1985). 7. Ibid. 8. For a helpful discussion, see Leeman L. Perkins, “Modal Strategies in Okeghem’s Missa Cuiusvis Toni,” in Music Theory and the Exploration of the Past, ed. Christopher Hatch and David Bernstein, 59–71 (Chicago: University of Chicago Press, 1993), esp. 60–61. 9. William J. Mitchell, Elementary Harmony, 3rd ed. (Engelwood Cliffs, NJ: Prentice Hall, 1965), 5.

Notes, pp. 143–146

255

10. Edward Aldwell and Carl Schachter, Harmony and Voice Leading, 3rd ed. (Belmont, CA: Wadsworth, 2003), 7. 11. Pieter Van den Toorn, The Music of Igor Stravinsky (New Haven, CT: Yale University Press, 1983), 460n14. 12. Richard Taruskin, “Chez Petrouchka: Harmony and Tonality chez Stravinsky,” 19th-Century Music 10, no. 3 (1987): 267. 13. Ibid. 14. Schenker, Counterpoint I, part 1, chap. 1, par. 5, pp. 20–21. The purpose of this digression was twofold: on the one hand, it allowed him to attack theorists who used modes and scales to explain the behavior of tonal relationships (for example, Capellan, Polak, Riemann, and Dittrichs); on the other, it allowed him to criticize composers who tried to expand the sphere of tonality by writing in modal or exotic styles (for example, Saint-Saëns, Busoni). 15. Scholars have generally underestimated the importance of Schenker’s comments on mode. Roger Sessions did broach the question of mode in his short review article, “Heinrich Schenker’s Contribution,” Modern Music 12 (1935): 170–75, but failed to appreciate Schenker’s great insight that the properties of harmonic systems do not depend on the properties of scales. So far as I can tell, nobody has ever stressed the significance of this claim. Nevertheless, several Schenkerians have undertaken analytical studies of modal music. See, for example, Peter Bergquist, “Mode and Polyphony around 1500: Theory and Practice,” Music Forum 1 (1967): 99–61; William J. Mitchell, “The Prologue to Orlando di Lasso’s Prophetiae Sibyllarum,” Music Forum 2 (1970): 264–73; Saul Novack, “Fusion of Design and Tonal Order in Mass and Motet: Josquin Desprez and Heinrich Isaac,” Music Forum 2 (1970): 187–263; Saul Novack, “The History of Phrygian Mode in the History of Tonality,” Miscellanea Musicologica 9 (1977): 82–127; Saul Novack, “The Analysis of Pre-Baroque Music,” in Aspects of Schenkerian Theory, ed. David Beach, 113–33 (New Haven, CT: Yale University Press, 1983); Felix Salzer, “Tonality in Early Medieval Polyphony: Towards a History of Tonality,” Music Forum 1 (1967): 34–98; Felix Salzer, “Heinrich Schenker and Historical Research: Monteverdi’s Madrigal ‘Oimè, se tanto amate,’ ” in Aspects of Schenkerian Theory, ed. David Beach, 133–52 (New Haven, CT: Yale University Press, 1983); Carl Schachter, “Landini’s Treatment of Consonance and Dissonance: A Study in Fourteenth-Century Counterpoint,” Music Forum 2 (1970): 130–86; Lori Burns, Bach’s Modal Chorales, Harmonologia Series, 9 (Stuyvesant, NY: Pendragon, 1995). Jonas summarizes some of Schenker’s arguments in his Introduction to the Theory of Heinrich Schenker, ed. and trans. John Rothgeb (New York: Longman, 1982), 27–31. 16. To quote him: “And thus countless systems are assumed in a situation in which even one ‘system’ in the strict sense of the word is impossible from the outset, since the all too modest tonal material is simply not differentiated enough. For that reason, the so-called systems—again exactly as in the earliest period of Western music—are of value at most only as mechanical-descriptive tools and can apply only to the horizontal dimension at that.” Schenker, Counterpoint I, part 1, chap. 1, par. 5, p. 21.

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Notes, pp. 146–156

17. Schenker, Counterpoint I, part 1, chap. 1, par. 5, pp. 20 and 39. In Harmony, he wrote, “Hence there is no violence against the spirit of History in the assumption that the old church modes, though they had their undeniable right to existence, were nothing but experiments—experiments in word and fact, i.e., in theory as well as practice—whence our art benefitted especially in so far as they contributed decisively to the clarification, e contrario, of our understanding of the two main systems.” Harmony, par. 28, p. 59. 18. Schenker, Counterpoint I, part 1, chap. 2, par. 1, p. 39. 19. Ibid. 20. Ibid., part 1, chap. 1, par. 5, p. 20. 21. Schenker, Free Composition, par. 4, pp. 11–12. 22. Schenker, Harmony, par. 41, p. 87. 23. These arrows are missing from Mann Borgese’s translation. 24. Schenker, Harmony, par. 48, p. 93. Schenker’s student Oswald Jonas went one step further to explain how Phrygian and Lydian effects can also be created. According to him, these effects can arise when diminished triads are added to the mix: in Phrygian mode the tonic and subdominant triads are minor and the dominant triad is diminished, and in Lydian mode tonic and dominant triads are major and the subdominant triad is diminished. Jonas, Introduction to the Theory of Heinrich Schenker, 28. 25. Schenker, Harmony, par. 40, p. 86. Used by permission of the University of Chicago Press. Translation modified slightly. 26. Ibid., par. 30, p. 76. Used by permission of the University of Chicago Press. Translation modified slightly. 27. Schenker, Counterpoint I, part 1, chap. 1, par. 5, p. 21. 28. Ibid., part 1, chap. 1, par. 5, p. 28. Emphasis in original. 29. Schenker, Harmony, par. 26–30, 38–52, pp. 55–76, 84–115. 30. Numerous other examples of modal works can be found in ibid., par. 26–30, pp. 55–76. 31. Barry Cooper, Beethoven and the Creative Process (Oxford: Oxford University Press, 1990), 56. 32. Schenker, Harmony, par. 29, p. 65. 33. Ibid., par. 29, p. 63. Used by permission of the University of Chicago Press. 34. Ibid., par. 29, p. 66. Used by permission of the University of Chicago Press. 35. Ibid., par. 29, pp. 62–63. 36. Significantly, Kevin Korsyn never bothers to mention Schenker’s discussion of the passage in “J. W. N. Sullivan and the Heiliger Dankgesang,” Beethoven Forum 2 (1993): 133–74. 37. Schenker, Harmony, par. 29, pp. 60–61. Used by permission of the University of Chicago Press. 38. Schenker, Counterpoint I, part 1, chap. 2, par. 4, p. 57. 39. Ibid., part 1, chap. 2, par. 6, p. 57. 40. Schenker, Harmony, par. 29, p. 67. Used by permission of the University of Chicago Press.

Notes, pp. 156–166

257

41. Ibid., par. 29, pp. 67–68. Used by permission of the University of Chicago Press. 42. Ibid., par. 29, p. 68. Used by permission of the University of Chicago Press. 43. Schenker, Counterpoint I, part 1, chap. 2, par. 1, p. 36. Schenker’s discussion can be found on pp. 34–39, Ex. 13–14. 44. Schenker, Free Composition, par. 251, pp. 95–96, Fig. 116. 45. Schenker, Counterpoint I, part 1, chap. 1, par. 5, p. 28. 46. Schenker, “Chopin: Etude in G major, Op. 10, No. 5,” trans. Bent, The Masterwork in Music 1 (1925/1994), 97. 47. Arnold Schoenberg, Harmonielehre (Vienna: Universal, 1911, 3rd ed. 1922), chap. 20, trans. Roy E. Carter as Theory of Harmony (Berkeley: University of California Press, 1978), 390–98; and Donald Francis Tovey, “Harmony,” in The Forms of Music (Cleveland: Meridian, 1956), 69. 48. Matthew Brown, “Tonality and Form in Debussy’s Prélude à ‘L’Aprèsmidi d’un faune’,” Music Theory Spectrum 15, no. 2 (1993): 127–43. 49. Burkhart, “Schenker’s ‘Motivic Parallelisms,’ ” Journal of Music Theory 22, no. 2 (1978): 157. 50. Van den Toorn, The Music of Igor Stravinsky, 73. 51. Ibid. 52. Carl Dahlhaus, Nineteenth-Century Music, trans. J. Bradford Robinson (Berkeley: University of California Press, 1989), p. 306. 53. Don Randel, “Emerging Triadic Tonality in the Fifteenth Century,” Musical Quarterly 57 (1971): 73–86. See also Robert W. Wienpahl, “The Evolutionary Significance of 15th Century Cadential Formulae,” Journal of Music Theory 4, no. 2 (1960): 131–52. Helen E. Bush, “The Recognition of Chordal Formation by Early Theorists,” Musical Quarterly 32 (1946): 238. 54. Randel, “Emerging Triadic Tonality in the Fifteenth Century,” 78–79, exx. 1–2. 55. Ibid. 56. Ernest T. Ferand, “What is Res Facta?” Journal of the American Musicological Society 10 (1957): 129–74; Margaret Bent, “Res facta and Cantare Super Librum,” Journal of the American Musicological Society 36, no. 2 (1983): 371–91; Bonnie J. Blackburn, “On Compositional Process in the Fifteenth Century,” Journal of the American Musicological Society 40, no. 2 (1987): 210–84. 57. Blackburn, “On Compositional Process in the Fifteenth Century,” 252. As she notes, Tinctoris gave two definitions of counterpoint, one in the Terminorum musicae diffinitorium (Treviso, ca. 1495) and the other in chapter 1 of the Liber de arte contrapuncti (1477). 58. Blackburn, “On Compositional Process in the Fifteenth Century,” 252. 59. Ibid., 254. 60. For details, see ibid., 253 (emphasis in original). Incidentally, Jeppesen also notes that repetitions are much more common in res facta than they are in noteagainst-note counterpoint for two voices. Knud Jeppesen, Counterpoint: The Vocal Style of the Sixteenth Century, trans. Alfred Mann (New York: Dover, 1992), 12.

258

Notes, pp. 166–173

61. Jeppesen, Counterpoint, 14, 81, 83, and 97–103. 62. Andrew C. Haigh, “Modal Harmony in the Music of Palestrina,” in Essays on Music in Honor of Archibald Thompson Davison (Cambridge, MA: Harvard University Press, 1957), 114. 63. John Clough, “The Leading Tone in Direct Chromaticism: From Renaissance to Baroque,” Journal of Music Theory 1, no. 1 (1957): 2–21; and Clough, “Indirect Chromaticism in the Renaissance,” Journal of Music Theory 3, no. 1 (1959): 147–50. 64. G. M. Artusi, L’Artusi, ovvero, Delle imperfezioni della moderna musica (Venice: 1600), trans. Oliver Strunk in Source Readings in Music History, 33–44 (New York: Norton, 1950). 65. Giulio Cesare Monteverdi, reply in Claudio Monteverdi’s Scherzi Musicali (Venice: 1607) trans. Oliver Strunk in Source Readings in Music History, 45–52 (New York: Norton, 1950). 66. Monteverdi, Scherzi Musicali, 47. 67. Panayotis Mavromatis, “The Early Keyboard Prelude as an Agent in the Formation of Schenkerian Background Prototypes,” Paper delivered at the Third International Schenker Conference, Mannes College of Music, March 12, 1999. 68. Mary Louise Serafine, Music as Cognition: The Development of Thought in Sound (New York: Columbia University Press, 1988), 56. 69. David Huron, “Interval-Class Content in Equally Tempered Pitch-Class Sets: Common Scales Exhibit Optimum Consonance,” Music Perception 11 (1994): 303.

Chapter 5 1. Schenker, “Miscellanea: Thoughts on Art and Its Relationships to the General Scheme of Things,” trans. Ian Bent, The Masterwork in Music 3 (1997), 71. 2. René Lenormand, Étude sur l’harmonie moderne (Paris: Monde musical, 1913). 3. Claude Debussy, “Music in the Open Air,” La Revue blanche, 1 June 1901, reprinted in Debussy on Music, coll. François Lesure, ed. and trans. Richard Langham Smith (New York: Knopf, 1977), 41. 4. Claude Debussy, Debussy Letters, ed. François Lesure and trans. Roger Nichols (Cambridge, MA: Harvard University Press, 1987), 76–77. 5. Claude Debussy, “A propos de ‘Muguette,’ ” Gil Blas, 23 March 1903, reprinted in Debussy on Music, 155. I have retranslated the terms parfait and imparfait. 6. Debussy, “Conversations, 1890,” reprinted in Debussy: Prelude to “The Afternoon of a Faun,” ed. William Austin, Norton Critical Scores (New York: Norton, 1970), 130. For a facsimile of Maurice Emanuel’s transcription of these conversations, see Léon Vallas, Claude Debussy: His Life and Works, trans. Maire O’Brien and Grace O’Brien (Oxford: Oxford University Press, 1933), 84.

Notes, pp. 173–180

259

7. Debussy, “Conversations, 1890,” 131. 8. The most important early attempts to explain the tonality of Debussy’s music from a Schenkerian perspective can be found in Adele Katz, Challenge to Musical Tradition (New York: Knopf, 1945), and in Felix Salzer, Structural Hearing: Tonal Coherence in Music (New York: Boni, 1952). For recent discussions of this issue see, Annie K. Yih, “Analyzing Debussy: Tonality, Motivic Sets and the Referential Pitch-Class Specific Collection,” Music Analysis 19, no. 2 (2000): 203–29; and Boyd Pomeroy, “Debussy’s Tonality: A Formal Perspective,” in The Cambridge Companion to Debussy, ed. Simon Trezise, 155–78 (Cambridge: Cambridge University Press, 2003). In my opinion, the main shortcoming of these studies is that they all fail to establish in any precise way what the limits of Schenkerian theory may or may not be. 9. Schenker, Free Composition, par. 162–64, pp. 57–60, Fig. 54.6. See also Schenker, Counterpoint I, Ex. 184. Incidentally, Lenormand cites this celebrated passage as well; see Etude sur l’harmonie moderne, 27. 10. Schenker, Free Composition, Fig. 53.3. 11. To quote Schenker: “Thus the preparation [of a suspension] itself can be elided and the dissonances placed on the strong beat in its absence. Dissonant chords thereby arise, for which in certain circumstances a purely implicit preparation though the preceding harmony can be assumed; otherwise the apparently free dissonance must be understood as the clearly established internal element of a latent passing motion. In the latter case, the elided consonance that would initiate the passing motion is to be inferred from and supplied by the harmony belonging to the dissonance itself. In this way we arrive at so-called free-suspensions, and it may be that the ultimate origin of seventh-chords is best explained with reference to the elision of a preparation or the consonant beginning of a passing-tone motion.” Schenker, Counterpoint I, part 2, chap. 4, par. 10, p. 283 (emphasis in original). 12. See Oswald Jonas, Introduction to the Theory of Heinrich Schenker ed. and trans. John Rothgeb (New York: Longman, 1982), 120ff. 13. Schenker, “Bach: Twelve Short Preludes, No. 12 [BWV 942],” trans. Hedi Siegel, in The Masterwork in Music 1 (1994), 62–66. 14. Ibid., 66. 15. Schenker, Free Composition, par. 249, p. 92, Figure 114.8. For a rather different account of the problem, see Howard Cinnamon, “Tonic Arpeggiation and Successive Equal Third Relations as Elements of Tonal Evolution in the Music of Franz Liszt,” Music Theory Spectrum 8 (1986): 1–24. 16. Schenker, Free Composition, par. 230, p. 82, Figure 100.6. 17. See Brown, “The Diatonic and the Chromatic in Schenker’s Theory of Harmonic Relations,” Journal of Music Theory 30 (1986): 1–34. 18. Schenker, Harmony, par. 156, p. 290. In Free Composition, he expressed the same idea as follows: “A truly well established tonality (or Tonalität) can guide even the greatest possible range of chromatic elements back to the fundamental triad.” Schenker, Free Composition, p. 5. I have changed Oster’s translation, which reads: “A firmly established tonality can guide even a large number

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Notes, pp. 180–186

of chromatic phenomenon back to the basic triad.” Oster, trans., Free Composition, Introduction, p. xxiii. According to Schenker: “Chromaticism is an element which does not destroy diatony (or Diatonie), but which rather emphasizes and confirms it.” Schenker, Harmony, par. 155, p. 288. Used by permission of the University of Chicago Press. 19. William J. Mitchell, “The Tristan Prelude: Techniques and Structure,” Music Forum 1 (1967): 203. 20. Schenker, Harmony, par. 89, p. 174. Used by permission of the University of Chicago Press. Translation changed slightly. 21. Schenker, Harmonielehre, par. 89, pp. 220–22, Fig. 173. Unfortunately, this extract is one of many passages omitted from Mann Borgese’s English translation. Schenker repeated his general criticisms of Reger’s music in a comment dated June 1911: “One thinks, for instance of Reger’s silly way of writing: with Reger, chord leads to chord, unsubstantiated by any sort of motive, and consequently the succession of chords has only an external effect. Insubstantial phenomena simply unload themselves at the outer doors of our consciousness. Only substantial on the other hand, are able to penetrate into the depths.” See William Pastille, “Review: Hellmuth Federhofer, Heinrich Schenker: Tagebüchern und Briefen,” Journal of the American Musicological Society 39 (1986): 673. 22. See Schenker, Free Composition, par. 244–45, p. 88ff. 23. See ibid., par. 244, p. 88. 24. Ibid., par. 307, p. 131. 25. See ibid. 26. For discussions of the term “directional tonality” or “progressive tonality,” see William Kinderman, Introduction, in The Second Practice of NineteenthCentury Music, ed. William Kinderman and Harald Krebs (Lincoln: University of Nebraska Press, 1996), 9. 27. Schenker, Free Composition, chap. 1, section 3, p. 5. 28. See Brown, Debussy’s “Ibéria”: Studies in Genesis and Structure (Oxford: Oxford University Press, 2003), 135–56. 29. Graham George discusses the concept of “interlocking tonality” in his book, Tonality and Musical Structure (London: Faber, 1970). For a brief discussion of George’s work, see Kinderman, Introduction, in The Second Practice of Nineteenth-Century Music, 9. 30. See, for example, Gregory Proctor, “Technical Bases of NineteenthCentury Chromatic Tonality,” (Ph.D. diss., Princeton University, 1978); and Kinderman and Krebs eds., The Second Practice of Nineteenth-Century Music. To quote Proctor, in classical diatonic tonality, “chromaticism is defined as the interaction of different diatonic scales,” but in nineteenth-century tonality, “there is but one chromatic scale from which all diatonic scales are derived as subsets.” Proctor, “Technical Bases of Nineteenth-Century Chromatic Tonality,” iii–iv. 31. For another response to these issues, see Robert P. Morgan, “Are There Two Practices in Nineteenth-Century Music?” Journal of Music Theory 43, no. 1 (1999): 135–63.

Notes, pp. 186–202

261

32. For a discussion of the text, see Margaret G. Cobb, ed., The Poetic Debussy: A Collection of His Song Texts and Selected Letters, 2nd ed., Eastman Studies in Music (Rochester, NY: University of Rochester Press, 1994), 102–3. For another reading of the piece, see Wallace Berry, Musical Structure and Musical Performance (New Haven, CT: Yale University Press, 1989), 144–216. 33. For a discussion of the text, see Cobb, ed., The Poetic Debussy, 122–23. 34. To cement the connections, both pieces contains the same chromatic inflection 5 –6 —D–E in “La mort des amants” and A–B in “Claire de lune.” 35. Katherine Bergeron, “The Echo, the Cry, the Death of Lovers,” 19thCentury Music 18, no. 2 (1994): 136–50. 36. Robert Baldick, Introduction, in Huysmans, Against Nature, trans. Robert Baldick (Harmondsworth: Penguin, 1959), 13. 37. Jean Pierrot, The Decadent Imagination, 1880–1900, trans. Derek Coltman (Chicago: University of Chicago Press, 1981), 123–24. 38. According to Des Esseintes, “[Verlaine] alone had possessed the secret hinting at certain strange spiritual aspirations, of whispering certain thoughts, of murmuring certain confessions so softly, so quietly, so haltingly that the ear that caught them was left hesitating, and passed on to the soul a languor made all the more pronounced by the vagueness of these words that were guessed rather than heard.” Huysmans, Against Nature, 186. 39. See Marcel Dietschy, A Portrait of Debussy, ed. and trans. William Ashbrook and Margaret G. Cobb (Oxford: Oxford Univerisity Press, 1990), 42. 40. See ibid., 49. 41. See Morgan, “The Dissonant Prolongation.” Several other writers have noticed this inconsistency, among them Carl Schachter, “A Commentary on Schenker’s Free Composition,” Journal of Music Theory 25, no. 11 (1981): 136–37; William Clark, “Heinrich Schenker on the Nature of the Seventh Chord,” Journal of Music Theory 26, no. 2 (1982): 221–59; Allen Forte and Steven E. Gilbert, Introduction to Schenkerian Analysis (New York: Norton, 1982), 244–45; Jonas, Introduction to the Theory of Heinrich Schenker, 120–22. The most notorious analyses appear in Schenker, Free Composition, Fig. 62, 1–4, and par. 215. 42. For example, in the Preface to Counterpoint I, he noted: “In comparison with the works of our masters, today’s compositions have to be considered musically too simple, even far too simple and primitive. Despite heaviest orchestration, despite noisy and pompous gestures, despite ‘polyphony’ and ‘cacophony,’ the proudest products of Richard Strauss are inferior—in terms of true musical spirit and authentic inner complexity of texture, form, and articulation—to a string quartet by Haydn, in which external grace hides the inner complexity, just as color and fragrance of a flower render mysterious to humans the undiscovered, great miracles of creation.” Schenker, Preface, Counterpoint I, xxi. 43. To quote Schenker, “The dissonant passing tone, including the passing seventh, is itself a means of composing-out. Therefore, as long as it retains its dissonant quality, it cannot at the same time give rise to further composing-out; only

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Notes, pp. 204–214

the transformation of the dissonance into a consonance can make composing-out possible.” Schenker, Free Composition, par. 169, p. 61. 44. Schenker, “Further Considerations of the Urlinie: II,” trans. John Rothgeb, The Masterwork in Music 2 (1996), 17–18. See also Donald G. Traut, “Counterpoint, Form, and Tonality in the First Movement of Stravinsky’s Concerto for Piano and Wind Instruments” (M.A. thesis, Louisiana State University, 1995); and Traut, “Revisiting Stravinsky’s Concerto,” Theory and Practice 25 (2000): 65–86. 45. Morgan, “The Dissonant Prolongation,” p. 53. 46. For a handy overview of the different positions, see James Baker, “Schenkerian Analysis and Post-Tonal Music,” in Aspects of Schenkerian Theory, ed. David Beach, 153–86 (New Haven, CT: Yale University Press, 1983), esp. 153–68. 47. Edward Laufer, “Review: Heinrich Schenker, Free Composition (Der freie Satz), translated by Ernst Oster,” Music Theory Spectrum 3 (1981): 161. 48. Ibid. 49. Ibid.

Chapter 6 1. See Matthew Brown, “Adrift on Neurath’s Boat: The Case for a Naturalized Music Theory,” Journal of Musicology 15, no. 3 (1997): 330–42; Douglas Dempster, “Aesthetic Experience and Psychological Definitions of Art,” Journal of Aesthetics and Art Criticism 11, no. 2 (1985): 153–65; and Dempster, “Renaturalizing Aesthetics,” Journal of Aesthetics and Art Criticism 21, no. 3 (1993): 351–61. 2. Schenker, Free Composition, par. 1, p. 10. 3. Fred Lerdahl and Ray Jackendoff, A Generative Theory of Tonal Music (Cambridge, MA: MIT Press, 1983), 5 (emphasis in original). 4. Schenker, Free Composition, Appendix H, p. 160. 5. See Schachter, “A Commentary on Schenker’s Free Composition,” Journal of Music Theory 25, no. 1 (1987): 119. 6. Schenker, Harmony, par. 18, p. 40. Used by permission of the University of Chicago Press. Translation slightly changed. 7. In response to my claims that Schenker’s generation of the major system is based on ad hoc and arbitrary assumptions, Suzannah Clark notes “In each of these cases, the factor Brown has missed is the Mysterious Five.” Suzannah Clark, “Schenker’s Mysterious Five,” 19th-Century Music 23, no. 1 (1999): 87. 8. Schenker, Free Composition, par. 16, p. 14. 9. Schenker, Harmony, Preface, p. xxv. Used by permission of the University of Chicago Press. 10. Joseph Kerman, Contemplating Music (Cambridge, MA: Harvard University Press, 1985), 85. It is worth noting, however, that Schenker did not necessarily believe that tonal composition was exhausted; see Schenker, Harmony, par. 8, p. 21.

Notes, pp. 214–224

263

11. H. Budge, A Study of Chord Frequencies (New York: Bureau of Publications, Teachers College, Columbia University, 1943). 12. Lerdahl and Jackendoff, A Generative Theory of Tonal Music, 337–38. 13. Ibid., 303. 14. Ibid., 111. 15. Ibid., pp. 304, 336, 40, and 41. 16. Ibid., 306–7. 17. Ibid., 5. 18. Ibid. 19. According to David Neumeyer, “Lerdahl and Jackendoff’s prolongational reduction . . . has achieved no success at all, to judge from adoption of its methods in the literature (outside of Lerdahl himself).” He continues, “the care of its grounding and the logic of its method are matched only by its aridity as an interpretative practice.” David Neumeyer and Julian L. Hook, “Review: Analysis of Tonal Music: A Schenkerian Approach, by Allen Cadwallader and David Gagné,” Intégral 11 (1997): 220–21. 20. Sloboda and Parker insist that such mental images are required for memorizing and performing as well, see John A. Sloboda and David H. Parker, “Immediate Recall of Melodies,” in Musical Structure and Cognition, ed. Peter Howell, Ian Cross, and Robert West, 143–67 (London: Academic Press, 1985). 21. Nicholas Cook, “The Perception of Large-Scale Tonal Closure,” Music Perception 5, no. 2 (1987): 197–206. 22. Robert West, Peter Howell, and Ian Cross, “Modeling Perceived Musical Structure,” in Musical Structure and Cognition, ed. Peter Howell, Ian Cross, and Robert West (London: Academic Press, 1985), 46. They cite the work of L. Cuddy and H. I. Lyons, “Musical Pattern Recognition: A Comparison of Listening to and Studying Tonal Structures and Tonal Ambiguities,” Psychomusicology 1 (1981): 15–33. 23. For some very perceptive remarks about levels of explanation, see David Marr, Vision (New York: Freeman, 1982). 24. This point has been made by David Butler in A Musician’s Guide to Perception and Cognition (New York: Schirmer, 1992), 162. 25. J. J. Bharucha, “Anchoring Effects in Music: The Resolution of Dissonance,” Cognitive Psychology 16 (1984): 485–518. 26. See, for example, Leonard B. Meyer, Explaining Music: Essays and Explorations (Chicago: University of Chicago Press, 1978); and Eugene Narmour, Beyond Schenkerism (Chicago: University of Chicago Press, 1977). 27. Diana Deutsch, “Delayed Pitch Comparisons and the Principle of Proximity,” Perception and Psychophysics 23 (1978): 227–30. 28. Burton S. Rosner and Eugene Narmour, “Harmonic Closure: Music Theory and Perception,” Music Perception 9, no. 4 (1992): 407. 29. Carol Krumhansl, “The Psychological Representation of Musical Pitch in a Tonal Context,” Cognitive Psychology 11 (1979): 346–74.

264

Notes, pp. 224–228

30. Rosner and Narmour, “Harmonic Closure,” 408. 31. Ibid., 407. 32. Bharucha, “Anchoring Effects in Music,” 507. 33. Carol Krumhansl, Cognitive Foundations of Musical Pitch (New York: Oxford University Press, 1990), 236. 34. David Huron, “Tone and Voice: A Derivation of the Rules of VoiceLeading from Perceptual Principles,” Music Perception 19, no. 1 (2001): 1–64. For a rare discussion of modal music, see J. E. Youngblood, “Style as Information,” Journal of Music Theory 2 (1958): 24–35. 35. Bharucha, “Anchoring Effects in Music,” 507. 36. For a general survey of prototypes, see Edward E. Smith, “Concepts and Thought,” in Foundations of Cognitive Science, ed. Robert J. Sternberg and Edward E. Smith, 19–49 (Cambridge: Cambridge University Press, 1988). 37. Diana Deutsch and John Feroe, “The Internal Representation of Pitch Sequences in Tonal Music,” Psychological Review 88 (1981): 503–22; Eugene Narmour, “Some Major Theoretical Problems Concerning the Concept of Hierarchy in the Analysis of Tonal Music,” Music Perception 1 (1983): 129–99; John Sloboda, The Musical Mind: The Cognitive Psychology of Music (Oxford: Oxford University Press, 1985); J. P. Swain, “The Need for Limits in Hierarchical Theories of Music,” Music Perception 4 (1986): 121–48; M. L. Serafine, N. Glassman, and C. Overbeeke, “The Cognitive Reality of Hierarchic Structure in Music,” Music Perception 6 (1986): 397–430; L. Cuddy and B. Badertscher, “Recovery of the Tonal Hierarchy,” Perception and Psychophysics 41 (1987): 609–20; Mary Louise Serafine, Music as Cognition: The Development of Thought and Sound (Oxford: Oxford University Press, 1985); Nicola Dibben, “The Cognitive Reality of Hierarchic Structure in Tonal and Atonal Music,” Music Perception 12, no. 1 (1994): 1–25. 38. Sloboda and Parker, “Immediate Recall of Melodies,” 160. 39. Serafine, Music as Cognition, 213–22. 40. Ibid., 222. 41. A. D. De Groot, Thought and Choice in Chess (The Hague: Morton, 1965); W. G. Chase and H. A. Simon, “Perception in Chess,” Cognitive Psychology 5 (1973): 55–81. 42. For a handy survey of the current state of research in expertise, see Michelene Chi and Robert Glaser, “Overview,” in The Nature of Expertise, ed. Michelene T. H. Chi, Robert Glaser, and Marshall J. Farr (Hillsdale, NJ: Erlbaum, 1988), xvii. 43. Sloboda, The Musical Mind, 116. 44. Ibid., 116. 45. Ibid., 102. 46. Lewis Lockwood, “The Beethoven Sketchbooks and the General State of Sketch Research,” in Beethoven’s Compositional Process, ed. William Kinderman, North American Beethoven Studies 1 (Lincoln: University of Nebraska Press, 1991), 8.

Notes, pp. 228–233

265

47. See Robert Winter, Compositional Origins of the String Quartet in C Sharp Minor, Op. 131 (Ann Arbor, MI: UMI, 1982). 48. For an extensive discussion of these issues, see Matthew Brown, “Composers’ Revisions and the Creative Process,” College Music Symposium 33/34 (1993/94): 93–95. 49. From Edward Holmes, Life of Mozart (London: Everyman, 1924), 255ff. See also Emily Anderson, The Letters of Mozart and His Family, 3rd ed. (London: Macmillan, 1985), xix. 50. Schenker, Der freie Satz, ed. Oswald Jonas, 2nd ed. (Vienna: Universal, 1956), par. 301, p. 198; and Otto Erich Deutsch, “Spurious Mozart Letters” Music Review 25 (1964): 121. 51. W. R. Reitman, Cognition and Thought (New York: Wiley, 1965). For general accounts of verbal protocols, see K. Anders Ericsson and Herbert A. Simon, Protocol Analysis: Verbal Reports as Data (Cambridge, MA: MIT Press, 1984); and Alan Lesgold, “Problem Solving,” in The Psychology of Human Thought, ed. Robert J. Sternberg and Edward E. Smith, 188–213 (Cambridge: Cambridge University Press, 1988). 52. Sloboda, The Musical Mind, 137. 53. Ibid., 149. 54. Mavromatis, “The Early Keyboard Prelude as an Agent in the Formation of Schenkerian Background Prototypes,” Paper delivered at the Third International Schenker Conference, Mannes College of Music, March 12, 1999. 55. Richard Hudson, “The Concept of Mode in Italian Guitar Music during the First Half of the 17th Century,” Acta Musicologica 42 (1970): 163–83. 56. Schenker, Free Composition, Chap. 1, Section 4, p. 6. 57. Ibid., chap. 1, section 4, p. 7. 58. Ibid., par. 301, p. 128. As he put it: “But genius, the gift for improvisation and long-range hearing, is requisite for greater time spans. Short-range hearing is incapable of projecting large spans, because it does not perceive those simpler elements upon which far-reaching structure is to be based. Yet the genius’s ability to encompass even the largest spans is not unduly astonishing. Anyone who, like the genius, can create the smallest linear progressions of thirds, fourths, and fifths abundantly and with ease, need only exert a greater spiritual and physical energy in order to extend them still further through larger and larger spans, until the single largest progression is attained: the Urlinie.” Schenker, Free Composition, par. 30, pp. 18–19. For an extended discussion of improvisation, see Schenker, “The Art of Improvisation,” trans. Kramar, The Masterwork in Music 1 (1994), 2–19. 59. Schenker, Free Composition, par. 51, p. 27. 60. Quine and Ullian, The Web of Belief (New York: McGraw-Hill, 1970), especially chap. 2, pp. 9–19. 61. For an interesting discussion of this and related issues, see Kevin Korsyn, Decentering Music: A Critique of Contemporary Musical Research (Oxford: Oxford University Press, 2003), especially 61–90.

266

Notes, pp. 234–236

Conclusion 1. William Rothstein, “The Americanization of Heinrich Schenker,” in Schenker Studies 1, ed. Hedi Siegel (Cambridge: Cambridge University Press, 1990), 202. 2. See, for example, Carl Schachter, “Rhythm and Linear Analysis: A Preliminary Study,” Music Forum 4 (1976): 281–334, reprinted in Schachter, Unfoldings, ed. Joseph N. Straus, 17–53 (New York: Oxford University Press, 1999); Schachter, “Rhythm and Linear Analysis: Durational Reduction,” Music Forum 5 (1980): 197–232, reprinted in Schachter, Unfoldings, 54–78; Schachter, “Rhythm and Linear Analysis: Aspects of Meter,” Music Forum 6 (1987): 1–59, reprinted in Unfoldings, 79–117; William Rothstein, “Rhythm and the Theory of Structural Levels”; Rothstein, Phrase Rhythm in Tonal Music (New York: Schirmer, 1989); Joel Galand, “Form, Genre, and Style in the EighteenthCentury Rondo,” Music Theory Spectrum 17, no. 1 (1995): 27–52; Charles Smith, “Musical Form and Fundamental Structure: An Investigation of Schenker’s Formenlehre,” Music Analysis 15, no. 2–3 (1998): 191–297; Janet Schmalfeldt, “Towards a Reconciliation of Schenkerian Concepts with Traditional and Recent Theories of Form,” Music Analysis 10, no. 3 (1991): 233–87.

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Index
IV/V Hypothesis, The, 45, 63–64, 180, 225 accented neighbor tone, 54–56 accented passing tone, 6, 54–56 accuracy, xiv, 2, 18, 19–20, 22, 23, 25–65, 234, 235 acoustics, 209–10, 211–15 addition (of a linear progression), 80–81 adjacent seventh chords, 53–54 Agmon, Eytan, 248n55 Aldwell, Edward, 143, 255n10 Anderson, Emily, 265n49 Anstieg. See preliminary ascent appoggiatura, 55–56 Aristotle, 245n51 arpeggiation (Brechung), 77–78, 83, 83–84 Artusi, Giovanni Maria, 167, 258n64 Artusi/Monteverdi Debate, The, 167–68 Audi, Robert, 245n51 Ausfaltung. See unfolding Auskomponierung. See composing out Aussensatz. See outer-voice counterpoint auxiliary cadence progression, 182–83, 188, 193, 197, Ayer, Alfred, 244n40 Ayotte, Benjamin McKay, 239n1 Babbitt, Milton, xiv, xix, 92, 239n6, 251n43 Babbitt, Milton, works by: Philomel, 14 Bach, Carl Philipp Emanuel, xiii, 13, 69, 229 Bach, Johann Sebastian, xiii, 13, 171, 185, 204, 230

Bach, Johann Sebastian, works by: Chorale, “Gelobet seist du Jesu Christ,” 158 Chorale, “Ich bin’s, ich sollte büssen,” 49, 89 Minuet II, French Suite I, BWV 812, 100, 134–36 Prelude in A, BWV 942, 175–78 Prelude in C Major, BWV 924, 126–30, 137 Prelude in C Major, WTC I, 217–18 Prelude in C Minor, BWV 999, 182–83 Prelude in C Minor, WTC I, BWV 847, 130–33 Prelude, Partita No. 1 for Violin, BWV 1006, 133–34 Sarabande and Double, Partita No. 1 for Violin, BWV 1002, 46–47 Bacharach, Burt, 14, 15 background (Hintergrund), 69, 91. See also tonal prototypes Badertscher, B., 264n37 Baker, James, 262n46 Baldick, Robert, 261n36 bass arpeggiation (Bassbrechung), 69, 73, 74, 86, 147, 182 Bassbrechung. See bass arpeggiation Baudelaire, Charles, 192–202 Baudelaire, Charles, works by: Les fleurs du mal, 192 Beach, David, 96, 239n1, 246n3, 252n47 Bechtel, William, 241n8, 243n19 Beethoven, Ludwig van, xiii, 13, 14, 27, 182, 228, 229

282

Index

Beethoven, Ludwig van, works by: “Appassionata” Sonata, 179–80 “Eroica” Symphony, 20, 46, 214, 218 “Heiliger Dankgesang,” String Quartet, Op. 132, 151–54, 157 “Kafka Papers,” 96 Piano Sonata, Op. 2, no. 3, mvt. 1, 48 Piano Sonata, Op. 22, mvt. 4, 3–4 Piano Sonata, Op. 81a, mvt. 1, 55–56 Piano Sonata, Op. 101, 70 Piano Sonata, Op. 110, 69–70 Schottische Lieder, 158 Six Variations, WoO 70, 2–4 Sonata for violin and piano, Op. 24, 96 String Quartet, Op. 59, no. 1, 63 String Quartet, Op. 131, 228 “Waldstein” Sonata, 9 Benjamin, William, xv, 100, 234, 240n9, 240n18, 252n5, 252n8 Bent, Margaret, 257n56 Berardi, Angelo, 27 Bergeron, Katherine, 196, 261n35 Bergquist, Peter, 255n15 Berry, David Carson, 239n1 Berry, Wallace, 261n32 Bharucha, J. J., 222, 223–24, 225, 263n25, 264n32, 264n35 Blackburn, Bonnie, 165–66, 257n56, 257n59 Blackburn, Simon, 243n25 Bononcini, Giovanni Maria, 27 Boyd, Richard, 20, 245n43 Brackett, John, 242n13 Brahms, Johannes, xiii, 13, 171, 185, 204, 229 Brahms, Johannes, works by: Hungarian Dances, 158; “Vergangen ist mir Glick und Heil,” 154–57, 161. See also Schenker, ed., Oktaven und Quinten

Brechung. See arpeggiation Bromberger, Sylvain, 10, 242n15 Brontosaurus, Theory of, 20–21 Brown, Matthew, xviii, 45, 63, 240n11, 240n20, 241n2, 244n35, 246n19, 247n21, 247n42, 248n46, 248n53, 248n64, 249n6, 251n27, 257n48, 259n17, 260n28, 262n1, 262n7, 265n48 Browne, Richmond, 247n20 Budge, H., 214, 263n11 building theories, 12–18 Burkhart, Charles, 251n44, 252n48, 257n49 Burns, Lori, 255n15, Bush, Helen E., 257n53 Butler, David, 263n24 Buxtehude, Dietrich, 230 cadence patterns, 29, 30, 31, 34, 36–39, 42–43, 58, 62, 69, 103, 116, 117, 163–65 Cadwallader, Allan, xix cantus firmus, 29–38, 68, 70–72, 117–21, 140, 146 causality, 10, 242n16 Causey, Robert, 245n54 changing note, 53 Chapman, Graham, et al., 245n45 Chase, William, 227, 264n41 Cherubini, Luigi, 49 Chi, Michelene, 227, 264n42 Chomsky, Noam, xiv–xv, 216–17 Chopin, Frederic, xiii, 13 Chopin, Frederic, works by: Etude, Op. 10 no. 5, 158 Mazurka, Op. 24 no. 2, 154 Mazurka, Op. 30 no. 2, 182–83 Mazurka, Op. 30 no. 4, 53–54, 173–75 Mazurka, Op. 41 no. 2, 62–63 Prelude, Op. 28 no. 2, 182–83 Chord Function vs Chord Derivation, 59–61

Index Chord of Nature (Der Naturklang), 210, 211 Chord of Nature Argument, The, 210–13, 214 Choron, Alexandre-Étienne, xiii chromaticism, 30, 33, 57, 58, 61–64, 68, 112–15, 141, 144–51, 152, 160–62, 167, 172–173, 180–82, 185–86, 224–25 Church modes, 29, 141, 146–58, 181–82 Churchland, Paul, 141, 244n42, 254n6, 254n7 circular reasoning, xvi, 75–76 Clark, William, 261n41 Clarke, Suzannah, 213, 262n7 Classical Theory of Concepts, The, 2–3 Clough, John, 167, 258n63 Cobb, Margaret, 261n32, 261n33 Cohen, Andrew, 248n46 coherence, xiv, 2, 18–19, 22–23, 209–33, 234, 237 combined linear progressions, 103, 123, 126, 130, 136. See also parallel linear progressions Complementarity Principle, The, 27, 56–65, 98, 136–37, 219ff., 225–26 complete/incomplete progressions, 172, 182–83, 186, 197. See also incomplete transference of the Ursatz; interruption (Unterbrechung) completeness of Schenkerian theory, 18, 82–83, 235 composing out (Auskomponierung)/prolongation, 64–98, 138, 147. See also transformations composing vs listening, 137–39, 210, 217–22 concepts, xiv, 2–3, 5–6, 12–13, 25, 209, 234 confirmation/disconfirmation, 12–18, 99

283

consecutive leaps in a single direction, 29, 32, 38, 39, 46 consistency, xiv, 2, 18, 21–22, 23, 99–39, 234, 236 Consonance Constraint, The, 31ff., 40, 55, 166ff. consonances vs dissonances, 31, 35, 36–38, 50–51, 173, 202–8 consonant non-harmonic tones and dissonant harmonic tones, 51–52, 204 context-free grammar, 217 continuous/discontinuous pieces, 182, 183–86 Cook, Nicholas, 219, 263n21 coolness, 18 Cooper, Barry, 256n31 counterfactual conditional, 7–8 coupling (Koppelung), 77–78 Cover, J. A., 241n8, 244n38 Covering Law Model, The, 8–10, 17, 242n15 covering laws/law-like generalizations, xiv, 2, 4–5, 6–7, 8–10, 12, 13, 25, 28, 29, 31, 32, 34, 45–46, 50–51, 58, 61, 62 64, 65, 72, 76–77, 82–83, 209, 235 chromatic generation, 61–64, 72–74, 83, 92, 207–8, 220, 224–25 global, 29ff., 72 harmonic classification, 58, 72–74, 83, 92, 207–8, 220, 224 harmonic progression, 58–61, 72–74, 83, 92, 207–8, 220, 224 local, 29ff., 72 main, 29ff., 72 melodic motion and closure, 28, 29–32, 38, 45–46, 55–56, 72, 74, 83, 92, 207–8, 220, 222–23 relationship between stable and unstable tones, 28, 31–32, 33–34, 50–55, 55–56, 72–74, 83, 92, 207–8, 220, 223–24

284

Index

covering laws/law-like generalizations (continued) relative motion between polyphonic lines, 28, 29–32, 38–39, 47–50, 55–56, 72–74, 83, 92, 207–8, 220, 223 subordinate, 29ff., 72 See also Covering Law Model, The criteria for evaluating theories. See accuracy; coherence; consistency; fruitfulness; scope; simplicity Cross, Ian, 219, 263n22 Cube, Felix-Eberhard von, 75 Cuddy, L., 263n22, 264n37 Curd, Martin, 241n8, 244n38 Dahlhaus, Carl, 163, 257n52 Darwin, Charles, 98 DeBellis, Mark, 241n7, 244n42 Debussy, Claude, 171–202, 237, 258n3, 258n6, 259n7, 259n8 Debussy, Claude, works by: “C’est l’extase langoureuse” Ariettes oubliées, 171–72, 186–92 “Clair de lune” (Suite Bergamasque), 193, 261n34 “Le matin d’un jour de fête” (Ibéria, Mvt. 3), 184 “La mort des amants” Cinq poèmes de Charles Baudelaire, 171–72, 192–202, 261n34 Prélude à “L’après-midi d’un faune,” 160–61, 197–98 “La sérénade interrompue” (Préludes, Bk. 1), 184–85 “Vasnier Songbook, The,” 202 Deductive-Nomological Model, The, 8 deletion/substitution (Vertretung), 82. See also implied tones demarcation problem, 16–17 Dempster, Douglas, xviii, 45, 63, 241n2, 247n21, 248n64, 262n1 description vs derivation, 137

description vs explanation, 2–5, 137, 146, 169 Deutsch, Diana, 223, 226, 263n27, 264n37 Deutsch, Otto Erich, 229, 265n50 Diatonie. See diatony diatony (Diatonie), 29, 57, 69, 147, 224–25. See also tonality Dibben, Nicola, 264n37 Dietschy, Marcel, 202, 261n39, 261n40 diminution, 34 direct chromatic successions, 61–62, 235–36 Diruta, Girolamo, 27 discontinuity. See Principle of continuity displacement (Der uneigentliche Intervalle), 10–11, 34, 54–56, 71, 82, 106–7, 117, 179 dissonance, 29, 31, 33–35, 207; contrapuntal origins, 50–55, 202–8; essential (suspension), 34; non-essential (passing tone, nota cambiata), 34–35 Dissonant Prolongation, The, 172, 202–6. See also Tonklötze divider (Teiler), 86–87 division of the Urlinie (der Gliederung des Urlinie-Züges), 87–90, 134 Doherty, M. E., 17, 244n33 Dubiel, Joseph, 245n2 Duhem, Pierre, 15, 16, Duhem-Quine Thesis, The, 15–16 Dunsby, Jonathan, xv, 234, 240n8 Dvorák, Antonín, 158 elegance, 18 Elk, Ann, 20–21 entrenchment, 14–15 Ericsson, K. Anders. 265n51 escape tones, 109, 116, essential/functional harmony (Stufe), xiii, 25, 26, 41ff., 57, 58, 64. See also Roman numeral

Index essential vs unessential counterpoint, 48 evidence vs system, 18ff. evolution of theories, 17–18, 24 exoticism vs tonality, 141, 147, 150–51, 158–62, 172, 181–82, 186, expertise/expert composers, xix, 138–39, 210, 217–31, 237 explain why vs explain how, 2, 4, 10–12, 137–39 fallacy of affirming the antecedent, 16 fallacy of affirming the consequent, 75–76 fallibility of theories, 14, 16 falsification/falsifiability, 15–17, 65 Ferand, Ernest, 257n56 Feroe, John, 226, 264n37 festgehaltene Ton, Der. See mentally retained tone Fétis, François-Joséph, xiii Feyerabend, Paul, 19, 244n39, 245n49 Feynman, Richard, 242n18 Flagpole Problem, The, 242n15 fliessende Gesang, der. See melodic fluency Fodor, Jerry, 244n42 foreground (Vordergrund), 69, 91 Forte, Allan, 261n41 free dissonances, 51–53, 172, 175–79, 259n11 free voice leading, 41 fruitfulness, xiv, 2, 18, 21, 23, 171–208, 234, 237 functional equivalence, 22, 57, 58–61 functional explanations, 9, 61 functional progressions (Stufengang), xiii, 57, 58–61, 179 Fux, Johann Joseph, 27–41, 50, 64, 67–72, 117ff., 166, 207, 246n4, 246n12, 246n15, 252n10, 252n11 Gaffurius, Franchinus, 27 Galand, Joel, 236, 252n48, 266n2

285

Gamut, 35 generative linguistics, xiv–xv, 216 genius, 64, 140, 179, 231–32 George, Graham, 184, 260n29 Gestalt psychology, 215 Gilbert, Steven, 261n41 Glaser, Robert, 227, 264n42 Glassman, N., 264n37 Gliederung des Urlinie-Züges, Der. See division of the Urlinie Global Paradigm, The, 69ff., 96–98, 183–86, 219ff., 226, 229 Global Rule, The, 28 Goldman, Alvin I., 5–6, 241n1, 241n4 Goodman, Nelson, 7, 14, 242n10, 243n24 Grieg, Edvard, Norwegian Dances, 158 Grim, Patrick, 243n32 gronal/gronality, 14–15 Groot, Adrian De, 227, 264n41 Grue Paradox, The, 14–15 Guilielmus Monachus, 27 Haigh, Andrew, 166, 258n62 Handel, George Frederic, xiii, 13 Hanson, Norwood, 19, 244n39 harmonic function as emergent property, 116, 136, 165 harmonic vs non-harmonic tones, 38, 204–8 Hassler, Hans Leo, works by: “O Sacred Head Sore Wounded,” 158 Haydn, Franz Joseph, xiii, 13, 27, 229 Haydn, Franz Joseph, works by: “Representation of Chaos” (Creation), 93–97, 180; Schottische Lieder, 158 Headlam, Dave, xviii, 45, 63, 247n21, 248n64 headtone (Kopfton), 73, 74, 89, 191 Heinrich Maneuver, The, 27, 41–56, 64, 98, 100, 219ff., 225–26

286

Index

Hempel, Carl, 8, 241n9, 242n12, 242n14, 242n15 hidden repetitions (vorgebene Wiederholungen), 92–96 Hintergrund. See background historical narratives, 9 Höherlegung. See register transfer, ascending holism, 15. See also Duhem-Quine Thesis, The Holmes, Edward, 265n49 Hook, Julian, 263n19 Hostile Witness Principle, The, 20 Howell, Peter, 219, 263n22 Hudson, Richard, 230, 265n55 Hume, David, 13–14, 243n22 Huron, David, 170, 225, 258n69, 264n34 Huysmans, J. K. 199, 261n36, 261n38 Hypothetico-Deductive Method, The, 12–13, 15–17 implied tones, 53–54, 82, 173 improvisation, 154, 230, 265n58 incommensurability of theories, 19 incomplete progressions. See complete/incomplete progressions incomplete transference of the Ursatz (Übertragen der Ursatzformen), 182s Inductive-Statistical Model, The, 8 informed listeners, xvii, 137, 139, 210, 231, 237 interpolations/parenthetical passages, 172, 183–85, 186, 197–98 interruption (Unterbrechung), 87–90, 134 intersubjective testability of theories, 2, 12, 16, 20–23 intersubjective testability vs objectivity, 20 inversional equivalence, 57–58 invertible counterpoint at the octave, 38–39, 106, 110, 124, 136

Jackendoff, Ray, xiv, 210, 215–19, 239–240n6, 262n3, 263n12, 263n19 Jeppesen, Knud, 166, 257n60, 258n61 Jonas, Oswald, 229, 231, 255n15, 256n24, 259n12, 261n41 Kassler, Michael, xv, 236, 240n7 Katz, Adele, 259n8 Keiler, Allan, xiv, 239n6 Kerman, Joseph, xvi, 214, 235, 240n14, 252n50, 262n10 Kinderman, William, 260n26, 260n29, 260n30 Kirnberger, Johann Philipp, 246n3 Kitcher, Philip, 21, 241n8, 245n47 Klee, Robert, 241n8, 243n28 Kopfton. See headtone Koppelung. See coupling Korsyn, Kevin, 154, 256n36, 265n61 Krebs, Harald, 260n30 Krumhansl, Carol, 222, 224, 225, 263n29, 264n33 Kuhn, Thomas, 18, 19, 21, 244n38, 244n39, 245n46, 245n48, 245n50 Kyburg, Henry, 241–42n9 Lagenwechsel. See register transfer Lanfranco, Giovanni Maria, 27 Laskowski, Larry, 239n1, 253n23 Laufer, Edward, xv–xvi, 204–8, 235, 240n12, 240n13, 262n47, 262n49 Law of Prägnanz, The, 215–16 learning and expertise, 220ff. Leerlauf. See unsupported stretch or span Leichentritt, Hugo, 158 Lenormand, René, 175, 258n2 Lerdahl, Fred, xiv, 210, 215–19, 239n6, 262n3, 263n12, 263n19 Lesgold, Allan, 265n51 Lester, Joel, 27, 28, 246n6, 246n7 level (Schicht), 66, 67ff., 83–91 levels of explanation, 263n23

Index Lewin, David, 28, 246n8, 246n9 limits of Schenkerian theory, 172–86, 202, 235 linear progression (Zug), 53, 62, 79–80, 83, 124, 126 Liszt, Franz, 205 Lockwood, Lewis, 228, 252n49, 264n46 long-range hearing, 265n58 Losee, John, 241n8, 243n19, 243n26 Louÿs, Pierre, 172–73, Lubben, Joseph, 75, 250n18 Lyons, H. I., 263n22 Maisal, Arthur, 240n18 major-minor system, 43 Manktelow, K. I., 244n33 Mann, Alfred, 246n5, 246n10, 246n11, 252n9, 257n60 Marr, David, 216, 263n23 Mavromatis, Panayotis, xviii, 168–69, 230, 236, 258n67, 265n54 melodic fluency (der fliessende Gesang), 45, 223 melodic prototype/upper line (Urlinie), 53, 69, 70, 73, 74, 75, 87–90, 123, 126, 130, 147 Mendelssohn, Felix, xiii, 13 mental representation, 217, 218 mentally retained tone or headtone (der festgehaltene Ton/Kopfton), 43, 79 Meyer, Leonard B., 223, 263n26 middleground (Mittelgrund), 69, 86–91; paradigms at deep middleground, 86–89 Mischung. See mixture Mitchell, William J., 143, 180, 254n9, 255n15, 260n19 Mitchell’s Axiom, 180 Mittelgrund. See middleground mixed species, 121, 122, mixture (Mischung), 43–44, 80–81, 114–16, 141–63, 180–82, 224–25; double, 43–44, 81, 116; secondary, 43, 81; simple, 43, 81, 114–15, 148

287

modality vs tonality, 35, 141–58, 172, 181–82, 186 model of music, xvii, 28 modus tollens, 16 Monteverdi, Claudio, 167–68 Monteverdi, Claudio, works by: “Anima mia perdona” (Madrigals, Bk 4), 167; “Cruda Amarilli” (Madrigals, Bk 5), 167 Monteverdi, Guilio Cesare, 258n65, 258n66 Morgan, Robert P., 202–6, 260n31, 261n41, 262n45 motion from an inner voice (Untergreifen), 62, 79–80, 83, 126, 160, 167 motion to an inner voice, 79–80, 83 Mozart, Wolfgang Amadeus, xiii, 13, 27, 143, 228, 229; Attwood Papers, 246n12 Mozart, Wolfgang Amadeus, works by: Piano Sonata, K. 280, 96 Piano Sonata, K. 310, 123 Piano Sonata, K. 331, 138 Piano Sonata, K. 332, 96 Piano Sonata, K. 333, 96 music psychology, 209–10, 214–16, 222–33 musica practica, 232 musica speculativa, 232 Mynatt, C. R., 17, 244n33 Mysterious Five, The, 213, 234 Myth of Scales, The, 140–70 Nagel, Ernst, 245n44 Narmour, Eugene, xvi, 75–76, 223, 224, 234, 240n16, 250n19, 263n26, 263n28, 264n30, 264n31, 264n37 naturalizing tonal theory, 209–33 nature vs art, 213–15 Naturklang, Der. See Chord of Nature Nebennote. See neighbor motion/neighbor tone

288

Index

neighbor motion/neighbor tone (Nebennote), 3, 51–53, 59, 62, 79–80, 83 Neumeyer, David, 75, 250n15, 263n19 Neurath, Otto, 17–18, 244n35 Neurath’s boat, 17–18, 235 nomothetic, 8 non-functional successions, 172, 179–80 nota cambiata, 3, 29, 33–35. See also dissonance Novack, Saul, 255n15 obligate Lage. See obligatory register obligatory register (obligate Lage), 73, 87 Ockham, William of, 140, 245n51 Ockham’s Razor, 22 Oppenheim, Paul, 8, 241n9, 242n12 Oster, Ernst, xv, 231 outer-voice counterpoint (Aussensatz), 75, 100, 104, 116, 128, 136 Over, D. E., 244n33 Overbeeke, C., 264n37 overtone series, 209–10, 211–14 Palestrina, Giovanni Pierluigi de, 154, 166 Papineau, David, 241n8, 244n34 paradigm shift, 21 parallel linear progressions, 123–26. See also addition; combined linear progressions parallel perfect octaves and fifths, 22, 31, 38–39, 47–50, 70–71, 84, 99–139 Parallel Problem, The, 101–3, 106, 116, 126–28, 132, 136 parallel triads, seventh and ninth chords, 53–55, 172, 173–75, 186 parallel vs convergent motion, 104 parenthetical passages. See interpolations Parker, David, 226, 263n20, 264n38

passing tone (simple, consecutive, accented, chromatic, leaping), 3, 6, 29, 33–35, 41, 51–55, 58, 59, 87, 147, 261–62n43. See also dissonance Pastille, William, 249n7, 249n8, 260n21 Pearl, Judea, 242n16 pedal, 100–139, 163 Peles, Stephen, 250n12 perfect vs imperfect consonances, 31, 35–36 Perkins, Leeman, 254n8 personal testimonies, 228–29 Petrouchka Chord, 161, 225 Phrygian II (Phrygische II), 62–63, 148 Phrygische II. See Phrygian II Pierrot, Jean, 199, 261n37 Plum, Professor, 138 polyphonic/compound melody, 43, 46–48, 226 Pomeroy, Boyd, 259n8 Popper, Karl, 15–17, 65, 243n29, 243n31 position finder, 44 prediction, 12–13, 14–17, 21–22, 99, 206 preliminary ascent (Anstieg), 89–90 Prima Prattica vs Seconda Prattica, 163, 166–70 Principle of closure, The, 216, 223, 231 Principle of continuity, The, 216, 231 Principle of inclusiveness, The, 216, 231 Principle of parsimony, The, 22, 141, 170 Principle of proximity, The, 216, 223, 231 Principle of similarity, The, 216, 231 procedures, xiv, 4–5, 10–11, 12, 13, 25, 209, 234 Proctor, Gregory, 185, 260n30 products vs primitives, 169 progress, 24

Index projectibility, 14–15 Prolog, 236 Prosdocimus de Beldemandis, 27 prototypes in general, 5–6, 226–27 Psillos, Stathis, 242n16 Putnam, Hilary, 244n38 Quine, Willard van Ormand, 11, 15, 16, 17–18, 19–20, 65, 99, 233, 242n17, 243n26, 243n27, 244n35, 244n36, 244n37, 244n38, 244n40, 244n41, 245n52, 245n53, 248n66, 248n67, 252n1, 252n2, 265n60 Railton, Peter, 242n11 Randel, Don, 163–165, 257n53, 257n54, 257n55 Rast, Nicholas, 239n1 Raven Paradox, The, 14, 243n23 reaching over (Übergreifen), 78, 83, 237 rectification of II (Die Richtigstellung der Phrygischen II), 62–63 recursion, xiii, 70, 81, 87, 141, 168, 207, 215 Recursive Model, The, 70ff., 83, 84, 96–98, 205, 219ff., 226 referential sonority, 205–6 refutabilty, 16 Reger, Max, Piano Quintet, Op. 64, 181, 260n21 register transfer, 77–78, 83, 126, 167; ascending, Höherlegung, 77–78; coupling, Koppelung, 77–78; descending, Tieferlegung, 77–78 Reitman, W. R., 265n51 relevance, 14 repeated tones, 29, 39 repetition (Wiederholung), 77–78 res facta, 165–66 Ricci, Adam, 252n7 Riemann, Hugo, 116 Die Richtigstellung der Phrygischen II. See rectification of Phrygian II Rimbaud, Arthur, 201

289

Rimsky-Korsakov, Nikolai, works by: Scheherazade, 158 Roman numeral, xiv, 43, 57. See also essential/functional harmony Rosner, Burton, 223, 224, 263n28, 264n30, 264n31 Rothgeb, John, 53–55, 247n40 Rothstein, William, xv, 124, 234, 236, 240n10, 240n19, 249n6, 250n23, 251n25, 251n26, 253n21, 266n1, 266n2 rule-preserving transformations, 22, 67, 70ff., 83, 205, 207 Rytting, Bryce, 251n37 Salmon, Wesley, 241n8, 242n13 Salzer, Felix, xvi, 23, 240n15, 245n55, 255n15, 259n8, Samarotto, Frank, xix Scarlatti, Domenico, xiii, 13 Scarlet, Miss, 138 Schachter, Carl, xix, 74, 143, 236, 249n6, 249n10, 252n48, 252n51, 255n10, 255n15, 261n41, 262n5, 266n2 Schenker, Heinrich, xiii, 6, 22, 23, 25, 41, 119, 121, 235 Schenker, Heinrich, works by: Beethoven neunte Sinfonie, 249n7 Beethoven Piano Sonata, Op. 101, 70, 220, 249n7 Beethoven Piano Sonata, Op. 109, 220, 249n7 Beethoven Piano Sonata, Op. 110, 69, 220, 249n7 Beethoven Piano Sonata, Op. 111, 220, 249n7 Der freie Satz, xiii, xv, xvii, 53, 69, 74, 83, 86, 87–90, 100, 122–23, 126–28, 130, 136, 138, 140, 147, 158, 173, 182–84, 202, 210, 211, 215, 220, 229, 231, 235, 237, 247n24, 247n26, 247n29, 247n31, 247n37, 247n38,

290

Index

Schenker, Heinrich, works by: (continued) 247n39, 248n54, 248n56, 248n57, 248n58, 248n59, 248n60, 248n61, 248n62, 248n63, 249n1, 249n2, 249n3, 249n4, 249n5, 249n9, 250n13, 250n21, 250n22, 250n23, 251n26, 250n28, 250n29, 250n30, 250n31, 250n32, 250n33, 250n34, 250n35, 250n36, 251n38, 251n39, 251n40, 251n41, 251n42, 251n44, 252n52, 252n53, 252n6, 253n14, 253n16, 253n17, 253n18, 253n19, 253n20, 253n22, 253n23, 253n24, 253n25, 253n26, 254n29, 254n32, 254n37, 254n1, 256n21, 257n44, 259n9, 259n10, 259n15, 259n16, 259n18, 260n22, 260n23, 260n24, 260n25, 260n27, 261n41, 261–62n43, 262n2, 262n4, 262n8, 265n50, 265n56, 265n57, 265n58, 265n59 Fünf Urlinie-Tafeln, xiv, xvii, 74, 89, 220 Generalbasslehre, 53, 247n41 Harmonielehre, xiii, xvii, 25, 41, 56, 69, 76, 130, 147–50, 151–54, 181, 210, 211–14, 220, 234, 235, 246n16, 246n17, 246n18, 247n45, 249n6, 250n11, 250n20, 254n27, 256n17, 256n22, 256n24, 256n25, 256n26, 256n29, 256n30, 256n32, 256n33, 256n34, 256n35, 256n37, 256n40, 257n41, 257n42, 259n18, 260n20, 260n21, 262n6, 262n9 Kontrapunkt I, xiii, xvii, 25, 27, 41, 51, 56, 58, 62, 69, 76, 99, 121–22, 140, 146–47, 150, 154,

157–58, 220, 234, 235, 245n2, 247n22, 247n32, 247n34, 247n35, 247n43, 248n47, 248n49, 248n50, 248n51, 248n52, 248n58, 252n3, 252n4, 253n12, 254n2, 255n14, 255n16, 256n17, 256n18, 256n19, 256n20, 256n27, 256n28, 256n38, 256n39, 257n43, 257n45, 259n11, 261n42 Kontrapunkt II, xiii, xvii, 25, 27, 62, 122, 235, 245n2, 247n33, 247n35, 248n48, 249n1, 250n20, 250n23, 253n15 Das Meisterwerk in der Musik 1–3, xiv, xvii, 122, 130, 133, 138, 140, 158–60, 220, 247n23, 247n25, 250n23, 251n24, 251n45, 253n13, 254n28, 254n31, 254n35, 254n36, 254n3, 254n4, 257n46, 258n1, 259n13, 259n14, 262n44, 265n58 Der Tonwille, xvi, 52, 70, 75, 126–28, 220, 247n36, 249n1, 253n23, 254n34 Schenker ed., Oktaven und Quinten, 48ff., 247n27, 247n28, 247n30 Shepherd, Roger, 216 Schicht. See level Schmalfeld, Janet, 236, 266n2 Schoenberg, Arnold, 160, 257n47 Schubert, Franz, xiii, 13, 205, Schubert, Franz, works by: Divertissement à l’hongroise, Op. 54, 158 ; “Die Stadt” Schwanengesang, No. 11, 205 Schumann, Robert, xiii, 13, scope, xiv, 2, 18, 20–21, 22, 23, 6698, 234, 236 scope of music theory, 232 Scriabin, Alexander, 205 Scriven, Michael, 242n13 sequences, 22, 75, 100–139, 157, 165, 197–98, 217, 236

Index Serafine, Mary Louise, 170, 226, 258n68, 264n37, 264n39, 264n40 Sessions, Roger, 229, 255n15 seventh-chords, 58, 259n11 seventh progressions (Die Septzüge), 126, 202 Shiman, Leon, 216 short-range hearing, 265n58 sigh figure, 188–92 similarity, 6, 11 Simon, Herbert, 227, 264n41, 265n51 simple counterpoint vs florid counterpoint, 28–29, 33, 68–72 simplicity, xiv, 2, 18, 22, 23, 140–70, 234, 236–37 sketch studies, 227–28 Sloboda, John, 226, 227–30, 263n20, 264n37, 264n38, 264n43, 264n45, 265n52, 265n53 Smith, Charles, 236, 266n2 Smith, Edward E., 241n1, 241n3, 264n36 Smyth, David, 134, 250n14, 254n33 Smoliar, Stephen, xv, 236, 240n7 Snarrenberg, Robert, 245n2, 249n10 Snell, James, xv, 236, 240n7 species counterpoint, 27–41, 64, 67–72, 117ff., 166 Stalker, Douglas, 243n24 Stimmentausch. See voice exchange Stimmführung. See voice leading Stimmführungs-Schicht. See level Straus, Joseph, 245n56 Strauss, Richard, 228, 261n42 Stravinsky, Igor, 228 Stravinsky, Igor, works by: Concerto for Piano and Winds, 204; Petrouchka, 161–62 strenge Satz, Der. See strict counterpoint strict counterpoint (Der strenge Satz), 25–42, 54, 61, 64, 117ff. 168 string divisions, 209 structural vs ornamental tones, 101

291

Stufe. See essential harmony Stufe Constraint, The, 41ff., 51, 55, 57, 165ff., 204 substitution or deletion (Vertretung), 81–82 suspension, 3, 8, 10–11, 29, 34, 51–52, 54, 71, 117, 119, 173. See also dissonance Swain, J. P., 264n37 Swinburn, R. G., 243n21 symbolist/decadent aesthetics, 186, 199 Taruskin, Richard, 24, 143, 245n56, 255n12, 255n13 Teiler. See divider Tepping, Susan, 250n16 Terzteiler. See third divider testing theories, 2, 12–18 theory reduction, 20–21 theoretical unification, 21, 66 theory-laden observations, 19–20 third divider (Terzteiler), 86 “Three Blind Mice,” 98 Thym, Jürgen, 246n3 Tieferlegung. See register transfer, descending Tinctoris, Johannes de, 27, 165–66, 257n57 Tinctoris, Johannes de, works by: Liber de arte contrapuncti, 166 tonal prototype (Ursatz), 66, 67–71, 72–76, 134, 136, 147 tonal theory vs tonal music, 2 tonal voice leading as transformation of strict counterpoint, xvii, 25–65 Tonalität. See tonality tonality (Tonalität), 69; classical diatonic, 185; vs diatony, 69, 147; directional, 182–83, 186; emergence, 142, 162–70; as family of languages, 185–86; interlocking, 185, 186; vs monotonality, 182–86; nineteenth-century chromatic, 185; progressive, 182–83

292

Index

tonicization (Tonikalisierung), 44, 80–81, 106–8, 112–13, 141, 147, 148, 151–54, 157–58, 158–62, 180–82, 225 Tonikalisierung. See tonicization Tonklötze (tone clumps), 175–79 Top Down/Bottom Up Problem, The, 101–3, 104, 116, 126–28, 136 Tovey, Donald Francis, 93, 160, 251n46, 257n47 transformational level. See level transformations, (Verwandlung ), 25, 66–67, 76–83; back-related, 77, 89; filling in, 78–80, 205; front-related, 77, 89; harmonizing, 79–81, 205; horizontalizing, 77–78, 80, 204–5; non-recursive, 81–82; polyphonic, 75; recursive, 79–81; reordering, 81–82. See also addition; arpeggiation; coupling; deletion; displacement; linear progression; mixture; motion from an inner voice; motion to an inner voice; neighbor motion; reaching over; register transfer; repetition; tonicization; unfolding; voice exchange Traut, Don, 247n42, 262n44 triad, functional. See essential/functional harmony (Stufe) triad, harmonic, 35, 39 Triadic Constraint, The, 35–40, 55, 165, 166ff. triads vs essential harmonies (Stufen), 41–43, 164–65 Tweney, R. D., 17, 244n33 Twentieth-century music/Post-tonal music, 172, 202–8 Übergreifen. See reaching over Übertragen der Ursatzformen. See incomplete transference of the Ursatz

Ullian, J. S., 11, 233, 242n17, 243n26, 244n38, 245n53, 252n2, 265n60 underdetermination of theories, 15 uneigentliche Intervalle, Der. See displacement unfolding (Ausfaltung), 77–78, 83, 126, 175, 189 unsupported stretch or span (Leerlauf), 74, 137 Unterbrechung. See interrruption Untergreifen. See motion from an inner voice Urlinie. See melodic prototype Ursatz. See tonal prototype Vallas, Léon, 258n6 Van den Toorn, Pieter, 143, 161, 255n11, 257n50, 257n51 Vasnier, Blanche de, 201–2 verbal protocols, 229–30, 265n51 Verlaine, Paul, 186–92, 201, 261n38 Verlaine, Paul, works by: Romances sans paroles, 186 Vertretung. See deletion; substitution Verwandlungsschicht. See level voice crossing, 33 voice exchange (Stimmentausch), 77–78, 83, 189 voice leading (Stimmführung), xiii voice-leading transformations (Stimmführungsverwandlung). See transformations Vordergrund. See foreground vorgebene Wiederholungen. See hidden repetitions Wagner, Richard, 196, 205 Wagner, Richard, works by: Parsifal, 196–97 Wason, Robert, 248n53 Web of Belief, 233 West, Robert, 219, 263n22

Index Westergaard, Peter, 74, 250n12, 250n17 Whittall, Arnold, xviii Wiederholung. See repetition Wienphal, Robert W., 257n53 Winter, Robert, 265n47

Yih, Annie, 259n8 Youngblood, J. E., 264n34 Zacconi, Ludovico, 27 Zarlino, Gioseffo, 27 Zug. See linear progression

293

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Explaining Tonality: Schenkerian Theory and Beyond Matthew Brown A wide range of music—from Bach to Mozart and Brahms—is marked by its use of some form of what is generally called “tonality”: the tendency of music to focus melodically on some stable pitch or tonic and for its harmony to use functional triads. Yet few terms in music theory are more enigmatic than that seemingly simple word “tonality.” Matthew Brown’s Explaining Tonality: Schenkerian Theory and Beyond considers a number of disparate ways in which functional tonality has been understood. In particular, it focuses on the comprehensive theory developed by Heinrich Schenker in his monumental three-part treatise Neue musikalische Theorien und Phantasien (1906–35). Schenker systematically investigated the ways in which lines and chords behave both locally within individual tonal phrases and globally across entire compositions. Explaining Tonality shows why Schenker was able to elucidate tonal relationships so successfully and why his explanations have many advantages over those of his rivals. In addition, it proposes some ways in which Schenker’s approach can be extended to tonal features in works from before Bach (such as Monteverdi) and after Brahms (such as Debussy and Stravinsky). Along the way, the book explores six methodological criteria that help in building, testing, and evaluating a plausible theory of tonality or, indeed, any other musical phenomenon: accuracy, scope, fruitfulness, consistency, simplicity, and coherence. It reveals how understanding the tonality of a piece can shed light on other aspects of musical composition. And, in conclusion, it describes some ways in which Schenkerian theory might fruitfully develop in the future. Matthew Brown is Professor of Music Theory at the Eastman School of Music, University of Rochester, and author of Debussy’s “Ibéria” (Oxford University Press).

Praise for Explaining Tonality: “Explaining Tonality is a cogent, concise, and eminently readable study of one of music theory’s most important subjects. Matthew Brown traces the philosophical and psychological contexts within which Schenkerian theory can be placed, and considers other relevant topics, such as strict counterpoint and nineteenth-century chromaticism, by way of a wealth of freshly observed compositional examples. Technically expert and critically evenhanded, this absorbing exploration of tonality in theory and practice sets new standards in its scope and authority.” —Arnold Whittall, King’s College (London) “Matthew Brown’s Explaining Tonality: Schenkerian Theory and Beyond carefully sets out a well-reasoned and convincing case for the scientific viability and logical foundations of Heinrich Schenker’s extraordinary approach to analyzing tonal music, revealing the solidity of its foundations. His work should be read by anyone who has an interest in the epistemology of music theory.” —Frank Samarotto, Indiana University