Structural Functions in Music (Dover Books on Music, Music History)

sfrucfura/ functions in music WALLACE BERRY Pnjssor q' Music Universiy Qf British Columbia

DOVER PUBLICATIONS, INC., NEW YORK

Copyright © l976, 1987by WallaceT. Berry. All rights reserved underPan Americanand International Copyright Conventions. Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable andCompany, Ltd. This Dover edition, first published in 1987, is an unabridged and corrected republication of the work originally published by PrenticeHall, Inc., Englewood Cliffs,Newjerseyg in1976. Correctionshave been made by the author specially for this Dover edition. Manufactured in the United States ofAmerica Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y ll5Ol Library of Congress Cataloging-in-Publication

Data

Berry; Wallace. Structural functions in music. Reprint. Originally published: EnglewoodCliffs, NJ. : Prentice-Hall, 1976. Withcorrections by the author. Includes bibliographical references andindex. 1. Musical analysis. I. Title. MT6.B465S8 1987 781 86-31912 ISBN 0-486-25384-8 pbk.!

fo Maxine

contents preface to

the do ver edition,

preface to

the first

xi

edition, xiii

introduction, 1

CHAPTER ONE

tonality, 27 Introductory notes Introductory comments concerning tonal and linear functions The concept of multileveled function further explored The ideas of primacy and hierarchy among pitch-classesand pitch-class-complexes; genericand particular tonal systems Tonal systemsof compositions: regions and interrelations The concept of a generic tonal system exploredin theory Primary and secondary tonicsand their structural and auxiliary functions; multileveled multiple! tonal-harmonic function; tonic and dominant forms Essential and auxiliary linear functions of pitches and pitchcomplexes, their hierarchic basis often determined by tonal and cadential factors Tonal order as an inflation of harmonic order and succession Tonal fluctuation and techniques of immediate succession by which it is effected Tonal organization as a pattern of relative stability opposed to relative flux Tonal rhythm The questions of tonal intersection, direction, and distance; intervals of fluctuation; high-level chromatic successions and nondiatonic tonics; the interchangeability of modes and the equivalence of parallel tonics Concepts of tonal progression and tonal recession Concepts ofharmonic and melodic progressionand recession; complementarities and counteractions of element-successions VII

viii contents Some examplesof quasi-tonal order in melodic and composite functions Melodic curve Some further observations concerningmelodic analysis Some particular issues ofharmonic analysis in later styles Procedures ofanalysis; symbolic representations ofmelodic and harmonic functions and afliliations Gregorian chant, Veni creator spiritus Two preludes from Fischer, Ariadne Musica Bach, Drauf schliess ichmich in deine Hande, from Motet, Komm,Jesu, komm Beethoven, SymphonyNo. 2 in D, Op. 36, fourth movement; representations ofthe tonal system 'andtonal rhythm Wolf, Das verlassene Miigdlein, from Gedichtevon Miirike Ravel, Le Martin-Pécheur, from Histoires Naturelles Bartok, String Quartet No. 2, Op. 17, third movement Webern, Four Pieces, Op. 7, for violin and piano, No. l Berg, Four Pieces, Op. 5, for clarinet and piano, No. 4 Summary notes on the universality and significance of the principle of hierarchic tonal order Concluding notes

CHAPTER TWO

texture, 184 Introductory notes Textural progression, recession, andvariation as structural factors Types of musical texture; problems of classification and terminology Some further considerations ofterminology and aspects oftexture Texture and style Textural rhythm Qualitative and quantitative values Density and dissonance Interlinear independence and interdependence Imitation, a universal feature of many polyphonic styles; multiple counterpoint The activation of simple textures The complementa.ryand compensatory dispositions oftexture in relation to other element-structures Some textural functions in delineation of form Textural processesin progression toward intensity, in recession toward cadence, and in anticipation of thematic statement

COHIBDIS Texture as space Motivic texture; the provocative effect of unusual textures Levels of analysis and of hierarchy in the textural structure Gesualdo, Or, che ingioia credea viver contento from Madrigals, Book 4! Telemann, Fantasy No. 4 for violin alone, first movement Bach, Denn das Gesetzdes Geistes, from Motet, jew, maineFreude Brahms, Symphony No. 1 in C minor, Op. 68, first movement Un pocosostenuto!, introduction Dallapiccola, Goethe-Lieder for soprano and three clarinets No. 1, In tausend Formen! Serialism and texture; texture as a product of chance operations Concluding notes

CHAPTER THREE

rhythm and

meter, 301

Introductory notes Fundamental concepts of rhythm Rhythm as activity and motion Rhythmic pattern as motivic The rhythms of element-successions A theoretical approach to the consideration of meter as accent-delineated grouping The concept of meter as, by definition, subject to fluctuation Meter as one manifestation of grouping in music Meter as opposed to the notated bar-line Impulses and their functional differentiations Functions of the cadential conclusive! impulse Criteria of accentuation Further comment concerning diverse impulse functions Levels of metric structure and analysis Metric irregularity: horizontal and vertical noncongruity The problem of preconditioning metric structure and syncopation Stability and flux; metric progression and recession; compensatory and complementary functions in relation to other element-structures Beethoven, Symphony No. 8 in F, Op. 93, first movement Chopin, Prelude in E, Op. 28, No. 9 Webern, Three Pieces, Op.ll, for cello and piano, No. 3; and Five Pieces, Op. 5, for string quartet, fourth movement Some twentieth-century problems in rhythm and meter; recent developments in serialism of durational units-theory and practice Concluding notes

X COI7f6I`lfS APPEND/X ONE editorial notes,

425

APPEND/X TWO translations, 427

index of

musical examples

index of

subjects, names,

and citations, and terms,

437

433

preface tothe doveredition

It has been more than a decade since Structural Functionsin Music was first

published. Thus, I should perhaps affirm at the outset of this new editionwhich is not a revision-that although my views and particular correlative analytical approacheshave undergonecertain refinementsand redirectionsof formulation evident in a number of publications during the intervening years, thefundamental ideasand methodsexplored in this book remain, for me, validand useful.These includethe important concept ofmusical structure and significative shaping as in one critical sense! an undulation between tendencies ofprogression andrecession. This books three essaysconstitute one testimony among many to the deepening preoccupationin the professional communityof musictheory with allied andoften parallelquestions aboutstructural unities, organic substance, and modesof expressivelydirected fluctuation in all of musics cofunctioning elements. Sinceits appearancein 1976,Structural Functions has beena factorin, and has directly stimulated, abundant discussionof various and frequently innovative approaches to studies of the shaped elementsof music. Such proliferating studies-complementary or refutative, and often speculativehave appearedin the form of conference papers,unpublished researchby young professionals,including many student theses,and a substantial stream of published work. Indeed, probing inquiries into musics melodic and harmonic dimensions, texture, rhythm, and other elements continue, as they must, without end. The complexities ofpreliminary and derivative questions about the expressive andintellectually provocative properties of music are such that the theorists quest is assuredly notfor final answers, whoseunattainability we know enough to know, but rather for deepened channels of understanding consistentwith our responses tomusic assensitive musicians. I wish in this brief prefatory statement toexpress thehope thatthis book will continue to be a basisfor respondentand pursuant investigations, asone of a growing number of theoretical works treating related issuesin disparate ways, yetalways to the end of better understanding ofstructure and effect in music. It is, to be sure, a matter of gratification to me that the general

xi

xii preface

to the dover edition

accessibility of Structural Functions in Musicwill besignificantly broadened as a result ofits republicationas aDover paperback. WALLACE BERRY Wncouveg 1987

preface tothe firstedition

Major premises and objectiveswhich are the basisfor thesestudies are fully set forth in the Introduction; the work can be characterized, simply and in summary, as an inquiry into tonal, textural, andrhythmic structures in music, and into conceptual and analytical systems for the study of these fundamental elements.While assuming the readers understanding of basic principles of -tonal form and analysis, this book carries ideas concerning tonality, texture, and rhythm-their processive functions and structural configurations-into the analytical discussion of many works and extracts ranging from early modal styles to recent times. Much important theoretical work that has provided foundation for my thinking about tonality conceived as a principle relevant far beyond the conventions of the eighteenth and nineteenth centuries!, and by which I have been informed and

motivated, will

be evident

to readers

who are

familiar with traditions, and with certain recent procedures of inquiry, in music theory. Such sources and points of departure which include the theoretical works of Schoenbergas well as important literatures embodying and reflecting concepts of hierarchically ordered structural elements! are cited at appropriate stagesin this book. At the sametime, I have soughtto bring about a presentation of ideas in conformity with certain essential underlying premisesand in the interestof consistentexposition ofa pervasive point of view which is, essentially,mine. The studies in texture and rhythm have few if any significant, direct antecedents; the work in those two areas Chapters 2 and 3! is largely in original formulation. Many of the ideasand approachesset forth in this book have been the bases forgraduate and to some extent undergraduate courses andseminars

in musictheory inwhich, atthe Universities of Michiganand BritishColumbia, I have beenprivileged to explore with inquiring and highly committed student collaboratorsthe lexical,methodological, andconceptual systems and problems of the three areas of concern. The responsiveness, butmore the challenges andinitiatives, of students in these studies,and in research stemxiii

xiv preface

to the first edition

ming from them, hasbeen ofthe greatestbenefit tome in the preparationsfor this book and the cultivation of

ideas central to its content.

A seminaron textureproved tobe thesource of at leastthree extensive research studiesof which I am aware, and that on rhythm and meter the source of others at early and advanced levels of graduate study, not to mention numerous searching papersof more limited scope. The dissertation of Dr. Anne Hall Texture inthe ViolinConcertos Stravinsky, of Berg, Schoenberg, and Bartok ; Ann Arbor: University Microfilms, 1971! brought to impressive refinement theoreticaland practical systems inthe realm of texture, applying them in probing analysis of significant extracts from the four subject works. I wish it were possible to list the names of other student contributors, some of them engaged in important research at this time. During crucial stages in this work the longer periods of its inception and conclusion! I have enjoyed accessto the resources andfacilities of the Music Division of the San FranciscoPublic Library and the Music Library of the University of California; my indebtedness tothe staffs of both is very substantial indeed. And I owe special thanks to Ilene Silverman, of the Music Library at the University of Michigan, who helped me in many ways during the year in which the book was completed. My colleagues, Professors IleneOlken, John DArms, and Otto Graf, advised me with respect to translations of texts in certain works cited as examples. A number of publishers have kindly granted authorization for the reprinting of copyrighted materials, and while these areacknowledged in onpage citations in the text I should also take this opportunity to express my appreciation to all of them, and especially toBoosey andHawkes, Inc., and Theodore PresserCompany, for permissions toquote from restricted works. Obviously, a book like this is quite literally impossible without such permissions.

My wife, to whom this book is dedicated, has helped in many ways, even in such tiresome chores as proofreading, but particularly, and critically, in her quiet encouragement through pressured times, and in her everlasting patiencewith my brooding preoccupationsduring such times. I thank her for a companionate, sustaining presence, and for sharing with me so overtly the exhilaration of achievement, whatever its measure, as this work has evolvedin many places towhich we have travelled, and where we have found composure in memorable natural ambiences, during these past seven years.

Finally, I am grateful to have learned a great deal, through much reading, discussion, and analysis preliminary to and concurrent with the preparation of this book, especially aboutJosquin, Beethoven,and Webern, and therebyabout whatI think to bebroadly relevantfactors inthe experience of music. WALLACE BERRY 1975

strucfura/ functions in music

introduction

Outside of simplest genres,it is unlikely that we ever attain full understanding of a particular musical experience, so complex are its elements, their actions,

and interactions.

This book

seeks to move toward

a better

understanding of structure and experience; it does soin systematic exploration of the elements of structure and their important interrelations, laying out a variety of approaches to the analysisof directed successions ofevents involving tonality, melody, harmony, texture, and rhythm-each of these treated throughout much of its range of potential operations. These four studies thusseek togive comprehensiveexposition to particular syntactic processes inwhich music can besaid to have meaning,and to illuminate in penetrating analysesof works of many kinds the procedures by which structural elements in nearly all music function expressively. Amajor concern is the consideration of relations between specificformulations and expressive eH`ect in significant works, and a number of concernsare treated in conceptual, methodological, terminological, and analytical systems relevant to such particular structural formulations. One sets out in a project of this kind in the hope, to use Edmund

Wilsons hauntingwords, ofbeing cured of someache ofdisorder, relieved of someoppressive burdenof uncomprehendedevents. I cite thesewords not because Ican hopeto have achieved with consistency soelusive anobjective, but becausethey reflect so eloquently the intensity of purpose and commitment to which a writer on serious mattersis dedicatedwho seeksto put forth systems ofanalysis and thought by which understanding can be induced. To a large extent, explorative emphasis centers here in theories of rhythm and texture, structural parameters towhich relatively little attention is given in the existing literature of music theory. A large-scale scholarly, philosophical, analytical, and systematic penetration of these two vital factors in musical structure and experience is an undertaking of awesome difficulty; yet, any broad, probing effort toward systemsby which rhythmic 1

2 introduction

and textural processes can be investigatedconstitutes, tothe extent that it is successful, aneeded anduseful stepin the direction of better understanding of theseand related critical factors in the musical experience. Although concernswith rhythm and texture are, in the somewhatmore pioneering efforts they represent,of central importance in this work, related and interactive tonal, harmonic, and melodic systems are examined as parallel shaping factors in musical structure. Moreover, by recurrent reference tointerrelations among element-systems, reciprocal and analogical correspondences areindicated in which the actions of individual elements are seen to project expressiveshapes of progressive, recessive, static, or erratic tendencies. Progressiveand recessive intensifying and resolving! processes are seenas basicto musical effect and experience. In the effort to subject to theoretical expositionand analytical explication a consistent, fundamentalview of .syntactic relations in music, a strongly permeative value is, by implication, embraced: a belief in the importance and necessityof logical analysis of the musical experience, of the study of objective data derived in the analysis of structure and experience. That experience canbe regardedas thesum of responses attributableto particular musical processes,the action-reaction complex set up whenever there is perception and enhanced whenthere iscognition! of musical stimuli issuing in contexts in which syntactic relations are cultivated and controlled as the result of disciplined creative acts. I strongly aflirm that belief in logical analysis, andin the necessity andworth of the pursuit of rational inquiry into the musical experience. At the sametime, I see that mode of inquiry as one in which conjectural hypothesisand intuition where intuition is the creative fusion of acquired knowledge and experience! are_ vitalin triggering necessary questionsand answers,and in suggesting interpretationswhich can be examined for plausibility and, at times, susceptibility to empirical verification.

Thus, the many analysesin which theoretical premisesare throughout this book brought into focus are directed to consideration of the vital question: How does musicspeak, andwhat is the nature of its language? Or, to what specyic irylection.r can we besaid torespond with understanding and _keling ?The study of particular syntactic techniques and procedures, howevertentative its conclusions,can render the expressivecommunication of musical meaning less mysterious than it is oftenthought and sometimes wistfullyhoped! to be; The distinction intended isthat betweentwo levelsof awareness in theexperience of music, twolevels goingbeyond simple hearing: thecognitive levelis regardedas thatinvolving understandingas opposedto that of simpler affective sensory apprehension. Thisdistinction isrelated tothat betweenknowing 'andfeeling, insightand innocentreaction; both kinds offactors arecritical in the musicalexperience.

introduction 3

indeed, the belief that logical insights can be had into relations between ltructure and effect underlies all productive aesthetic inquiry. It is a belief proudly affirmed in this book. The business ofmusical analysisis to consider the nature of functionsand expressive #set in the tones and rhythms of which music is made.

Surely it is clear that any serious investigationof structure and effect, to the extent of its validity, must significantly inform the critical evaluation, stylistic understanding, and interpretation of music, while laying important bases fordecision in musical creativity. Beyond this general statement one can do little more than affirm with the most positive emphasis that the understanding ofstructure and effect in all musical elements andin their interactions isof very decisive importancein all serious, professionalendeavor in music, andto the listener-participant as well. Moreover, a theoretical system which affords useful commenton the relation of structure to expressive effect will suggest many significant and necessary parallels linking music with other, especiallytemporal, art forms. " The performer, for his special part, must fully understand the functional-expressive basisand significance of energizingand subsidentelementproeesses which it is his challenge to portray and project. However, I believe the path from analytical insight to interpretive decisionis oneof considerable complexity. Conclusions drawn from the analytical undertaking do not lead directly and unequivocally to particular interpretive decisions; rather, the performermust often-perhaps usually-make difficult judgments within a range of plausible solutions for example, whether in performance to underscore bythe slight adjustment oftempo anessential recessive process!. For now, in the interestof achievinga rigidly defined, circumscribedfundamental purpose, I have had to be content with a restricted number of suggestive references toproblems of interpretive decision in the examples towhich this book refers. Nonetheless, suchreferences occur,especially in Chapter 3. I believe a great deal of understanding of musical process,in its essential terms, to be accessible tothe involved layman or amateur. Indeed, many of the most persuasive factorsin musical effect and function are delineative of shapes andprocesses thatcan be demonstrated, given necessary theoretical and analytical calculations, relatively simply. A related view holds that many of musics most immediate and compelling strengthsderive from the shaped actions of elements of primitive substance and effect e.g., those of dynamic changes, or timbral differences!. Nevertheless, the thorough analysisof all the elementsof structure in their confluent and con2See also relevant statementsin the Preface tothe authors Form inMusic, 2nded. Englewood Cliffs,N ._].:Prentice-Hall, Inc., 1986!.

4 introduction

tiguous operationsat all relevant hierarchic levels isan issueof sophistication and complexity-and one that must constitute a significant and essential part of the competenceand experienceof the professional musician. In an important sense, this book is concerned with groupingof many kinds and at diverse levels~event-groupings goingbeyond thoseof form,the articulation of units by formal cadential, thematic! procedures. In general, structureis discussedas toits functional and expressiveconsequences within an intensity curve delineated by groupings and controlled associations ofevents underlying nearly all composed music. In music that is composed asopposed tomusic of random operations or random consequences!, actionschanges, events!involving various elements lines of pitch change, tonal and harmonic succession, rhythm and meter, texture, and coloration! are so conceived and controlled that they function at hierarchically ordered levels in processes by which intensities develop and decline, and by which analogousfeeling is induced. Element-actionsmay converge in collaborative, directed lines of change; or, probably more commonly, certain prevailing lines of change function toward a particular expressive endwhile others, subordinate to the essentialfunctional tendency, are counteractive.

Examples of universal musical processes canbe indicated in observations ofthe sort: the music iscoming to a close, or the music is advancing toward a peak of intensity, or the music is in a tentative state, with something of 'special consequence about to emerge, and comparable cognitions. In the first of these instances,the processof closure, predominant elementactions of necessity followa course of decline but at the same time others may and often do resist that essential structural tendency. For example, crescendo, or increase in textural density, or, in cases of tentative internal closure, even increased textural diversity, may result in relatively active cadential process: li92

Essential to this concept is the principle that the senseof motion in music passage from one event or state to another! largely, although not exclusively, dependson change within one or more element-successions. From this premise, it follows that where there is not change there is in critical respects immobility. For example, in the fixed repetitions of an ostinato pattern at the samepitch level there is motion at only the lowest level of structure-that of profiled configuration within the pattern itself Eventfulness in music demands that iteration without change, at any specified level, occur only within

limits: to

the point of such limits iteration

can be

introduction 5

intensifying, while beyond theselimits, diflicult to define, tedium and tautology result. Motion in music may, in principle, be erratic, but its usual condition is one of directed activity-courses of change-in lines of _growth ordecline at various levels. A line of decline at one level maybe observedto be subsumed within a broader line of growth at another.! The punctuation and proportionalization of such linesof changeconstitute animportant aspectof rhythm i.e., are expressive ofelement-rhythms and meters which may or may not converge!. The successions of element-events trace shaped, controlled profiles-a concept arising analogously in the sense oftemporal paths and, wherelines ofaction crossfields ofpitch, of shapes in space. Musical structure may be saidto be the punctuated shaping oftime andspace into lines of growth, decline, and stasishierarchicalbv ordered. Of course, not all elements actin commensuratesignificance, or even act at all, in given instancesof suchconcurrent linesof change.For example, it is clear that the processof digressionfrom and resolution into a primary tonal systemwill have little or no significancein certain styles; or the action of melodic line in rising and falling, conjunct and disjunct, pitch successions may in particular instances be in part irrelevant or utterly inapplicable. In the analysisof the interactive and hierarchically related! element-actions in expression ofparticular functional processes the initial question is, thus, that which seeksto determine within whichparameters contributive actions occur. As to general relative importances of such actions in perception and cognition of functional-expressive effect,we will make conjecture from time to time~speculating, for example, that such immediate, direct, and primitive actions as changesin dynamic intensity or textural density have more palpable persuasiveforce than a relatively elusive factor like that of tonal fluctuation. But, in general, the questionof relativesignificances ofcorrelative actions in cognitive experience is a difficult issue which this book does not attempt to explore. Element-actions in music function in certain basic formal processes or activities which can be rather simply stated, assumingit is understood that there is an infinite range of possible andpalpable degreesof change, rates of change, and specific configurations of interrelations among contributive elements-and hence of expressiveeffect conceivable and potential within such formal processes. Such fundamental classificationsof formal process are those of ! introduction-often involving dissonant prolongation as a primary factor of expectant intensity in precarious balance with quiescent, 3Presumably, the limited capacityof pure iteration for intensification isaccountable to the rising expectationof inevitabledigression.

6 introduction

tentative conditions such as relative textural sparsity and slow tempo; ! statement-normally in a context of 'relativestability; ! restatement-often involving variation, but again in relatively stable conditions; ! transition and development-both normally relatively fluctuant; and ! cadence-in an environment of relative decline and deceleration. Someaspects offormal technique are characteristic of particular styles and, of course, particular systems; thosemost representativeof broadly prototypical formal procedures are explored in many existing works. The present work addressesitself to theoretical discussionand analysis of certain growth-connected and decline-connected! intensity-activity shapes inwhich functional and expressivelines ofchange arevery commonly conveyed. Although certain formal processes areby definition associated with particular functional tendencies e.g., that of the cadential processwith recessive action!,in general, progressive andrecessive linesof change occur variously in diverse formalcontexts. Indeed,music isconstantly involved in both concurrent and contiguous event-relations! in a dialecticin which opposing tendenciesof growth and decline and their correlatives up-down, far-near, dense-sparse,simple-complex, etc.! are in continual interplay at different hierarchic

levels.

Formal processesand element-actionsof growth and decline are thus both discretely identifiable and interdependent asaspects ofform and structure, and the study of lines of changing intensities yields significant understanding ofparticular compositional techniques bywhich formal processes are carried out. The primary concern of this book is the analysis of structure apart from formal processas such,but it cannot fail at the same timeto note that particular functional tendencies are critically allied to essential and universal techniquesof form. The idea of music as dialecticalbin balance between intensifying and resolving tendencies, and involving complementary or counteractive relations, brings to mind a moving, terse statementby Pousseuron the third of

Weberns ThreePieces, Op.ll, for cello andpiano discussed as to metric structure onpp. 397-400of thisbook!. . . . it is through the positioning of well-defined and clearly perceptible units, articulating themselves bymeans of reciprocal effect, each one limiting and defining the nature of its fellows. Thus an available space is established among the units, an emptiness within which they may reign calmly in all their abundance. Symmetry and asymmetry, determination and indetermination, equality and inequality, are thus more than simply contradictory principles excluding each other in an absolute manner; rather, they are complementary properties, each conditioning the other and mutually dependent. An excess inone sense These includethe authorsForm inMusic, citedmrlier.

introduction 7 or the other leadsto the same pathologicaldisorder. Only a correct proportion, a balanced tension which may be realized in an infinitely varied manner! can engender afree order, vital and significant, representingmultiplicity and communication, individuality and recognition?

Pousseur istalking about relations of units; yet, the remarkable spectrum of expressive possibilitieswithin the range of musical techniques is immediately apparent when one-thinks of further potential relations among units of dwrent but concurrently functional! elements, and of the cofunctioning, complementary or counteractive, relations of such element-units at diverse hierarchic levels of structure. The musical situation and its experiencepose inthis light a challenge of great complexity, yet one of beguiling interest and inescapable significance. To see istructural function in music as it concerns lines of intensity

change is to see,in general,three possibilities: that ofincreasing intensity to which we apply the term progression!, that of subsiding intensity to which the term recession is applied!, and that of event-succession involving unchanging degrees ofintensity stasis!.° Of the latter, it is noted that true immobility is probably inconceivable at the lowest structural level; even a successionof sound events of equal qualities can be viewed as consisting of attack-decay shapes in each of which it may be reasonable topostulate recessionin theory and experience. At the same time,the principle of stasis,even whileit cannot be said to have absolute applicability throughout a leveled structure, is of practical importance. For example, repetition of a motivic unit in which parametric factors by which the motive is characterizedremain unchanging to the extent that this is possible in performance which is not mechanical! is, for all practical purposes, staticat levels beyond that of the motive itselli The concept of musical motion iscritically allied to the concept of progressive, recessive, and static events andevent-complexes. Tothe extent that motion is a useful concept in musical experience,it may be said to reside in factors ofthree kinds,of which the mostimportant is that involving changing qualities in contiguous sonorousevents.

Henri Pousseur,The Question of Orderin NewMusic, in PerspectivesNew qf Musab, V, l 966!, pp. 104-5; translationby David Behrman. °It is assumed thatcertain structural conditions generallybut in random contexts only accidentally!are expressive of intensity,others ofrelease. Such conditions areevocative of feeling-of anxiety, of calm, of exhaltation, etc.,not really explored hereas such,but implied. _ This underlying assumption isin somedegree subjectto conditionsof probability, predictability, andfamiliarity within the respondentsrelation to a particular style, system, or rhetoric. Thus, anyassumed intensity scale orstate within a givenstructural parameter depends inpart bothas tokind anddegree onthe understanding or intuition of probabilities within the terms ofa particularstyle andsystem.

8 introduction

First, in a purely temporal sensea successionof even equal! sound events, suchas onethat consistsof evenly spaced strokeson a drum lacking dynamic or other changes, is felt as moving in time. Thus, passage through a temporal Held is felt as a kind of motion-a stream of eventfulness whichcourses throughtime. Even when a stretch of time is vacant

unpunctuatedevents, by inasQ !, itis bounded by events at itsextremities whose connection-a relation reaching across the vacant temporal field-denotes passage as motion of a kind. A secondand far more critical factor of motion in musical experienceis that associatedwith successionsof sound events havingchanging qualities. As suggested, suchchange in element-actions isnormally traceable in, at given levels, consistentdirections of increasing or receding intensity,but it may, at given levels,be or seem! arbitrary and erratic too. In any event, the succession ofa sound eventby another qfdW:rent qualiy'or qualitiessay, two events of different degrees ofloudness! conveysan analogicalimpression ofmotionof a distance between disparate qualitative states havingbeen traversed. Thus, in a sense,in music, change ismotion. A third factor has to do with the illusion of a spatial Held in music delineated by the pitch ambitus inherent in the spectrum of perceptible frequencies; this ofcourse can be viewed, within the scope ofthe secondfactor described above,as motion implicit in change within the element of pitchline, of melody. But obviously it has special significance and is usefully regarded asa distinct factor. To the extent that succeedingsound eventsof changing pitches are felt as coursing over an analogical spatial field a broad leap between twopitches isfelt as going farther than a succession linked by a conjunct intervallic relation!, the eventscan reasonablybe saidto describe motion within that field. Thus it is that lines actively cross or are related diagonally or triangularly in musical texture. With reference to all of thesefactors, this book embracesa concept of motion in music presupposing,in its direction and control, a value which inheres in significant musical expression ofall times. It was earlier stated that directed, functional-expressive elementsuccessions havethree conceivable tendencies: progression,recession, and stasis that of aimless or arbitrary lines of change in some music would constitute a fourth, although its consequenceis likely to be one of relative stasis atgiven levels!. In any real musicalsituation severalelements arelikely to be active, and these do not necessarily-in fact are not likely to-correspond with consistency indelineating progressive,recessive, orstatic concurrent lines of change. Moreover, when element-successions move in parallel, or complementary, directions, theyoften do so, assome analysesin this book will show, at differing rates of ascent or descent in the intensity lineat differing rhythms of element-change.

introduction 9

Within these concepts, musical structure if form is seen as the thematic-developmental scenario! can be regarded as the coryluenceof shaped lines of element-succession which either agree are complementary! in intensity direction or disagree are mutually counteractive, orcompensatory! in direction. Within the range of musical elementssubject to control within these procedures,some are of course of greater relevance to certain styles than others; but the underlying concept of the confluence of functional element-actions as fundamental to

musical structure

and effect is of crucial

significance in all styles. We may posit, for example, certain common kinds of element-action having function analogous tothat which is clearly pertinent within the loudsoft - 1 ;! spectrum of effect.Some seemrelatively clearly demonstrated in identifiable musical experience; othersare of more conjectural functional significance. Une thinks of rising pitch, shrinking units acceleration!, increased textural quantity, increased compressionof texture, crescendo, harmonic succession ofincreasing tonal distance, increasein the overall spatialHeld, Huctuationtoward morepenetrating coloration,increased rate of attack, and dissonance, amongother musical states commonly associated with

increased intensity.

Such atentative and preliminary discussiontakes noaccount, ofcourse, of the question of rates ofchange, which may be gradual or sudden, or anything within these extremes, with profound consequences for expressive effect. And it has beennoted that confluent actionsoften differ in rates of changein angles of descentor ascent.The factor of rate of changein element-successions, ofthe degree ofincline or decline in the angleof ascentor descent, is a highly important and pervasive aspect ofrhythm in music, one of the most telling factors in the perception of motion and eventfulness. As a matter of fact, temporal

segments in music are identifiable, in

one sense, as

discrete stagesin lines of element-change, as rhythmic units of one kind, their progressive or recessive shortening or lengthening acceleration, deceleration! expressing intensification and release of further functional significance within the rhythmic parameter. Implicit in all of this is the thesis that contiguous sound eventsmanifesting changewithin any parameter and in any degree resultin functionalexpressive effectof, to that extent, intensity change. Put another way, the premise is that no change distinguishing contiguous sound events can be neutral with respect tointensity. Thus, pitch change, however slight, is suggestive ofmodification in the degreeof intensity, as areany and all changesin tonal reference, harmonic content, rhythmic activity, textural complexity 7It is by now clear that intensity is regarded asa product of qualitiesiorstates involving usuallymany concurrentlyoperative parameters;as suchintensity can only be defined inthe listing,conjectural ordemonstrable, of such qualities.Particular intensifying conditions ofmelodic, harmonic,tonal, textural, and rhythmic structures arerecurrcntly discussed as a fundamental,continuing concernin this book.

10 introduction

and quantity, metric structure, and coloration. Of course,in accord with the necessary conceptof hierarchic levels .of structure, it can be stated that change ata relatively low level is of inferior structural significance relatively neutral! in contexts ofa higher level-i.e., in broader implications. Thus, the pitch successiona-b-a is in a sense, atthe level of the total unit, unchanging; the progression to b, in the total context, is in a sense neutralsubsidiary to an underlying, fixed pitch event. But at the circumscribed level short of resolution a-b- ! the modestascent inpitch and minute tonal departure! has identifiable functional-expressive significance which can be felt as important in a leveled structure and experience. A theoretical formulation in which vital factors of musical meaningare attributed to controlled directions of growth and decline in lines of elementchange requires,as suggested above, hypothesisand observationwith respect to the connotations of various qualtities in the dialectical intensity-release scale of values.

Further such assumptions arestated, in a summary way, in the following outline; a number of them will be regarded ashaving a kind of commonsense logic e.g., complexity = intensity, or the forte-piano dialectic!,others have the support of traditional acceptance e.g., dissonance= intensity!, while others emerge asof more tentative significance.At the same time, the theory of functional-expressive significancein controlled lines of change within structural element-actions with shaped tendenciesto and from points of intensity at different levels hasintrinsic importance apart from any specific formulation of connotative valueswithin the intensity-release scale. While any premature statement in the direction of classification of intensity values within elements of music is inevitably too simple, it can be useful ifread assuggestive only,and of very restrictedscope ofconcern. These ideas are extensively developedin the three studies to follow. In Fig. 0-l specific manifestations of recessive tendency are not indicated but are of course oppositeto the kinds of events characterizedas progressive. Reasoned premisesconcerning the functional-expressive connotations of events and kinds of events! and concerning their interrelations and confluent parallel or resistant tendencies andinteractions are indispensable in analysis. And since inthe complexity of real music thelikelihood of comprehensive verification, or even susceptibility to verification, is uncertain, analytical judgment is commonly one of carefully reasoned, interpretive The question of ambiguityin particular element-qualities requires distinct, special consideration. Inambiguous conditions, by definition, the cognition of structure must be considered uncertain, and functional-expressive significance would depend onthat cognition. Where apparently ambiguous conditions are understood as ambiguous, in a context in which ambiguity is not a normal state, the effect is presumablyintensifying. Thisproblem is treated further,but not extensively, inChapter 3.

introduction Fig. 0-1. Somepremisesrespectingintensity valveswithin the spectrumof qualitiespertaining to eachof certainfundamentalelementsof musicalstructure. Element Melody,a line of contiguous pitches Harmony,the line of harmonic succession Tonality~the line of tonal reference Meter, the successionof accent-delineatedunits

Progressive action: Up; leap expecting closure, especially when dissonant; instability of tonal or other felt tendency Away from tonic; dissonant; inverted; complex forms; chromatic deviation from primary diatonic resource! Awayfrom primarysystem, in relation to tonal "distance" and assumingreferential adherence of primary I; chromatic succession and expansion Towardshorterunits; asymmetry and fluctuation; clarity of more frequent accent acceleration!; toward instability, departure from relational unit norm Acceleration in rate of occurrence at given level Greakr interlineardiversityand conflict; increased density; wider spatial field

Tempo,or rhythmic "pace" Texture,the line of changes in numbers and interactions of components Increasedsonorousweight and penetration strings Timbre,eventsinvolving did b .!;I d coloration, dynamic level, higher registerssharper "focus" of intense registral change, articulation color; more percussive,stressedarticulation

choice. Within any associationof partly counteractive confluent elementactions,the analytical judgment of prevalencei.e., what tendencydominates and is germane to broad structural efFectand intent, is of course extremely critical.

Stasis is nonaction, to whatever extent this can pertain, within any parameter at any given level. We have noted that stasis,being "unnatural" and expecting the intervention of change, is short of the point of tautology intensifying, and usually counteracted by eventfulnesswithin someopposing parameter.

Within tonal structure, for example, stasisdoes not necessarilydenote neutrality of intensity: indeed, prolongation e.g., of V! is common as intensifying technique. But intensifying stasisof tonal condition almost invariably involves dissonance,often gradually increasinglevels of dissonance,as a distinct factor significantly complicating an apparent condition of tonal immobility. Dissonanceis, by definition, active and mobile and must be regarded, in one sense,in its discrete significance. One seriousaspectof the oversimplicity of Fig. 0-1 is its incompleteness in the listing of parameters of potential action; at each stage in this book primary attention is given to the fullest possible statement and illustration

of

12 introduction

areas of potential functional-expressive action within each broad element classification. This is carried out with exhaustive attention to quoted examples, and to the presentation, development, and illustration of many techniques of analysis applied to these examples atall stagesof discussion. Lines of element-change controlledand directed along profiles of appreciably increasing or decreasingintensity are common to all music except that limited literature of very recent time in which concurrences and conHuences of sound eventsare unforeseen,and of purely orsignificantly arbitrary association.

Of course, the associationof tonal distance with intensity would have radically diminished relevance tothe extent that a style isatonal; dynamic progression and recession in loudness levelsare of unlikely significance in fifteenth-century polyphony; or, in twelve-tone music,the predetermination of pitch-class PC! content in simultaneous aggregates mayalthough it does not necessarily! result in diminished significance in functional-expressive contextual control of, say, dissonance-consonance values within that parameter and, correlatively, increased significance in other element-structures. Moreover, that the mannerin which elements are articulated i.e., the techniques ofelement-change! differsamong stylisticapproaches canbe noted, for example, in strikingly different applications of the rhythmic factor of tempo change in Baroque asopposed toRomantic styles,or of melodic and textural change invocal asopposed toinstrumental polyphony. These areonly a few examples ofparticular relevances of particularactions inparticular styles. Still, it is a soundbasic premisethat the shaping oflines ofelement-change isan all but universal factor in musical function and expression;and certain elements are applicable across virtually all historical-stylistic boundaries, notably those of texture and rhythm.

Techniques ofprogression andrecession inlines of change are,then, in different ways significant in the relation between structure and expressive function in some importantdegree inall music in which controlled contextual function is a compositional objective and aesthetic desideratum.With this assumption, analytical effort must be directed to those elements which are functional in the particular style or medium, or particular work in question. The concept of progressiveand recessiveactions within confluent element-structures suggeststhe useful basic principle that, in an important °It is clear, too,that the particular mediumof expression significantly conditionsthe prevalence ofcertain elements over others,or to the exclusionof others.Thus, forexample, in solomedia thetimbral spectrumis limited to registral,dynamic, andarticulative differences, inaddition tosuch devices as mutingor otherpreparations ofthe particular medium -a fundamental distinction ascompared with the broad spectra ofmore heterogeneous media..

introduction 13

sense, there are dissonancesana' resolutionswrithin allqfmusics parameters. The clear relevanceof this idea is apparent with; even a little thought: thus, for example, anupward leap in the pitch-line, a highly active perhaps imitative! texture of competing lines,relative ambiguity or in relation to a referential norm! asymmetry of metric relations-any of these can,like other, comparable conditions of instability and expectancy of restoration of simpler states, beconceived asdissonant, as evocatiive ofintense feelingand subjective involvement in potently absorbing actiomsand interactions taking place within the projected medium of sound events. These actions and interac-

tions, pertinentto the elements of tonality, lharmony,melody, texture,and rhythm, asto furtherparameters within each ofthese, area principalsubstance offunction and expression inmusic. Levels ofpitch structure, as of other erlement-structures,can perhaps usefully be conceived asanalogically of relative distance: the most fundamental as background in an imagined three-dimensional field, the most immediate as foreground, in focus as one regards the structure at close range. The metaphoric conceit of relative focus is helpful in engendering the image of structural depth in this sense, and in conceiving of various middlegrounds coming into increasingly sharp exposureas detailsblur, and as increasingly comprehensiveevents are the objects of attention. The idea of referring to identifying! such levelsas to spans fy'context is at times also useful:half of a bipartite form thus can represent that level at which its own inherent, delimiting funda.mental elementsare in focused exposure-that broadly middle level which is nearer than superior elementfunctions extending into contexts beyond that of rwrence, in this case one of two major divisions. More foreground levels are thus identifiable as to more limited temporal units, or spans. Inthis sense,the level of the phrase would mean that level at which those events of primary structural basis in the phrase arein exposure-that level whoseprimary events go out of focus in favor of elements of broader significance as one regards broader contexts more basic levels! through events of the relative foreground. Or, the level of the phrase is the level of those eventswhich delimit the phrase as toits most basic content while more fundamentallevels-extending ultimately to that of the whole-are comprised of eventswhose implicationsreach beyondthat oi, in this case,the phrase,or other, limited referential context. In the succession oftwo phraseshaving harmonic content I, IV, V - I, IV, V, I, the phraselevel action structure at the level of the phrase! isI, V - I, while the action at the broader, two-phrase levelis I-I, the comprehensivetonic prolongation. It is my belief that levels ofdepth, providing the basisfor important analytical constructs, haveexperiential validity within all elements ofmusical structure. The idea of leve1 also arisesin other connections: for example, asto the abstracted, leveled hierarchy by which a tonal system, particular or

14 introduction

generalized, can be represented, or the multilateral leveling of pitch events asnoted in Chapter l. Moreover, a conceptual basisfor identifying structural levels in meter as to temporal spans one of which is taken as the referential metric unit! is discussed in Chapter 3; in that respect, metric structure at the level of the phrase is that whosemetric unit is the phrase. The concept of hierarchic levels ofstructure is of great importance. To put it in the light of the above formulations: every individual element-event has immediate local, foreground! implications; those that are more fundamental have broader implications as well. Hence, the concept of hierarchic levels in all element-structuresarises naturallyand inevitably, and it must be seen asapplicable to the functioning of eventswithin any given confluence of action.

The theory of hierarchic levels is not only of very critical importance in the study and analysis offunctional-expressive effects of element-formulations, but it also has important implications for theoretical treatment of individual structural elements, especiallyin matters of classification and terminology." Thus, for example, the difficulty of modulation as it concerns theextent and perceptual significanceof tonal reference tonicization! is greatly attenuated when it becomespossible tospeak oftonal shift having referential significanceat agiven level and not at others. The problem of tonal shift remainsone ofcomplexity and, at times,ambiguity; but in at least many instances it proves to be of significant clarification to seethe tonicization of secondary pitch factors asrelevant to particular levelse.g., thoseof the phrase and smaller units! but not to higher levels. A like approach to the discussionof fluctuation as manifestat given, identifiable levels

of structure can be found useful

in the

theoretical treat-

ment of other structural parameters too-for example, meter, coloration, and texture. Consider, for example, broad potential implications within the symmetry-asymmetry polarityin metric structure, asopposed topurely local effects of subordinate fluctuation within this particular parameter; and a comparable polarity within the realm of texture is that of the homorhythm-

1°Thc conceptof 1eveled structure derivespreeminently fromthe analytical approaches ofSchenker topitch structuresin tonal music, butfrom thework ofother theorists too-for example, that of Hindemith in his conceptof step progression or Schoenberg in his importantconcept ofmonotonality. Schenkerian approacheshave nostrict or direct representation in this book, butsee Felix Salzer,Structural Hearing: Tonal Coherence in Music New York: C. Boni, 1952! orSalzer and Carl Schachter, Counterpoint in Composition New York: McGraw-Hill, 1969!,in addition to many other sourcesin an expanding literature,some ofthem cited at appropriatelater stages in this booksee page 113!. Hindemiths concept of step progression is setforth inThe Crajt J Musical Composition New York:Associated Music Publishers, Inc., 1937!, Book I; Schoenbergs theory ofmonotonality ispresented inStructural Functions ij HarmonyLondon: Williamsand Norgate, Ltd., 1954!. Other sources of comparable interest are noted in appropriate, later contexts.

introduction 15

polyrhythm dialectic, with its varying levels of significance.! In analyses which make up much of the substance ofthis book, the issueof hierarchic structure is recurrent and of fundamental importance. We have noted the importance of hierarchy in such qualitative distinctions asthat which observes stasis at one level e.g., the level delineated by a pedal point! and activity at a lower level e.g., that of fluctuation over a fixed or elaborated, prolonged pitch event!. Similarly, application of the concept of hierarchic level can bring about decisive and critical refinement of a theoretical idea like dissonance: in comparison, for example, of dissonance function of purely local importance e.g., the foreground melodic event in

conflict with surrounding pitchfactors! andthat of broader importance e.g., the expansive dissonance prolongation that often preparesa stablearea of thematic statement!. And the underlying concept of hierarchic levels serves to inform and establish certain functional distinctions

between elements

themselves: note,for example, the clear significance ofthe conceptof hierarchic level in the frequently differing implications of harmonic as opposed to tonal events,the latter a hierarchically superior extension or amplification of the former.

It thus follows that recessive andprogressive linesof element-change must finally be evaluatedas to level offunctional signyicance ; within this necessary approach,broadly viewed actions ofone classificationwill be seen commonly to contain lower-level actions ofan opposing or corresponding,but subordinate! classification. Let the reader contemplate briefly some of the element-structures in which the extension ofimplications to the broadeststructural level can most easily be grasped-conceptually and often perceptually. Thus, in any multimovement work the succession of primary tempi constitutesa broad line of changewhich is the highest manifestation ofthe tempo-structure.Or, the broadest levelof metric structure in any work of suchmajor formal divisions i.e., movementsor comparable units! is evident in the proportional differences andinterrelations manifestamong suchdivisions. And the coloration of a multimovement work constitutes,at broadestlevel, aline of color changein differences involving areas morethan detailsof spectrum! of critical functional-expressive effect.More diflicult is the fact that the broadest lines of tonal successioncan be seen in many instancesto correspond to to be an augmentation of! those ofrelatively local tonal or harmonic and melodic! successions.

It seemsto me that there is no reason todoubt the perceptual, experiential as well as conceptual, significance of these broadest configurations in

element-structures, especially those functioning within element-structures of most immediate and uncomplicated apprehension-within the soft-loud, dark-brilliant, slow-fast, long-short, and comparably apprehensiblelines of change. In carrying out expositions and analytical illustrations of major

16 introduction

categories of element-structure and element-content I have at times been compelled to stop short of discussionand application at broader in a sense simpler! levels, those for example of intermovement relations. But it is my assumption thatthese conceptsare applicable and apprehensiblethroughout the structuralhierarchy, especiallywhere theyconcern musicsmost directand palpable areasof action and interaction. In any concern with the functional consequences of contextual associations and interrelations, there

arises the issue of randomness, whether overt or

the result of the blind predetermination of musical events. Kfenek embraces unpredictability as a premised value underlying total serialism; in a somewhat ambivalentstatement aboutinspiration Einfall! as spontaneous and uncontrolled therefore of random consequences!, Kfenek correctly notes that it is in fact conditioned by prior understanding and experience.In that light, he saysof the composer of totally serialized music: In order to avoid the dictations of such ghosts i. e., recollection, tradition, training, and experience!, he prefers to set up an impersonal mechanism which will furnish, according to premeditated patterns, unpredictable situations.

But the surprise which is thus assured isone in which no contextual norm of probability is demonstrable or relevant; hence, an environment is established inwhich all events, and no events, are surprising.The affective value of surprise,important to any composer,must derive from a context in which a particular event is not what the percipient is conditioned to expect.! Roger Sessionsasks oftotal serialism: . . . what is being organized, and according to what criterion? Is it not rather a matter of organizing, not music itself, but various facets of music, each independentbr and on its own terms or at bestaccording to a set of arbitrarily conceived and ultimately quiteirrelevant rulesof associationIP

The affective value which inheres inisolated from contextual relations! sonorous events should notbe overlookedor discountedin any comprehensive view of musical expressionand experience.Whether isolated-sonorousqualities canproperly be said to convey meaning," there seemsno doubt that "Extents and Limits ofSerial Techniques,in TheMusical Quarterly, XLVI, 2 960!, p. 228. 12Problems and Issues Facing the Composer Today, in The Musical Quarterly, XLVI, 2 960!, p. 169.Italics added. "This dependson whethermeaning is considered torequire syntacticrelations, or whether a kind of meaning inheres in any singleevent or element-complex eliciting affective orother! response. Meaning in the experienceof art is commonlyconsidered to depend oncontrolled contextual interactions ofelements: once two eventsoccur intemporal

introduction 17

phenomenal-perceptual attestation and experience speak persuasively of the affective force implicit in such eventsquite apart from contextual interactions of the kind which are the prevalent concern ofthis book. The primitive impact of the naked musical event-its intensity, its timbral quality, its general pitch locus, itsdensity if it is a tone complex!, and the like, divorced from syntax, must be regarded as an aspect of musical effect which is far from adequately understood, yet of a level and substance surelyinadequate to full experience ofa temporal art form in cognitive, intellectual dimensions. Implicit throughout these studies,and from time to time made explicit in them, is the belief that controlled contextual function of element-events isof

fundamen/tal value and, even,of necessity in musicalcommunication where the respondent is engaged in apprehensions of functionalordered relations as aspects of experience. The view that purely superficial sound qualities in individual isolation-or in adventitious association-have aprimitive communicative potentialsuggests some kind of noncontextual significance,even though an important aspect ofany meaning in the isolated event say, a big tutti attack in the orchestra! is normally the expectation it arouseswith respect to likely subsequent events. At the same time, however, the most important aspect of musical experiencederives from the interaction.: and interrelation; of contiguous and concurrent events of diifering qualities within contextual procedures determined by controlled lines ofprogressive andrecessive successions. Such a concept of meaning is not applicable where event-confluencesare random and arbitrary. A subtle factor in this area of critical judgment is the problem of element-change ofa rateand degree so extremeas to, in effect, nullify the effect and perceptionof controlled change, evenof changeitself A useful analogy here isin the rapid spinning of a color wheel,which becomeswhite beyond a certain speedof rotation.! Where change e.g., of register! is fast, constant, and extreme, many percipients testify to the dulling ¢ct of uneventfulness which is the ironic consequence ofsuch action. This point is quite independent of whether element-successions in a situation of

this kind

are, in

com-

positional intent, random or arbitrary, but it is a relatedquestion becausethe phenomenon ofuneventfu1 change is in fact characteristicof many products of total serialism.

This book includes within its range of musical illustrations several instances ofrandom and highly serialized i.e., serialized beyond the single contiguity, thepercipient assumes a contextualsyntax to be thebasis fortheir association, and fortheir further role in a largercontext. Toldthat suchfunctional contextualinterrelations arebeside thepoint, heis left to the limited experienceof individual events whose association hasno appreciablerationale-events cooccurringin an apparently contradictory situation in which they have noappreciable functionalinteraction or interrelation but are nevertheless run together.

18 introduction

element ofpitch! operations in order to provide some expositionof specimen techniques andin order to provide the basisfor. analyticalcomment. But it is important to concede at the outset 'the bias by which successions ofevents having no controlled or even foreseencontextual associations,relations, and interactions are regarded as antithetic to systems ofmeaning upon which the musical experience, in virtually the entire history of the art, critically depends. Theissue mightbe characterizedas oneof starting-going-stopping in a given musical structure as opposed to beginning-prog'ressing-recedingending.! When a soundevent succeeds another purelyfortuitously, contextual interrelation is precluded unless it is accidental, or unless preselectionis within a narrow range of planned consequences!; andchange which leaps far acrossa spectrumof possibilitiesin rapid succession oftentends to nullify the effect and the effectiveness! ofchange. It is an interesting fact, too, that music in which important decisions are left to the performer is guided, and often contextually directed, by whatever creative gifts andintuitive experience the performer may bringto hisparticipation. But music in which major elements are prescribed in predetermined serial operations i.e., in which few if any significant elements are left for determination on the basisof the needs ofcontext! is the most random of all, in that contextual relations arelittle if at all foreseen, muchless abasis by which events are determined. The highly serialized pieceis thus in a critical sense morerandom than the overtly random piece in which fundamental decisions areleft to the performer. Of courseit is recognized thatthe intentionsand creativepurposes ofthe total serialist may havenothing to do with determined-in-context associations of events,nor can he be censured forfailure in an aestheticdesideratum and objective hedoes notembrace. On the other hand, it must bestated that two common protestations-that performer-directed chance music issimply an extension andfurther manifestation of historical improvisatory procedures, and that extreme serialismis merely another systemicbasis forcontrols analogous tothose ofmodality or tonality-are decisively rejected. In no traditional system are contextual associations adventitious or predetermined in any critical degree: in traditional improvisation, structure is rigidly planned and specified in crucial, prevalent parameters; and in historical systems of modality and tonality creative choices asto contextualaffiliations ofrhythmic, textural, and relatively foreground pitch events and event-complexesare of broad and decisive latitude,with functional consequences incontext, rather than predetermining systemic influences,the considerablebasis for creative choice and the determinants of content in the particular piece. Tonality as a systemthus predeterminesthe content of the particular musical instancein only the most general terms within which expectations are aroused and fulfilled!, and only with respect to certain structural elements; and contextual associations

in tonal music are determined as to the

introduction 1.9

functional-expressive needs of specificcontiguous andconcurrent conditions, not predetermined. Even while an underlying assumption ofthe value of contextual determination is conceded, thepresentation ofspecific worksand extractsin which this value is called into question will inform the reader of disparate compositional practices,while calling attention to works and ideas that may by their own persuasion leadhim to conclusions verydifferent from those stated in the foregoing commentary,with respect to an area ofconcern fraughtwith heavily contentious

and controversial

issues.

It is/irnportant to emphasize that theserialization ofsome elements in no way precludes the contextual determination of others. Moreover, any predetermination in classical twelve-tone procedures is commonly of PC pitch-class! rather than of specific pitch, so that, for example, melody is preconditioned in only one of its aspects. The serialization of PC content in classical procedures in which rhythms and other element-actions areleft for contextual determination has produced a vital literature, even at times with tonal bases, as examples inthis bookdemonstrate. Importanttwelve-tone works are cited and discussed in illustration of this books fundamental premises andin documentation of the values it espouses.' To the extent that the composer does not determine or significantly condition or even foresee! the concurrencesand contiguities of events in planned contextualrelations, analysiscan only describe proceduresand show their applications. It can, in theory, attempt to describe expressive-functional consequences of random procedures,but any effort to do so, in critical comment basedupon premisesof value likely to be contradictory to thoseunderlying suchworks, isirrelevant to the extent that the contextual confluencesof events are arbitrary and

uncontrolled.

The problem of the apprehension ofsignificances ofevents whoseassociation in particular contextsis purely or essentiallyadventitious issometimes discussed as aproblem of overload in the channel of communication ofinformation, with reference tothe concepts of information theory:. . . the absence of a stable stylistic syntax, archetypal schema, audible compositional order, andpatent natural patterning resultsin a level ofredundancy so low that communicationis virtually precluded. Leonard Meyer, Music,The Arts,and Ideas; © 1967 byThe University of ChicagoPress; p.290; all rights reserved.Emphases added.! See tooEdward Conesdiscussion ofthe problemof what he refersto as synoptic comprehension inthe musicalexperience Musical Form and Musical Peyizrmance; New York: W. W. Norton andCompany, Inc.,1968; pp.88-98!. Atone pointin thisexcellent essay pp. 95-96!, Conemakes thestatement that. _ _ non-determinedmusic, whetherthe sequence of eventsis left up to the performeror to pure chance,may imply a continuumthat often seems tocombine thepurely musicalwith the quasi-dramatic. Thetwo determinismsof formula andfortune leadto the same result:the arbitrary way in which suchproducts begin and endand thefortuitous nature of theirinner connections ensure thatthey canat bestbe experienced onlyas surfaces. Emphasis added.!

20 introduction

This books discussions ofstructure and analysis in harmonic and melodic successionsare less extensive than one might wish, but there are many sources,reflecting a variety of approaches andpoints of view, to which the reader can refer in these connections, and it has seemed important to concentrate heavilyon problemsof structurenot widely treated in the existing theoretical literature, especially thoseof texture and rhythm. Moreover, the treatment of quasi-serial cellular associations ofpitch and PC complexes islimited, and the treatment of this aspect of structure centers primarily in complexes having tonal or quasi-tonal significance; again, the reader is referred to supplementary resources. And it has provedimpossible toextend this book to the point of including a discrete study of structural functions of coloration of timbral differences, of orchestration, etc.!, although there are many parenthetical references to these factors in connection

with central

issues of concern and

treatment, especiallythat of texture. In view of this, it seems wellto make a point of the fact that one vital aspectof coloration, that expressedin dynamic levels, often goes very far indeed in accounting for the nature of musical experience andmeaning. If its attention in these chapters is not equal to its importance, it is largely because itsfunction is direct and obvious, and because ofits comparative accessibility offunction and significance. In the organization of this book, it is true that the treatment of discrete elements in individual chapters imposessomewhat arbitrary lines of distinction. But overlapping concernsare explicit, for example, in the books concept of rhythm as including the rhythms of all element-changes e.g., tonal rhythm!, or in the obvious relation of texture and harmony, and in many other instances and connections. With the understanding that this book is about music, none of whose structural

elements often assumes inde-

pendent manifestation,it will be acknowledgedthat there is a methodological advantage and common didactic purpose in individuating at provisional stages identifiableand classifiableelement-actions forsingular emphasisand for explication of conceptual and practical ideas relating to each such element-structure in

turn.

On the other hand, a very important manifestation of the conceptual unity and interdependence ofthis entire range of studies canbe noted in the conscious effortto treatin anabfsis certain recurrent examples from chapter to chapter, and to refer asoften aspossible toassociated events involving cofunctioning elements otherthan thoseof immediate concern. To some examplesan effort is made to apply comprehensive, althoughsummary, treatment at various stages; each chapter cites examples of substantial scope treated in full at least asto operationsof the particular element under primary consideration. The most important ofthese havingbroad applicabilityis GeorgePerle, Serial Composition and Atonality: AnIntroduction to the Musicof Schoenberg, Berg, andWebern, 3rded. Berkeley andLos Angeles:University of California Press,1972!.

introduction 21

The procedure by which a basic, highly diverse, set of musical references wasestablished foranalytical at times recurrent! treatment had been an essentialpremise fromthe earlieststages ofthis study. In fact, the selection of works to which to devote attention was the first problem undertaken. Implicit in that initial undertaking was the challenging assumption that a thesis ofbroad importance in the structure of music had to be, and must be shown to be, applicable to an ample samplingof works representing a great breadth of chronology, mediumlsystem, and stylistic disparity. Only occasionally has it been necessary to omit the quotation of music to which significant referenceis made; usually this is becauseof necessary reference to lengthy excerpts whose quotation has proved impractical, or because ofrestrictions of copyright. On the other hand, referenceis often to music readily available in widely circulated, published anthologies. While there are within the range of twentieth-century practice some illustrations of new techniques, this book would be highly untrue to itself if it conveyed animpression of special advocacyof contemporary or recent music in which ncvel means areevident at the expenseof twentieth-century music oi, in the particular element of concern, more traditional persuasion. It is somehow aparticular mark of the present time that musicians tend to devote disproportionateattention to works that constitute manifestoes,or are in some sense apparentlytrailblazing, when in fact there is much of great interest, force, and resourcefulness in music of traditional confidence within conventional means-music that is

bents and assured unfortunately less

studied in view of its relative accessibility. Again, one can only plead the limitations of space. Whatis ofinterest isthe discoveryof relations pertaining between structureand expressiveeffect in all music that is strong andinteresting; indeed, the restlessassertion of novelty often finds its outlet in mere notational appearances.

Notational problems are a primary reason for the lack of attention to electronic music-not only the problem of the accessibilityof scores,but the absence ofany generally intelligible mode of graphic representation of specific elements of structure. But I wish to state at the same time my conviction

that the principles of contextual shaping to which this book is devoted are fully applicable, within a context of vastly expanded resources notyet well understood in perceptual and cognitive implications, to electronic music. Analyses ofmusical extractsand piecestake manyforms in these studies, and that breadth of technique of application is altogether intentional. I "These includeBurkhart, Anthology _hr MusicalAnabsis, 2nded. New York: Holt, Rinehart andWinston, Inc., 1972!; Davisonand Apel,Historical Anthology of Music,2 Vols. Cambridge: Harvard University Press,1949!; Starr and Devine, Music Scores Omnibus Englewood CliH`s, N._].: Prentice-Hall,Inc., 1964!;Parrish, ATreasug ofEarbv MusicNew York: W. W. Norton and Co., 1958!; andBerry and Chudacolf, Eighteenth-Centugw Imitative Counterpoint Music : farAnabrsis Englewood Cliffs, N._].: Prentice-Hall, Inc., 1969!.

22 introduction

strongly rejectthe conceptof any fixed orthodoxy of analytical procedure and representation, believingthat the approach dependson the nature and basis of inquiry, and may in specific areas take,equally, any of a number of forms. Thus, someof the analyses inthis book take form in descriptive statements of an expositorynature; others are graphic or symbologicalwithin a substantial range of possibilities-some innovative, arising necessarily out of newly formulated concepts.Throughout this work I have soughtto fulfill the analysts obligation to make clear all lexical and symbological basesfor graphic, expository, and other methods. Analytical sketches, of which there are many and many kinds, go beyond the text in analytical comment and statementand must be reviewed in necessarysupplement tothe text. The question of how to prepare synoptic sketches ofpitch structures is oftentreated informally, again within the intent

of latitudeand fiexibilityof specificinode ofexecution; there is, however, an abundance of illustrations, and suggestions ofsymbols and procedures are given. In general, it should be kept in mind that sketches ofpitch or PC! content representfor eachexample, sometimes within a limited, specified context and restricted realm of inquiry, an interpretation of structure and functional processand affiliation. This applies not just to pitch structures but to all the elements examined;in Chapter 3, too, the hierarchy of metric functions is represented with careful explication of devicesemployed. In any study of this kind, the problem of terminology is a difiicult and considerable one.It has been my hope to counter this problem at least in significant degreeby the best possibleconsistency andclarity of terminological selection and definition.

The index lists all crucial and problematic termstogether with numbers of pageson which their definitions can befound. In many instancessignification is probably self-evident, given a few stated, underlying terminological assumptions, butI have tried to give the reader thebenefit ofany conceivable doubt

Brief expositionof certain most essentialand recurrent terms andusages might be given herein passing,but it should be read astentative and preliminary. I use the term elementwith reference to any of that set of primary structural parameters within which events oflike significanceand character take place in music; thus, the element of tonaligy isprojected in tonicizations; that of harmony in chords orvertical simultaneities! ; that of meloafy in contiguous pitches orattacks occurring within an identifiably continuous linear stream, voice, or stratum; that of colorin timbresor sonorous qualities" within a given "These are determined by physical characteristicsof the sometimes alteredor manipulated! soundingmedium, by registral locus,by articulative mode ofproduction of sound, andby degreeof loudness.

introduction 23

vocal or instrumental spectrum; these arefor the greater part conventional usages.

The element of texturemight be said to consist in events bywhich the interrelations of lines or other cqfunctioning components areconditioned, but the

textural element is alsoregarded as inclfuding such factors as density and space. The most complex elementof all, that of rhythm, isin a sense theproduct of actions of all other elements; constituent parameters of tempo andmeter are seen asof great importance. Naturally, all of these termsand conceptshave full presentation in the chapters which follow. The terminological premises notedbriefly above,and givenfull explication at later stagesin this book, yield certain derivative terms which I have adopted ordevised inaccord with underlying theoreticalconcepts. Ahyphenated term like element-structure refers to the shape delineated by changing qualities and intensity levels assuming thereis change! within a given element, or -to theflat shape of unchangingevents within that element.Terms like tonal structure aresimply specihcations of the generic form, elementstructure; but comparable specificationsare also used to denote structures arising within constituent parameters. Thus, tempo structure, and related terms, are employed from time to time. Similarly, terms like element-action, element-event, element-change yield the more specificforms of which metricevent and harmonic action mightbe cited as instances.[The hyphen occurs in generic forms, and in terms, especially noun-noun forms, which I have devised e.g., pulse-tempo!, and where the associationof component terms might not be self-evident.] Equally fundamental are the terms progression, recession, succession, and to a lesser extentstasis; in fact, these terms representin the simplest consolidation a primary conceptual basis for this book, as we have seen.Forms like element-succession, harmonic progression, tonal recession, and the like, are in constant use.

I use the term function to refer to the processive,structural role of an event or succession succession denoting a generic conceptsubsuming progression and recession!. Ifthe concept of structural function is in terminological reference commonlyallied to that of expression, is it becauseI cannot regard them as distinguishable in the object-subject relation of actual musical experience: functionis the role, or nature of participation, of an event in the import of expressive contentand significance; element-function is the basis for source of! expression; the functional-expressivemeaning and substance of the event are its perceived, cognized characteristics or those presumably intended! and relations to affiliated eventsin a given work at a given level. It is my view that every musical eventin contextsto which these theories are applicable i.e., all except those in which contextual contiguities and simultaneities are fortuitous and arbitrary! has discernible, rationally interpretable, functional and expressivesignificance. The expressive content

24 introduction

or potential of any musical event rests very decisively on its functional role within a contextual process;an events functional significance in the structure! is indissolubly allied to, and the primary basis for,its expressiveimpact in the experience!. It seems to me that the only sensein which these are experientially separable concepts assuming consideration not merely of objects, but of experience! isthat in which the isolated musicalevent whether it is in isolation, or is construed in isolation in the absence ofapprehension ofcontextual relations! might be said to have purely expressiveas opposedto _functionalexpressive import." Since this book is concerned with the analysis of contextual relations in shapedcombinations of events, it employs commonlythe hyphenated formfunctional-expressiveaddition in to didactic useof the individual terminological components. When I wish to refer to the eventwithout necessaryregard to its expressive contentor functional role I often usethe term prejection, denotingsimply that which issues fromsource to percipient in the musical experience-the stimulus itself, whatever its functional-expressive contentand significance, and apart from such significance. Finally, with respect tothe confluent successions of concurrent elementactions, I employ fundamentally the terms complementary and compensatoy sometimes such associated terms as parallel and counteractive!to denote relations betweenelement-actions whichhave, respectively,allied or resistant functional tendency at somelevel. Within this concept, the term neutral isat times necessary,having to do with stasis or nonparticipation-nonfunction within agiven element-structure, with respect to agiven directional tendency, a atgiven level. In description of situations characterizedby the confluence ofseveral element-successions, multiple terms suchas tonal-harmonic succession texturalor metric intensificationare useful, even unavoidable.

The above summary of some basicterminological propositionsis, to be sure, not intended as a glossary; for explicit discussions ofthese andmany other terms the reader is referred to the text of the book itself and to the index.

The foregoing is intended simply to call attention to lexical items of most general importance. While discussingparticular structural elements andconsequences ina very broad selection ofworks, this book points out and illustrates many approaches to analysis. It should pose no unusual difficulty for the reader equipped with understanding of fundamental conceptsof music theory, and it is my hope that thesestudies succeed in laying out in theory and in application somevital ideas and methodsin supplement to the work of many others Compare Cones immediate apprehension as opposedto synoptic comprehension in the sourcecited earlier footnote 14!.

introduction 25

along distinct but often complementary linesof inquiry. Important, relevant theoretical sources are cited from time to time.

fl*

By no means do I regard extramusical systemsand communicative structures, applied analogously or adopted assources oftheoretical models, as lacking important usefulness in the study of music." But the present studies byand large follow from direct encounterswith musical works themselves, fromwhich observationsare drawn with respectto functional elements and expressiveeffect, without recourse to parallels in extramusical systems and without employment of systemic analogsand models of other kinds. Understanding is sought in direct discoursewith the object itself-the score page andthe sound images ofwhich it is the symbol, and in exploration of the experienceof what psychologists callthe stimulus object, the musical work transpiring in time. Of course much work remains to be carried further; the problem of logical considerationof pathsfrom analytical insight to decisions ofperformance isonly one area of need that comes quickly to mind. I have also implied that, in -my view,thorough studyof subjectresponse to the single musicalevent in isolation in the widest possible rangeof combinations of parameters! must ultimately play its part in the understanding of the musical experience; for, if the syntactic operations and relations of confluent eventsare expressiveof meaning at the most sophisticatedlevels of apprehension and comprehension, the naked qualities which inhere in the single event unconfused bycontextual disorder! presumably haveevocative powers in themselves insome pure and primitive sense. Adancer will respond with related movement triggered by the affective content of a single sound.! The qualitative impact of the individual event is intrinsic rather than syntactic-with this I mean notjust the event out of context, but the event forwhich no plausible functional relation to surrounding events can befelt or deduced, orthose aspects of the event, contributing to its import and impact, heard and felt apart from contextual associations. Thus, the qualities of single events e.g., articulation, density, loudness!often have in themselves unusualaffective importanceat low levels ofstructure. I have the feeling that purely surface qualities-those divorced from or regarded and felt apart from! contextual implications-are underestimated in the study of musical experience,probably becauseof the apparently unintellectual nature absence of the intuitive or deliberate, cognitive act of relating! of the appreciation of affective valuesin isolated events ofpure, local surface. Of further areas of potential exploration, one feelsthat, for example, subjective association of musicalevents withextramusical experience-association triggered by analogical parallels of various kinds-may have greater "Perhaps themost importantcurrent sources of suchanalogs areset theory,probability and information theory,Gestalt psychology, and linguistics.

cH/wen onus

ton olity /n troductory notes

The concept of the hierarchic ordering of pitch content has in one mani-

festation oranother served as abasis formusical structure since theearliest

stages the in Western traditiorg is It theintent of this chapter to introduce some ideas about tonal order in music and to illustrate

and discuss certain

approaches inthe analysisof tonality and tonal successions according to the concepts presented. The practice of hierarchic systems oftonal order, by which the concept of tonality is engendered,is of truly monumental significance. Mans cultural history is marked by few achievementsof comparable magnitude; indeed, it can be argued persuasivelythat nearly all music of the Western tradition is structurally conditioned by some kind of expression oftonality. The discussion that follows restsupon the broad definition of tonality necessary tosuch an argument.

Tonality may be thus broadly conceivedas aformal system in whichpitch content is perceived asfunctionalty related toa specijic pitch-class or pitch-class-complex ey resolution] often preestablishedand preconditioned, as abasis for structure at some understood level of perception. The foregoing definition of tonality is applicable not just to the tonal period in which the most familiar conventions of tonal function are practiced roughly the eighteenth and the nineteenth centuries!, but through earlier modality and more recent freer tonal applications as well. The tonalsystem consists of a hierarchic ordering of PC factors, with the tonic final, axis, center, etc.! the ultimate point of relationship which tonal successions arecontrived to expcct. In the tonal period of conventional common practice the primary system consistsof hierarchically oriented degrees of the diatonic scale and the tertian harmonies erected on these The terms pitch-class and pitch-class-complex symbolized PC and PCC! are used to denote pitchindependent ofspecific registraloccurrence, ora complexof suchpitches generically understood. 27

28 tona/ity

degrees. Inmodal styles,corresponding systems are the modes, with normative points of tentative and fi-nal cadential repose. In more recent stylesin which tonality is relevant a system may but need not! consist of specific scalar formulations PC collections! of these or other kinds, with derivative

melodic andharmonic configurations disposed insuch away asto express and give primacy to a particular tonic or, tonics. Often

in fluctuant contexts, particular

such tonal content is reminiscent of conventions of the tonal

period, and its validity in experience may in significant part be based on powerful conditioning in the all but universal practice of systemsof tonal order in Western music.

There are of coursemusical idioms, especially ofthe current period, in which tonalityl as aformalized systemis lessrelevant-or irrelevant, and the less relevanttonality in- a particular style the more musical structure must depend on other factors, even extramusicalfactors, exceptwhere considerations of structure are put aside altogether, asin somealeatoric music. More-

oviiftyles in which tonalfluctuation isconstant and extreme inlatitude may find the structural function of tonality impaired, resulting in a changed experiential significancefor the sense oftonic or vacillating tonic. Although no amount of argument cansettle thequestion ofrelevance of tonality in music describedas atonal, it is important that the question be raised in the light of specific musical instances. The many examples of

twentieth-century sources in the following series of analyticalpresentations, and the discussion oftonal referencesor allusions affirmed to be implicit in them, will amplify the question and suggest possibleinterpretations. These analyses thusinclude referenceto music later than as well as earlier than! that in which conventions of tonal-harmonic order prevail. Questions of the establishment of tonal reference by harmonic-melodic

succession, andof the nature of tonal fluctuation, like that of conjectured tonal allusion in antitonal styles, are necessarily somewhatsubjective, and the concernin the analyses whichfollow will be to cite methodsarising from theoretical propositions stated, and to present plausible conclusionsarising from such methods and hypotheses, rather than to achieve any kind of inflexible truth about the examples andideas explored.Effort is directed here to an understandingof the techniques bywhich tonality is expressedand to formulation of interpretations of tonal implications which are demonstra2The termsconvention and conventional will refer specificallyto triadic and othertertian harmonies, and melodic successions having these as theirframing bases, as wellas predictable successions of harmonies bywhich othersare seenas exceptional or deceptive in the period of major-minor tonality.Conventions of tonal expression are thusparticular means of aflirming the tonal hierarchy in practice asopposed tothe broad principle of/zierarchic order in thesense inwhich it can beseen'to begenerally applicable.

tona/ity y

29

ble in the circumstances discoverable in the analysis of a given work. In discussion of tonicizing factors those which support the emergence or existence of a particular tonal center, or central PCC! primary interest resides in techniques by which the basic tonalorder iscfirmed andestablished.

/ntroductory commen ts concern/'ng Iona/ and //'near functions Melodic and

harmonic functions

are of two kinds: the first of these has to do

with position, identity, and hierarchic statusin eachof the system components of the particular tonality in question tonal function! ; the second isthe role of the event in the melodic-harmonic linear stream linearfunction!, also hierarchically defined with respect to a given level of reference.Linear function is the relation of an event to the structural relatively essential! linear frame or basis, orits auxiliary subsidiary, elaborative,embellishing, prolonging! relation to an event ofhigher order. We shall suggest thatthese functions are multidimensional-tonal function as to relatively deep or superficial levels of tonality, and linear function in a comparable sensebut also in the fact that a given event is associated oftenwith more than one other event in different ways, even at the samestructural level of reference. Of linear functions, the neighbor auxiliary embellishes astructural point in harmonic or melodic succession and is oftenderived in voice-leading ofone or more stepwise adjacencies, common-tone associations, or other proximate relations". Often a neighbor separates two appearances of the structural harmony it elaborates. A second typeis the passing auxiliaiyoften a succession of passing events!in which movement, up or down, links two harmonic or melodic points in an action filling space beyond that which is normative in the neighbor association. In a passing configuration, implicit in one or more voiceswill be relatively conj unct descentor ascentin a passing stream normally filling intervals of a 3rd or greater. Distinction between neighbor

e.g., /92 or 92/ !and passing e.g., ,___---"' ! configurations involving linear auxiliary functions is, then, a matter of voice-leading; where harmonic"as opposedto monophonic textures are concerned, theconsideration of voice-leading gives primary attention to outervoices, and especialbf to the bass. An event may be auxiliary at a broad level, essentialat a more immediate level; or a neighbor auxiliary at a very local level may be seenas part of a broadly passingstream at a higher level. For example, aV cadencewith

Q second scale degree! in theupper voice is theessential structural aim of succession atthe level of the phrase it concludes; at a broader level the ca-

30 tona/ity

dence isperceived as auxiliary toits I and l! consequent. Thus emerges the concept of multileveled linear function paralleling that of multileveled tonal function.

Example l-l points up the issueof linearfunction asrelated andcompared to tonalfunction, andit argues that the two kinds of function do not reflect, as is sometimessupposed, mutually contradictory or exclusive concepts.The analysis isbased on the premise that except in relatively rare tonally non-

functional elaborations, tonal function is of significance and in evidence and an experiential reality! in foreground, immediate contextsat the same time that the seriesof harmoniesmay beof significanceas aspace-filling linear stream of passing orneighbor auxiliaries. The exampleis a severe reduction and abstraction of harmonic content in mm. 3|-12

of the fourth movement

of Beethovenssecond symphony.The essentialsuccession is iv-V, the former appearing in two positions linked by a complex succession ofpassing auxiliaries related by tonal function aswell asby chromatic successionthe in lower voice and other, comparable linear successions!.Following the process ofsubdominant elaboration, V undergoes prolongationby neighbor auxiliaries, chromatically related, above and below. This sketch, vividly illustrating the two aspects of harmonic and melodic! function which are a principal premise for this study, is indicative of the fact that linear andtonal functionsnearly alwaysappear incomplementary conjunction in tonal music. In the Beethoven thelinear force of chromatic ascent is of course inescapable,but at the same time the tonal functions indicated are plausible, perceptible, and never far removed from the primary system; The concept of multileveled linear function is unavoidable,and itscomparability tothat of multileveledtonal function will be readily apparent.The multileveled aspect oflinear function hasto do, as wouldbe expected, with the diferingfunctions pitch of eventsat various levels ofstructure-a multiplicity of functioncrossing structural levels. Forexample, apitch event whichis of central essential!function within a phraseis seento be auxiliary to more fundamental factors at broaderlevels, asin the instance described above; oran eventwhich is a neighbor auxiliaryat a very immediatelevel isseen tobe a factor in a larger, passing complex ofevents. Theconcept ofmultileveled linearfunction thusconcerns the nature of the function itself as viewed or heard! within differing levels ofperspective. Linear functionhas afurther aspectof multiplicity, or ambivalence:that in which a particular eventis afiiliated with more than one related event.For example,a particular pitch might function asupper neighborauxiliary to another andat the same timeanticipate be a factor in the prolongationof! an essential pointwhich follows.Or a particular pitch event mightfunction asauxiliary to two relatedpitches inthe succession g-a-f-g-a, thef is part of the neighborencirclement ofg andat thesame timea lowerauxiliary ofa!. Thiskind of dual,or multiple,functional relationmay beoperative ata singlelevel ofstructure; it has to do with the fact that an eventin a linear succession can bemany-sided inits roles.This might be termed multilateral linear function in keepingwith its character, asdescribed. Multilateral linearfunction thusconcerns the diferent events qfqjiliation ofa givenfactor ata given level. Themultiplicity of linear functionis, then,of two dimensions. See pp.37-40 forfurther referenceto theseconcepts; added treatment ofthese issues is alsoto be found in discussion of examples tofollow. -

tona/ity 31 Ex. 1-1. Beethoven, Symphony No. 2 in D, Op.36. Abstraction ofapassage in the fourthmovement. 111.303

;ZZI~¢ 92" 92'!

I `e' ' 92./V i

_; .,

I _ __ __p_.

e 92/

_

iv, IV *D,d2 iv

IV rv/Iv vii N2

V iv/ iv vii l

V/ iv iv VI

IV V A6!

g: i

V/iv IV

vii/V lg

iv vii/V

v? V

`i

V/ iv! C,c: V

I vi

Vi

vii/ii? ii?

Specimen identifications of multileveled tonalfunction seepage 52!. *Some romannumeral symbols are simplified. m.302 fa /"0 Jf Fl.

3

;I

/"0 f_`

_ »_ f

O.

=£C==§== Ob. :in-ll"l -H

3- _

lan, l

=a"I:Ii =

?_

I i==-_-_I §== i= Q!-_I

'° Bn.

_

»» - 1 1£1 1 »_ . __

zz '|.= °

.» _

- .7

- ff

._ '° 0' _ =Bi==i==i:11i, h1'li==

~

= i;hm~vI'nIi1;=i==r~ I U 1111! I Z

1o

Vln.

I

=:==§:===§===

.¢

-I7 °

if °

=Hiv.:==1'_1i,I=11'1s= I I 1 I Q.Ie I *o -0 lmhmwzgr-11==i==r-i1_1i:r'

' ff

~ ° *f ~'

:';:i|T1|'Q1H2I I ll !=1__» :1 Vla. ::t'x~'1l_;n:=l==l1»1 I

$111114 . nr 1111.1

il Ve., 3'-£;fI1U|l=1JI'Y1l=== _ ' Db. §-_|92'7 r

=-f7'_-.--!5r'2--22ll -

` if

'Q

:Q-I;

` sf

tona//ty 11

ntinued.

Boi /*___*

//_é /_X

¢ fm

1:1-` E'

uf

A2 _

il J

1? `

= ?_.E¢=-.,,.;_i '.;191 4' 4? °

ff

ff

Eg Q f ,E :r..E=:rE

f'- 3 =g;. u-,---rr-= r==:r * ff

ff ff

ff

tona/ity 33

Fl. Elf" T a #V ob.

f/F J

i"

5V

il T

Hn.! 2/ if

62

i; 9

;

u Q

Timp- 'G39¥l;,;3&192:JZ

V., jllfj Effie ji @ 5is at at 1 wa' @ if Vi V f f i VC-» f

f ll

ll fvci

Dbwgag é.; fqi Db-JT!

they include brief, but felt, local~tonicizations ofiv/iv c! and V A!, in addition to that of iv g! itself, the primary harmonic basis f`orthe entire ascending passage. Romannumeral analysesare given at severaltonal levels,including tonic and subdominant, the former of broad significance, the latter of signiiicance atthe immediatelevel of the passage itself, reflectingthe principle of multileveled tonal function. The example demonstratestwo fundamental classes of harmonic and melodic! activity: that of prolongationof iv,IV, then of V! The conjunctionof twolike symbols of upper-and lowercase, separated by the comma, denotes asituation inwhich majorand minorforms orsystems are conjoined, andin which both are relevant.

34 tona/ity

and that of movement in this case, recessive succession, overall, from iv to V!, as well as linear functions both passing andneighboring. The reader may wish to make comparable study of other passages fromthe same movement, harmonic fluctuations of comparable features-for example, mm. 220-21, 317-21, or 240-44.

A further example ofharmonic abstractionis givenin Ex. 1-2 in resumé unfortunately without quotation of the too extended extract on which it is based! of the reaffirmation of T; I5 and V following developmental fluctuaEx. 1-2. Beethoven, Symphony No. 3 in El., Op.55. Synoptic harmonic abstraction ofan extractfrom thefourth movement. m.277 m.292

m.303 m.3l5

.£%='?=¥¬1'I'|T¥1='»{§'T?'==3 lT?= .IL ..92|vf 1-1| _-_¢ 3,- _ ...ID T 1:-'Ti 2 ' "I -M 1 ,- _ _-l3__ | lil-', ,_ -m 1111 _=

`,v ~._.v 92..v 92.1 92.v ' le F

IILQ - L . 3-1--1:0-_ lf* -'VIYQ -Il _ C -ll!I BI C :ll-1.1

Q

§'5 92-V E 92.v

[sm SD

Wedge-like configuration toward V

st SD

D]

,l

note chromatic role ofmodal variant of tonic!: @§_'=-%2 ,.

Q *Undergoes 21measures of unbroken prolongation.

I

I

-IIEQH

tion in the final movement ofBeethovens Eroica Symphony. T: I is stated powerfully in a seriesof utterances,given in the one-staffsketch, elaborated by neighbor auxiliaries someof which tonicize thediatonic harmoniesii, IV, and vi, with ultimately a more extended referenceto Ab, the subdominantnormal medium for direct reintroduction of T: V-and with linking, passing auxiliary harmonies embellishing the recessive harmonicmovement from IV to V. All

of this is summarized in the sketch, to make theessential harmonic

directions and functions visible, and thereis addeda brief synoptic representation of the contrapuntally potent, largely chromatic, wedge-like configuration toward the V. Although it is not representedexcept by an inserted note, the V undergoes no fewer than 21 measures ofinsistent prolongation plus fermata! in a typical atmosphere ofgrowing intensity before the I is reaffirmed in thematic

statement.

The Beethovenextract showsa primary tonal systemon Eb expanded by referencesto the diatonic tonics f, Ab, and c. Again, purely linear factors are obviousand extremelypersuasive andthey are accounted for!; but tonal 51. e.,I of the tonic T! system,here Eb.

36 tona/ity Ex. 1-3 continued. 1,117 l Y.1:"l1:Y'i" ,_ lrnlllll I lil 11 lN 'lLlP'_ g,/ 92_,

7 lrtv' 1 v -f

"__, Li

-'_ i

'1 |

-HI: , I.gn-»';'_,*.,,, :1 4 im.: ini

|

'

gf: _

92_,_ ¥_/v v1 v/v t: R/t: V

III? I!

prolonged andembellished bya seriesof chromatic auxiliaries. They become, in mm. 29-32, virtually nonfunctional,°-i.e., their positions aspassing chromatic harmonies in embellishment of the structural harmony does, in this context, eclipse any potential inference of tonal meaning in the stream of diminished 7th-chords. This is because of the parallel succession inwhich any inherent potential for contrapuntal interaction and afliliation, or leading-tone action in support of potential tonics ofreference, isavoided. This is very different from the Beethoven examples,in which despite the force of consistent linear

succession in elaboration of

essential harmonic

content,

tonal tendenciesare fulfilledor suggested in foreground actions atimmediate levels of structure.!

If relative significances of linear as opposed to tonal functions are a matter of degree, it is nonethelessa valid analytical judgment that at some point tonal function is superseded inexperiential eH`ectby the force of relatively straight linear function in which potential leading-tone relations are unfulfilled at any structural level of perceptible connection, and not at other points, where suchrelations arecontextually demonstrable.The juxtaposition of the Beethovenand Chopin examples isuseful in illustrating this distinction.

The Chopin analysis mustalso giveattention to the root succession ata more essential level: BI;-Gb-Eb-a

triadic formulation

with its

own logic.

The sequentialrelation at the outsetof the succession F-Bb,Db-Gb! imparts °Again, with respect totonal significance; thus, in mm. 29-32,despite thepattern of tonal shifts D ?-G?-A ?-D?-E ?!inferable fromand inherentlypotential in the succession of diminished 7th-chords,and the .logic of 5th movement in that succession, thepassage involves direct, parallel chromatic succession from one diminished7th-chord to the next. The chords areof aspurely linearfunction asit is possible toimagine, lackingtonal significance of evenimmediate functionalvalue. Thus,the parallelismof voice-leadingpreempts and precludes expression or suggestion of tonal function inherently,if ambiguously,potential in the chords.Such pureparallelism ofvoice-leading isa primary technique oftonalb nonfunctional harmonic usage; itis to be foundin examples,especially offantasy or improvisational style, inthe eighteenthcentury, andof coursebecomes common as anelaborative device in late-nineteenth-century impressionism.

tona/ity 37

a further underlying, logical unity.' The chromaticism of the vocabulary of the example is enhanced by chromatic succession ata number of points. In Ex.l-3, to set the stage for the series of passing diminished 7thchords, the primary tonic is impaired by cancellation of its leading-tone in the intriguing, repeated allusionto the Gb passingtonic t: III ?!. Compensatory reaflirmation of the primary tonic at m. 33 isconsequently ofenhanced, strongly resolving effect.

The concept of mu/t/Yeve/ed function further

exp/ored

The conceptof tonal functions asambivalent, in accord with that of tonics of reference ofvarious levelsof structural significance havingrelatively local or broad implications, is of continuing necessity in the analysis of tonal-harmonic function.

Similarly, the linear interpretation of functions of pitch materials their dispositions in lines-their horizontalness and contiguity in time! concerns aleveled structure except in the most microcosmic contexts.Often many levels can be characterized, ranging from the extreme background the total form as an all-encompassingimpulse!-the broadest and highest hierarchic architectonic level, through numerous intermediate grounds, to the mostimmediate, local context ofreference treatedin traditional theory as foreground, the actual surfaceof music. It is vital in

the consideration

of leveled structure in

music that

the

surface, orforeground, be kept always in view, while regarded in its proper hierarchic status.

The reference to hierarchic

status as low in

level bears

no implication of relative importance. Indeed, one might well observe that in music of normative stylistic terms i.e., music in which norms can be identified-nearly all music!, background structure isof moregeneric content the more broadly it is viewed; theHavor, character, and unique expressive consequence of the particular work resides in very important respects in the particularities of elaboration in the foreground by which broader structural factors are realized and amplified. Analysis is far too often content with delineation of broad structure as THE structure, leaving untreated the vital, arresting patternsby which the great work of music projects and extends in time the prototypical basis. We have been concernedin discussionsof pitch structure to this point, and continue to be concerned, with functions-including tonal functions and identities_of pitch complexes within very local areas ofreference aswell as more essential functions, insistingthat immediate tonal implication is very critically a part of the experienceof music. 7Consider as well thehorizontalizations F-Bb-Db, Bb-Db-Gb, andGL-Bb impliedroot of the primary dominantat the end of the passingstream ofdiminished 7ths!-Eb.

38 tona/ity

The concept of multileveled, or multiple, tonal function is the basis for discussion ofthe concept of tonal fluctuation modulation!, in which it is asserted thatthe signyicance of fluctuation in any given instance corresponds to the breadth of structural level over which an emergent referential tonicprevails. Similarly, the concept of chromaticism, as it concerns alteration of the factors ofthe diatonic scalein melody and harmony, depends fordefinition on the concept of multiplicity of level and of tonal reference: thus, a particular PC may be chromatic in relation to the tonal system at one level e.g., the primary system C,where an F# appears in tonicization of the V! but not at another level in the same hypothetical situation, not at the secondary levelrepresented asthe immediate context of the tonicization of G, where the Fll is diatonic!. Important, then, is a procedure ofanalysis whichexamines aparticular harmonic or melodic line in stages ranging from the particulars of its appearance on the score page to the general structure described by its most essential points. For example, even a small unit of melody can be viewed varyingly so that at one level a particular factor appears relatively essential while at another level relatively auxiliary subsidiary, elaborative!. The differing conclusions derivedin this way are not of course contradictory; rather, they are complementary and interdependent views.! To put it another way, a melodic line of severalformal units reveals, inanalysis, anincreasingly complex hierarbhy of functional significances: anessential pointwithin a phrase may appear to have an auxiliary relation to a more basic point when the phrase is viewed in a larger context, and the high point in pitch within a single phraseis likely to be interpreted asa subsidiaryhigh point in the analysis of that phrase in combination with others. Two examples follow, each involving a kind of linear analysis of pitch functions having 1eveled significances. The high point of the first phrase of Ex. l-4 is subsidiary to that of the second phraseof the group. This the expansion ofspatial compass!is, along with motive mirror, the chief means by which the content of the opening phraseis varied in the second. Thereis thus a conjunction of formal units into a larger unity by relative superiority here of pitch, but often of duration, stress,density, etc.! of one event over a related eventin a linear sequence.The significance qf bothpoints intonal function

l,

should also be emphasized; and ahigh-level conjunct successionhigh of

points and structurally important cadential pitches, clearly implied despite registral variance,is shownas part of the representation in Ex. l-4. In all references to linear functionof pitch materials, theterm essential is orcourse to beunderstood as meaning of the essenceand notin the more restrictedsense of necessary.

tonality 39 Ex. 1-4. Brahms, Trio in C minorfor piano,violin, andcello, Op.101, thirdmovement. -1 Andantegrazioso

J J Q92_»Lfl'

P

/A

4 /-J: T' "` +. ,_ T

~ ~_._ __ ____ ____ _____. _~_..

_____________

. TE

' EE

5! 1

~@ I-

The phrasefrom the Sequenza IIfor harp by Luciano Berio° Ex. 1-5! is a concise melodicunit, its cadence achievedin part by rhythmic means. The exampleillustrates afurther aspectof hierarchicrelations amongmelodic units within a single phrase. The structural basis is felt in the succession of principal and subsidiary high and low points toward the cadential D, which can be interpreted as analogous tothe traditional tonic. Ex. 1-5. Berio, Sequenza I/ for harp. m.l4* __

g gl

I

I Iii //PP 1 lPP f iff' #ff | gi/,

Pf

PP

II

*Each measure is a fixed durationalunit at 40MM. © 1965, Universal Edition. Used by permission the of publisher. Theodore Presser Company, sole representative United States, Canada and Mexico.

The quotation, with supplementary explanatory symbols,makes these functional interrelations visible. The melodic line assumes akind of wedgelike character seen in this light, its course very purposefully controlled. Several observationsneed to be made in extension of the analysis: both °In the index will be founda listingof citedcomposers and their years. Each vertical bar-line corresponds to onemetronomic pulsationat 40MM. Time distributions withinthe unit are represented spatially, andapproximately, inthe notation of pitch events. Beriouses accidentals only to indicate pedalchange. Accidentals given in parentheses inthe quotation are thoseapplicable by virtue of already established pedal settings.

40 tona/ity

cadence PCs D, A! are immediately preceded by leading-tones whose resolutions arefulfilled in octave displacement;the recessionof low points is diatonic, the sequence ofhigh points chromatic; the stressed notes,marked forte, constitute in a middle register a descent diatonic! toward D, complementing that of the succession oflow points; each subsidiary high point is preceded by a rest and/or upward leap; and a rest precedes theprecipitate cadential succession. It is of course vital that

these relations, functions, and

interactions be understood in performance. Some ofthe examplesof recentmusic tobe presentedin later analytical discussion in this chapter will be concerned with instances of hierarchic linear arrangement and order in which tonality is not, or is less, relevant.

The ideas of pr/macy and hierarchy among pitch-c/asses and p/rch-c/ass-comp/exes; generic and part/cu/ar tona/ systems

The discussion and analytical exploration of tonality, however broadly conceived, tends to center in those particular conventions by which the tonal period is identified. Propositions respectingexpressions oftonal order in earlier and later styles areoften introduced in analogous relation to that most immediately understood conventionalpractice. In the projection ofa primary factor within the PC contentof a musical work or style! much of the weight of historical practice suggests thechief importance of its relation to the pitch a semitone below and, in some styles, above! and the pitch a 4th below or 5th above. The leading or leaning force of the proximate pitch in semitonal relation, and the strong natural relation of the 5th and its inversion, emerge throughout tonal practice of all kinds as preeminentaffiliations by which the central factor in the hierarchic order is approached and understood. The primacy of the tonic is a function too of its role as the ultimate point of cadential arrival. The hierarchic tonal .gzstem can be referred to as oneof genericapplicability-i.e., the tonal systemof a style might be conceivedas that embracing, for example,a diatonic set togetherwith vertical PCC derivativesof common usage and expanded by subsidiary collections of chromatic affiliates usual in such a given style. Thus, the tonal system ofa style would represent the normal range and ordering of PCmaterial reasonably to be expected. In the same way, the tonal systemof any class ofworks might be theoretically conceived. Of more immediate concernis the concept of tonal systemas it applies to a particular musical instance, aparticular work and the experience ofthat The word tonic willbe freelyused to denote thePC or PCC at the centerof the hierarchic orderin all styles in which suchorder is a relevant principle, including those which predateand postdatethe periodof major-minortonal conventions.

tona/ity 41

work, as opposed to the theoretical system genericto the class of which the work is a member. The generic systemmight be, for example,the phrygian scale together with predictable secondaryleading-tones attending and inflating, or tonicizing! certain of its degrees, orthe diatonic scale ofC comparably expanded. The tonal system specificto a composition representsthe total resourceof PC content basicto that work, and the hierarchic ordering of the content asexpressed the in formulations cy' thatparticular work. We will see ofcoursethat particular tonal systemsreflect stylisticnorms; but at the same time the analysis of the tonal system ofa composition will illuminate its tonal structure by making visible the particular directions, range, and byimplication! rates of _fluctuation by which its primary system isexpanded and enriched. Such a particular system may include chromatic factors e.g., Neapolitan harmony! as significantly projected secondarycenters, may be seen to extend a particular modal

ambitus beyond the norm, may be seen to

have some kind of anomalous second-order component e.g., recurrent cadential G within the phrygian context, or predominantly plagal action within the framework of eighteenth-century tonalstructure!, or may be seen as, for example, a primary system consisting of a trichordal or tetrachordal collection rather than the more extendedresource commonto a given generic system. Methodswill be suggested herefor the analytical representation of tonal systems,methods by which the tonal image of a work can be portrayed and its latitude and directionsof fluctuation shown. In the representation of a particular

tonal system the rate of fluctuation

may be inferred in

a general way; but the rate of fluctuation is a rhythmic factor to be treated precisely in subsequent portionsof this book.

Tona/ systems of compos/'t/'ons.' reg/'ons and /'n terre/at/'ons

The specyictonal system may be defined as the collection of tonics tonicized PCS and PCGS! and the supportive encircling, leading, preparing! pitch factors oriented toward each of these."

Example l-6a is agraphic representationof the tonal systemunderlying the expositionand developmentin the first movement of Mozarts Sonata in F for piano, K. 332. It shows aprimagf system on F, and secondary expanding, auxiliary, inflating, embellishing! systemson C, Bb!, d, Eb, G!, and a!; those shownas parenthetical are of distinctly inferior status. Thetonal range expanse, scope,latitude! is immediately evident. Although the ascending order of the chart represents thechronology oftonal rwrencein the piece, it could also be arranged to representa hierarchic ordering based on such factors See pp. 134-38 forseveral representations of thetonal system ofthe fourthmovement of Beethovenssecond symphony.

42 tona/ity Ex. 1-6a. Mozart, Sonata inF forpiano, K.332, firstmovement. Representation the of tonalsystem andextracted key tonal eventsin expositionand development.

a! '

A

, 92__/,

G, g!

A 92/'

-1, P 1f|nuu _;: ' I 1' nr v1 E -inn;

Eb 6

s

C,c gigs: as I7

2 11-rf; -:-ai-'

R/d ` n B/-B

Bb! |

Q 92.Jr

FQ

§___""Dr"°°5

I,

'

| 3;

92/v

IY s

T SD

D,d sm r!

' #2

p

Orcéer refegence

Allegro Z_`

5@ 9292ll ' n I1

ff~ 5

__92

J =» 1 _ 9292.J .- V ass }`-

5=

/=

EEEEEEEEEEEEEFEEEEEEEEEEEEEEEE!

m. 22

Ei? 35%

E

tona/ity

l

1g7é/ f_92 ll ll

f /'ba

f V

Jihizjr fb; bf

ESFC VPJ|'P_| If E fP

fP

[if

44 tona/ity Ex. 1-6a continued. m.1l7

g7

fP

fP

fP

f' J

fP

f

Eggs-f."i% 3 I m.l27 92

°s

l

1¥'

1

"' "E

Pf

as frequencies and durations of tonic occurrence cadential and prominences of such occurrences. Curvedarrows representdirections and paths of especially dominant-tonic! tonal affiliation within the system. The whole is a visible, synopticimage of the tonal resource; it could ofcoursebe extendedto represent the total harmonic

resource as well.

Example l-6b is a further kind of representationof the same example and its tonal system. While relative importances of tonics are suggested by note values, specific durations of tonal reference arenot represented. It is important to emphasize ina graphic exercise ofthis kind that the notation represents PCsonly, not specific pitches,and should of coursenot be read as a pitch-line.

The underlying tonic and dominant components" are offirst-order and second-order significance, respectively, asmight well be expectedin the style of which the exampleis an instance. A tentative representationof fluctuation to and from points of relative distance isgiven below the exampleas further indication of the structural importance of fluctuation within the expanded tonal system. Specific typesof tonal function embellishing, passing! arein evidence here and will

be discussed further later.

Of course,

the relative

superiority of any given tonic of reference isa questionof structurallevel: the central tonic, F, is thus preeminentat the broadest level;C, the dominant, is The term component .9»stem or component refers toa level within the total system-a related system,itself hierarchicallyordered andcapable ofexpansion bysuperimposed inferior systemcomponents. Thus, in the Mozart Ex. 1-6! ,d is a systemcomponent ofrelatively inferior status.

46 tonality

The concept of an emergent secondarytonal system can be seen in a representation of the modestly expanded primary system asshown in Ex. 1-7a. It indicates a tonicized iv as second-ordersystem componentand Ex. 1-7a. Josquin, Tu pauperum refugium. Synopsis oftonal system.

Q uSMa» E I

I'

g

r1== uli 1 1 |19 ll! ; 5 f ¢_ GP=_ 1|

1| CXO |10 li

__ Qsd ilhi10| ,

t

suggests themildly tonicizing effect of the plagal approaches tothe cadential VI as well. The secondary regionon A is tonicized in the second phrase and elsewhereby application of the G# leading-tone. The applicability of the principle of tonal expansion bringingabout limited excursion beyondthe fundamental, primary system is obvious, and its representation as shown provides animage of hierarchic tonal order which can properly be viewedas a foreshadowingof complex tonal structures of later music? In Tu pauperum rqfugium, tonal structure is of course also evident in cadential functions,

and the

hierarchic order

of cadential

functions is

of

fundamental significancein study of the tonality of the piece Ex. 1-7b!. Examples in this chapter reflect theconcept qf tonaligf broadly dqfined, as is evident in the most cursory examination of sources.That concept applies to early music predating the tonal period! no lessthan to tonal music of the twentieth century: it is implicit and essential inthe idea of a scalar or other PC resourcehierarchically disposedin cadential and other focal applications. Indeed, in that senseit is evident in liturgical chant no lessthan in sixteenthcentury polyphony. Beyond that general concept, much music of modal systems canbe seento foreshadowthe conventionsof later tonality as well: the functions of root relations of chords and of primary and secondary

tonality 47 Ex. 1-7b. Josquin, Tu pauperum refugium. Succession and hierarchic order of cadentialcenters. Pronounced texturalcontrast Varied reprise

§¢l.l..% .

--ll Strong i

elaborations

Hierarchic orderof cadentialcenters: third-C

second-a first-e leading-tones in the applications of musieajicta,which show in incipience the means of later, broad tonal expansions by chromatic extensions of fundamental diatonic collections. And the tonal principle of opposition of areas of fluctuation with stability, and of ambiguity with focus of tonal orientation, is evident in many highly provocative sixteenth-century works.

The concept of a generic tona/

system explored

in theory

Since the intent of this chapter is chieHy to explore tonal principles and specific tonal systems manifestin individual works, the concept of a generic tonal systempertinent to a classof compositionsis discussedonly limitedly in a forthcoming illustration. The deduced norm-defining tonal system of a class ofworks, while not a major concern here, can constitute a useful point of referencein the study of stylistic trends, terms, and evolutions. T wo relevant, interestingsources mightbe mentioned,one concemedwith early manifestations oftonality, the other with later manifestations.Edward Lowinskys paper, Tonality and Atonality inthe Sixteenth Century Berkeley: University of California Press,1961! illuminates aspectsof tonal order, as well as what the author calls atonality" and floating tonality in sixteenth-centuryworks, in provocative studiesin which tonality is conceived asa tonally centered organization p. xii!. Examples of sixteenth-century tonality are discussed often in the light of emergent devices common tolater major-minor conventions. Atone pointLowinsky notesthat a net of cadences on varying degrees related to the tonic and organizinga wholework into various sections comes closer to definingtonality p. 15; emphasis added!. Another sourceof substantialinterest andrelevance tothe concept of tonality broadly viewedis RoyTravis, Towards a NewConcept ofTonality ? in journal qfMu.ric Theory, III, No. 2 959!, pp. 257-84. "The theoretical presentation whichfollows is related to,but significantlydistinct from, the chart of the regions suggested byArnold Schoenbergin StructuralFunctions of Harmony London:Williams and Norgate, Ltd., 1954!, Ch.III. The reader isreferred to Schoenbergs theory as astatement ofmajor importancepromulgating atheoretical, generic concept oftonal systemapplicable toderivative representations of the tonal systems of individual works.

48 tona/ity

What follows Ex. 1-10! is a theoretical representationof a chromatic, panmodal tonal system onC, capable of courseof transposition throughout the equal-tempered pitch resource." It is a theoretical system of proposed generic significance as opposedto one Ex. 1-6! in which systemic content derives from a particular context.

A symbology of tonicdesignation akin to Schoenbergsis adopted here. It represents tonicized factorswithin an expanded tonal system by symbols which go beyond the usual letter designation ofPC to indicate the relation of a secondarytonic to the primary tonic. Thus, such symbolsas T tonic!, ST supertonic!, M mediant!, R relative!, and like symbols, as well as lowercase correspondents to representminor systemcomponents, will be used.The affix /T or /t will be used whereconfusion might result as in usesof the sym-

bol D as dominantto distinguishit from that standardletter designation of pitch or PC! ; otherwise itis understoodthat the symbol is one of direct reference tothe primary tonic e.g., st is understood asst/T! _The slashalso shows other kinds and levels of afliliation within the system: e.g.,r/D relative minor ofthe dominant!, etc., as wellas relationsfarther removed:e.g., r/D/st,etc. Again, /t or /T is understood as the concluding term, explicit only when necessary topreclude ambiguity. The symbol r/T or r = relative minor of the primary major tonic; R/t or R = relative major of the primary minor tonic. M or M/T = E, GH, Bin C major; M/t = Eb, G, Bb in C minor; etc.! It is important to recognize that in a representation of this kind these symbols representtonics-actual or potential centersof systemcomponents or in Schoenbergsterm! regions of tonality, not merely harmonic factors.In that sense,SD can be viewed as an extension of the principle of IV and a succession T-SD-Dan extension or inflationof thesuccession VI-I V. Similarly, to indicate relations as well as simplePC identities, symbols like smziv in C, iv/vi or aziv! or D/Tzii in C, ii/V or Gzii! are sometimes used. Relationsof considerableremoval from the primary system SD/SD/N, where N denotes theNeapolitan! are sometimes shown;although such relations can be adducedwithin an infinite breadth of systemicpossibilities, it is important to make distinction between relationsof a theoretical nature like those adducedin the following chart, Ex. l-10! and thoseof a contextual, or practical, basis which are the chief concern of particular analyses!. Example l-10 includes alisting of relations bywhich all PCs within the chromatic primary system canbe relatively closely identifiedwith the central tonic, C. These are listed as nearest relations,i.e., relations cjjifwest terms. Although the N is noted occasionally,in view of its common membershipin many practical individual systems, the relations given are otherwise through diatonic links.Thus, for example, the identification of f il as st/D/r or m/D/D "Other, related generic systems could beadduced aspertinent to phases ofstylistic experience withinthe tonal period: e.g.,that of the unexpandeddiatonic majoror minor collection andits derivativeharmonies, orthat of the bimodal collection withoutfurther chromatic expansion.

tona/ity 49

suggests that its nearestrelation to C easiestdiatonic accessibility!is asseen in Ex. 1-8.Again, note that Ex. 1-10 represents tonal, notmerebf harmonic factors ,°the summary given as Ex. l-8 should similarly be read as consisting oftonics-tonicized harmonies, diatonic system.

Ex.1-8.

rather than

simple harmonic

successions within

a

3 kI iw, QQ., E o ;9292_ ___§____ j

T r/

T D/

r/T st/D/r/T

gs j W s 92/

T D/T

s

D/D/T m/D/D/T

Key signatures are listed as including the tonally vital leading-tone of the minor system a practice shown in Ex. l-6b!. Although

key signatures are

one indicator of tonal distance, thisfactor alone can be deceptive, asis noted in the discussion oftonal distance which follows later in this chapter. A very critical factor in tonal distance or proximity! is the immediate adaptability of a major triad as dominant to T or t-what Schoenberg calledthe interchangeability of mode"-a principle responsible in very significant degree for the expansibility of tonal systemsin conventional tonality. Implicit in Ex.

l-10 area theory of chromatic harmonic succession and a theory ofmultileveled or multiple! harmonic function, bothof thesetreated further as thischapter unfolds. Since theconcept isinvolved with questions oftonal ratherthan purely harmonic range, all degreesin the primary chromatic system areregarded as of tonic function; hence, eachis consideredthe root of a major or minor triad and its theoretical significances andaccessibility derivedfrom that assumption. The minor triad has suchpotential significancesas r, sd, st, m, or sm. The major triad may have tonic significanceas R, D, SD, or SM. Where potential system components are hypothesized, diminished and augmentedtriads 3Schocnberg, Structural Functions, Ch. VII.

50 tona/ity

are notconsidered be to ofpotential tonic function, i.e., to be capable of functioning as system centers. Analyses oftonal fluctuation, as suggested in the chart, might well show the particulars of relation demonstrated in specific contexts;the symbology proposed has the advantage of showing ultimate relations to the primary tonic as well as immediate functions within secondary tonal systems.Such a representation mightbe asin Ex. 1-9, whichincludes abranched diagram in which the dominant systemcomponent region! appears asan outgrowth in the given context! of the primary system, and the third component as an outgrowth again, in the given context! of the second. Thefirst-order system primary! is shown at the base as T. The designation of relations by this principle avoids the often dubious identification oi, say, a Bb triad in expansion of the system of C as lowered subtonic and suggests for it such a plausible, precisecontextual relation as SD/SD. Ex. 1-9. T: V D/T: IV r/D: iv

-I

V/V! -V

etc. V/iii! - vi _ ii

- etc. V/vi! etc. -V - etc.

Finally, in extension of a chart of this kind Ex. l-10!, or in representation of any modification or qualification of this hypothetical, generic, tonal system, further relations-relations of greater complexity, of greater distance, of more terms-could of course be adduced. For example, one might go further, in theory and without reference to any particular contextual derivation, to note such potential relations asr/SD/SD for the minor triad on G, or SM/r/ D/r for the major triad on A, etc.-all such relations potential and plausible in functionaltonal contexts. Implicit in the functional chromatic system is the principle that the chromatic factors have a diatonic basis, i.e., derive as alterations of factors in the basic diatonic collection on T or t.

tona/ity Ex. 1-10. A view of a theoretical, generic tonal system,and hypotheticalproximate dlaton/e re/ations linkingeach ofits potentiallytonicized degrees with Cc Mode of Tonic Examples Tonic Triad Degree Signature

of Close Relations of Diatonic Succession

mo

m/R/t, st/SD/T

Mo

D/T

mU

_W ' _ m/T

-1 e

MU

D/sm r!/T -1

M -l-

R/t

m ii

sd/sd/sd/t -1 e

m

st/T

M ""-;_`__

D/D/T

m ;,

__

M 92

r/D/r/T -1 __

nl D

e

¢

SM/sd/t +N!

'__-_'_-l-___-__ I

M

T

m I!

33'---`_`__

M M_

.2

IT! U1

! ____._______

M-. m M m ` M m M.

m/D/T

!

lg 55' EL li

I! II _

Z D/D/r/T SD/SD/T, D/R/t sd/sd/¢ r sm!/T -1 D/st/T, D/D/D/T m/D/f/'r sm/t

eI

°

52 tona/ity Pr/mary and secondary ton/'cs and the/r structura/ and auxi//'ary functions; mu/ti/eve/ed mu/I/Qo/e! tona/-harmon/c function; tonic and dominant forms

As implied in Ex. 1-6b, tonicscan berepresented inmuch the same waythat structural and auxiliary embellishing harmonies and, in melody, individual pitches! are represented incommon techniquesof linear, reductive analysis. A fundamental difference must, of course, be kept in mind: a stream of tonics in such a graphic, hierarchic representation canin no sense beinterpreted asa sounding line of actual pitch events. Vital to this considerationis the concept ofstructural level. The primary, or central, tonic is of course always of first-order importance as the term primary suggests!. At the same time,secondary tonicsof referencemay be structural inimplications within low-level contexts over whichthey prevaile.g., F at the level of a single phrase over which it has tonal predominance!. But the idea of low-level predominance doesnot pertain to the cadential tonic representing within the sameunit of reference tentative a departure from the primary or other superior tonic, where departure expectsreturn and conditions the interpretation of the tonic of departure, if hierarchically inferior, as submissive, andthe superior tonic asgoverning." The concept of level of tonal structure yields that of multiple tonalharmonic function, or multileveled tonal-harmonic function, a concept by which it is proposed that the true analytical interpretation of tonal function in any given harmonic or melodic! event is often one of numerous terms. The multiple function of tonal harmony is suggestiveof an ambivalence not necessarilyambiguity! in the significanceof any single harmonic event or successioninfluenced by the emergenceof a secondary tonic or complex of suchtonics. At one level,tonal meaning is traceableto the most immediate tonic of reference; otherfunctional meaningsderive from references toother systems operationalin the given context.A number of the examples analyzed in the following pageswill illustrate this concept. The full explanation of a secondary tonalfactor involvesa rangeof tonal functions which, with experience and understanding, can be felt in a way that lends harmonic events within multileveled systems anextraordinary depth of signyicance,and consequent expectationof ultimate, sometimes veryremote, fulfillment within the primary, expanded system. If in a specific tonal structure there is one, and only one, primary system Schoenbergs monotonality!,2° it follows that there is among "Some of the examplesto follow will demonstrate,however, particularcontexts in which the tonic of departure ultimatelyprevails-tonal structures, that is, which setout initially in subordinate regions ultimately absorbed within a culminating, differentsystem e.g., somesongs ofWolf!. 2°Schoenberg, Structural Functions, Ch. III, especially p. 19.

tonality 53

secondary systemcomponents a hierarchic arrangement particular to the musical instance see Ex.1-7!. Given the termsor norms of a particular style it can be predicted that, for example, the dominant or relative might well emerge as second-order systemcomponent. In certain tonal styles other orders are somewhat predictable, often occurring at intervallic distances of the 5th or in other idioms the 3rd!, just as in modal systems PCsof likely cadential prominence can be preinferred. But beyond suchnorms the particulars of tonal system are a vital factor in the character of the individual work and its expressivepotential. The concept of secondarytonal levelsis of course thebasis foridentification of specific harmonic events as secondary dominants, secondary subdominants, etc., and for such a term as secondary leading-tone. All of these havetheir counterparts: primary subdominant, primary leading-tone, etc. We have already made useof the terms primarytonal gystem, primary tonic, secondagf system and .gystem component, etc., as well as expanded tonal gfstem,the latter denoting all primary and secondarysystem componentsparticular to a given composition or class of compositions!, the system componentscollectively viewed.

Those triads of the diatonic system most likely to assumethe role of secondary tonicin music of the tonal period are of course theconsonancesthe major and minor triad forms. Such functions as T:vii°, t:vii°, or t:ii° and III* 2' will function as secondary tonicsonly when modified to appear as major or minor triads.

When t:vii° and t:III+

are tonicized as major triads they normally

involve thelowered 7,a violation of the primary leading-tone,and have a consequent weakening effect on the primary tonic. It is for that reason especially thattonal-harmonic functionsof the minor mode commonly derive from the harmonic form of the scale. I. e., the evolution of major-minor tonality out of modality involved more than any other singlefactor the inflec-

tion ofthe7 asa leading-tone.! For thesame reason tonal change toward the minor tonics relative major R/t! can be regardedas far more disruptive of the primary tonic and its predominance than fluctuation into the dominant region, whose tonic triad unless tzv! has solid diatonic status within the primary system." The leading-tone triad in the majorand minor modes, respectively, and thesupertonic and mediant in the minor mode. 22In otherwords, fluctuationin the direction of R/t t:III tonicized!, exceedingly common intonal forms,is considered here topose theparticular problemof unusualdisruption of the primarysystem incancellation ofthe primary leading-tone! ascompared with the equallycommon fluctuationin the direction ofD/t or D/T, whosetonic isdiatonic, and highly supportive,in relation to the primary center. This will be discussedlater as one manifestation ofdistance between relative systems as comparedwith others of greater apparent distance. Prevalence inthe tonalperiod offluctuation andtonal expansionin the

54 tonality

just as particular chord forms function as tonics tonic forms:major and minor triads! , soparticular chord forms functionas dominants.Dominant forms are those in which the two chief factors of dominant action and potential relation are in evidence or clearly implied: ! the leading-tone and ! the potential for root relation a 5th above the affiliate tonic form. It must be recognized that when the leading-tone, i.e., the note a semitone below the tonic affiliate, is the apparent root of the chord, the particular dominant form in question isa variant of the very closelyrelated form built on the note a major 3rd below that leading-tone.Thus, is

virtually indis-

tinguishable fromand interchangeablein function with .! A resuméof the group of dominant forms is givenin Ex. l-l 1, arbitrarily relating to a potential tonic affiliate D. All have potential dominant function, primary or secondary. Listed forms of III* and I6 are of course 4

understood as highly dependent chords-tentatively delaying the usual dominant major triad of resolution and, in fact, inseparable from auxiliary anticipations of! the dominant triad. To these forms could be added still Ex. 1-11 . A resuméof commondominant forms.

* * ai Q V V1

vii° vii?

Vs

Vo III*

i or U2

*Often considered to havethe dominantdegree as implied root,thus analogous to the corresponding V7 and Vgforms. 1'Usually infirst inversion,the dominantdegree aslowest note. Iln second inversion: delaying,dependent on,succeeded by the V.

others: for example, the dominant 7th-chord with its 5th raised or the dominant triad

with the

same alteration-a

favored device

of the

later

nineteenth century, especially ofBrahms. Of the following dominant forms, the dominant 7th-chord and the diminished 7th-chord on the leadingtone must be recognized as the most important ; it is understandable that such dissonantforms would have priority over the simpler dominant triad because of their strong drive toward tonal resolution. The diminished 7th-chord is favored over the half-diminished because of its versatility in direction ofR/ tmay wellbe aresidual tendency remaining frommodal traditionsof cadential Huctuation lackingdependence on the later leading-tone functionby which major-minor tonality is largely defined.

tona/ity 55

resolution to both maj or and minor triads; other aspectsof its versatility are treated later.

In Ex. l-12 certain factors of apparent tonal significance are quickly evident: there is reiteration of the major triad on C at the outset; there is fluctuation from the diatonic system Fit in m. 2, Bb in m. 5!; both of the accidentals introduced

in the Huctuant portions

are soon cancelled; and the

original C is reaflirmed in these cancellations and by appearance of its dominant in

m. 9.

Ex. 1-12. Beethoven, Sonata inC forpiano, Op.53, firstmovement.

Allegro con brio Z-$_

_. =i¢n|m:_ ll..1V l _ 1 ' -I

1 I3 _

1 H Q: 11 _H _ _ -'=== li!

1_ Z _

|¢1111;111_ 1 .-at-1-7-ri-1-1-71

1- 1:-1-1-1-1-9-I

m.4

/'i

:E _

-' - -_----= -. Ei. _

e=

m.7 /_92 KT- is r92 ~I!-11'!1$1l92:~l .4 IH 1 §|1.

U; xi1=g_:$ :=

E=

|513-D-sennnzrsnnnn-:Hui

'Q

° ====

» li

1 -= H l 1:1 _ _#_-' _- 1

:Er -=

--=-=1 1'-- '= == 1 111-1-1

111-1 1

A 25 -

#nh ° __Il 1

- '-4

- 5:2

IQ

1_

_:lx-_ --_Z-._

'

CTCSC. "'°' Ui I i-1-_-Q.;-_TQ-1-1--_-_--_1_-Ifiuiijnii 1_---I*_------ini!--Z!-= l1lllll1--

-1-I H-Q!-IQ

b 5l'= Tonal Huctuation thus occursprecipitately, and the early threat to the primary tonic, C, is one of the factors conveying a sense of unrest in the opening passages of this work. The first accidental, Fil, is in a veg!local sense a leading-tone, resolving to G, its tonic, as canbe seenclearly if mm. 2-3 are, solelyfor analysis,considered in the contextof the secondary system of G. Since theprimary system is G, not G as examinationof the entire movement would confirm!, the F il must be interpreted asa secondaryleading-tone, theG asa secondarytonic;

56 tona/ity

thus, theseevents havetonal importin somedegree. Theprimary tonal system, on C, is thus expanded andembellished bythe inflated significance attached to its dominant i. e., by the presenceof a secondary dominant,by the tonicizing of C:V!. The above analysis ofmm. 2-3 as expressinga secondary tonic, auxiliary to C, suggeststhat the opening harmony, the tonic of C, has an ambivalent, dual function-again, in a very local sense making possibleby simple diatonic meansthe referenceto the secondary level,G! ; it is also a subdominant in the secondary system on G, hence a secondary subdominant.

The progressionfollowing m. 5 is analogous in important ways: it is a sequential repetition of the opening, a step lower; it is a reference to the secondary tonic,F, involving the samesecondary functions IV, V!. There is an important difference, however:it is not derived diatonically. That is, the introduction of Bb involves chromatic succession mm. 4-5, Bll becoming Bb!, expanding the systembeyond the preceding tonicizedV. The consistent chromatic line in the bass within and surrounding this successioneases the approach, indeed makes it seem inevitable. In broad analysis ofthe Waldstein opening, theseprocesses mightwell be capsulatedas momentarydeviants swiftly brought into focus asembellishments ofthe primary tonic. An analysisof the tonal structure and systemunique to this work accounts for the inflation of that systemby system components auxiliary neighboring, encircling! in relation to it--D/T and SD/T, .gymmetriealbf ordered around the primary C, each prepared by a secondary IV the SD/T:IV alien to the primary diatonic system, moredistant, and chromatically derived!. The entire process leads to an unequivocal T:V7 by

persuasive and inexorable chromatic descent inthe basslinking l and 5. Example 1-12illustrates auxiliarysecondary tonal systems in expansion and embellishment of a primaryrystem. Secondary systems canbe seenin general to have functions of two discernible principal classes: some embellish the primary system, with such embellislzing secondary sjystems succeeded by reaffirmation of the primary tonic; otherscan beseen tolink disparatesystems. Such passing ortransitional secondary tonics alsoembellish and expand, of course, but they have the particular function of participating as connectivelinks in transitional, fluctuant successions. parenthetical A gfstem, which can have either function, is one which has a tentative presencewithin the procedural relations of tonal expansion,but which does nothave morethan a very nebulous, implied manifestation e.g., Bbin Ex. 1-6!. A parenthetical systemcan often be identified as important in defining the relationsby which tonal fluctuation occurs, while its perceptual signficance is uncertain or clearly negligible; see, amongothers, Exx. 1-39 and 1-47.! 23The Al;of m. 8 hasno tonalsignificance; itsimply altersthe modeof thesecondary tonic, F, continuing thebasss chromaticdescent tothe dominant by which the primary tonic ispowerfully reaffirmed.

tonality

57

In the Beethoven,G and F appear as embellishing secondarysystems symmetrically neighboring the primary C. The issueof the levelof structure taken as referential is essentialto the characterization of function of a system component: thus, for example, in the Beethoven,G is passingat one level, embellishing neighboring, surrounding, encircling at a broader level. At the broadestlevelof structureall secondary systemcomponents can be interpretedas expanding theprimarysystemandelaboratingtheprimarytonic. The third of Liszt s Transcendental Etudescontains, just before its conclusion, a provocative expansion of the primary tonality conditioned by a largely chromatic descent in the basslinking F:I with F:V Ex. 1-13a!. Ex.1-13a. Liszt,Transcendental Etudes,No. 3 in F Landscape!.

ritenuto ed appassionato assai

tona/ity 59

In the Liszt successionreferences areremote from the primary system and the means of tonal fluctuation extreme. There are firm primary tonal pillars framing the passage I, V! and the fluctuation is swift and brief; yet the auxiliary tonics emergeas distinctsecondary passing functions, enriching the primary system chromaticallyin the works final developmental episodes. As examples for analysis reflect progressivelylate styles in the tonal period, increasinglyprovocative questionsmay arise,since extremesof range and fluctuation

are more common in

later nineteenth-century

music. The

parameters arethose offrequenqy qf tonal change the quantitative facet of tonal rhythm! and distance and volume qfehange-both vital questionsof style. Tonal expansion, especiallyas it concerns thesecond ofthese considerations, might be symbolizedas in Fig. 1-1. Fig. 1-1 . Symbolic representation of tonalexpansion. 61-! primary

Expanding resources within_the system and within other individual system components

+ Expanding range of fluctuationinto secondary systems; increasing-number and distances of components

Tonal expansion

In more Huctuant stylesof highly chromatic idioms there aresometimes extremely remote and rapidly driven fluctuations in the tonal structureeven at such critical stages asthe beginnings of works, creating tonal forms which might be representedgraphically asin Fig. 1-2, thelevel portion showing the ultimate stable expressions relativefocus! of tonality after widely ranging fluctuation at the outset. Indeed,this isa shapingprinciple by which a great deal of later tonal music is conceived. Fig. 1-2. Symbolic representation of tonal structure broadlyreceding towardfinal pointof relative focus.

L

tona/ity 61

unlike many instrumental as well as vocal works of highly chromatic idiom less commonand generally less severe Classical precedents of courseexist! in which the primary tonic is very richly expanded-its prevalence even impaired-at such crucial points, or at great length in the forms internal stages.

It must be emphasizedthat the sketch belowthe Brahms quotation is contrived to show tonal, not merely harmonic, succession.

Essential and auxi/iary //'near functions of pitches and pitch-comp/exes, the/T hierarchic basis often determined tonal and cadentia/ factors

by

A central premise of the present study is evident: harmonic and melodic analysis is,in an aspect complementaryto that of concern with tonal function, the interpretive identification and evaluation of auxiliary and essential linear functions which make up the stream of successions. While of course not the only approach to harmonic and melodic analysis, norits only important aim we might note, for example, the importance of study of style characterization!, the identification and evaluation of linear functions are nevertheless basic. The identification of some harmonic or

melodic factors

as essential

basic! and others asauxiliary elaborative! is not a matter of absolutes.The conclusions reached,based upon a complex of factors less objective than those by which tonal function is determined, constitute an interpretation of linear .functions. Linear function is strongly conditioned by tonal function as well as, at lower levels

or in nontonal contexts,

rhythmic prominence,

reiteration, stress,cadential position, and like characteristics ofevents. There are, of course, someinterpretations which are readily seen to be ill-founded while others derive from persuasive logic.And in the midst of such extremes many analytical interpretations represent a range in which right and wrong are in somedegree inapplicableconcepts. Onlyin some instances is there reasonfor insistence upon a right interpretation to the exclusion of others.

In the identification and analysis ofessential harmonicfunctions weare inevitably much concerned with the analysis of cadencesand their relative strengths. Cadentia]events areinvariably of fundamental, essentialfunction at somelevel, thelevel ofsignificance dependingon thestrength ofthe cadence in relation to others, its strength a product of such subtle cadential qualities as registral placement, distribution of harmonic factors, metric prominence, linear approach to the cadence point,and especiallythe positions ofthe cadential harmony in relevant primaiy and seeondaty tonal systems and the place ofthe cadence

62 tonality

in form conclusive asopposed to preliminary!. A predominant preoccupation of harmonic analysis is thus the analysis of cadence-its functional identity and character, the motions toward it, its relation relatively affirmative, tentative, etc.! to cadences which precede and succeed it. These are vital, crucial considerations in all harmonic analysis of whatever nature or premises, andwith respect to whatever musical styles. In very much conventionally tonal music, harmonic analysis will reveal a broad prevalence of the primary often triadic! harmonies: tonic, dominant, and subdominant i. e., I and its chief auxiliaries a 5th removed!, and especiallyof the first two of these.In such tonal music the vast majority of harmoniesdesignated asessential in the basicframe of structure must be I and V-the latter, when tonalmusic isviewed inbroadest terms, an auxiliary support andembellishment ofthe former, for which it is the principal medium of tonicization. Conventional tonal composition thusgenerally manifeststhe central structural functions of I, IV, and V; in fact, we tend to regard others as tentativeor allied forms of these. Thus,we think of and hear! II as functionally interchangeablewith IV having considerableoverlap of PC content,

and oftenoccurring infirst inversion with fl as bass note!; weregard VI, preceded by V, as a tentative substitute for,

I-a delaying

action which

brings aboutresolution ofthe leading-tone and introducesl but otherwise suggests furthermotion toward more conclusiveresolution. And we consider VII analogous in function to V-almost indistinguishable from V,, and III, often III,, as V-like in function because ofan overlap of PC content which

duplicates theleading-tone andthe 5 itself The centrality of I, IV, V in conventional tonal composition is necessarily acrucial basis for distinctions among relatively essential and relatively auxiliary harmonic events in the linear stream; and the discussions whichmake up much of this chapter are concerned with this underlying premise. While any and all dimensions andparameters of musical articulation are potentially signiiicant in the interpretation and eH`ectof essential and auxiliary functional distinctions, precluding any simple,general ranking and accounting of such factors, the following broad observations can be useful if they are not read as absolute and universal in their implications. ! Cadential harmonies areessential inthe linear functional hierarchy, although at relatively broad levels ofstructure weakercadences areauxiliary to more affirmative, tonally and formally more decisive cadences towhich they are related.

! The

primary tonic and dominant are very often, in conventionally

tonal music, the essential substance of the structural

frame. However,

indi-

vidual tonic and dominant harmonies can, in low-level rhythmic formulations analogousto thosein Ex. 1-15, functionas highly subordinate auxiliary events.

tona/ity 63

C 'ESM-tt EI I Y' iV

! Many

V

factors extrinsic to actual, specific PC content can at lower

levels affect, even determine,

the structural

value of a harmonic or melodic

event: e.g., stress, reiteration, duration, etc. Ex. 1-15 is pertinent to this point.!

! In nontonal styles,such factorsextrinsic to actual PC content, and the factor of cadential occurrence, areof increasedsignificance. The general superiority of tonal primacy, where tonality is relevant, to other manifestations of structural value can be illustrated and empirically demonstrated in the imagination or in improvisation of a frequent situation in tonal music: that in which a dissonantfactor a dominant 7th-chord, an appoggiatura, a dominant triad over tonic pedal, etc.! is of great relative duration i. e., of decisive agogicsuperiority! in comparison withits cadential resolution. No ear would deny the superior structural value of the higherlevel resolution, the ultimate aim of motion, the medium of absorption of preceding dissonantenergy and intensity. Finally, all evaluation oflinear like tonal! functionmust restupon evidence intrinsic tothe particularcontext inquestion. The first prelude of Bachs Well- Tempered Clavier, Vol. I Ex. l-16! illustrates harmonic prolongations and auxiliary embellishments. The issue of distinctionbetween superiority of tonal and linearfunction andmetric accentual! valuemust beintroduced verytentatively atthis point, to be developed further in Chapter 3. We are not, in the present context,saying thatmetric oraccentual value necessarily inheresin thechord ofcadential resolution by virtue of itstonal primacy.In Chapter3 we shallwish tomake afirm distinctionbetween metric structure, inwhich unitsare delineated by accentdissonance, density, duration, stress, etc.! and the kind of structuralbasis whichis our concern in the present chapter.In the present frameof reference,the superseding cadential eventultimately I! is thesupreme tonal-harmonic-melodic factor uponwhich all other eventsof pitch and pitch-complexare suspended in elaborating, prolonging actions.

But it is notnecessarilymedium a of metric accent,or a delineator initiator! of metric

unit. Indeed, within the elements ofpitch content the primary tonal event I, or 1! is the object ofrecessive action. We shall take theposition in Chapter 3 that articulationsin which elaborating PCand pitch events areseen as grouped infunctional associations around more basic, structuralevents interact,as do purely formal-cadentialgroupings ofevents, interestingly andvitally in musical structurewith often counteractive delineations of metricaccentualb' pnyected! units and unit relations.

64 tona/ity Ex. 1-16. Bach, Prelude No. 1in C We/l-Tempered Clavier, Vol. 1!. Representation of linear andtonal functions. Sequential patterns, each repeated once

m.1 ,-

_ --.,I_"

m.19

i/% -é

Q 7%

VQ

TI E

: II Q 92

Q ..

if-`: Q

ti 92.Z T!

_;_.:;_|. 1*r-PP : Q.,_IV p '

»1

R ~ .' _

. ~_,

E 92__'

Xp: `/1

'2

/

§92_di "_llrl!!!_ E

E_ 92-' _aj !` ' Z '_

i C T!:l V!

I vi

v/v v

ii/V! G D/T!: IV ii V

v ii

I! ii

Vl

II [d st/T!: N vii° i] representation of multileveled tonal function

*Note bassdescending diatonic scale cl-c!.

I i_. AC-I ze; I uii v

iii f I-ll _iS II

ES; 4 11 I' I_

7 U*/V

I_

ii; Y' 4 Y'

iii 1'

___l7_i' 4 --_ UI

____ ___ --I II

»

Ya _lf I__-

m.3 --.4 1' Y. 1' 7

_IP2'GT' Y_

Lia ei ;__,,»r, ° Yi o' 3, I

--- ____ _Y I' _I/ 2 I

_ I.

' 1:

17-9;"|' 7_9;"f

92-/ |57

.

tonality

m.7

m. 11

m. 15

66 tona/ity Ex. I-16 continued. m.l7

II

2° I

. J ."J;'ilg ° ` The function of iii and V, in the prolongition-embe1lishment ofthe 25

opening Iis particularlyclear inthe voice-leading, in the easy adjacency of pitches of the two auxiliaries. Reading ahead fromthis relatively brief pro-

longation whichserves to establish the primary tonic!,we cansee thatthe ultimate aim of the ensuing succession is the toriie of m, 19, toward which

there isgeneral descent, made visiblein the analytical sketch,The factof linear descentsuggests a melodic recession complementary to that of the tonal-harmonic coursetoward I.! The diagonal lines drawn through the bas; of the Sketchindicate the succession ofpassing auxiliary harmonies linkingth; essential point; at either extremity in this overall succession. Auxiliaryelements ofrhythmic-metric emphasis might be indicated by the symbol 1 , used in other, comparable

situations.! Twosequences the in descending, passing stream are bracketed and they of course represent ahighly standardized convention of the style. Harmonic content beyond m. 19 consistsof an expansive aflirmation of I

by a broad cadentialformula encirclingI with its primary harmonic supports: I V/IV!-IV-V/V-V prolonged

for nine measures, 23-31, eight of

these over the 5pedal!-I. There are two secondary tonal references,one to the dominant, the

other to the supertonic,the latter derived chromatieally,There arealso subsequent, ephemeral secondary references to the snbdominant region, e.g., mm. 20-21, and again to the dominant, m. 22.!

The romannumeral symbols show whathae. emerged as amost important factor in the theories ofanalysis developedhere; the deep tonal significance multiple, or multileveled, function! of harmonies, For example, the

tona/ity 67

minor triad on A, followed bythe C: V/V, is shownas viin the primary system, as ii in the secondary system,and as ii/V in the primary system; indeed, it has allQf these signifcances within a leveled perspective. The later D minor triad is both i in the secondary systemand ii in the primary system-a tonicized primary supertonic. The dual or multiple meanings ofharmonies in tonal contextsare thus here reemphasizedas a fundamental premise. The designation by roman numeral symbol of a harmony in two or three lights e.g., the Czvi, ii/V, Gzii! is not an inconsistency; it is a realistic representationof the range of functional significancesattending a harmony of both primary and secondary contexts of tonal reference.The identification of such multiple functions is a reflection of the depthof tonal-harmonic meaning and of the way harmony is heard in tonal music. The concept ty" multiple function isnecessarythe in fullexplication qftonal-harmonic signyicance. Tona/ order as an inf/at/'on of harmonic order

and succession

lt- is well to pause here to make explicit the concept of tonal order and hierarchy as directly analogous to, and an inflation of, harmonic order, hierarchy, and succession. Harmonic rhythm thus has its counterpart in tonal rhythm and all parameters ofharmonic structure their counterparts in tonal structure.The inflationary process bywhich IV or 92 VJ IV

becomes SD is oneof tonicization; fl becomes the basis not merelyfor a harmonic factor, but for a secondary tonal system expanding the primary tonal system.

Thus, there is an important sensein which, as a hypothetical example, a five-part rondo might be seenas an inflation extension, prolongation! of the contrived

harmonic structure

illustrated in

92 ,D/t: vii-I isdz I

Ex. 1-17.

I

68 tonality

An example of tonal expansion isgiven in Ex. 1-l8a. In Ex. 1-l8b, the succession oftonicized factors tonal events!can be seen asa broadly, richly inflated underlying harmonic succession. Ex. 1-18a. Chopin, Prelude No.9 in E. Largo *

...S /'T

|_

I'

- _ _lla _

_

""*""5§5§£5?£5§ =EEF-§E5§§5§é= 33 33 cresc. ------

_ - ,,_,,,q

lsrmlli / XBl|'ll!Il 'lil "I'

J°

-!

-oo

--

69

tonality

theless,it is IV of the primary system.The restoration of the primary tonic is accomplishedby enharmonic-diatonic successionseep. 73! in which a/:i acts pivotally as E:iii. Example 1-18b shows without fidelity to original voice-leading! the entire passageas an elaboration of E: I again with linear expressionof the factors of the prolonged tonic its root, 3rd, 5th!. The passingsecondary tonics are identified as embellishing tonal functions. Ex.1-18b. Enharmonic-diatonic succession

A I',a~/gg R/sd/SM /t m /T! F,f SD,sd/SM/t! C SM/t! F.,e T,t! .

The "vagrant" diminished 7th-chord is important in the progression: the harmony at the end of m. 7, notated with E!, looks back to the tonicized F through which the music has passed;notated with F>, it looks ahead as a dominant to A>.

Tonal fluctuation and techniques ofimmediate which itis effected

succession by

Tonal fluctuation is, of course, denoted by the appearance of accidentals, whether they represent shifts to primary, embellishing, or passing systems. tially tonic or dominant harmony may occur purely as a sonorouslinear factor having no significanttonal function. On the other hand, if a dominant form is suggestiveof a tonic prevalentin local or broadcontext,or if it hassufficientdurationalor other emphasis,it may havefunctional effectwithout associationin the immediatecontextwith the expectedtonic of resolutionor its deceptivesubstitute,i. e., without normal root movementor expected resolutionof the leading-tone.The conceptof nonfunctionalityis applicableto only low levels ofstructure, and thejudgmentthat a particularfactor is "nonfunctional"is, inevitably,often interpretive.

70 tona/ity

In theory, T: IV can be tonicized by I without accidentals, but this is rare

and, without7, notreally convincing.2"! Accidentals may, of course,appear l»

without tonal significance, as when the normal raised7 of the harmonic minor scale appears, whenpurely modal changes occur,when altered nonharmonic tones are used, or when there are cancellations of any of these alterations lackingtonal significance.The special concern of tonal analysisis the change or set of changesintroduced in the harmony so as to have the effect of making referenceto a new tonic. Alterations through which the primary system persists unimpaired in its hegemony i.e., those of no tonal significance! will be cancelled shortly after appearanceexcept whenthe changeis oneof more or lesslasting modal shift.

The identification of specificplaces wheretonal referenceshifts isbased upon objective criteria, while again the force of the shift is often a considerably subjectivequestion to some listenerseven themost elusivechange may upset anestablished tonalsense!. Toput the issue anotherway, the question of the existence of tonicization relevant at somelevel isone of substantially objective determination; the question of a hierarc/zic ordering fy .system components, likethe judgment that a tonic emergesonly parenthetically, is one in which determination is in some part subjective. Our definition of tonal_fluctuation willbe simple and inclusive: it is a change of tonal reference, secondary or primary in its implications, and however immediate in its significance. We will consider that a secondary tonic is as a rule supported by the appearance of its leading-tone i.e., is tonicized by a dominant form! or, occasionally, byother means", normally including but conceivably without the expected tonic resolution or its substitute, since the dominant alone may have strongtonal implications. Tonal fluctuation involves tonal events-harmonic successions irylated by tonicization;every tonicization is of tonal significance in some degree. Tonal fluctuation is thus significant with respect toa level of function: for example, a secondaryleading-tone may have purely foreground impact. Moreover, thesignificance ofaparticular event in the tonal structure depends in part on the environment in which it occurs. Thus,if a tonicization occurs Wl denotes the lowered seventh scale degree. b 2°The theoretical problem ofmodulation is, again, entirelyone ofthe level of analysis perception, observation!. Any changeof tonal significance-any tonicizationother than that of prior incipienceor prevalence-ismodulation in some degree ranging frompurely parenthetical tovery substantialexpansions of the primarysystem. Thesignificance oftonal shift isthus measurable in accordwith the level ofstructure overwhich itseffect isrelevant; it is true ofcourse thatthe retentiveinfluence ofa prior system, likethe extentof its supersession byan enteringone, isa partly subjective issue. °'I`onicization canof course,although lesscommonly, bebrought about by plagal action andeven, intheory, bydurational orother insistence on aparticular tonicform without supportof attendantharmonic functions.

tonality

71

within an atmosphereof fluctuation e. g., during the courseof what we shall refer to as accelerated tonalrhythm!,it has more persuasiveeffect as tonalchange than one that occurs within a stable closed! context in the tonal structure e. g., one that occurs over a pedal point, or one surrounded by persuasive referencesto an underlying system having predominance over a broader structural level!. For example, the tonicization of D/T at the end of an antecedent phrase will have significance in its cadential affirmation at the level of that phrase,while at the level of the entire period of which it is part the function of D/T will be seento be tonal embellishment of the prevailing T which the consequentreaffirms. Finally, this study continues to make the point that the analysisof harmony, including tonicized harmony, is ideally comprehensiveand multileveled, with the multiple interpretation and identification of harmonic function a necessary consequenceof the fact of structural levels in music.

Secondary or renewed primary! tonal references shifts, fluctuations, modulations! involve particular techniques of derivationWhile.such techniques are of less interest to many theorists than are the broad outlines of tonal

structure, we would argue for their importance, like the importance of many sharply felt eventsthat are of primarily local impact, in the expressiveeffect of the musical surface.How a tonal inflection takesplace the processcan be decisive in the quality of its expressiveimpact. Processes,or techniques, of tonal change can be classifiedin a way that provides a basis for useful inferencesand conjecturesrespecting such effect. A diatonicsuccession occurs between two pitch factors pitches, PCs or PCCs! which coexist in a single diatonic tonal system" not

necessarily the

primary system in whose context the successionoccurs, possibly even a parenthetical system. A diatonic tonal fluctuation occurs through a diatonic succession Ex. 1-19!.

Ex. 1-19.

Diatonic succession

~ IW ~C-F 30Forthesedefinitions,"diatonic A tonal system"in the minor modeis consideredto be the harmonicA scale,with the raised7, sinceit is this scaleform which expresses tonal function. The lowered7 is a melodicfactor tendingto subvertthe tonal senseby a leaningtoward the relativemajor.

72 tona/ity

A chromaticsuccession occurs between two pitch factors which do not coexist in any single diatonic tonal system Ex. 1-20!. It often involves a chromatic inflection G-Glt, B-Bb, Db-DU, etc.!, but this is not necessarily true consider an augmented-6th chord and its resolution!. A~chromatic tonal _fluctuation occurs through a chromatic succession. Ex. 1-20. Chromatic succession

;VJJ-gJ1inflection Chroma J

V` 92

VV

V

The hypothetical generic tonal system discussed earlier implies a possible, interesting theory of chromatic successiongermane to its apparent experiential effect. That is the theory that chromatic succession, whenof tonal significance,is in effect tonalelision Ex. l-21!. Thus, for example, the Ex. 1-21. r_*-f0l1iCS'ii| tonic ,c

Q 1 Q tg

*I TD

D/D D/D/D

tonic

z an

*I T D/D/D 92 tonal elision

,

modulatory processby which SM/T = D/st/T, etc.! is derived, when one of direct chromatic succession, maybe regarded as an elision of the process T-D-D/D-D/D/D = SM!. This theory of chromatic fluctuation proposes that the fact of omission of stages terms! in a potential diatonic series of underlying relationsin somedegree accountsfor the effect ofapparently wide tonal thrust in relatively little time. Whether relatively distant regionsemerge directly chromatically! or gradually from T is an important facet of style." Since very nearby regions can bederived chromatically e. g.,approach tothe SD of C by theinflection B-Bb!,the presence of chromaticsuccession does not of course initself imply tonal distance.

tonality

73

An enharmonic succession is one in which enharmonic equivalence pertains and enharmonic change occurs, explicitly or implicitly. It the succession! may be diatonic or chromatic, so that it is proper to distinguish between an enharmonic-diatonicsuccessionand an enharmonic-chromatic succession.Where

tonal significance is involved, these are further techniques of tonal shift." In Ex. 1-22a, diatonic relation pertains between the two harmonies, assumingthe necessaryenharmonic interpretation; in Ex. 1-22b,a chromatic nondiatonic! relation pertains. Ex. 1-22a, b. Enharmonicdiatonic successions

e7 Note: x p

el»

g!'.

y an enharmonicrelation; z a diatonic relation.

Enharmonicchromatic succession

atic inflection

+

E7 Note: x p

z E/y ~ p an enharmonicrelation; z a chromatic relation.

ssAnalysts sometimes speakof an "abrupt modulation"asa classification of technique, but it mustbe seenthat if the term is construedto denotean "unprepared"entry into a new tonal region, the conceptis contradictory and illusory. While a modulationcan when associatedwith the entry o
tona/ity Ex. 1-24. Schubert, Quartet in D minor forstrings "Deathand the Maiden"!, D. 810,first movement. Allegro m. 123 J. :;.-. ' 7./°9292 |/__ gp 92

,_ s

H rz,r-ci_|= 3 :pgs ,.

» l.

u/

92 92 -V

PP dzm. :L-

o= _____f 92 °V PP ° lIT92 _ Yf 1 "'?', ° 1. =|.§:? Zllfl

=ii¢l I

o

pp dim. 'Q A A i X' g=

A- 5'

_ dzm.

lu i92

#1 1

``

?Ȏ i

_!

_ Q l,r'l1 _ gr

g IQQ:

-lT _ -lrI=? §'7_g'

e

PP dim. Enharmonic-chromatic succession

| f ul ._

92 _ Ulf] 21. llw .4-|»| i,

gy " 92 Y

vI V1=Ge*

. S#/.»__-» *The German augmented 6th on iv! in gil/ab is enharmonicallyequivalent to a:V-; more locally derived as a diminished 7th on 2 in F!; the augmented6th, with its resolution, alwaysconstitutes achromatic succession.

Tona/ organization as a pattern of re/at/ve stab///'tyopposed to re/at/ ve f/ux

While it may be obvious both in principle and in application, it is well to make a particular issueof the importance of the conceptof tonal structure as alternating between,and opposingin various ways, areasof relative stability and areasof relative fluctuation. It is vital too that the concept be extended beyond developmentalas opposedto expositorypractices ofthe tonal period, despite the significance of these practices.Thus, applications in earlier and

later practices are relevant:in a modal form,a series of phrases culminating on thefinal, ascompared withothers expressing fluctuation toother degrees; and in late stylesof the tonal period and in the twentieth century, the delineation of tonal structure by projection of, quite apart from specific tonics, areas

76 tona/ity

of relative instability ambiguity, lack of clarity, nonresolution! contrasted with areas of relative stability focus, clarity of tonal direction, resolution!. Some examplesin this chapter pp. 59-60, 142-47! illustrate procedures in which tonal structure derives very importantly from the ordering of successions whose overall function is gradually to bring the tonality into focus, or the reverse ofthis, with the final cadence left undecided in opposition to clear tonal orientation at the beginning.Of course,we mustalso beconcerned with the comparable opposition of stability and instability as a critical shaping elementin the music of Classical tonality. Consider, for example, a traditional form in which the principle of alternating stability and fluctuation is a fundamental point: the eighteenthcentury imitative invention or fugue. It is in the nature of the episode in the invention-fugue genrethat it is fluctuant see Exx.1-25a,b!, often serving to bring about transitional tonal references in preparation for the level of a following, stable thematic statement and, usually, its imitation!. That an entire form can be conceived asderiving in an important sense from this principle as well as from the principle of ultimate tonal rounding or reaflirmation of the initial, primary system! is evident in examination of any instance offugal procedure. One has only to consider awork of this kind as auditory experience to be vividly aware of the fundamental significance of the alternation of tonal conditions, and a further look into the areas of fluctuation normally reveals a number of elementsfunctioning in complementagr support of the organic principle of tonal fluctuation: for example, an acceleration in surface rhythmic activity, or the complementary activating force of imitations at relatively brief intervals

of time.

An example of Classicaltonal procedurewill serve as abasis forfurther techniques ofanalysis andrepresentation of tonal structure and system,and tonal processesof expansion,elaboration, and fluctuation. It has been notedthat the fugal episode,like transitional and developmental passagesin tonal forms, can be conceived as an area of accelerated tonal fluctuation. Tonal successionand the fluctuative processare illustrated in two episodesfrom the Mozart Fugue in C for piano, K. 394 l-25a,b!. The episode quoted and summarized in Ex. l-25a reveals a passing stream of harmonic successionsinvolving chromatic relations-transitional and auxiliary in linear function but nevertheless oflocally signyicant tonal TWTence. Theanalysis showsthe operation ofa trileveled tonal structure in which system componentsare representedby symbols indicating their relations to T-relations not merely theoretical but of demonstrable contextualvalidity. Although it is not carried out thoroughly, a multileveled harmonic analysis is indicated, onein which it is suggested that,for example,the D major sonority

tona/ity 77 Ex. 1-25a. Mozart, Fantasy and Fuguein C for piano,K. 394. Andante maestoso m. 9

i,_g-I-EIT1-_1 l;Y'Q! f.i §Y'l ITT'-ll Fil l:Y'l,Ll#.Ll ¥~1="' "*f'

'.I~F{f'111-_lu 'TIT'

1

"1 i

!!I!

alili- §.J'l

=lY g

i=é;Il|||l: =

Passing, locally functional, chromaticallyrelated harmonicevents: -; ,-§ 1 /LY. ll.. mu!' i

| /_` 92 'Y ll `"5 i H7-11-1 nl 1,

U1 0 'A HEI

la T

I il

ir? %!'

/

. -1 92 : ' » i*7

L2 iz |+*1

C T!: vi a r!: i

ii IV/vi

V V/vi

II

iv IV

; D/T!: xg, V 'vii° I

I

iv V A5!

II _ vii?

d D D'd/D/T!'

i vii

I IV?

A6! Concept

._

of hierarchic

tonal systemin operation Concept ofmultileveled harmonic function

can be and must be! heard in vagfingfunetionalimplications as it is related more broadly orloeallv tohierarohiealbr ordered operational _systems relevant tothe example. A second quotation from the Mozart Fugue Ex. 1-25b! reveals a comparably fluctuant environment reflecting again a multileveled tonal system. But here the process is entirely diatonic:

i. e., the successions are

diatonic first in SD, then in SD/SD. At the sametime there is an element of chromaticism in

evidence in the function and relative distance of one nondia-

tonic tonic:Bb, SD/SD.! As in Ex. 1-25a, the tonal references areof swift and ephemeral effect,serving primarily to fill tonal space and achieveforward linear motion with factors of harmonic richness beyond the primary tonal

78 tona/ity Ex. 1-25b. Mozart, Fugue, K.394. m. 19

_f1i;;1 §l §--lT7'l I " ll._1'I " l92l|_{

J

`I

/ "_lf] I'-I___ I 1 I

1|,F'| L5 ' HY'

~"

v. II|9292 .l1_l.l X11-_--1-_-11 Iii-.J-ILIU 1'l'1"/_iii-_-Z2--Ql'?7Y._-

_ "92 | 1 .f"l li -if IZY .___ nv-

'V

-I

1- »

Passing, locally functional, diatonicallyrelated harmonicevents: 92

`A

I £1 *Il

`J

/I

_1

/

92_/' ;_Z, C T!: wiv msn/T!: v

IV V/lVlV 1 vii#

nb SD/SD/T!:

V vi, ii

wiv ivv

vi ii?

v 11

iv viii;

Nondiatonic secondary tonic as elementof chromaticism

resource; yet, their implications are locally felt as expansions ofthat tonal resource, and thus an important subject of inquiry. Without such inquiry neither the directions of expansion of the primary system nor the meansof such expansionwould be apparent.

Tona/ rhythm

What we have statedin the preceding section,put another way, is that it is fundamental to tonal structures that they are characterized by controlled distributions of changing tonal reference, that the time intervals separating tonicizations contractand expandin functional ways. Thiscontrol of the rate of tonal change constitutesan aspectof the rhythm, and rhythmic life, of a composition. Tonal rhythm, characterizedat active, developmental stagesby substantial acceleration with surface rhythmic drive and other elements often acceleratedand intensified in complementary actions!, is an important

tona/ity 79

part of organic substancein music. In this sense, wecan seethe traditional principle of development transition, episode, etc.! as, in an important respect, one of tonal rhythm. See Ex. 1-45 and the attendant discussion.! The strength definiteness, clarity! of a tonicization is certainly an aspect ofits rhythmic function; in other words,if a tonal referenceis a rhythmic event,it must be true that the degree ofinsistence inthat event, what one might call its accentual force, is a necessaryfactor in tonal rhythmic#ct Again, onehas only to think of the acceleration intonal fluctuation as experience. andof the extent to which that experience isenforced incircumstances in which tonal changesare strong and emphatic. A tonicization is relatively strong if it is effected by devices that lend clarity that carry the ear along without equivocation into an emergent tonal region!; other factors in the accentual strength of tonicization are the degree of distance traversed the extent of change representedby a given tonal event! and the duration of the predominance of a systemcomponent. Of course, emphasisof any kind of timbre, of texture, etc.! is relevant here.

The questions of Iona/ /htersection, d/rect/'on, and Cl'l;9I3/`!C6,` /hterva/s of fluctuation; h/gh-/e ve/ chromatic successions and nond/aton/'c tonics; the /7716/'C/73/lgeab/'//lyof modes and the "equ/'va/ence" of para//e/ tonics

We havereached, with the needto discusstonal distance and direction, a particularly critical and difficult stage in the theoretical treatment of tonality. For while a concept of tonal distanceis at least inferential in much of our thinking about tonality we speak oftwo systemsas closely related-e. g., C and G!, the evaluation of distances betweentonal systems is a very complex problem. The tonal distances traversedin a given musical form are surely of great significancein structure and style, and -inexpressive effect.While it is doubtful that any absolute scale of relative tonal distances e.g.,list of systems relatedto any given systemin the order of distance fromthat system!can be achieved, we can make some useful observationsof certain criteria by which a theory of tonal distance would be informed. One problem, of course, is: What does onemeasure? Forinstance, if accessibility ofone systemto another through diatonic harmonic succession is a reasonablecriterion, we must recognizethat the outline given in Ex. 1-10 a theoretical chromatic tonal system of hypothetical, diatonically nearest, 33Tonal rhythm is comparableto but not to be equatedwith harmonicrhythm, which may or may not be parallel.

tonality

80

relations! showsthat all of the chromatic degrees,even as roots of major and minor triads, are fairly readily approachable in tonicizations by diatoni'c succession. But that has to do only with process i.

e., how do you derive a second

system without recourse to direct chromatic succession?Another basis of criterion must be the question of urhereyou are the extent of difference betweentwo systems.This questionis probably more crucial in the experience of fluctuation, sinceno distanceis significantly inaccessibleeven diatonically, and especiallyallowing such common nondiatonic devicesas the Neapolitan harmony. The criterion of extentof changebetween two systemsis probably decisivein the "effect" of being at a given point tonally "remote" from another point. The following is an outline of somebasesof criterion and measurement of tonal distance, letting system,for the presentpurpose, mean diatonicsystem in exclusionof evenvery common chromatic factors. Obviously, tonaldistance is in somesense a functionof theextentof intersection between diatonicPC collections of tonalsystems. What is problematic is that tonal distancecannot be conceived merely as the extent of such intersection; the particular factors which intersect

since some are more crucial to tonal expressionthan others! are of necessary consideration. l! The extentof intersectionbetweentwo systems:hey signature Corre-. spondenceof key signature is one indicator often cited. If the key signature indicator is to be of use, it must account for the raised 0 of the harmonic minor scale, since this is vital to tonal function. Moreover, the "distance" of

three Hats or sharps between signaturesof parallel keys e.g., C and c! is surely in an important senseillusory. ! Theextentof intersection: scaleandprimarydegreesOne eas.ilymeasurable factor is the extent of intersection between two diatonic

scalar sets of

PCs; yet, it must be that the lack of intersection involving such fundamental

degreesas 1, 5, and 7 wouldhaveespeciallysignificanttonal consequences. ! The extentof intersection:triadic diatonicsystems,and primary triadic functionsAgain, the .intersectionof diatonic triads coexistingin two systemsis easily measurable; but the intersection involving those of primary tonal lfi lfie e 1 t 1 ly ltl 1. The 1 te etio IV=V ll hl fi, y,G dfil, ldp +hlye tlt tel thl "1

" ltl

th

ldthelot

tlo

linking, say, e and C. The foregoing considerationssuggestthat T and t are very closely related systems,despite the considerabledifference of signature. In fact, the crucial correspondenceof fundamental functional harmonic content I,i; IV,iv; and especiallyV! suggeststhat in practice it is reasonableto consider

tonality 81

T and t as distinct modal forms of the sametonal .gistem and to distinguish sharply and fundamentally between tonal change and modalchange. These observationsalso leadto intriguing reconsideration ofthe issueof distance betweenso-called relative keys in practice of the tonal period, a problem to which referencewas madeearlier. Thus, T and r are very similar

in signature there isone importantdifference ifthe inflectionof 7 is taken into account!, and an analogous situationpertains in the relation between t

and R, whose signatures are alikebut for the crucialquestion of7 in the minor mode.

This is the basis for earlier

comment that

T and

D are sig-

nificantly more closely allied than the pairs T, r or t, R. F luctuation in the direction of R from t, a commonplace inconventional tonal practice perhaps deriving from modal traditions," has far more drastic or emphatic effect as tonal change than does the easilyinterchangeable succession T-D. In summary, the issueof intersectionis fundamental,provided accounting is made of the particularfactors which do and do not intersect. One such factor is of courseV; and the importance of harmonic intersection in evaluation of tonal distance is further evident in the likelihood that we experience greater remotenessfrom a system whenthere is fluctuation to another tonic having nondiatonic relation to the first-i.e., when there occurs a high-level chromatic succession between two tonics, even though the second may be derived in a low-level, step-by-step diatonicprocess. ! Diatonic accessibiligf ofthe second V, or aV preparation in thesecond system, a particular, vital aspectof the question ofintersection. Tonal distance might be conceived in terms of the capacity of one system to yield diatonically without recourse to direct chromatic succession!the V of another, or the V preparation. That is, when V of one systemis containedin another, an especially significant kind of relation pertains; and the intersection involving a potential V preparation commonly I, IV, or II! is a significant relation not much removed. Chromatic necessityin derivation of the critical V implies and conveysa senseof distance.The issueof diatonic intersections involving V, or common V preparation, is summarized in Fig. 1-3. Here, the summary takes as an assumption the equivalence of enharmonic relations and tonal The problem more acutelyconcerns T-r R-t! than t-R r-T! since in the latter process R:Vis immediatelyaccessible from t:i e. g., azi,being equivalentto Czvi,can introduce C:V without chromaticinfiection!. No doubt a better understandingof the evolution of major-minor tonality especiallyof seventeenth-century music! would illuminate bases

and origins of theconventional, standardized practice of fluctuation between relative keys. Three factors must certainly be ! the ambivalence of 7in evolution of theminor modeits

lowering inclining t inthe direction of R!, ! the prevalence 'IZ-with of same the resultin avoidanceof the awkward melodicinterval of the augmented2nd, and ! the attraction toward potential tonal function in the relationof ii°-III as secondary vii°-I! inherent in the diatonic aeolianmode.

tonality

fluctuation in conventional styles often proceedsin 4ths and 5ths. When that happens,it is useful to distinguish further betweenplagal actionsthose in which successionis a 5th down or a 4th up i. e., in which one tonic is, at least in this important sense,dominant to the succeeding tonic! and the reverse. That is another way of referring to the critical V intersection cf. C ~ F and C ~ G in Fig. 1-3!. ! The numberof termsin any harmonicdiatonicsuccession leadinginto the second system.Here, referenceis again made to Ex. 1-10. For example, the relation of T to D or SD might be conceivedas one of two terms there is an immediate diatonic relation between them!, but T to SM of three terms [as

in the diatonic successionT-st-SM in which st = sd/SM!]. In the discussion attending Ex. 1-10 a number of variables are noted, including the fact that this basis of measuring tonal distance accounts neither for relative frequencies of diatonic triadic functions nor for possibilities of succession including such common nondiatonic functions as the Neapolitan. All of the above must be understoodas a setof observationsgermane to perhaps in the direction of, but far from constituting! a theory of tonal distance. The concept of tonaldirectionis related: theorists speakof the "flat direction" counter-clockwisein the circle of 5ths! and "sharp direction" clockwise!; as suggested,for example, in connection with fluctuation in 4ths and 5ths, reversals of "direction have significant implication in structural function in music and, often, in the perception of tonal distance as well i. e., the nature and extent of intersection, and the severity of change!.

Concepts of tonal progressi on and tonal recession To the extent that distance and direction can be inferred on the basis of

stated principles, we shall regard tonal fluctuation away from the primary tonic asprogressive and fluctuation toward the primary tonic as recessive. The conceptsof tonalprogression and tonalrecession are thus opposed tonalsuccession applies to both tendencies,denoting change without accounting for relative distance!.Theseconceptsare in keepingwith a fundamentalidea of this book: the idea of progressive fluctuationin thedirectionof intensifyingconditionsexpecting resolutiondissonance,complexity, ambiguity, instability, distance, acceleration, etc.! opposed to that of recessive fluctuationin the directionof conditions achieving or tendingtotvardresolutionconsonance,simplicity, clarity, stability, proximity, deceleration, etc.!. These concepts require the assumption that in any particular tonal structure, if there is fluctuation," there are points of maximal tonal distance and "best" tonal stability the latter, at broadest .Where tonal fluctuation is lacking, there is harmonicfluctuation of restrictedbut analogouseffect.The conceptof harmonic distancefrom I! within a singlediatonic systemor

tonality

85

levels, usually terminating! against which tonal progressionsand recessions are evaluated. A tonal structure is thus seen as fluctuant between conditions

of tonal "dissonance"and "resolution" distance from and proximity to the primary system.The progressiveand recessiveactionswithin the tonal structure may conform to have complementaryrelation to! thoseof other elements, but the tendenciesof concurrent elementsare often opposed in compensatory, counteractive relation!. These tendencies and interrelations of elementactions, of which tonal fluctuation is one, are regarded as fundamental to structural function and expressiveeffect in music. While spacedoes not permit their quotation, we may refer in passing illustration of theseconceptsto two songsof Wolf, both characterized by the unusual condition of tonalfluctuationat thebroadest level On.eof these,to which referencemust be made without quotation, is Seligihr Blinden No. V of the ItalienischesIiederbuch,Vol. I!, which begins in E/ and concludes in A/. Surely the tonal relations so describedconstitute a tonalrecession overall from D to T rather than, say, a progressionfrom T to SD!; but the tonal structure is complicated by the fact that E> yields to more distant regions G> R/d?, and B! D/D/T!

before its role as D/T is directly fulfilled. Thus, the tonal

structure is first progressive,then recessive;the path of recessionmust by any reasonableestimate of tonal distance be seento begin with the reference to B>, m. 16. The function of E! as D/T is first clearly suggestedin the appearance of E>:I as a dominant form with D>! in m. 19, after which it undergoes elaborate and extended prolongation before T arrives in m. 27, itself then prolonged and embellished in a fourteen-measure,strong verification. A Wolf songof comparably recessive,active tonal form is Der Mond hat cineschwere Klag' erhoben, No. VII of the samevolume, also unfortunately discussedwithout reprinting. Its three tonal systemcomponents whosecenters comprise the notes of T: I.! are shown in Fig. 1-5 with their relations to T

Fig.1-5. Outline of recessivetonal structure in Wolf, Der Mond hat cine schwereK/eg' erhoben No. Vll of ItelienischesLiederbuch,Vol. I!. Recessive tonalsuccession:e/

s>/f >

C/ = components of C/:I

expandedsystemis of courserelatedto the issueof tonal distance,and is implicit in standard theoriesof commonpracticeharmonic successione. g., VI-IV-V-I! as well as in observed probabilitiesof succession and cadentialformulation in modalsystems.The inflation of such a principle of harmonicdistancecould provide a further basisfor measuringtonal distance note the proximity of V, I as comparedwith that of D, T!.

86 tona/ity

adduced onthe basisof plausiblecontextual function. In describing the overall structure as recessive, the systemon Gb is, in view of the tonally powerful I = V intersection, regarded ascloser to Cb than is eb.

Concepts of harmonic and me/od/'c progression and recession; comp/ementarit/'es and coun teract/'ons of e/ement-success/'ons

Melodic and harmonic successions are likewisecapable of classification asto progressive andrecessive orstatic! action. It follows that an attempt at such classification, on a theoretical basis subject to empirical verification, is a vital necessity once a premise is undertaken to the effect that harmonicand melodic changes are generally not neutralwith respectto theintensity-release scale of expressive efizct. Of the following observations,some arerelatively conjectural, some are easily demonstrable, and a few are manifestly reasonable and even selfevident.

! Progressive harmonic action, action in the direction of increasedintensity, might be summarizedas: a! Action away fromI to V, thenceat somelevel to II, IV! ; thesuccession I-V-II IV!-VI-III can be regarded as oneof increasing progressive tendency providedthe closeassociations ofVI and I, and II

and IV, as well as in certain contexts III and V,

are taken into account whenever pertinent; a general consideration of progressive actionaway from I, in the sensein which it is noted here, would of course be of relevance particular to conventional tonal systems,and the concept away from I in freer, or other particularized, tonal systems requiresadaptation to the terms unique to such systems or individual works. b! Action in the direction of increased dissonance ry' an implicit kind increased dissonanceof a tonal kind would presumably be synonomous withincreased distancefrom I!. c! Action in thedirection ty" increased densigf and ofincreased spatial jield a concern of texture explored in Chapter 2!. d! Action in the direction ofacceleration of harmonic rhythmnoted in Chapter 3!. e! Action in the direction ofmore intense coloration, or of relative intensity within other cofunctioning elements. ! Progressive melodic action,action in the direction of increased intensity, might be summarizedas: a! Upward succession, which in intensity is increasedby the extent of leap i.e., rate of ascent! and by the duration-span of continued local or essential ascent.

tona/ity 87

b! Action away from f-a consideration of tonalbases eff the above criterion of harmonic action away from I!. c! Action in the direction ofincreased dissonance of intervallic relations between pitch events at a given level of structure. d! Action in thedirection cy" acceleration melodic of rhythms or moreintense coloration, etc.

The conjectural observations listedabove haveto do with the specific structural parameters dimensions! of harmony and melody, respectively,or with related, cofunctioning element-actions. In the characterization of certain successionsas progressive and others opposite to those listed! as recessive, variousrelevant factors must at some point be regarded independently. Factors such as those listed do, in fact, Wenfunction inparallel com-

plementay! ¢Wct-i.e., a melodicsuccession may move awayfrom l and upat the sametime; but it is obvious that a line may berecessive tonal in fubillment and progressive in intensmwing upward movement at the same time, andsuch counteractive tendencies are in fact very commonin music and of important paradoxical effect. Similarly, for example, in a harmonic successionthere may be concurrent, complementary

effect in

parallel actions

toward increased

disso-

nance and away from I; but the reverseis common-e.g., harmonic action toward I but involving intense, implicit dissonance in often prolonged! V on the brink of in very close proximity to! I. In such a situation, the counteractive relations of coincident element-actions areof veg:powerful qfect indeed, anddissonance oftenincreases progressively e.g., V-V7-V,! in the course ofprolongation. What we are talking about is of course tightly related to conceptsof tonal progression andrecession notedearlier, except that considerations herehave to do with possibilities inherent in a purely diatonic, unexpanded tonal system. One can readily hypothesizeor cite further situations in which distinct melodic and harmonic actions which are coincident are of complementagf effect or of counteractive effect: for example, onemight imagine, cite in music, or improvise an ascendingmelody over static harmony, or any comparable or reversesituation. While observations inthese studiesare not always made specific with respect toprogressive andrecessive tendencies, the implications of such expressive actionsshould always be kept in mind.

Some examp/es of quasi- Zona/ order in me/od/'c and composite functions

Melodic analysis of initiating subject statementsis shown in extract from a setting of the Deprofundis textby _]osquinEx. l-26a! . Three of the statements are seento fluctuate, essentially, betweenthe final and cofinal of the mode, as would be expected, while one of them-that of the altus-is comparably structured at the level of the 5th above,the cofinal G its most essentialpoint.

88 tonal De profundisc/amavi Motet!.

I*J92_§JJ 15 VHQE' _"

@ I-I

- ~I

5% I

'I '

'I

5,1 JE;

_II

§rI'I9b'IJ @§Lr__I I" I9 I

I 'I 'E

I'I

-I.H3I%

JJ.I-

--

V5 r' ma -

vi,

ad

If I

- vi,

ad te,

V

IV

De

pro - fun

J92

.IJ

is

cla - ma

I

I cla -

I HI

tona/ity 89

m. ll

gr'-' E'1.f' te Do

----- ml

- ne;

E

@1f-Vg-EVV @,!li£E @;m,¬i ,K , P*1'VVVE'ftf Q5 JQI

SS31 upcdu _[E ; Jw

J l i in lil*

90 tona/ity

The relatively tentative characterof the superius statementis evident in the primacy, within

its own terms, of G. The other three statements have, in com-

parative melodic structures, relativelystronger aflirmationsof C, the final, as essential basis. They can

be seen in a composite sketch as reflecting very

clearly the V-I motion

Ex. l-26b! which further underlies the entire

Ex. 1-26b. Josquin, Deprofundis. Composite sketch of initial entries showingessential motion toward C "l"!. | e /"`

0_

_, 0

_

¢

|| `92_i_, §

92 _ /T?! ,|-|92 ; |I

92`____

V RI ____/v

composition. Inthe composite sketch onlyessentials, degrees l, 5, and Q,are represented, with the auxiliary neighbor and passing indications of the melodic reductionsof individual subjects omitted for a more fundamental view.

In the Berg melody Ex. 1-27a! certain notes, by virtue of tonal conditioning, actual resolving motion, and/or subordinate status in relation to more essentialmelodic points to which they are proximate and rhythmically oriented, are of clearly auxiliary function. Some are of especially strong tendency. Thesecond note,dbz, isapproached likeand resolveslike the classical appoggiatura. The subordinate g at the end of m. 2 is a neighbor auxiliary to the prominent ab preceding; it has resolution in the following measure, butthen reappears,lacking resolutionexcept aspart ofa descending passing group.The fb of m. 3 expectseb which does occurin the piano!, but it is releasedunfulfilled an important aspect of expressiveeffect!, even in its second appearance at the end ofthe samebar, where the senseof nonresolution is intensified. Its felt relation to Eb has to do with the prior appearance ofeb in the line, and the impact of an Ab tonic expressed in the clarinets first

measure.

Although rhythm as suchis not at this point a matter of attention, we can briefly note how ideally rhythmic effect complementsthe melodic structure. Essential points have agogic emphasis al»2, m.2; el»3,m. 5! as well as emphasis byrepetition mm. 3-4!. As the line ascends toits high point the

.92 tona/ity

exposed pitchis the a, or bb b, of m. 4, projected asa subsidiarylow point of some prominence,awaiting resolution.! The descent of the line finds compensation for greater distanceand fastermotion in more conjunct succession in which even 3rdsand 4ths give the impression of steps in a sequenceof motion-stasis-motion ordescent-plateau-descent. Ex.1-27b. Al»=°°|"?

g I,E T T1f . .°iif-1 '-1 VI

2P

___ ____________ __

A synopticsketch oftheBerg melody,with auxiliary notes orientedtoward the basicoutline, revealssome persuasive factors of unity and ambiguity Ex. 1-27b! .What emergesmost strikinglyis atriadic basison Ab,with a two-octave ascent to eba!, the high point occurring almost exactly at midpoint, then a descent of approximately three octaves in accelerated, recessivemotion of line. Ambiguity is expressedin the conclusion, which is ironically contradictory to the established tonal implication, arriving at the structural d, manifesting a tritonal relation with the otherwise fundamental Ab. The tonal structure of the line is thus open. The sketch suggests arelation between that isolated al bb b! of the clarinet m. 4! shortly before thehigh point and the pitch notated as bbb in Ex. 1-27a! appearing later as upperneighbor to the essentialab; the earlier is interpreted as an anticipation of the upper auxiliary, unresolved until the subsequentmotion to the ab' tonic. That is its eH`ect. The characteristic 37 The piano toohas anopen tonalstructure, insistenton D! but contrastingto that of the clarinet, moving,in chromatic bass descent, from the D tonic of the opening toa concluding Bb.In this light, a composite alignmentof bitonal tendencies canbe seento intersect, forthe apparentlydeviant outcomeof the clarinet line achieves D,the tonic of the pianos initial repetitions.In fact, those repetitionsconsist ofthe major 3rd, d and f 3, duplicated inthe final utterances ofthe clarineton the same pitches exactly.

tona/ity 93

anacrustic material of the opening, lendingstrong accentualvalue to the immediate goalofthe line, ab 2,is representedas associated with the ab 2,but with slurs and arrows suggesting lower-level orientations within the anacrustic group. Essential prolongations, high points, and other factors are represented as in other examples, and a second sketch shows a further reduced resumé in which the most essential motions and directions are extracted.

The Webcrn example Ex. l-28! is perhaps an exaggerated instance of melodic tonal expression,as thesketched analysismakes abundantlyclear, showing auxiliaries of strong leading-tone function in afhliation with a structural tonic. The lower leading-tone, Gil, provides in addition to strong support of A the neededlow point in a line whosecurve is beautifully molded, moving in its most basicoutline from a to al, with the range of the line extending slightly above and below these points. The motivic unity of the melody, involving sequence, is representedin the analysis in a limited way. Ex. 1-28. Webem, FourPieces forviolin andpiano, Op.7; No. 3. sehr iangsam Ji- caao! mn Dimpfcr ' °'

S'°' ' `1

Qe

%%* E

Hllolme cresc. 6 collegno ""23f'=" -,_ *"-'92 ' ;'f¢=°"S:-° 11=¥¢=2

.'

`3

HP;'=i

-J "'?¥"" ._ `- _ *

°

HW

.'

n"'

-h

'

kaum héirbar g|'||S[¢s_ ________ _ _`

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94 tona/ity Me/od/'c cun/e

The curve pitch-profile! of a melodic line can be representedin many ways, showing upward and downward directions with whatever specificity or generality seemsappropriate in accord with the aim of inquiry, and revealing the graphic image of its tonal succession orbasis. Forexample, in a senseand at one level of consideration, Ex. 1-29 revealsa seriesof up and down movements; in another sense it is a static melodic element. Ex. 1-29. Brahms, Intermezzo in A, Op. 118,No. 2,for piano. Andante tenerante phi lcnto! . m.57 _. . &JlL&1='==`1=JI-:-§l='f=5- .1 .H _E gl E |=ll§-'.=='==I:{§5'.?§= in fl ' 1 =2=213:22= **=2=$==% i===S== *'=|==" '== -~~'

A

t

The graphic representation of the melodic curve would show clearly where the melodic activity is concentratedin the range of the line and temporal durations of its active and static stages,illustrating the comparative breadth or narrowness ofits pitch compass range,ambitus!. Such a representation could be designedto illustrate the symmetryor asymmetry of its units, and the disjunctness or conjunctness of its intervallic successions, andthe balance and distribution of skips and steps. A very important part of the analysis of curve must derive from the particulars of melodic movement the broader strokes ofwhich are described bythe essentialpoints of the line. Example l-30a shows thegraphic representationof a melodic curvein a period theme. The essentialdirection of the first phrase is down, that of the second both down and up. The second phrasereaches beyond any point established inthe first-thus, while built upon the same motives,establishing a new high point. In the graphic picture of the line, vertical units on the left represent divisionsof pitch each having the value of a semitone! while those across thetop representdivisions of time in this instance,each representing an eighth-note or eighth-rest!. The diagram shows wherethe melodymoves, whereit is stationary, and in what directions and by what distances it moves; it shows whether the melody moves in large or small intervals, and where the activity is slow, where accelerated-all important factors in its effect. High and low points in pitch are immediately apparentand it is clearwhether or not they are repeated and how they are approachedthrough subsidiary high and low points. It

96 tona/ity

If we isolate the essential notesof each phrase of the melody to which Ex. l-30a

refers, we arrive at a reducedcurve, that of a more basic dimension.

The analysisof curvewill thus be simplified in a way that is usefulfor perceiving the most essential directions ofthe line-a higher level ofstructure Ex. 1-3Ob!. Ex. 1-30b. Synoptic representation of curve inthe Haydnmelody.

This processis analogousto that in which structural points are identified and outlined-those most strongly assertedand/or fundamental i.e., in tonal and linear function! as opposedto thosewhich are ornamental and auxiliary-as a means of getting at the most fundamental basis. The principal notes of the melody in the interpretation suggestedare f2, with which the first phrase begins andwith which it continues after brief digression throughmore than half of its length; cz,with which the first phrase ends and toward which

the movement of the line is aimed after the initial f 2

is abandoned; the f 2 with which the second phrasebegins the same pitch as that with which the first opened,thus manifestinga prolongation of that note!; the caat mm. 8-9, climax note of the entire melody; and the f 2 withwhich the second phraseends, still another manifestationof a pitch central to the entire period at the broadest level. These points have special significance with respect to one or more of the following factors: formal articulative function e.g., initiating or cadential!, durational value by repetition or sustained length, metric prominence, dynamic projection, prominence by virtue of high or low pitch, relatively emphatic manner of approach i.e., by leap, and by interval of leap!, and most important of all, centraligf of function inthe tonaligf, adecisive criterion in all tonal music. It is of interest that the principal notes are the dominant and supertonic dominant of dominant! degrees. The essential tones of the melo@ are factors indominant harmony. The eschewingof any point of tonic repose in the melody quite beiits its introductory role. The intensity occasioned by the

prevalence ofdominant feelingin the introductory theme, Q or 5/5! is ultimately resolvedonly in the Allegrowhich it prepares, andthe melodic line of the two-phrase themeof the Allegro carriesout this intent. Identification of structural pointsin the consequent allegro theme wouldshow that its melodyis anexpression of tonic harmonyjust asthat of the introductionis anexpression of dominant.

tonahty 97

A melody of Boulez,radically different in character and substance,and atonal, appears as the final subject in this series; it is useful in illustrating analytical representationof a melodic curve of extremefluctuations posing particular questions see Ex. l-3la!. Ex. 1-31a. Boulez, Structures for two pianos,part la. 23

45

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98 tona/ity

The melody in Ex. 1-31a issounded without counterpoint or accompanying voices.Its curve is oneof greatextremes ofrange in a singlephrase it encompasses almost the entire piano keyboard! and angularity of movement. The leaps are of such extreme distance that smaller relations of intervallic succession d-Eb,e2-fit, or the concluding F-b! appear almost conjunctin the established context.In general, changes arequick and of radical scope. The high and low pointsare temporallyadjacent, occurringjust beyond the middle of the phrase. It is possibleto imagine discretely perceptible'groups or units within the phrase-the Hrst three notes,four notes, or five notes, definedby the inherently punctuative effect of the longer third, fourth, and fifth notes, as well as any comparable formulation, although the composer prescribesa legato execution of the entire phrase.

The sheer extremes of movement within the line impart, certainly, a sense of drive, and

the absence of tonal function contributes

to an effect of

unpredictability and mobility!. Lacking repetitions of any kind-of motive, of rhythmic value, of pitch or PC, the melody is one of constant renewal. The example,

like others of its kind, poses problems of coherence and

apprehension, despite such unities as achieved by the shape of line [the wedge-like succession and interrelation of high and low points, clear in the graphic representation Ex. 1-3lb!, settling at a relatively middle ground at the endof the phrase] ,or the consistency ofdynamic level,like other elements a product of serialcontrol. The problems arenot merelyof the absence ofany sense ofdivisibility in the phrase into smaller units that can be grasped as such and of which any point is made, or of the absenceof hierarchic function lending any kind of tonal order. More basic still is the fact that in its disposition of pitch and rhythmic differentiation-the primary factors by which melodic form is perceived-contrasts are so extreme and soconstant that the %ct of contrast is, ironicalty, radically attenuated. Thereis a paradoxical feeling of randomness in the melody-paradoxical in view of the thorough predetermination of

all its elements.

Henri Pousseurdiscusses the Boulez, referringto this same phrasealong with another

from the same work:°

"One recognizesthat thesejudgments, inwhich somedegree ofconjecture mustbc acknowledged, are based onvalues whichcan bedeclared irrelevantto the aesthetic ofthe work in question. Butthe aestheticitself is the subjectof the questions posed. The issueof more orless total serialization ofelements andelement-events and the comparablequestion of random operationsin musical composition aretreated in this books introductory section, wherethe importance functional, expressive! of thedeterminationevents of and eventrelations on the basis of contextual interaction argued. is 4°Henri Pousseur, The Question of Order in New Music, in PerspectivesNew of Music, V,1 966!, pp. 93-l l l; translation byDavid Behrman.There isno assertion,in the quotation of these passages from Pousseursexcellent article,that his concerns necessarily parallel thoseof this writer. The second melodic example towhich Pousseur refers isthat of Piano I, mm. 57-64 Tresmodéré!, in Structures, Part Ia.

tonality

99

Ex. 1-31b. Graphicrepresentation of the Boulezmelodicphrase. Note: Each graph unit representsone semitone vertically and one

horizontally.

~Anomalous:a unit of 11would be expected seederivedduration series,Ex. 1-31a!.

Our difficulty in making a precise comparison between the two figures when we listen to them is caused,among other things, by the fact that each one is organized in the most irregular, least periodic fashion possible. Both passagesmight be likened to what are called "Brownian movements," i.e., movementslacking from the observers viewpoint! in all individual signification and therefore offering a high degree of resistance to unified over-all apprehension and to distinct memorization.

100 tona/ity Thus the rigorous serial procedures which determine all the detail do not seem to have a positive function. Far from establishing perceptible symmetries and periodicities, regularity in similarity and in differentiation-in other words, an effective and recognizable ordering of diverse figures--they seem insteadto hinderall repetition and all symmetry: or to put it another way insofar as order and symmetry may be assimilated one with the other!, all true order. The effect of statistical disposition differentiation of dynamics, tempo, and attack . . . , or differentiation in density betweenthe two and other moments of the piece! upon the over-all form is to guarantee a permanent renewal, andan absolute degree of unpredictability, at this higher structural level.

Some further observations concern/ng

me/od/'c ana/ys/s

The foregoing discussions ofmethods of melodic analysis are concerned chiefly with melodic curve and linear functions. We find that any example of melody has a certain essentialstructure-not an objective absolute, but something of presumed empirical validity that analysis seeksto discover and interpret. The basicstructure consistsof pointsin the line which we have classified asessential, these points elaborated and prolonged by points classified asauxiliagv-some dissonantand nonharmonic, others consonantwith immediate harmony, some stressed,some themselvesembellished by auxiliaries of a lessfundamental level. The perception and understanding of such melodic functions are of importance in how a melody is heard, and of course in interpretive performance. In modal examples, underlying,essential structureis determinedlargely by systemic functions in significant ways analogous to those of later tonal systems. In modal melody the final and cofinal are primary degrees often

l, 5! on whichthe melodyis conceived and onwhich it most oftencomes to rest, in graduated stagesof finality. Underlying structure in tonal music is,of course,similarly conditioned by the hegemony ofthe tonic and its main support, the dominant. We have observed that tonal melody is qftenthe horizontalization ry' harmonies, and that structural points often expressthe primary triads and other tonal harmonies. We have noted other qualities by which low-level and nontonal! essential structureis projected,by which in analysiscertain points are regarded as essential,others as auxiliary, both classifications relevant to hierarchic level. These have to do with such factorsas duration, prolongation by subsidiary embellishments,dynamic stress,metric prominence, cadential function-any characteristic imparting significance to particular points in the line and greater relativesignificance tocertain melodic events. Still, the factor of high-level prirnaqf qftonal function surpasses all others, where it is relevant to the

style inquestion, so that, forexample, adissonant appoggiatura §, however long orintense however accented!, is structurally subordinate to the l of

tona/ity 101

its resolution. This is a consideration that becomesparticularly important in later tonal styles, inwhich dissonancesare sometimesof great agogic value." Many examples studied in this chapter reveal that even when tonal expression isunorthodox, in comparison with norms of established systems, tonal function can bea decisivefactor in establishing andprojecting essential structure.

Some features of melodicstructure andform motivicunity, sequence, inherent contrapuntal implications in compound line, balance of conjunctdisjunct movement,and others! are treatedonly peripherally in thesestudies, although they are discussedextensively in many existing books on form, counterpoint, thematic process, and comparable subjects. Factors which lend motivic unity to a line, or which motivically relate the various formal units of melody, are sometimes elusiveyet significant in effect! and discovered only when analysispenetrates beneaththe surface.In Ex. 1-32, the esEx. 1-32. Brahms, Trio in Eb,Op. 40,for piano,violin, andhorn, thirdmovement. Adagio mesto

° If-92

H111

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"An aspect of this problem is the necessarydistinction betweensignificance inthe sense of linear andtonal functionalprimacy asopposed tometric accent,often counteractive

tonality

702

sential successionof tonally structural points is seen to involve important relations expressingsignificant deep unity between the two formal segments shown. Certain repetitions are deleted in the analysis.! These two units are,

in fact, identical in essentialsuccession,while importantly varied in color register, etc.! and embellishing configurations. To go only slightly further in thesesupplementary commentsconcerning underlyingmotivicandproceduralunitiesin melody factors revealed only in the analysis of structural baseswe might well note the very general relevance of underlying step successionin melody of all times. Some theorists believe this a virtually universal condition having implications of "necessity" and value."

This characteristic can be observed in Ex. 1-33.

Ex.1-33. Boulez,Le MerteausansMaitre " The Hammerwithout a Master"!, on texts of RenbChar;third movement,for alto voiceand alto flute. J

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52!

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This example is the conclusion of the third movement of the Boulez work fiute part omitted!. The ear is very likely, especiallywith familiarity, to hear the conjunct or nearly conjunct succession of high and subsidiary high points:

particularly

those of

to the structuralsignificanceof tonal "absorption"of the kind characteristicof theauthentic cadence.Of course,in stylesin which tonality is diminishedin importance,other factorsof primacy of projection,including many which are externalto specificPC contentaswell as registralsuperioritiesof somepitchesoverothers,becomeespeciallyimportant in delineating essentiallinear structure.In thesecircumstances, linear and metric functionsaremorelikely to coincide. SeeChapter3 forfurther treatmentof theseissues.! 42Amongtheseis Hindemith. SeeTheCraft of MusicalComposition, Vol. I, translated by A. Mendel New York: AssociatedMusic Publishers,Inc., and London: Schott and Co., Ltd., 1937!,Chap. V, especiallypp. 19396.

tona/ity 103

the final cadence, andperhaps tointerpret them as supportiveof Bb as something analogousto tonic, or point of tonal objective andquasi-tonal resolution.

Some part/'cu/ar /SSUGS ofharmonic ana/ys/s

/h /ater styles

While basic procedures and aims of harmonic analysis are appropriate to many twentieth-century idioms, it is obvious that the harmonic analysis of some worksof the recent period dictate or suggest specialpremises, objectives, and approaches. Wereject any assertion orimplication that harmony, properly and comprehensively defined,is fortuitous or irrelevant except in certain extreme

instances of randomness.

We have already seen,for example, that tonal functions, quite apart from other manifestations of harmonic progression,are of vital importance and relevancein many twentieth-century styles.Where tonal function is less relevant, or irrelevant, as a significant organizing principle, the fluctuation in degrees ofharmonic dissonanceas a means towardmotion and punctuation, and progressiveor recessiveaction, may be an important object of analysis. See thediscussion ofa cadentialextract from Schoenbergs Op. 23, p. l07ff.! In other idioms the textures ofchords, thequantitative factor of texture often referred to as densigw, may be an important object of harmonic analysis, with harmonic intensity and release achievedsignificantly by fluctuations in harmonic density. Factors of textural fluctuation are treated in Chapter 2.! In some instances harmonic succession hasa basis of logic in voiceleading itself i.e., of individual strata!; voice-leading the persuasions of linear function! can bevery important as agoverning factorapart from, or in some degreeapart from, considerations ofspecific tonal function or dissonance or density fluctuation. Voice-leading is thus an important harmonic determinant in all styles,with such considerations asadjacencies orproximities linking harmonic events, textural resistance of opposing directions a progressive tendency!, and semitonal and quasi-leading motion, as the bases forharmonic content and major conditioners of harmonic substance and control. Voice-leading can emerge asa trub: critical factor in contextsof pronounced tonal ambiguity. Such a succession as in Ex. l-34 must be consideredto derive in signihcant degree from the pure logic of voice-leading. Although the example is contrived, it points up the doubt that semitonal voice-leading can achieve harmonic successionindependent of quasi-tonal allusion; this is because of Quasi-tonal effect is enhancedby the penultimate Bl] the samePC beginsthe final phrase, m.43, voicepart!, constitutinga chromatic,leading-tone factorin octavedisplacement, enforcingBb, aswell as by the prominence ofthe C-Bbrelation in the penultimate phrase mm.41-42!.

104 tona/ity Ex.1-34. fh

Q5

l 3;-

$ Lf`92~° #U

the relation of semitonal succession toleading-tone function, as in the Dtl-E succession or,to a lesser degree,the Gil-A relation. Other factorsextrinsic tospecific PCcontent canbe ofgreat significance in the sense ofharmonic progressionand resolution.For example, a harmony whose contentis not of appreciablychanged dissonance or density can convey the senseof cadential release by virtue of its registral location e.g., the culmination of descent!, its dynamic level e.g., the culmination of diminuendo!, orrhythmic factors likeduration, often the indefinite duration of the fermata.

The operation of factors extrinsic to specific tonal or linear function, and specific PC content, can be demonstrated with reference to Ex. 1-35. Ex. 1-35. /A

52

HA

L4 §/

.-we-" If the dissonant harmonynotated in half-notes isplayed with any of the bass notes playedafter, the entire sonority held as though with fermata, it will be perceived that the total complex assumes cadential feeling because ofagogic value and textural isolation of the final note, and despite severe dissonance and considerable overall density. Moreover, the bass note,whichever it is, may assumethe characterof root, even tonic, because of its temporallyand registrally isolated,final appearanceand consequentcadential prominence.»° Perhaps the best root-tonic should besaid tobe Fil,forming a lowest perfect5th and a major 3rd with the top noteofthe chord. A comparablesituation is the subject of reference footnote e! with respect tothe pianopiece, Accenti, from DallapiccolasQuademo Musicals diAnnalibera, in which dense,dissonant harmonies find resolution in singletones which often take on the effectsof quasi-tonicsby virtue of the emphasis inherent in their cadential isolationand in the recession into so radically reducedtextures afterheavy density.

tona/ity 105

Traditional or quasi-functional harmonies,embellished by linear harmonic functions, are often concealed byirregularity of position, notation, or resolution. The augmented 6th to the primary dominant of mm. 2930, in Debussys l0mbre des arbres,is an example of this Ex. l-36a!; its function as a passingauxiliary is readily apparent, but its tonal relation like that of the V/V following it! to the dominant at the end of m. 30, while significant, is less evident. Ex. 1-36a. Debussyg /0mbre des Arbres "The Reflection of Trees"!, No. 3 of Ariettes Oub/iées "Forgotten Melodies"!, onpoems ofPaul Verlaine. See Ex.1-36b forrelevant excerpt. _f|Fmix;n| -Ill' f.1l|1Zl'92| § lllill llutihavi. lw 1* g,-`_ 92/ iv?

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Voice-leading is essential tothe understanding of the series ofpassing chords starting at m. 24. This succession, representative of some of the features and problems of harmonic analysis in impressionist styles, is worth extracting for special focus. Two half-diminished 7th-chords occur in the direct parallelism of successionwhich is one of the styles best known techniques, constituting the most severe denial of traditional contrapuntal interaction and an extremenegation ofimmediate tonal function in the passing, potentially functional, individual chords. The harmony following the two half-diminished 7th-chords retains three of four pitches to form a diminished 7th-chord seeming thereby to move in the direction of clearer tonal focus andfunction, yet still obscure!. This harmony, on All, takes onthe feeling of altered submediantto a parenthetical secondarytonic C, in view of the dominant 7th on G which follows in m. 27. But the reference is slight indeed and the dominant on G mm. 27-28! is of course areminiscence of the recurrent harmony of m. 2 in a changed, passingcontext which prepares the primary dominant of m. 29 Ex. l-36c!. ln the abstraction Ex. 1-36b!, elements of pure parallelism and of common tone retention are encircled. Example l-36c is a sketch ofthese closingbars in two distinct representations; in both, the strong and simple functional

basis is evident.

106 tona/ity Ex. 1-36b.

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tonality

107

i. e.

+The horizontalizeddiminished7th-chordis identical to that recurringin the passing series.

The chief concern of this chapter is the presentation of analysesof diverse extracts in which interpretations of tonal and linear functions can bc suggested.And it has been noted that aspectsof harmonic processextrinsic to specific PC content questions of harmonic rhythm, and textural-spatial fluctuation are reservedfor trcatrncnt in morc appropriate contexts. We might well, however, digressfrom central purposesbriefly to consider the issueof dissonancefluctuation. The question of dissonance is of course dificult and problematic, and little understood in connection with expressive process in music. Nevertheless, few would deny that it. is a very

vital factor in musical expression;we are inhibited in thc theoretical exploration of dissonancefluctuation as a controlled, purposeful processbecauseof the diAiculty of objective understanding of dissonance"values" degrees of severity in dissonanceas experienced.We are certain that any scale of dissonancevalue must relate to the stylistic context in question, but studieshave been inadequate and inconclusive in dealing with the problem of relative dissonancevalues. Nor is it clear how far techniquesof scientific inquiry can ultimately lead in the direction of objective understanding of dissonance effect in the musical experience.Still, somethingcan be learned in the exploration of the issue of controlled dissonance fluctuation

on the basis even of

selected not, perhaps, implausible! premises; let us consider, for example, questions andpossibletechniques of inquiry with respectto dissonancefluctuation in a brief extract from Schoenberg,the final six bars of the secondof Five Pieces for piano, Op. 23 Ex. 1-37a!.

Our concernhere is with evaluation of the final cadence,particularly in the matter of dissonancefluctuation."' Before turning to considerationof thc harmony, it would be good to enumeratea number of other elementsserving the effect of cadential release: decline in pitch, clearly recessivein general tendency toward the very low descentof the final, bassmotive; deceleration in the qualities of rhythmic motion seeC,hapter3, p. 309! which, while not within the range of immediate concern, is probably the most persuasive single factor in the achievement of cadence; reduction in textural density 4sTheconsiderationof dissonance hereis dependentin part upon an interpretationof normsassociated with the tonal period out of which Schoenberg's style evolves.

10S

tonality

«. 1-37a. Schoenberg, FivePiecesfor piano,Op.23; No. 2 Sehrrasch!. allmahlich langsamer werden

m.2P

desletzten Taktes

non legato m.22

dolce

Reprinted bppermission of G. Schzrmer, Inc.,agent for 8'ilhelmHansen Edition.

tona/ity 109

toward the final counterpoint of two voices; the recessive diminuendo. Each of the _/Yrregoing is rj enormous importance in complementary relation to the harmonic element.

A factor in dissonanceand its fluctuation is the predominanceof diminished triads, in some degree of dissonant effect, in the first two measures quoted, extending into m. 20, and their gradual abandonment. In this process ofrelease, themajor triad emerging at the beginning of m. 20 right hand!, and such increasingly consonant melodic motives as the three-note units appearing in the bass of mm. 20-21 following the initial diminished triad outline, are of importance. Example l-37b is an abstraction of the intervallic relations occurring in the last fourbars. Enharmonicequivalents aretreated asinterchangeable. Ex. 1-37b. *I /92 |~'.5|~6.,. "3 i3c~»§`»a» wow S G H;ef=§,=- Esaisfee E52 ess

rf "

,` f92 1~ 66 555, EE

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WJ iltll.ttQi 92./92./ 92"

f`

8

*Intervals ofdisparate sizes ofthe sameclass IC!-e.g., M2 andm7-are distinguished on the assumption ofdifferent dissonance intensities. However,and arbitrarily, compound intervals aredistinguished fromthose within the octave only to the extent of the 9th, although morerefined distinctionswould undoubtedlybe pertinent on the same basisof assumed milderintensity for larger intervals.

In any evaluation of dissonance fluctuationit must be borne in mind that intervallic

content is not alone decisive. Such factors as the extent

of

simultaneity of pitch events, the registral levels at which they occur, the distances betweencoincident pitches, the relative timbres, the dynamic levels, the manner of articulation,

the rhythmic values of the notes-all

of these, and

the list could be extended, haveto dowith therelative severity of dissonance eject. The evaluation of dissonancefluctuation is thus a vastly complex question, of which the comparisonof intervallic differences isonly an important part. Even the single question of relative dissonance severity intensity! is itself highly complex, and agreement is difficult concerning any proposed scale for intervals and chords. There is agreement, however, with respect Efforts to understand theexperience ofrelative dissonance intensities goback to Plato andEuclid, extendingto Helmholtzin the nineteenth century,Hindemith inour own, and manyother musicians, philosophers, and physicists; inthe twentiethcentury dissonance is a fundamental concernof the psychology ofmusic in which one concept ofdissonance intensity would trace it to relative degrees offusion of tones in the phenomenological

1 10

tona/ity

to the existence ofsignificant functional differences indissonance severityand consequent fluctuations in most musical contexts!and as to the importance in most music of the considerationof dissonancequalities and functions. The profile emerging from controlled dissonance fluctuationis especiallyvital in twentieth-century music in which tonal functions are diminished, although the conclusion may at times seeminescapable thatdissonance fluctuationis arbitrary, with form deriving, presumably, from purposeful fluctuations in other element-changes. The following attempt to evaluate dissonancerelations takes into account none of the extrinsic elements enumerated above, and it is intended to

suggest only a further dimension in the analysis of Schoenbergs cadence, and a methodology for its consideration within one parameter of harmony. At the left is a scale of hypothetical values, with those intervalsof assumed greater intensity listed at the top of the scale Ex. l-37c!. The tritone, listed as an interval of considerable intensity because of what are regarded as inescapable traditional implications of ambiguity, is sometimes considered of neutral dissonance quality.! The graphic representation of the dissonance curve shows a line of detailed changeswhich is a jaggedone with frequent fluctuationsoften among extremes. Superimposed over this line are two essential curves showingrelatively low dissonance intensityat the start, relatively severe dissonancein the middle of the succession, relativerelease atthe end-a profile which, in complementation to other factors noted earlier, must be regarded as of harmonic significancein the process ofconclusion ofthe piece. Whatever the value of conclusions derivedin a study of this kind, there is usefulexercise inthe exploration of premises,questions, andprocedures for experience of intervals andchords!. Animportant, if not widelyaccepted, effortto establish a physicalbasis forgenerally applicable theoretical approaches to theanalysis andcompositional usage!of dissonance fluctuation, aswell asa scaleof dissonance values andbroadly encompassing theory of interval and chord roots, canbe reviewedin Hindemiths Cn# of Musical Composition, Vol. I. The questionof dissonance values isalso treatedby ErnstKfenek in Studies in Counterpoint New York: G. Schirmer,Inc., 1940!. Kfenek suggests no physical basis forhis conclusions, listing theunison, m3,M3, P5,m6, M6, and 8as consonances; the M2, m7 as dissonancesof lower tension ; and the m2 and M7 as dissonances ofhigher tension. He states thatthe degree of tensionmay beexplained byvibration-ratios, combination-tones, orother acousticalphenomena; yet,the decisionof whatshall beconsidered a dissonance and how it should be handledis an arbitrary assumption emphasis added!inherent in a particular musical style,for it depends exclusively on aestheticconcepts. He suggests that the P4is consonantor dissonantdepending onits context a traditional view!, consonant when the precedingintervals areof highertension, dissonant when thepreceding intervals areof lower tension. Heregards thetritone as a neutral interval, dividing the octave intwo equalparts. Kienek goes onto discussthe tension-degrees of chords on the basis oftheir intervalcomponents e.g., chords ofthree consonances, of two consonances and one mild dissonance, etc.!. Hindemiths conceptsof dissonance and dissonance fluctuation are far more elaboratelystated, takinginto accountphysical factorsand complextheories of theefl`ects of spacings and vertical alignments in chords,and moreabsolute inconclusions.

1 12

tona/ity

! Even where there are not appreciable or perceptible tonal-harmonic functions,orthodox or unorthodox, there may be controlled profiles of dissonance Huctuation,of relative intensity of dissonance andrelative consonance, creatinga senseof harmonic progression andrecession within this parameter.

! Harmonic successions mayhave significant logical basis in principles of voice-leading even pure linear function in adjacencies and other factorsof approachand departure!, in supplementto or independent of other harmonic principles. ! The harmonic qualities may be vitally conditioned by factors rhythm, dynamic intensity, and others mentioned! extrinsic to specific PC content of harmony. The control of harmonic rhythm is often in itself a critical shaping factor. ! Particular harmonic factors, by virtue of structurally essential status underscoredin agogic,iterative, dynamic, cadential, or other modesof emphatic projection, may be heard as basic, embellished by surrounding auxiliary neighbor, passing, elaborative! harmonies, apart from tonal significance. ! Harmonic techniques ofany of these, or of other kinds, may conceivably prove irrelevant, or of no significant relevance, in the particular instance in question. [In many musical situations, structure must be seen as shaped decisively by certain element-actions which predominate while others arenot of shaping relevanceto structure; in serial music, forexample, PC content of harmonic and melodic events maybe determined largely by factors external to immediate contextual function and process within recessive andprogressive tendencies, while other elements rhythm, registral distributions, densities, etc.! are contextually determined and of decisive importance.]

Procedures of ana/ysis; symbolic representations of and harmonic functions and aft?//'at/'ons

me/odic

In the application of the following symbols andmethods to analysis oftonal music, tonalorder including modal and other systems ofquasi-tonal order! signyicantbi governs the structural hierarchy. Pronounced agogicsuperiority of an event in a nontonal style might well be decisivein establishingits hierarchic first-order status, while in a tonal order emphasis ofthat kind is submissiveto the predominance of tonal considerationsat higher levels. The procedures outlined in these pages areapplicable to lines of both melodic and harmonic events. The systemsof symbolic representation of linear functions and affiliations are useful where reductive _graphic sketches arerelevant toparticular analytical

tona/ity 1

13

purposes and objects of inquigf. Many of the symbols andprocedures givenhere are related to those of existing studies especiallyof Schenkeriantradition!; others are not."

Since the relation of symbology given here to those systemsallied to Schenkerian tradition and thought will be evident, it should be noted that there are important conceptual differences betweenanalytical approaches pursued in this chapter and those of orthodox Schenkerian procedure. A chief difference is visible in the mere dioersigfof sketching procedures em~ ployed in these pages,and the underlying premise that works and stylesof music suggest and require! varying modes of inquiry and procedure even within the consideration of hierarchy of pitch materials. Thus, for example, in many instances harmony is abstracted and represented as an element apart from its specificapplications in voice-leading andtextural distribution -a procedure that is legitimate and useful whentonal apart from linearfunction is theprimary object J inquiry. Other procedures,or procedural modifications, areadopted in line with other modes and purposes of inquiry. Moreover, a number of conceptual interpretive departures from traditional Schenkerian method including the

interpretation ofhierarchic structure in contextsother thanthose ofconventional tonality! will be apparent to the readerwho is aware of that tradition, by which at the same timethe presentstudy issignificantly affected.Naturally, an important aim in this context is the presentation of concepts and procedures whichare deemedvital and which do not significantly duplicate those of widely available existing sources. One hesitates to add yet another system of symbology to the many which exist some ofthem cited in the note printed below! ; yet, the system presented hereis innovative where particular approaches arethought useful as representingconceptual pointsof departure basic to this study. No dogma is intended, no universally applicable assertionof a number oflevels common to music, nor is there insistenceor evennecessary consistency in the modes of treatment of various subjects,methodology subjectrather to particular con-

textual problems of differentkinds anddifferent demands. Our purposeis to explore a range of questions andpossibilities of useful analytical approach.

At the same time,many of these analytical representations do employ the "Despite differencesof conceptualsubstance andemphasis, one is reluctantto offer a redundantand necessarily superficial reviewof techniquesof analysisof voice-leadingand directed motionassociated withthe now firmly establishedtheoretical literaturederiving from Schenkerand alliedanalytical systems; that expandingliterature speaks powerfully, if often withinunfortunately narrow biases, foritselff Schenkers most importanttreatise, Der freie Satz, isavailable ina translationby ErnstOster, published under thetitle FreeComposition New York: Longman, l979!. Anda comprehensive and systematic presentation of Schenkers theories and methodology of analyticalrepresentation Introduction is Schenkerian to Anabwsis by AllenForte and Steven E. Gilbert New York:WW Norton and Co..l982l.

1 14

tona/ity

technique of sketching harmonic and melodic essentials; wherethat is true, the following symbols areoften usedand will be usefulin carrying out further analyses oflike purpose and comparable basis. ! Representations of pitch factors ingeneral, wherehierarchy ofstructural function is germane to the analysis, are notated in the following, descending order of significance within the perspective qf the example itself!:

.°

°°

_ Procedures vary as to

how many, and which, of theseare used. ! Beams by which pitch eventsare linked represent hierarchicstatus and prolonging aliiliations by various means:at times, an outermostbeam is representative ofhigher-level structurewhile inner beams representprogres-

sively lower structural levels: | l____ ''__| or |

ti? I

|

; at other times solid beams are contrasted with

dotted and/or lighter beams ofrelatively lower structural level: I' ' "' ' '

! An

' "1

events notationmay beenclosed in a rectangular form forany

stated reason

on the

basis of which attention

is drawn

to it:

_

! Emphasis extrinsic tospecific PCcontent dynamicstress, duration, etc.! may beindicated byan accentsymbol overthe notationof the

event: ! Passing configurations are often shown in summaryby a diagonal line, with actual pitches omitted: ! Formal

punctuations of relative significanceare indicated by one,

two, or more vertical dashes throughthe top line of the staff: i*i - .

tona/ity 1

15

! Attention is drawn to particular events, oftenhigh and low points in melodic line, by arrows overor under the eventin question: the representation of such points by dotted or parenthetical arrows denoteshierarchic leveling of signiHcance comparableto the devices given in item 1 above:

2 |

!

8! Brackets over or under a sketched exampleare often used to draw attention to comparable often sequential or iterative! groupings of events, or for other purposesindicated with the particular example:

9! A wavy line may indicate excision ofmaterial which is redundant or extraneous to the particular purpose of the illustration in question: or ,

etc.

0! As with passing configurations item 5, above!, particular very local elaborative contours may be representedwithout inclusion of specific pitches: . ll! Arrows linking pitch events indicate dominant relations, the arrow pointing to the tonic of reference atan appropriate level of representa-

tion: olgl

or !

2! As in numerous examples alreadygiven, harmonic alignments of multiple tonal implication functional significance atdifferent levels of tonal reference! areshown by vertical rectangular enclosure, byvertical brackets, or by braces: V

IV OI'

OI'

3! Upperand lowercase distinctions G, g; VI, vi; M, m; SD, sd; etc.! are generally observed to distinguish between major and minor elements in conventional tonal

situations. But these distinctions are abandoned

a! where the referenceis to an abstractconcept apart from any specificity of application-e. g., V asa conceptapart from any particular form or manifesta-

tonality 1

17

An events representation asto hierarchic level of linear function often changes as the levelff referencechanges-the broaderthe level of referencethe fewer essential factors, locally essential events subsumed and seen as auxiliary in the more comprehensive view.We have described thisaspect of linear function

as multileveled.

With enumeration of the foregoing suggestions of symbologyfor representation oftonal and linear essentialor auxiliary! function of harmonic and melodic events,it should be repeated that in many of the graphic sketches given in this chapter actual voice-leading among harmonic events is not given, nor is everypitch ofa given harmoniccomplex necessarily represented. Where emphasis of attention is on tonal identity and function-usually multileveled-it is

often useful to abstract harmonic factors without consid-

eration of linear alignment and distribution; no singleillustrative representation can portray every functional significance, and there is value in isolating for analytical attention single aspects of suchfunctional significance. At times, too, voice-leadingis faithfully represented withrespect toone voice, or two often bass, or outer voices!, while inner voices are represented in summary abstractions,or not at all. At the same time,we should make very explicit the desirable, ideal end of comprehensive analysiswhich takes all pertinent factors into account. Certain of the proposedtechniques arein evidencein examplesalready given. The analysis ofa melodic conception ofCorelli Ex. 1-38! will further illustrate some of the methods outlined.

Typically, the melody outlines the primary notes of the tonality. The line is particularly striking for a number of reasons: its wide range for so few bars, the melodic dissonances diminished 4th, diminished 7th!, the sequential repetition suggested butnot fulfilled, and the failure of certain tendencies toresolve asexpected-the g of m. 4 and the e' of m. 6.

The prolongedl with which themelody begins might otherwisehave been indicated by connection to the underlying beam

,

or by other, comparable devices. Affiliated embellishing groups appear beamed and oriented to structural pitches to which they refer; they might also berepresented byslurs orby other techniques ofself-evident significance. Melodic analyses in this chapter aredevoted primarily to considerationsof pitch relations withinthe linearstructure andtonality anddo notusually treatquestions of motivic connections and processes; that is againonly becausechoices haveto be made inassigning limited space.Nothing is of greaterimportance in the experienceof melodic unities than motivic relationsand permutations, but theseissues are treated extensively in existingsources and to a markeddegree instandard musicpedagogy. [See, for example,Rudolph Reti, The Thematic Process in Music New York: MacMillan and Co., l95l!.]

1 18

tona/ity

Notes attached to the underlying beam are here selected to show essential pitches whichhave superiorprojection andwhich comprise the iharmony of which the melody is, asoften within tonal conventions,a linear expression orhorizontalization. The sketch includes most of the actual pitches except for symbolized passing groups!. Further excision or symbolization e.g., `! image. Tendency

L,

would yield a broader view-a more

essential

tones having accentual emphasis are indicated as

but their resolutions andmore primary pitch afliliations are

represented astonally-structurally superior, as arepoints of cadential arrival. Were the analysis extendedto include subsequent phrases, further reduction would be of increased significance in showing essentialmelodic functions in the illustrated bars, from f "2 to c2", expecting long-range resolution on the tonic degree!. Ex. 1-38. Corelli, Concerto Grosso inF minor,Op. 6,No. 3,third movement. Grave _

.:=%=:===-:=::=:= %:.- ==-..@=,

--~-~- L.

II

»i

l-*

e92_, ` H

5 is

_"" ----- --f'F IT

_-

__ =

f-

e

Ed' ____ i

'_;:iT_°l*'___

iT

*The f 2 isrepresented as expected becauseof ! the precedentin the initial fivenote motive,! the unfulfilled leading-tonee2 precedingcf. the el of the secondphrase!, and ! the otherwiseconsistent conjunct descent fromthe high point, aL»2, to c2.

Example l-39, also from the eighteenth century, is a four-measure extract from the binary first part of a dance movementby Couperin, that portion in which the secondary tonic,b, emerges.The relative major of the secondary tonic D major, R/d/t! appears as a brief parenthetical reference linking the two chief tonics. The essential progression,from ezi also bziv! to b:V also e:V/V!, is through a passing formulation of roots a 5th apart, in sequence.The aim of the progression, b:V, has prolongation and embellishment well beyond the point quoted before its resolution in b:i at the conclusion of the section.The strong tonal allusionsare fortiiied by the fact that the underlying, structural progression goes to b:V from its subdominant,

tona/ity 1

19

a further, important functional meaning of e:i. This broad view is included in the representation of roman numeral symbols of multileveled harmonic function. Example 1-40 is an extract from

the twentieth

century, a piano piece

of Kienek of which the final cadence, instructive in a number of ways, is shown. The factor of voice-leading adjacenciesis clearly of decisive importance in the derivation of auxiliary harmonies, andthe strong sense ofEb as root and tonic at the end imparts to the penultimate bass D the senseof leading-tone in a quasi-dominant. There are at the same time functional dispositions ofboth dissonanceand density in the direction of release. The example serves,in a limited context, to illustrate further some of the techniques of analytical representation outlined. Ex. 1-39. Couperin, La Bondissantefrom Ordre21!; ornamentsomitted. Gayement m_5A A

=:;:: ..- -::r' g!l l

Ei' "

L., -_ii "

5-EE*

'l~=::;====='='=5§£=@ "==%§E:::, -55-'I--_ ! 5=:: "2'LZ

X-

»

-

I. _ 5_

+; i c

T af* 92 g li

e92.f i #Ta 92 l

_.-

,e r

ee

at

____ T_

____ ________|; r ..

-» fiso .

me can

l. ¢tx D R/d!: ii b d/t!: iv

W1 etc. VI

V/V lst V/vi? th

IV? ii/vi? III? VI

biiv-V y

ii V

order! nd! order! rd! nd order!

lst!

Hierarchic order Hierarchic order in themovement inthe phrase R/t also appears!

tona/ity 121 Ex. 1-41a. Chant, Veni creator spiritus authorship uncertain!. ..

|'

45. -

a fl

a°

' at

In Is

'I

I

a Pg

f °°

I

~ E - ni Cre-- ai tor Spi-ri-tus, Méntes tu-or - urnvi-si- ta: _92 »-92 ¢ I_¢ I

9I 1. Ve-ni

a - tor Spi

i92 1.

2. I_-

rum vi

- si - ta:

3. O-

/°"92

5

Cre -

:g I9"'

Quae tu

I *" o- I-

cre -

4 I-

- ri - tus,

}` I. Z

I

Men- tes

tu -

'92 ' -O g_| .92- 'O'I_` - .

-0-

Im - ple su

| "92 , /*_*c c

-0° Q!

lo

a - sti pec

- per

|.

- na

gra- ti

'_

/T

- to - ra.

A-

o-

-0- '_ -a

men.

These pitchesare, of course, of central importance in the melodic structure and we regard them as analogousin importance to primary degrees inlater tonal systems. [The Hrst phrase makesit abundantly apparent that premises oftonal conventions of major-minor usage must be put aside in understanding and perceiving the relation between modal final and coiinal. Heard in the terms of later practices ofClassical tonality,the progressionfrom G to C, describing the interval of a 4th, suggestsdominant to tonic; Similarly, the cadenceof the secondphrase, in the senseof later tonal usage,has the effect of a domi1935!. Adiscussion ofthe history and traditionsof Venicreator spiritus, and a translation of its text, will be found in Carl Parrish,A Treasury q/'Early MusicNew York: W. W. Norton, 1958!, pp_. 16-18.

122 tonality

nant cadencein C, with the final cadence,if we submit to the conditioning of later conventions,seeming inconclusive.The confusion that can arise in the conflict between modality in the Medieval-Renaissancesense! andtonality in the eighteenth- and nineteenth-century sense!has largely to do with the conditioning of modern ears to regard the leading-tone to tonic succession

-l! as a necessarily semitonal relation, so that in theexample chant of given here the succession F-Grequires deliberateacceptance ashaving conclusive function to the modal final tonic!. Thus, a deliberate effort must bemade how deliberate depends onthe listeners experience andunderstanding! to hear

the chant

in modal

terms and

in accord

with modal

functions.

Although the foregoing is somewhat digressive,it seems important to urge upon the reader aconscious dismissal of the norms of Classical tonalityin approaching the problems of analysis of the example] Certain general conventions illustrated in the chant should be noted briefiy: the overall curve of the melody consistsof a rise to the high point e, at almostthe exactmiddle, anda return to the starting note, the modal final, thus describing an arched profile of wonderful grace. The first phrase consists ofan essentiallyascending progression, from g to cl with d' as auxiliary neighbor!; the second describesa further essential ascentto d; the third moves down and up again-an inverted arc within the established pitch range ;and the final phrase, with its more afiirmative function, is essentially a recessivedescent fromc' or b?! to g the latter having an auxiliary lower neighbor, f!, the opposite cf the_first phrase. Part of the perfect unity of the melody is inherent in the musical rhyme of phrases I, 2, and 4, which employ the same cadential motive, mirrored in the final phrase. The cadential degreesare dominant cofinal! for phrases l and 3, and final, as expected, for phrase 4, while the second phrase concludeson d', the cofinal of the authentic form of the mixolydian, functioning here as, at the broadeststructural level, an auxiliary to c. Example 1-4lb is an interpretation of melodic functions.Some passing notes areomitted, and hierarchic levelsare representedby leveled beams running below the sketch. It shows the essential progression of the first Ex. 1-41b. Sketch of linearfunctions inVeni creator spiritus. Cf.

|§ 1,

==lF92i " __X /_N 'T;:;__ Q ._ - ln_ _-v :P E' »`~_ -1 _ _..-'e le ___ `----Y-' .n ,Y 'ii 1.11 .i __ , _, , _i I I ""' I ;1 Levels of hierarchy

`929292929292 n,

1

l -_l_.iii _ _ ' SQ _. _ , .__._,_ -_.__ 92`_L_/

tona/ity 123

phrase from final to cofinal, g to c. Each of these pointsis seenas embellished by auxiliaries: the g is embellished byupper and lower neighbors,the c' by its upper neighbor. The final is established with considerably greater prominence in the first phrase, havingfour appearances;on the other hand, the cofinal emerges stronglyin its cadential placement. The analysisof phrase2 exposesa primary essential progression from c' there is a subsidiary, elaborating deviation to g and back during the prolongation of c! to d', the latter embellishedby its upper and lower neighbors the final, C, thus having an auxiliary function in the second part of the phrase! andidentified asa significant structural point because ofits cadential role, yet more broadly of auxiliary function in its strong leaning toward the more basic c which

surrounds it."

Indeed, the

d resolves to c'

in direct

succession intophrase 3. While C, the cofinal, is clearly the primary structural basisfor the third phrase, thereare severalembellishing deviations-passing formulations up to auxiliaries e' an early repetition of the high point! and d, and twice down to g. The vacillation of the line throughout between g and c' is of course conventional in the application of the mode, with the expected structural primacy accorded 'the final and cofinal. The fourth is the only phrasenot beginning with final or cofinal; it has a particularly striking effect, starting as it does with the leaning, lower neighbor of c'. The analysis showsthis auxiliary to have emphasis. It is of course notactually stressed,but its position asthe first note of the phrasedoes mark it for attention.

Phrase 4 is shown to have an essential structure mir-

roring that of the first phrase. Again, each of the two essential points has embellishment. Thenote b appears forthe first time as auxiliary to a degree other than the cofinal-in this caseauxiliary to a, which in turn is auxiliary to the final, manifesting again the dimensional implications of linear 5'I.e., dl is clearly of greater hierarchic valueat thelevel qfphrase 2in view of its cadential rolethan at the broader higher! levelof phrases2 and 3 conjoined;in the larger context it is seenand felt as clearlydependent oncl. There is, here,as in numerous other references self-evidently comparable, aninstance ofthe multileveledsignificance oflinear functions dlof phrase2 isof differentsignificances at different levels!.The first dl of phrase 2 canbe seen in ambivalentmultilaleral function at the same time:it is upper neighborauxiliary to the essential cl, aswell asanticipation ofthe cadentialdl part of theprolongation of

that pitch!. It might be symbolized thus:

.

"One can see againat this point the interdependence of structural leveland linear function: inthe localcontext ofphrase 3,the final G is clearly subsidiaryto the cofinal C; at the more basiclevel of the total melody the reverse hierarchicrelation is evident; the linear functionsof bothpitches inphrase 3are thusmultileveled. Theg mightbe symbolized

,or-?--,andthecl I t .

124 tona/ity

function. Thecontrasting tendencies of bin thesetwo environments-/' and 92-is a striking factor here.! The final cadencerepeats, ata much removed point in time, the melodys low point, ff It scarcely needs tobe said that it is of the utmost importance in the execution of a melody of this, or any, kind to be aware of these essential directions and linear functions, of the degrees of repose or lack of it in the cadences tentative, more tentative, tentative, conclusive!, of the nearly contiguous reiteration of the high point, suggesting a slight difference of inflection. This will be affected in some degreeby the text setting; the first el hasits own syllable, while the second,except in the sixth stanza, continues the syllable of the preceding note; on the other hand, the more dependent auxiliary role of the first e, and the exposure ofthe secondin its échappéelike departureby leap,suggest furthercriteria of interpretive decisionmaking.! Having observed in careful analysis both the overall and individual curves ofthe melodic units, and hierarchy of functions at a level in which all notes are represented, wemay characterize the melodic structure at more fundamental levels.

Example 1-4lc summarizes, in a more fundamental analytical interpretation, the two primary structural points, the modal final the more basic of the two, with d, the cadential note of phrase 2, shownin this more basic sense asan auxiliary of c, the elaborating and supportive cofinal. Ex. 1-41 ¢.

at 1 Carrying the implications implicit in thisprocedure to iheir proper

conclusion, l-41d Ex. shows the most f`undamen`tal essence of thkmelody as an extensive prolongation of the modal final, G, with the two other most important pitches, in a hierarchic ordering, shown as auxiliaries: C as auxiliary to G, D as auxiliary to C. Ex. 1-41d.

92" 3

tona/ity 125 Two Preludes from Fischer, Ariadne Musica

Two of the preludes of the Ariadne Musicaof Fischer 715, a forerunner of Bachs Well- Tempered Clavier! are treated next in reductive analysesof two dwrent kinds andpurposes. ThePrelude in E phrygian! is seenas, at its broadest level,a prolongation of the tonic in elaboration with A, an important cadencepoint in conventional applications of the phrygian mode. The sketch Ex. l-42! is conceived torepresent, primarily, the interior manifestaEx. 1-42. Fischer, Prelude in E phrygian! fromAriadne Musica for organ. 1 --*i'

T 'gg-FF' -H l

Il In V -ll' I °lfllim-_`I_92-2 liU _

' ' -I * "I-fl-'='E==l !_._92I __ I

! l'!

_ nu; ..f'l ll

J- A

J»

_ P--__'--.l-lu

PW _ I! 1*

nl-5 "I L-

_ "_ i_

'1'

'ifl

-- --

ll Ill; IF ll 0' fi r er

sd/t! in

I m.9 E" " or

"" "'*'- """

lu: J - V-

Interior manifestations of iv

Y3

eb » -I l

1 ¢_ 3¢_¢U *°?3

"

l_ l!E H_ ,

Vl

rw

§ =? *Dl= $= " 'fill ' phrygian i

-1

broad plagal elaboration

5-

92, QE

ij-R H

e f

92§==|--ri5f==Hl3}-=If¬.l'1|}-¬l|§¢§1E=f|;: Q `- 92.f 92 i `-' 1 7 »1

'I r

iA_

1, _ 92.1 -____ `d'

_- _-_;;::--Z--,__

e Twice-tonicized R,M/t! G lower neighbor of second-order a

Reprinted from Liber Organi: Deutsche Meisterdes 16. und 17. Jahrhunderts, E. Kaller, editor, published by Schotts Sbhne, Mainz. Usedby permission U Belwin-MillsPublishing Corporation.

tions of the repeatedly tonicized iv as the chief neighbor auxiliary of the prolonged i, in a broad plagalaction ofembellishment. Third-order manifestations of G in later tonal evolution the R/t! are also represented;G can be seen inmultileveled function both aslower neighborauxiliary of A and, more broadly, as a passing linear factor linking the elaborative A with E. The

126 tona//Ty

broadly elaborative A is represented bythe upper beam and upward stems, as indicated.The exampleincludes acomplete sketch,but omits the concluding three bars i prolongation! of the piece itself. The Prelude in D minor from the samework is sketched Ex. 1-43! to represent, overa broad harmonic fundament, entries of the persistent moa

tive i

_which is sketched

as ,

in representation of its essential structure as con-

sisting of an anacrustic wind-up from the 5th below the anacrusis note embellished! and extending to the upper neighbor auxiliary also embellished!, both of theseelaborative ofthe arrival note, which has agogic,cadential, tonal, and metric prominence. Seenin this reduced sketch,an interesting V-expressing relationof essentialfactors in the top voice andthose ofthe tenor emerges: the former rests on the PC successionA-E-A-E-Cil, the latter on the succession A-E-C#-E-A. The bass, absolutely without elaboration, underscores in a series of pedal points the fundamental I-V-I structure. Typical elaboration of the concluding I with the auxiliary iv, of which it is dominant! is again in evidence. In Ex. l-43 the quotation of the piece itself is necessarily abbreviated;however, the sketch iscomplete. Ex. 1-43. Fischer, Prelude in D minor fromAriadne Musica. 3s 1_1 :I-1. I ..921 l1.l '_'-1 1*-2----

'e

all f-

§;:= -

"_

m.4 3 f_ I/. 1 h-ll|_I-1 929292'

"'"

33 7° ll-ililf' 2. H-7232: ' _ 5 192l

' ==F#IiF== l -1 _ l'LA| mul l.l11 'I V' _

_I

--l

._ ul

' !""'=

rn ' _J -

T7

IF

,|

3 11 _ IQ _U __

r-92¥% 3 [hill

Reprintedhom Liber Organi: DeutscheMeister des16. und17. Jahrhunderts, E. Koller, editor, published by Schotts S6/me, Mainz. Reprinted by pennission Belwin-Mills of Publishing Corporation. "Both preludes,with others,are includedin WallaceBerry andEdward Chudacoff, Eighteenth-Century Imitative Counterpoint: Musicjbr Analysis Englewood Cliffs, N._].: PrenticeHall, Inc., 1969!.

tona/ity 127 IIL4 I ,§1 '

V! ¢_ 92

Q ra _P~9292 -A___.--' °'°" "" ' "`~-_

I Y!

`

--------------

7 1V

I V/iv! iv

_W |v

lV

Bach. "Drauf schliess ich mich in deine Hiainde" from Motet. Komm, Jesu, komm

In the quotation of a choral movement fromBachs Motet, Komm, Jesu, komm Ex.l-44a!, parts for two choruses arecompressed intonotation on two staves, withthe text given abovethe stavesexcept wherenecessary toshow a different setting for the bass voice.

Ex. 1-44a. Bach, Choralaria "Draufschliess ichmich indeine Hénde"from Motet,Komm, Jesu, komm. Andante Drauf schliess ich mich

:Zim -='°

in dei-

ne Hiin

ef =I

___ 4 .1.95 Yf -r l __;/' *_

n|l:n»:11 c

in dei

- de

'!=... ._

»

r_

hi J

___

- ne!

m.5 und sa ge, Welt, zu gu - ter hiacht! 4"

f

f_

f at

. L.

l!$= -

n,92_;¢.1.92.1»92 £ PI r ~ =§i"¥i' gu - ter! W 54For a translation ofthe text,sec p.428.

__

asf

128 tons//ty Ex. 1-44a continued. m_9 eilt

gleich mein

Vf

Le -

bens -

- lauf

zu En

- de,

"92,J J~ e7'-=§§EE il "v ' :V

V" "5 I

Z*

:___ 1

-92'92-I J

Le - benslauf - zu! m.l3

J

ist doch

der Geist

wohl an

- ge

92 eJ/'I-& mm:

,.

xe e

e ,lv 5 _

e 9292q- V =

_

1. an -

Er soll

bei sei

- nem

.92.»

- bracht.

L .o

e _ e §%':::

/V ee

@

= ge -

Sch6p -

bracht.!

fer schwe

.1 /7 -

- ben,

~J J evvf

1 ` I'

_#J J"5;J.|

21 weil_Ie sus - ist

m. -

-flu; Jz'-=1|? Je H _F ag'

und bleibt

J1

_-- l :g 5

J; ` 4 ,_ _ Qje -

sus lst

der wah -re

und bleibt

55-55 lu

,

L :Il der wah

- re!

tona/ity 129

Weg u

zum Le - -

-J--

.EF .,L_ I s

4 3 Z"

ben.

__ °

o I' 1 . |

'gli A SJ .:'

.

In as conventional and ordered a tonal style as this the essentialharmonies are those of primary tonal importance-the tonic and dominant cadential harmoniesare one or the other, although of coursefrequently at other than the primary tonal level!. Theseare theharmonies ofmost frequent prolongation, reiteration, rhythmic-metric prominence, and key positions in the articulation of form. Auxiliary harmonies are, logically enough, very often those one step back in the tonal order: dominant auxiliary to tonic, subdominant andsupertonic to dominant, submediant to subdominant, etc., and the intensity of relation is frequently underscoredby chromatic alteration Ex. l-44b!. Ex.1-4-4b.

if #lil

e

. &/'f'j . .3 F

s

I

""~-9 %~___/'I "`~_--w §/

`~-- *

The following series of analyses takeseach phrase in turn, sketching some linearand tonal functions and relations. Roman numeral analysesare included to show many not all! of the tonal functionsin primary and secondary tonal systems. Finally,under each of the sketches theanalysis iscarried to a more fundamental

level.

The sketchof the Hrst phrase Ex. 1-44c! reveals alinear expressionof i, one in which the background of the soprano pitchesoutlines that harmony as doesthe bass!and in which i occurs atcritical points including the beginning and end. Asi has an outer prolongation, there is inner prolongation

130

tonality

Ex.1-44c.

g t!: BbR/t! F D/R/t

of iv, alsounderscoredby upper voice factors,asrepresented.Most individual harmonies are symbolized in accord with the concept of multileveled tonal significance. Primary i and V are linked, in hierarchic order, to an underlying beam, and harmonic recurrencesare shown in association by established means. Where necessary,actual explanatory or identifying notes are inserted. The value in this procedure is in the analytical reasoningby which it is derived aswell asin the image of structure which it immediately affords. In the further reduction given below the first sketch, the essentialharmonic factor is seenas i, supported by its chief auxiliary, V. In the secondphraseanalysis Ex. 1-44d!, it is shown that, especiallyin referenceto a secondarytonic, the primary tonic has auxiliary function in a context of its tonal and rhythmic subordination." It is incidental at thelevel At the sametime that it points toward the cadentialBIs:I, the g:i contributesto prolongationof the precedingcadentialharmony a multilateralimplication! and assertsa more tenuouslink with the underlying tonal basisevenwhile it participatesin the linear encirclementof B!:V a multileveled implication!. Functionsof the harmonywhich begins phrase3 arecomparable.!This analysisof the Bachmovementis not sufficientlydetailedand exhaustiveto accountfor all suchmultidimensional implications.

tona/ity 131 EX. 1-44d. Phl'3SC 2 _'ll' I '1-` f1

Phragc 3

11 2S

R/ISV _g if "-_ ' HZ/ ;: :_ 2:

:_

! °92

e 21- e -

Fila* l:°

5

-- e

' 9"¢

aee z*

"" i' ' _" '° `|--1-D/R/t:V -J

3

'" IV v

I1 rv vii°

nf-_"a`::":::::::`i` " it ; ie

as

F

lg

_13"- :¢ -`7i55` _ _t

V/lil? m?

B5 R/t!:vi v F D/R/¢!= V/lv?

"""'

: ~;~ 21 2 I" . " °-** §92"Ll_QQ___5_`f'l§1'_'T__592f °

_2

-» g 5* I;-L-R/t: I in bass--I s 92!= i

g D/R/ul _ ,` `_ ll uftlf 7 1,5 E_ "2 :"

Iv if a *im* _ ':::;;:::;::_

bass v/vu?vu?

-~~ WV/Ill?! V/V? v Iv v

_§.T`___ _-.___.

I °*"' --~-'~+ 'iifffz

:ii if1:;:;

92 __T_;:i§i§f;:;i:E§§}f;_-_ __::§?§§f:~ -31 T132

92-vi __-___;m__`;__"m'Hm;f»vT'l;»§: __ _'W

of thesecond phrase, functioning as part of the embellishmental group in the secondary tonal expression ofBb. In the analysisof phrase 3 Ex. l-44d! a number of harmonies and a few in the second phrase!are given questioned interpretation in the primary tonal system. Because bothIII and VII, as secondary tonics,involve the cancelled primary leading-tone, it is considered that their effect asfunctions in the primagfsystem is substantially attenuated. In phrase 3, the harmony containing EH is reasonably interpreted as vii°/V in Bb ; the question refersto the fact that the tonic of referenceis in dissolution, submitting

to F.

Phrase 4Ex. l-447e!is directedtoward the tonicized primarydominant. Its first appearanceas aminor triad is thenatural result of the preced-

ing secondary tonics, bothof whichinvolve Fl, butthe phrase concludes with the triad in major form; with this, its reference to the primary tonic is of course sharpenedin affirmative effect. In phrase 5 Ex. l-44e!, directed toward the primary subdominant, there isa singledeceptive succession in which c:V moves to vi°, indicated in the functional analysis as a provisional, extremely local tonic substitute. Questioned functional interpretations have the same bases asthose in preceding phrases.They are included becausethe tonics of reference, while in Certainly this event abridgesthe distancefrom the primary tonal system, despite the apparentproximity ofBb to g R to t!.

134 tona/ity

analogous contentbetween phrases1 and 4, and between the pairs 2-3 and 5-6.

Beethoven. Symphony No. 2 in D. Op. 36. fourth movement Representations ofthe tonal system andtonal rhythm

Figures l-6a and 1-6band Ex. 1-45 showvaried representationsof tonal structure in the fourth movement ofBeethovens second symphony, themusic of which is not reprinted but might well be consulted.Figure 1-6a is an abstract viewof theentire tonal system, aview generallyof akind suggestedearlier. In this casethe systemcomponents arenot hierarchically ordered, and they are not in chronology of appearance otherversions ofthe chart could show theseor otherarrangements!; rather,the systems are arrangedin an order of increasingsharps inone directionand increasingflats in the other.However, there is limited implication of hierarchic significance in the enclosure in horizontal rectanglesof the most important primary systems they are D, f tl , A-the factors of the tonic triad!. Important intersecting relations linking the systemsin contextual operations areshown by vertically aligned rectangular enclosures.Roman numeral symbols not so enclosedsimply list harmonies which occur significantly as part of the tonal resource but do not participate in intersecting relations.The indication of modal variants in the listing of tonal systemcomponents suggests, as doesthe inclusion of roman numeral symbols of both upper- and lowercase atcertain intersections, the importance of modal variation in facilitating the intersectionsby which tonal iluctuation is

often achieved.

Some minor mode tonics emerge as variants within major contexts, while others appear independently e.g., f 8, g, and c!. Although the chart does not give symbols identifying specific practical relations of secondary systems tothe primary system, suchrelations can be inferred on the basis of actual contextual functions and derivations asin Fig. 1-6b. Example 1-45,a further representation ofthe Beethovenmovement, is the result

of a diflicult and

tedious exercise

which is

at the

same time

extremely revealing as a digest of actual tonalrhythm. Again, it must be remembered that the digest is not to be read as a pitch-line in any sense at all; the notes arestaff representationsof PC tonics in the order in which and in the durations forwhich they emerge and,even momentarily, prevail before a supersedingfluctuation.! Structural and auxiliary distinctions, i.e., relative importances of systems, areevident in some degreein relative durations, but a continuousbeam is usedto link all occurrencesof T, and subsidiarybeams tolink the important tonal systems D/T and r/D =

m/T!. Items

in parentheses are of most fleeting

cE !0 E O s-

o

~c ~tA CVg! 0C 2 0 + C c.

0 a0 C

N CO c O cICA 0

C I O . EO I CI LL

SU~Sesmuy

QSg SUISEMDU$

136 tona/ity Fig. 1-6b. Beethoven, Symphony No. 2 in D, fourthmovement. Secondary system components andtheir contextuallyimplied relationsto the primary tonicD,d. A D/T

Second-, third-order systems Nm orr/D/,IV Decreasing sharps, increasing flats De

Yf

E D/D b sm r!, or st/D st, or sm/SD G SD C SD/SD a r/SD/SD F R/t g sd/t Bl: SD/R/t c sd/sd/t sd/sd/sd/t

effect; theaccent symbol v is here usedto indicate emphasis notnecessarily of duration ; arrows denote dominant affiliations; the symbol ? indicates uncertainty of tonicization effect; and a wavy line indicates an excision of repeated material. The circled numbers over the sketchare for reference to the list of notes following. In this list of notes PC letter names are used in addition to symbols of relation to T; where D might otherwise be unclear, it is shown as D/T if the dominant system componentis meant. Finally, a list of durational note-values arbitrarilyemployed inthe sketch,but precise in their applied relations, is given at the left in Ex. l-45. 1. Note within the first stable area very slight tonicizations of f ll and b, perhaps predictive of subsequent strongtonicizations of these factors.Typical symmetry of tonal reference, herein 3rds. 2. Transitional phase characterized here by D-A vacillation: highly significant tonal-rhythmic ordering in increasingvalues qfA. 3. Passing secondary tonicsin movement in 5ths. 4. Symmetries within D/T area: E-A-D, C-A-F. Note again a rhythmic ordering in the tonal form-decreasing values ry" A tonicizations. 5. As predictable in conventional tonal practice, D/T has emergedas structural tonic of second-order importance of course auxiliary to T at broadest level!, and is in decline as noted above. 6. Longest pure manifestation of primary tonic. 7. Note passing tonicsin symmetrical formulations. 8. Emergence of strong reference to mediant m/T-fll [Note embellishing, mild tonicization of D/m Cll ?!.] 9. Compare the beginning of the exposition, but here the primary tonic is reaflirmed as expected in single-movement sonataform.

138 tona/ity 16. Note traditional, almost universal emphasison SD seen in incipience in sixteenth-century examples! in coda as in elaborations of authentic cadence in general; very strong here. 17. Note changes in values of G and D SD and T! in the coda: again, rhythmic ascent in values of T, descent in values of SD-controlled changes inthe tonal rhythm having clearly functional implications. 18. Note final tonics as outline of SD:I; cf. 16, above. 19. While relative prominences as representedhere would clearly be a factor in any broad, hierarchic ordering of system components, other factors would qualify any mere computation of relative rhythmic values: e.g., degree of uninterrupted prevalence as of m, f il, at 8!; and degree of independencemaniof festation of secondary systemcf. m, f ll, at 8, with SD, G, at end of movement, which has much stronger dependence onthe primary tonic undergoing elaboration!; and, among other factors, theposition ofoccurrence of the tonicization in the form.

Particularly important is the controlled change fy' tonal-rhythmicvalues, mentioned a number of times in the above notes. Wolf. "Das verlassene Magdlein"

from Gedichte von M6rike

A further example of striking tonal expansion isextracted from a song of Wolf, one that is much discussed. Itstonal structure is closed tonally rounded!, unlike the open tonal structures of the two Wolf songs cited earlier pp. 85-86!. The point at which the extract begins rests upon the tonic PC expressed atthe beginningof the song, A.It is followed by a seriesof successions, some ofthem chromatic, referring ephemerally to passingtonics. A number of the dominant harmonieshave the form of the augmented triad. As a dominant form, the augmented triad has a particular wayward-

nessview in its of ambiguity of root. It can resolvemight as III; t: or'Q/+! 2

to a normal tonic consequent Ex.l.46a, mm. 20-21, 26-27, 28-29!, but it can also act in deviant ways. In mm. 22-23 there is evasive,nonfunctional i.e., tonally nonfunctional! chromatic succession from one augmented triad Ab: V+?! to another a semitone below. While the notation of the augmented triad of m. 23, on Gb, would suggest it might function as a dominant to Gb, there is no realization of that potential. Rather, there is a further chromatic successionto still another augmentedtriad, notated as a dominant to Bb. This time the suggested tonaldirection is confirmed by the resolution at m. 27, repeated at m. 29. Analogous processes continue until, at m. 34, anaugmented triadon G which could function asa dominant of G! The entire song isincluded inBurkhart, Antholog_y_jor A/Iusical Analysis, 2nd ed. New York: Holt, Rinehart andWinston, Inc., 1972!.

tonality Ex. 1-46a. Wolf, "Das Verlassene Magdlein"from Gedichtevon Morike. Langsam

m. 21

m.26

etwaslebhater

140 tona/ity Ex. 1-46a continued.

H1 30 etwas mr 'Y

ruhiger

0o

~

r9

Kna -

be, dass

-f-gl .

ich die

_ .f ---=--` P

i

rn. 34 »_

55 °5 [QV af ha -

Nacht vondir ge

9

- triiu

u'

fllffl _ YG: _;'/ :I `

°

~ met

U PP T

gl

wie zu Anfang

@

'

' E'

ggi

be. Tri

II

Q4 - ne auf

fI

m

- =

5¥tlsl1ls:!l_e'eE!!!1|s 9292l'1:92¢_-.a]m QI.! _nj '2|.L.¢

'Q W

assumes thefunction of mediant as dominant! to e. The e of its resolution is then inflected

to become

a dominant,

then a

dominant 7th-chord,

for

reaffirmation of the primary tonic. The above has perhapsseemed somewhat digressive forthe immediate purpose ofdescription oftonal structureand fluctuation, but the ambivalence of the ubiquitous augmented triad between tonal function and nonfunction is an important aspectof tonal structure in the song.What is presented here, between m. 17 and the clear restoration of the primary tonic at m. 37, is an extended passage of characteristic tonal fluctuation and uncertainty. One is almost tempted to assert that the tonal structure

is best characterized not

in

terms ofspecific tonicsbut, rather, in terms of the pattern tripartite! stability_/luctuation-stabiligy. Still, one approach does not exclude the other, and both Ab and Bb, in one senseembellishing neighbors qf A, are tonicized in significant actions.

The passageof Huctuation is extreme in a number of ways: ! in the recurrence of the vagrant augmented triad sometimes a V+, sometimes

tonality 141

IIIJ, sometimes nonfunctional!, ! in the complete denial of functional expectation arousedby the dominant 7th of m. 18 A:V, of vi ?!, which gives wayto seeminglyirrelevant chromatic succession tostart the stream of fluctuation, and ! in the tonal distances traversed: there arereferences toAb, Bb, andiinally e, which becomesthe primary dominant in function. Especially the relatively longer references toAb a semitone belowthe original tonic! and Bb a semitone above! seem to impair, yet in a purely linear senseso logically elaborate, the initial tonic A. The Ab and Bb references thushave a persuasive linearlogic as semitonal neighborsof A, and thus act as tightly related linear compensations to the errant tonal progression to remote regions ofwhich they are centers.In summary, two leading-tones,Bb andAb, act asencircling tonalnot merely harmonic! embellishments. A fundamental view of the entire passage, illuminatingthe chromatic derivations of two passing tonics, functional and nonfunctional augmented triads, and implicit chromatic linear successions of the voice-leading, isgiven in Ex. 1-46b. A telescopic viewreveals a stepwise succession of augmented triads linking two structural appearances ofA,a:I,i. Ex. 1-46b. Synoptic sketch ofthe Wolfexample.

measure numbers: 17 20

23 24

31 32

36 38

t: I, i ;-Y/:PIII _ l ._Y'1||921Y' /._ I

Ill1-ll II Ill

_21 If

_ ll _I.' Ill 'lf m/'

I Functional dominantsl As complement to the foregoing discussionof pitch structure in an extract from Wolf s song, Das Verlassene Magdlein, there is given in Ex. l-46c a reduced sketchof the entire harmonic structure in abstraction. Successions in and profiles of individual voices arenot representedexcept for some fidelity to essential movements in outer voices.! Again, an exceedinglysimple basis V-I! underlies an elaborate superstructurein which there is chromaticism of immediate successionas well as of tonal expansion. A synopsis is given on a singlestaff abovethe example,representing in the midsection the tonicized chromaticneighbors-Ab and Bb-of the primary A. This provocative material is seenin a further light in the one-staff representationbelow the example: theplausible interpretationof bassfactors asexpressing, horizontally, the primary dominant, with the primary leading-tone represented enharmonically.

142 tonality Ex. 1-46c. Wolf, "DasVerlassene Méigdlein." A reducedsketch ofabstracted tonal-harmonic structure. Chromatically encircling neighbor auxiliaries!

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uiméz-n ill 'H--f 92, I V?

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/N .F 1 :

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*Substitution ofaugmented formajor triad on Ab primary dominant III"'! ? Ravel. "Le Martin-Pécheur" from

to provideenharrnonic formof

Histoiras Naturelles

A Ravel song isthe subjectof referencein the following discussion! In the song Ex. l-47a!, a primary tonal systemon F# is expressed,richly colored, and confirmed at its conclusion.Example 1-47bis a synoptic rendering of the entire structure, only a few details of which can be discussed. The final six bars see Ex.l-47b, where key measurenumbers aregiven! are a prolongation and embellishment of F ll:I even in its final appearance activated by static dissonances, G# and Dil! ; prior to this there is modulation from the principal secondary tonic, E, to F tt. Measure 15affords an excellently clearillustration of the prolongation and embellishment of a structural harmony here, E:V,,! by parallel auxiliary chords of similar here, identical! intervallic content. While these are for the most part representedonly symbolically in the sketch, the dominant on B is embellished by an auxiliary dominant on A and these are,at a more immediate level, linked by chromatic passage amongnonfunctional parallel chords, the root of each yielding an ornamenting bassnote. The process of parallel embellishments continuesin m. 16. That bar begins with the sameE:V,3 and the sameauxiliaries but instead of returning to the E:V they continue a step farther to a dominant 13th-chord on G shown in the sketch!.The shift toward the primary tonic now takesplace by

143

tona/ity Ex. 1-47a. Ravel, "Le Martin-Pécheur," No. 4of HisioiresNature//es. On ne peut plus lent :flm:r.||:| r.¢||»l;m1na |/._192|»»1 P uv li

QA

/'

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tonality Ex. 1-47a continued. Tres cahne

m. 15

r 3w

Je ne res-pi-raisplus, tout

m. 16

tona/ity 145

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ion qf the publisher, Durand etCie., and Copyright 1907, Durand etCie. Usedby permiss Elkan- Vogel, Inc., sole representativethe for United States.

148

tonality

Ex. 1-47b. Synopsisof tonal structurein the Ravel.

,'Ff:!"[" E!:

"y/ "y/y/

[Fs

92>+» >[> ~~

>> y>~

>[>~ ~>y>~ » y» >y/y»7

chromatic successionto Ff: "I". It is interesting that the harmony of m. 18 is a shadowyreflection of thoseof the preceding bars; it is thus of somewhat dominant character, although its minor 7th, E, is delayed to give the harmony relatively greater stability. After a seriesof auxiliary harmonies on E, D, B in a descendingstream! the Ff harmony is returned as the ultimate tonal resolution.

The techniqueof tonal shift which had earlier introduced the secondary tonic E mm. 9 12! is related and should be compared. Measure 9 begins with a quasi-dominant of F$. This harmony is "horizontalized" note the bassnotesCf, Gf, B not shownin the sketch! until at the secondhalf of the samebar the basssettleson the primary tonic note; but the chromatic successionof parallel auxiliary chords above the bassnow settlesinto an harmonic complex making of that Ff a dominant 9th-chord. The dominant 9th on Fll becomesa minor 9th-chord in m. 10, the Fx so it is notated in the voice part! resolving into E:I after tantalizing delay, and the C serving as a chromatic link to the B of E:I. Despite these passing, nonharmonic factors, the progressionat its basis is diatonic, the two essentialharmonies having diatonic relation in B, representedas a parenthetical system in the structure. One further element in the succession described above must be noted:

it is an harmonic elision of a sort typical of the style. The dominant on F$ "expects" B, which would in turn be a potential dominant of E. But the movement here omits the B harmony which would resolve the one and prepare the other. Viewing the successionin this light helps to clarify its nature as a diatonic rather than chromatic succession, and further accounts for the

"parenthetical" function of B. Finally, attention is called to some summary concluding notes which

tona/ity 147 have direct reference to the sketch in Ex. l-47b. The reader should note: the

extent ofchromatic relation in contiguousharmonies, functionaland nonfunctional, especiallyin embellishing successions; stylistically characteristic resistance to semitonal successions in the voice-leading reflected in the overall tonal successions F# -E-Fd!; the significance of an underlying motive,

,

as bassline determinant; the types of embellishing

chords e.g., major dominant 9th-chords and derivative half-diminished 7th-chords!; means of tonal stabilization opposing the activating power of dissonance-the broad F il-E-F# successionof prolongation and neighbor auxiliary; comparison of the I of m. l and its recurrence atthe end; m. 15 as avery visible prolongation and embellishment ofsurrounding, functional, structural harmonies e. g., retention of gil!; mm. 18-22 asprolongation of F it:I; the recurring e of mm. 18-20 asa backward look toward the secondary tonic E; and most fundamentallythe underlying closed and explicit manifestation ofthe primary system, whose expansions areunorthodox in the quasi-modal prominence of the system on E, but nevertheless, intheir secondary tonics, entirely diatonic more strictly, pentatonic! in their relation to F it .5° Bartok. String Quartet No. 2. Op. 17. third movement

The Bartok movement posessome relativelycomplex problemsof tonal analysis, butits approachesto tonal structure are stringent and definite: it is a structure whose chief foci are diatonic to A. Consider, for example, the tonal implications of the succession inEx. 1-48a,which constitutesa recurring Ex. 1-48a. Bartok, StringQuartet No.2, Op.17, thirdmovement.

Lento JS63-60! -m.20

Y # jf ry P 12-Llf

Copyright920 I byUniversal Edition; Renewed 1948. Copyright and Renewal assigned to Boosq CQ Hawkes, Inc. for the U. S.A.!.

The collection of secondarytonics, togetherwith the primary, havea very characteristic formulationwhen renderedin a chordal configurationas PCssuperimposed over E: . Compare the formtaken byFil :I pearances.!

in openingand concludingap-

148 tona/ity

harmonic motive in the movement. While some fragmentsof Bart6ks music are quoted in connection with the following discussion, the reader should, ideally, have the scoreat hand.! The succession is stronglyaflirmative of A,a,°° the tritone F,B acting as a quasi-dominant of the sort discussedearlier. With the tonic A in mind it is established early in the movement!, consider thetonal implications of the lines in Ex. l-48b, each ofthem directed toward or otherwise giving prominence to leading-tones of A, the tonic resolution explicit in the bass following in each instance. Ex. 1-48b. m. I5 _ :;r1i _

P My

difn. _. QYZ 3. -1 I

o

#0 or

rn. 23 ff r P'

Like other freely tonal music and this work is part of a vast body of music of comparable aestheticin the twentieth century!, that of Bartok relies upon someof the techniques oftonal expressionnoted earlier, especially the following: emphasis upon the tonal center and its supportive functions by iteration, prolongation, stress, etc.;affirmation by quasi-functional harmonic and melodic elements resemblingconventions ofthe tonal period the quasi-

dominant actioncontinues torely heavilyupon theconcept ofleading-tone tendency and accustomed rootmovement!; and by linear direction toward and indirect or direct encirclement of the tonic PC, especially at points of formal punctuation. °°I nthe Bartokmovement, the tonal system is typicallyone inwhich modaldifferences are freelyexploited. Symbological distinctions between A and a despitethe movements conclusion on the minor3rd sonority!,or betweenl and i, etc., are notobserved inthe discussion of this example.

tona/ity 149 The initial

two motives in the cello form, with the violas E,

a kind of

evasive tonic -|- dominant! sonority in which the root, A, is tentatively suggested byencircling motions above and below: B-Bb and G# are pivotal in these movements cfl A: VH!. Thereafter, the cello phrases consistently conclude onA, which is alwaysa longer note mm. l 1-12, 15-16, 21-22! and often directly supported by leading-tones aboveand below. Its descending triad of mm. l8-l9- may be said, moreover,to act as V to the I minor 6th, C# and A! which follows. At the same time, the long major 6th

in the

viola mm.

9-12, then

15-17! resolvesas a V to the later octave C, eventually absorbed into A: I. Note that this major 6th is also part of the traditional A:V-,, just as the tritone of m. 20 is both suggestiveof C and an element in the traditional dominant 9th of A, which of course prevailsat the cadence. One of the most compelling of the melodic formulations of-quasi-dominant character is the first violin motive of mm. ll-12, beginning on one leading-tone of A, ending on the other, strongly expectant of the tonic PC, which is held in reserve. The same is true of the second violin motive,

mm.

l-2, which accompanies thetonic note. The opening measure thusboth establishesA and sets againstit a Vlike motive primary dominant augmented 6th?! in the second violin. Thereafter, within the first section, the melodic units often are projective of and/or directed to G# and Bb leading-tones toA, sometimes,as a reflection of ambivalent tendencies, notatedenharmonically! and D#/Eb E is established earlyby emphasisin the viola, andquasi-leading auxiliariesare strongly felt in the recurring Eb of viola, mm. .13-14, andsecond violinthereafter; in the first violins active tritonal successionof mm. 1-2; in the major 2nd of m. 5, etc.; and especially at the releaseof the first violin, m. l9'!. While these interrogative leading-tone gestures and motives occur, the cello part continues tobe directedconsistently andrepeatedly towardthe A of the cadence. The longest notes inthese opening 22 measures are E, D, C, B, and A, their functions in the tonality obvious. It is interestingthat despitesuggestive leaning toward E, and early emphasison E in the viola, that PC fails to emerge cadentially, omitted from the quasi-tonic sonority at m. 22, perhaps because of its subsequent prominencein the tonally contrasting section to follow. Immediately before the cadence,there are two tentative suggestions of quasi-tonic harmony: the end of m. 20, where the harmony is activated by a suspended Bin the bass, delaying the root; and the second half of m. 19, where thelower instrumentsexpress thetonic at the sametime that the upper instruments expressthe inconclusivedominant.°' Preceding this is a quasi"Quite possibly,the Cil, Dil, G8 sonority ispredictive ofthe C8 secondary tonicto come.

150 tonality

dominant which includes the upper leading-tone, Bb, notated as All! of exactly the kind referred to in theory earlier in this chapter. In Ex. 1-48c are some of the melodic and harmonic materials to which

we havereferred, aswell as others that are supportiveof the tonality of these opening bars. Measures 23-27 repeat the cadential formula, reestablishing the A tonic, coincident with the now familiar G# -Bbleading-tones, thento leadinto and prepare the contrasting lento whichfollows. It is impossible to examine every detail, but let us at least point out further manifestations ofthe cited techniques oftonal expression. Again there are clear examples oftonal direction and orientation. The first violin of mm. 23-27 articulates a phrasebeginning and ending with the two A leading-tones, stronglyemphasizing Bb.Later it comes torest on G# mm. 33-36!, which resolvesto A, then conjunctly climbs to and insistently reiterates E, as part of the cadence at m. 46. The final harmonies of this section are ambivalent in tonal meaning. The violins, one sustainingE, the other reiterating A and its upper leadingtone, are supportive of A, and the F of the inner voices seems notto have departed far from the primary tonal system. But the ultimate thrust of the cello toward c and

clk' foreshadows, in the environment of

an inconclusive

cadence, theC# tonality to follow. The more intense outer voices piano,as opposed tothe pianissimo of secondviolin and viola! project the C#,E seeonday quasi-tonic sonority.

The Ct#has beensuggested harmonically,of course,much earlier. The sustained, motivic harmonies at the beginning of this section first reiterate and reconfirm the quasi-dominant and quasi-tonic functions in A mm. 25-27!. But at mm. 29-31 they effect a tonal shift pointing to Cl, the latter tonic sonority having the same form as that of A which preceded. From here to m. 46 the melodic lines arein essenceexpressive ofA and Cif, sometimes ambivalent

between the two tonics,

like the first violin

from m.

31,

which has quasi-dominant suggestiveness of Ci# at the same time that the resolution of the long G8 to A is reminiscent of earlier tonal reference. The second violin from m. 27 also tends to expect C8 ; it prepares perfectly the eventual C8,E resolution at m. 31. In the climactic, essentially rising passageleading up to the cadence, thesubtle hints of C# combined with its careful evasionare of enormous importancein the effectiveness ofits ultimate arrival in the cello. Consider,for example, the substitution Q/'Bit _/or C itin theotherwise diatonic rise ofthe jirstviolin fromgit to es. The transitional passage towhich we refer above is sketched in Ex. 1-48d in a manner intended to show in larger notes, by enclosures,by arrows, by connecting lines,and by other symbols! some of the expressionsof tonality we have notedas wellas othersof importance and comparablefunction and technique.

152 tonality Exe 1°48d|

Tonal fluctudderivation ion Cl:ofl"! mm.23-36

I

=" ""*_1'.. :Qi-.iw ~ ` 9292|' .AW

_ rr

IQY'_ na U1 ri1

--" _/` "ir II

--'- "-------

if / 1 '

.| il/ ul

/ ll

~ 92

as -. 92_/ `_/"

"`L""iF1?'|S&If-§5".§f15C ' i, 4|"F='l|ll I ff* [L_

A "7

|| _II -1111 / l . I -iv ./ I

' --~»

Increasing of cl factorspredominance

Structural emergence of cl , Gll, E.

-11; ""' J "',~~ y

mm.37-47 '

=,_ "`*°§f"ii ii

92 l

t ' #;_ _;ff_; __ __,"£

Largely chromatic linear successions toward¢- A, CI i+ , andoi l-1-----»

It is necessary fromhere on to limit discussion to the most essential points, leaving for the readers independent investigation a number of elements of tonal expressioncomparable to those alreadyidentified in earlier portions of the movement. The closerelationship of C# to A, the fact that G# is both the dominant root of one and the leading-tone to the other, and the coincidence of the notes C#and E in both tonic triads, make it possible forBartok to express a certain ambivalencein the contrasting material following m. 47, to establish tonal variety by functions suggestive ofCH! without relinquishing implications of the original tonality. Note too the interrelations explicit in the forms the quasi-tonic harmonies take: the occurrence of A:I,,, with Cll as its lowest pitch, in the motivic harmonic succession, as well as the implicit relation between the two tonics in the M,m triad on A, consisting of superimposed minor 3rds which are the basic ingredientsof the two chief harmonies of resolution:

.

Melodic analysisof the first violin part from m. 47 to m. 67 would show very clearly that it is built around E at the beginning a member of the Cll:I and at the same time a dominant preparation for the ultimate destination of the line, A! : its first nine measures, in fact, are a prolongation

tona/ity 153

of e, after which it prolongs e3. Followingthis high point in a broader sense, a resumption of the e of mm. 41-46!, the line descends emphasizingBll, the C# leading-tone! in a quasi-dominant formulation from m. 60 to m. 63~. The descent of the line after this point, sequential, is a condensed,abrupt, reaffirmation of A, the note of its cadence,through an intriguing diversity of modal implications. The secondviolin part also requiresexamination in similar connections. It is, of course, conceived to a considerableextent as a parallel doubling of the first violin; much of it is a prolongation of the note B lending a modal quality to the tonic Gil!. Shortly before itsfinal resolution, on an emphasized A, there are reiterations of the upper leading-tone Bb, notated as All, in the repeated successionB-All!, and the lower leading-tone, Gil, has a single tentative appearance in m. 64, important nonetheless inits propinquity to the arrival point. The strongest aiiirmation of G# is in the cello, fundament of the harmonic structure, whose melodicanalysis wouldshow a prolongation of c# in the first several measures,then describing an octave descent to Gil, through Gil and Fil. With the cancellations of sharps except for the bass note of A:I° in its familiar form and context at m. 65! at mm. 64-67, the cello comes into agreement with the other voices in the shift which restores the primary tonic. We do not take space for the melodic analyses suggested, although the reader may wish to do so in confirmation of the tonal expressions ofthe individual lines in this passage;perhaps theconsequences of such analysis are,in general, preevident. The harmony is, of course, conditioned to a considerable extent, like the individual lines, by Bart6ks technique of mirroring the outer voices and filling the texture with the parallel 4ths which accompany them in the opening bars of the Lento assai.Nevertheless, harmonicfunctions, especially quasi-dominants andquasi-tonics, are of decisiveimportance in the expression of tonal feeling and coherence.The first harmony of m. 47 is a recurring Gil :I. Some ofthe auxiliary chords by which it is embellished,incorporating one or more leading-tones, actas quasi-dominants e. g., very potently at m. 50, third beat!; theseare markedin the sketch in Ex. 1-48e. The cadence of mm. 54-55 prolongs the I approached through a quasi-dominant in mm. 52-53! by alternating the root position with the first inversion. Elided

with this,the violinquintole figures describe triadic functions suggestive of A, alluding to the later restoration of that primary tonic. The sketch indicates essentialquasi-tonal functions in the modulatory section of mm. 56-67 Ex. l-48e!. The C|l:I is seen to persist in upper voices and there is a G# :V on G8 following. The Gil, as suggested before, has an ambivalence in the tonal dichotomy of this passageleading-tone ofA, dominant root of CB.! The cello reiteration of F it denotes a prolongation of harmony on Dil, coincident with the prolonged leading-

154 tona/ity Ex. 1-486. Cl; "v"

A="1"

mm.47-64 --.- - /E --. ..

:I "

' ,fl I-r

`"

v Y -_

`92 92~

_ inn.

-./' /_ " M- ""-""`il---~. lr____ "

_ if

92 -

l *__-:flags

92J l

|92!l. .f1V§.l lil! FSIC921.¢ ,' -_- I unix'--uqfuvn lT"'l7l11"lL_ 1».¢|u 1;-uf: 11 r qw _-¢/1.4-~_|u/-vi|. »r»_|_I1 nltniir1.-1 m|z.|.;i|1-r p aw?

-' cz

-_ _--_/'

Cf:l" prolongation, elaboration ---. .... _

`;92 »;_" l

|ig!=-Sllniiiér/ 1|~-C ll"' ¥ nn 1

V n

/-

ll' .

clwv

DL V » I"

mm.64-67 ____ iii"---~;92` *Ii

"* "-~---i1f;;~.

*_ ll-

.*'-`-`L=9292" `_--

2 "*--11. __

"3

__| "`---_i " """

A: "V"

Nl" V

V gf ||I||

tone of C# in the top voice; the All of this C#:II, as we have seen, later turns, asBb, toward the cadential A, so that the Dil triad also hassomething of an ambivalent function. The final cadential harmonies in the primary tonality are by now familiar, although in this instance the viola F of m. 66 makes of the A:I, a I-I-VI, the F anticipating the recurring quasidominant of

m. 66.

The developmental section which follows mm. 68-87! is of great interest tonally,and of considerable fluctuationand someambiguity, ultimately directed toward the SD level, D see mm. 84-85!. It should be regarded from two viewpoints. In the beginning of this section the two violins reiterate a motive derived from m. 47, except in m. 85 consistently a3rd apart. At first, the 3rds move twice to a D root, forecasting thePC of culmination at m. 85. The pattern is as shownin Ex. l-48f; the points designatedby accentare thosewhich are repeated or, in the case of the conclusion, stresseddynamically and otherwise prominent as highpoints in the riseof pitch. Note that the sketch notates theupper note of the penultimate interval, the only tritone, as Eb; it does indeedact asan upper leading-tone ofD in a harmonic complex which is a strong quasi-dominant consisting of the normal dominant root, A, and the upper leading-tone, Eb.! Also striking is the relation of adjacent interval pairs ascompared with that cy'the movement s two chiqf tonicsonorities. The secondstratum of tonal progression the two are initially in a kind of antiphonal relation, but come increasinglyinto confluenceas thedevelop-

tona/ity 155 Ex. 1-48f.

' ts / 1"'

Y

Y

Fi Q

[Y 47'

92,_..-"'f

J_

92__,vj Progression ofupper tessituratoward D tonicization culminating atm. 85

[.f.

1

ment progresses!is that of the viola and cello, in which the second violin also participates. This series ofharmonic successions begins mm. 69-70! with the familiar cadential formula in A, with its familiar resolution. The progression is then repeated fourtimes, with extension andvariation, deceptively moving to harmonic sonorities of changing tonal implications. The A: V by this time in the movement clearbf expects the A: I , realized only in the first succession of the series.The deceptiveresolutions createa senseof fluctuating tonal direction: each is a dissonance, each has one or more specific tonal implications seeEx. 1-48g; possibly theyare quasi-functional in C# and A!, but the sum total is one of considerable,uncertain fluctuation until the final clarification at

m. 85.

If the flats like those ofmm. 74 and 77! are heard enharmonically, the range of tonal fluctuation is lesswide than it seems.Against the precedents established earlierthe secondharmony of m. 74 is interpretable as Cl#:IV and that of m. 77 asC# :V. The sketch in Ex. 1-48g showsthat the second Ex. 1-489.

WT W» P 716 »...~»»» -. etc. f

FJ #Q

PQ

156 tona/ity

of these,the Ab/Gil harmony of m. 77, now recurs asconsistent penultimate

harmony inthe motivicrecurrences which followf. Whileit has the effectof recalling earlier quasi-dominants of A and Ci, iit resolves indecisively to deceptive dissonances until finally, abruptly on D. In the course of this the expansion ofthe motivic interval is of coursedeliberate and intensifying, and the tonal implications of its placementsare visible:with the extraction in Ex. 1-4-3g. Example 1-48his a sketch, like those seeneazrlier, intendedto represent schematically some of the observations of the albove discussion. The piano appearance ofthe motive in mm. 86-87, somewhat whimsical, tends to subvert any tonal decision achieved by the strong sauccession preceding it. It serves both to establish inconclusiveness of cadence and to recall in a subtle

inflection the primary tonic, at the same timeeasi_ly interpretablewithin the secondary tonalsystem onD, just aflirmed, neither of which is supportedin the measures which follow.

Ex. 1-48h. mm.68-88 un

_ ,, _--_ /` _-----~--_` ---__ ' `~ "_-----_ 4 .r |aw/ rf? 7 '1" v-3 /h.a's-|'92s»:Lf1qf 'I ,rg: ."» _ ."lui'_in,_/`°'°"8 . , -_____--'3`~._____-.A I I- =='ii _ ;» v V v Ir ' lI.~ ":" 'Y lI 9 ||:f.! ' .'T||..§|.§"..|'_ll. T ll ' f C7 V ||ll 1 ,ru-1.: gli' llff -dill' _IU if Ilmil. 1' fi r,' 4 I |w 'I 924 I *_* | ,, . 4, . fx ,` U r v ¢-92 M 'Q _ __ ____ '_ 'I ' H l'. h I " Is ' 'A ._ `:_ U Lil ' _ll H/ I 1 Z :fa =IY-1§{'!

ll: V

__ '

»= -I

~-_.______ _?_---

¢"

After A: V-l" in normal" succession, increasingly ` deceptive, fluctuant consequences A:V:of ambiguous. transitional tonal references linking A, D.

The Bartok movement approaches now asecond stagein developmental procedures whichwill lead back toclear reaflirmation of the primary tonality. The sourceof the materials developedafter m. 88 will be found in the movements first bars first violin takes the cello motive fof mm. 3-4 as its point of departure, asof coursedoes theviola; the opposite voices-second violin and cello-take for extension andvariation the top voiczeof mm. I5-I6 , which in turn relates to that of mm. ll-12, or the secondviolin of mm. 7-8! . The tonal Huctuation of the section followilng m. 88 begins with a strong allusion to F, F and C# symmetrically relate:d toA; it is a tonal reference in which both voices join.

tonality 157

Following the first few bars of this section Ex. l-48i! tonal meanings are again relatively ambiguous. The Bl»,E tritone so suggestive ofF in the above-mentioned extract gives way now to rapidly changing tritonal relations in which the lines do not concur!-each of them having certain tonal implications but none of these realized, as tonal allusion shifts from dissonance to dissonance. In the brief sketch in Ex. 1-48j mm. 92-97! the EX. 1-48i.

9 #jf-5 F. Ex.1-48j.

Q . l*

r1 T

git; rf-3-

indeterminate tritonal

fluctuations are

Fw !

1

ref-1

bracketed; the sketch also makes very

apparent the essential chromaticdescent ofthe bassfrom the BI; of the initial F reference down to F il, a point of brief prolongation. It must be acknowledgedin passing that the above-noted passagecan be broken down into more explicit tonal allusions if the individual lines are examined. For example, the top voice of mm. 93-94 expressing Eb! is sequentially related to that of mm. 89-90 expressingF!. But the difference is that in the caseof mm. 89-90 the lower voicesupported the reference toF; in mm. 93-94 the lower voice is more independent, although the Eb :VII is briefly formulated in the bass notesD, Cb, Ab at mm. 94-95. Another significant difference is in the fact that in the earlier passage the prolonged E leading-tone of F! moved down to C F: V!, while in the latter the El»,D succession moves to A in one of the tritonal movements!, tending to disrupt any senseof Eb which may have beenfleetingly conveyed. Again, the passages treated above havetonal implications which can be felt; that they constitutean interval of relativetonal flux and obscurity at the

158 tona/ity

same time is anecessary conclusion of analysis, revealing an important aspect of theidiom andthe form.What isevident here is theshaping opposition of relative ambiguitywith relativeclarity oftonal reference, a principleprobably applicable to all idioms of tonal music despite differencesin the means and extent of ambiguity.

In mm. 97-991 the lower voicetraces theleading-tone harmony of Cl# and theupper voice underscores that reference in an elaboration of thequasitonic harmony; here, C8 is a clear passingtonic in which the voices concur. It is broken up with the bass Gof m. l0l_ andthe tritonal leap from G# to D in the upper voice of the same bar. Either note could of course function

ornamentally inthe C8tonal system, but theysignal atonal departure at this point.

Measures lOl-ll.,in which the materialsare developed climactically, are thesort ofpassage sometimes described as pantonal, although tonal specificity andunanimity become relatively morepronounced inthe approaches to the cadence. Asthe climax develops thereare brief tonal allusions within

the linesbut thegeneral tonalef'l`ect largely is indeterminate. At onepoint a suggestion bitonality of arises, when thebass expresses a quasi-dominant, then tonic, formulationin imitation of a similar configuration in the upper voice expressing Gb. The tenselydissonant clash of tonalreferences Ex. 1-48k! is calculated to serve therising intensity. EX.1-QUIK.

. a- J

,

ii/J 3'

K,¢"'|

tg

The approach to the cadence atm. _lll, is also bitonal, the first violin persuasively aflirmative of AI; primary leading-tone ?!, the lower voices joining in quasi-dominants andquasi-tonics suggestive of B. At the cadence, interestingly, the upper voiceends with a quasi-dominantformulation in Al» , the lower voices statingthe first inversion ofthe B tonic. All of this isshown in the sketch l-48m! of mm. 104-lll . Finally, it should benoted that while the cadential successions into the B: I are underway the bassvoice is itself of extraordinary fidelity in the

tona/ity 159 Ex. 1-48:11. Concurrent, transitional references to A* and B

mm.l04-1 ll

L

/&l is

__ ll*

....

..___

__

expression ofthat harmony in a linear succession which includes a complexof leading-tones mm. 107-ll; see Ex. 1-48m!. Bartok has manifested inthe abovepassages the traditional tendency to establish in developmental areasa pattern of tonal fluctuation. Following m. ll4 the primary tonic is restored in harmonic and melodic functions of greater stability, recollective of those of the first part. The fluctuation into the primary tonal system retonicization of A! is brought about in a number of ways. The iirst violin part of mm. 112-13, the only voice occupiedwith significant melodic motive, is itself revealing, as indicated in

Ex. l-48n.

[;"".f 1 A: KW"

NI!!

At the same time, the harmony is a quasi-dominant m. ll3! which is an expectant preparation of the A of the cello. That harmony is built on B, which has throughout the movement beena vital factor in quasi-dominant functions within the primary tonal region. The V contains the upper and lower leading-tones the latter notated as Ab!, and it is anticipated in the first violin of the measure preceding.The D of the viola might be saidto have its premature resolution in the C-C# of the first violin motive. This quasi-

160 tona/ity

dominant see Ex. 1-480! is itself introduced by a harmony of semitonally related auxiliaries except for the bass,which is tied!-suggesting again the kind of linear determinant

of harmonic auxiliaries which

is so crucial in this

and other late tonal styles. TheV and its prior auxiliary harmonic complex are shown at the outset of the concluding sketch Ex. 1-480! of tonal functions and aH'iliations in the Bartok movement.

Exo 1'48o|

mm. 1 2}I I /:___`92

he IUX

rv -. 51;

on '

,- f

Ia `

~i

/

~+ F-

IN 1

l:P1I' il; Y '

fnIrt, : h

T

92» l -° A; ~v»-»» *i

|28 d 9¢ml¢°*| ¢8hb°"

::...~@-=.;.: =: l-92 92 Af. , l .921||_I"i1

/K

` i fl /I

__ -

__

_

us"

C|92°|l¢t¢fi9'-i¢_3¢i|i°fY °f

mm -en L --1 '_.."§¢:¢f '. f

wg 7/

'

_ I-

r

Underlying chromatk bass

successionl and linking 5

-_ , #I-_"~_

A

n~;';|~1.av192./ __:92.r :gil ",now Q _J r ,r92 aE |¢ | ~--' , 92_"Y'L|l`Ll'| ri.; l.| -¢ +" a na I* W 92` 1 II ;` -,_ ____ ____---" ' &/4 O 92 ,ir 92é4' 92.»r nv/vu? nv" "ln

V, -_ 92 » 92` 9292 ;,

The phrase which begins at m. 114, with the first violin imitating the cello abortively, provides in the unaccompaniedA the resolution of the preceding quasi-dominant. The diminished triad motive thereafter occurs at various levels: on A and All, on G and Gil a momentary sounding of A: VII!, on F and Fll _The tonal references arethus free and fluctuant but two factors are of particular importance: the ultimate quasi-tonic six-four over the bass dominant E, toward which the entire successionis directed, and the chromatic descentof the bass melody with two register shifts! from the ton-ic

A to

the dominant

E-the latter

sustained for

three bars.

Both factors are represented in Ex.l-480. The quartal Ighas considerable prolongation, embellished by quartal auxiliary chords again consisting of semitonally related neighbors i.e., leading-tones! in a rising sequence of pitches. The prolongation of the Ia traditionally a dominant auxiliary! 4

is a striking functional preparation forthe _final resolution. One of the embellishing auxiliary chords, an enharmonic notation of those preceding,is itself prolonged asa quasi-dominant-the harmony built

tona/ity 161

on the bass All at mm. 127-29; this harmony is embellished by auxiliary neighbors ofits upper members, but the bassAll remains constant. That All can again be understood as the upper leading-tone of the tonic A and, in accord with its notation, as the ambivalent lowerleading-tone tothe B prolonged withit, the B also resolving to A mm. l28-29 and l3l in Hrst violin, and mm. 130-32 in cello! in a functional recession whichhas beenrecurrent

Ex. 1-48p!. The harmony is, likethe preceding I%, fullyquartal in construction. Ex.1-48p. /~, 'T'

Ut * EK. 9292 9292'

r

. #»'9292. 9292

In a quartal harmony the identification of root is problematic; the analysis suggested here givestonal, functional precedence tothe bassnote of the harmony on A# as doesany reasonabletonal assessmentof the tradi-

tional Ig!. The harmony occurring over All canbe construed as quasi-dominant twice removed V/V/V!, introducing the V/V on B, the All acting in its ultimate course as an upper leading-tone to A see Exx. l-480, p, q!. The cadence is consummated by the motivic quasi-dominant tritonal sonority F, B! and the quasi-tonic harmony A, C!, preceded bya dissonantanticipation of I combining C with C# in characteristic modal ambivalence. The entire succession might be conceived asanalogous tothe following traditional sequence oftonal harmonies, a hypothesis best examined in actual, attentive listening toBartoks cadence in comparison with the conjectured prototype, which is reprinted in Ex. 1-48q in explicit demonstration of the significance of the concept of quasi-functional tonal harmony. Ex. 1-48q. _

=, __.n

=*" BW

gg

92 !|V 92_2

Cf. -Q-!

Q it _ v/V/V

~"~ Q

lfgvg Eg

ew

,,.

;

'Q

V/V v

B

= _

162 tonality

While the cadential formulation is underway there is a most important manifestation of tonal expression in the first and second violin parts. The earlier prolonged B in the former, with its A# auxiliary embellishment, now descends toA az!, after which the line falls two octaves intoa cadential resolution on a. The second violin,moving in parallel 3rdsdown to G, a modal auxiliary, underscores the arrival on the tonic in a concurring resolution. The primary tonic is now well reestablished the ; restis its prolongation for conclusive finality. The sketch in Ex. 1.480 shows the motive of rising 3rds, only mildly evasive in tonal allusion, subtly animating that concluding tonic elaboration. These motivic iterations supply A: V once m. 132! and otherwise recall the secondary tonicsC# and D, as well as the B of the cadential function just realized. The motive occurs onceagain, on Eb, in a slight stir of tonal evasiveness immediatelypreceding the final two tonic statements, this time lacking the quasi-dominant tritone. Further enhancing the sense of tonal resolution is a melodic tonal reference-that of the first violin in its phraseat m. 135 ,again beginning on Gil and ending on Bl, compare mm. ll-12 or 23-27!. We have sought in the above discussions toexplain the overall and specific directions of Bartoks tonal structure in this movement, identifying tonal functions in melodic and harmonic extracts, and pointing out areas of relative tonal stability and relative activity as delineating structural factors and giving analysis ofthe meansby which tonal referenceand tonal fluctuation are carried out. The discussion is not of course exhaustive, and it is confined to factors

of tonal structure and technique. A tonal .gzstem of the sort described and at times charted with reference to earlier works could of course be derived and

graphically represented: it would reveal the remarkable extent to which tonicized PCS by which the system is expanded are diatonicalbf related loA. Webern, Four Pieces, Op. 7, for violin and piano, No. 1

An important example from the works of Webern is given next as Ex. 1-4-9a.It is the first of the pieces for violin and piano, Op. 7. The tonal structure isof truly compelling effect.There can be no insistent suggestionof consciouscreative intent, although Webern is said to have afiirmed the importance of tonal relations in his works; the tendencies which seem unequivocalin this piece and it is representative ofmany! may well be intuitive. The important concern of analysis isexperiential efzct as plausibly described and underscored in objective evidence. Such evidence of course requires interpretation, and verification in experience i. e., listening!, with such experienceconditioned by the understandingwhich follows from analysis.

tona/ity 1

63

Ex. 1-49a. Webern, Four Pieces forviolin andpiano, Op.7; No. 1.

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164 tona/ity Three sketches will illuminate

tonal functions

of different

kinds, all of

them oriented toward a primary tonic Eb. The tonal system of the piece is simple suggestive secondarytonicizations of the leading-tone D could be inferred, but the primary Eb is virtually without enrichment of expanding secondary systems in this very brief, concentrated work!. Indeed, there is no significant fluctuation: no real deviation from insistent referencesto Eb at a number of levels.

The first sketch Ex. l-49b! shows the descentfrom structural manifestations of eb to eb and finally to el», root of the final chord. The first of these pitchesis simply held asa violin harmonic, establishing the tonic PC. The second is reiterated, and preceded by a succession derivedfrom a chromatic set illustrated! which signyicantbf avoids Eb. The repeated figure containing the eb is released, significantly,on eil and d2,creating an intense expectancy ofreturn of the establishedEb. Ex. 1-49b. Tonal tendencies in the Webernupper stratum. chromatic set

lacking El'

J

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Q 9292 9292 _,

92 .

At what might be described asan internal stratum in the pieces spatial field, a quasi-functional harmonic recession takesplace, as shown in Ex. l-49c. A further, implied step in this succession,expressed asa melodic configuration, might well be included; it is representedin the third sketch, Ex. l-49d. When onehears thissuccession its functional implications are very persuasive. Thetwo neighbors of the V are shown by the symbol N ; one of these is stated in two

registers, and

is embellished

by its

own lower

tona/ity 165 Ex. 1-49c. Quasi-functional harmonic successions in theWebern. NN --+` 9292 i

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re;

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neighbor asindicated in Ex. l-49c. A dissonantform of I precedes the final resolution. Throughout the succession,the importance of voice-leading, its smoothness aresult of linear adjacencies,is apparent. Finally, an ascent inthe lowertextural stratum Ex. l-49d! complements and is a counterpoint to the descent notedearlier Ex. 1-49b!. Again, Eb isits center; the succession is Eb, eb, e|». The harmonic succession noted above is

included parenthetically,and tendenciesof leading-tonesand otherauxiliaries are indicated by arrows. Particularly effective in function is the penultimate event, Fb Ell!, which functions like the d2 of the violin, shown in the Hrst sketch! as a leading-tone expecting, and directly antecedent to, the final tonic root.

Ex. 1-49d. Further especially lower stratum! tonalexpression in the Webern. J

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A summarysynoptic sketch,superimposing andincorporating principal factors representedin these three, would project a comprehensive imageof tonal tendencies and relations in the piece as a whole.

166 tona/ity Berg. Four Pieces. Op. 5. for clarinet and piano. No. 4

The works of Berg, strong and resourceful in their expressionsof tonality, within both serialand preserialprocedures, aregreatly rewarding in study of this kind. The following discussionrefers tothe Four Pieces forclarinet and piano, Op. 5. The reader would do well to study others of these piecesas to manifestations of tonal order; we refer here to No. 4, in C. In this piece, expressions ofC are of numerous kinds. The opening C basspedal is of courseitself highly suggestive andpreconditioning, especiallyas theroot of a major triad to which is added two pitches that might be construed aschromatic auxiliaries of the G and E of the triad!. During the statement of this reiterated harmony consistently of the

duration of 3,f!! theclarinet sounds two motives one the extensionthe of other! which tend to encircle the C tonic, one of the motives beginning on B, the other on Cll. This principle of encirclement is made extremely compelling and explicit in the return of the chords atmm. 1 1-12 Ex. l-5Oa!. Ex. 1-50a. Berg, Four Pieces for clarinetand piano,Op. 5;No. 4 clarinet notated at sounding pitch!. Langsam

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In the approach to this reafiirmation of C, there isa strongly suggestive and functional succession whichin some ways expects C and at the same time powerfully manifests a quasi-dominant sonority in the piano in mm. 9-10. The piano harmony consists oftheconventional Gand F with the upper

tona/ity 167

and lower chromatic auxiliariesof C-all of this symmetrically disposed around c2, which appears only fleetingly in the clarinet and is otherwise deliberately withheld, intensifying the expectation of the C resolution of

m. ll and its anticipation inthe clarinet,as ca,in m. IO. SeeEx. 1-50b.! Of major interest in this passage is themotion of the clarinet line toward

ci. In m. 8, the clarinetbegins thisprocess withtremolo repetitionof two pitches: b|»and g. Once theline moves,it doesso with insistent implications ofbileveled compound! structure functionally directed towardca, as summarized in Ex. l-50b..The lower level of the compound succession moves

toward cz,as does the contraction in thepiano, whose chromatic neighbors of C are transferred unresolvedto the clarinet in m. ll, at a lower octave. Of Ex. 1-50b. Berg, Four Pieces for clarinetand piano,No. 4. Mm. 8-10and sketchof recessive actions towardC. /

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course, thereis tentative resolution in the arrivals on c insistently repeated in slowing tempo in the clarinets upper stratum!, then C in the bass ofthe piano; the absent c' does appearas root of the pieces linal sonority. Other factorsof quasi-functionalsupport of the primary C can be noted briefly. For example, thefinal recessionof tonal structure toward C involves an underlying chromatic ascent of the clarinet toward ga e-f 1-f li-ga! in a progression bywhich that dominant is projected with great emphasis.The progression isopposed toa piano descent toCC in mm. 13-17, after which the C triad with B! is first outlined in the piano bass, thensustained in a chord of piano harmonics and, subsequently, repeatedin unstopped notes over c. The function of the pianos gradually accrued penultimate harmony echoed in the clarinets final phrase! as a complex of leading-tone auxiliaries resolving to the final tonic is apparent in the sketch of PC content in the two harmonies Ex. l-50c!. Ex.1-50c.

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tona/ity 169 Summary notes on the un/'versa/ity and s/gn/Hoance of the pr/nc/p/e of h/erarchic Iona/ order

Any analysis of liturgical chant demonstrates,like that of Ex. l-41, that a critical factor in its structure is the quasi-tonal hierarchy which yields, at points of cadence, to preestablished componentsof the modal scale as elements of relative finality, the ultimate being the final itself, which is in this sense unmistakablyanalogous to the tonic of later, formalized tonal functions. Moreover, we have seen thatin modal polyphony groupsof auxiliaries tend to revolve around notes most basic in the hierarchic, modal or quasimodal structure, and this too mustbe recognizedas analogousto the systematized tonal functions of later styles. We therefore take the position that modality whether pure or in later evolutionary stagesis a manifestation of the fundamental principles essentialto tonal organization in music. Indeed, in much modal music the later specificconventions oftonality are in vivid evidence. And while the transition from purely modal to emergent tonaldevices isgradual-and overlapping among different coexistent styles, it is clear that the conventions of the tonal period are incipient and

sometimes very significantly manifest in applications which long predate their formalization in Baroque and Classicalliteratures, and that, in modality, the concept ofhierarchic tonalorder is trubf relevant.

Along with the wider applications of leading-tone functionsby the uses of musicajictain contexts of modal derivatives, onemust regard as indicative of later tonal conventionsthe increasinglysupportive role of the bass inearly multivoiced compositions--a role characterizedmore and more by the useof 4ths and 5ths of quasi-dominant-tonic succession,less andless bythe intervallic and rhythmic equality between bass and other voices which is a Renaissance ideal.

An element in this trend is, of course, the bass line ac-

companying sixteenth-centuryvocal declamation in, for example, the practices of lutenist composers;its function is substantially that of defining root progressions ofclear tonal purpose, asultimately is that of the figured bass of the Baroque, an important embodiment of harmonic conventions in the tonal period. It is primarily the practice of musicajicta, ofcourse, with the increasing emergence ofleading-tone relations,which contributed in the Renaissance to "Of _]osquin,Donald Groutwrites thatwe becomeaware ofa consistent organization of theharmonies along the linesof ourown commonpractice. Thoughthey arestill rootedin the modalsystem, the chord progressions and thegeneral harmonic plan in most of_]osquins works are to a large extent governed by dominant-tonic relationships.One sign of this organization isthe conductof the bass line:more andmore it has begunto bedistinguished from theprimarily melodicnature ofthe othervoices andhas begunto assumethe function of a harmonic support;consequently itoften movesby fourthsand fifths.This kind of harmonic organizationand this type of bass movement become increasingly prevalent in the course ofthe sixteenthcentury. [A History ofWestern Music New York: W. W. Norton and Company, Inc.,1960!, pp.243-44-.]

170 tonality

the gradual dissolution ofpurely modal conventions, especiallyin polyphonic music, and to the increasing expansionof tonal systems and consolidation of the conventions ofthe tonal period. At times densechromaticism is practiced," constituting one of the most intriguing sixteenth-century stylistic currents, to a degreethat eventuatesin a great flux of tonal reference Lowinskys floating tonality!, presumably as a conscious expressiveresource. It is noteworthy that most highly chromatic passagesare homorhythmic in texture, to

facilitate and

focus attention

on chromatic

successions without

competing textural complications. In other than chromatic styles, too, the increasing use of simpler textures, often in modified polyphonic contexts, is another manifestation of the growing emergenceof harmonic formulae of the major-minor system. It is easy tounderstand that more homorhythmic textures, linkedwith a bass line less andless independent,increasingly given to defining codified successions in relation to a central tonic function by movement of4ths and 5ths, would be associated with advanced trends in the rise of major-minor tonality. Moreover, one of the fundamental differencesbetween thepolyphony of the Renaissance andthat of the Baroque an issueof style,ofcourse, not of value! is in the latters firm conditioning of individual lines toward conventional tonal ends and of their confiuence in the Baroque to serve prescribedtonalharmonic functions.

In many transitional works the blends of modality and major-minor tonality are extraordinarily provocative and often of great charm. Where this occurs, modal vestigesshould not be regarded as inhibiting tonal order although many do in the specificsof major-minor terms!, but as conditioning the mean.;by which tonal order is expressed. In the Baroque, the use ofold melodies-chant, Lied, chorale,etc.-as cantusjirmus in polyphonic compositionsor assubjects forvariation in compositions of highly provocative ambiguities markedly in phrygian derivatives! exacerbated the modality-tonality ambivalences and conflicts, the melody imposing upon evolving harmonic conventions of the rising major-minor system vestigial inflections of the past, the harmony increasingly pointed toward the later conventionsof the tonal period. Music theory awaits a deeper understanding of this evolutionary processin, for example, seventeenthcentury music; at the same time analysis attempts to determine structural consequences of tonal referencein theindividual work,and seeksto understand how tonality, broadly conceived,functions counteractivelyand complementarily in its relations with other elements in the musical work. Historical factors of ambivalence and evolution concern the devicesof tonal aflirmation 3 not the fact of tonality itself See Lowinsky, Tonaligy and Atonaliy, for analytical discussionof music of Lasso, Gesualdo, andother sixteenth-century chromaticists.

tonality 171

The technique of juxtaposing tonally fluctuant areas ina musical form with areas of relative stability or ambiguity with clarity!, while dating from very early practices, isfundamental to larger forms of the tonal period. It is of particular importance in the late nineteenth centuryand in the tonal music of the twentieth century, where fluctuation may be over an extremely wide range. We have noted that in those twentieth-century styles in which the identity of tonic is less explicit, the expressionof relativerepose and relative mobility of tonal allusion is a particularly vital, even necessary,and often clearly felt, aspect ofmusical structure and structural function. When one or more of the essentials of dominant function are retained in a succession, the sense of that function, as we have seen, with the implica-

tion of affirmation of tonic, is likely to be preservedin some degree-whether significantly must, again, be in part a matter of subjective response and judgment. Of importance in such proceduresis the expansion ofthe concept of leading-tone: with the important technique of tonicizing encirclement, especially potentwhen the encirclement is by chromatic auxiliaries, the upper leading-tonetakes onan importancecomparable tothat of the traditional

lower asobserved, for example, inthe Bartokand Berganalyses!. Such leading-tones act in encirclement of tonic factors at disparate levels, and in encirclement of the individual factors ofquasi-tonic harmony, in practices of voice-leading involving frequently quasi-dominant! harmonic derivatives comprised of semitonal neighborsto harmonic structures of resolution. The extent to which such functionsdenote tonal reference depends,of course, on their relative strength as experienced in a particular context, as established by rhythmic, metric, dynamic, and other means, as well as on the supportive or contradictory terms ofthe surrounding particular context.It is frequently possiblein textures ofconsiderable complexityand freechromaticism to establishtonal feelingby suchapplications ofsubtle inferencesof traditional functions, as we have seen in quotations from relatively recent works.' The question of when tonality is expressed andfunctional! and when not-i.e., that of atonality or nontonality!, is among many that are probably incapable of insistent verdict. Some theoristsinsist that any configuration or concurrence ofPCs conveyssome inevitabletonal sense,especially in cadential actions, while others consider tonal coherence obliterated

in contexts of

relatively mild complications. Still others regard any consistent application of a twelve-tone setas akind of tonality. Perhaps the truth is that examples of altered, diminished, possibly extinguished reflections of tonality in recent musicare subjectto classificationinto a number of levels ofsignificance of tonal effect and function Fig. l-7!. The tonality-atonality question arises particularly with respect to serial music. Tonic function can of course be obscured by contradictory events, but it is clear that serialized associations of PC materials cannot,

on

the basis of the prescriptive technique of composition in and of itself, be

1 72

tonality

Fig. 1-7. Conjectural set ofclassifications of levels ofsignificance oftonal function. Absolute atonality? Atonality as a relative tendency? Irrelevant tonality? Multitonality? Pantonality ? Tonal flux extinguishing, or severely attenuating, tonal function? Tonal flux within broad, prevailing tonal unity Extended expanded! tonality Tonality of quasi-functional manifestations Conventional major-minor! tonality Tonality of ambivalent conventions Tonality of modal conventions Purely melodic tonality Primitive pedal! tonality

pronounced necessarilyexternal or

irrelevant to

the experienceof tonal-

ity. In connection

with his discussion of cadential functions

and other fac-

tors which can be interpreted to point to E as a tonal factor in Schoenbergs Klavierstzick, Op.33a, Edward Cone makes the following observations, of significance forour considerationof the relevance oftonal allusion and tonal analysis! in certain twentieth-century styles: More controversial is the attempt to find tracesof tonal form in avowedly atonal compositions; yet I do not see how music like Schoenbergs, with its usually clear cadential structure, can fail to arousecertain traditional associations and responses. To the charge of irrelevancy, I answer that one who cannot indeed hear such cadentialphenomena in this music must judge the analysisto be prescriptive and inapplicable. But one who does hearthem must admit to that extent the validity of the approach. He may counter that one ought not to hear the music in this way; but he is then criticizing the music, not the analytical method. Unwanted cadential effectswould be as great a Haw in atonal music as thechance appearanceofa human Figure ina nonrepresentationalpainting."

We would embrace Conesconcept ofrelevance as vital and necessary in analytical discussions of tonality. In serial works,tonal manifestationsmay well appear as often in Schoenberg, for example! in the light of techniques already much discussedin examples of preceding pages. In others, atonality or irrelevant tonality may indeed be found to pertain. But hierarchic tonal order may be implicit too in a twelve-tone set itself see the following discussion ofthe Schoenbergsong, Tot! ; somespecific twelveEdward T. Cone, AnalysisToday, in The Musical Quarterly, XLVI,2 960!, p. 185.

tonality 173

tone setsof obvious,potentially functionaltonal possibilities are quotedin Ex. 1-51.

Ex. 1-51. Twelve-tone sets:Berg, ViolinConcerto; LuigiNono, CantoSospeso forsoprano, alto, andtenor soloists,mixed choir,and orchestra.

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The tonalimplications of the Nonoset areespecially evident when asin No. 6b,Corn éduro dire addio! thenote Ais stronglyestablished at beginning and end as a referential PC. The opening of Part 6b has the tenor voice sustaining boecachiusa! a' for two bars, after which the upper and lower leading-tones notes 2 and 3 of the set! enter. The final cadence of the section takesall twelve notes mm. 407-1 l! in order from 1 to 12, then returns the first note, now a, in the first violin. There is, it seems, afirm sense of tonal resolution.

In the song, Tot,by Schoenberg,a tonic element _ is suPPorted b Y

two leadin

formin E-tones 8

°a

u

.the C1 uasi-dominant com P

lex

_ 5 , . _ Schoenberg thus expresses a tonal structure means by that are free ofstrict tonal conventions, yet insistent. Examples 1-52a and 1-52b show the composers applications ofthe abovequasi-tonal functions. In the excerpts, some suggested primary and secondary tonal implications are indicated. While there are comparable tonal functions at other levels, expressed both melodically and harmonically, those notedare the most vital and pervasive.

In Schoenbergs song, thetwelve-tone set is sodisposed that when the voice partis basedupon theF ,Cquasi-tonic PCC and itsleading-tones, the piano is not, and vice versa.The entire song isan excellentexample ofquasi-

tona/ity 175 m.I I_

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tonal functions in a twelve-tone work, and the final cadence_significantly realizes tonalexpectations, yetis imperfect. Tonal orientations, which of course would not require realization in actual applications, are potential in the set itseltf Schoenberg exploitsthese particular possibilities it will be noted that, for example, thefinal three PCs of the set could have been applied to tonicize FI# as well! in deliberate expression ofthe primary tonic F and secondary tonic D, in much use of Ex. 1-52b. Twelve-toneset forthe Schoenberg song, "Tot,"with potentialquasi-tonal affiliations suggested.

Note that the right hand of the pianopart in the final two bars,taking thesecond tetrachord oftherow in retrograde orderG, Ab,E, Bb-8, 7, 6,5!, comesout onthe lowBb as finalnote, lendinga functionalambivalence tothe F which has been sopersistent and nearly constanta tonic center of reference. Atthe sametime, theleading-tones FII,B and the tritoneE,Bl» have in relation to the F ,Csonority stronglytonicizing effect.

176 tonality

relevant tetrachordal segments, andin avoidance, although the set isused in retrograde, qf any transpositions. The second tetrachord has the potential of leaning, dominant-like, toward F, a potential exploited as shown in the quoted excerpts Ex. l-52a!. Tonal referencesmay thus, more or less significantly,follow from certain specific properties of the twelve-toneset andits applications, when compositional procedures submit to the implications of such properties: the set fully or partially disposed around a particular axis or axes which, symmetrically applied, may become spatially central to the pitch content as at times in Webern; see Chapter 2, pp. 249-53!; intervallic relations implicit or explicit in the set such that potential is established for quasifunctional relations dominants, leading-tones, etc.! supportive of a recurrent PC as tonic ; such thematic usesof the set asformal recapitulations or other recurrencesin which a particular variant and transposition of the set takes onquasi-tonal significancein terms of recurrent thematic direction and orientation e.g., Webern, Op. 22!; and the application of the twelve-tone set

such that

invariant functional

relations recur-relations

grouping together in disparate set formsinvariant adjaccncies of two, three, or more PCs having structural predominance; and, of course, rhythmic and other applications in any way lending PC primacy and, conceivably, hierarchic arrangementcentered in that primacy! to a particular element of the set, or elements of the set.

One of the problems in tonal analysis of music in which tonal centers are supported only ambiguously is of course the extent to which analysis should goin the pursuit of and explanationof tonal allusions thatare suspected seep. l7lf.!. Any music, reduced toits most microcosmic units,obviously has tonal references-i.e., if one considersa work interval by interval nearly any analyst will find in each the experienceof one tonic root! or another. Any work, then, can theoretically be regarded asconsisting of the constant fluctuation of such tonics so that fluctuant tonality is presentin every work, even the most atonal.

The termparztonal can refer toa musicalsituation ofthe kindsuggested above, in which a kind of all-tonality prevails, a constant shifting of tonal reference within the smallest units, and a free combination among textural strata of diverse tonal implications, creating what is analogous to the whiteness ofthecombination of all colors. In the extreme, sucha pantonality must be a neutralization Qf tonaligg theextreme spreadof tonal reference causinga blurring to the point of extinguishing the effect of any single tonal implication. Pantonality is a concept suggestinga particular perspective forregarding what are also described asatonal situations. This conceptmay be pertinent to Ex. 1-53, a much-quotedexample, in each microcosm of which a tonal center could be said to be implied, but

the

tona//ty 177 Ex. 1-53. Webern, SixBagatelles for string quartet,Op. 9,first movement. D? G?

_,m l -/T Hal" H2

Dlw? F?

l?

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Elv?

r 92q i / i c /I _ §__./

, sord. 11! I 5:1 ' lui I

3-1 g

,'

v7

am Steg ___. 1__.. PP sord. §_*1'

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.l

.

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sum of which is so fluctuant as to be a neutralization of tonal feeling. The individual tonal allusions suggestedare inherent in Weberns row, which is included in the example. Whatever results are realized in particular analyses, and these are, again, often in some degree subjectivelydetermined, there is certainly music of the twentieth century in which the aim of the composer isto resist tonal leanings andto achievewhat Pousseurcalls a distributive equilibrium-an equilibrium which is directly opposed tothat of tonality becauseits specific harmonic weight is equally distributed at all points with a resultant harmonic homogeneity and mobility which, rather than the thematic rigidity of the seriesof Schoenberg, are the function of the twelve-tone series.66 These words seem to state very well the intent and eH`ect ofatonality. Henri Pousseur, The Question of Order in New Music, in PerspectivesNew of Music, V, 1 966!, pp. 107-8; translation byDavid Behrman.

178 tonality

Almost since music began there have been, in given style contexts, complexes ofpitches in linear and vertical arrangements by which listeners are conditioned to expect certain responsive succeeding events.These are sometimes conduciveto a range of expectations, sometimesvery specific in

their implications:

It is thus virtually impossible to exaggerate thesignificance of tonality in the structure of Western music.

Tonal reference

and fluctuation

are the

chief pillars upon which the standardforms of the eighteenthand nineteenth centuries rest, and we have seen that music of the Western tradition

before

and after what is described asthe tonal period is very often dependent on orientation of musical events in relation to a tonic toward which melodic and

harmonic elements perhaps quasi-functionalharmonies! aredirected. In the vast majority of works of this monumental tradition the languageof music is substantially that of dissonancein which cadential andother formal articulative functions are characterizedby fulfillment, nonfulfillment, or limited fulfillment of expectations arousedand conditioned by prior assertion ofa tonic and its structural, hierarchic system. Foregoing exampleshave demonstrated that many twentieth-century composers havesought to achieve new ways of establishing tonal feeling, without altogether dismissing the idea of order around a tonal center, or complex of centers, or tonal axis, with tonal homogeneity a fundamental source ofunity and tonal contrast of variety, or with relative stability of tonic opposed in formal delineations to its controlled fluctuation and, at times, relative obscurity.

While the many examplescited in this chapter, diverse asthey are in style and historical orientation, give testimony to the vast importance of tonality in the determination of musical structure, our chief purpose has been to show how in various idioms tonality is establishedand made to Huctuate, and to demonstrate techniques the in anabfsis qf tonaliyfwithin awide gamut qf genres and styles. Theanalytical techniques incorporated in this study are of an intentionally heterodox scope;their aim is the discovery oftonal structure, range, andsystem pertinentto the individual work or at times morebroadly to classes of works!, and the specialmeans bywhich tonal structure isexpanded and modulated in accord with expressive needs. When tonality is abandoned,such factorsas therecurrence and variation of melodic-rhythmic theme become basic,as do the shaping effects of rhythmic elements, dynamic contrasts, and functional contrasts of texture. The extent to which tonality is resisted or dismissed in music is thus normally the extent to which other elements-color, texture, rhythm, and

tona/ity 179

others-have ascendant significance, with the controlled distribution of contrasts and aflinities among these elements, and among their individual manifestations inprogressive andrecessive operations, determining structure.

Conc/ud/'ng notes

The foregoing analyses dealwith the question of tonal andlinear functionsin melodic and harmonic contexts in which tonality is of essentialimportance, with a few exceptionsin which particular issuesare exploredapart from such primary concerns. We have seen that in most music linear and tonal functions coincide, and a central point has been that harmonic and melodic analysis isproperly viewed from the perspectives ofdiffering structural levels in which events havediffering implications. One of the main points of discussion hasthus been that analysis must discover the

various levels of structural

function. The

rnost immediate

is

that level which is the object of analysisseeking toidentify tonal and linear implications of all the notes thecomposer haswritten. But an event that is of essential functionat a given level, e.g., a cadential note,is often seen to have at other levels auxiliary function in relation to structural factors of higher hierarchic order.

It is this kind of consideration that leads to the identifica-

tion of structural profiles which are more and more fundamental, and which underly that of the harmonic-melodic surface. Still, in the surface reside many, probably most, of the characteristic features bywhich the uniqueness of expressivepower is felt. It might well be that all the melodies ofMozart, transposed to C and reduced to fundamental structural bases, would look very much alike.! In some of the foregoing analyseswe have been concernedwith supplementary, important approaches. Harmonicand tonal rhythms are such a concern, as are considerations

of dissonance-consonance fluctuation and

of

density fluctuation, especially in styles in which tonal implications are obscured. Melodic

curve, motivic

unities, and

rates of eventfulness are other

such important supplementary concerns.And in music in which harmonic colors are an important factor in themselves, sometimesindependently of tonal function, we have noted that auxiliary chords are often introduced for relatively pure linear function and for their sonorousqualities, evenin parallel, nonfunctional streams or complexes of like sonorities in elaborative formulations.

Analysis cannot hope to point to a single, correct conception structural functions.

of

To make this clear and to remind the reader from time

to time that in most music a number of analysescan be shown to be admissible, wehave repeatedlyused theword interpretation to describethe resultsof

tonality

180

analysis.The skillful analyst learns,however, to make the most valid possible choicesamongconceivablesuggested interpretationsof a givenexample,on the basisof proceduresand criteriaofjudgment of the mostpersuasivepossible substance.

Chapter 1 will have to be seen,ultimately, in the total context of which it is an aspect.The concernsof the remaining chaptersare with texture and rhythm in music; fundamentally, this book regardsmusical effect and experience as deriving from the complementary and counteractive functional associationsof all element-actions. The comprehensiveanalysis of music must take all into account, and must see,ultimately, their progressiveand recessiveactions in interrelations by which expressiveeffect is realized.

NOTES In the samecollectionoccur manyother examplesof potentialvaluein theseconnections, and while spacedoesnot permit their investigationhere the readeris advised that he will find this basic,rich anthologya fertile ground for study of early tonal structures.For instance,a later Renaissance composition,the delightfully spirited chanson of Lasso,Bonjourmoncoeur,p. 159,might be analyzedfrom the standpointof a tonal structureon G which can soundto modernearsambivalentin its modal Ff and consequentleaningtoward C, in tonicizationsof D and evenA in tentativetonal expansions a stepbeyondthat of Ex. 1-7 foreseeing the later tonal principle of expansionto the level of the dominantbeyondthe dominant!,and relativehomorhythmin which the lower voicemovesa great deal in 5ths and 4ths. The tonal systemof the Lassoexamplemight well beseento havethefollowing,symmetricalimage: FC

GD

a.

bMeasures19-37of the samepiecereveala comparableexpansionof the tonal system in the prolongationof T:I. Secondarytonicsb/ and D! both diatonic factorsin t! occur. Moreover,a synopticsketchof the bassshowsa linear expressionof t:iv

The D/ systemcomponentis stronger than

any of the fluctuant tonal allusionsto th.'s point; it is continued, with its own auxiliary subdominantand dominant functions,for sevenmeasures. The two Liszt examplesshowtonal expansionin elaborationof harmonicprogressionT:I-V! and prolongation T: I-I!. ~A comparableexampleof the principle morebroadly applied is Brahms'Intermezzo in A minor, Op. 118,No. l. In that piecethe primary tonic a! is stronglyaffirmed at the end, supportedby a strongdominantpedal.But the first primary dominantis in m. 11,the first tonic very brief! in m. 19.Thus,the primary tonic is embellishedby fluctuation at the outset,and in a sense the fluctuationis moreradical than in Op. 53 sincethe first measureis itself a secondarytonal reference.There are passingrefer-

tona/ity 181 ences toF, G, and C before thefirst primary leading-tone appears. Even thereafter the musicobscures the primary tonic more thanit supportsit. The structural importance ofthe primary tonic is that it draws intofocus followingthe tonal ambiguity which is the chieftonal featureof mostof the form. Thereare, in the Huctuantareas of the Intermezzo, certainhints ofthe ultimatetonal course:the substitutionof a sixfour onA in m. 2for theexpected resolution of theF:V-1; theaugmented 6thin m. 5, which expects A:V but docs notproduce it; and otherabortive references to functions which, in their normal contexts,would be supportive of the primary tonic. Similarly, thereare afterthe final,long dominantpedal, reminiscences of earliertonal deviations-e.g., the BL of m. 35 part of a dominantof the subdominant!, andthe F-C appoggiaturaover the primary dominantof mm. 37-38. A synopsis ofthe bass succession might be asfollows; it shows againthe principle of linear expression of t : i, and the tonal system components are a,F, C-factors in the F :I F = SD/R/t!, F beingthe systemof referencewith which the tonal structure setsout. Thus, of the two most fundamental triadsof reference,the overall bass lineexpresses one and a resumé oftonal systemcomponents expresses the other:

' E: -an" ='

|p

c D/SM R/t! -, .1

and e-

SM

SD/R/t!

A remarkably comparable example for melodicanalysis can be foundin mm. 12-19 of the first violin part, second movement, of Bart6ks Divertimento forstring orchestra: its essential course runs from all established and prolonged differentlybut with comparably strongeH`ect! to da, againin a tritonal essentialmovement. Thereader will find in the Bartok melody thetechnique ofsequence repeated motivic unitstransposed ina conjunctsuccession of subsidiary highpoints oflargely diatonicrelation! : U _i,»"~-___;_--- `92_¢ __ gf '

Q1 1 3

| » I

In Accenti, second pieceof the piano set Quaderno Musicale di Annalibera,by Luigi Dallapiccola, thesingle notewhich emerges out ofthe harmonicallydense motivethe final noteof theform ofthe serialhexachord used in the given instance!often takeson a tonal meaning simplybecause, inits severeisolation anddynamic stress, it hasthe effect ofresolving the preceding, intense dissonance and density. Whenthere are harmonic ormelodic elements that supportthe feelingof tonalresolution atthe same time, that feeling is of courseenormously strengthened. Thus, at the pieces final cadence a strong sense of tonalfulfillment arisesfrom thequasi-dominant character of the outervoices ofthe harmonyofthe final measure B and Dil! as wellas thechromatic succession inmoving voices: C18-D-D#-E. Whether this is the composersintent cannot besaid, butthere canbe little question thatthese factors impart tonalimpressions and,thereby, strongerimpact to the powerfulcadential thrustwith which the piece ends. Another example useful forstudy inthe sameconnection isthe firstof Weberns Five Piecesfor orchestra,Op. 10, in which a concluding,reiterated F,in an atmosphere ofseverely reduced texture, hasthe effectof quasi-tonalresolution. Thepiece should bestudied for evidences, ofwhich many could be cited, of quasi-functional projections of or expectantsuggestions of!F, and A and C, in the pitch materials of the piece. [For example, lastnotes of the trumpet-F, G, A, F; trumpet and trombone ofmm. 6-7, centering onF; the leading-tone E in the flute motive of mm. 8-9; clarinet phraseof mm. 4-7, centering on F and A; or the succession E

82 tonality flute!-F-F if-G violin!-A-G1f cello! preceding andsupportive oftonal feeling in the final, isolatedF.] Certainly it must beacknowledged thatthe analystwill find many instancesof apparently arbitrarily and rapidlyfluctuant change,or staticdissonance effect, in recent styles. Thethird of Schoenbergs set of piano pieces, Op.19, isa casein point: dissonance seems relatively fixedin intensity,as isharmonic densitypitch factorsof four and five in chords throughout thepiece!. At the sametime, the pieces structureis compellingly shapedby other factors-quasi-tonal function and harmonic rhythm are ofparticular importancewithin the harmonic element, not to mention theshaping effects of other element-fluctuationsspatial field, movements ofline, rhythms of attack, directionsof dynamicintensity, etc.!.It must not beassumed, then, that dissonance fluctuation is necessarily a significantfactor in the shapingof structure.The discussion ofthe Op. 23 extractinvolves onlytwo-note simultaneities; where chords are concernedthe evaluationof dissonance quality and process involves appraisal of such qualityas thesum ofinterval propertiesin chords.This can be calculatedin a number ofways, andagain ajudgment mustbe madeas tosignificance ofdissonance quality in intervallic relations between thelowest in relation to upper voices,the upper in relation to lower voices,or all intervallic relationsamong all voices inthe texture. Uln the third of Eight Etudes and a Fantasy for woodwind quartet, Elliott Carter develops hisform entirely out of coloristic contrasts,since theEtude is entirely an expression of the D major triadwithout changein its position ordistribution andwithout auxiliary notes orharmonies. Onlythe assignment of notesofthe triad among the four instrumentschanges. Itis perhapsthe mostelemental expression of tonality that is possibleto imagineand it is, in this piece,of strikingeffectiveness, although it is of course adevice thatdoes notbear repetition. When early modal scalesare usedas resources in melody and diatonic harmony in twentieth-century styles, as theyare not infrequently, especiallyin earlier music of this century,they createthe samemodal/ tonalambivalences one often associates with

works of] say, thesixteenth century, in which,for example, the mixolydian lowered 7 can to modern ears effecta leaning toward the region of the subdominant.For example, thefirst thirteen measures ofBart6ks Divertimento for string orchestra, third movement,sustain, inintense tonalreference, aquasi-augmented-6th harmony of primary dominant function.It is followed bya V-I root succession setting offthe principal thematicelement, amixolydian melodywhich, while on an F tonic, leans toward Bb. In this piece, thereader maywish to evaluate dissonance fluctuation in the light of the composers own specifications of dissonance in thesource cited on p. 1 10.A number of cadentialsuccessions show what appearsto bea verypurposeful dissonance control: mm. I4-15, in the direction of milder dissonancenote, too,the voice-leading determinants: outer voice contrarymotion, and more conjunctinner voice!; m. ll, an appoggiatura figure whose pattern of dissonance release is very evidentwhen thetenor and sopranovoices areregarded independently; and m. 6, both of whoseharmonies contain sharpdissonances, but in which there isnevertheless sense a ofrelease occasioned bythe factthat in the firstharmony thedissonance occurs between outer voices while in the secondit is relegated toan innerposition withthe consonantmajor 10th between outer voices-a good example of the mitigating injiuence qfspacing and distribution inharmonic dissonance effect. In mm. 18-22, the general increase in dissonance intensity seems tobe a compensation forrelative rhythmic inertia; and toward m. 30 there is a distinct mounting of dissonanceand steady increase in density! towardclimax. JRave1s Surgi de lacroupe du et bondNo. 3of TroisPoémesMallarmé! de mightbe citedas an examplein which tonal structureis notconfirmed, i.e.,is left open, at the conclusion. Theearly, relativelyconcise tonalexpressions of D major! deteriorate atits

tona/ity 183 conclusion, where there isonly theslightest hintof the original tonic. The final harmony isa combinationof dominanton C, to which the basshas progressed in chromatic descentfrom E, and an A major triad with F 8added andnot resolved. This latter, if construed asa primary dominant, isthe only,faint allusionto the initiating D system.!Tonal clarity thus substantially dissolves in the finalcadence. Inthis sense, the tonal structure is illustrative of opposition ofstability and instability relative clarity and ambiguity! asshaping features. The earlierexample fromBrahms l-14! and theBrahms Intermezzo, Op. ll8, No. I, cited infootnote description on pp. 18081, reveala patternin whichinstability oftonal reference gives way to ultimate,strong clarification; the Ravel doesjust the opposite:

*It wassuggested earlier that the two mostprominent tonic sonorities, harmonically conjoined, providethe intervallic basis for the recurrent motive of superimposed 3rds:

1

The collectionof thesemotives atthe endof the movement canbe seenas a wholetone seriesin which, as noted,basic tonalfactors arerecalled: .-I. lla f.92 & .. 92___..,

ILJT _92 92 J92.||92

L, »-lnL, &:?

_

The lastof themotive statements, that ofmm. |38-39,is tonallythe mostdeviant, but it is strongly functionalin reemphasizing, at a penultimate point,the Bbupper leading-tone with which the violin phrasehad concluded,at the same timeproviding a tonally distantperspective against which the ultimate cadentialrelease isstrikingly enhanced ineffect:

---_____""~-__ -_--____" _-__` `§_1```` _"V/ 92 /*92 92{1-l/§H"§/_'" I" I If Relatively anomaloustechniques oftonal usagelike bitonality and polytonality are not treatedas suchin this study, but in a bitonal situation e.g., Stravinsky, Symphonies qf Wind Instruments, 1947 revision,following rehearsalNo. ll, where different tonics areactively anddeliberately opposed! the expressionof tonal center at each stratum is likely to be in accord with techniques andfunctions likethose notedin this chapter.Indeed, it is characteristicof mostmultitonal usagethat each stratum asserts itsindependent tonal center ina verydirect, evenprimitive, manner,as in the Stravinskyexample cited.

CHAPTER TWO

texture

/n troduotory notes

Certain of the qualities and classifications of musical texture have been treated abundantly in the work of music theorists; references to relative densities andsparsities oftexture, to categories ofdescription such as polyphonic or monophonic, and to many other features and types of musical fabric are common. But adequate formulationhas notbeen givento analytical treatment of processesinvolving textural events and changes, or to the significances ofthese in the structure of music. What is musical texture? The texture of music consists ofits sounding components; it is conditioned in part by the number of those components sounding in simultaneity or concurrence, its qualities determined by the interactions, interrelations, and relative projections and substances ofcomponent lines or other component soundingfactors. Density may be seenas thequantitative aspectof texture--the number of concurrent events the thickness of the fabric! as well as the degree of compression of events within a given intervallic space. There is a vital relation between density and dissonance; the relative intensity of a highly compressed textural complex say, three components within the range of a minor 3rd! is a product of the severity of dissonance as well as of density. Furthermore, density clearly has a relation to coloration; thus, two simultaneous pitchessounding in tight compression say, amajor 2nd apart! will project varying degreesof intensity depending on relative homogeneity of coloration e.g., two clarinets, forte! as opposed to dissimilar coloration in which the proximity in musical space seems attenuated by the separation implied in the disparity of color. Similarly, intense dynamic levelsexaggerate The term concurrent or concurrence is used inthe sense of occurring together -coexisting, intersecting,overlapping inreal time. Two lines are in some degree, and at some specified level, concurrent if they overlap inany part. Concurrence ofindividual sounds is onlyof themoment ofsounding together, unless such sounds are parts ofconcurrent lines. 184

texture 185

the effectof spatialcompression. Relations between coloration and density could be traced far beyond thesesimple observations. The nature of interactions and interrelations

within the musical fabric,

apart from calculable density, might be said to constitute the qualitative, as distinct from the purely quantitative, aspect of texture. Thus, in conventional broad linesof classification,monophony isa type of texture, and a condition in which certain qualitative features oftexture apply. Unlike textures of two or more concurrent componentschordal, polyphonic! monophony isexplicit in its condition of minimal density. Examples which follow will be much

involved with the demonstrationof musicalprogression and recession as shaped by changes bothin densigfand in the qualitative interactions qf the components ¢y musical texture. Reference will be made as well to the applications of such changesin the delineation offorms-as, for example, inthe emergenceof polyphony in developmental contexts,the stabilization of textures in cadential formulations, 'or the common association of relatively uncomplicated texture with

thematic statement.

Of necessityin the analysis oftextural qualitiesis the evaluation of various kinds of interrelations and interactions among textural componentsthe degreeand nature of interlinear concordance agreement,lack of conflict! or coincident factors of relative intensity and variance counterpoint!. Changes in relative independence and interdependence among concurrent components in a given musical texturewill be seen toconstitute someof the most decisive and subtle! factors in the expressive shapingof structure. Much of the attention of this chapter will be devoted to considerations of textural progression and recession shaped as bysuch changesin interlinear relations and in qualitative and quantitative textural conditions, and to the convergence or contrast between a textural structure

and those of other elements.

The complementary and compensatoryrelations of texture with other elements of structure will be evident in examples analyzed in this chapter. A simple instance maybe cited: that in which, for example, heavier,greater densities seemfrequently to require relatively reduced rhythmic activity. Other relations of complementation and counteraction compensation! between textural and other factors are suggested, forexample, in foregoing references to coloration and

texture.

If a single pitch is sounded,a texture here, oneof maximal simplicity! is established. If a second pitch is sounded in simultaneity, the texture is altered-its density is increased. If the two pitches are a 2nd apart and they are succeeded bytwo pitches a 6th apart the upper moving up a 4th, the lower down a 2nd!, a textural event takes place-a succession involving not pure quantity density-number! but involving densitycompression and a number of important qualitative factors in the texture viewed independentb' of otherelements. An incipient quality of interlinear indepen-

186 texture

dence is asserted in the opposition of direction of movement-a quality subject to enhancement if rhythmic differentiation in the relations is introduced.! Two lines moving in parallel 3rds may in an important sense besaid to constitute a single real textural factor consistingof two components. At any point at which differentiation is established-in rhythm, in direction of motion, in the distanceof motion, or in any other sense-a texture initially consisting ofa singlereal factorof two sounding components!becomes atexture of two realfactors or at least progresses inthe direction of such differentiation!. Progressions and recessionsinvolving changes of these kinds or of analogous effectare decisivein the shaping of musical structure. The foregoing extremely simple illustrations are suggestiveof proposed distinctions among implicationsof factorsof quantity and quality in musical texture, both subject to a vastrange ofsubtle and constantly changingfeatures. The sectionsimmediately following are concernedwith the theory and terminology of musical texture, and with further fundamental considerations of the means bywhich textures undergo quantitative and qualitative modifications in functional events

Textura/ progression, factors

and in the delineation

of forms and structures.

recession, and var/lat/on as structura/

Example 2-la serves asa very simple, but exceedingly vivid, instance of progression andrecession withinthe element of texture. In the example, thereis progressivedevelopment oftextural complexity toward m. 4 and recessive decline in that complexity toward textural accord and simplicity! in approach to the cadence atm. 7. The changesin degrees of density and textural diversity-the succession toward maximal interlinear independence andthence toward textural homogeneity-can be traced by evaluation of the numbersof sounding and real components ateach point of significant change. The exposition of the brief subject, presentedin overlapping imitative entries at harmonic intervals of the 4th and 5th, is conventional. Lessstandard in procedure, but important in the expressive quality as an element of asymmetry, is the variation in time intervalof imita-

tion: 2al , 4J ,2J The . accrual of voices, their relative independence in the directly vertical sense!asserted byrhythmic and/or directional differen2The termcomponent may refer genericallyto any textural ingredientor factor as indicated in the immediatecontext of consideration, andas qualified by such adjectival modifiers asreal component, inactive component, doubling component, etc.

texture 187 Ex. 2-1e. Milhaud, Six Sonnets for mixedchorus No. ; 3,A peinesi lecoeur vous a considérées images et figures lfthe hearthas scarcely considered you, images and impressions! on textof J. Cassou.

J = 92/ "ff

-1 pei-ne si

le

016 F A pei-ne

si le

coeur vous

a con~si - dé

V .I

H

A pei- ne si

le coeur

vous a con - si - dé-rées, vous

i "2

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'Q .L _LQ Q3J is coeur vous

IV rees vous_a

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92 --Z . si -

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ma-ges F1 - gu

- res

ma-ges fi

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- gu

6-

gu -

res

188 texture

tiations, canbe represented as follows:l _l_ if l_ _L!}_

the lattercondition of

ll 1

maximal diversity achieved at m. 4, with the entry of the fourth voice. At the end of m. 4 and the beginning of m. 5 the upper two voices become associated in parallel rhythms and directions we are overlooking minor differentiations in interval of motion at the most foreground level!; this 2

condition mightbe represented by thesymbol _L-showingthree realcom1

ponents and four soundingcomponents-the first significant decline in the textural diversity. Immediately subsequently the lower voices are disposed in comparable association: 2 , representing a further decline in textural 2

diversity and complexity but none in density-number!. Finally, at the approach to the cadence, all four voices becomestrongly interdependent in both rhythm and direction of movement with octave duplications!-a vital factor in the expressionof cadence. Two extremely foreground factors might be noted in extension of the analysis in Ex. 2-lb. First, in the recessiveprocess distinctionsof intervallic Ex. 2-1b. Qualitative and quantitativetextural progression and qualitativerecession inthe Milhaud excerpt.

1lllg2g24

1 21,

lg gl

1 _1_

2 1

1

92 Quality curveconditioned as changes by in independence-interdependence!

T Quantity density-number! curve

3In thissymbolization, theactual verticalalignment of voices isnot necessarily represented. Parentheses may denotea componenthaving independence or substancein some way restricted. The term density-number will be usedto distinguishthis parameterof density from that of density-compression,briefly notedearlier. Thesetwo aspectsof densityconstitute texturesquantitative dimension;both havefurther treatmentlater in this chapter.

texture 189

content amongthe variouscomponents continueto occasionmodest diversity so that, for example, in a situation of almost consistenthomogeneity m. 6! as manyas threedifferent ascendingleaps areprojected? A second very1oc_al factor can

be seen in the cadence itself, where in one sense there are four

components four different pitches and concluding successions!, inanother sense onethe agreementof all four voices in rhythm and direction!, and in another sense two octave duplications of soprano-tenor and alto-bass!. The analysissuggested inEx. 2-lb traces thetextural processmore broadly, omitting any accounting of such local, minor differences,but it is important that they be mentioned as indication of the complexity of even modesttextural development and analysis beyond relatively categorical observations. The foregoing example illustrates very simply the concept of textural progression and recession, thecontrolled shaping of textural events here,both quantitative and qualitative! in speci/ic structuralfunctions here, cy' development and

cadence!. Changes in texture-surely quantitative changes,but thoseinvolving textural qualities as well-are :Wen among the mostreadily perceptible and appreciable inthe experience of music. An example from _Iosquinis drawn from a setting of the De profundis Ex. 2-2!.

The example is a beautiful illustration of the controlled sloping of textural structure. The phrase is characterized,in its texture, by significant interdependences ofthetwo linesat the beginning andend, andby significant independences in its central portion. The complementary relation of the textural processesto the melodic structure of the superius is stunningly effective, as is the complementary application of rhythmic technique-syncopation anddotted rhythm at the climactic point. The associationof the two lines at the phrase extremitiesis not without diversification: at the outset,there isconsistent rhythmicidentity but oblique, then contrary, motion; and there is some directional opposition within the homorhythmic approach to the cadence.The internal portion is characterized by rhythmic and directional differentiation. The phrase is thus an illustration of progressive diversificationtoward a point of relative intensity, and of the cadential function of` increased interlinear interdependence. In the kind of symbology which has begun to emerge in these discussionsthe

example mightbe saidbroadly toprogress, then recede, from2 to L to 2. l

These twoexamples expose, of course,only someof many parameters of textural shaping; other exampleswill pose other kinds of issuesand reveal other functional, expressive, processive operations. It seems clear that what we have described as textural diversity complexity, activity-like density anaspect ofintensity in the texture! seeks 5This texturalcondition is identified asheterophony.

190 texture Ex. 2-2. Josquin, Deprofundis c/amaviMotot!. m.36

'f

V if

:°~*°JHiU

1, f

_

Errflffrg Reprinted _/rom the Smgers edition, published by G.Alsbach and Co., Amsterdam, by permission qf Creyghton Musicology-Musica Antiqua, Bilthoven, Netherlands.

release in what wehave described as reduced interlinear independence i.e., textural interdependence, accord, homogeneity, simplicity, inactivity!.Thus, progression toward increased levelsof diversity and interaction creates the sense ofneed for reconformity, an expectation that the trend toward com-

plexity will be reversed in cadentialexpression. Like dissonance, rhythmic acceleration, ascent in pitch,etc., theprogressive complication and diversification ofmusical textureare assumed to be evocative ofthe impression of rising intensity,an intensitywhich increases the more the progression is prolonged.

texture 191 Types of musical texture; terminology

prob/ems of

c/assi#cation and

Texture isconceived asthat elementof musicalstructure shaped determined, conditioned! by the voice or number of voices and other components projecting the musical materials in the sounding medium, and when there are two or more components!by the interrelations andinteractions amongthem. Except, of course, in monophonic textures,the evaluation of interrelations and comparative substanceof motive, of activity or stasis,etc.! in and among thecomponents ofthe texture is the all-important problem of analysis both toward the characterizationofthe textural class andtoward the analysis and understanding of functional successions oftextural events. Rhythm is surely the most critical factor in interlinear relations; thus, within the independence-interdependence scale of textural values it is the most decisive factor in the assertionof interlinear opposition diversity, resistance, counterpoint!. Attention is essentialtoo to interrelations of motivic material including imitation!, intervallic content, directional opposition, and dissonance,as well as number and degrees of proximity in the vertical alignment of events, i.e.,density. One is reluctant, as always, to make the problem of terminology a central issue; nevertheless, apart of the businessof music theory is lexical, and words along with some concordance of understanding as to what they signify! are essential todiscourse. Complexhyphenated terminologicalforms which follow from the system outlinedhere arerarely needed but are nonetheless logically plausible and fitting! ; and it is always possible simply to describe events,even though terms denotative of textural conditions and classes are a useful convenience.

If allowance can be understood forthe difficulties of iirm and arbitrary distinction within any spectrum of textural classifications, andif it can be well understood that any musical instance,seen beyondmost local events, is Density is defined asthat textural parameter, quantitativeand measurable,conditioned by the number of simultaneousor concurrent components andby the extent of vertical space encompassing them: densigr-number and densib'-compression. is interesting It that the two aspects of texturetermed density-compression and texture-space developed later as theoverall fieldor ambitusin which events takeplace! arenecessarily contradictory in progressive or recessive inclinations, assuming afixed density-number. Thus, theinflation of texture-space is regarded asan intensifyingfactor expecting recontraction! andincrease in density-compression similarly intensifyingcf. dissonance, and theexacerbated interlinear conflict of highly crowded events!. Thefact isthat cadentialprocesses in music aremore often thannot operationstoward spatialcontraction and,consequently, normally increased density-compression relation in to which suchrecessive factors as consonance and textural simplicity, aswell aslinear declineand anynumber offurther recessive element-successions, function compensatorily.

192 texture

likely to represent a mixture of such classifications,there will be value in the statement of proposedterminology. Certain widely used terms of relatively_firm conventional signyieation canbe accepted asgenerally understood. 1. Pob/phonic,while literally meaning many-voiced, can serve to denote, as conventionally, multivoiced texture of considerable interlinear independence, oftenimitative; it is thusgenerally understoodto havequalitative implications beyond its literal, limited meaning. 2. Homophonie would literally denote a condition of interdependent voices, but its traditional connotation is that of texture in which a primary voice is accompanied bya subordinatefabric sometimesinteractive in tentative ways, the bass normally in a contradirectional or other contrapuntal relation to the primary voice or voices!. 3. Chordalis aperfectly acceptable,and very useful, conventionalterm referring simply to texture consisting essentiallyof chords, its voices often relatively homorhythmically related. 4. Doubling can denote lines homorhythmically-homodirectionallyhomointervallically associated see thefollowing definitions of these terms!. 5. Mirror association, usuallyunderstood asstrict, involves a relation that is homorhythmic-homointervallic-contradirectional; again, it is a term in common usage.

6. Heterophonie is understood to denote a relation that is homodirectional parallel in contour! but heterointervallic see below!-having minor diversification of

interval content.

7. Heterorhythmic is a conventional term adopted in accord with its conventional signification see below!. 8. Sonorigfmay be defined as the overall sonorous character determined by texture including doublings! andcoloration including articulation and intensity of dynamics!. 9. Counterpointcontrapuntal! denotes a condition of interlinear interaction involving intervallic content, direction, rhythm, and other qualities or parameters of diversification. The traditional usage of contrapuntal is comparable to that of polyphonic.! ` 10. Monophonicis taken, as conventionally, to mean single-voiced monolinear! . Beyond the above, it seems appropriate to formulate certain terms which can be used to apply with some precision to specific conditions of interlinear relation in multivoiced or two-voiced! and multilinear textures." 7In thisstudy theterm linerefers toany texturalcomponent inwhich horizontalrelation and configuration canplausibly be traced as a logical continuity-an identifiable stratum inthe textureat somegiven level.The termvoice willnormally denotea linehaving distinct relativeindependence; itmay thusbe a complex ofdoubled lines,but is not itself

texture 193

A system-scale spectrum! of textural conditions or values from simple e.g.',monophonic! to complex e.g., ultimately multivoiced, contrarhythmic-contraintervallic-contradirectional,3 the highest manifestation of polyphony! can be establishedas a lexical basis for description of relations among textural components which are concurrent or simultaneous at some given level. In the following system the prefixes homo- uni-,or eo-!,° hetero-, and contra-'° areadopted to refer to conditions of identity, mild and veiy local diversification asin the conventional heterophonic!, and morepronounced contrast, respectively.Moreover, three specific parameters aspects, dimensions, spheresof reference! are adopted as relevant to the evaluation of textural conditions: these arerhythm specificallyrhythmic pattern!, direction of melodic succession!, andlinear intervalliccontent." l. Within the parameter of rhythm, the terms homorhythmie, heterorhythmic both of these in conventional usage!, and eontrarhythmie all emerge as potentially applicable and useful. 2. Within the parameter of direction, the terms homodirectional, heterodireetional, andeontradireetional motion in a straight line exists asa possibility along with motion up and down! all have potential applicability to relations among components oftexture.

capable ofdoubling. Theterm multilinearalso bilinear,trilinear, etc.! can thusbe usedto denote textureof morethan onesimultaneous or concurrent component. As arule multivoieed two-voiced, three-voiced, many-voiced! has qualitative implications.Polyphonic, asnoted earlier, strictly means the same thing and has conventional implicationsof qualitative diversification.! Withinthese distinctions, which seemnecessary and unavoidable indiscussion ofmusical texture,monophonic single-voiced! texture can of coursebe multilinear; line isthe moregeneric concept. While such hyphenated terms are rarelyneeded, theymight beused inprecise reference toa complexof specificconditions ofinterlinear relation.It is more usualto refer to single parameters of consideratione.g., homorhythmic,or heterointervallic!,or perhapsto two conjoinedparameters of reference inhyphenated terms like contraintervallic-homodirectional. Applications of the prefixhomo- are simply anextension ofconventional usage, as in homorhythmic. 1°While the term polyrhythmic cf. the possible birhythmic, triintervallic, bidirectional, etc.! is areasonable and perhaps admissible synonym forcontrarhythmic, it is often usedin a sense more properly expressed as polymetric a distinctiondeveloped in Chapter3!. Beyond the terminological system outlined here,phrases likerhythmic contrast, intervallic conformity, directional opposition, etc., arefreely usedto describeparticular relations. An intervalclass IC!includes anygiven interval within the octave togetherwith its inversion complement!and all compound extensions expansions by one or more octaves! of the given interval or its inversion. If enharmonically equivalent forms areconsidered of the sameclass, thereare sixinterval classesthe unisonexcluded!; theissue ofenharmonic identity is,of course,one oftheappropriateness of functional distinctionsamong enharmonic equivalents withinany givenidiom.

194 texture

3. Within the sphere of intervallic content, the terms homointervallic [applicable to IC interval class!, intervalcomplement, specificinterval, etc., as areall terms within this group-a distinction to be clarihed when appropriate], heterointervallioregarded as synomous with the conventional heterophonic!, and oontraintervallie can all be used to describe particular textural situations and relations, usually applying to specific intervals rather than classes.

Of vital, general importanceis the need in all analysisor description of musical texture and textural conditions to maintain awarenessof the level of structure towhich referenceis made. Thus, two lines may have contradirectional relation in a local sensebut be seen, significantly,to have homodirectional associationat a broader level.! Example 2-3 is a list of very brief, contrived examplesin illustration of the foregoingsystem ofterminological classification. Ex. 2-3.

Relations the at levelwithin the temporal context! illustrated

§: i# RR homorhythmic F heterorhythmic contrarhythmic

I'

QP##Fl, Fl, ll-1' -|ll---ull'

_-__I--1:2 I i

J. contradirectional *Applicable atlevels ofone, two,or threebars.

homodi ional r¢¢¢ heterodirectional#

texture 1.95 Relations atthe levelwithin thetemporal context! illustrated

homointervallic'

heterointervallic'

contraintervallic'

'As to specific intervalof motion.

Some further considerations of term/no/ogy and aspects of texture Doubling does not result in interlinear interdependence of any certain,

absolute degree: the intervalof doublinghas muchto dowith the extent of association betweentwo doubled components, evenwhen the doubling is absolutely strict. This fact has to be taken into account in evaluation of a textural situation in which doubling occurs, and it can be appreciated if the differences in degree of interdependence or f`usion! are considered when doubling is at the octave[duplication of PC pitch-class!], with texturespace andsonority affectedbut with the qualitative association ofthedoubled lines virtually complete, as opposed to a situation in which the interval of doubling is dissonant e.g.,2nd, or 7th!, where individual components assert a greater degree of independence. The factor of interval of doubling and its impact with respect to interlinear independence isone ofdissonance and of size or distance between the doubled components!. Of course, any doubling at whatever interval greatly compromisesthe relative independence of lines." 12It isnot clearwhether considerable space between two linesis conduciveto greater independence, or whether, onthe otherhand, a small distancewith dissonance,as in doubling in 2nds! occasions a heightenedinterlinear resistanceconflict = independence! than does alarger interval.The complexityof thisproblem isfurther evidentwhen oneconsiders the impactof specificinterval sizeas opposed to IC in, for example, arelation whichis contraintervallic but homodirectional, andwhen the disparity of interval of motion hasone component movingby, say,the compoundform of that of the other: a l0th with a 3rd. Presumably, therelative independence manifest in the larger interval of motion is considerable, despite the affinityof IC. No doubtone mightgo onto hypothesize situations posing

1.96 texture

It is noted above that there is one aspect of texture which is affected significantly by doubling at octave or compound-octave distances. Attention is givenlater to the rangeof the sounding complex its space,and the profiles delineated inspatial changes!as anaspect oftexture particularly important in certain styles.In the analysis oftexture-space, andin the evaluation of progressions andrecessions formed by its increases anddecreases, octave duplications are of course significant. We shall adopt the term texture-space or simply space! in referring to this aspect range, field, compass,ambitus! of texture.

Example 2-4 consists ofthree brief extracts from the Stravinsky Variations for orchestra, a work of importantly texture-conditioning structure. Our concern here is chiefly with illustrating some of the conditions and distinctions of terminology with which we are currently involved; the examples also indicate some complexities of the problem of evaluation of textural relations, and some of the features of textural process to be developed in later analyses. The initial utterance ishomodirectional resultingin considerableinterdependence ofcomponent lines!but contraintervallic, a factor to which any degree ofinterlinear independenceis attributable. The subsequent,responsive motiveof flutes and stringspits oneconcurrent choir against theother in a contradirectional relation denoting a qualitative increase in textural diversity!. The line between heterointervallic heterophonic! and contraintervallic texture is tenuous but significant: the strings asa group are in heterointervallic association, the flutes contraintervallically related. Within the strings, while celli and violins both have ascending tritonal leaps a homointervallie relation as to class!, they are differentiated in PC-the doubling is in fact at a highly dissonant interval, as well as in typical for Stravinsky! heterorhythm. The texture of the flutes is homorhythmic and homodirectional, that of the strings heterorhythmic-heterodirectional. That of the entire sounding complex is heterorhythmic-contradirectionalcontraintervallic." comparable questions; but the underlying pointis that each texturalsituation will, beyond certain importantgeneralities, pose issues particularto itself. It is assumed that doubling byperfect intervals,especially theoctave, andto a lesser degree consonant 3rds and6ths, producesa higher degree ofinterlinear interdependence than otherdoublings. Doubling at theunison ofcourse does notaffect textureat all-not even its spatialaspect, butonly sonority. Within each such relationis likenessflutes 1-2,and betweenviolas lower line and cello! involving duplication ofIC but disparity of distance-i.e., occurrenceof simplewith compound formsof the same intervals. The overall texture of flutes andstrings togethermight be described asof eight sounding components lines!, whileof the following numbersof realcomponents withineach ofthe threeparameters of differentiation: inthe rhythmicrelation, l ; in directional relation, 2; in intervallic relation,5 or 4 asto IC!.

Ex. 2-4. Stravinsky,Variationsfor orchestraAldousHuxleyinmemoriam!.

2 Fls.

Alto Fl.*

2 Ten.Trnbs

BassTmb.

Harp

Piano

Vln.

Vla.

Vc.

~All instrumentssoundaswritten. 797

texture 199

Both motivesof m. l are activated by devices ofcoloration 11° the articulative doublings of chords in piano, then harp. It is clear that, in summary, the relation between thetwo utterancesof m. 1 isone ofprogressive tendency toward textural diversification and expansion of texture-space! in what is perceived as a textural event ofimportant, although local, structuralsignificance. The discussionof m. 1 isindicative of the potential complexities ofanalytical treatment of apparently simple textural conditions, all of them significant factors in textural eject. The second quotation m. 6! is given in illustration of a single voice heavily underscoredin coloration and sonority. It is, of course,an extremely potent monophonic statement, with a variety of colors and articulations. In this instance,all doublings but one are unisonduplications-of PC and pitch and intervalof motion, as well as ofdirection ; the texture-spaceis thus affected no more than its monophonic class. One lower octave doubling does momentarily extend the space, as the double-bass launchesthe ascending motion with its initiating attack. And there is some heterorhythmic diversification, e.g., in the activating role of the harp.! The beautiful phrase quotedas thethird of the extracts, mm. 20-22, is a further useful illustration. It is strikingly interesting, here, to compare the successions of distinct

element-events: that

'of ascent and descent

of

melodic line, and that of rhythmic deceleration. But the comparison of textural and color successions affords vivid illustration of necessary distinctions betweentexture and color or sonority. The texture changesneither in density nor in class. But the coloration is curved in a manner beautifully complementary to the downward-upward curve and rhythmic recession of the line. The distinction-conceptual, terminological, practical-between coloration and texture isvery clear in this brief excerpt, as in numerous other passages inthe Variations. Concepts and techniques oftextural activation,also with reference toStravinsky, have extensive discussion later in this chapter. The Variations for orchestra are highly interesting in the study of texture. One avenue ofapproach wouldexplore thefunction oftexture asthematic, asa primeelement by which the form is ordered inlieu of the traditional concept ofmelodic-rhythmic theme and its transformations. Claudio Spies, inan analysisof the Variations [Notes on Stravinskys Variations, in Perspectives of NewMusic, IV, l l965!] gives onp. 63! an outlineofthe Variations form with divisions variations! marked by changesof tempo, texture, and coloration. Spieswrites pp. 62-63!: Of Variations thereare twelve,separated fromone another, inmost instances, by a measured pause, a fermataor a change oftempo .... There is, however,no Theme on whosemelodic, rhythmicor phraseological characteristics these variations areconstructed. Instead, a sectionaldesign ispostulated onvarieties ofchange and contrast, ratherthan onactual transformation-diversityin phrase-structure, for exampleon subtleor abrupt textural shifts,on rhythmic variables, tempo-relations,as wellas ona beautifully calibratedsystem ofrefrains anda perfectrecapitulation. Sections II, V, and XI constitute the structuralrefrain . . . , and the musicin these polyphonic measures provides thesingle mostarresting features of the Variations. Some sentences omitted.!

200 texture Texture and style

Characterization and evaluation of the qualities of musical texture are important means to the understanding of styles and style periods. A few parenthetical examplesof the kinds of questions havingto do with textural phenomena and their relevance to style classification and description will underscore this relation.

l. It is well understood that,for example, the role of the lowest voiceis a major consideration of style classification. Thus, a primary distinction between polyphonictextures oftheRenaissance and the Baroqueis the higher degree ofbass equivalence in the former, ascompared with its later increasing restriction to intervals of succession whichbest underlie and support harmonic content within a

tonal order.

2. The unique texture of organum, of two or more highly interdependent lines is the feature by which that style is primarily defined. And the evolution of stylistic tendencies within organum has to do essentially with the gradual progresstoward modestdegrees ofinterlinear diversification. 3. The radical style change which characterizes musichistory around the beginning of the seventeenth century, the development of monody, is significantly a textural phenomenon characterized by the advent of sung declamation with a subordinate _fabric onlyoccasionally vitalized by emergent, briefiy competing strands-a development perceivedas revolutionary largely because of the prevalence of polyphony of supreme contrapuntal values during the Renaissance. 4. The conceptual attitude termed pointillism, a dominant feature of some twentieth-century music, is a matter of texture. Such music is stylistically characterizedby distinctive, uniquely transparent textures. 5. The evolution of style and craft in the art of many a composer is most comprehensivelyand revealinglytraced in the evaluation of his textures at various points. Beethovensmusic is an obvious example. 6. Often the distinction between musical genres is most persuasively drawn on the basisof textural differences. Thus,it is possible togeneralize to a remarkable degree concerning textural distinctions between sacredand secular musicsof many ages cf mass movementsand chansonsof _losquin or Lasso!, or in the latter of thesecategories betweendance and other kinds of music.

7. Texture is often not, of course, invariably! the key to dramatic developments inthe resourcefulshaping ofother elements:thus, for example, it is often in relatively simple textures that the most expansive and adventuresome treatment of harmony is seen, as we have noted. Or consider the homophonic contexts in which highly embellished melody is to be found, as in keyboard music of the Rococo, or the fiorid operatic styles of the eighteenth and nineteenth centuries.!

texture 207

8. Any description of the style which we know by the term impressionism will necessarily givefundamental attention to texture. An obvious example of this is the technique, in impressionism, of ornamentation of a fundamental harmonic scheme by the idiosyncratic, uniquely

characteristic,

parallel movement of auxiliary chords within dense texturesof interdependent lines.

Textura/ rh ythm

This book suggests thatprogression andrecession withinelement-structures, events pacedand shapedin a way that is interesting and consistentin functional relation to an apparent expressiveend, are basic to musical effect. These studiesfurther suggest that a primary facet of rhythm is manifest in the pacing and qualities of the changes constitutingsuccessions of events at various hierarchic levels and involving the various elementsof structure." just as tonal, melodic, and harmonic events changes!express inqualities of extent of change! and pacing what we can describe as tonal rhythm, melodic rhythm, and harmonic rhythm, so the changes in texture are expressive, intiming andin thenature ofchange, of what is properly termed texturalrhythm. In a logical extension of this idea, reference canbe made to color rhythm in music-the consequence ofthe distribution and nature of

changes in structure as

delineated by

coloration, color

events.!

Naturally, and significantly, the idea of textural rhythm is extensibleto that of rhythms of its various aspectsdensity rhythm, spatial rhythm, etc.!. The importance of textural rhythm, and of other element-rhythms, can hardly be exaggerated. It may be realized, for example, that the rate of accrual of the voicesin the imitative exposition of a fugal subjectis a rhythmic phenomenon i.e., the pacing and assertiveness of such events in the quantitative-qualitative textural unfolding!, this specific aspectof progressiona factor of textural change asof no other elementand, hence,a fastor slow, strong or weak, succession in textural rhythm. Within structural segmentsboth large and small, the rate at which texture changesin the course ofprogression and recession isa vital aspect of expressive effect.Textural rhythm is of most obvious and immediate effect where changesin density are involved, and when thechanges aredecisive; butsubtle qualitative changes aresignificantly "The conceptof concurrentelement-rhythms, mutuallycounteractive orcoincident in articulation, is pointed out recurrently in this book. The comprehensive treatment of tonal, harmonic,melodic, textural,and metricrhythms notto mentionrhythms ofelementparameters-or subelemcnts-like density, dissonance, aspects of coloration likedynamics or timbre, etc.! would require a broad contextdevoted essentially to theseproblems; but Chapter 3gives summary emphasis tothe conceptof element-rhythmsand their functional relations, andto therange ofissues withwhich significantdevelopment ofthe conceptwould have tobe involved.

202 texture

expressive subtle of rhythmiceffect. Withthe concept of texturalrhythm we move toward a comprehensive concept of rhythm asthe combination and interaction

cy' allelement-rhythms-indeed, parametric of rhythms of individualtendencies within elementsas well e.g., oftexture, butalso ofits individual aspects!. Example 2-5ais a basis forcomment concerning textural rhythm as expressed very simply inthe recurrence of a signilicant characteristic, even motivic! textural event in a Webern song."

The pianopart isthroughout an alternation between two events sharply distinguished intexture asin other elements!: these appear inEx. 2-5athe triplet sixteenth-notes witheighth-note leap this motive is shortenedand

varied in its many appearances! and the punctuatingchord sometimes reduced toa singlepitch ofanalogous articulative eH`ect, as in m. 4 or m. 9!. Reference inthe followingcomments isto the recurring chord,clearly a motive of primarily textural definition. Although the evaluationof intervallic

content andits changes, as wellas functions of suchchanges, is an intriguing and importantproblem ofstudy with reference tothe recurring chordal motive, and one ofrelevance toconcern with texture, it cannot beundertaken comprehensively here.!

Ex. 2-5a. Webern, "Wiebin ichfrohI" How joyous Iaml!, No. 1 of Drei Lieder,Op. 25, on textof H. Jone.

12-tone set: i|

* ll*

Langsam J = ca 60 rit.

-.

P 9-L07

tempo rit.

_ll' _

Wie bin

I3

1 ll

ll.!'_IF- l uv; in IF

92 g

ich frohl

i 6 I _ =i§-v 1grl!1 ,VIZ /1 U

» pq f ll _k

u

9; q

_-

,-

_I

'.

F1

The songis givenin full in Burkhart,Anthology Musical _/br Analysis, 2nd ed. New York: Holt, Rinehart andWinston, Inc., 1972!, pp.497-99.

texture

205

Ex. 2-6a. Carter,EightEtudesand a Fantasyfor woodwindquartet,No.2.

Ob.

Bn.

«Soundingaswritten. Reprinted bypermission of Associated MusicPublishers, Inc.

ture with that of coloration!is largely thestructure.Textural progression and recessionfunction in a broadly axed space;" texture is accumulative in intensity to m. 12 and subsequentlyundergoesrecessiverelaxation to the s Despitebroadly fixed space,densityof coursevaries,in compression aswell asnumber, at local rnotivic levelsasstatementsarrive, proceed,exit, reenter,etc.

206 texture

cadence at mm. 17-18. Following this cadence, thereis quick reaccrual of the textural components precedingthe final, more abrupt cadence. These are quantitative considerations. Since the motive is fixed in length, the varied spacingsof its entriesthat is, the varied distances or time intervals! of imitation, are crucial in the structure; they constitute the principal qualitative element of textural change. The control of changesin the distance ofimitation taking into account contiguous entriesonly, although any entry is in a sensean imitation of all preceding entries! can be represented asshown in Ex. 2-6b. The figures represent distancescalculated in eighth-note units. Ex. 2-6b. Analysis ofthe qualitativetextural progressions and recessionsas theyconcern changing distances of imitation! in the Carter Etude. Measure number of imitative entry:

34

l5$6 5

56

89

a I3

-_ if

10 ll

4§4§3§3§3 1

--fi 1

12 I4

15

4§6§ f Maximal textural intensity m.l2!

What is striking here is, of course, the progressive complexityof the texture, the progressive horizontal compression achieved by the increased stretto. Thistechnique is of basic importance in qualitative textural progression in a greatdeal ofmusic, assubsequent examples will further demonstrate. The progression toward the smallest time interval m. 12! is, as one would expect in a subtly contrived art form, less than consistentand predictable. Measure 12 is followed by a succession ofrelease in which two subsequent entries follow at increasing distances in preparation of the strongly punctuative cadence of mm. 17-18.

The concluding bars 9-22; see Ex.2-6a!, which are a rapid, intense resumption of the motive, contain imitative entries at extremely close dis-

tances-2IJ!, 5, 2¢b-in a culminative, sharply progressive operation, followed by relatively abrupt cadential recession.

The Fantasies for stringsby HenryPurcell are, as awhole, atruly splendid sourcefor the analysis oftexture in all its implications. The quoted excerpt Ex. 2-7a! is the third of five sections typically

texture 207 Ex. 2-7a. Pu ell Fa t yln C mlnor Z 738, forst g fn.34 slow! J1'll'§ A lf % I° Ylfi -m l929292':111l2 ..92i1Slli7

EEF _ _ 3

I II |Jl" Vfjlli I l|.h"1J I l'1§'ll-IW11

fff f°f

V V3? 'V'

II NJIT' llI' Vfjll gil! I 'H

0

YCYI1 1 lIl'l; Up°:J_

92 f 92-9* m.36 7 I ¢§

J EE

F Wfff e

f 6!

==

J /,-15

5J

/' if'

__ F`.IV

1192

lull' In-'Y -!1 ll __-||

|!'||771 -_

fe e-=»» f 92J__,1. J

--

E e: !,.

%VF IIJII. _ lfl' . 11- l11 -l-_-_-22-_UI IVZLIH ll.4"ll| _.l1|_.l1_1i.11 IITIQQZZ

EQJ .I

% qg

H -1]

o 2

208 texture Ex. 2-7a continued. m.40

A "E' §'r

I

f'

F' V

I

_ .J

J

delineated by changes in tempo and motivic content. The qualitativequantitative progressionis striking. Ex. 2-7b. m. 35!

3* 3

mm. 38-39!

21

T [Tl

3

4

L 1! *As to rhythmic interdependence.

Following the process symbolizedin Ex. 2-7b, involving qualitative progression and recession, arediversification takes place in resumption of progressive qualitativechange asrepresented inEx. 2-7c. Ex. 2-7c. mm. 39-40!

42

L 2 l_

L 1

texture 209

Gne has only to give attention to thesemeasures asa listening experience to appreciate the persuasive effect,vitally functional, of the textural events represented. Time interval fluctuation is functional too: there is

vacillation first at between the intervals of 6,l, 2J anextreme, early contraction!, again 6,l ,again 2J , andthen imitation between voices 2 and4 at theintermediate distance of 4,|. Time intervals2ofJare associated with tentative stages in thefirst qualitative progression; of that 4J is associated with and somewhat compensatory in relation to! the area of maximal diversification and complexity of interlinear independence. The reader is urged to pursue questionsof textural structure and process inother sections of this fantasy, and in other fantasies ofPurcell, proceedingto consideration of overall textural shape to which those of individual sections contribute.

Density and d/ssonance

Density asthe number of sounding components isthe density-number; density as the ratio of the number of sounding components toa given total space is the densigf-compression.

Despite thesedirect and apparently simplepropositions, thequestion of density is, like those of other aspects of texture, very complex indeed. For example, while there can be no doubt that the proximities by which components areseparated invertical alignment the degreeof compression!constitute an aspectof density superimposed 2nds make upa denser textural complex than superimposed 5ths!, the issue of dissonance hasto be regarded as a related, conditioning factor. It is reasonable,for instance, to ask whether an harmonic tritone is in some sense moredense than a major 3rd. At the sametime, it is a convenience toregard the evaluation of dissonance as 2'Density, inthe secondsense inwhich wehave definedit that having todo with the extent ofcompression of the texturalcomponents-i. e.,their spatialboundaries in the vertical field andtheir distributionwithin suchspace! canalso beseen inits interdependentrelation to dissonance in the higher probabilityof dissonance, and itshigher intensity,in contextsof greater density-compression. I t is assumed that dissonance effect canbe evaluated and itsimpact asa texturalfactor subject therefore to analysis. On pp. 107-ll there is referenceto the problem ofclassification of dissonance severities.! Whatever scale of dissonance values onechooses to accept, or however itis consideredto be modified-as it must be-according to the historical stylein question, itis probablytrue that the minor 2nd and its compounds m9, etc.! or inversion M7! are in generalexpressive of relative dissonance and that of thesethe smallestinterval, the m2itself, isthe dissonance of highestintensity. Texture density-compression! clearly is a factorin this judgment, andin the subsequent statement that among the forms within anyIC dissonance properties are more severe the smaller the form. Whetherthe M2 is moreintense adissonance than the largerM7 is however problematic;these aredistinct classes.! The position of the m2 in the scaleof dissonance values, inany style,that is, in which the dissonanceconsonance profile is apertinent structuralfactor, seems doubtless; whatis moresignificant as astyle factoris the manner ofits resolution.

texture 2

11

Figure 2-l gives the key points in a quantitative progression in the Schoenberg song,Tot, in a statementof the density-number curveexpressed in numbersof simultaneouspitches. SeeEx. l-52a, pp. 174-75,which quotes most of the extracts to which reference is made.!

The graphic representation in Fig. 2-1 is explicit in description of the quantitative progression of that aspect oftexture which we havetermed densitynumber, and its subsequentrecession. Atthe microlevel, of course,there are further gradations; for example, the extreme opening of the songhas adensity-number 2,

and the final sound is of density-number 1.

Fig. 2-1. Functional curve ofdensity-number in Schoenberg, "Tot," from Three Songs, Op. 48. Measure numbers: Density-number :

l-6 6 4- 5-6

9-10 ll-12-13 78

l3-14 I5-18 63

The successionis spacedin a rather common linear ordering in which there is accrual toward a point slightly beyond the temporal middle of the song.The high melodic linesof m. l l are a complementary factor in this progression, asis the striking coloration in that bar, with the voicepart heard entirely below the piano. The dynamics, on the other hand, are at times complementary to the density-numbercurve m. 6! and at times a compensatory element m. ll!. The foregoing discussion doesnot but could be extended to!take up the problem of fluctuation in density-compression. In the texture of the cadential harmony from the third of Weberns Pieces, Op.7, forviolin and piano Ex. 2-9! ,dissonance anddensity, whichare clearly interrelated factors here-one an intensification of the other-are of primary signihcance

in effect.

The structure

of the

piece is, in part,

a

progression towardincreased densities: the openingis relatively dense incompression butnot in number-a sustained note in the violin with staccato,ppp, impulses inthe piano a semitoneabove never prolonged! ; but subsequently the textures are relatively open and uncrowded until the conclusion. The piano texture immediately preceding the one quoted is a cluster of three pitches two semitones, within a diminished 3rd!.

texture

212

Ex. 2-9. Webern,FourPieces,Op.7, for violin and piano,No.3. Example1-28, p. 93, quotesthe completeviolin part.!

Sehrlanasam/=ca.60!

m.6

col legno,tyeichgezogen

m. 12

Copyright 1922,Universal Edition.Usedbypermission of thepublisher.Theodore Presser Companysokrepresentative UnitedStates, Canada andMexico.

texture 213

The density of the final sonority, a striking and emphatic contrast to the prevalent textures of the piece,is of course relativelyambiguous in pitch content becauseof its low register-the register, however,enhancing therichness ofovertones andthereby the sense ofcompression. Thereare six pitches and PCs! within the compass of a minor 14th or augmented l3th!. The tritone is the greatest distance separatingany two of its components. The sonority is one of severe dissonance even though use of the pedal and the low register are calculated to counteract any explicitness ofpitch!, and the dissonanceis intensified by the density; the harmony contains tritonal relations G-c# and Gb-C! as well as semitonalrelations or their compound or inverted forms! to each ofthe threefactors inthe major triad at its center. Density-number in the piece as a whole, measured in simultaneities alone, progresses along the following consistent order: I - 2 - 3 - 4 - 6, punctuated of course byreversions tolower orders.It is impossible to escape the conclusion that these element-progressions aspects of density! are decisive in the structure.

1/nter//near independence

and interdependence

lMuch of the concernwith texture is directly involved with what is undoubtaedly themost fundamental and significant criterion of textural quality-the :relative independence and interdependence of its components.

Illustration has already been made of essential factorsby which interllinear relationsof independenceor its lack areexpressed: directional, intervallic, rand rhythmic corybrmity or disparity-all of these ofcourse concernedwith relattions among concurrent lines further subject to complementary or counter:active factors of dissonance, imitation motivic parallelism in temporal sseparation!, color,spatial distanceand compression,dynamic or articulative distinction, and any of other parametersof projection by which independence iis assertedor minimized between two or more components. Example 2-10 is presentedfor the readers consideration at this crucial jpoint. There should becareful scrutiny and evaluation of the various qualitaitive changeswhich occur in this beautiful movement,and of the contrapuntal values it embodies without prevalent imitation. Of course, directional conttrast is applied in degrees it cannot exist absolutelyl! so that it is comproimised as the counterpoint gets underway note relative intervallic and tdirectional-but not rhythmic-conformity of voices l, 2, and 4- atthe outset, 22A symbolization of density-compression in theWebcrn sonorityin questionwould be :22 pitches withinthe spaceof 22semitones!. Although compression within a portionof lthe total spatial fieldis, aswe havenoted, oftenvitally pertinent, distribution is relatively eeven here.

214 . Ili

certo Grosso in F minor, Op.6, No.3, thirdm

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or between voices 1 and 2 in mm. 6-7!.23 Rhythmichomogeneity studiedbf is avoided except where functional necessities bring the voicesinto rhythmicaccord in recessive processes. The function of dissonanceshould be examined in the greatestpossible detail throughout the movement.The particular value of viewing dissonance as a textural quality is that this view induces one to seeall dissonancerelations. For example, the opening note of the second violin is seennot only as a suspended dissonance against the bass but in its relation -th! to the upper voice as well. This view of dissonance as a textural quality leads naturally to the conclusion that the intensigf ofdissonance #ict has todo inpart, and significantly,with the number ofsimultaneous dissonance relations withinthe texture as well aswith theirrelative severities. The mere computation of the number of such relations is, therefore, a significant analytical procedure in the evaluation of

dissonance effect and fluctuation.

Attention should be given to manipulations of texture in cadential approaches. At m. 9, for example, there is of course increased interlinear accord; but some vitality persists thisis characteristic of internal cadential expression inwell-made music!in the separation of voice l from the others, to all of which it is dissonant, and in the directional opposition of voices 2 and 4. In the final cadence, too, the complementarity of textural recession with those of tonality left open, one step short of the tonic!, harmony, and line is evident and balanced. Along with the tendency toward total interdependence oflines there are compensatoryevents, again expressing vitality within the cadential process:homorhythm is achieved gradually, preceded

by texturalgrouping of2; and considerable contrary motion andintervallic 2 23In numerous instances inthis chapterthe textural components willbe numberedfor convenience of reference; voice number l is alwaysthe uppermost.

216 texture

opposition are maintained. Again, dissonance exertsits influence of interlinear tension and resistance-especially the six-four and the diminished 7th are vital in this regard."

/mitat/on, a universal feature mu/t/Q0/e counterpo/nt

of many polyphonic sty/es;

The near universality of imitation in polyphonic styles in Western music and its frequency in homorhythmic, homophonic, and other textures! is evidenceenough of its paradoxical value in asserting the individuality of voices. It can be regarded evenas the supreme manifestationof interlinear independence, normallycomplementary tosimultaneous rhythmicand, often, directional opposition: the expression rj contrapuntalcompetition implicitin the enunciation fy" like motivicmaterial separated in time. Two textural components projecting like musical substance at a distance assert their relative independence byvirtue of the sqzaration-andconsequent diagonal interaction-of clearlyidentifiable materials in time. If interlinear interdependence accord! is the ideal of repose toward which textural process ultimatelystrives, imitation is the most persuasiveopposition to that tendency because of the explicit potential for parallelism denied by the imposition of a margin of temporal separation--the distance of imitation, the time interval. Thus, the shorter the time interval the more intense the con-

flict arising out of the contradiction of motivic affinity and temporal separation. Progressions in adjustment of the time interval are an important technique already demonstrated in the intensity-release curve of textural shaping. See Ex. 2-6b.! Actions of dissonance ofthe sort to which attention is called neednot be seen as purely, or even essentially, matters oftexture; dissonance is an issue ofrecurrent concern throughout thisbook. Butviewing dissonant relations asto their impact in textural shaping broadens, andcomplementarily sharpens, the understandingof this vital aspectof musical effect. The intensity associated withimitation requires clear distinction between that technique and dialogue in which twovoices exchange motivic material,the leaderdropping out while the follower responds, withno concurrentactivity. And the deviceof melodic sequence, inwhich motivic recurrence iswithin a single voice,differs fromboth imitation and dialoguein that it implies no texturalinteraction between voices. 2°The timeinterval of 0 is of coursethe ultimate accord, or resolution, ofimitative conflict pursuit!.Up to the pointof suchaccord, theassociation of progressive intensity with contraction ofthe timeinterval seems clearly appreciable;any valueother than0 expresses intensity greaterthan that of wider imitative distance. "Imitation, of course, isvery frequentlya strict duplication ofrhythms andintervals and directional successions; but every musician will recall instances inwhich imitativestatement duplicatesintervals onlyin oppositedirections, orrhythms in relative durationsbut longer or shorter values. As the time interval changes inimitation in augmentation or diminution, theangle of diagonal relation changes continuing in fluctuationof theintensity curve within this particular parameter.!

texture 217 The distance of imitation,

or time interval, is

thus seen as a factor in

textural intensity in the sense thatthe more distant the imitation the more leisurely the pursuit of one stratumby another; fluctuation in time interval is functional in many more contextsand stylesthan generally thought. The closer the imitation the smaller the time interval! the more intense is the competition; it is reasonableto assume,and experienceconfirms, that the awareness ofexplicit motivic affinity put out of joint by temporal discrepancy mustbe heightened by arelatively small margin ofdistance. This is the basis for the useby composers-demonstrated in countlessinstances-of stretto, often late in a given form, as part of the expression of intensity toward which textural structure, with other element-structures, progresses. A mental image of comparative relationscan be suggested inthis connection. Thus, the following changes areof strikingly different effect in the intensity-relaxation curve, the effect of eachreadily appreciable: two lines in simultaneity, one doubling the other; a separation of the two such that a diagonal relation, an imitative pursuit of one by the other, is established; a closerimitation, a narrower time interval. It seems impossible to doubt that the most intense of these situations,and the texture of highest urgency of expected resolution,is that in which the temporal disparity is close enough to give stark exposureto the fact of parity temporally denied. Fluctuations in intensity, occasionedlargely by changes inthe distances of imitation, are illustrated in graphic style in Exx. 2-1la,b,c, basedon three imitative sections from Bart6ks Quartet No. 5 for strings; these sections alternate in the Hrst movement with areas ofreduced textural complexity.

For an account of many important, prototypical musicalforms and procedures in which imitationis offundamental importance, and fordescriptive statements concerning the imitative proceduresinvolved, the reader is referred to the authors Form in Music: An Examination of Traditional Techniques of MusicalStructure and Their Application in Historicaland Contemporary Styles, 2nd ed. Englewood Cliffs, NJ.: Prentice-Hall,Inc., 1986!,Chapter 12. Although the harmonic intervalof imitation, on the other hand, cannot easilybe seen asa factorin textural intensity it in no way determinesthe harmonicor otherinteractions at the pointsof interlinear vertical coincidence!,it is of courseprobably a significant aspect ofstyle; the preponderance, exceptions notwithstanding, ofimitative entriesat perfect intervals octave, 4th,Sth! in music upto recenttimes isevidence ofthis. While the memory and effect of] say, dissonantintervals of imitation are perhaps plausible aselements ofintensity, theharmonic effect of thisseems tobe superseded by that at the point of vertical coincidence-i. e.,the intervallicrelation betweenthe entering voice follower!and the leading voiceat thepoint ofentry iswhat is decisive, itwould seem, for intensityimpact ofinterlinear relation.Furthermore, adissonant harmonic interval of imitation doesnot find resolution ina consonant harmonic intervalof imitationin general practice. 20The Bartokquartets areextremely fruitfulsubjects ofstudy in this regard.See, for example, thefirst movementof the fourth quartet: mm. 8-13 or 14-26 as to progressive interlinear complexity,then release;or mm. 14-22 asto progressiveintensification inthe shrinking temporaldistance ofimitation; or mm. 27-37; etc.

2 78

texture

Ex. 2-11a. Barték, Quartet No. 5for strings,first movement. J: 138-132

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Copyright 1936 by Universal Edition Renewed ; 963. Copyright and Renewal assigned to Boosey Co' Hawkes, Inc. for the U.S. A.!.

Ex. 2-11b. Bart6k,QuartetNo. 5 forstrings,first rnovernent.

15

17

18

........ Resolvent contractionto time interval0; in m. 16,prepares ostinato;in

rn. 18,prepares 1 simplificationwithin ostinato. 2 1

Copyright 1936byUniversal Edition;Renewed 1963. Copyright andRenewal assigned toBoosey

P Hnwkes, Inc. for theU. S.A.!.

Ex.2-11c. Bart6k,QuartetNo. 5 for strings,first movement. poco rit...

5$

a tempo

P1ZZ.

45 46 47 48 49 50 51 52 53 54 56~ C & ~g ~C

Copyright 1936' Universal Edition;Renewed 1963. Copyright andRenewal assigned toBoosey

H Hawkes, Inc. fortheU. S.A.!.

texture 221

The illustrations show thefunction in a line of intensity of the degree andrate of changein the time interval in relation to measurenumbers acrossthe top of each graph, and, on the left, a scale oftemporal valuesexpressed innote durations!. The intensity line is of course drawn to connect points of change in the time interval. Most illustrated changesare progressivetoward a point of maximal shrinkage of the time interval; at other times there is fluctuation within relative stasis, broadly viewed. As often in comparable discussions, itis possibleto quote only minimal extracts from the work.! Multqble counterpointusually double, triple! is a particular species of imitation in which, if two or more different contrapuntal alignments are contiguous, linesof direct imitative relation intersect, asrepresented in Fig. 2-2.

Multiple counterpoint in which the treated motive is reduced in size in contrapuntal realignment can be somewhat analogousin effect to the procedures of contracted time

interval discussed above. Moreover,

vertical lineof coincidencee.g., from to

-

a shift in the

! alongwith

multiple contrapuntal realignment couldhave functionalconsequences analogous to those of time interval change. But variation among appearances of motivic factors in multiple inverted! counterpoint such that their vertical coincidences arealtered isrelatively rare. Nevertheless, multiplecounterpoint is in the simple fact of contrapuntal realignment in vertical distribution an important technique of textural variation. Fig. 2-2. Diagonal imitative interaction

" '- -' Triangular

imitative interaction

-li interaction

Intersecting imitative as in multiple counterpoint!

3°The intervalof inversionis probablya usefulcriterion of style distinction,but the extent to which this may generallybe true is uncertain. Certainly, the preponderance of examples before the twentiethcentury areat the 15th oroctave!, 12th,or 10th-intervals of inversionin which dissonance-consonance relations are relatively not entirely! stable.

222 texture The act/'vat/'on of s/mp/e textures

Musical textures are often activated by dynamic, articulative, rhythmic, coloristic, and other means.Techniques oftextural activation are applicable, of course,to any circumstances, butthey are of specialimportance in vitalizing relatively simple, fixed textures. A few examples aregiven here in illustration of some of the primary techniques oftextural activation; others will appear in subsequent analyses. The works of Stravinsky are particularly replete with examples of heterophonic, heterorhythmic, dynamic, articulative, and other types of activation within simpler, often inert, textures. We shall return to some of these questions in analysis of Stravinsky, referring for the moment to an excerpt from one of the symphonies Ex.2-12!. This extremely vital passageconsists oftwo major textural complexes in an antiphonal relation, both chordal and considerably homorhythmic. Neither undergoes significant qualitative change: one of them, that in the higher brass and woodwinds, is only moderately progressivein harmonic, melodic, chromatic content; the other, that of the strings, is static and its influence extendsinto the wind sonorities, where low brass andhorns persist in the pedal A!. Within this fundamental context of considerable tonaltextural inertia, an extremely important range of activating devices isinterposed. The heterophonic doubling heterointervallic, homodirectional! between, forexample, thetwo bassoonsor cello and viola! in the first quoted bar B in one, C in the other! gives a momentary, particular thrust to the ostinato pattern. A comparable heterophony occursin the next bar between double-bass andtimpani. The same techniqueis seenat other points-e.g.,

Provocative questions would beconcerned withthe extentto whichgeneral statements might bepossible incharacterizing practice within givenstylistic contexts. It can be suggested that in tonal counterpoint,that of the eighteenthand nineteenthcenturies, theintervals of inversion octave,15th, 12th,10th mostcommonly! arerelatively predictable within a given range. To what extent and how! is this true of earlier styles?To what extent are the resources extended in morerecent styles ? Theanswer tothe secondof thesequestions would seem perhaps more apparentthan it actually is: inversion atthe octaveand itsmultiples, and at the 12th and 10th, is surprisinglyfrequent in many twentieth-centuryliteratures, perhaps forreasons of the relativestability ofintervallic relations.The intervalof inversionis an important textural factor since it determines theimmediate intervallicrelations among concurrent voices. "A common device intraditional music,especially thatof the eighteenth andnineteenth centuries, is the local arpeggiation qf chordal factors eitherin purely chordal texturesor in texturesin which there ischordal, or partially chordal, accompaniment toa dominant thematic element. The Albertibass, ubiquitous particularly in the latereighteenth and earlier nineteenth centuries, is acharacteristic, exceedingly familiar exampleof this.One mightsay that rhythmic activation vitalization!of this and comparablekinds actstoward compensation for the absence of functionaltextural eventfulness.

Ex.2-12. Stravinsky,Symphonyin ThreeMovements, first movement.

2 Fls. 2 Oba 2 Cls. inBb Bass Ci. inIL

4 Hns.

2T ts.

in Tuba

Vln.

Vc.

Ex. 2-12 continued. 2 Fls. 2 Obs. 2 Cls. inBb Bass Cl. inB'~

2 Bns.

4 Hns.

5 Tpts. inC

Tuba

T imp.

Piano

Vln.

Via. Vc. Db Reprinted bypermission of Belurin-MillsPublishing Corporation.

texture. 225

two barslater betweendouble-bass andcello andin the following bar between the samelines. Heterorhythmic relations amongpiano, timpani, low woodwinds, and strings, in the ostinato texture of which they are components, mustalso be cited. A further

factor in

the activation

of both textural divisions

is the

enormously resourcefuland potent array of dynamic-articulative emphases devices of orchestration, of coloration!: the frequent, heavy doubling of A in violin and viola, the low octaves ofthe piano, or frequent marks of articulative stress.

'The entire fabric is, then, oneof relative textural simplicity, directness, and minimal textural or tonal, harmonic, or melodic! development-a context in which devicesof the sort describedfunction significantly in compematogw activation oflargely inert textures.° In Ex. 2-l3a, also from Stravinsky, a number of points will be seen as of crucial importance and significance in the analysis of texture. Concern isdirected hereto identifying and appraisingdevices bywhich textures are activated in atmospheres ofrelative inertia of essential pitch structures." Looking at the example very generally one can seethat the texture is layered in groups of components differently colored, each qualitatively distinct, so that the entire context can perhaps be said to consist not merely of linear components i.e., of lines! but of somewhat distinct subtextures in a polytextural complex. Tonal relations among thesestrata shouldbe consideredindependently -the various implications of the PC materials ofseparate texturalcomplexes, the essentialharmonic structure within each stratum see for instance Ex. 2-l3f!, the general C-emphasisof mm. 324-26, and the final harmony as a tonal synthesis of the preceding PC elements. Beyond tonal-harmonic interactions and relations, subtextural complexes aresharply contrastedin rhythmic and articulative character, coloration, and diatonic asopposed torelatively chromatic content; the interaction among thesecomplexes isthus oneof vital counterpoint. Activating elements within each .stratum-complex, especially vital in a context in which tonalharmonic succession is restricted, are heterorhythmic and heterophonic impulses within bass doublings cf Ex. 2-12!, points of imitation, and various techniquesof ornamentation within each stratum. Specific examples of techniquesof activation which are of vital effect butwhich do not materially affect qualitative textural content are given in extensions of Ex. 2-13. All occur in the opening measures ofCanticum, PartV 2-l3b,c,d,e!.

Attention is called toan interestingand insightfuldiscussion of Stravinskys textures: Edward T. Cone, Stravinsky: The Progressof a Method, in PerspectivesNew qf Music,I 962!, pp. 18-26.

texture Ex. 2-13a. Stravinsky,CanticumSacrum,PartV llli autemprofecti!; text from the Vulgate, St. Mark 16:20.

Chorus

2 Bns. Cbn. +

Tpts. inC

Sass

Tmbs. Bas ~Contrab

Vla. Db.i ~Soundingoneoctavelower.

texture 227

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228 Ex. 2-13a

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230 texture Ion as to essentiallinear movements!, Ex. 2-13b. Activation by articulative differentiat a species ofheterorhythm.

IIL307 Bn_12

9- -4 5f

_

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Jk

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924444 4444!

texture

231

The above techniques recurrent in a movement of rondo-like restatementsof thematic material! are motivating in contextsof counteractive forces of inertia, as are those illustrated in Exx. 2-13d and 2-13e.

Ex.2-13d. Heterorhythrnic techniquein the Stravinskyall notationat soundingpitch!.

Cb.Tmb.

Db.

Ex. 2-13e. Heterorhythmic and heterophonictechniquein the Stravinsky.

2 Bn.

Cbn.

The rhythmic activation of a textural massof considerabletonal-harmonic inertia is evident as well in the organ solo Ex. 2-13a, mm. 327 35! by which the Illi autemprofectiis punctuated. A sketchof its pitch content is given in Ex. 2-13f, one in which its relative tonal stasis is clear. The texture is

Ex. 2-13f. Relativetonal-harmonicinertia of rhythmicallyactivatedtextural massin the Stravinsky summaryof structuralpitchesand neighborauxiliaries!.

232 texture

moved within a limited tonal-harmonic spaceby devicesof rhythm, diH`erences ofornamentation of the fundamentally fixed structure, and the like. The techniques of controlled textural change and activation of relatively inert textural massesby such devices asthose noted are of recurrent importance in Stravinsky, whoseworks, as suggested inthe foregoing references, -are invariably rewarding objects ofstudy of shaping aspectsof texture. Analysis of the examplefrom CanticumSacrum wouldideally go much further than hasbeen possiblehere, reachinginto other areas ofinquiry: for instance, the transfer of harmonic elements over boundaries in the form where there is

severe and abrupt quantitative change e.g., between mm. 311 and 312!, where polytextural structure gives way to monotexture with corresponding pitch factors transferred from soprano and alto voices to the organ, right hand, thoseof tenor and bassto the organ, left hand, and that of the second trombone to the organ pedal in palpable links of pitch material bridging, in counteractive functions, severe textural and color changes. Throughout the Canticum andother works of Stravinsky complementarity of such elementsas dynamics and tempo not to mention others! at points oftextural changeare of great importance.

The comp/ementary and compensatory dispositions or texture in re/ation to other element-structures

Some examplestreated earlierhave takenup the questions ofcomplementary and compensatoryinteractions amongelement-events infunctional contexts. For example, Ex. 1-37 is a discussion ofcadential processin one of the piano pieces of Schoenbergs Op. 23 see pp. 107-9!. In that analysis a point is made of textural recessionas one aspect of that process: texture is reduced in density and qualitatively simplified in the closing bars. The substantial thickness accruing in the broad course of the piece out of the lean two-

voice texture of m.1! isfinally reduced to L, thenl. In the qualitativesense, l

the insistent imitative interlinear interactions of, for example, mm. 18-19 are slowly relinquished. These interactions among textural components continue into mm. 20-21 but in less explicitforms; and in the final two bars

they dissolve almost totally.There isthus agradual qualitativeas wellas quantitative! textural recession. The element-structures involving linear descent, rhythmic deceleration slowing tempo aswell as longer durations!, dynamics, articulation A then - , as well as longer legatounits!, and meter as to recessively longerdurations of event groupings! all functioncomplementaribi. In Chapter l it was noted that a compensatory factor in prevailing decline isthe maintenanceof considerabledissonance ininterlinear relations; but here too a recessive curvewas noted as a probability.

texture 233 A choral

work of Purcell, a double canon in which

there are concur-

rently two mirror canons at the gl !and two strict canons at the o!, is helpful in analysis ofcomplementary functionsin another instance ofcadential process.

Measures ll-13 of the Purcell piece Ex. 2-14! represent an area of maximal textural vitality" in the broad context of the entire work. ProgresEx. 2-14. Purcell, Gloria Patrl m.ll I ; in nt , ` |1l= _ _ i¥§--I1 1 __ -i :nt U't;1 lIi.`h -. 'M _ _' _ _ |-¢!_m92`_-I uni; 1 1 ~_ _ I 92 'igx 1' ~9 2'"_'_lv 9292 I/ /, 9292 - lo- rum, se - cu- ,, sc cu - -rum, A /H- men,A men, se cu - I 92 92 I/ ' 1 92» !f_ , I i-' _" ' ' "_ 92 92_, ' _ _ 92_ /'92 __ 9292 -1 l nF.£,i.1i=:=§=i|-1..i wil y; il; §=1»1=l1£n:r1'=1 = 9292l'- -I-'I~./_-_ Y'll!1!*lF'F$ 1il1`;Ql|1 -'_-|13 f'l" ln __ 92 92 _ 9292 / , 92, 92_ 92 92 se-cu-la se-cu-lo-rum, se-Cullo ,/' , se-Eu92l m,A men,A _92 / . `92 92 _ / _ ` c _ 9292 _ _ _ _ -'||I- ' 92 _ I' -1 2 '5I Q _ |$-1|1 itll# _" 1.1711 1in1»1 iii 31_ .' l. _ Q' __ s '. ` 92 / _ 92 et in se- _cu secu- _Iorum, , se-c`u~_ lo-rum, A__ -.la se-cu- lo- rum /. 92 _` `92_/_ I 92 92 _

._ W_ _L °"""'~__-;'i' __A-;"* _ :_ " V _VI__II-I !-.!ll-_-._*7'|-_ 1, sem-per et

in se

=| i| 1' ll -_Zin 1 T- _ /1, _ ! _ _ __ I ll - cu- la se-cu-lo-rum, se - cu- lo-rum, se cu - -

33The setof terms adopted forclassification ofinterlinear relations on the basis of directional, rhythmic,and intervallic aspects! canalso beused todenote variouskinds of imitative relation,with the term imitationunderstood asgeneric andsubject to modifiers of many kinds. The term homodirectional imitation opposed to contradirectional! thus refersto interaction in which leaderand followerconform inthe directionof motion,a situationfor which nogenerally accepted term isavailable. Homodirectionalimitation may or may not conform intervallicallyand rhythmically; hyphenated termshere too are usefulin comprehensive denotation.! Contrarhythmic imitation can ofcourse befurther qualifiedby common terminology in augmentation-usually understood as homointervallic,rhythmically free, etc.! and contraintervallic imitation issubject tofurther, more precise standardclassification tonal, by complementation, by compound expansion, etc.!. Free imitationor the term _/ieeb: imitative! may denote imitativeprocedure inwhich the aspect ofrelation Huctuatesat somestated level. The commonterm strictimitation can denote likereal imitation! a followerwhich conformsin all aspectsexcept, usually,that of PC e.g., imitation at the 5th!. Mirror imitation is homointervallic andhomorhythrnic, butcontradirectional, andthe term rhythmic imitation, or rhythmic canon, identifiedby specificreference toa singleparameter, isunderstood to exclude theothers i.e.,to pertain to imitation which conformsin rhythm only!; this is suggestive offurther plausibleterms likecontour imitation directional conformityonly! and pitch-class imitation or interval-class imitation as in manyserial procedures!. One couldcontinue in this process ofdeveloping andclassifying termsof potential usefulness, incorporating language incommon usage wherever ithas generallyunderstood meaning. It is unlikely that any setof simpleterms canor shouldbe intendedas applicableto all situations; descriptive adjectival modifiersare almostinvariably usefulexcept insimplest conditions. 34 The Gloriais includedin Wallace Berry and Edward Chudacoff,Eighteenth-Century imitative Counterpoint Englewood Cliffs,N._].: Prentice-Hall,Inc., 1969!, pp. 36-37. The reader isencouraged to trace progressive actions towardmm. ll-13.

234 texture Ex. 2-14 continued. m.l4

»- f V is ~lo92 - 92 rum,

nT A-

'QH men, Ji?A

men, A- -

+__._9292 !

as e men, et _,lvl i

_1 e Z . 9292 =_=.,,=;r_z;_ n .2?-=

Jl

--

- men.

tj' 92 V6

in se - cu la se - cu- lo- rum,A "" ' 92 /_92

H

- men.

e -»~ - °" f

I men, A -

men, se - cu - *lo

rum, A

I1 ~°:;:"___°____L:_i ,

kall 92 N -lo -rum, _E' , A - men,A -

- men. _

men, A

9292|§ H - men.

Reprinted from The Worksof Henry Purcell, A.Lewis andN. Fortune,eds., by permission qf thepublisher, Novello and Co., Ltd.

sion to that area is characterizedby complementarysuccessions withintonal, linear, and rhythmic structures, and by progressive developmentwithin qualitative textural parameters aswell, culminating in the complex web of interactions at highly contracted temporal distances! partly shown in Ex. 2-14. Following this climactic area a number of recessive tendencies, in com-

plementary function, can be noted immediately: the final dissolution of the canons and of imitative interaction constitutes textural simplification of important consequence, although considerable contrarhythmic activity persists, with voices settlingat differing times exceptfor the homorhythmic relation of inner voices; the important rhythmic deceleration can be seen at a glance, as can the recessive, gradual essentially conjunct! descent of upper voice against modestascent ofthe lowest-tendencies in which their canonic followers of course concur. A complementary factor which is less obvious is metric, and can be seen, for example, in the settings of Amenin the outer voices: thoseof the soprano having recessive valuesof 3, then 5,

then IOJ5, and those of the bassthen 3, then 5, 14.° Example 2-15is an illustration of quantitative recessionin which qualitative factors are only negligibly affected and in which the sense ofdecline is extreme!. With a situation of tonal stasis,the exampleshows howpersuasive a textural recession,with complementary dynamic and melodic decline, can be.

Of course, with the linear sloping of the upper voice, it is clear that the recessionis one of texture-spaceas well as density.The function of dissonance as a textural factor in interlinear independence is particularly

IEXIUIB

235

Ex. 2-15. Barték, Divertimento for stringorchestra, second movement.

,"" " » tr 5

.fi

Vln.1

-=

.. '

jf dim.

Q"1""=1.=-?=E

.... K

jf dim.

__

__ _

...----- "{

a I

dim. __.._

@

_ #4

_

§gn5%F:§_g§i% Ejf dim.-` ---- -"/% ,J Q Edim, J_...Lp jf m.46 vw

,J

.`/_ -

7 d____ ;== ii

i-

lGi=i2|»5" _ 5X,gl¥' 4..5pl ____ `4'= _ ___ 92/IJlFI7F _M,92~ ._

- I- 4 -r-

i

i

'I

17; on

con sord.

' EJ

J

M

__ .1 J ---==-== -» » »-fr";'1i= i

92 `_/

1 it

___t l

2l_|;r' |_-;r'_|Y I

3

Copyright 1940 by Hawkes éi Son London! Ltd.;Renewed 1967. Reprinted by permission of Boosey 6'Hawkes, Inc.

236 texture

apparent: the directional opposition of lines throughout the passage adetermined one, the two essential componentsin an oblique relation! is intensely complemented by a bitonal situation, the two tonal implications merging at the point of resolution. The texture is, strictly speaking, onlyvery incidentally of morethan two real components, and both remain active until the cadential measure, 49. In that sense thedecline is of sonority as well as densityand space.Textural qualiyf thus remains relatively constant and insistent while the range of doublings is gradually reduced. The disappearanceof the lower stratum of the first violin mm. 44-45! does, however, constitutea true modification in textural quality as distinct from the sonority or density by which the most severe shapingof the succession is carried out. There could scarcely bea more lucid example of the complementarity of elements thanthis in which density, space, sonority,dynamic coloration, melodic lines,and tonal relations achieve by their confluence apervasive anddetermined functional effect. We are thus cognizant here, as in parallel studies of other elementprogressions andelement-structures, ofthe necessitythat texture be seenin its relations to other elements. Wheresuch relations are not made explicit, the reader is urged to posefor himself the appropriate questions andto seek their answersto the extent that other elementsare functionalin the particular processive contexts observed.

Some textural

functions in

de//heat/on of form

Texture its class, itsqualities, itsdensities! isofcourse anessential elementby which thematic

statement is rendered distinctive

and expressive. For exam-

ple, the stark exposureof a line to be emphasizedin and for itself may be in monophonic orvery restricted,transparent texture.Or, the contrasts between opposing thematic elements orgroups within a sonata or rondo movement are often textural, with textural differences playing a role analogous and complementary to that of changes of tonal reference, dynamic level, and the like. Or controlled textural diversification may be an important variational feature within a single thematicunity, and among itsconstituent parts. Textural progressionand alteration are of course fundamentaltechniques in thematic developmentas inany manipulation of motivic materials in a generally unstable contextf Among earlier literatures, one of the most beautiful and often cited! examples which come to mind in illustration of the principle of textural

texture 237

change functioningin delineation of form is _]osquins Tu pauperumrqfugium, subject of extensive earlier comment see pp. 45-47!. The brief extracts quoted Ex. 2-16! show the strong textural differences bywhich, along with contrasts of tonal order and other kinds, sections in this recapitulative form are contrasted. The first part, only a portion of which is quoted, is in itself a very expressively disposed textural progression setting out from a condition of perfect homorhythm which, in the second phrase, is gradually diversified with tentative imitative interactions, then largely restored in the brief third and fourth phrases not quoted!. Complementary to change in cadential pitch upper voice: g, a, g; bass voice: e, A, e! is this sloped succession shapedby textural diversification and resolution.

The second part elided with m. 20! is contrapuntally imitative, although imitation is not predominant. Actually, imitation enters very subtly within the restricted diversification of the fifth phrase, not quoted.! At the same timethat imitative contrapuntal texture emerges, incontrast to the first part, density is reduced to a consistent two-voice pattern with contrasts of coloration!. The reduction in density serves the function of contrast and brings into vivid exposure the interactions of increasingly independent voices. The third section isrecapitulative, yet a variation note metric change, for example!, a technique in which Josquin showsadmirable inventiveness. Its beginning is quoted in Ex. 2-16; the homorhythmic condition and the thematic material! of the opening return here. A look into the complete example will reveal subsequent return and variation of the third and fourth phrases. Thesection precedingm. 60 is, in fact, one in which the composerdemonstrates aconsummate craftof remarkable economyin which the simple motivic substanceof the original phrase 3 mm. 12-13, with anacrusis! is widely extended anddeveloped. Throughout this process,as in the varied return of the opening two phrases,there prevails an attitude of essentialhomorhythm in which there is occasional,restrained diversification by slight rhythmic differentiations usually productive of animating dissonance. The premature cadence onthe modal final, E, at m. 60 setsoff a closing section of cadential elaboration again marked by significant qualitative contrast of texture. In opposition to the preceding, there are now once again points of controlled imitative interplay on earlier motives! by which the texture is rendered contrapuntal in modest complexity opposing the stabler, predominantly chordal textures of Parts I and III! 3Apel and Davison, cds.,Historical Anthology of Music,Vol. I Cambridge: Harvard University Press, 1949!, p.92.

238

t6XfUl'6 -.

Josquin, Tupauperum refugium, from Motet,Magnus estu, Domine.

; ii -J

E E51? ';, ° 5 ? gf Qi 5 #12 |

5, fi f; U f m tu lan-

-wi;-mmj;

é QP?§ffE '

J

T4 la -bo -ra.n s-ti -

L; .ILA um, vi

--

-a

J

/¢, m,,° Tf f' 92

°I

II

5T 1f_ II

II

f8Xl'UI'&

JI-|:|

239

i

- C88

MQ# H '¢t

et

Ea _i'_'

#gr-f; 'LF' 8 Em# Jw-# J.

'O'

- tor

Do

- mi

- ne,

3.

ob -

dor -

Q

?

mi

Diff? 9? i ;1 i%' ob -

dor -

mi -

at

in mor-

-

i ? H5 %

0

- or-mi

in

240 texture Ex. 2-16 continued.

T 1?T T V T' _ te la; -

T T' J-

C4 Ji

-ma

_'

A, -L; 'é-__?l`;gll_i92_g _F T T -.f-'

92e=Q'

Fiiliuiv F `=i=»f_ `;=s=

Copyright 1946, 1949, by the President and Fellows of Harvard College. Reprinted by permission of HarvardUniversity Press.

Dominant textural projection is thus one of chordal homorhythm activated by diversification subtly and often gradually introduced and sub-

sequently relinquished, the overalltextural structuremarked bythe change to imitative counterpoint in mm. 20-33 and, to a lesserextent, in the last

nine measures. The mannerin whichJosquin anticipates the texturalquality of the Hrst of these contrasting passages notablyin the second and fifth phrases! isfurther testimony to the subtlety andcontrol by which his creative procedures are characterized.

Any discussionof texture in the delineation of form and forms! could

of coursego on indefinitely. Theprocedures by which, in so manyprototypical designs,formal delineation is one of relativebfuncomplicated texture in thematic statement setagainst subsequent relatively diversified, sometimes intense, textural activity in developmental andvariational processes,will be recalled by any experiencedlistener.

texture 241

And the literature is full of examples ofvariation forms in which progressions andchanges within the textural element are basic techniques. Examples of variation sets are an extremely fertile resource forstudy of textural progression and recessionwithin individual variations and along the broad lines of overall form encompassingthe variation series. Of course, textural events

at all levels of structure are

of inestimable

significance in the delineation and processive molding of all prototypical forms, and of all forms beforeand following the tonal period except of course where texture is limited to monophony without even the implications of contrapuntal diversity and interaction in compound melodic line. Textural stasis, progression,recession, andvariation are basic in the functional processes bywhich forms are shaped,and by which expressivefunctional events climactic, cadential, introductory, expository, etc.! are projected see Ex. 2-27!.

Textural contrasts, more broadly defined, do of course serve as well the function of broad delineation, in complementarity with other relevant elements, in intermovement relations

in multimovement

forms, and in inter-

sectional relations in highly diversified forms. This factor is perhaps too obvious to require illustration think, in one of countless manifestationsof the principle, of the Baroque suiteand the role of texture in the traditional distinctions among dance movements-between, say, the sarabande and the gigue!.

Textural processes /'n progress/'on toward intensity, in recession toward cadence, and in antic/Qoation of thematic statement

A factor of which a good deal of mention has already been made, that of the shapingof textural change inthe direction of increasedintensity in which other element-structuresoften work complementarily! is carried further in amplification of earlier referenceto ajosquin setting of the Deprtjundis. The De prqfundis settinghad prior reference pp. 87-90! in connection with linear

melodic and harmonic functions

at its outset, as well as limited

discussion oftextural processespp. 189-90! in a different extract. The ,portion of this magnificentwork quoted as Ex.2-17 isclimactic, and its climactic effect might well be said to be moresigng/icantlyproduct a oftextural progression See cantusfirmus, ostinato,and other textural proceduresdiscussed asvariation technique in Robert U. Nelson, The Technique of Variation LosAngeles: University of California Press, 1948!, pp.l0fl`., 51f.,58-60, etc.,and Berry,Form inMusic, Chapters 8 and 9.

texture Ex. 2-17. Josquin,Deprofundis.

Do - mi - ne,quis su-sti-ne - bit?

Do

mi - nc, quis su- sti- ne- bit?

Sland2 J

m.47

/ ti> J!

Oo u! Reprinted from theSmijersedition,published by G. Alsbach andCo.,Amsterdam, bypermission of Creyghton Musicology-Musica Antiqua,Bilthoven, Netherlands.

texture 243

than of any othersingle factor,although of course complementary rhythmic acceleration and tonal progression play important roles." In mm. 42-54 decisive factors in the texture are noted graphically in the quoted example. The time intervals of imitation are progressivelyshorter within the section as well as between this and earlier

sections: the imitative

entriesthe of words Domine,sustinebit? quis initially are oat,then at J and

againo;atand the rising quarter-note motive,§ J J ,l s , the motivic drop of a minor 3rd, ,

and the cadential drop

of a 2nd in half-notes, participate in a number of brief but functionally

important imitative interactions mm. in 45-46 at time intervalso,dof ,

and ,das indicated theinexample. Finally, in mm. 51-52, which are highly animated rhythmically and texturally, but not harmonicalbf;note the compensatory stasisof harmonic rhythm!, imitative interactions among appearances of the dotted-rhythm

motive appear in anumberinterlinear of relations at the distance _ The J textural complexity is enhanced here, moreover, by contradirectional imitation.

This section is the only one to this point in which four independent voices emerge incontrarhythmic activity. They do so earlyin m. 44, two bars after the passageis launched in the imitative entries described above. There is,

of course,interim release in, for example, the2 texture of sucha point as 1

m. 49, where the internal voices become interdependent homorhythmic, homodirectional, almost homointervallic!, or even simply in the withdrawal of the tenor at mm. 45-47 or the superius at mm. 48-49. Such factors as the relatively stable bassusat mm. 50-51 cf. the bassrole of later styles,and its restrictive submission tothe demandsof tonal-harmonic support, cadential definition, etc.!, the rhythmic identification but directional opposition! of tenor and bassus atm. 52, and the ultimate interdependence of voices to which the textural structure finally turns at not long in advance of ! the cadence arevital factorsof recessive process. The progressiondescribed above,and the larger higher-level! progression of which it is a culmination, are thus excellentlyillustrative of a number of the textural factors with which we are concerned. And this highly active, diverse, andcomplex passage, the section of greatest vitaligv inthe motet, is thrown into perfectly calculated perspective by the almost entirely homorhythmic phrases whichfollow-the only such phrasesin the entire piece, significant "The sectionof maximalintensity oftextural interactionand diversificationEx. 2-17! has anear parallelin the final sectionof thepiece, towhich theinterested reader might refer for comparativestudy.

244 texture

in their maintenance of the full density of four components presented in the forms simplest textures.

The reduction in textural quantity density! as a cadential processis surprisingly rare in music, especially where Final cadenceis concerned; it is morecommon in tentative, internal cadence, especiallywhen suchinternal cadence ispreparatory to important events to follow, when such events, as often with thematic exposition, call forth relatively full densities see the Brahms reference,Ex. 2-20!. The procedure of final cadence, wheredensity of texture is concerned, seems generallyto favor the emphatic substance ofquantity, as a rule, while com/Jensatovy action of recessive simpwication withinqualitative texturalparameters takes place. Musical examplesshow that greater densityof compressionas well as of number is

characteristic of

many cadential

formulations; relatively

ample density-number is, especiallyin tonal contexts, of value in the clarity of consonant tonal-harmonic function at its most crucial point, that of resolution, and increaseddensity-compression is a usualconsequence ofdescent ofupper lines. Nevertheless,examples canbe found in which dissolution of density is an important aspect ofrecessive, finalcadential process. A cadence from Sessionspiano work From My Diary Ex. 2-18! shows a functional, comparable density-numberrecession 4-3-2 as well as acontraction of texture-space from29 to 13 semitones. Ex. 2-18. Sessions, From My Diary,third movement.

Larghissimo emisterioso J= 32!

EH' 2

tempo _

Auf; fm

`*

fa g

morendo g

1*-U rfb P 'ii

g_

tr--D lr' _-T""=lf-'Z

Reprinted permission by Edward of B.Marks MusicCorp.

Even wheretextural qualitative complexity is maintained to a very late point, cadential expression maydepend altogetherin its preparatory stages,

and predominantly in itsultimate formulation, on processes of declinewithin other elements. In such circumstances cadentialfeeling is often relatively indccisive.

Example 2-19 is memorable in the twentieth-century concerto literature, and an excellentexample forstudy ofthe actions oftexture in cadence, in this instance thefinal cadenceof a movement. Themotive stated twice by

texture 245 Ex. 2-19. Schoenberg, Concerto, Op. 36,for violinand orchestra, first movement. Lento J 76! m.263

;=.:=.:_ -

1- 1,iE; y con sord. 'V

Hn.1 '

Tuba- 92

si _

rn

JE

__=;_ __

n

- +--f e__==____

"

_

l g`

/' -M/J

~ nv , ,_

Timp. -

'."

rP

rx

'g lf?-

_.

,_

5*

f ______"`

'

'° `

1.2. _

I _F: W //I .

In 'V__ sgmdo

" -e-e,I if

'

'_ 5:'__ " ___' --3

via E- Q iii? ii*

pp

_l/J

e

_M A

' ` ='

1 Ql g . J 5 P* 1

'l

ff i

' pizz.

' ,_

E ,QE

I

ma .ta *All parts sound aswritten. Copyright 1939 by G.Schirmer, Inc. Reprinted by permission.

the soloviolin is of primary importance in the entire movement. The radiation of its influence throughout the texture is seen in the indications drawn into the example: there are varied imitations of the characteristic semitonal succession inalmost every instrumental part. The significant and necessary

observation which must followis thattextural complexity, activity, incipient and abortive but impressive interlinear independence, and considerable density are compensated forby the lack or tight restrictionof linear, harmonic, and tonalactivity andespecially the complementary decline in tempoand dynamics. Cadence isoften, aswe havestated, markedby realization of substantial density, evenits full potential in the particular context, with other elements expressive ofcompensatory recessive stasis anddecline. In the Schoenberg

this principle goes beyondthe accrual of density useful for emphatic effect! in a texture active withinterlinear, diagonal relations Q'a considerable number, while the stasis of relatively leve1 lines, dynamic reduction, and gradually slower tempo to the point of fermata are of`essential recessive function, dominant in their overall eH`ecteven in the atmosphere of a cadence

ultimately slightlytentative infeeling. Tothe extentthat textural activity persists, cadentiallinality is qualified.!

246 texture

Of course the deliberate minimizing of the natural tendencies ofinterlinear interdependenceat the point of internal cadenceor preliminary stages in Hnal cadence! isa universal and vital technique of texture for suppressing too punctuative or conclusive an effect. And the device of preparation the process ofestablishing anticipatory atmosphere! can be an important factor in internal cadential process, usually involving dissonance ina universal technique by which expectation is heightened. The shaping of texture is requisite and critical to cadential and anticipatory effect in Ex. 2-20. The pattern is of course oneof decline in density-number, a very common technique by which anticipatory feeling is enhanced complemented by the unsettled tonal-harmonic situation! in preparation for resumption of thematic statement.

Measure 64 is a point of maximal density: the texture is essentially chordal, strongly activated by heterorhythm and intertransference ofchordal PC factorsamong very active voices.Dynamic level jbrtissimo! andextremes of pitch texture-space! are of complementary effect. Quantitative release begins in the following bar; there is somediversification here-a qualitative

Ex. 2-20. Brahms, Quintetin G, Op. 111, for two violins, two violas, and cello, second lT920V6lT928l1t.

Vln.l

/m.63 /-` Ad X ,X /_92 92

Vln.2

é on

N g Ag g/` ,`

CZ?! X F _ /,_ 1

" /._

/` `

Vla.l _ Vla.2

C1722 ` Vc;

.n

V'

.1 .11 .1

Common modification density reduction, simplification! of texture jbllowing cadential punctuation, one ofthedevices bywhich cadence is throwninto reliefand resumption announced, is anomnipresent factor in music.

texture 247

m.64 /A

nw /`

.J A ZVH 1.1.' ll.. Luv

'92

;/`; FJ _/` '9292

A

-Y| I

~~

A /$

=~;~ i

92 /1

_

,-5 _

5

wr'

_ :=. `ii;§

_

``

92

9. .22

EL F"-5

== , ?"T' 3 ° 3 3 3*

i W5

92 J'

m.65 _;, _

J

f _tiv

92, __,

in TJ f "/PR ii?= ==» 3 f'

fP

e f fJP

7

!

J kph 1.

. ""92

,b

»

f 1;

92 __

-f X

3_

,1

=§§a@§§_§g§a :Z ;:_-_:i_':

.54-~Jf -92 dim.

.I

.Jg

dim.

/

248 texture Ex. 2-20 continued. m.67 Jlf: , 1. ll.. _' nw

-' P espress

|_ 1. _1 lfl in-1'

'

ll.. t

P espressO

| I,|li i I_ m_ 4- 92 -fn Y I-m Y l 1-I . _ . Q ; 92 I/._ll11h.||1.3 l:r1Zr1|.4I_:|.1 9292-92vl:~ll111¢:|.A;| ll_| _Q * `i -;`_Q__& *i92 * i_ | tn i |i'r| 7 »` ,»-'Y _ ||.;1v ||'1 Q-f&` Ya I in » ,_ F `92

I

92 1,

I

/f

_

'Y °

e

'

r .O pizz lf

-i

compensation inthe quantitative decline-in a limited projection of motivic substance. 1

The subsequent stage inthe process might besymbolized as R3!, with all voices but one now reduced to stasis, anddeleted gradually. The surge and accelerated activityof the viola line is compensatoryto continued decline, holding the recessionin balance, as isthe maintenanceof dissonance intensity. Gther recessive factors are the continuing dynamic reduction and the total cessation inharmonic rhythm. The ultimate cadential measurecompletes theprocess, recedingfrom l

to l, with rhythmic values lengthenedgradually asthe viola line falls in a complementary gesture, andwith harmonic inactivity maintained within the dissonantframework. The entire passage is extremelyuseful asan example of textural succession calculated in expressionof cadential and anticipatory structural function, and of the complementarity and compensationof other, relevant element-events. The manner in which persistent instability 'of tonal-harmonic content on the brink of resolution! counteractsall tendencies ofrecession istypical of tonal contexts.

Texture as space

The following examples showtexture in its spatial dimension and indicate particular textural dispositions ofvery distinctive, even motivic, forms.

texture 249

There is a sensein which melodic successionin the extremely high or low components expresses animportant, functional aspect of texture-a modulating field in which events take place. The shape and compass of that field, which we are describing astexture-space,two-dimensioned a fieldsetting out horizontal and vertical boundaries enclosingthe element-successions which constitute the musical work, are functionally effective in circumscribing and governing much of structure, and it seems usefulto regard this space as a distinct aspect of musical texture. The spatial factor texturespace! might be defined as the field enclosed by lines tracing the pitch successions of outer componentsin addition to the two vertical, or diagonal, lines linking these componentsat left-right extremities at some given level ryf structure. All of these referencesare of course analogousto physical structures on a plane." One of the examplestreated at some lengthin the discussion oftonality is the first of Weberns Four Pieces, Op.7, for violin and piano, discussedas Ex. l-49, pp. 162-65. Let us attempt to see that piece in its spatial texture, as defined,by describing the field within which its pitch events arebroadly circumscribed and then by attempting to identify some of the important characteristics of changes in the spatial field as they might constitute an expressive succession within this particular parameter. In expression and prolongation of its EI; quasi-tonic the piece sets out with two functionally significant occurrences ofthis PC four octaves apart Ex. 2-21!. The violin descends thenceto a prolongation of activity on eb while the bass, firstrepeating Ebin m. 4, ascendsin the penultimate measure to eb, after which both recessionsconverge onel», the root of the final triad. Obviously, there are deflections in that spatial field, like the low F ll of the piano in mm. 6-7, a prominent local expansion of the space; but the foregoing comment describes theessential linesof the spatial contraction as to a marked extent influencedby the tonal centrality of the PC Eb, and the axial centrality of ebl.! Seen in this sensethe texture-spacesuccession isone of recessive contraction! The resultant focusingon the root of the final, cadential triadic resolution is of coursesupportive of tonal function, and the analogously spatial 3°It wouldseem reasonable to extendthe conceptof two-dimensioned musical space to include a further dimension,that expressed by the hierarchic levelingof element-structures from the foreground of the mostimmediate, detailedsuccessions to the remotestbackground-the structural level atwhich thebroadest, most generalized lines of succession are discerned. Hereagain, inthis referenceto a third dimension in musicalspace, terms like foreground and background are analogous toterms usedwith referenceto the qualities of physicalspace. Themost vital dimension, time,is symbolizedby the left-right factor in this image, whilethe up-down factor symbolizesthe field of pitch frequencies. The third dimensionto which the abovereference ismade is,then, one of actual depth of experience ofstructure. Otherparameters ofmusical experience-thefield of intensity, or that of timbre, forexample-can of course be conceived as further dimensions.!

250 texture Ex. 2-21. Contraction oftexture-space defined by recurrences of tonal PC inWebern, Op. 7, No. 1. mm.l-3 mm.6-8

l_ m.9

anticipated

_ in

mm.-4

m.6!

m.8

formulation described by these lines of succession fromthe extremesof the opening to the compressionof the final sonority is an important aspect of the pieces structure, constituting a factor significantly complementary in functional eH`ectto that of tonality, to which it is inseparably linked. Concurrently active element-structures for example, that of rhythmic motion, gently accelerative toward the middle and restorative of longer durations toward the end; or upper voice linear descent! are complementarily or counteractively shaped." The idea of space as anaspect ofmusical textureis especiallyrelevant to theworks ofWebern. In certain of that composersworks spatialsymmetries are formed by the distribution of inversionally-related set-forms in equidistance around a central axis. In Ex. 2-22a, an extracted major segment of the first movementof the Quartet, Op. 22, there can be seen aprogressive expansion of the texture-spaceto an extreme point m. 22, a critical point in the movements overall structure!, at which the synopsis givenin Ex. 2-22b is terminated. The point of extremity to which the spatial progression movesis underscoredin a number of ways: there are complementary progressions ofqualitative interactions of texture, and color and rhythmic structures are directed intensely to this same point, from which the movement, in a number of complementary element-successions, recedes. The As noted earlier, final increased density-compression is inevitably counteractive, while clearlysubordinate, torecessive spatial contraction and,in the Webern, ofrelatively and potently!consonant implications.

texture Ex. 2-22a. Webern,Quartet,Op.22,for piano,violin,clarinet,andsaxophone, first movement. 12 tone set:

Sehrmassig J.= ca.36! tD-

f

Vln.

Teno sax.*

Piano

m.4 Vln.

C1.

Ten.Sax

Piano

«All pitchessoundasnotated.

P1ZZ.

texture Ex. 2-22a continued. arco

Vln.

C1.

Ten.Sa

Piano

Copyright 1932,Universal Edition.Usedbypermission ofthepublisher.Theodore Presser Company,solerepresentative United States, Canada andMexico.

Ex.2-22b. Summary diagrammatic representation of thetexture-space in thefirstmovement of Webern's Quartet,Op.22, mm.1-22.

m.22

texture 253

functional actions of all relevant elements to and from m. 22 constitute

a fruitfuland criticalarea ofstudy inthe movement. See also Ex. 3-1,p. 307.!

Only theopening bars, and theimmediate approach to m.22, aregiven in Ex. 2-22a; Ex. 2-22b, however,is a synoptic renderingof the entire passage. Itis contrivedto showexpansions in the texture-Space, disposed equidistantly around the fll /gb' axis,and maximallyinflated atm. 22. A further example goes back to a far earlier historicalreference, the

Prelude to a Handel Suite for harpsichord. The shaping of the texture-space is vividindeed; and this particular kind ofconfiguration space of cflthe Webern examples!, achieved byalmost consistent, high-level directional opposition betweenouter components,projects a characteristic contradirectional counterpointwithin an essentially simpletextural context.

Example 2-23a shows brief a extract and Ex.2-23b is a condensed represenEx. 2-23a. Handel, SuiteNo. 3in D minor forharpsichord. Prelude.

m 6 Presto fl Y.-fi |l.. ' Ku'

*1' X

ll? #Ll =n'33 l rl

li-1 Zi

llf l-=

rl: =

1l

r'§.llPi111 »| _.atv

"_,

c

r35/

i §

_i_l_t _f

-8 f

=

»:-1_92L_ =r"' |92~l_ _ ,q HI

,,' ii ,,f

-I

il ' aui1;=' _U---I*-4'ii 11. .Ji _92"_/ ¬i.A7'/ jg 3 `

m.l0 / .I _e Y Hgi?.uf r921:;:.z:r;'1.a1 . ||n1uI;..J1.A_', _ f' 92 Q.

~' fi 'xli' _92 i

_-1; _

1 I1 I __ Q

H!__92 .f

11`_ ___ 11

~'

92i / hi J

__-92_ _».92/-nrnl Ir' .. ' 3 :.:1.4! 1 92 3 P;

3 92 lf /:ri l -

254 texture Ex. 2-23b. Synopsis ofinflation andcontraction inthe texture-space as expressed in contradirectional relation of outercomponents.

/+

I"

2 92

'.'/_E 9

|3

92 .»-fy,

3 929292

li'

92i ----

£55 lf

-*/-S

as

¢9292

~»

5'-.... ..-..----""»"__i...-'P

iv V

i

tation ; it is evident that there isa single,forcibly directed succession progressive, then recessive! in which spatial inflation and subsequent contraction must be regarded asmost critical elements in the structure.

Mot/'v/c texture;

the provocative

effect of unusua/ textures

There are many times in music perhaps especiallythat of the last hundred years! whenmaterials areof suchdistinctive textural cast, andwhen the particular qualities of texture are sovital a factor in the identity and interest of thematic-motivic material,that it seems plausible to think and speakof texture as motivic-or of a specific texture-motive. A thematic melody in voices interdcpendently doubled in 3rds, where this textural feature is a necessary and recurrent aspect of the material, would be properly conceived asthematic texture in an important sense. In Ex. 2724, the motive is incapable of adequate characterizationwithout reference to texture as wellas to rhythmic and linear formulation: one can scarcelyimagine representingthis materialsay, in a thematic index-with one line alone. A very famous exampleof motivic texture is the initial sound of the Stravinsky Symphony qf PsalmsEx. 2-25; seep. 256!; it is a recurrent motivic factor of arresting quality primarily becauseof itsarticulation and anomalous, distinctive spacing.It is not significantly defined merelyas an E minor triad as might e:i in a Classicalcontext!, nor with respectto its orchestral colora"Consider, for example, thefirst movementof StravinskysConcerto inD for violin and orchestraor the opening thematicstatement inBeethovens Sonata in C, Op. 2, No. 3, for piano.

Ex.2-25. Stravinsky,Symphonyof Psalms,first movement. J 92

5 Fls.

4 ObL Eng.Hn.

S BnL Cbn. 4 Hns. Tpt. in D 4 Tpts. inC 2 TmbL B.Tmb. Tuba Tlmp.

S. A.

Chorus T. B.

Harp

2 PianosllOh ItpCc gg div. piss. div. piss.g Db. Copyright 1991byEditionRusse deMusique;Renewed 1958.Copjeight andRenewal assigned to Booscy S Hamkes, Inc. Revised EditionCopyright 1948byBoosey O' Hamkes, Inc. Reprinted by Permission.

texture 257 Gesualdo. Or, cha In gioia credea viver contento Book 4!

from Madrigals.

An example from Gesualdois given below for relatively comprehensive analysis oftexture. Major attention is given to changing interlinear relations within the range from maximal diversity to total accord. Changing, func-

tional patterns of imitationare, however, also ofgreat importance and interest. Thus, while all four sectionsare imitativc at distancesranging from

lgl tol5¢l !,the shortest time interval, briefly suggested in m.4, becomes appropriately prevalent in the section Fuggesi . . _ flies away!, an area of relative rhythmic and textural animation conditioned by the text ; and

imitation persists, mostly at 4J , in theintensely chromatic final section Oimé, vienmeno; alas, it wastes away!. A representation of textural progression and recession with limited symbolization of` qualitative relations as well as density! is given as Ex. 2-26b. Example2-26a isa quotation of the entire primaparte ofthe madrigal. Investigation showsthat nowhere in this example isa diversity value of 1

I 1 achieved.toward Developing ofpoints maximal textural are comple T T

two progressionsof persuasivegrowth from one voice alone, or from two lines in interdependence!. The intensity profile of the entire structure thus rises quickly within the First severalmeasures, subsequentlyreceding and progressing againto a point of maximal diversity at the immediate outset of the intensefinal section. Example 2-26brepresents thestep-by-step processes by which the textural structure is shaped. It is important to note that progression in density-number is significant at more foreground levels that of the phrase especially, where there is progressive accrualexcept in the Hrst two units, with the cadence usually formulated in full density! while the progression of textural diversity significantly features lower degrees tj complexiy but notqfdensigy-number!the in internal phrases. Itis thus in this second parameter of textural structure that the overall shapeat the highest hierarchic level is bestseen: thefive voicesare engagedwith considerableconsistency throughout the piece while changes in their degrees of interdependence shape the broadly defined textural succession? Telemann. Fantasy No. 4 for violin alone, first movement

That the study of functional textural events canbe critically important

258 texture Ex. 2-26a. Gesualdo. Or, che ingioia credea viver contento from Madrigals, Book 4;first part ./

lI

l Or, chein gio -

II

lE

JV

,

Or, chein gio - -

lI

I _I

ia

-; Cre~dea vi- vercon-

'-"_ '~

Or, che in gio - g__&

ia cre-dea vi - ver con - ten - to,

l" 1

31 1

`t Or, che in gio - -

JM _WE_ , L_ ____ ia, or,

_e e

cre-dea vi /92

- if t V-VVI

E rE

ia cre-dea vi -ver con-ten- to,

'che ingio -

cre -denvi -

ia cre- dea

vi -

5_4

ver con-ten-

Lo

or, chein gio - - na cre-dea vi- vercon-ten - to, `°""_l%"_°1;__°"i"_ '_""_' ___ ' ' l __ VW _ Q1.l -I I1 _:lQ: 1 1 p. 92 gi/ IV ten - to, or,_____che ____ in gio -ia cre-deavi - ver con - ten - -1. r r ===. frm" 7'1"-I 25 - ver con-ten - to,

If

or,____* chein gio -

II

ver con-ten-

vi - ver conten 7,

mr*-r:.;=;=a= J

vi -

E*?__1rto,

ver con -ten - to, cre-dea

m.8 -/]'|4_ _ Y ,t,, /--*_§ , N' ' e ' to, M'a-pre la gio - -

Q* - ia.

m'a - pre la gio -

M'a ~ pre la

gio -

ia, |'n92

- pre |a

pre la

gio -

ia, |'l92'l

- pre |l

to. M'a

&_ to. M'a Np

ia cre-dea

-I 1

M'a - pre la gio

I !° pre la

L er

gio -

ie, m'a

r1

- pre ln jo

t

-

texture 259

m.ll

4 »'92 .I - iall se - no, Fug-ge-si

:1 1I J

lal-ma el cor, c'lcor,oi-mé, oi-

7O

l

I 92

gxo - - 131| se no, Fug

- SC-Sl lal - mael cor,fug -gc-sl lal-mae]cor, ol-me-

gio -

-ge-si la|- maelcor, oi

- iail se - no, Fug

- ia

il se - no,

-

Fug-ge-si -laI mael cor, fug -ge-si lal- mael cor,

oi -

92c c il se - no, m.l5 4 __

Fug-ge-si lalma el cor, fug- ge-silal - ma e|cor, oi-mé,__

,

@*1{_F3 gf? mé, vien me - no, # il:-»

oi - mé, vien

. *;Z;f;_

me -

oi - mé, vienme -

=» I - f=

- mé

e F ,_ , , - mé, A 1»=1' 7 _

it r

-1 -

mé _.__ vien me - no.

D vien me

- no

oi -

mé, vienme - no.

'_ JQ oi -

_

LE

no, oi

'8

mé, oi

, it

° ¥==92 JH - no. ..

oi mé,

£:=92 H

mé vien me -

I-

- no._____

oi -

mi, vien me -no.

Reprinted fiom the UgrinoEdition, W. Weismann, ed., by permission qf Associated Music Publishers, Inc., as agents Q' Deutscher Verlag fur Musik, Leqrzig.

texture

260

Ex.2-26b. Texturalprogression and recession, quantitative and qualitative,represented at phraseandbroader levelsin the Gesualdo example. m.4 m.7 m. II Qualitative changes:

Numbers nf real vniccs:

II I2

2I

m.9

m.I I

I I2 2,2

I2 2 !,2

3!,

2 5,2

ni.14I Oime... I I I2 2 II I2 2 22 I2 2I I2 I5 2I 3I

Qualitative textural profile:

Numbers oflines:

2-27a. We can only scratchthe surfacehere,again; but textureand textural structurein virtually all its aspectscould well be the basisof inquiry in comprehensive explorationof this movement." The movement'sform is basicallyone of recurrentstatementsof an openingthematicidea separatedby episodicdigressions of a more fluctuant character.One objectof inquiry might thereforeconcernthe applicationsof textural variation in thematic recurrence.The reader will perceivethat the openingmaterial Ex. 2-27b!, like other instancesin episodicas well as expositorypassages, posesa line of compoundimplicationsof two strata in oblique relation, with sequentialrepetition. The textureof the firstappearanceof the motivemight be symbolized as ,

where I! isan "implied" independentvoice.At the secondentry, in

mm. 34 35, thereis variationin the doublingof the lowercoinponentin the secondpart of the sequencebut not in the first. The statementat m. 57 is analogous to that of m. 15 exceptfor restorationof the primary tonic.! These variations,graphicallyrepresentedin Ex. 2-27c, show texture undergoing transformations complementaryto thoseof fluctuant tonal reference.The 42TheTelemannmovement is brieflydiscussed and quotedin full in Carl Parrish,A Treasury ofEarlyMusicNew York:W. W. NortonandCo., 1958!,pp. 297301.

texture 261 Ex. 2-27a. Telemann, Fantasy No. 4for violinalone, firstmovement.

/.l 1111--I--2.1 `l!,--Z1 `Z

Iii1 I-111 111111 bl--1 l__-Qu# _111 111 i1 ll-.1--H --1 ` Q

If -1-!-ul-H

m6 _ W _____ l |;r. 5i'11_-ld

92 __ -l 1: 'Till-U'1Y'1Y'1-I r~!1!!1_1$-$_1_1

F rr l!l!_El -

.Gln 1 /.th VI I

_ 1-111 r-11-1 _I

1l

V V TTT

1

1

ll _Z

'°'

.=. !

1 2| _2

I1

m.33 tr

a !""""'

1: E

11 Reprinted from Georg Philipp Telemann, Musikalische Werke, G. Hausswald, ed., by permission Bdrenreiterof Verlag.

Ex. 2-27b. Compound implications in theprimary motive, m. 1. hm-. ___ .un-mi -f /_.min lj __ ilix

1 Il'

.1 D

_ .u 11 nnnlIQ;1ll' » -

1|

'-'-F etc.

Ex. 2-27c. Texture inthematic recurrences. mm. 1-2

for J

mm. 15-16

]E

J J

mm. 34-35

[J

]

progression constitutedby this series ofappearances isone in which an initial 1_ _. texture T! of isenhanced subsequent in appearances by atrue, sounding doublin com 8P

onent°»where this au mentation of 8

densit Y is Y onl artiall PY

262 texture

carried out mm. 34-35! there is at the same time extension of the unit by repetition of the two-measurepattern. Most vital to the analysis oftexture in this as in any example is again the consideration

of the concept of

textural progression:

the quantitative

increase, orthe progressiveintroduction of qualitative changesin the direction of increased complexity and intensity in interlinear relations. It is remarkable, and typical of this movements unusual interest _within the restraints ofthe medium, that there are qualitative and quantitative textural shapes whichconstitute an important aspect of its structure in complementarity with other elements. The obvious but critically expressive profile of the opening phrase, considered in the light of textural change, isan example. The accrual from a condition of two implied voices, one relatively static, to two sounding voices in considerable differentiation, to the four-voice density of the dissonant chord at m. 5, to the releaseinto cadence,with a second soundingvoice inactive! momentarily introduced, is expressive ofa persuasivetextural shape beautifully complementary to that of the line of pitch events, and to that of the occurrenceof tonal and implicit dissonance aswell. The textural structure of the digression at mm. 21-34 Ex. 2-27d! requires special study: it is a crucial element in the climactic, chromatic EX.2-27d. '°'92 /°`

.

.w:=§=='§;:§§.:==L_E5== =;_53= ='

e=g5,=;.-=-' ~-=;_,._,....

m. 25 'J

1A '

=f

l§l@ e e k

A-92 ,.92

/`

13

m _ . /_T - _:Ji G5 "=--.!....! =!E=-.EE-%'{'°§;E=; -'-E I'-I lllll Ill' ~ s'

f92 on

-I

~

Reprinted from Georg Philipp Telemann, Musikalische Werke, G. Hausswald, ed., by permission Bdrenreiterof Verlag. The textural structure ofthe first digression partiallyquoted in Ex. 2-27a!is very different from that of the theme: it consistsof an alternation between contrasting textures, an alternation inwhich changeis variationalrather thansignificantly climacticallydirected or subsident. Measures 42-50 ofthemovement are comparable.! Space does notallow detailed

texture 263

development precedingthe m. 34 cadence.Here are stages verysubtly contrasted indeed. Compare, for example, the implied two-voice texture with which it begins, onein which only one of two implied voices is significantly active in pitch-line; from this tentative diversification there is progression reaching a level of determined opposition of the two albeit still implied! strata after m. 30, and the subsequent, late recession into a condition of relative interlinear accord m. 33!. Further insight into this extraordinary progression canbe suggestedin the comparison of analogous measures23, 25, and 29-32. In what important sense ism. 27 different in textural qualities from m. 23 ? How, throughout this process, istextural releaseachieved at punctuating intervals as the directed motion is otherwise continued ?! The importance of interlinear independence achievedby the determined, insistent, opposing demands of concurrent textural components isnowhere more potent than at the summit mm. 30-32! of this progressive shapean intensity of texture momentarily resumed at m. 4lHf!. This fact is critically suggestivewith respect to an overall, broadly defined textural progression at the highest architectonic level. Finally, a spectrum of textural values and conditions might be identified for this piecein a symbolization of content of quantitative and limited qualitative distinctions Ex. 2-27e!. Ex. 2-279. 11

l1

34

T 'I

Densities 3and 4 represent, ofcourse, valuesof infiated importance in so restricted a medium. Qualitative diversity of three voices, one of them implied, is maximal for this particular context; yet, even the importance of

%!, which in is there determined differentiation components of the by means noted above, must be emphasized if Telemanns structure is to be understood. The movement is, in summary, a seriesof textural progressions and recessions,the former developing at precadential points! toward maxianalysis ofthis passage, nor of the textural structure ofthe final digression mm.63-70!actually aninterruption in the final theme statement, one ofsubtle, restrainedcontrasts. Interlinear intensity in the climactic portion of the movement issuggestive ofa textural featureof importance,that in which diversityis achieved by projectionof lineswhose relative independence is heightenednot only by directionaland rhythmic opposition butby dramatic contraintervallicrelation aswell, i.e.,by theprojection ofintcrvallic featuresofthe highest possible dijerentiation-that of semitonal asopposed toprevalent succession, in the competing stratum,by leaps.

texture

mal quantity and/or maximal diversity and activity in which the texture is significantly complementedby other intensifying factors, notablytonaljfuctua tionanddissonance. Intense qualitative developmentin the extendeddigression at mm. 21 34 Ex. 2-27d! is crucial in textural structure at the broadest hierarchic level. Bach, "Denn das Gesetz des Geistes," from Motet, Jehu. meine Freude

So excellent is Ex. 2-28 for the study of textural structure and process that it is quoted in full. The opening nine measuresare a counterpoint, occasionally imitative: 2> !,4' 4sbetween the alto and soprano components,the latter in a consistent doubling in 3rds; the texture is thus one of two real voices within a density of three lines with restrained imitative interactions and generally contrarhythmic-contradirectional relation between the two essentialcomponents. Someimitative interactions are traced asdiagonal relations in Ex. 2-28. Following m. 9 there is a progressionin which total interlinear diversification emergestentatively m. 11, m. 15K! but in which this tendency is restrained by intermittent recessionto the textural conditions of the beginning m. 10, mm. 12 14; note how the textural component consisting of paired interdependent lines undergoesfrequent recoloration!. The incipient tendenciesin the qualitative progressioncan be analyzed further in observation of the logical extension of the trends noted above. The consistencyof two real voiceswith occasionalimitation in mm. l 9 thus representsan initial stage; the following bars, with compromising interdependencesand consistentimitation, constitute a secondstagein the direction of maximal, but not yet achieved, interlinear independence; after m. 15 there is interlinear independenceto the end. In an extreme foreground the rhythmic identification with considerabledirectional contrast between outer voicesin m. 15 is a still intermittent step in the processof diversification, as is that of m. 17.!

Quantitative contrast in the final stage and momentary relief within the parameter of density-number! is achieved by the opening of the texture to a single voice, with reaccrual of the others, after the m. 19 cadence,the cadential effect of courseenhancedby this textural device. Increasing rhythmic interdependenceas well as relative inactivity of the lower voices especiallyalto! help to prepare the final cadence,which is nonethelessa relatively suddencessationof competing contrapuntal activity, enhancedin its provisionally conclusiveeffect by the fermata and the metric 4 The symbol denotesimitative interaction.

texture 265 Ex. 2-28. Bach, "Donn dasGesetz des Geistes" fromMotet, Jesu,maine Freude.

S°P'-1 :¢=:=

' ::

=l_=v::=;=-

"-" "-*

Denn Gedas setz - des Gei stes, -da- le der

swf-2

JJ

5*

J = ===-

Denn das Ge - setz "`des

Alto vu; __ .5 ='i|'{ 5=F=F -1 :wiQ11 l 11 N

Gei -stes,_de_r` _-_da"le _hen - - ` dig__

3 ' 1- __J* 92 JT 1E

Denn, denn

-/In i

5E§§§§5:-IE f 0

as Ge - su;

1.11 _ .""' -|

_Q

_Ie -

rf - lr:=:=.== I/»:__|!|!"-I Nl- _ 1i 1

--

des Gei

___ V, V1 =e su, in

°*' =

Chri -

-j

e

__

- sto

me =

_, ,_ .i_. 1 _le - su, in Chri -sto =,"' ._ =E 7 c == §Z.111 7 '' |/..'i1i|1»1 11111 !'Q§_jl!92-'2§l*92i_ 9292|'mz§'lQ|-I-:-2-_Q "_ In "Y-2i;_ 1 _ _-Z £5 - stes, der dale - ben -- dig mach et - in chfi - sto m.9 -J In_ _ =i"-' 'l

:u °

je -

:" ' -

LG; 1

su, hat

e

_Ie :" I/._' _I-'D- Z____j 9292-92' I je -

.;= '

,_ ' F=

F=~ '

mich 92 frei 8°

!i

3°- macht von

~-_I1======:i:=5 e : '

su, hat

mich frei

a=

` frei

:' =l-|!P =-_- -g-f 0 su, 1

hat mich frei ge

m.l3 4I'|. ! U. U-l!7_-_-_-||lZ 2 §l-.-I la. IIN9292-92' -7--!_2!_YQ

ge -macht, _'

- macht,hat mich :nf -L_

5:

frei ge

- D-' _ Y_ __

hat mich frei ge - machtvon dem_ Ge 1'| Q -bf-- -:_--I! _ Z.. e E_2{F=§i'$===-?=F:'2{=='|i== l'=| 9292l' lY1'l1 _u _Q il-IP_ l hat mich frei _ gemacht von dem Ge -setz,von dem Ge-setz der Siin - de,

I! |! - macht, , _

der

266 texture Ex. 2-28 continued. m. I7 J 4V _

i_

i: '*-.-.u;=2f" E

I"

- de und des

To - des, = ` _== -.1if3

.--li-To ~ _"|` lu. lj-C ._* I ._-I"

` -'l- mr

,_

12

=' .aéesi fi /I i

_

vo%Ge-setz der 92 ` '

-~--- des, _ L1- lQh1

_ -Btu'--'Q

QJ

Siin-de und des To

"Y-

hw-ll:l|.°: ! - des,

von dem Ge-setz derSiin de

D

und des

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strength of the final accord. The final phrase, with its stream of imitations

at the, isof course a lower-level textural progression in itself; like the prior section beginninghat mic/zfreigemacht. The reader will be interested, in further referenceto this extraordinary example, to consider therelation of other elementsto the textural structure. For example,consider theoccurrence ofthe highestessential pitch in conjunction with maximal density and complexity of texture, and the concurrence of the tonal structure, which is modulatory and whose shiftof tonal reference also occursin conjunction with the area ofmaximal density and intratextural activity. The persistence oftextural complexity, after the noted progressions, to a late point might well be felt as complementaryto the open tonal structure.!

Brahms, Symphony No. 1 in C minor, Op. 68, first movement Un poco sostenuto!; Introduction

Of necessity,discussion centershere on a portion only of Brahms first symphony, the introductory first thirty-seven measuresof the first movement." It will be possible in this context to consider in an orchestral context a number of featuresof texture as theypertain to a broad structural function, that of preparation of the main body of the movement. As so often in orches-

texture 267

tral symphonic introductions, the tempo is slow in preparation for the allegra main body of the movement. The example is, at the broadesthierarchic level, anacrustic, consisting of a powerful initial attack and subsequent growthand decline to a condition of minimal textural substance,low dynamic level, halted rhythms, relatively settled pitch-lines, and prolongation on the dominant, the most likely harmonic areaof anticipatory function. Broadly speaking, thereare two textural progressions of compelling spatial and qualitative growth, the second of these havingits own, lower-level, progressiveanacrustic preparation. There are thus two releaseareas: the first of these ismonophonic, then chordal in texture, the second more active and diverse texturally but with dialogue rather than overlapping interlinear imitation. To a considerable extent, the senseof releaseis accountable to dynamic reduction and a decreasein sonority, aswe shall see, but the resolution of textural complexity is vitally complementary. The entire introduction is quoted in Ex. 2-29a; then the features of texture and textural progression andrecession ineach of its sections aretreated in detail. 0ne striking fact to be emphasized inthe analysis is the unmistakable manner in which textural and complementary! structures are reflected in analogous shapes at varioushierarchic levels from the broadest, that of the entire introduction, a single motive.

to the most microcosmic,

that of

It seemsdesirable to pause briefly to seethe introduction as a macrounit, a broad anticipatory gesture shapedalong the lines suggestedabove and below. These linesrepresent a path of growth and decline, an intensity curve, asrepresented inEx. 2-29b; the graphic treatment includes asummary curve and shows theoverall metric function of anacrusis in a discrete symbolization.

Detailed analysisof textural events andprocesses inthe introduction is best treated with reference to its four divisions:

mrn. l-9;

9-20; 21-24

and

25-29 i.e., 21-29! ; and 29-37. Certain basicfactors shouldbe kept in mind: the broad sweep of the introduction at the background level in which its anticipatory function is preeminent and in which it is broadly seen asa I-V progression preparing the Allegro; dramatic technique in which elementchange is at times violently severe and abrupt, as when texture explodes under the pressures of developed intensity; distinctions between real voices as opposed to doubling lines, and among textural parameters of density, space, and qualitative interlinear activity; and the operations of complementary forces,to which adequate, consistentattention cannot always be given-events of coloration dynamics, articulations, orchestration!, tonal-harmonic factors

such as dissonance and chromaticism, the

drive of

melodic line powerfully inflating the texture-space!, and rhythm in all its manifestations e.g., surface accelerations, as of m. 7 or m. 24, metric elonga-

tions, like that of m. 8, or the constant useof syncopationand metric displacement, etc.!.

268 texture Ex. 2-29a. Brahms, Symphony No. 1in C minor, Op.68, firstmovement. Un poco sostenu to

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Ex.2-29b. Intensitycurveat two levels,asshapedby texturalandotherevents,in the Brahms introduction. 24 13

68 57

Intensityof sonority andtexturalactivity density

10 12 14 16 18 20 22 24 26 28 30 32 34 36 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 Areaof Broadestspaceand tentativetextural forceof anacrustic interactions, build-up;areaof intense reduceddensity Areaof densityaccrual interlinearactivity andsonority

Secondprecipitaterecession within all texturalparameters

Precipitaterelease; areaof low density, minimaltexturalactivity

In Section I there are three voices,powerfully stated, each doubled and reinforced. Betweenvoices 1 and 2 there is very little coincidence of movement i.e., a high degreeof rhythmic and metric differentiation!; vigorous contrapuntal conflict is apparent too in directional opposition, this factor applied very consistentlyand fundamentally. Voices 1 and2 movewith great and inexorable drive, intersecting as directions of motion change, dramatically opposedto the rigid insistenceof the fixed basspedal. Voice 2 woodwinds, horns 3 and 4, violas! is fortified by doubling in characteristic 3rds and later 10ths and 6ths!, providing increaseddensity with interdependence, a technique largely responsible for the richness of sonority. The sonority of this component and of the texture as 6 whole! b

is of course further enhanced by octave doubling: pedi

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texture 275

complex is thus one of three sharply differentiated, heavily doubled components.

The pedal voice is activated powerfully timpani, double-bass, contrabassoon! by repeated notes in throbbing regularity-a rhythmically insistent factor against the variegated movements of other componentswhile the first and second horns sustain the tonic PC. The pedal, while a tonally-melodically static component in and of itself, thus constitutes astrong contrapuntal factor against thedrive ofthe other voices,to which it is obliquely related. The senseof rigidity opposed tomovement the latter in surging chromatic successionsof great dynamic intensity! implies an impact of severe tensionnote suchdissonances as C, C# ; C, D; C, Bl- ;etc.; and finally C, Fl?!which builds to a really febrile state, rupturing under its own pressures at m. 9.

The progression, as to texture, maintains and does not compromise the firm independence of the three components; in this sense there is no qualitative change. The vital parameter in which there is significant change in the texture is that of space. Thedetermined, largely chromatic rise of the upper voice and subsequently that of the secondvoice! creates, as against the fixed pedal, a tense inflation of thetexture-space. very A acute factor in the increasing space,and increasing conflict of dissonance aswell, is the final chromatic movement of voice l-violins, celli-from F to F# against the dissonant pedal.! The texture-space, as to its upward component, is in its foreground a complex interaction of conflicting tendencies: rise of voice 1; rise of the original! voice 2 against descent ofvoice l; final rise of voice l against descent of voice 2 into urgently indicated, transitory resolution in m. 9.

The stark contrast of Section IIs condition of relative serenity is an aspect of characteristic romantic effect. The expression ofrelease is,with the confluence of complementary element-changes, considerablya function of radical change of texture and sonority, but cf mm. 29-37! with the allimportant, persistent, compensatoiy intensity ty" tonally dissonant prolongation. The relative prevalence of homorhythmic texture is an indication of textural recession,and in itself a radical contrast to the texture of the preceding phrase. Actually, of course, chordal texture accrues,with sustained woodwinds; and sonority is enhanced, incompensatory relationto the severe dynamic and textural reduction, by such techniques asfour-octave duplication in the initiating voice. At phrase level, this initial development has to be seenas a gradually expresseddensity-number progressionpreparing the chordal, homorhythmic texture which becomes explicitat mm. 11-12. The tonally distant harmony of m. ll, derived enharmonically, is of complementary effect in relation to pronounced textural change to explicit homorhythmic conditions.! The only contrarhythmic factor is the continued,

276 texture

residual, now motivic, iteration in the low voice: a subtle activation spilling over from the preceding pedal voice. The accrual represented in Ex. 2-29c is repeated in sequencein mm. Ex. 2-29c.

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13-15. But mm. 15-19, which correspond to but are an extension of mm. l 1-12, achieve moresubstantial progression:texture and sonority developin complementation to crescendo, the pitch-line rises, there is strong chromatic leaning, etc., and the harmony of m. 15 is itself still more distant tonally than that of m. ll. Again, there is doubling in enhancement ofsonority, but more significant is the inflation of texture-space cfi Section I! which is the consequenceof pitch rise; tracing the progressively increased distance between upperand lower components revealsan inflation of the spatial field

i

from to

bv

The abrupt dellation of space m. 19! is analogous indrastic, summary effect to that of m. 9, although of course within smaller dimensions.The complementarity of linear descent and diminuendoare strikingly persuasive. Measures 9-20are, in summary, aset oflower-level textural progressions and recessions moving,at the level of the section, essentially to and from the syncopated attack at the end of m. 18. As a whole, this section represents a significant contrast to the preceding tumultuous, contrapuntally active conditions; thus,

at the

hierarchic level

of the introduction as

Section II is an area of relative sparsity and inactivity of texture.

a whole,

texture 277

Section III serves, inone of its functions, as areturn of thematic materials of Section I, but it is condensedfrom eight to four measures andnow is essentially an expression ofdominant rather than tonic harmony in keeping with its position in the broad functional tonal-harmonic curve of the introduction as a whole. The anacrustic portion by which it is prepared mm. 2124! contains, as dosurrounding sections,references tothe movements principal thematic material. Somewhat like Section II, whose materials contrast, it is built texturally of a compromised monophonic statement with an accruing chordal background and, like recurrentpatterns in the introduction, a large spatial inflation by which intensity is mounted and in relation to which dynamic and rhythmically accelerative progressionsare again complementary. The main body of SectionIII mm. 25-28!, to which the preceding four bars are anacrustic, alsorepresents intensespatial expansion, as in Section I, but without the deviating subsidences enroute hence, the condensation referredto above!. The technique of inflation of the texture-space by persistent, determined rise of the upper stratum against a fixed pedal Ex. 2-29d! is thus a recurrenttechniquein; Section III it is fundamental to the expressivecharacter of both anacrustic and subsequentportions. EX.2-29d.

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The development of progressively inflated and expanded texture and sonority in the sustainedchordal background against which the ultimately rising and accelerating line of the strings movesis representedin Ex. 2-29e. Exo 2'29el

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278 texture

Here againis asubsidiary progressionseen tohave important expressive effect within lower-level functional shaping at the level of the phrase. Once again the complementarity of other elements mustbe noted; and once again there is the dramatic device of stark contrast of pitch level and range, dynamics, texture, sonority, etc., between Section III and the following. But note the compensatory continuing stasis oftonal-harmonic condition-static in its fixed content, but intensifying in the prolongation of dissonantinstability.! With the repeated severeand abrupt change at m. 29 a fourth section emerges, likeSection II one of relative tranquillity within all e1ement-structures except that of tonal function.Here, for the first time in the introduction, there is significant qualitative as opposedto quantitative and spatial! tex-

tural development. If this is paradoxical,it is not uncommon:at the point of quantitative decline with complementary reductionof dynamics,deceleration of rhythmic motion, settling of pitch-lines-horn, bassoon, clarinet motive, and final cello line, etc.!, Brahms interposesthe activating stimulus of interactions among components-the only point in the introduction at which significant imitations arise in a brief, abortive animation of generally declining textural states. Attention is directed to the individual components ofSection IV. Again there iscontradirectional relation, a characteristic, potent feature of Brahms counterpoint, between the descendinglower motive derived from the inner voice of the symphonys opening bars! and the ascending ana' compound upper motive a mirror relation freely derived from Section II ?!. Texture is again comparable to that of the first section and subsequent manifestationsof the same, unifying principle: that of afixed element counterpointea' against two compo-

nents forcibly driven in opposing directions: --""__""__"_:The fixed component IS the lower stratum of compound lme: texture thus is remarkably analogous to the very different opening of the movement.!" It is suggested that further attentionbe givento comparisonbetween Sections I and III. The main bodyof SectionIII the recapitulative mm.25-28! isagain characterized by determinedly opposed upper voices,severely contradirectional and contrarhythmicin relation, againstthe fixedpedal beneath. Here, however,the originalsecond voice fails to make ascent subsequent to its initial descent.It levels off; hence,the voicesdo not intersect as before, because of theformal condensation as wellas thehigher pitchof theopening m.25, violins andflutes! andconsequent restrictions of range.In compensation for this restriction, imposed as well onthe originalupper voice,the illusionof consistent rise isattained byoctave transference by doubling violinsat dwérent times. "Reference hasbeen madeto the compound linearstructure ofthe primary motive of SectionIV see the oboepart, mm. 29-32!. Here is a fascinating studyin texture and textural progressionat a microcosmic level. For this line is, in its microcosmic projection,a manifestation ofthe shapingof texture-space like that of the monumental SectionI and other segments of the introduction asdescribed inthese pages. Within that line, and at its lower impliedlevel, a pedal is established bythe recurringdominant pitch; against this another impliedlevel effectsthe lines primary risinginfiection toa high point from which

texture 279

A number of imitations should be traced. The motive taken up by the

oboe, flute, and cello imitatedGJ., at then twice IJ.! at effects an upward linear projection in the oboe portion and subsequently an implied linear ascent of PC, but in descending octave registers, in the succession from 1.

to to

, a striking

use of compensatory progression of higher pitch within the opposing context of recessively lowerregistral placements. The other motive is taken by horn, then bassoon andclarinet, first at

a distance of 3 twice!, thenaatdistance of SJbetween clarinet the entry of m. 32 and the final hint of the motive, the viola descent ofm. 34. In the progression ofimitative interactions noted above a modest one! it is to be noted that the function of stretto, restrained though it is, in mm. 32-33, is significantly supportive of the 'achievement at the same time of a peak in the linear rise beforethe final cadential settling. All of this constitutesa lowlevel shapeof coursesubsidiary in relation to the ultimate textural-dynamicpitch releaseof the introduction broadly heard. Section IV might, in textural structure, be representedas in Ex. 2-2911 Ex. 2-29f. A representation of qualitativeprogression and recession, and the curveof densitynumber changes, in SectionIV. 29 30

3| 32

33 34

35 36

37

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22

!

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| Maximal interlinear independence. maximal density

*Symbols notin vertical ordering ofvoices. '|'One unisondoubling: a differentiation of sonority and coloration, not of densitynumber. 1In m. 34, twounison doublings. melodic motion and with it, the spatial relationbetween thetwo implied strata! recedes. 'I'his issymptomatic oftheextent towhich analysis can goin considerationofa singleelement or characteristic of structureat variouslevels ofprogression, as well asthe remarkableparallels thatcan pertainamong suchlevels.

280 texture

There is increased intensity within this subsidiary shapein densiggin space, and inqualitative interlinear interactions; itis illuminating to see this progression-recession inthese foreground details and then to stand back to see and hearl! it in its broader context as a releasearea yet anticipatory of the following Allegro, with its vigorous resumption of energy. In Section IV, as throughout the introduction, a basic issueis apparent: the achievement of textural diversity by the rhythmic and especially directional differentiation of lines, an aspect of which is the movement-against-stasis relationof pedal to moving voices. In relation to this fundamental technique, theimitations of Section IV are of ancillary importance, nevertheless significantlyfunctional within the foreground level of that section itself Finally, we should recall in broad summary the significance of all textural parameters in functional contribution in the expressive effect and character of the introduction as a whole, and in the often strikingly analogous shapes of its parts at various levels.Texture functions in complementary relation to the structuresof other elements, in the end achieving the release of spentintensity in a number of aspects,texture included, a releasewhich is compensatory to the persistently static finally dissonant! tonal condition resolved only in the main body of the movement.

Dallapiccola, Goethe-Lieder "|n tausend Formen"!

for soprano and three clarinots No. 1 .

Dallapiccolas Goethe-Lieder, like many of his works, are without exception of special interest for the study of textural conditions and processes. Primary attention is given in the present analysisto the first of these songs Ex. 2-30a!. The works twelve-tone setwill be found within a brief supplementary referenceat the end of this chapter.° Brief observation suggests thatthe song is divisible by cadential punctuation andby disparityqf technique into three sections: thefirst, from m. l into m. 10, is divisible into two subsections, with the second after the m. 7 cadence! presentingthe voicesof mm. 1-4 in mirror variations and with all four forms of the row presented in a context in which imitation is free and occasional ; the secondsection, m. IO to the upbeat of m. 14, is altogether based on the three-note cellular unit formed by order numbers 1-2-3 and 6-7-8; see the twelve-tone set!, so that there are constant imitative interactions on varied motive

forms at close distances; the third

section, in which

the disposition of the row is entirely linear m. 13, following the cadence, to The variation in mm. 7-10 isessentially textural, with thevoices restating material in multiple counterpoint aswell asin mirror relations.

f6XfUf6

281

Ex. 2-30a. Dallapiccola, Goethe-Lieder No. 1,"In tausend Formen"!.

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texture 283

the end! features acontinual concurrence of inversionally related set-forms Bb clarinet with bass clarinet, and voice with Eb clarinet! producing two rhythmicalbf free mirror canonsthe relations are contrarhythmic-contradirectional-homointervallic! with a considerable number of imitative relations, those onceremoved° close enough to have significant impact. There are at various levels important and expressiveprogressions in the textural structure and these will be summarized. Already clear in the above prefatorycomments isthe function of texture in delineating the songs three major divisions; naturally other element-changescontribute to this delineation note rhythmic or dynamic shapes,for example!, but the contrasting implications of texture classes andtechniques areof great importance in this respect. It is of course generally true in a serial work that where the row is deployed in horizontal distribution, imitation, if the texture is of more than one voice, will result in some way granted that it can be elusive if rhythmic nonconformity is extreme and persistent!; in this sense the first three measures where the prime form of the set is distributed in vertical and diagonal linesthroughout the three-voice texture!contrast with, for example, mm. l4~l5 where each voice presents a given set-form in linear ordering-I'°, P°, and PS, then I5 in the Eb clarinet!. T/zefunctiom qftextural change in cadential ddinition are particularly evident in this piece, notably in the final cadence. It is significant that conditions of severe dissonancetend to be maintained at points of cadential resolution; release is critically a consequence of element-changes other than that of harmonic content. To some extent cadential expression is rhythmic e.g., longer valuesat evententative cadenceslike m. 4, clarinet parts; at m. 7; and at the more decisivecadences ofmm. 9-10 and the end!. Dynamic change plays a part too: except for the punctuation at m. 13, very much suppressed in the interest of preparation for the climactic section following, cadence is characterized by diminuendo-particularly deliberate in the final bars, where there isrecessive colorationdenoted bysuch instructionsas psubito, pit!p, ppp, a boccasemic/ziu.sa,"° and perdendosifl And melodic descent is also palpably functional in cadential expression. If three voices ormore! participatein imitation on asingle motive,the earliestfollower isin direct or contiguous!imitation while the next follower, in direct imitation with the secondentering voice,is at the sametime in imitation once removed with respect tothe original statement.Other voicesfollow in imitation twice, three timesremoved, etc.,with respect tothe originalvoice andin closerdiagonal relations to precedingvoices subsequent in entry to the original.This is a significantfactor in imitative textures,resulting in triangular and multiangular relations, because, especially where varied subjectforms appearin a single exposition, thelines ofsignificant relationassume very complex patternsin perceptible not just theoretical!interactions whenthe distanceof imitation is not extreme. Whenthe imitation onceremoved iscloser inshape andintervallic contentto theoriginal subjectthan isthe direct imitation, triangular relationsare particularlypersuasive. Note the Dallapiccolaafter m. 13, where imitationsonce removedare homodirectional,as comparedwith more proximate contradirectionalimitations.! 5°With half-closed mouth. 5Vanishing.

284 texture Related to the factor of melodic descent in cadential formulation

is that

of the contraction of texture-space aasdevice in cadence. The reduction of texture-space tothe compassof a minor 6th in the final cadence is an indication of this: in fact, the final chord achieves theworks smallest space. Comparable recessionsin which the contraction of texture-spaceis an important factor can be seen in the synopsisin Ex. 2-30b; the superimposed pointing bracket is intendedto draw attention to composite recessive contraction of the third,

fourth, and

fifth units

at a broad hierarchic

level.

Ex. 2-30b. Texture-space contraction in the Dallapiccola phrases and ata broaderlevel.

=f` WG.

I 'Zi f- '92

FI!

'-a t /"/ L.

In other respects too texture functions in the expression of cadence. While the cadences tendto maintain a density of 4 except for the initial progression fairlyconstant in the work in general!, they aremarked by increased interdependencecomponent of lines-againwith the exception of m. 13, where cadential stability is not appropriate in view of the approaching, climactic progression. The tentative cadence of m. 4 is marked by the accord of all three voicesand is prepared by qualitatively recessiverhythmic association between uppervoices inmm. 2-3. That of mm. 6-7 is prepared by rhythmic

identification between upper clarinetsat m. 5 andamong allfour voicesat the start of m. 6. The cadence intom. 10 is ofcourseanalogous tothat of m. 4, since it concludes themirror variation of the opening bars,but it is enhanced here by the interdependent bass clarinet and singing voice, each of which pauses ona note of long duration, the voices coincidentin attack. And the

interlinear interdependence of thefinal cadence is themost marked of all; and its homorhythmic, homointervallic, homodirectional except for mirrored outer voices-a final surviving factor of textural vitality! conformity is again gradually prepared by qualitative recession. Interlinearinterdependence as The texture-spaceconfiguration ofthc introductoryphrase, mm.I-4, and that of its mirror, mm. 7-10, arcof specialinterest; thespatial shape of thefirst phraseis onein which the initial compass isthat of a major 7th, expandingto the same intervalin its compound form, andcontracting tothe originalinterval transposed but in a proximatearea.

texture 285

expressed byDallapiccola is often marked by considerable directional and intervallic conformity as well as rhythmic. In a context of unusual rhythmic diversity, interdependenceswithin the texture units of 2, 3, 4! have particularly strong effect and are obvious factors of shaping control. Brief attention should be given to the factor of time interval and to the considerationof possiblefunctions of changes inthe distancesof imitation. In m. IOHI, the first section in which imitation is prevalent, indeed constant, the trichordal cellular subset mentionedearlier appearsin numerous transformations. If we take its original occurrence as order numbers 1-2-3 as definitive and take as referential the IC succession 1-2: a rising minor 2nd, then a descending major 2nd!, we can note the following specific variations, with rhythmic formulations generally involving two sixteenth-notes with the third tied into a longer note or, later, eighth-notes throughout or triplet sixteenth-notes with the third tied into a longer note: mirror e.g., Eb clarinet, m. ll!, retrograde e.g., Bb clarinet, m. 12!, retrograde inversion e.g., bassclarinet, m. 12!, and occurrences in which compound and complementary forms of the intervals are used in expansion of the intervallic content e.g., bass clarinet in m. ll, or voice setting of Zauberschleiern, also a distinct augmentation!. The grouping is easily identified and the interactions among voices unmistakable, although of coursethose imitations in which two like forms or closely similar rhythmic and intervallic formulations follow directly bring the effectof imitative interaction suddenlyinto clearestfocus: e.g.,Bb clarinet and Eb clarinet, beginning of m. 13; or the samevoices in mm. ll-12. The effect isone ofdense entanglementof texture in which four voices,all of them constant in activity, project a singlelimited idea obsessively inreferences that relate diagonally to other voices and,if one considers imitationsonce removed see dottedlines in Ex. 2-30c!, a setof interlinear relations of great complexity. It is in this senseperhaps morethan any other that this section is a pronounced contrast to the others.

In mm. 10-13 the distance of imitation does not undergo significant change. There is imitation at 4 3 between

between upper clarinets, m. 10! and at

bass and Eb clarinets, across thebar-line separating mm. ll

and 12!, but otherwisedirect imitation.: are consistentlyat a distance of2 -in

one sense a preparation for the tightest stretto, atm. 13, at 105 . This does not take into account the augmented forms of the motive in the singing voice, which while involved in the web of interactions seem at the same time some-

what removed

in relation to the faster-moving clarinets,! Thus, the pro-

53The lineof text set here,Du mags!mit Zauberschleiern dick bedekken You mayclothe yourself inmagic veils!,seems a very suggestive basis forthe musicstexture.

texture 287

and lab Bb and bass clarinets!. In the consideration timeof intervals in this sectionthere seemto be three important conclusions tobe drawn regarding the functional effect of imitative distance and its changes: ! there is constant modification of the time interval-any regularity of spacing seems insistently avoided in favor of constant flux ci the preceding section!; ! the rhythms of the component lines are so contrived that absolute concurrence time intervalof 0! is achieved only incadentialjormulation, as noted earlier; and ! the abruptly and maximally tight stretto withwhich thesection begins lower

clarinets lgb! at contributes complementarily climactic toand energy, coincident is with _/brteand crescendo, the performance direction appassionato, and other factors.

It is noteworthy in the texture as a whole that density changes are slight; in general the four component voices arerather continually engaged apart from the progressiveaccrual at the beginning and the repetition of this pattern in variation a few bars later!. A charting of the texture throughout, however, revealsa seriesof qualitative successions in the direction of maximal diversity and subsequently in the direction of resolution; it reveals maximal textural complexity and interlinear interaction in the songs climactic measures 3-15! and the section approachingthese. With reference to the presentation given in Ex. 2-30d, we can make the following summary observations:at point a!, progressionto and through

Ex. 2-30d. A schematic representation ofinterlinear independence and interdependence throughout theDallapiccola song. l2

34

56

78

9 10

ll°l3 14-17

18 I9-20

llll°l3.l_l.l!l_l.ll|llllL°""Lll"l'l4 liz 232 llllll Ill'-" l!ll'l`l 11

92, I

1 lll2l

/92 b

/92 2

11- --2

1 1-

/d

11

g/92 c

*Symbols ofinterlinear association-e.g.,!, denote rhythmic interdependence and often directionaland intervallicconformity aswell.

The time interval8ofAoccursdirect in contradirectional imitation between bass clarinet andsinging voiceacross thebar-line betweenmm. 13and 14.

288 texture l2

briefly manifest diversity Land density 3!, recession through diversity T 1

to cadential accord; at b!, with density 4 constant, fluctuation between three and two real components to the cadential accord, 4; at c!, density progression from3 to 4, qualitative progression tomaximal diversity-activity in m. 8, then a recession ; at d!, quick but graduated! progression todensity l

4 anddiversity T,maximal diversity continuing into the cadential bar with

T T

punctuation a function of rests-i.e., brief opening oftexture by the releaseof the singing voice and bass clarinet;at e!, again, quickly progressive accrual to density 4 and total diversity as above,then a recession tothree real components, followedby highest degree ofaccord at the cadence. Again, analysisat variousstructural levelsis indicated-e.g., at onelevel the considerationof the shape within a phraselike that of mm. 7-10, and at a broader level the consideration ofthe function of this shape andits quantitative and qualitative states in the course of the directed line of intensity reaching itsculmination in subsequent sections. In thebroadest view, the texture is seen to attain through the various stagesnoted a condition of maximal complexity, vitality, and density in the internal, climactic portions: the 1

diversity T,tentatively developed in m. 8, is increasingly persistent in mm. T

T 11-16.

The final cadence isachieved primariiytexturalbf despitethe actions of complementary element-structures; it is the point of resolution of highly complex and active texturesprogressively developed,and it is characterized by almost total interdependence of components, an interdependence compromised onlyby the contradirectional relation of outer voices in a persistent but subordinate manifestation of final textural vitality.

Seria//sm and texture; texture as a product of

chance operations

What might be referred to as classical serialism has of course important implications for the content and character of musical texture: the usual avoidance ofoctave doublings,with consequences for maximal heterogeneity of simultaneous PC content, and the frequency of imitative polyphony

texture 289

induced by twelve-tone procedures are two such implications. A further implication is in the predetermined interlinear heterogeneity of PC content resulting from combinatorial applications of serialism. Densities andother aspectsof texture registral placement, therefore its spatial configuration! have been prescribed in serialization as extensions of the serial prescription of PC and other elements!. Of course, therecan be no generalstatement about the methodof serialization of densitiesand other nonpitch elements; one can only cite this or that method adopted by a particular composerin a given work. The PC set or derived interval set! can be made to yield a numerical set, of course, by any of a number of means: the set of numbers representing the intervallic distances between the initial pitch of the set and each remaining pitch; the set of numbers derived from such intervallic distances betweeneach pitch of the set and the next; the set of numbers denoting PCs as ordered in the set usually 0-l l, 0 = C and ll = B! may be used as the set, or basis of derived sets, of densities and durations; or the composermay establish an independent density or duration! set, or an arbitrary or designed setfor any element ofhis work. Texture in serialism means quantitative aspects susceptibleto purely numerical analog.! One exampleof specificmethod is provided by Kiienek in discussion of his Sestina,a work for soprano, violin, guitar, flute, clarinet, trumpet, piano, and percussion.Ki'°eneks derivation of densities concerns both the original row and the rotational principle of the poetic form, sestina." No presentation of combinatorialityis possiblehere, butthe readeris referredto the following sources, especially toBabbitt, whois ofpreeminent importance in the development of combinatorialprinciples andapplications, andto other sources towhich thesemay lead: George Perle,Serial Composition and Atonaligy, 3rd ed. Berkeley: Universityof California Press, 1972!, pp. 97-98; Milton Babbitt, Some Aspectsof Twelve-ToneComposition, in The Score, 12 june 1955!, pp.53-61; Babbitt, Set Structure as a Compositional Determinant, in _journal qfMusic Theory, V,1 961!, pp. 72-94.Schoenbergs often quoted statement oninversion oftherow transposeda 5th below, combinedwith the original form so as toavoid PCduplication within each pairof correspondinghexachords, occurs on p. 116 of Styleand IdeaNew York: Philosophical Library,1950!. Incontrast tocombinatoriality of this kind is proceduralfixing or invariance of segmental content among setassociations; see, for example, Babbitt,Twelve-tone Invariants as CompositionalDeterminants, in The MusicalQuarterly, XLVI,2 960!, pp. 246-59. 5°To quote Ki'°enek [Extentsand Limitsof SerialTechniques, in The Musical Quarterly, XLVI, 2 960!, p. 223] :The Sestinais oneof thepoetic formsdeveloped bythe Provencal poets ofthe twelfth century, itsoriginal specimenbeing ascribedto Arnaut Daniel. It may well be called aserial formof poetry, and its essential formative principle isrotation. The poem consists of six stanzas of sixblank verses each. Ithinges uponsix keywords which appearat the endings ofthe individual lines. If in the iirst stanzathe orderof these words is1 23 4 5 6, the wordswill appearin the second stanza in the order 6 l 5 2 43. The principle of rotation which is applied here consistsin switching the positionof every two keywords equidistant from the center of the series,proceeding fromthe end toward the middle. Accordingto the same principle,the positionsof the keywords inthe subsequent stanzas are 3 6 412 5; 5 3 2 6 l 4; 4 513 6 2; 2 46 5 31. The processends here,since the next rotationwould producethe originalseries. Thesix stanzas are followedby a Tornado of

290 texture

Ki'eneks consideration of density as serially determined is that of the number of components vertically coincident; in the following extract from his analysishe citesdensity-compression ourword! as another parameter, the location of the tones within the gamut of six octaves designatedas the ambitus of the work, subject to a different governing mode of operation. Ki"eneks twelve-toneset isgiven in Ex. 2-31, and below is further quotation of his discussion ofSestina. Ki"eneksgroup A is the sets first hexachord.! Ex. 2-31_ Twelve-toneset forK¥enek's Sestina, with deriveddensity series. /-Group A-l-92

Iz'I:#"_l, Derived densityseries: 6 3 5

4l

2

Reprinted by permission Bdrenreiterof Verlag. three linesin which the keywords,one ofeach pairin the middle andthe otherat theend of the line,appear inthe order2, 5,4, 3,6, l. Reprinted bypermission ofG. Schirmer,Inc.! Ki'°eneks self-composed text follows in quotation of the first two stanzas only,the translation asgiven in the cited source, butwith the rotation of keywords indicated.None of Sestinas music canbe quotedhere; Ki'°eneksarticle quotesfrom it only very briefly p. 227!. Vergangen Klang und Klage, sanjler Strom. 1 Die Schwingung der Sekunde wird zumMass. 2 Was inGeschichte lebt, wars nurein Zufall? 3 VeU'all, Verhall, zerronnene Gestalt? 4Die Stunde zeitigt Wandel, wendet Zeit. 5 Das Vorgeschrittne ordnet sich der Zahl. 6 In Schritten vorgeordnet durch die Zahl gestaltet sich Gedanlce, doch zum Strom wird strenge Teilung, uhr-genaue Zeit. 5 Ist esvermessen, solches Mass von Mass 2 dem Leben aufzuzwingen, der Gestalt? 4 Der Zwangzerrinnt, erzeugt den neuen Zufall. 3

6 1

[Bygone are sound and mourning, tender stream. l Vibration of the second becomes measure. the 2 What lives in history,was itonly chance ?3 Decline, fading sound, vanished shape? 4 The hour causes change, turns the time. 5 What looks ahead subordinates itself tonumber. 6 In stages preordained number by 5 thought takes shape, but a stream is theresult of!strict division, of clocklike, precise time. Is it presumingforce to such an extent J measure on IW,on shape? Force vanishes, brings forth new chance.]

l 5 2 4 3

Reprinted permission by Biirenreiter-Verlag. of The composers English translation, included in the articlecited above, reprinted by permission of G.Schirmer, Inc.

texture 291

D¢nSitY is the Nxt Pa1`am¢tC1` b¢todetermined serially. Thereare six degrees of density whose successiondetermined is by the positionof thepitehes in groupA. Againthe lowestC! is called 1,the highestGi! 6_ Consequently the initial series ofdensities is 6 35 4 1 2.In density l the two tone-groups A andB run off simultaneously in a sort oftwo-part setting in whichthe dural1l0I'l ofthelI1dlVlCll.lal 1165is dCtCl°II1iIlCbyl [bg mgghanism dgsgribgd____ In

density 2 the iirst and secondtime segments of groupA run eoneurrently with the first segment of groupB. In density 3 two Sggments of eachgroup are developed simultaneously, and so forth,until in density 6 six segments of each group, i.e. twelve all together, run off at the sametime_

Another parameter is thelocation ofthe toneswithin the gamut gfsix octaves designated as theambitus ofthe wc,r|¢_ The serialstatement adopted for thisarea reads that thetones ofeach segrnent should runthrough asmany octaves as there aretones. Thedirection ofthe motionis determinedby the d.iI`CCtiOI`lthC of COI°I'CSpOl'ld.lI`lg lfltCI'V2.l ill the originals¢|°ie5_ Since many Sgg.. ments contain less sic! than six tones, they eoyer less than six oetaves and

therefore could extend overvarious bands of theeornplete ambitus This too, is regulatedby specialserial statements. Needless to say thatall theseserial organisms are subject torotation accordingto the sestina pattgrn,which is th¢ 5UPf¢m¢ law S°V¢fni8 CVCTY m0V¢ Of everyvariable within the whole composition."

In thereference from which theabove istaken, Kifenelr explains further devices of density serialization in hisSeah; Vernzsssgne, a setof sixpiano pieees_ A later serial work of mine is a set gf six piano pieces, ealledSgghj Vermessene. This German titleis a play on Wgrdg, sincg vermessm German in means completelymeasured aswell aspresuming, a pun that eannot bg reproduced in English. While the time mechanism is similar to that gf the

Sestina, the construction differs from it in that for thefirst threepieees system a of five layers isset upin which the first has density 1 i_e_,one noteat a time!, thenext hastwo tonestogether, the third thrgg the fourthfour, andthe fifth six t0I'lCS. The tiITlC II`lC3Sl1I'ClT1C!'ltS fOI` thgVarigus layers arg 3 rgsult Qf summing up the interval magnitudes involved in the eonseeutiye toneeom-

binations. Forexample, thetone series of thiseomposition being; .

the first combination of tones in density 2

_

is; _

The

numerical values derived fromthis progression are 3 a minor third frgm G to and l. half-step fI`OI'fl E to F!. COnSCqu¢ntly first the timgSggmgnt of the firstlayer hasthree units,the firstof the ggggnd has fgur _|_ ]!_As the density ofthelayers increases, the numberof simultaneously sounding intervals and thus the numericalvalues oftheir sum; heeome higher_ Therefore the .$`8g77l¬l'ltSlonger become ,m8anS #18 Chgfdy,t0ne-¢-lujtgfs, of incfeasfng 57Ibid., pp. 225-26. Reprinted by permission of ;_Schirrner [ne_!

2.92 texture thickness are spaced farther apart, while the singletones Uthe first layer _follow each other more rapidb. Computations of this kind form the basis of the whole composition.

In illustration of Ki"eneks basic methodof serialization of layers and densities inthe first three of these pieces,an extract is givenfrom the first Ex. 2-32! _Although the quotation is limited, the density progressionis wellunderway, as is the progression ofincreasing durations.Identifications of these are included in the example.

Ex. 2-32. Kienek, Sechs Vermessene, No. 1.

SD .

9.60 W /92 /-92 ||

J

_______., 2 °""

I 'ff 1:fP |Q +l

LI

|I

| |2

"' f

V| Density 4

t

+7 2 | NPC!!

3:22223 ;___I" , Density 5

|1g +5

IL

|

_ fi

.....-.- ----a T

*Notes withwavy stemsand numbersindicate subdivisions of the eighth-note. Numbers withinslurs showdivisions intoequal parts,numbers withinbrackets thegroupings of the unitsof subdivision. Reprinted permission by Barenreiterof Verlag.

In another operation, procedurally opposite yet often paradoxically analogous asto effect," and commonin certain contemporary styles,textures

are theresult ofchance bywhich theperformers choices are madewithin often considerable latitudes.

53Spaced farther apart in attacks, i.e.,in relativelylonger durations,not in separation by rests. 5°Ibid., p.229; emphasis added. Reprintedby permissionof G. Schirmer, Inc.! °°While in the Kienek piano piecea contextually functional, palpablysignificant successionserially is predeterminedthe progression of densities having increasing durations!, Kieneks methodis unusuallysimple inconsequence, with accrual ofdensity andtemporal! values ofdirectly increasingconsecutive orders.

texture 293 Cono/ud/ng notes

The foregoinganalyses anddiscussions illustratethe modesof textural process as a structure-delineating element in music, and they are a demonstration of the significance and vitality of this element of structure. Although texture in music is relatively little studied albeit much referred to!, its effect is almost always an important factor in structure and expressive effectin some degree and at some level. Even a slight changee.g., theabrupt and momentary engagementof the bass oran inner voice in interaction with another melodic entity-can give significant vitality, however abortive, to the texture at the foreground level. One of the evidences of the significance of textural structure is its immediacy of effect. If this is particularly apparent in quantitative progression e.g., the graduated accrual of textural components as in so many musical works!, it is also true, although subtler in effect, in qualitative changes-for example, the overt passage fromhomorhythmic to contrarhythmic texture, the fleetingvitalizations of texture by momentary diversification, or the rhythmic and other devices of activation of textures of persistent simplicity of character and content. The textural class of a musical instance is almostcertainly one of the first attributes of which we, in listening, become aware; and textural progressionor recession or in static situationsthe obstinacy of unchanging texture! can be especially compelling and direct in expression ofeffective, affectivemusical process. A specific indication of the significance of musical texture is seen in the apparent fact that complexity or simplicity of texture is important in distinction betweenwhat is commonly referred to as art music as opposed to folk, popular, or commercial music. Clearly, one of the primary factors bywhich art music is,at its best, relativelychallenging andinteresting is its broad range of textural attitudes and attributes, and modes oftextural complexity. The textural uneventfulness which, in part, characterizes so many commercialand popular genres sets them clearly apart from the elaborate, provocative, sometimes monumentalprocessive texturesof Western art music. So evident is this that to cite examples seems pointless. It cannot be suggested thatthis factor is altogetherdecisive andconclusive insuch distinctions art music of small forms, and unpretentious intent, is sometimes relatively, even extremely, uncomplicated in texture!, but it may well be that in those instancesin which art works project little vitality of texture they most approach in that respect! the realmsof popular forms. Where the range of textural possibilitiesis, in a great art tradition-say, that of liturgical chant-severely limited, it must be said that ! the limitation is that, a boundary which circumscribes theexpressive potentialof the

294 texture

literature by restricting the parameters within which processivedevelopment can take place; and ! the subtlety of structural delineation, and range of expressive projection,affecting and characterizing otherelement-successions is signQieantlycompensatory-and probably of magniyiedvalue ana'importance-in relation tothe imrnobilityof textureseen asan independent factor. Both of these observations are valid with respect to liturgical chant, the most obvious instance of highly expressivemusic in which texture is by definition simple. That great composers insubsequent traditionshave sorarely devoted themselves toworks of monophonic texture is evidenceof the inherent limitations of monophony and the critical importance of textural vitality in highly developed formsof art music. As this chapter concludes, it should be said again that literatures involved with the study of musicalelements otherthan tonality and harmony especially thoseindependent of specificity of PC content! leave much yet to be done.Texture and color timbre, articulation, dynamic intensity, registral coloration, etc.!° have been much too little explored in their structural implications. We need especiallyto understand process better,exploring all musical eventsas to their expressionsof such fundamental structural functions as progression and recession; this has of course, with respect to the various dimensionsof texture, been theconcern ofthis chapter, and if examples to which we have turned have, with accompanying comment,served to point out some approachesto this kind of understanding, they will have served animportant purpose. Rhythm, especially meter, manifest in the functional, proportional interrelations amongaccentually articulated groups of events, isthe concern of the chapter which follows.

"Unfortunately, referencesto functional and expressiveevents and processes of coloration havehad, andwill have,only tentativeexploration inthis book. But of all processive functions of element-structures, those ofcoloration areprobably mostaccessible to analytical identification,even thoughrelative intensityvalues oftimbral differencesmust be attimes thebasis forhypothesis of plausible sometimes seemingly self-evident! but notyet empirically demonstrated function andeffect.

wnwe AH NOTES Study of the varyingspecific intervallic content as to relationsbetween voices l-2, 2-3,and 3-4 numberedfrom top down! in the recurrentchordal factorreveals thatthe M7 ll semitones! is alwayspresent exceptat m. ll , whose firstchord nevertheless contains overlappingM7s with the m3 the intervalof overlapping.Relative dissonance intensities in the chords are complementedby dynamic stress m. 6! or counteracted bydynamic underemphasism. 3! in what can beinterpreted assignificant functional relationsin articulation of structure. Generally highdissonance value in the chords is underscorednot only by prevalenceof the M7, but by its frequent occurrence betweenvoices l-2 and 3-4 it so occurs in half of the chords!; the interval betweenvoices 2-3P4, m6,m3! is generally relativelyconsonant, although the crucialforte chord of m. 6 projects the M7 between theseinner voicesas well, standing out as the only exceptionto a prevalent normof intervallic content and distribution. The chord of m. 6 is the most dissonant,with three 7ths M7s! superimposed; ithas the formal function of launching the songs second part at midpoint. Measure 6 is unique too as the only point of two consecutive chords note thedissonance progression between them!.Least dissonantchords areprobably thoseof m.2 piano!and m. ll pianissimo!at relativeextremities inthe form.The content ofthe chordsis ofcoursegoverned inpart bythe setsthree equal trichords3 r_° l--1

7 if

0-he 3

3 , and related

_,_ ____i_i

IC 1

31

31

of order numbers 6-7-8or the 9-10-ll.

33 configuration of order numbers

3l Fluctuations inthe spaceor density-compression! of chordalevents arealso of interest inevaluation offunction orfunctional possibilities. Again, theattack at m. 6, _/brte, stands out asthe chord of broadest space and necessarily least density-compression! as wellas highest dissonance value; on the other hand,the pianissimo chord of m. l l has by far the mostrestricted spaceand necessarily greatest density-compression!. These two events,of 32-semitoneand 19-semitonecompasses, respectively, are distinctly anomalous; otherwise spatial fluctuationamong chordsis limited to the ambitus of 5-26-27-8 sernitones. The relation of PC content in the chordsto that of concurrentvocal phrases is alsoof interest:the degreeto whichthe chordsfunction responsivelyi.e., punctuatively!. There is considerable intercomponent interaction of 'thiskind at m. 3, less at m. 5, most ofall at m. 1l, wherethe chordsdo seem responsive-punctuative especially in a reduced tempoin which such relationsare verydistinct! in an aspectof the cadential process. bl tis interestingto notein theforegoing thatwhere fluctuationis relativelystatic rather than progressivee. g., mm. 19-22!complementary effects of ostinatoare to be seen. Or, in mm. 21-22,the compensatory effect ofmotive truncation-analogousto time interval contractionin effect,is adevice ofprogressive rise in the intensity curvewhile

the time interval change is Huctuant between 2, l,and once only! SJ! The . device

of progression illustrated here,which hasimportant implicationsfor metricstructure, is highlycharacteristic ofBartok asother passages in the fifth and other quartetswill show!, asof many other composers of imitative polyphony in all times. °It is noteworthy in the abovedescribed circumstances that there is a series ofveg' localbf progressive accumulations of sonority inthe strings,which arechiefly responsible for theostinato factor.Thus, complementary to localprogressions of pitch and hence of texture-space! within eachfragmentary unit,is thegrowth in sonority ascello and

296

f6XfUf6 viola, thenviolins, enter.The texturecannot besaid tobe materiallyaH`ected within these low-level progressions, but sonorous and spatial growthis in the immediatesurface afactor in the vitalizationof stronglyrestricted tonal-harmonic content. In the ostinato texturein which the strings prevail the essential PCfactors areA and C occasionally E,and later D is added!; of these Ahas thegreater tonalimportance, and thehorns andtuba, echoingthe ostinato,concentrate altogether on A. But last bar of the quotation!as thetwo texturalcomplexes become concurrently veryactive, the A-C distribution is altered in a direction of instability see thelocal, tentative, contradirectional relationbetween double-bass and upperstrings-a slight hint of the technique ofdisplacementessentially of staticPC factorswhich can contribute to the activation oftexture!. dNo impression is intendedthat Stravinskystextures are generally qualitativelysimple, activated byrhythmic, articulativeand otherdevices, although this isoften true.An instance ofvery complextexture highly diversified qualitativelyand quantitatively is found,for example,in the remarkable canons of Part III of the Canticum, where the delineation of formal units is very significantly oneof differentiation in extent of textural complexity.A caseof texture of highly individuated, intenselyinteractive, voices isthe canonicweb beginning Ego autem humiliatus. ..! at m. 219 in Part III. The factthat eachline isa linearapplication ofthe twelve-toneset indicates of course a necessaryconsequence of interlinear relations;if one considers these relations as extending to,for example,retrograde imitations, and imitationsonce ortwice removed like that of contrabasstrombone and third viola!, the entanglementof diagonal, multiangular, andintersecting linesof imitation becomes an intricate maze.Several

kinds of canon are in process, including mirror canon l f !, retrograde canon Z !, strict canonf f !, as well as imitations of other kinds homorhythmic-contraintervallic-first and third violas;cf. mm.224-H1 and 22 7H1; homodirectional-homointervalliccontrarhythmic-second violaand contrabass trombone; etc.!.Rhythmic differentiation is controlled determinedlyin interlinear individuation aswhen theviola voices enter atvarying metricpositions inotherwise strictimitation! aswell asby expected, usual nonduplicationof PC content in vertical coincidences. In the passage there is progressive accrual of density-number and complexity,with recessive process into the cadence m.236! achievedby gradualinterlinear rhythmic accord without density reduction. Thereader shouldpursue theanalysis ofthe canons,and of the striking usages of textural contrastin formaldelineation throughout Part III as inthe heavily sonorous butmonophonic statements which begin and end the movement!. The example isone oftruly organictextural shape and content,and intensetextural energy and motivation. Comparable examples of complementary element-successionsexpression in cadential of and other functions will easily befound. The anthology whichincludes thePurcell Gloria seefootnote! containsmany excellentobjects forstudy of textural factorsin functional processes. For example,the factor of increasinglyintense contrapuntal interaction amongepisodes infugal andother imitative forms isitself a fruitful area of investigation;this canbe evidentand unusuallyaccessible in works ofeven minimal textural complexity,as in the Telemannmovement fromSonata inE minor for two

Hutes, whose imitationsincipient are mm. in 5-6 at J.,then J,progressing ex-to

tended interaction at Jfollowing25 m.and at the metrically anomalous 2 J as well as J and4 J!in the Final episodic development following m. 33; a number of element-structures function complementarily inthese progressive operations. fSo commonis developmental, functional texturalprocess in tonal worksthat examples for referenceseem almost gratuitous. Thereader might,however, bereferred tothe first movementof Haydns Symphony No.102 in Bl. One of this movements most striking passages for studyof textureoccurs inmm. 122-85in the development. Ina process of powerful density progression out of asingle voice, there iselaborate, gradual

texture 297 qualitative progression culminating in an intensecanon onone ofthe basicmotives. Complementary processes of otherelement-structures are of greatinterest: tonalflux is themost evidentof these,but analysisof concurrentelement-actions should include review ofchanges inthe motiveitself-e.g., in the sizeand dissonance of its interval of anacrusis-and consideration ofthe functionsof such changes.! In the canonic segment, voice 3 oftenfunctions asa nonconforming, deviant presence, adding density and impact, asserting thecontrast of relative unpredictabilitywithin the canonic regularity, andmaintaining, especially rhythmically, thecomplete motivetruncated in much of the canon. The canon ispunctuated, thenresumes atmm. 168-69; at m. 176voice 3doubles voice2 in qualitative recession at thecadential approach, a recessive event in which sonority is maintained. Considerable canonicenergy spills into the c:V cadence ending this sectionof the development, instriking texturalcontrast to the extremely simplethematic statement which follows.In studying this example, attention shouldbe directedto analysisof varioustextural eventsin theseprogressive and recessive processes, and perhaps toother passages of functionaltextural change sometimes intensely, often abortively,imitative and polyphonic, asat m. 2l0ffi!. gln this finial texturaldiversification thereare severaldegrees intcrlinear of independence andinterdependence: heterorhythmic-heterointervallic-homodirectional relation, g

, occurring, for example, within a spectrum of

textural conditions extending to true contrarhythmic activity mm. 62-63! and resolving in final textural`accord in which modestdiversification evensuggestive, abortive imitation! is maintainedto the antepenultimate bar. Much further comment might be made: the textural progression andrecession partially by which thefirst fourphrases are unified arean instructiveobject offurther study; or, for additional exploration, onemight considerthe meansby which the texture ofmm. 27-33 partially quotedin Ex. 2-16!, still strongly imitative,are contrived toproduce, withinthe imitation,a momentaryhomorhythmic relationbetween the two voices inpreparation forthe variedreturn of Part I. An example of textural structure in this broad sense, sharplydelineating severely contrasted sections of a freely structuredmovement i.e., highly variegated, nonrecapitulative! ofbroad design,is thePrelude toBachs PartitaNo. 2 in C minor. Its first section, slow, is essentially chordal-a largely homorhythmic accompanying texture underlyingan upper voice melodysometimes doubledin 3rds! and occasionally setting off brief imitative responsesin other voices. The second section, moderately slow andante!, isan embellishedupper-voice melody against whichthe bass voicemoves ina relation considerably subservientand far less intricate! but having contrapuntalvalue infrequent contradirectional relation toand in occasional imitative interactionswith the upper voice.The third section isovertly polyphonicconsistently imitative,a kind of invention,the sectionof far greatest contrarhythmic and contraintervalliccomplexity andintcrlinear independence. The lastsection ispresumably fast;tempo thusemerges as the mostapparent complementary element of delineation.With texturaland tempostructures, apalpable tonal structurelinks thethree sections into a broad progression centering ontonic and dominant levels;moreover, thetonal structureis oneof progressive degrees of fluctuation through the Preludesthree sections. There isthus complementarityamong the various relevantelement-structures atthe broadestarchitectonic level,as often at more foregroundlevels; thevarious structures so projectedconverge infunction to serve parallelends of formal delineation and progressiveintensification and subsequent release. JA furtheruseful example in this connection isthe third of SchoenbergsPiano Pieces, Op. 19,in all four phrases of which although minimallyin the second! contractionin

298 texture space isa significantfactor in structure andin cadentialformulation aswell asin the pieces overall shape smallest spaces occur in the third and fourth phrases!.The following showsthe spatial configurations ofthe four phrases withup-down distances given insemitones; note recurrences of 37 and27, ascompared withcadential spaces of 14, 12, ll:

If, I

Tl l.@4.,f ifji* b bv

ll

b*92l1'I/

*In the Bartok Divertimentofor stringorchestra, aninstance ofstriking verticalordering at mm. 58-61of the second movement! is attributableto texture,sonority, and coloration. Theaccompanying fabrichas tight density-compression close in spacing between double-bass and viola; and this thick textural backdrop, givena rumbling activation byrhythmic andarticulative diversity,is astriking foilindeed forthe void, two-octave doublingof the motive, presentedin muted violins. Here, the densitycompression can be significantlydescribed onlywith referenceto the distribution of events within the texture-space.Cf. other instances ofprovocative texture, e.g., mm. 66-68,in the same movement.! The madrigal, Moro, moro, first part, is suggested for furtherstudy ofsimilar questions. Again, formis delineated by linesof text,and themanipulation oftextural components in and out of cadential punctuations should beconsidered. The text inducesat Corre volando, literally runs flying! functional devicesof animationand texturalintensity comparable tothose ofFuggesi fiees,flies away! in Or, che ingioia. Aparticularly striking featureof thetextural structureof Moro,moro, perhaps recurrent inGesualdo, is thetendengr of each phrase to progress to evolve! texturalbr, both quantitatively and qualitatively, with interdependent relationsqualifying these paths of progressive diversification. Atm. 3,for example,voices 3and 5start thefirst process of diversification with homorhythmic but contradirectional relation;voices 2 and 3 are often doubled inrestraining theprogressive tendencies at phraselevel, asat mm. 4-6.! The tendency ofeach phrase to progress in thedirection oftextural diversity nowhere fully contrarhythmic-contraintervallic! isassociated with the settingof theintense opening words in typically arresting chromatic manner.These words I die, I die . _.! are, with heavy chromaticism, set inrestrained andsimple texture:they arethe focus of anguishedfeeling outof which the entirepoem andcomposition! arises. As commonly, the intense chromaticismsuggests atexture of relative simplicity-near homorhythmic, chordalsuccession. This restraint oftexture isfunctional notonly in exposing therich chromatic vocabulary, but in exposingthe crucial words of text too; in each phrase,with this initiating precedent,subsequent diversification of texture follows from this condition in a highly important, and intensely perceptible, aspect ofshape. The important textural issueof compoundline in this movementcannot really be developed inthe space available; butideally thisaspect oftextural diversityshould be pursued independently.Note, for example, theextraordinary contrapuntalinteraction atmm. 40-41,at dramaticallyopposed registers:

l

F

!

texture 29.9 In suchinstances there is truetextural diversitywithin ostensiblymonophonic texture, with interactionsof vital directional, rhythmic, and intervallic opposition inwhich dissonance strongly is felt.Understanding ofthis kindof concealed textural diversity and interactionis of course absolutely vital to performance, ifdissonant andother interactions ofthe sortsketched above are to be broughtout. F or further studyof Brahmsin theseand relatedconnections attention is drawn,for example, toconditions oftexture andtextural process in the shaping ofphrase structure in the openingof the Quintet in G, Op. lll, for strings. Anotherexample of great interestfor thestudy oftexture isthe Adagio of theTrio in Eb, Op.40, for piano, violin, and horn. Somefactors representative of theextraordinary rangeof significant textural processand functionin this movement canbe suggested in hasty summary. The role of textural events indelineation ofform is vitally important throughout. For example,the entry of contrastingthematic materialat m. 19 isthe subjectof startling exposure lgv virtue qf thechange texture, in quite probably themost arrestingof a complex of complementary element-events, and textural change isthroughout an important variational device in thematic developmentand recurrence.! Shaped successions textural of eventsare ofcompelling significance: for example,that of the progression from m. 19,which canbe tracedin variousstages, including that of contrapuntal interplayin which a time interval of imitation undergoescontraction from

12 ,D to 6f; continuation of the process, counteractive to diminuendo but complementary to extreme tonalshifts, achieves at m. 30 a time interval of 3 , with the motive undergoing complementary truncationin a manifestation of intensifying metric function.With the time intervalfluctuating between3 and 4- , the motive is variedin sizeand in placement inrelation to the bar-line:complementary actions are thoseof rhythmic motion e.g.,stringendo and increased activityin the piano!, metric instability, intense coloratione.g., morebrilliant violin and horn registers, and crescendo!, and especiallythe intenserise in the pitch-line; tonality undergoes compensatory stabilization but with rich chromatic embellishment maintained as in the Neapolitanharmony ofm. 35.There isgradual recession into theoriginal themes restatement eightmeasures afterm. 35, and thereare like processes and events of textural, structuralsignificance throughout the movement.But note,at m. 59ff., the condition ofextreme texturalstasis-absolute homorhythm and nearlyabsolute homodirectional association of horn and violin, as wellas virtualinactivity of the bass,all complementary to slowed harmonic rhythm, tonalstability, andrelaxed dynamic level. Toward m.74 starklysimple textureis counteractive, on theother hand,in relationto intensifying events of pitch-line, rhythmic acceleration,and especiallydynamic level, texture remaininginsistently andrigidly uncomplicated.! °No. 2 is a strict canonbetween thesinging voiceand the Eb clarinetat a distance of seven bars; and thecontinuation ofthe leaderis, fromm. 9,a retrogradeofthe original canonic subject. Thus, simultaneous with strict canon isa retrogradecanon between clarinet, asleader, andthe singingvoice concurrentwith it; the retrogradecanon is

at 3J . No. 6 of the Goethe-Liederbasically is a mirror canon, the canon in a sense framed by the tonally suggestive pitchrepetitions ofthe beginning singing voice on a! and end bass clarinetssettling onc!. Adjunctive to the canonic procedure are developments of the IC succession l-2as in Eb-Eh-D!, twicea trichordal subset of

the row:

IE' k', t t . The occur`;__l rences ofthis trichordalcellular setwill be quickly identifiedin analysis.With these recurrences, thetexture isof greaterdiversity andgreater complexityof interlinear interaction thanthat of No. 2; the anglesof subsidiarydiagonal relations change con-

30 0

f8XfUl'6 stantly evenwhile thecanon isin course.In fact, Dallapiccola concentrates altogether on the three-note subsetorder numbers1-2-3 or6-7-8! with the resultthat concurrent with the broadercanon isconstant imitationon that cellular unit--imitation in mirror andretrograde duplications and inmany rhythmic,directional, andintervallic variants. It is strikingthat only in m. 7 nearthe middle!do immediatelyconcurrent intervallic and directional formulationscorrespond, althoughin a contrarhythmic relation. Theprinciple of progressive, then recessive process in the framing of the very activecanon andimitated cellularunits, withmelodically andtonally morestatic sections of simpler texture,is functionallyimportant.

CHAPTER THREE

rhythm andmeter /ntroductory notes

All element-processesare rhythmic. In an important sense, the study of rhythm is thus the study of all musical elements, theactions ofthose elements producing the effects ofpace, pattern,and groupingwhich constitute rhythm. To study rhythm is to study all of music. Rhythm both organizes, and is itself organized by, all the elementswhich create and shapemusical processes.

While there are many compelling factors suggesting the critical importance of rhythmic and metric analysis, one of the most persuasive isthe fact that metric analysis, in its proper range of implications, is a vital basis of construction and interpretation of phrasingand articulationin performance. The study of rhythm, and especially meter, thus proceedsfrom questions which are truly indispensable to critical, insightful interpretation. Some of these questions canbe stated in the following ways: What are the chief events in the hierarchy of musical impulses in a given work, the points of intensity toward which others are oriented? Or, where are the points and where is thepoint! to which and _#om which musical processesare directed at various levels? A related questionwhich puts the issueof meter in a different way is: Where are

the true

bar-lines at

diverse levels in the

musical structure?

The present study inquires into many aspects of rhythm beyond that of meter, but questions like the foregoing are a recurrent, underlying preoccupation in the discussionsto follow. It may well be that rhythm and meter, seen asa part of rhythm, constitute the most persuasiveand immediately perceptible quality within the range of musical effect.The rates at which events changes!take placewithin the various structural parameters, and the patterns into which eventsgroup themselves, areof decisive significance in expressive effectin the musical experience. 'Grosvenor W.Cooper andLeonard B. Meyer, The Rhythmic Structure of Music © 1960 byThe Universityof Chicago!,p. 1; all rights reserved. 301

302 rhythm

and meter

The analyses and expository materials which follow will showbeyond somewhat ancillary references made in preceding chapters-how the operationsof rhythm and meter permeate andinfluence the entire range of elementsconstituting the musical projection. It has been suggested throughout this book that contrasts in the operations of various structural elements the element-structures! and the progressive and recessive linesof change in those elements underly morphology and meaning in music in one important sense. Rhythm too undergoes changeswith functional consequences formusics intensity scale, playing an essential andtelling role in the delineation of processes ofgrowth and decline, climax and subsidence, stabilityand flux. In this sense, itis no absurdity to speak, for example, of a metric rhythm asthe rate and pattern of metric change, or a temporhythm in music. Contrasting rhythmic events function with levels.

those of other elements to delineate musical structures

at all

A relatively recent effort toward comprehensive theoretical treatment of rhythm gives necessary acknowledgmentto the subjectivities and complexities of rhythmic interpretation. Rhythmic grouping is a mental fact, not a physical one. There are no hard and fast rules for calculating what in any particular instance the grouping is. Sensitive, well-trained musicians maydiffer. Indeed, it is this that makes performance an art-that makes different phrasings and different interpretations of a piece of music possible. Furthermore, grouping may at times be purposefully ambiguous and must be thus understood rather than forced into a clear decisive pattern. In brief, the interpretation of music-and this is what analysis should be-is an art requiring experience, understanding,and sensitivity.2 Because every rhythm isunique, havingits own organization and hence its own particular analytic problems, no selection of examplescan possibly cover the rhythmic permutations and analytic problems which may arise. The task of organization . _. becomesmore difficult once one leaves the hothouse variety of example behind and ventures forth into the world of real music. Here, variables do not operate singly. Nor is it generally possible to classify the rhythm of a given example under a single simple category. Not only do groupings vary from one architectonic level to another, but particularly on lower levels changes of grouping are the rule rather than the exception.3

We shall have to be continually aware that the importance of a theoretical problem is not invalidated bythe dwicultiesof approaches to solution 2Cooper andMeyer, TheRhythmic Structure of Music,p. 9. Cooper and Meyer, op.cit., p. 60. Secalso Howard E. Smither, The Rhythmic Analysis of20th-Century Music, in journalof MusicTheory, VIII,l 964!, pp. 69-70.

rhythm and meter 303

or by the uncertainties or equivocations of proposed solutions. Indeed,the significance ofa question is at times inverselyrelated to its simplicity of treatment. In many instances inprior studies, we have had occasionin analysis to note the factor of interpretive judgment, not of course capricious or arbitrary, citing criteria of analysis in support of such judgment, as opposed to the determination of right answers to the questions from which analysis proceeds. The range of significant, plausible interpretations of rhythmic structure is often of particular breadth and diversity; and the possible

validity ofdiffering conclusions must benoted asan importantobject of analysis. The subjective and often elusive criteria at the root of particular rhythmic interpretations are especially evident in the study of accentdelineated metric

structure.

The awesomecomplexity of problems ofrhythmic structure and analysis canbe seenwhen one appreciates thatrhythm isa generic factor, one aspect of which is meter. Yet meter is onb one ofnumerous manyestations ty' grouping. And meter, as conceivedhere, is dependent on accent-a phenomenon whose existence noone would deny, yet in which many qualities of impulse event, attack! interact variously at different levels of structure. Thus, questions of rhythm lead at some point to questions of grouping, and in turn to questions ofmeter, which rests uponthe difficult questions ofaccent. Inevitably, one manifestation of the complexity of rhythmic theory is the problem of terminology, the problem of the medium by which theoretical discourseis carried on in expression ofbasic concepts.The meanings of such words as rhythm, accent, and meter, not to mention terms like syncopation or even duration Does the duration of a rhythmic impulse include surrounding silences?!, are variously usedin the literature on music. A constant effort is made in the present study to establish fundamental premiseslinking essential terms and concepts, with terms defined so that their use, if not standard or conventional, will

be clear.

The broad basis for the present study might be shown graphically as in Fig. 3-1, in which rhythm is presented as a generic classof pacing,patterning, andpartitioning events in music;a facet of rhythm is grouping, a subcategory of which is represented asmeter. The position of meter in the graphic representation in Fig. 3-1 must not be misconstrued asindicative of subsidiary importance. The fact is that a very significant, often perceptually immediate, functional grouping is delineated in music by accent, by an events superiority of content and projection in which surrounding anticipative and reactive! events are absorbed into an accent-governedmetric unit. When accent-delineated grouping is firmly established, opposing accents, which are very necessary andusual in interesting music, may be felt as syncopated against counteractive to! a prevailing, precondition-

OV tp Cg hs V he N

V lO0 EO lg

VV

~~

Ul !C lO EO tO E La

0 O. tD

V 0C .

I-I IL

rhythm and

meter 305

ing, still underlying metric basis. Or counteraccent, imposed against a pattern of units not referentially establishedand preconditioning, may bring about fluctuation in the metric structure: contiguous and concurrent units thus emerge in Huctuant,

Fundamenta/ concepts

asymmetrical relations.

of rhythm

While it is sometimesbroadly described as ordered time or meter, it

seems essential that rhythm

narrowly as

be viewed as both manifold and

specific: the sum of a broad range of factors eachof which is in some way a manifestation of pace and grouping, the former a product of relative frequencies of events, the latter of their relative qualities and the means by which they are unit-ordered. The factors of rhythm can be detailed as follows. Rhythm is:

l. Tempo, which has two aspects: the eventfulness of music degree to which the temporal continuity and flow are filled with articulate impulses or related silences! andthe frequency of pulsation at some given level. In the latter aspect, note the recurrent concept of a norm of tempo-e.g., that of a physiological manifestation like heartbeat or stride.! We shall refer to these two aspectsof tempo as activity-tempo and pulse-tempothe degree of eventfulness andthe rate of pulse succession, respectively!; tempo is thus the quality of rhythmic motion and drive. As to controlled, functional changes in activity-tempo, see Ex. 2-22, pp. 250-52, one of many extracts cited in which the factor of shaped degreesof eventfulness is described as essential infunctional effect.! 2. Pattern or motive cfi rhythmic mode!, as expressed indurational and other strong-weak combinationswhich have in a given context motivic significance at some level, or characteristics of pattern by which a style, genre, or work can usefully be identified. Rhythmic pattern is explicit Pulse will be understoodas thefelt, underlying,at timesregularly recurrentunit by which musicstime spanis measuredand its divisions feltat somespecified level-the basis for counting, or conducting, and for metronomic indicationsof tempo. Pulses recur regularly only at certainlevels inmost music,conspicuously at the levelof the notated bar; some music however, notablyliturgical chantof theMiddle Ages,must beregarded as indifferent toregular pulsation. Impulse isregarded hereas the event itself attack, stimulus,integral silence!superimposed onand relatedto the stream ofpulsation. 5As toshaped controlof pulse-tempo relations, see Robert Ericksonsdiscussion of his Duo for violin andpiano in_journal of Music Theory, VII, 2, 963!, pp. 174-92. Inthis poignant andarticulate article,entitled Time-Relations, Erickson states and enlargesupon his creative interestin thecontemporary relevance and usefulness of aconcept ofaHuid incommensurable! time field of which the traditional fermatais representative.

306 rhythm

and meter

especially in the sequenceof attack and changeby which the musical line is delineated; an important item within this area of rhythmic effect and identity is that in which motivic units become recurrent within broader units and fundamental to the delineation of musical form e.g., isorhythms, etc.!. 3. The prqfiles expressed in element-changes manifest individually and in confluence! as thesechanges involvepattern, rate, and degreeof change. Thus, melodic rhythm normally seen asthe rate and pattern of pitch change or attack without pitch change! within a line, harmonic rhythm, textural rhythm, and other element- and subelement-rhythmsconstitute this aspect of rhythmic experience. 4. Grouping,or partitioning of musics time span by associationsperceived within and among punctuated or articulated unit-orderings of events. One mode of grouping is meter; it is, like tempo, one of rhythms most telling aspects.

Rhythm as activity and

mot/'on

Although it is not to be a subject of extensive attention in this chapter, rhythm as the motor aspect of music pace, drive, activity-tempo!-the energy dispelled in relation to the extent of eventfulness in the succession of impulses-is exceedingly important and fundamental in musical structure and eifect.

The control of rhythmic activity is among themost elementalof musics shaping forces,one of the most direct of functional devices, comparable in immediacy to the control of dynamic intensity.° The experience ol, for example, activity-tempo acceleration in the direction of developmentalclimax, or deceleration signalling release,is a fact to which everyone conditioned to interesting musical experiencewill testify? An example of the shaping power of controlled changes in rhythmic motion activity-tempo! is afforded by the first movement of Weberns Op. 22, to which reference was made in Chapter 2 Exx. 2-22, and 3-l!. Analyses of other element-progressionstoward the climactic mm. 22-23 can be complemented by illustration of composite activity. Pulse-tempo is constant, but there is a distinct progressive curve in activity-tempo mounting toward m. 22 and receding toward the fermata between mm. 27 and 28, the subsidentaim of recessive operations following m. 22. In Ex. 3-1 °The quality of rhythmic motion can be an important and essential factorin the identification andcharacterization of style. Forexample, many literatures ofthe Baroqueare marked by relatively constantactivity-tempo ascompared with,for example,the more romantic, changeable impulses ofcertain otherstyles!.

rhythm and meter 307

there are three viewsof the rhythmic progression.The first of these,perhaps more a representation ofsignificant textural accrual, showsthe rate of attack of two- and three-note motives which are a basic unifying material; the second and third portions of the example represent in different ways the composite motion or rate and spacing of attack. The upper line shows this

by indicating the frequency of occurrence of rests separated abby distances of 3, 3, 2, ll, 15, 6, 5, 2, 3, 1, 1! and the lower line by indicating frequencies and durations of notes anywhere in the texture. The example is characteristic.

Ex. 3-1. Representationof activity-tempocurve ina passage from Webern's Quartet, Op.22, first movementsee Ex.2-22, pp.251-52, forextracted quotations!. m.l9 m.22

Motive 3

4

83

?§l2f.'2..¢, »» » R3

3 ll

E »»

»» »

35

|!» -1!!

3 fa

e eensusul-ns ~ -1 ' |! I6

gene!:/ 92 est

frequency

I units -.

i2

2 if _'_

liiiniiiiii liiiiiiiii-if-iii

. ./.

3I I

_L -ng*

lliiil _T

.

Recessive and progressive rhythmic actions and relations, tightly controlled and directed to and from relative levelsof stability, can be seen in works of Elliott Carter. Carter is known for a procedureof controlled change in tempo described as metric modulation by which pulse andpulse rate change gradually, often at frequent intervals, in a procedure determining an aspect of tempo-structure at various levels. This method of covert tempo change is illustrated in a passage from the String Quartet No. l Ex. 3-2a!. One factor in the quoted passage is contractionin the low-level metric unit. Compare the changes in activity-tempo as well as=. pulse-tempo; of the two aspects oftempo, it can beseen thatfirst they run parallel, then while there is acceleration in pulse-tempo activity-tempo is held back before the progressive actionsof m. 22.! Example 3-2b traces the process toward an

accelerated Pulse 4| from 72 MM to 120 MM.

308

rhythm and meter

Ex. 3-2a. Carter,StringQuartetNo. 1, first movement.

Maestoso J - 72

m. 15

Reprinted bppermission of Associated MusicPublishers, Inc.

rhythm and meter

309

gx. 3-2Q. "Metric modulation"in the Carterexample,

J=G.! i. e., MM 72

288

3=!! 9'.,

or 360

360

sP D 120

10 In a changed metersignature, >6. 6

Some composershave used notational spacesbetween sounds as they are representedon the printed page as impreciseguidesto the temporal distancesby which they are to be separatedin performance.One such example of proportionalnotation,but in which mensural temporal intervals are very specifically indicated, is the Berio Sequenza II for harp. Here, durations within notated "measures" are suggestedonly by relative distancesseparating the printed notes, but each measure is to correspond to a metronomic pulse at 40MM, so that tempo is rigidly controlled at that level. Metric units at phraselevels are controlled too, in significant degree,by their consequent relative durations, differences of pitch content, differences of intensities, punctuative devices like specified silencesof specific or proportional duration, or abrupt dampening of sound, etc.!, and other shaping factors. The element of chance in this work is, then, relatively minimal, amounting to a modestlatitude of possibilitiesin which basic rhythmic structure is relatively inviolate.

In the first phrase Ex. 3-3a!, within a stasisof pitch-line and opposed counterpoint of intensity changes,the rate of attack is one of acceleration, then deceleration in a recessive cadential action of palpable functional effect. This can be expressedvery simply Ex. 3-3b! as a progressive,then Ex. 3-3a. Berio,Sequenze II for harp. 40MM~

! 1965, Universal Edition.Usedbypermission of thepublisher.Theodore Presser Company, sole representative UnitedStates, Canada andMexko.

310 rhythm

and meter

recessive rateof increase and decrease in events per stipulated, regularly recurrent unit

of time

at 40MM,

the distribution

of events

within the

mensural unit suggested bygraphic proportioning but not specifically controlled." Comparable processes characterizeother phrases. Ex. 3-3b. Accelerative anddecelerative rhythmic changes inthe first phrase ofthe Berio Sequenza ll.

Number events of + per 40 MM unit:

10

22

+ + 2 -2 24

66

-2 42

l _._

Rhythmic pattern

as mot/'v/'c

Especially at more immediate levels of structure it is readily apparent that durational and other weak-strong combinations and distributions have great and constant importance in the projection of distinctive, identifiable, associable motivicand thematic ideas which are exposedand developed in the music of most Western styles. Rhythmic patterns have in various systems undergone theoretical classification, often as rhythmicmodes corresponding to and derived from the classes and terminology of standard feet of ancient Greek poetic theory. The most common of these modesare six: trochaic long-short, or strongweak: - U!, iambic U -!, dactylic U U!, anapestic U U - !, spondaic - - !, and tribrachic U U!. Other modes e.g., amphibrach: U - U ! are noted in some systems. 7At the same time,the metricformulation within the phraseclearly hasto beseen as one ofvertical noncongruitybetween thetwo linearcomponents of pitch activity, the lower stratum havingaccent atits inception,the upperhaving anextensive anticipative wind-up in preparationof its later, noncongruentaccent. 8For briefdiscussion ofespecially thirteenth-century!rhythmic modal theory, and some bibliographicalreference, see Willi Apel, Harvard Dictionary qf Music,2nd ed. Cambridge: TheBelknap Press of HarvardUniversity Press,1969!, pp.535-36. Whilerhythmic modal theoryis of great importancein certain music for example, inthirteenth-century motets, wherethe applicationof suchtheory isoften highlysystematized!, its applications in music broadlyviewed caneasily beoveremphasized, with a resultantsimplistic viewof what is commonlyhighly complexand diversifiedrhythmic organization.This is not to say that analysis andidentification or conjecture! respecting the substance of rhythmic pattern at various levels has novalidity; but it is a cripplingpreoccupation when it is taken asthe end of analysisin which rhythms areconsidered invariably reducible toone ofa fewtraditional classifications, ormodes. Musicalsituations ofreal interestpose acomplex variability of patterned associations not readily or usefully classified withina limited range of assumed referential norms.Nevertheless, the classifiable modal rhythmic pattern may at times be relevant togeneric combinations upon which thematic substance and structuralprinciples are based ; andthe traditionalterminology andsymbology of rhythmic modesare oftenuseful.

rhythm and meter

311

There are, as might well be expected, instances of applications of other systemic "modes" indeed, the theoretical possibilities are without limit. In certain of his works, for example, Olivier Messiaen, a figure of considerable importance in contemporary explorations of rhythmic technique, employs rhythmic patterns derived from exotic systems.In Ex. 3-4 the music is based on three Hindu rhythms two of which undergo pre-

scribed transformations eachattack ofthefirstincreased bya sIIateach repetition, durations of the second decreasedby the same value at each repetition, and the third reiterated without change. In the brief extract, the threemotivesare circled; in their immediaterepetitions,prescribeddurational modifications are indicated.

Ex.3-4. Messiaen,Reprises parinterversionNo. 1 of Livred'orgue!.

Man.

Ped.

Reprinted bypermission of Alphonse Leaguc 8 Cie.,OwnersO' Publishers for all count's.

312

rhythm and meter

The concepts of rhythmicostinatoand isorhpthtathe latter denoting a recurrent set and ordering of durational values, commonly in tenors of

early motets! relate to another practiceof rhythm applied motivically. Every student of music is familiar with isorhythmic structures, for example,

in many fourteenth-century motets,'althoughmore recent applicationsof analogoustechniquesare of comparable interest and, perhaps, significance. The third movement of Webern s Variations, Op. 27, for piano illustrates the isorhythmic principle: thus, the opening four bars plus one undergo immediate isorhythmic repetition. That is, the initial twelve attacks are duplicated in rhythm by the next twelve Ex. 3-5!. The application of rhythm as thematic ' is again explicit and significant as a parallel to other relations in concurrent effect in the Webern example e.g., that of the twelve-tone series,occurring as R" on EI in the first twelve attacks, then as RI on El> in the next twelve!.

Ex.3-5. Webern,Variations,Op.27, for piano,third movement.

RuhigfliessendJ - ca.80!

Copyright 1937,Universal Edition.Usedbypermission of thepublisher.Theodore Presser Company,solerepresentative UnitedStates, Canada andMexico.

SeeHistoricalAnthology of Music,Apel and Davison,eds. Cambridge:Harvard UniversityPress,1949!,Vol. I, Nos.43 and 44, for examples.A fascinatingstudyof isorhythmic andsymmetricalproceduresin Machauts MesseNotre-Dame, including discussionof remarkably consistentcenter-axial"horizontal" symmetriesin the work and extendingto treatment of comparabletechniquein other works,is Otto Gombosi,"Machaut's MesseNotre-Dame," in TheMusical!uarterly,XXXVI, 2 950!, pp. 204-24.

rhythm and meter 313

One could of course continue to cite relevant examples of motivicthematic rhythm at various levels ofstructure throughout the vast historical span of Western music. Consider, for further reference, the fifth symphony, or the sixth, of Beethoven as to rhythmic motive, in extreme foreground manifestations, having vital and consistent significance. And the entire range of techniques associatedwith the serialization of rhythms and rhythmic relations is of course relatedto the principle of motivic rhythm; some of these arenoted near the end of this chapter. But concerns ofmotivic pattern are recurrent too in plentiful studies of thematic process in musical form, especially in literatures dealing with the tonal period. It seems important, then, to move on to more problematic areas of rhythmic theory.°

The rhythms of e/ement-successions

We have reiterated the important principle that every structural element is, in its distributions andqualities qfevents, expressive qf rhythm. There is thus a rhythm of pitch-line rhythm seen ofcourse asincluding meter!, a harmonic rhythm, a tonal rhythm, and a rhythm of each of the other elements and parameters ofmusical events.These rhythmsare preeminently: ! of pacing or tempo-the rate of event articulation and change; ! of pattern, as manifest in varying durational combinations; ! of proportions, comparative durational relations amongunits, or groups ofevents; and ! of relative qualities of events and event-successions-degrees distances! of change, of accent."

The extent to which element-rhythms function concurrently in complementation to or compensation for progressive and recessive,underlying tendencies is invariably striking and illuminating as to function and expressive effect.Take, for example, a brief example from the Diabelli Variations of Beethoven Ex. 3-6a!. Quickened, sometimes chromatic, Men double-dotted thusmotivic! harmonic rhythm in the Beethoven example reveals action within this parameter decisively complementary to an overall rise in pitch-line, a tonal progression toward the mediant typical of much Beethoven, and a variation on progression toward the dominant in the theme!, and a driving creseendo. Balanced against all of this is the extraordinary poise of constant melodic and iterative motivic rhythms within a slow tempo-the entire complex of element-structures a confluence of deeply affective character. A comparable pattern pertains in the second halfof the variation. As to the harmonic-rhythmic parameter, the reader should note in detail the progression out of broad tonic, dominant! values of the beginning, characteristic of the theme, into controlled

acceleration.

Let us focus briefly on mm. 4-8 of the variation. Of harmonic rhythm

314

rhythm and meter

Ex. 3-6a. Beethoven, Thirty-threeVariationson a Waltzof Diabelli,Op.120,VariationXIV. Grave e maestoso

it can be said that all changesare pronounced in degree; roots are usually a 5th apart, once a tritone, once a 2nd the latter in the cadential iv-V!. In the harmonic rhythm the rate of successionacceleratesfrom to

etc. The tonal rhythm is characterized by references to F, and e the F-e successionrelatively strong in distance!; a brief reference to a is also crowded into thesesame bars. Hence, there is tonal-rhythmic eventfulness of strength and frequency. Within the element of coloration, as it concerns dynamic intensity, there is an increasedrate of change

f

so< P f

P ! inmm. 5-8 ascomparedwith the

broad pace of dynamic change in mm. 1 4.

316 rhythm

and meter

over, while density undergoesonly slight changes,- theshape delineatedby space and texture-compression! is demonstrably functional: note, for example, the essential stepsuccessions of upper voicesin mm. 4-8 as compared with the ascending successions by leap in the bass. In summary, the general activity profile within various elements biseets the example ina manifestation of the role of shaping element-rhythms to projectgrouping in a broad sense analogous to thoseof meterand phraseolog_y.° The vital importance of study of the conjoined interactions of rhythmic element-structures in music is evident in references to complementary and compensatory relations of concurrent element-successionsthroughout this book. At the same time, it must be acknowledgedthat these considerations, and with them the considerations of individual

and confluent element-

and subelement-rhythms," require exploration far beyond the possibilities of this book.

It is apparent that element-rhythms correspond in diverse ways: some are of course totally or substantially inapplicable in particular situations while others are of primary assertiveness andimportance; some may be accelerative and others decelerative or passive, relatively inactive, or static! in particular contexts, with the weight of movement in one or another of thesedirections, or in balance, or in inert condition; and someare clearly more applicable at lower levels of structure e.g., pulse-tempo!. It is also apparent that proportional groupings or units manifest in areas within individual element-rhythms, and their interrelations of asymmetry or symmetry and relative strengths of projection, constitute a fundamental aspect of felt rhythmic effect. Moreover, all element-rhythms are subject to analysis and interpretation at various levels; thus, for example, harmonic rhythm functions at the level of most immediate detail but also at the level of broad changes. For example, theessential prolongation of a dominant may represent in one sense broadly inactive harmonic rhythm and in another, more local sense a foreground harmonic rhythm of activating, embellishing changes. An ideal and fully comprehensive analysis, if such were possible, would deal with all element-rhythms individually and collectively, and with their interrelations-all of them projecting and the analysis of all of them illuminating! that complex, many-faceted, element-structure which is the rhythm ofthe musical example in question. 1°See pp. 78-79 ontonal rhythm; Ex. l-45, p. 137,on tonalrhythm in the finalmovement of Beethovens Symphony No. 2; pp. 201-4on textural rhythm; Ex. 2-5, p. 202, as to certain aspects oftextural rhythm; and other recurrent references to element-structures as rhythmic.

rhythm and meter 317 A theoretical approach to the consideration accent-de/ineated grouping

of meter as

An iterative or neutral! rhythm pertains among successive, regularly spaced impulses stimuli! in a series lacking parametric changesof any kind. But functional rhythmic shapes involve grouping occasioned by dWrentiations among contiguousevents. The question of meteris the question ofaccent, since metric units are initiated by accents, impulses of relatively strong projection. The question of accent is a very complex one indeed; nevertheless, itis an indispensable issue for the analysis of this vital aspect of rhythmic structure in music." In embarking on a study of meter, we shall make the assumptions! that in the vast majority of instances musicaleffect rests upon the experienced orpreconditioned, imaginedsensation ofa seriesof pulsespunctuating and articulating the time continuum as a psychologicalfact not necessarily manifest in physical events, such pulsesreferential as a mental imagery in the musical experience, anda basisfor perception of the rhythmic relations of events; ! that impulses-the actual sounds andsilences ofwhich musical projection consists-are superimposed on the established, felt stream of pulsation, and commonly groupedby distinctions of variouskinds; and ! that metric partitioning is one of severalkinds of grouping in music. Thus, the analysis of meter involves certain primary questions: ! Which stimuli are perceived as accentual? ! What is the nature of the structure effected by metric grouping and the interrelations among its units? ! What is the weak-strongorganization by which the metric group is characterized at any given level of structure? ! What is the relation of metric partitioning to other modes of grouping, and of metric structure to other element-structures? From these diflicult but fundamental questions, of course, many highly important subquestions issue; moreover,as implied above, all such questions are subject to diverse implications of different levels ofstructure ranging from the most immediate apparent in the smallest manifestations ofgrouping! to the most broad in which it is at least conceivable that an entire form might usefully beregarded ashaving a palpable organization-e.g., weak-strong~weak-of significant, essentialeffect!. "One trusts,in usingthe termaccent, that it is clear nosuggestion of overt emphasis in performanceis necessarily implied; indeed,the propertiesof accentare generallyinherent in the various parameterspitch, duration, etc.! of the eventitself Sometimes,especially at .higher levels ofmetric structure, subtle and restrained interventionin performanceis necessary to bring out important metric functions. Butoften nointervention isindicated at all, onlythe avoidance of counterinclinations of timing,articulation, andpunctuation. These are vitaljudgments required in performance;but to say thata particularimpulse event!has accentual value is generallyto makean observationrespecting itsinherent properties.

318 rhythm

and meter

Meter, then, consists ofunits large and small at various structural levels! _farmed by d@rentiations in the musical events inwhat we shall describeas diverse impulse functions. If there is diiferentiation it is expressed in some parameter or complex of parameters. Meteris thataspect of structure articulated as accent-delineated groupings within the attack event!sequence, and the proportional interrelations of such groups at all levels. Metermay be symmetrically ordered, asymmetrically ordered, or ambiguous in its ordering. Moreover, it may be symmetrical at one level, asymmetrical at another; and of courseindividual textural componentsare oftenof contrastingmetric ordering at the same level. Thus, the analysis of metric structure, whose application in examples in various styles is the principal concern of this chapter, is the evaluation, identification, and/or interpretation of its two primary factors: ! pattern within the metric unit, its weak-strongcomponents andassociations, and! the accentual

articulation of

the units

themselves, with

consideration of

their proportional interrelations. The functions of meter in these respects are a necessary, constantconcern to which analysis should ideally proceed, involving descriptive conclusion or conjecture respectingthe role of meter, and the functions of its varying qualities, in shaping the expressive content as well as the structural unity and diversity of music. To retrace in summary the broadeststeps thathave led to this point in the consideration of rhythmic-metric structure, it seems desirableto note again that rhythm is regarded as a generic class of techniques affecting pacing, patterning, and grouping of all events at all levels; that meter is one of this classof techniquesand phenomena,defined asaccent-delineated grouping whose structural and expressive functions are manifest in proportional relations of metric units whether contiguous or concurrent! and the weak-strong associations internal to such units; that meter, like other aspects ofrhythm, is felt at various structural levels, perhaps even at the ultimate macrolevel; and that meter is not to be equated with regularity, so that metric fluctuation,

however extreme, is not meterlessness."

The concept of meter as, by dehn/tion, subject

to fluctuation

True metric structure is neither necessarily regular nor necessarily coincident with notated bar-lines at the mensural level." A great deal of It is possible thatthe term arnetric has useful applicabilityanalogous tothat of the term atonal,with referenceto structuresof relatively extreme instabilityand ambiguity. This study will adopt and employthroughout certainterms whichrefer to classes of structurallevel in which thenotated measure is regardedas aconvenient objective point of reference:mensural refers to groupingapproximate to, or derived hom, or otherwise demon-

rhythm and meter 319

interesting and expressive musicis of irregular accentuation, ry' irregularmetric grouping. In situations of altered interval of accentuation, meter fluctuates, and the idea of Huctuant asopposed to regular meter is absolutely-necessary to the understanding of music other than that of unequivocally regular proportions. It is on this basisthat the assertion ismade that in highly fluctuant contexts like those of recent stylesmeter is not extinguished, but assumesa character more often far more! Huctuant than those of certain traditions. We shall see as well that even much music commonly regarded, in submission to the deceptive impression of the notated signature and bar-line, as regular in metric grouping is indeed highly variable atleast atcertain levels. We thus avoid any suggestion ofa terminological equation in which meter = regularity of unit grouping.

Meter as one manifestation

of grouping

in music

We have systematically revieweda number of factors as rhythm-aspects of musical structure generically classified asrhythmic. One of these has to do with relations and interactions among, and weak-strong orderings strably corresponding in scopeto thatof thenotated measure without ofcoursenecessary conformity ofaccent tonotated bar-line!;intramensural refers to smallerunits andlower levels; intermensural refers to broaderunits andhigher levels.There isnothing absolute,as weshall see, in the identification ofthe mensural unit; often, to describea particular unit or accent, or level! as ofmensural significance is tomake judgment a of comparability relations. cy' Inthat judgment, complementary motivic groupingis oftenrelevant, asmay bethe notatedbar-line. "Compare, forexample, concepts of meterstated orimplied inEdward Cone,Musical Form and Musical Performance New York: W. W. Norton and Company, Inc.,1968!, p.82; Cooper andMeyer, TheRhythmic Structure of Music,p. 4; or Smither,The Rhythmic Analysis of20th-Century Music, in journalof MusicTheory, VIII,pp. 7| and 72.The Smitherand Cooper-Meyer comments in this regard describemeter asonly primarily, 'not necessarily, regular; andof courseHuctuant meteris at times considered rhythmic, not metric. In an interesting study entitled Extra Measuresand Metrica1Ambiguityin Beethoven [in Beethoven Studies, Alan Tyson, ed. New York: W. W. Norton & Company, Inc., 1973!, pp.44-66], Andrew Imbrie makes on p. 53! the following statement:The idea that oncean organizationin time becomes flexible rather thanmechanical itthereby becomes rhythmic rather than metrical stems from the notion that meter has to be absolutely regular. Imbries own proposal as to a useful distinction betweenrhythm and meter proceeds fromthat important qualification, inwhich heasserts thatmeter isnot necessarily regular. The identificationof meter with recurrent,regular denominators renders extremely problematic the understanding ofmetric structure in highly Huctuant low-levelcontexts before and afar the tonal period;moreover, it induces asimplistic view of interestingtonal music, whosemetric structureis sooften morevaried andfluctuant than meter signatures would suggest, especially athigher levels.It is argued herethat a concept ofmeter ispossible and valid! which isapplicable tomusical experience in general,even tocontexts ofhighly fluctuant metersuch asthose ofRenaissance polyphony and the present century. Compare the increased ratesand distancesof fluctuation in tonal reference, in melodic line,and in other elements.

320 rhythm

and meter

within, units manifest in grouping of a number of kinds. One of the phenomena by which events are grouped is that of accent, relative impulse superiority, in relation to which surrounding impulses at various levels can be seen as reactive, anticipative anacrustic!, and conclusive Meter is thus an aspect of grouping, or partitioning, which is in turn a vital aspect of rhythm. We might identify the following as factors ly' which, ingeneral, events in music are perceived as grouped into leveled units of structure. 1. The grouping of class-ajiliated element-events is one such mode. For example, eventssubsumed withina particular tonal system,embraced within a given tonal reference,are grouped in this sense, as are, for example, events within a particular kind of textural activity e.g., imitation!, within a particular timbral unity, within a generalized melodic inclination, or within an harmonic complex of associated factors,etc. In Ex. 3-6, bisection within certain element-structures-e.g., tonality-is on this basisas wellas onthose of phraseology and other modes of grouping.! 2. The grouping ry'tendency-ajiliated events crossing distinctionsbetween elements isanother mode of unit delineation. For example, a concentration of events an element-complex! functioning in the direction of intensity within a given, unified process atsome level, is a factor of partitioning. In this sense,the Beethoven excerpt cited earlier Ex. 3-6! is bisected in the marked acceleration of conjoined, intensifying element-eventfulness in its second half This factor might be characterized asgrouping by the delineating effect of the composite profile of change in degreesof activity, in the line of element-rhythms; a segment of that line, where it can be seen to have unity in relation to a given progressive or recessive tendency, is an important area of grouping in this special sense. 3. The most subtle factor of grouping and partitioning, and perhaps the most elusive of perception and penetration, is the groupingof pitch structures at various levels in associations of linearfunction. Such grouping is expressed in the orientations of relatively auxiliary pitch events toward and around more essential ones, the sortof associationmuch discussed in Chapter 1. At one level of structure this can be seen asthe punctuation of broad time spans as to relative durations of structural prolongations-manifestations of predominances! of underlying, background events-the areas of hegemony of certain, elaboratedevents. This concept isof courseclosely related in much music to item 1, above, where the specific element of tonality is concerned; but it has to do as well with grouping around structural pitch factors in melody and harmony whether or not such factorsare supportive of, and predictable within the norms oi, given systems oftonality.! The unit delineated hereis one consisting ofthe essentialpitch event again, at some

rhythm and meter 321

given level! together with affiliated eventsby which it is elaborated, and by which its manifestations arelinked in passing motions. This mode of segmentationis particularly important to distinguish from what we regard asmeter: thus, the event which is central to the intrarelated linear complex, like the event central to the set of tonally related events, is often central to a specific tonal system; but that event is by no means necessarily thebearer of accent. Weshall find, for example, that the cadence

on l, which has of course that degree as central to the linear-functional grouping of eventspreceding it,is very often nonaccentual-recessivein the metric sense. That is to say that the cadential event is always central within the unit to which the cadence pertains!to the group definedas tofunctional, linear pitch affiliations or, in tonal music, to the tonally centered unit!; but it is often not a point of accent central in delineation of the metric unit see Exx. 3-7 and 3-8!. Ultimately, the tonally-cadentially central event is metrically recessive atpertinent levels, a point on which we shall dwell at a subsequent stage in further theoretical discussion ofthe distinction between primacy of tonal and linear function as opposed to metric accent. Within the concepts developed here, an often assumed necessary interdependence, evenequivalence, betweenbackground tonal and metric structures issharply denied. 4. Extramusicalfactors may condition, even determine, the grouping of musical events in still another sense. Textis of particularly critical importance in much music in unit delineation at diverse levels. See Ex. 2-14, where the setting of text is functional in bringing about a recessivesuccession of units expressedboth textually and in musical meters.! The kind of condition shown

in Ex. 3-7, in which textual

and accentual

musical inflec-

tions exactbtcorrespond, is by no means universal in vocal music. Textual accents on thou at the broadest

level, on me at another level,

and on stole

at the lowest level are in perfectly balanced concordance in relation to musical accents of pitch and duration, as can be seen in the example as represented at three different levels of metric structure. Other modes of grouping, too, are in correspondence, but the distinction between the pri-

mary linear function ofGzl and I, as opposed the to metric accent with which the phrase begins, isclearly evident.! That the issue of textual infiuences in grouping is one of substantial complication will be evident in other examples to be cited. One literature in which this issue is of very great and not well understood! significance might be mentioned in passing-that of Renaissance polyphony, where grouping of all kinds is often of very greatly Huctuant character and complexity, the nature of fluctuation of meter in an important degree the outcome of prosodic infiections which often have to be decided in editing and in

performance, and

which are commonly for

example in josquin! in

rhythm and meter 323 Ex. 3-8. Haydn, Symphony No. 104in D,fourth movement. m_3 Allegro spiritoso

l J,

# J.

J ,J 2/ 3,/` oo.»-- TJJ J

g ,Q

J

*See discussion of contrastinggrouping oflinear functions, pp. 320-21,and below.

In the Haydn excerpt Ex. 3-8!, as in the Dowland phrase quoted earlier, phraseology corresponcls meter: to thatis to say that the phrasebegins at the same timeas thephrase-level metricunit." The initial event is regard-

ed asaccentually superior for eachphrase-as initiatinga metricunit at the same time that it initiates the phrase. We shall considerother examplesin which this correspondence does not pertain.!

Group unity,moreover, is simply andclearly evidentin the afiiliating relations of all eventsin the Haydn theme to eachof severalelement-classes: tonality, texture, and color.

The metricand phraseological ordering themetric structure at phrase level evidentin thefact thata bar-line for eachphrase would immediately precede the first impulse!are inopposition tointernal orderingof grouping as to linear functions, here of course tonally conditioned. An analysis of the phrasesas to orientations of tonally subordinate events to essential ones

would regard the cadentialnote of each phraseas fundamental-points toward which other eventsincline in underlying stepwisedescent. For

the entireperiod, thelast note,l or harmony, I!is the essential event toward which all others, at the level of the period, incline, the cadential

Q V! of thefirst phrase a broadlyviewed auxiliary. But theultimate cadential eventis recessive at all but themensural and lower levels of metric structure. "The symbolsused inexplication ofmetric structureat variouslevels roughlycorrespond tostandard conducting gestures, except that the final impulseof the metric unit is represented differently depending onits functionin endingthe unit as opposed to functionin anticipation of thrust toward,anacrustic preparation of! the next unit. The symbolsare in no wayprescriptions forconducting, norare theyconsidered tohave anycomparable practical applicability;but theyare usefulas away ofthinking about relations within the metric unit. BT hecorrespondence of metric structureto phraseologymay well be characteristic of Haydnmore than,say, Mozart.

324 rhythm

and meter

A great deal of distortion in musical performanceis accountable,we believe, to failure to appreciate this usual metric function of cadential arrival."

Meter as opposed to the notated bar-//ne

It is fundamental that meter is often independent of the notated bar-line, so that a necessaryquestion in all analysis of meter is: Are the determinants of metric grouping in accord with the notated bar-line, and if not what is the real meter? Or, where is the true bar-line ?! A simple kind of accelerating metric change is illustrated by Ex. l-3, pp. 35-36, to which reference shouldbe made.Through much of this Chopin prelude the triple division of the notated bar is upheld by accentual mensural

factors of various kinds including the J anacrusis at theoutset!. But at m. 29 the triple unit yields to a duple division in intensifying contraction of the metric unit complementary to the rise in pitch-line and crescendo to m. 34, after which the normal situation resumes. One view of the metric structure might be represented asin Fig. 3-2. Fig. 3-2. An interpretation of metricstructure inthe Chopin Ex. 1-3!. m. 26 3+3 4

28 30

32 34

VF

2+2+2+2 2+1

Anomalies Restoration

/V 3+3

of normal mensural unit resolution

of metric dissonance!

Whether the mensural unit changes inactual as opposed toincipient metric fluctuation! or simply briefly admits an acceleration in accent frequency

in relation to which the original F and $1 units arefélt stillprevailing as and referential is a question ofperception to be taken up later as to the concept of preconditioning metric structure. A further example Ex. 3-9! is shared with Cooper and Meyer. It is necessary that, like many examples in this chapter, it be given in incomplete notation.! 1°Feminine cadentialarrival ismetrically strongat onlya stilllower level,at thelevel of somedivision ofthe notatedbar. 2°Cooper and Meyer, TheRhythmic Structure af Music,p. 90.

rhythm and meter 325 Ex. 3-9. Mozart, Symphony No. 41in C, K. 551 Jupiter!, secondmovement. cantabile AAndante ` IIHIQMMI EM'1llllI.11Mi'FMl.M

_Af '592 YIMII-IQ_M11"_`

B? P M II' I 1 1: E:=:*=-§»=- : 33 VV

.':E 'i

QV '

= v-iI

'' °3

P `M 921_= 1

'=

.i ..='

3 5 .3+2! VV

,»":5 ~5 v

V

'iv I

v

1

Broadened, J asymmetrical unit Point of resolution

What is provocative in the example is of course the accent given the second beat in every one of the first six measures:by durational superiority, by dynamic intensity stress!, by texture, by anacrustic support, by superiority of pitch, or by combinations of thesefactors. Uncertainty, as between the real and notated meters, is at least for a time resolved by the strong afiirmation of the downbeat of m. 7 intensity of dynamic level, duration, harmonic change, initiation of new motivic idea in complementary group-

ing, changed motion, leaps, §tc.!, but what of the first six notated measuresLet ? us assume the thatVnotated meter has no preconditioning effect, forgetting that we know the piece and its notation and ultimate metric order.

Cooper and Meyer insist that the triple meter must be expressed in performance. However, the melody is, as Mozart makes clear, definitely in triple, not in duple meter. Nor should the theme be thought of or played as metrically

ambiguous orvague. Thereare tworeasons for this. Froma historicalpoint of view, the rhythm is that of a sarabande, with its typically heavy secondbeat. Since this rhythm is normally precise, it would be stylistically wrong to perform it ambiguously. From the point of view of internal structure, too, the triple meter should be decisively articulated. For if the latent duple organization is permitted to obscure ordominate the manifest triple meter, the meaning and character, not only of the theme itself but of the whole movement as developing out of the theme, are considerably weakened."

We shall take a very opposite view of the example. Let us argue that the first two notes are anacrustic to the double-dotted a,

and that Mozarts

2'Cooper andMeyer, TheRhythmic Structure qf Music,p. 90. References toexamples are omitted. "

326 rhythm

and meter

anomalous accentsdo shift the bar-line-i.e., do impose a real meter out of accord with the notated bar-line. What is expressed isnot a latent duple meter but a triple meter which is temporarily displaced in relation

to the notated bar-line. Cnly harmonic rhythm accords with the notateél

bar-line establishinga subtlecountergrouping!. There is asymmetry in the size of the unit just preceding m. 7; Mozart makes this very persuasive by continued melodic descent, while faintly recalling the preceding irregular accent with harmonic rhythmic change underthe Bbof m. 6 partitioning the 5 3+2

unit as'O !. The entire point ofthe metric structure the of passage

seen inthis light is asa succession toward metric resolution in which the shift in real bar-line and consequent asymmetry of the unit just described are stabilized and brought into accord with the notated bar-line! at m. 7. Dynamic stress must not be overlooked as contributively meterconditioning, nor should the real metric anomaly be thought weakening. Mozart has created a provocative structure involving recession out of a condition of metric ambiguity dissonance! and it is vital that this be expressed; m.7 arrives as a point of emphatic clarification resolution!, not of course decisive for the entire movement.

/mpu/ses and the/7 functional different/ations

The analysis of meter requires the evaluation of differences amongmusical events, thedistinctions by which somemusical impulsesare felt to be strong and others weak, and by which weaker impulsesare grouped in relation to strong, metrically initiating impulses. In these connections, questions like the following arise: How do impulses events,stimuli! differ in character, strength, and function? Can such functions apply analogously to silences? How might these functions be classifiedand, perhaps,symbolized? Canan impulse have more than one function-a duality elision, conjunction! of function? We shall take a fundamental position that musical events, which include silences through which established pulsation continues and whose functions are preconditioned by established grouping, are capable of classification on the following exclusive basesof functionalidentity anddjzrentiation. In the concept ofmeter developed here, afactor ofmetric dissonance and resolution is essential; this concept isperfectly illustratedin the Mozart. It is now clearthat meter and metric here referonly toaccent-delineated grouping, andto noother type.In the unfolding discussion of meter,this essential understanding is no longerexplicit.

rhythm and meter

327

l. An impulse may initiate a metric unit cf. "downbeat"!: it is then an initiativeimpulse accent!. The initiative impulse is strong at the level of the initiated unit.

2. An impulse may conclude a metric unit: a conclusive impube.The conclusive impulse, the last in a "reactive" seriesat a given level, is weak at the level of the unit which it concludes.

3. An impulse may simply carry forward, within the unit, accentdelineated thrust motion, energy!, in a senseabsorbing the force of the initiative impulse, reacting to its accentual "energy" and predominance: a reactiveimpulse R.eactive impulses relatively passive, absorptive! are increasingly weak within the unit." 4. An impulse cf. "upbeat" or anacrusis! may direct energy toward an initiative: an anticipativeimpulse.The anticipative impulse is weak at the level of the unit initiated by the impulse it prepares, or, put another way, at the level at which it is anticipative. The meaning, or sense,of the anticipative-initiative anacrusis-thesis! relation is that the former weaker! impulse is in someway allied to, thereby

supportive of, the latter stronger!: intemporal proximity ofattack cf. 3! and;

in actual articulative aftiliation e.g.,;

e.g., tonal !,

in systemicrelation-

probably only contributive; or by established

precedent e.g., when a series beginseI ~,etc.!. Contrarily, the absence of this anacrustic-thetic! relation means that in some way the two impulses are functionally separate. The initiative impulse determines the position and occurrence of the metric unit; the nature and number of associated,weaker impulses condition its character. The presenceor absenceof anticipative impulse the question of iamb or trochee, anapest or dactyl is fundamental in determining the character of the metric unit.! Whether a particular impulse is seen as weak or strong greatlydepends on the level within which its positionis viewedits position in the structural hierarchy taken as the context of reference.Think, for example, of the opening two measuresof Brahms' third symphony: each of the two chords is metrically strong in the most local sense,yet clearly anacrustic to m. 3. Essential to the foregoing theory of four impulse functions is the idea that "impulse" at a higher level is a complexof lotverleveleven-ts. The concept of impulse thus expands to the extent that the level of referenceis broader. 24Thephysicalanalogyof an object set in motion, impelling another object, which setsin motionanother,etc.,is tempting; the diminishingforcein sucha series seemsdistinctly parallel to that perceivedin the relationsof impulseswithin the metric unit.

328

rhythm and meter

We can illustratethis very simply by referring to an anticipativeimpulse beyondthe foregroundlevel. In Ex. 3-10, the initial group of four sixteenth-notes is clearly anticipative to m. 1. At the level of the, the groupformsa low-level intramensural!metric unit of four events: initiative s !, reactive .-~!,

reactive~-"!, conclusivet !. But the entiregroupis anticipative; thus the "anticipative impulse" at the mensural level consists not of a single event but of a complex of four events. The principle goesfurther in theextendedanticipativeimpulseto thephrase-level initiative at m. 4; here the anticipative impulse is an extended complex of events consistingof metricunits at severallevels.In Ex. 3-10, symbols are varied in representation of impulse functions of such different levels. Ex. 3-10. Beethoven,Sonatain B!, Op.22, for piano,first movement.

We can seein the foregoinga modestillustrationof the ambivalences of impulsefunctionsat digeringlevelsof metric structure i.e., the multiple, or multileveled, functions of impulses!. An initiative impulse may initiate more than one unit at different levels !. Or an event may be initiative at one level, while subsumed within an anticipative group at another ~!, or initiative at one level, while subsumedwithin a conclusivegroup at another !. There are functional dualities conjunctions, elisions! of the samelevel too, as when the final impulse of a metric unit is both conclusive and an-

ticipativeat thesamelevel Q, or conclusive andinitiative at thesamelevel

~, seeEx.3-17andothers!. Conclusive andinitiative impulses arethus exactly conjoined in formal elisons,situations in which the inherently weak conclusive assumesvicariously the accentual strength of an imposed initiative again, Ex. 3-17!.

330 rhythm

and meter

Although tonal function can of coursesupport metricfunction as in the Beethoven motive!, it is in and of itself metriealb neutral:

is

not, apart from rise in the upper voice a discrete parameter!, more or less

plausible gR l J than cf e !. The I l!orcan be initiative, as when preceded by anacrustic factors or otherwise accentually superior; but it can also bemetricalbr weak, as at the end of a metric unit at phrase level see Exx.3-7 and3-8! _The allegratheme ofthe Haydn Symphony No. 102 is a useful further case in point Ex. 3-l l!. Ex. 3-11. Haydn, Symphony No. 102in BL,first movement. Allegro vivace I= 00 oo v '_" Z_` rn»z1.41u===1=111:_.|.a.;.?.;.;I.; I o1.v .Cfi'HI1l 1 l22_12!!|1__]2l!-1-1-I I 1§l._l1 I..li ' _ 5 Y * l_1 ' 11l11lI111Z111.llIl -I2--lm 92|| `°¬ iu V I 0 / ,92 9292 '|lI$_§ll11l11l1§_I.l.L..l.l.l..l_-I I l U/ .Cfiii31111111-1ll1l1'Ill-1114.1 /__1' r *~ |1i1 qv *-1

00 '1

00

»'92 _ Iii 11 .

I1

ff "

i. e., metric structurein the consequent phrase:

II *v ;Il'$.L.§_.l I .__'-!--I---uf |-'/ --1 l..92iYil-_1!-!--92|'_l`;!!-H!-_ YTI*-»_ ul 92

Z'*§..} 11.1 Y .J I or I--II-2-I--1.1 "Il-lf?-QQ! H_.l..1 I-

The Haydn phrase isa metric unit analogousto that of the level of the notated measure.The phrase begins with a strong anacrustic thrust toward the highest pitch: from that phrase-level metric initiative impulse there is consistent descent decline within the melodic parameter! in a pattern in which it seems inconceivableto interpret the cadential recession settling, resolution! as metrically strong, despite itstonal primaqy.Again, the example suggeststhat the cadence ismetrically recessiveexcept when it has the vicarious accent of elided interjection i.e., is simultaneous with initiative accent!, or is-as rarely-the actual point of superiority within its unit, and

rhythm and meter 331

except that it commonly occurs within a preconditioned scheme of intermensural units

so that it initiates

a unit at a lower level as a means of local

enforcement. [The Haydn cadence Ex. 3-ll! is initiative at the local mensural or intramensural level; more broadb,it is recessive metric in function.] Central to the concern enunciated here is the concept of downbeat, of metric accent, as initiatingthe metricunit. A further way of regarding the cadential arrival as metrically recessive is to consider that it is, after all, the culmination of a process in which element-actions of decline necessarily surpass thoseof growth; unless this balance is effected in the sumof cadential actions, thereis compositethrust, which is contradictory. This does not mean that progressive forcesdo not often attend the cadential process; indeed they do, and they often lend compensatory emphatic affirmative force to cadential resolution, or they convey an active quality in cadential expression-a sense of instability occasional even in Hnal cadences. But it means that in balance, to the extent

that thecadence stable is andconclusive, recessive tendencies prevail,and cadential action is normally strong at only locallevels ifmetric structure. Let us point to one final example in which this principle is manifest Ex. 3-12!, considering the extent to which misconception of the arrival as metrically strong would resultin grossinterpretive distortion. Ex. 3-12. Beethoven, Sonata in C minor, Op.13, forpiano, second movement. Adagio cantabile m5

uv t,r l - e 11 nil é f'92--*

/'%-Y

No principle of cadential metric function can be conceived inflexibly, and the foregoing discussion,amply illustrated in accompanying and earlier examples, canperhaps bestbe further amplified in treatment of a situation which is exceptional to the stated principle. There are times, for example, when the final event in the phrase canplausibly be interpreted as the object of anacrustic thrust on the part of every preceding event i.e., by the entire complex ofpreceding eventswithin the phrase!. Whenthat situation pertains,

332 rhythm

and meter

it is to be expected that the late phrase-level accent in an end-accented phrase! setsin motiona balancingmetric unit,often an anticipative group of parallel content in the phrase following. Consider, for illustration, the slow movement of Mendelssohns SonataNo. 2 for cello and piano Ex. 3-l3a!. Ex. 3-13a. Mendelssohn, Sonata No. 2in D, Op. 58,for celloand piano,third movement. Adagio l'¢Y1. 7 X Ulf?

"

__

| _' _ 5 171 Y.|-`:§lll..1Y1 LVQZL- in!

111 _ l"§ H4 _ Qlf I¢ -2 Q .lllnl it , i'

,I _;. " l'.'lS -11 vI

I _-I /_

,_ l'92YII_ I YUII¢f1"*2'1. _ I - -'_ e --

| _ _ 'ITD ml Ill L .l'-'1!'_T ' __ .il I ¬'='-'Q QW Qlll 2| li;_ ll _ uw 1 _.._'_=1_ -H II ¥../

sempre arpeggiando con Pedale" / m.5 [1122 Bn: " _

_

A »°92 1721 I 92 /5 Y' $~ Ki? _ 92./ ' Y¢YfIPl' 92an:;92~ 941.114 _ 92./ `

h

''

-V

1. -_nli 11

cre . - 1- Z 1275 . , 1:1 1

=' izjrfj-11:55 V1 / _ll _-s in/i Zz 1 "

H,

= _1

! :nm inA-ll 1Q..¢il 1

71

Z1 l

_

-ii .

W _92

mcg M m'

fappmszonato edanimato

. »-_

ee s __ dam.

1 li ?"C i!11'l4l 2c

# g_/

1.1 _ X' _

-I _ Q-

111

rhythm and meter 333

The theme has a guileless quality of great simplicity, and in some respects e.g., texture! it is simple indeed. In others e.g., meter! it is intriguingly problematic. That its units at phrase level are asymmetrically related is immediately apparent in their varying lengths, and it will also be noted that apparently like events havevarying placement in relation to the notated bar-line.

The basic questions towhich, in analysis, anumber of possible answers might be suggested arethese: Where is the primary initiative accent in the first phrase? Does the theme phraserecede from an initial accent? Doesit advance toward, in an extended anticipative gesture, initiative impulse at its conclusion?! And, what are the relations between metric structure in the first and thoseof succeeding phrases? An interpretation with respect to these problemsis essentialto illuminating performance. Interpretation of the phrase as end-accented is supported by less equivocal structure in the units which immediately follow, especially the third phrase, where the concluding impulse has agogic, and pitch, and dynamic accent. The entire Part I of the ternary Ex. 3-l3a! could then be viewed in analysis as given in Ex. 3-l3b, embracing a recession in metric unit sizes i.e., a broadening! from 3 to 4 to 5, the latter quantity maintained to the

end of the initial

thematic statement.

The anticipative complex can thus be regarded asa strong preconditioning factor throughout the theme, and Mendelssohn brings into increasingly sharp focus its metric function in an end-accented phrase. Its specific form in phrase 3 where there is a four-note motive repeated in powerful ascent to the high point! has especiallydetermined preconditioning effect in structure of the succeeding phrases. The accentual value of the concluding attack is of course severely compromised in phrases 4and 5; phrase 4 undergoes reductionin dynamics "Let it be emphasized again thatno interpretationof metric structure impliesgratuitous, bruteaccentuation ofattack, althoughslight underlininginfiection, orunderplaying of potential accentsnot consideredmetrically valid, might be appropriate. An underlining inflection can be in timing asmuch as,and in place of,articulativc emphasis. In the first phrase of the Mendelssohn firstof four antecedents inan enlarged period!, experimentationin varying modes ofgrouping will render immediatelyclear the fact that profound differences of structuraleffect canbe effectedby evenslight differences of intensification andtiming; if it is decided thatthe first beat ofthe secondnotated measure is not ofinitiative importanceat any level, theperformer mighthave to underplay that attack, not merelyavoid itsintensification. Onthe otherhand, constructionof that event asprimary would seemto require for its projection somekind of intensification or other deliberate underscoring. End-accent of the first phrase isalso supportedby its recurrence atm. 29, which should beconsulted. While the firstphrase unlikethe others!has arelatively staticcharacter, beginning andending onthe samenotes throughoutthe texture,its repriseat m. 29 hasthe concluding bass moving decisively, the sense of redundancyat the cadence thereby lessened in expansionof the texture-space. Atthis reprise m. 29! the final impulse isalso longerin duration.

334

rhythm and meter

Ex. 3-13b. Representation of phrase-levelmeterin the Mendelssohn asend-accented.

4 +1!

5 +S! I' 5 broadening intervals separating phrase accents!

Weaker, but conceivably preconditioned as phrase accents

and linear recession,while maintaining the inherent motivic grouping of phrase 3; and phrase 5 assumesa background linear stasis now on g'! comparable to that of phrase 1, and it undergoesdiminuendo. Moreover, its concluding impulse forms a simultaneousinitiative for the subsequentunit. End-accent at phrase level may neverthelessbe preconditioned in the final two phrases." The notion of extended anticipative grouping, and a balancing subsequent unit containing the analogousgrouping in preparation of the next initiative impulse at the samelevel, can perhaps best be seenin a notational diminution of representative phraseswithin appropriately ordered, revised bar-lines. Ex.3-13c. A renotationof the Mendelssohn first phrases.

soWhilethe analysisseesphrases2 and 3 asconditioningothers,and confirmingtendencieslogically deduciblein all, an interpretationoffluctuationin the locusof phrase-level downbeatcannotbe ruled out.

rhythm and meter 335 Criteria of accentuation

Accent is a theoretical term denoting the relative projective, qualitative strength of a given impulse ascompared with others which precede andfollow it and, with it, form a metric unit at a given level. There are of course times in music when accentual impulses are of relatively unequivocal effect and function; no one, presumably, would dispute or fail to appreciate the mensural metric structure in Ex. 3-14.

Ex. 3-14. Haydn, Sonata in D for piano,third movementBreitkopf andHarte! No.37!. Presto, ma non troppo

-» 4

§'§[.li92=92Qa _ Yll|fi|-I I |l..]l1-11 LV' ' 92_.f. 7 YQYTI __ Yf1fI92f1"i1f- -;J!l_ 1 IQll ~ --

A' ' -I '_ 1I

I 2?

/ ` -l _ | _ I2 2

_ 121'-2!'!l'!

._

' 92

I

-._

1T

,

The mensural units in the Haydn phrase in conformity with the notated bar-line! are clearly delineated by relative accentual superiorities at intervals of

2.

: accent here is expressed in anacrustic thrust, pitch, and

duration. More subtle factors of texture, and such complementary factors as harmonic rhythm, could also be noted.! But music is by no means universally so explicit even at mensural levels, and surely not at higher levels of structure. In fact, the Haydn theme requires some interpretive thought as to the question of metric functions at the level of the phrase: it is not clear, for example, that the initial downbeat al is metrically initiative for the phrase-level unit. Is it, rather, part of an anticipative complex underscoring superior accentual value of a subsequent impulse? The question concerns metric structure internal to the phrase-not formal delineation of the phrase by cadential punctuation at four-measure intervals in another, often counteractive, and in the Haydn probably simpler, mode of grouping. In some degree, metric structure emerges in a purely neutral interpretation in performance-i.e., accents which are the result of such factors assuperiority of pitch and duration will, presumably, be felt in some degree in their central referential functions, and there are many instances in which metric grouping, whether in accord with notated bar-lines or not,

336 rhythm

and meter

derives from inherent accentuation of such clarity and decisiveness thata neutral approach is clearly in order. When it seems necessary to bring out the metric structure in instances deemedof relative ambiguity!, it is done in a variety of discreet ways: by slight intensification of accentual attack, usually no more than a kind of gentle urgency in projection; by the slightest hurrying of the anticipative impulse toward the subsequent accent; by understating the conclusive impulse, which may be a low-level accent, creating a perspective inexposure ofanother event; by nearly imperceptible adjustments in timing of attack; and by comparable means whichhave been implied but which, unfortunately, cannot be explored fully. No meter at any level calls for crude, overt accentuation, unlessin parody; at no level is the analysis of metric structure to be understood assuggesting gratuitous intensification of any attack. Although intramensural groupings of particular importance e.g., asymmetries of necessary vitalizing function! at times require awarenesswithin proper restraint, subtle performed intervention is more likely to be indicated at broader levelsof structure. We are obliged to put aside any extensiveinvestigation of these problems, but it is important that we recognize their ultimate significance and something of their nature, difiiculty, and scope. The above comments and questionsare indicative simply of the kinds of problems encountered in performance, the kinds of decisions andjudgments a performer must make; he must also decide, of course, the extent to which explicitness of metric structure is stylistically appropriate compare, for example, Wagner and Debussy; or Stravinsky and Webern!. If we accept the Cooper-Meyer definition of accent as a stimulus . . . which is marked forconsciousnesssome in way3°-and it is hard to imagine a more suitable one-we must then endeavor to understand the means by

which, in music, certainstimuli are so markedor are perceived asemphasized assubjects ofaccentuation. It is true that relatively little can beunequivocally known about accent, and that few studieshave approached the problem of accent perception in Cooper and Meyer, TheRhythmic Structure qf Music,p. 8. The authorsdo not go far in developingthe issue of criteriaof accent,stating p.7! that . _ . sinceaccent appears to be a productof a number ofvariables whose interaction isnot preciselyknown, it must forour purposes remain a basic,axiomatic conceptwhich is understandable as an experiencebut undefined interms ofcauses. How much isprecisely known about interactionsof any musical events?! Peter Westergaard notes thatCooper andMeyer aresubject tothe relatedconcern that in their analysesanalytic choice at lower levels isevidently largely determined by metric position even thoughoriginally meterwas definedas beingproduced byaccent!, at higher levels intuitively, bythe feel of theharmony. [Some Problems inRhythmic Theory and Analysis, in Perspectives qf NewMusic, I, l Fall 1962!, p.183, fn.l4.]

rhythm and meter 337

laboratory conditions. Those that exist dealwith the problem of accent and grouping in only relatively primitive contexts i.e., in circumstances concerned with rhythmic grouping as conditioned by few parameters of accent usually operating singly!. Nor can we with present understanding hope to deal objectively with the question of the influence on thesefactors of various degrees of preconditioning-the preconditioning structure of the musical work as a whole, or the preconditioning understanding most percipients have in some degree of the particular musical rhetoric and its norms. 32 Accent in real music usually involves many element-actionsoperating together; and while such actions cannotbe objectivelymeasured andcomeasured, there is no way out of the need to evaluate their cqfunclioning ifmetric "See, for example, JamesL. Mursell, The Psychology of Music New York: W. W. Norton and Co., Inc., 1937!, Chaps.IV, V. Mursells bibliography on the psychology of rhythmic experience is auseful source of reference, referring backas faras, andin significant degree to,the classicexperiments ofHerbert W. Woodrow: for example, A Quantitative Study of Rhythm, in Archives ofPsychology, No. 14 909!, and The Role of Pitch in Rhythm, in Pqchological Review, XVIII, 6 911!, pp. 54-77. Woodrows studiesconcern groupingas perceivedin seriesof stimuli variously differentiated induration, intensity,pitch, andinterval ofspacing. Inexploration ofdurational and intensityaccents, heconcluded thata subjectinterprets theaccent ofintensity loudness! as group-beginning and the accentof durationas group-ending, while heconcluded that pitch differentiationproduces neithera group-endingnor agroup-beginning effect. The Role of Pitch in Rhythm, p. 77.! Woodrows findings arefundamentally qualifiedand disputed inlater research,all of which appearsto becomparably farremoved fromthe complexity of real musicalconditions. Within the conceptspresented inthe presentstudy, of course, thequestion ofaccent isnot oneof group-beginningor group-ending,since accent is defined as metrically initiative; rather, theissue isone ofexistence of anticipative impulseor lack

of it,in the pattern l f*'! or &!| l &, every metric group, by definition, beginning withaccent. Theissue ofiamb ortrochee is,then, oneof thefunctional affiliation by

temporal proximity, by preconditioning, by articulative relating, etc.!of animpulse toa

subsequent, stronger ode, as in J I J

IJ

I J . Or,the trochee differs from the

iamb in that its weaker component is somehowfunctionally discretein relation to the following accent. It shouldbe stated,too, thatstudies ofpsychoperception of grouping seem often to be impaired by a failure to accountfor the perhaps potentiallydecisive! mannerin which a series ofdifferentiated stimulibegins. Sources ofrelevant bibliography,in addition to Mursells, are: Woodrowsreference in the first of the abovelistings, Chap. I! to still earlier experiments; andM. Schoen, Bibliography of Experimental Studiesin the Psychology ofMusic, in Proceedingsthe of Music Teachers National Association, Theodore M. Finney, ed.,December 1940, pp. 524-27. 32A toneor groupof tonesmay appearto beaccented, notbecause of any particular distinction whichit possesses per se,but becausea previouslyestablished grouping tends to perpetuate itself,making thistype of organization thesimplest. [Leonard Meyer, Emotion and Meaning in Music© 1956by TheUniversity ofChicago Press!, p. 104;all rightsreserved.] The notion of preconditioningstructure, whichhas itsbasis inGestalt concepts of psychological perception,arises laterin the present study;Gestalt approaches are alsothe basisfor the ideathat ultimately indivisible metricgroups areof only two orders-those of two, and those ofthree pulses.

338 rhythm

and meter

anajlysis isto be carried out. No penetrating approach to the studyof rhythmi¢ structure

is conceivable without insistent

efforts in

the direction

of a

wmuprehensive statementof criteria by which accent is evaluated. As often in seeking to understand the musical experience, onestrives in metric analysis to achieve plausible, reasoned hypothesison the basis of demonstrable features of the experiencein question where empirical verification is unlikely gr attainable in only limited degrees. In approaching such a statement of accentual criteria, it is important to recall the concept of accent initiative impulse! as initiating the metricunit at some level, the energy of the accent having a kind of radial impact of diminishing _!%rce through the reactive and conclusive impulses which followwithin the mgrfic unit.

The effort to understand accentual criteria in the experience ofmusic is, apart from the problem of incommensurability, the effort to enumerate

famvrs whichappear tocontribute to and condition the perception of groupingby aa-entuation certain of impulses as metrically initiative. It

is of course imperative,

gspgcially inthe listingof abstractcriteria, to remember thatthe necessity gf their contextual evaluation in real musical situations is ultimately decisive

indeed, criteriacan onlybe seenor expressed as tocontextual relations!. We must point ultimately too to the necessity of evaluation of accentual

impulse inrelation to a given structural level tj rehrence. The necessaryconcern with contextual relativity of accent means, of CQUISC, thatany change within any parameter of musical events can be

significant withrespect toaccent-delineated grouping in an atmosphere of rglative parity of events;hence, thelist of potential accentual criteria is long and comprehensive." At the same time, the list excludes mentionof factors

S¢¢ pp.320-22! expressive of other modes ofgrouping suchnonmetric or gxtramusicalfactors, notedearlier, as text, cadentially punctuated phrase, ¢¢¢_!; thesemay be mitigating or counteractive at times in concurrent

cxpressions of grouping notstrictly metric in nature. Nonmetric factors of grouping e.g., motivic resumption!, whenconformant in relation to mgfric grouping, may give the impression of _hrtyfyingthe accentual initiative

impglse and intensifying itsquality of accent. The following list of accentual criteria is punctuated by brief examples which are contrived for the purpose of focus on individual parameters in artificially simple conditions. Within the above twoessential concerns, context is normally viewedas thatof the Worx inquestion, butit may also beof styleor of genre, etc. It can be arguedthat everychange withinany parameterof structureis metrically initiative i.e., accentual! atsome level, although relatively minor changes i.e., relatively nonxccentual events! can affectmeter atonly verylow levelsof structure.

rhythm and meter 33.9

I. Element-changes toward accentualb' superior valuesconstitute aprimary class of criteria of accent.

l. Changeto fastertempo evident, : in inumerable examples,in the broad formal pattern with slow introduction; or a tempofollowing ritardando, or relatively low-level piu mosso,or a particular tempo event as the object of accelerative drive, may have metrically initiative effect Ex. 3-15a!. Less commonly, changeto slowertempo may have accentualeffect; seeEx. l-3 7!. Ex. 3-1Sa. . piu

in--' §"'

-"- l

rit. _ _ _ ,atempo ls

2. Pronounced change ejpitch : high pitch especially is of accentual effect, but the exposure of an event-distinctly high or low in pitch-by reason of pitch, in a context of lesser change in pitch-line, must be seen as potentially accentual; superiority of pitch denotes the usual accentual factor of higher frequency, but pitch exposure is an accentual value involving uncommon projection by reason ofpitch, high or low Ex. 3-l5b!. EX.3-15b. 1.

0 35 i

Q|

l`

I

3. Related to the above, approach by leapsin lines, especially leapsupward: compare thc factor of anacrustic approach, most often in a rising inflection; if a leap has accentuatingeffect, it accents theevent following the leap, or contributes to that accent Ex. 3-l5c!. Ex. 3-1Bc. | I

t,, I,

4. Longer duration agogicaccent! .°relatively long impulses often have initiating effect Ex. 3-l5d!; note, again,that the longer impulse in the iam-

340

rhythm and meter

bic foot is regarded as initiative of the metric unit, which contains the impulse anticipative to the next agogic accent." Ex. 3-16d.

5. Articulativestress:e.g., the goal of crescendo, or any relatively loud attack, may function initiatively with respectto metric structure Ex. 3-15e!. Ex. 3-16e.

6. Changeto moreintensetimbre: i.e., changes in orchestration from relatively neutral timbre e.g., clarinet in medium register at moderate dynamic level! to more intense, penetrating timbre e.g., high, muted trumpet, even moderately loud; Ex. 3-15f!. Ex. 3-16f.

' Ob. s Smitherdiscusses relativeagogicvaluesasto particular kindsof contextualassociations, treating "absolute"and "relative" agogicvaluesand implicationsof the positionof a note precededand followedby shorteror longernotes.The perfectlyjustified qualifications in Smither'sappraisalof variouskinds of contextualpositionand relation for agogicaccent indicateagainthe complexityof the problemof analysisof metric structure,evenwithin the concernsof a singleparameter hereof duration!. The readermight well referto Smither'sdiscussionof accentualcriteria asa supplement to in somewaysan extensionof, and in someconceptuallyopposedto! that undertaken here. " The Rhythmic Analysisof 20th-CenturyMusic," in Journalof Music Theory, VIII, pp. 61-69.! re In somecircumstances,e.g.

, durationof an event

beyond its immediate locusat a broaderlevel! may haveaccentualsignificance;but this factor is of unlikely relevanceasfelt accenti.e.,asa factorin meter!whereelaborativeprolongationis over very broadspansof time. Comparelinear functional associationasa nonmetricmode of grouping,notedon pp. 32021.!

rhythm and meter

341

7. Changeto denseror otherwisenoreintensetexture;thus, accompanying voices or other voices
Ex.3-1$g. ??

8. Tonal or harmonicchangeof unusualdegreeor distance:e.g., chromatic progression Ex. 3-15h! may impart accentual significance to the second event in such succession.

Ex. 3-16h.

9. Dissonance in a given harmonic event Ex. 3-15i! contributes, by virtue of intensity and instability, to its accentual value."

+Smither op.cit., p. 66! groupsaccentsof timbre andtexturewith "dynamicaccents." s~Specific tonal-harmonicor scale-degree! contentin a harmoniceventhasbeennoted as,apart from suchqualitiesasimplicit dissonance intensity,a neutralfactor with respectto accent.Evenwheredissonanceis involved,its potential for accentualsignificanceis readily submissiveto more dominant factors: for example,in the following instancesupper voice risein pitch, a verymodestfactor, seemsclearlyof preemptingsignificance:

seems"naturally" strong-weak,while

seemsweak-strong,both implicationssubjectof courseto counteractingor preconditioning opposingforces.

342 rhythm

and meter

Ex. 3-1Bi. -O !

II. Certain associations of impulse functions constitute a distinguishable class of accentual factors.

1. A metric initiative is signilicantly enforced in its accentual properties by a precedinganticzpative anacrustic! impulse orcomplex, and by any ornamentation analogousto that of anacrusis Ex. 3-l5j!. Ex. 3-1Bj.

* IL. §§

31

e' ,_ "l' 92-1/

'

'°

2. Temporallyvery proximate events following the initiative,and elaborating it, as in certain kinds of ornamentation, contribute to accentual effect Ex. 3-15k ;elf the possible durational impact of repetition and elaboration of a

pitch event near enough its immediate level tobe perceived as afactor inits agogic value!. Ex.3-15k.

.2 i- '

¢

2 I 3" I l

'

III. Accentual factors of particular conjecture, but of very probable significance, must be accounted for in any comprehensivelisting. 1. One might suppose that, other factors being equal, primacy of accentual valueaccrues tothe jirst in aseries of contiguous eventsEx. 3-l5m! ; thus, the first in a seriesof identical units e.g., impulses at whatever levelsingle attacks,phrases, etc.!is in this sensesuperior, and subsequent equal stimuli may thus be perceived as increasingly recessive redundant!. For example, it is unlikely that a percipient would sense ananacrustic or iambic!

pattern two in equal stimuli divorced from any larger context, bgt he rgight well perceive the two stimuli trochaic. as the Or perception of V V V V where the two accentsare equal! as a group of four would derive from the sense of the first accent as primary. Similarly, the second in a pair of identical phrasesappears weaker,subordinate in accentual value,because it is second.!

rhythm and meter 343 Ex. 3-15m. I

: Il I I u

II

'I

2. The relative proximity qf unitsmay be a factor; for example, accentual

force maybe mitigatedin its effect bythe sheerproximity of a preceding unit by which animpulse isabsorbed in some degree-aquestion related to that of the delineation andperception oflevels ofmetric structureEx. 3-l5n!. Ex. 3-1Bn. in

:li "_

ti-rf-» I I

3. Any quality in an event which is in a particular individual or stylistic! context unexpected-the sudden intervention of an event in a context in which it is not likely-may have accentualsignificance; that is, accentual value presumably accrues to an event which intrudes upon the normal or establishedcourse ofevents Ex. 3-150!. EX.3°160» I 'Q

I I

aa,/9292 =

me ; |0

t °7

I

|f|;-¢

I I

4. An exceedingly diflicult but necessary point, one to have further

344 rhythm

and meter

understanding whenand whether a given unit is, except in the most blatant cases, preconditionedand/or preconditioning. 5. Finally, whatever the apparent accentual qualities of an event, and however strong and persuasive they appear to be, it would seem that the position ofan event as theobject rj'accelerative progression is a strengthening factor while the position of an event as object of a decelerative recessionmitigates against accentual value. In the illustrated contrived successionEx. 3-l5p! the loud, long, and relatively high final event is clearlyfelt ascompromised in accentual force by the preceding metric recession, i.e., succession ofunits of increasing size, a broadening and thus decelerative tendency.

cf. if

No doubt the list of factorscontributing, whether decisively or modestly, to accentual inflection in musical successions couldbe extended and qualified in various ways; such a list of premises isin any case a necessary basis for the analysis¢y" metric structure, whetheraccent is in a given context considered to delineate meter or to function counteractively in relation to somehow otherwise established metric structure.

The problem in the analysis of meter i.e, the location of real barlines at various levels! is of course not merely one of the identification of accents. Rather,it is an issueof evaluation of usually many coincident factors by which contiguous and concurrent eventsare differentiated. The analysis of accentuation must weigh, therefore, and not merely enumerate the various factors as objectively as possible in the terms of context. It is unlikely that any accentual factor, even of stress or duration, is necessarily of primary effect, invulnerable to the preemptive or preconditioning! action of another. A summary, very brief chart of some of the broad, general characteristics relative to accentual effect might be usefulat this point, but it must not be read as in itself a sufficient statement of criteria see Fig. 3-3!.

rhythm and meter 345 Fig. 3-3. A briefsynopsis ofsome qualities of accent. As opposed to Strong Weak Intense Relaxed Unresolved Resolved Active Reactive High Low Loud Soft Long Short Unique, unexpected Common Object of acceleration Object of deceleration First Second, third Preconditioned Uninfluenced by prior events Anacrustically underscored Unsupported by associated impulses ornamented! Exposed Assimilated Dense Sparse

Further comment concern/hg d/'verseimpu/se functions The anticipativeimpulse isthe clearest, most unequivocal of all rhythmicmetric functions-weak, tentative, subservient,a ubiquitous device of thrust toward and support of the metric initiative! The anticipative function extends of course to all levels, so that ultimately an entire form may be seen tohave broad anticipative-initiative relations: e.g., the introduction and exposition taken as wholes! in many works, or the first part of binary design in relation to the second, mayexist in this relation very broadly seen. It seems likely, in fact, that the broadest metric shape of many most ?! musical structuresis as summarily and simplistically! represented in Fig. 3-4. For analysisof anticipative, initiative, reactive, and conclusiveimpulse functions and determining factors, we take as subject a single phrase from Corelli Ex. 3-l6a!, the bass of a portion of the movement quoted in Ex. 2-10, pp. 2l4-15, noting the accentual criteria which seem applicable. The first impulse could be seenas initiative for the entire unit: highest pitch Q' the essential line, beginninga descendingessential recession from tonic to tonic; contrasting interpretations of the phrase are discussed presently. As to initiative impulses at lower levels, andreferring to the numberedevents, it can be observedthat ! is strong, entering after rests seem. ll of Ex.

rhythm and meter 347 3

unit of F which includes The !. first five impulses form subunits of 2+ 3

-|- 3.! Impulseemerges ! a3 further assubsidiary ingiative ¢y"plau.sible high level function it initiates a final ,V subunit ofthe first V by ! virtue of duration and strong approach by dissonant leap; ! seems distinctly anticipative in its durational inferiority and position prior to leap. Impulse' ! is comparable to !; and the preconditioning effect of the 3-4-5 grouping is a factor in support of the anticipative function of !. Impulse 0! is the longest note of the phrase; further, it is supported by a strongly tending anticipative impulse 9!; 2! is conditioned as a mensural initiative by the preconditioning precedenceof the 9-10 motive, as well as by duration and

pitch superiority within final theof unit .E

Proportions in the resultant metric structure might be represented as in Ex. 3-l6b.

Ex. 3-16b. An interpretationof metricproportions inthe Corelliphrase.

ZF [s

+ §-4 , "I I

1;

+ g-9

EE i i 4* :-|.|`

I

F [2+ 3+ 3]

,

]

4 -I '|_'|~ 4 I

'il' +4`!

g F ;t![ 4 + 5 ]

|I |l

*Accent-to-accent grouping. |'Segments including anticipative impulses.

Asymmetrical featuresare of great importance in expressive,activating effect; but unifying symmetriesand balancesshould benoted too.Thus, there are severalmotivically related units of 4 + 3, in which the first impulse is anticipative! and the outerunits ty' 8 and .9 from which the internal 4 unit has significant registral separation! are of important balancing $0 and close interrelation.

Moreover, another

way of

regarding the

asymmetry-symmetry

characteristic? to consider is the regularly spaced mensural-level accents followingVaantioqzative complexinterpretation an discussed below! as yielding the pattern in Ex. 3-l6c. The question of chief accent structural downbeat! at the level of the phrase should be carried further. In connection with this question,

348 rhythm

and meter

Ex. 3-1ee. 5+4+4+4+!

note the two essential streams of pitch succession f-F ; and the chromatic

succession A-Bb-Bll-c! as well as usual criteria of accent. 5 An essential question for performance follows: Is the first group of anticipative in its functional relation to the first skip accent, A? In considering a question of this sort it is sometimes usefulto speculateon relative effects of hypothetical varied transformations Ex. 3-l6d! for the light that can be shed on functional relations

and actual

creative decisions.

Ex. 3-16d. !* !

QJ ll

11

! poco

§E°*r@J §'¬rrb@J 22 22

acceL ? !g 2I

*Results intotal phrasesymmetry.

The expressive value of Corellis solution is evident in its less predictable, less symmetrical result as compared with No. 1! and in the intensifying 2-2-l, low-level metricprogression withiiz this anticipativegroup as compared with the relative rigidity of all three alternate writings!. In testimony to the range ofconclusions towhich evidenceof accentual criteria and impulse functionsmay lead, Ex. 3-l6e provides asketch pointing out the two different, reasonable metricinterpretations implied in the above discussion, thesecond ofthem construing the first five impulsesas an anticipative complex at the phrase level. In the second construction, the halfnote A, start of the chromatic stratum in the compound line, is interpreted as phrase-levelinitiative accent, its accentual qualities those of departure from diatonic precedents, dissonance,leap the first!, and agogic value in its immediate context. No doubt other interpretations could be argued. Consequences forcorrelating tendenciesin performance e.g., treatment of the first note as up or down ?! are crucial indeed.

rhythm and meter 349 Ex. 3-1Ge. Two interpretations of metric structure in theCorelli phrase.

2: /I

l ¢--/

1:1 "" [--- --

"'92;

°O

---- essentialf-Fmotlon---°--"

~f

'

el e 1-

°°

.

- - chromatic motionof - - - - - --' lower stratum

I

Levels of metric structure and 308/YSIZS The concept of levels of structure has been treated at every stage in this book, and it is of great importance in the consideration of meter. As we proceed, someterminological clarifications should be recalled, and othersestablished. Metricunit canrefer to the grouping at any level, as indicated in a given context of reference. Thepulse isthe felt unit of counting as identified!; it is a matter of level-i.e., it changes in relation to the level of structure to which reference is made. Pulses are a psychological frame of reference-undifferentiated, and generally of regular, evenlyspaced occurrence except in anomalous situations or at higher levels of structure. The beat is the pulse of the mensural level-the denominator of the meter 26

signature admitting aswell as D incompound meters!, hence specific as to level ; it can change, to be sure, in the course of a piece, but when that is the case a common denominator relating the two beat values can often be identified. The impulse isthe event itself attack, or integral silence; stimulus!; it is the factor subject, by differentiation and preconditioning, to grouping and weak-strong associations. Theimpulse at higher levels is an entire formal unit

two or three of which constitute

the level of reference:

these range,in theory, from the phrase to the largest formal divisions. In determining levels of structure in meter, vital and necessaryanalytical judgment concerns theeomparabiliq qfunits, impulses, and accents;no objective, absolute specification of structural level is possible. To identify a higher level of metric structure is to identify in analysis l! a stronger,more fundamental and far-reaching,aecentual impulse and ! the aeeentual initiative rj a largerunit in which subordinate accents preceding and following are subsumed as anticipative and reactive!. Leveled metric structure thus has, like other element-structures,implications both of breadth

350

rhythm and meter

as to the temporal span over which a given primary accent prevails! and of depth in the penetration of a primary event's metric function through more foreground levels!. A higher-level accent conditions and delineates a broaderunit, and it is initiative as well for all lower levels throughout the structural depth.

The principle that a pulse seriesat a higher level is roughly analogous to the accent seriesof a lower level is useful in construction of the concept of metric levels.

It is theoretically and experientially valid to refer to an ultimate primaryaccentprimary initiative, structural downbeat! for an entire structure: it is thepoint to whichenergyis directedandfrom whichit recedes in lines punctuated by lower-level metric initiatives. See Fig. 3-4.! Figure 3-5 is a set of symbol distinctions representing impulse functions and bar-lines! applicable to graphic representationof metric functions and structure at various levels. There is nothing hard and fast about these, and any ordered selection can be applied to analytical representation in a given example. Examples differ, and analytical ranges of inquiry differ, in the number of levels relevant or selectedfor reference.But such symbolic distinctions are of great usefulnessin the analysisof meter, as demonstrated in examplesin this chapter. Fig. 3-6. Symbolicmodesof representation of metricfunctionsandstructureat hierarchically relatedlevels. Accent

Bar-line

etc.

rhythm and meter 351

The question of a terminology of reference toparticular levels of metric structure as to levels in other element-structures! alsoarises. Conceptsof pulse, or principal subunits of reference, are necessarily subjective and especially diflicult beyond low levels. We could in theory refer to a period or other phrase complex as metric at the level of the phrase, a measure at the level of the beat, or a phrase at the levelofthe measure,or mensural unit ; but this mode of reference isextremely problematic at higher levels, whose units are divisible in various ways at various subordinatelevels. On the other hand, 'once thescope ofthe unit at the level of reference itself is identified, it becomes possibleto refer to that level Qyr¢:rence tothat unit. This method of terminology yields such referencesas these,where the term following of is the broadest formal unit defining the given level frame! of reference-the most comprehensive metricunit of reference: level of the phrase, of the notated bar, of three notated bars, of one-half notated bar, of two half-notes,of Part I , of the period, of the motive, of the sequence pattern, of the ostinato, of the talea, of three eighth-notes, of the hemiola, of the whole, and the like.

The Haydn theme usedby Brahms for variations will serve as a useful, parenthetical subject of reference Ex. 3-l7a!. In the theme, the question of phrase-level initiative is not at all unequivocal, i.e., not immune to alternate interpretations. Ex. 3-173. Brahms, Variations, Op. 56A,on aTheme ofHaydn; theme. I I ' -~-~»

lair? Two interpretations are suggestedin Ex. 3-l7a: one of them regards the first impulse, d, as initiative for the phrase; the other suggests the possibility of anticipative function for the initial group, in approach to the higher, longer eb as phrase-level accent.The analysis which follows adopts the first of these interpretations, recognizing the essential structural importance of the d and its validating immediate recurrences in prolonga¢ -0--__ tion :

_ `¢

In the following representation of metric levels intramensural, mensural, intermensural or hypermetric! the lowest are of course, while

rhythm end meter

352

theoretically demonstrable and analogous in weak-strong associationseven

internal tothe/, ofnorealexperiential orpractical significance; theyare representedfor comprehensivenessEx. 3-17b!. Meter at the level of the phrase is revealed as a magnification amplification! of a five-unit mensural structure, and it is viewed as parallel in weak-strong associations. The renotation in diminution facilitates the apprehension of this idea, and is a recurrent device in this chapter. The question of the recessivefunction of cadencearises here, and of the weak-strong relations of impulses at broad levelsa question pursued further as discussioncontinues.But let it be noted that the consequentphrase in the initial period represented as one "measure" in the renotation! is Ex. 3-I lb.

Renotationof the Haydn-Brahms theme,with impulsefunctionsat variouslevels symbolicallyrepresented.

S omdp

looel ofwhole!

rI

level ofphrase! level of2 notated bars!

level of l notated bar! level of$ notated bar!

etc.

L e~

rhythm and meter 353

recessive tonaliy and melodicalb in relation to the antecedent, as is the final

phrase, although the latterhas anelided interjection of renewed phrase-level initiative lending it vicarious accent. A chamber work of Brahms Ex. 3-18a! provides the basis for some

extremely absorbing questions ofmetric structureand process. Ex. 3-1Ba. Brahms,Trio inC, Op.87, forpiano, violin,and cello,first movement. Allegro

m.343 /_#F/~92

,...,,,,,.,,,," 'Ei gif

I

/""'*92

l"

"V Ffej

,.,,..,,..,,., e

r`|

¬iE E:E::;E=::= i§iEiEEE5EE

$3/o_o 4 ali; m.3-1;/_9292J/92'_ EE V/-9 Y; _T Z"".-_-__ _92

="f' ei

'

li dr

-LJ

If /92

.:__==_==-_..i.-__ -__-_.ig fa get

@~ i V#

il ,

-;._!=_==A -g

a

e 4 is ;.../

,»?

Ex. 3-18a continued. m.349

rhythm and meter 355

In Ex. 3-l8a, intermensural units are ofvery clearly delineated both in metric ordering and in phraseology. Against this preconditioning, irregular agogic and registral prominences ofviolin and cello betweenmm. 351 and 358 might well be felt as phrase-level syncopations;but the piano at these points maintains, in the bass, the preceding intermensural shape 4

with strong accentthe at start of the unit.!A representation of metric structure in the entire cited excerpt is given in Ex. 3-l8b. Ex. 3-18b.

l 2§

4_

w@f~ 4 r"""

f~ -~» 2A

*'i_92 ri*

L

2g

2E Ti?

~» l cf.§

4 I , fl

f" i

n.! !

_I

+1 l

Il

f

a

2

II

sw

lf

. Al

i

L

3f

;

V gn; Ill. 343 344 345946 947 848 349Sw 35|

852 353 354 355 356 357 358 359

I *Violin-cello phrase-levelmeter very explicit againstpianos basssyncopations. TPiano bass reiterates phrase accents against upper voicesyncopations, but with First

and third impulses equal-an accelerative tendency in more frequent equal accents. IConclusive impulsenow renderedpowerfiilly anticipative at phraselevel. §Second unit at halllphraselevel accentuallyequal tofirst in piano bass!or strongerin violin, cello! at mm. 351-54 ascompared withmm. 343-46.

Example 3-l9a_ is one whose metric structure involves important questions of level beyond the mensural. In the present example there is one has only to conduct the first 4.

..

.

Eve measuresat o to senseit! a strong intermensural unity with acceleration in rhythmic motion! in the first phrase; a second intermensural initiative begins the theme of m. 5. This initial four-! regularity becomes the

rhythm and meter

358

Ex. 3-19a. Mozart,Concertoin D, K. 218,for violin andorchestra,first movement. Allegro 2 Obs. 2 Hns. inD

SoloVln.

Vln.l Vln.2 Via. Vc.,Db.

Ob. Hn.

Vln.l Vln.2 Via. Vc.,D

357

rhythm and meter

m.9 _

_°

ob. iiil92;' I _ .:

_1 Q

-

"nn 'il'

Q_

Y] 0 J' .4

v. 92 Q"l`i92-£531 92 mI_¥l ml I..-Q1 V92l.: U 1- 1 _ ___

V :=:

2_5 '

° /92 _§==gi===l.!l.1 l.;1.J

LI--I ll-.1

I-H H ll

» ° _ °1 1

1

'1= i 1 'l=ilr1|i'i.§

'Q F _ -

2_

I'

_'

»

ll-if-i~.

Ll l.L11.1

VC-.DM

..

nl ___

-7

'1='|5"1'°'-`§|_§-='.= P!

_

// ob. -

1-

Hn.

1-

I0QI

'

mu I? 92 cnsc.

cnsc. _ 1 ___ f.

"92 ° v.=..m.. _ 1 V

__

fP

V, 4 :k.1==-.~ ._.:.., . .'.

VI; E

. Oj .

f_

. =?.=E";- E.?;.. =?.=E:- ?.=?;.':"

___ ___

.. 9 1 _yill j

f

___ _J

._ .0

1

°_

. - - Q !- _ '*

___ .J _. '

1.P

P

'_

358 rhythm

and meter

Ex. 3-19a continued. m. I5

-

r"r F

0.

Fr F

fn n

Hn. '

-r

r

W"-11' i5= _ ; " *=§§§§i5' 3l§£?5iFE§ .l " " __ _`

v»»»2l §§§f§$ ___ _ § s s§§z¢=5a:§;§g;=ss:E '_ U vu... E

1 ,Q

v¢.,m.. E

§;.5E;'i.-=§;§

f_

H_

Q 1 lisgii-__i'_i§=g

/

1... . Ob.

'

Hn.

_'

Vln.ll

W I_ P

ig - *-==:======='§ =

_ mf]

Www » . Pj Vc.,Db.-° -

° ,.

_

1.% f°~ IE f

basis for provocative divergences: the dynamic-textureaccent of m. 6, like itscounterpart twobars later,heard assyncopation withinthe established four-measureunit. The initiative accent with which the phrase begins ameasure before is, byvirtue ofhigher pitch,anacrusis, and striking exposure asa piano,therefore contrasting! thematic beginning,a strong perspective against which thesyncopation ofm. 6 is felt, quite apartfrom the question of preconditioning in mm. 1-4. "I t is interesting to note,in looking ahead, that the subsequent theme of m. 19it has four-measure intermensural units! features an analogous, syncopated, dynamic-texture accent

rhythm and meter 359

Measure 8 poses a more complex question. The extreme thrust of pitch accent, in tutti, at the middle of m. 8 is compelling as anew mensural and intermensural! initiative not only because of the sheer strength of

accent and consequent disruption of 3 phrase meter, but because of the strong senseof resumption conveyed by introduction of a new motive, a motive which occupies one measure crossing the notated bar-line. This theme unfolds in consistency with its altered intermensural placement: note the corresponding initiative at the half-phrase level at the middle of m. 10, and the recurrence of the irregular m. 8 initiative at its counterpart in m. 12. An asymmetrical horizontal noncongruity of I

Qi. is

so far established.

The restoration of conformity with the notated bar-line a recession! is of equal interest. Compare, first, the beginning of m. ll with its counterpart in the repetition, at m. 15. Measure ll is recessive inrelation to the accent at the middle of m. 10; but the variation at m. 15 beginsa process of reassertion of the notated bar-line by aooentingby upward leap and decisive pitch superiority! the corresponding motive at that measures jirstbeat, leading into a recessive-conclusiveintermensural impulse at m. 16. The

metric structure of the irregularly placed theme from m. 8, third !Jthus turns out to be f;f§`6&§'_2 The. notated bar-linepowerfully is reconiirmed at

m. l7H`.

The entire passage isoutlined in Ex. 3-l9b as to primary events in the phrase-level meter. One is inclined to comment again on sometimes relatively obscure! asymmetries in Classical works, and on techniques of syncopation, and of metric dissonance and resolution, observed at early stages inmany movementsf A problem. incapable of conclusive objective resolution is the weakstrong relation between antecedent and consequent factors in periodic relations and extensions ofthese-e.g., binary, ternary, etc.! having tonal resolution in the second part. We have argued that tonal resolution is, in itself, a recessive processwhile accentual impact is rather associated with tonal distance and dissonance,and that the factor of specific tonal-melodicharmonic content is relatively indecisive and submissive asto accent. Of references cited up to this point, Exx. 3-8, 3-ll, 3-12, and 3-17 are clearly of strong-weak - U ! structure evident in the weaker at thepoint of its secondmeasure second o pulseof thephrase!. Ineach case the anomalous accent isin a context ofdecline in pitch--a factor which contributesto its perception as syncopation inrelation tothe phrase-level initiative of the barpreceding. Syncopation as to preconditioning structure is discussed on pp. 372-76.

360 rhythm

and meter

Ex. 3-19b. A representation of phrase-level fluctuations inthe Mozart. Phrase downbeats at middle of bar, creating symmetrical level units Phrase downbeat lestued to conformity with notsted bl~ line: strongly assertedlr:in sgogic value and mscrusls

i "t..-T"I

I 325746 IS 2l5I7I4 I6IB New th¢m¢s

conformant initiative _ I9 20

II

II

II

P .f

II

II

II

I-- Phrase-level syncopation: texture-dynamic accent cf. pitch accent m.5 of and preconditioning of mm.l4!

II

II

II

2|

II

I Alterationmotive of to establish pitch accent at notatcd bar-line cf. m. l 1!

lntermensunl

syncopation cf.m.6! properties of the consequentimpulse; Exx. 3-6 and 3-13 appear to project the opposite relation. The issue of weak-strong association internal to higher-level hypermetric units in tonal contexts is thus dependent on accentual intensity, activity! factors largely extrinsicto thatof specifictonal content. Even in so conventional a procedure as massively amplified V-I, introduction-allegra high-level anacrustic-thetic impulses!, the universal downbeat character of the consequent part is a product of accentual conventions beyond the tonal content as such:stress, tempo!!, texture, coloration, etc., rather than the necessarily resolving, releasing action oftonal movement toward I.° The concept of the metric neutraliyof tonal content apartfrom factors of dissonance, chromaticism, distance, etc., gives rise to important corollary conclusions. The prolonged V of the formal introduction like its analogs atlower levels! requires for anticipative function such nonaccentual properties as practice demonstrates: textural sparsity and simplicity, slow tempo, lower dynamic levels, etc., or 'shaping towardsuch conditions, its rhythmic understatement heightened in relation to an accented consequent see Ex. 2-29!. It must be remembered that in situations of broadly weak-strong or strong-weak relation the conception of impulse is that of the entire con°We have stated that,in principle, the accentedV is not lessplausible ornatural than the accented I;in fact,given theusual dissonance of V, just the reverse mightbe dcmonstrated.

rhythm and meter 361

st°t1 uent umt: t heH accent73 is ' an entire antecedent or consequent segment, not of course itscadence orany other single, subsidiaryevent." In discussion of leveled metric structure, we accept and further promulgate the view that even at the broadest levels structural relations long-short, soft-loud, solo-tutti, thick-thin, low-high, fiuctuant-stable, etc.! constitute significant metric order. Most of the music with which we shall be concernedis architectonic in its organization. That is, just as letters are combined into words, words into sentences, sentences into paragraphs, and so on, so in music individual tones become groupedinto motives, motives into phrases, phrasesinto periods, etc. This is a familiar concept in the analysis of harmonic and melodic structure. It is equally important in the analysis of rhythm and meter. As a piece of music unfolds, its rhythmic structure is perceived not as a seriesof discrete independent units strung together in a mechanical, additive way like beads, but as an organic processin which smaller rhythmic motives, while possessinga shape and structtue of their own, also function as integral parts of a larger rhythmic organization.

In reasserting such a principle, we are in part regarding rhythmic effect apart from the problem of its perceptibility and intelligibility in the musical experience. Far too little is known about the range of musical perception, or the extent to which macrorhythms are perceptible and intelligible.

Yet expresseddoubts regardingbroad rhythmic experience apply equally to the question of relations of any kind over the broad expansesof a large work-relations of theme, even,not to mention those of tonality or tempo, or orchestration!. In the following, Sachs alleged limitation in a twelvesecond period of distinct and immediate awareness wouldin itself preclude In formal relations oftwo parts period, binary,etc.! in which strongermetric accent inheres inthe second, the appropriateness of theterm initiative accent might be questioned. I.e., what does theultimate initiative accent initiate ?-a necessary question especially at the highest level.!Yet, asone conceives of impulses as ofincreasing breadthat higher levels onebecomes aware of a principle by which theaccent theinitiative impulse!with its function over a longer time spancontains the process of its owndecline: itis a sloped event within andof itself;moreover, the inherent process of decayhas timeat higherlevels inwhich to function. Thus, whenaccentual valueresides inthe secondof two, broad impulses,the process and fact ofdecline, orrecession, functional is inthe course ofthe impulse, manifest finally in the recessive cadential action in which initiative energy isultimately dissipated.The possibility within longer timespans forthe impulseto havesloped recession within its own duration isa majorand vitaldistinction in the natureof impulsefunctions atdifl`erent levels. Actually, anyimpulse-accentual ornot, and at whateverlevel-declines Mthin its own duration except when deliberate progressive intensification is imposed;but thisis aperceptually significantfeature onlyat levelsin which impulses have significant duration. "Cooper andMeyer, TheRhythmic Structure of Music,p. 2.

362 rhythm

and meter

deep assimilation of any temporal work of art; it fails to allow for the capacity of memory, memoryas it applies even to inarticulate feeling as well as cognition. Everybody is entitled to call the ABA of a da capo aria a rhythmic structure or, if he so chooses, even the four movements of a symphony .... But the accessibility to the senses isopen to doubt. For any longer piece is very definitely at variance with' the findings of modern psychology that the maximum filled duration of which we can be both distinctly and immediately aware is twelve seconds,which is the reason why the ancient Greeks limited the length of a verse to twenty-five time units.

There is much testimony to meaningful rhythmic experience even at ultimate structural levels, and the sense of hypermetric structure at intermediate levels can surely not be disputed. Thus, the concept of multidimensional metric structure is developed here as of entirely conceivable practical significance, not as mere theoretical artilice. To an appreciable extent,of course,intermensural and intramensural! meters reiiect the characters of mensural meters within a given style. That is to say that, in a very general sense, asymmetricalintermensural divisions are more likely in styles in which irregular mensural units are prevalent. But there are important exceptions tothis principle: in surfacesof apparent symmetry, critical asymmetries are often imposed within overall metric structures of considerable variation. The principle of the variability of metric content and proportions at different levels ofa multileveled structure is, then, of great expressive importance,as in certain of the most stimulating of Classical literatures. See Ex. 3-19, and others.! Metric irregularity: horizontal and vertical noncongruity We shall refer to metric asymmetriesas horizontal and vertical nonoongruities and to symmetry as eongruity: conformity, agreement! in precise parallel to the geometric sensehaving to do with the potential for exact alignment or nonalignment! of figures in superposition. Noncongruities in metric structure have much to do with musical effect. The present concern is with fluctuation and asymmetries ofvarious kinds, a primary aspect ofinstability in metric

structure.

Examples of asymmetries amongcontiguous units horizontal asymmetry or noneongruity! are evidentin someof the preceding citations Exx. 3-13, 3-16, and others!; further illustration will follow. Curt Sachs, Rhythm andTempo NewYork: W. W. Norton and Co., Inc., 1953!, p. 17.

rhythm and meter 363

Disparities among metric units occur not infrequently at simultaneous and overlapping points in the musical texture, creating polymetric structures the opposite might well be termed homometric-conditions ofparity among concurrent, vertically related metric units at comparable levels!. Disparity and aiiinity of metric unit concerns,of course,not only the size,or duration, 6

of the metric unit but its contentwell; as thus, for example, a Dunit in one 3. _. voice combined with V a meter in another would correspond in size at the mensural level! but not in content; and at a lower level within these units there would be noncongruity of size too. In polymetric situations, there is instability and almost certainly asymmetry as well, manifest at some given level s! of structure-an effect of metricdissonance in many styles. The nature of noncongruity typically varies in ways that constitute progressive or recessive tendenciesof functional significance, a particular aspect ofHuctuation which remains to be discussed.At this point it should be notedsimply that metric instability, or diversity, can be manifest through the texture or contiguously from unit to unit, or in any combination of these tendencies.

In all of these connections we are, again, talking about real meter which, aswe haveseen, isfrequently not in accord with notated bar-lines and signatures. And again, it is necessaryto emphasize that we are treating as meter organizations which may be irregular in unit relations and, even, organizations which do not rely upon a constant beat, like the groupings of chant, certain kinds of recitative and declamation, improvisatory passages, and the like.

The nature of relations of proportion among metric units is allimportant in the structure and rhythmic life of a musical work. Such proportions relative durations! of metric units can be reckoned in a number of ways: ! in total real time durations, accounting for tempo changes; ! in real time durations calculated from initiative to initiative; ! in the relative numbers of active pulses i.e., pulses marked by events!, with reference of course to the same unit of pulse in the units compared, and The terms pobrmeter and homometcr must apply asextensions of the termmeter and are generally understood to haveto do with texture, that is to concernvertical relations;that is the sense in whichthey areadopted here. The homometer-polymeter spectrum ofpossibilities applies toconcurrent metricunits at various levels, and to their relationsof contentand size. Smither makes a numberof relevantobservations based on theproblematic equation of meterwith regularity. For example,in discussionof an example fromBlachers Ornamente _fiir Klavier, Op. 37,Smither notesBlachers prefatory reference tovariable meters and says: Meter for him evidentlymeans eitherregular or irregular groupingof a constant unitof measurement, rather than regularity of accentuation.The Rhythmic Analysis of 20th-Century Music, in journal qfMusic Theory, VIII,pp. 80and 82.!

364 rhythm

and meter

discounting held as distinct from reiterated impulses-in a sense ameasurement of activity-tempo; ! by comparisonof numbers of such pulses whether active or not cf l, above, except for the factor of tempo change!; and in other ways. Such disparate modesof calculation of proportional content and relations yield information of different kinds, all of potential significance in metric function

and relations.

F ermatas and other phenomena involving indefinite durations are obvious examples of asymmetry among contiguous segments, assuming they are notinterpreted as even multiples of established units, andthe functional meaning of such devices is invariably of interest. Meters two primary facets weakstrong associationswithin the metric unit and relative proportions among units! can be distinctly functional in a context in which the regular pulse succession isinterrupted. Of course meter does not cease toexist in such a context. Moreover,

in free, cadenza-like situations,

it defeats demonstrable

functional purpose to project a continuing, regular pulsation and no sensitive performer would do so. The opening of the second movementof Beethovens Symphony No. 8 is a celebrated instanceof contiguous asymmetry horizontal noncongruity! in conflict with the notated bar-lines; naturally, it has been approached in analysis in many ways. In view of asymmetrical relation between clearly corresponding eventsoutset andm. 4, second beat!there isclearly fluctuation of some kind in some degree.This movement begins in a situation, then, of metric dissonance cfi Ex. 3-9! subject to later resolution. It is of special interest to carry somewhat further the interpretation of the middle of m. 2 as mensural and phrase-level! initiative. In that view, Ex. 3-20a. Beethoven, Symphony No. 8in F, Op. 93,second movement. Allegretto scherzando . | Q O I UI ?T~45-=I I2 3 ==== i'li'L11."1¬.

O C O OI l I 0 I U C O. . . . . . . . . 0 . . 3 31:L == S = U ='='='}'=='=========

' Wi=£:£:':=== il1==&'-=§==i'{=£££$=¢===£ eH 7."$ JlJ=ldfJ=iJ=|' ¢=!YEJ..;-¢5m_-_ilrvw .J l' ,$." i*'_:"'=l!:=!3 '==1'!!=='=;" * " i -'__ °4+2l of! _

f}1".;¢q2-.'-4.17-.4 .-.1 .-In ' '=="-3==':" i'=f= ';" "7==!==f==

+ S!

lr 3; + ""';+2! + 3 Mcnsural units

rhythm and meter 365

a corresponding initiative is heard at the beginning of m. 6;° these two 7.

accents are separated by the asymmetrical interval of

-, the temporal

span by which the beginning iteration is separated from its resumption at the middle of m. 4, and by which the latter is separatedfrom the corresponding event at the start of m. 8 not shown!. Other factors are representedin summary in Ex. 3.20b, including some resultant questionsof performance. Of chief importance in the two phrase-level accentsare agogic and anacrustic factors.

Ex. 3-20b. Dissonant. leveled metric structure in theopening ofthe A//egretto of Beethovens eighth symphony.

/" l

71

__ , »;.;L¢A;;.|-1 lr! , _ If ' e |_l.J»l l_1 I-'~_e 'ff If _ -fl* |n_i-17-===v 1 F1 fi l?*1$""'__'_ .|-_l_|lI:!lZ.|92.¢ 1 nl mf Iran! ~' II-Ill-I III-li "'~ il' 3 92 .__-ni I1-_llf ' ||lISll_lll1»»l~*'l""S ' 1_ L `-1 'L I' Ii I1 1-7 _1 , |.J ?I *_gr; . --~.

:r

.___ I

I ' '*

1 I~

1~

, ,r r *Extends toevent corresponding to that of middleof m. 4.

TCL strong accentlater. 7d Higher-level horizontalnoncongruities oftennot immediatelyapparent! are exceedingly common in music, especiallybeyond period levels in tonal music, but in general in most music other than of unsophisticated genres. The concept of vertical noncongruity, in which structure is polymetric, is that of a real bar-line which is disjunct, notperpendicular to the line of temporal succession. In the extract from Stravinskys lHistoire du Soldatquoted as Ex. 3-21 the diyunct nonperpendicular! bar-lim can be drawn at a very low level with Reference to measure numbers is alwaysto notated bars.

366 rhythm

and meter

relative unambiguity; it reveals a highly diverse, activesituation resting on

J and eb pulses of fixed durations." Ex. 3-21. Stravinsky,/'Histoire duSoldat; PetitConcert.

.|-|2o ,

m.60 : | ' .I -. .' .I Cl' Y-1§§1_i2I1* !";J!.l 1u1||i1|1||7i|»_: 221|-1lJ1|lll1|l1§1l 1 nl! Z V niililti ° A* I/Q _ IIQlo'_H . lsiliq ln mv; In _ |PP .¢_` | » 92 -___ I9 `l ' I Hn- rivlm mfg; ~ZKil|'*l .92 lirringm _u o:'l_l|r'uni..lI¢"'r*l-1 _ 921|» 92¢ .1u _ n:~l_l »_ _ KL§l1"lf92Q.""'K 1:11 ~ * =92;'1i'li11¢92ll|o_l',!l1l |l.r-Quilt!! lf.-1-QIOQJIKIIQ l.I_|-_92I'-|lll_f-ll-l°_§YT--|ill92... IDA ' 1.|l » :lo _ sl -1111 'up ' .0 '. |' .' 0 °o |' OI |' |I OI | '92 |* 1,1121-1l_Ql|lll11ll_1`921l_ilJZ'111lll11' 'fu-I' __ _'Taz' _IT--1| -!AYUII _---!" |/at Ziilli _Ili intuiui 11| ihmi ii lnlnqml ll1l|l mv; i Q-liulblijjili1i§1_ij1l§I'1i§2lJ1§i .pp k | 92` la 92` ° lZZ. ' I' `| l¢rx1_l_;i4l_1uil.Alu1.1i.Jillu;.A .;n;1 Db° LltI1h.D1li_'1'i1n7l1r1n1il_:';7lniiilu17 U-Il.-QQ-.lZllQ22._|Q_Q'_.-__l°!§$.-IQ" lTt1;i"";_llni1l$llu11--_!l71ioi1ll»ix-92 lI 1I ,_ Z

EPP : I-ll Restoration

Ii of perpendicular

bar-lineat asm.66. *All parts sound aswritten.

Reprinted by permission qfj.W. Chester, Ltd.

The reality of resolutionof metric dissonance isalso strikingin Ex. 3-21, with the nearly perpendicular bar-line at which fluctuation converges there is full convergence atm. 66!. The questionsofthe exact nature of fluctuation, noncongruity, and resolution of metric dissonance,depend on interpretation of the low-level grouping of the double-bass motive;while other motives are

relatively unequivocal in metricstructure, that of thebass poses possibilities Qv rv

of or

I /""

i,

with interpretive consequences

as to shaping of the motive itself and the precise nature of total polymetric interplay. The foregoing is a very common situation in polyphony, if not often as unambiguous in mensural ordering. Metric analysis is of particular complexity in music before Baroqueinstrumental conventions; not only is much unknown about performance There arisesin theseconnections the intriguing andtroubling issue of thesignificance of the conductors presence, his gestures an indicationof a given notated!metric structure where fluctuationis an expressive desideratum highly functionalin the structure; theconductor oftenimposes inpolymetric situations the appearance of conformityand compromises, presumably, theexperience qffiuctuation, or he may in some degree have thateffect.

rhythm and meter 367

especially articulative, punctuative practices! of earlier styles, but the metric meaning of a musical language in which no norm of bar-line regularity can be assumedis elusiveand problematic. Especially in contexts

in which preconditioning metricinfluences arehighly unlikely we are often dealing with meter which is almost constantly nonoongruentall in directions, while cadentially punctuated by areas of relative stability and felt against a perspectiveof often steady succession at the level of the beat. The frequent presence oftext is a complicating factor in metric fluctuation in pre-Baroque polyphonic and other genres. The extraordinary rhythmic fluctuation of this music, assuming expressive,functional significance in relations of Huctuant to more stableareas, explainsas muchas anything the fascination and arresting power of much of it. The _]osquinDe prqfundissetting cited for previous analyses Exx.l-26, and 2-17! is of further interest in connection with metric structure." The analysis of meter in _]osquin proceeds from certain basic considerations: . l! the considerable irrelevance of notated bar-line; ! the decisive importance of text e.g., metric structure in the initial subject, reproduced in Ex. 1-26, would, presumably, give superior accentual value to the syllable zfun- even though the starting note is longer and equivalent

in pitch, the in Deview of its textual subordinacy interpreted! as/¬"!; a likely, almost constant noncongruity in horizontal and vertical relations-a frequently disjunct real bar-line; ! the functional resolutionin the occasional concurrence of perpendicular! bar-line; and ! the crucial implications for performance, in which a discreet, unexaggerated fidelity to accentual values, including those of textual inflection, results in a remarkable vitaligr of metric_/lux. Let us, for example, consider metric structure in the initial exposition of the De prqfundissubject in superius andaltus Ex. 3-22!. The agogic value of De is suggestive oflow-level initiative accent-a manifestation of the works strongly assertive,decisive beginning-but the textual importance of prqfundis,whose secondsyllable is the normal recipient of emphasis, lends superiority, at a slightly broader level, to the initiative of the second notatedbar. It has relative agogic accentand pitch accent in its immediate context.! Setting of the following word, clarnavi, isbeautifully A comprehensive analysis of the important, related question of activity-tempo rhythm asmotion! wouldinquire intothe tendencyof phrases to beginwith relativelylonger values, accelerate, and thenrestore longervalues atcadences-a matterclosely linkedto that of texturalfunction. It would alsoseek toanswer thequestion ofcomplementarities between faster rhythmicmotion i.e.,in shortestdurational values!and otherforces ofintensity rise in pitch, dissonance, etc.!as well as textural interdependences and other compensating factors ofkinds treatedin prior discussion. Ajust considerationof rhythmic motion would consider itsrole inshaping increasingly large segments and, ultimately,its functionsin determining the entire tempo-structureof thework.

368 rhythm

and meter

Ex. 3-22. Mensural and higher-level accents deriving frommusical andtextual factorsin the Josquin.

FF

l5

:

: _ nn. =

"dna ZITI!;-"=-";`h'==

'l!f__..-"" pro- ful3_-,----' 'elif cls

De pro

F -:fun : - dis :els: ;===='l===;_$-= =l=.?.== 13= -ma - vi, :ad i te ., °i

»2

4 in

1°+s!

VI

FV

Area of resolution of upper voices against contermetric relation g_ of lower voices. ' 5 2 l1'|.6 l

i _J

vi, acl

-

32

.F

If

s. is

.0

,_ p

te Do Do -

J = =' *-::== l -:==':

- mi - ne, els Qms mi -

ne, cls -2 me- -

_,--.-»- '°°° lDe pro

:

De pro

- fun - dis

- ; vi,

vi, i

- fun- .els -ms -

-

Reprinted _fiom the Sm§ers edition, published by G.Alsbach and Co., Amsterdam, by perrnission q/`Creyghton Musicology-Musica Antiqua, Bilthoven, Netherlands.

expressive ofnormal accent on the second syllable,in relative agogic superiority and in pitch, although the first syllable has agogic value as well, establishing aforeground mensuralunit. In a very local senseacccnt-determining units are as expressed inEx. 3-22, with typically fluctuant, asymmetrical metric structure; in a broader sense, andat a higher level of metric structure, -ma- appears in its purely musical parameters as well as in its textual importance a centralpoint of initiative orientationin meter. The extremely fluctuant character of the music is already vividly apparent, and this opening subject exposition is one of relative simplicity. Assuming the editors underlay of text, Ex. 3-22 reveals somethingof an apparent coniiict between textual and purely musical influences in the

rhythm and meter 369

setting of ad and te; as indicated, it is the former that has musical accent supported by leap to higher pitch, and in agogic emphasis!, while te is subordinated musically in every instance exceptone-that of altus in m. 5,

where itis the object ofa slightlift in pitch andof melismaticelaboration. There is sudden initiative accent onDo- in the upper voices atm. 7, although this prosodicallyimportant syllable fails to have emphasisin the tenor of m. 12 quoted in Ex. I-26, pp. 88-89!. Moreover, -vi has emphasisin its musical setting in the upper voices ofm. 9 and ela-in the tenor of m. 10. These apparent discrepancies in musical realization of normal textual inflection can only be discussedas ! instances in which the composer deliberately subordinates textual to musical impulses, asis sometimesthe case one has only to think, for example, of Stravinsky!, or ! instances in which the underlay of text is subject to question. It is significant that, again given Smijers underlay, textual-musical aeeents are inconsistent through thecourse qfthe work i.e., in recurrences of words, accents of musical inflection vary; cf. settings of Domine inmm. ll-l2!; that would suggest consciouslyfluctuant treatment on the composers part. The problem of metric structure, as it concerns theseinfluences of text, is one of very decisive significance, only suggested here.One imagines that these are issuesrarely consideredin performance, even though the extent to which performance violates or underscores natural textual inflection or projects words utterly indifferently, if that can be done! critically affects the experience of metric structure in such a work, and the experience of complementary and counteractive groupings governed or conditioned by textual accents. In any event, no interpretation could possibly beliethe fact of pervasivediversity and vitality in _]osquins metric structures and processesas delineated in textual and musical qualities! It is important to make an effort to see somethingof the sources and nature of the inexhaustible rhythmic vitality of Brahms. The slow movement of the Horn Trio Ex. 3-23! illustrates many highly interesting techniques ofmetric and rhythmic diversity and controlled fluctuation. A very visible instance of noncongruity at the level of the notated measure is evident at m. 26; but throughout the first part of the quoted excerpt numerous potent means of mensural variation can be seen even while the notated bar-line retains steadying support: the motivic, articulative and low-level metric! grouping of the horn in mm. 23-24; in the violin at m. 25, the irregular five-ordering of eighth-notes an aug-

mentation theoffourab motive of the horn the in preceding bar!; or the further irregularity in ordering of the motive in the bar

see the violin,

and piano right hand, with its hemiola!; and others. Moreover, there are

Ex. 3-23a. Brahms,Trio in E!, Op. 40, for violin, piano,and horn,third movement.

23 Vln.

Hn. in E

Piano

m. 29

Adagio mesto

rhythm and meter 371

instances inwhich vital interpretive decisionshave to be made and worked out in performance, involving possibilities of underscoring implicit metric __

__

noncongrutty-e.g., the same four-note,mottvtc entity !

/nvnnigi

to which we have called attention often crossesthe notated bar-line see especially thepiano part of mm. 23-24!. All of theseare mattersof significant potential for diversity in the metric structure and all require careful consideration of structural function and interpretive consequences. Especially in mm. 27-30 there are many factors of rich interest in the ._

.3

metric structure. Note, for example, during the p1anos

._

.

hemiola opposition

manifest at one- and two-measure levelsl!, Brahms ingenious exploitation of the same four-note motive discussedabove, with its subtle potential for 4. .. . a superimposed D inan intriguing, v1tal_ context of constantly fluctuant

polymetric organization see Ex. 3-23b!. Items of comparable rhythmicmetric interest arise throughout the movement. 4

Ex. 3-23b. Potential superimposed for grouping Dmotive at inlevel the Brahms passage, with consequent fluctuant, accelerative effect.

5 4i U

F! r Fl f

7

rr

|r V. I

P

O ; W

etc.

rr r

]

372 rhythm

and meter

The prob/em of preconditioning syncopation

metric structure

and

The concept of preconditioning metric structure must now be taken up beyond its incidental references inforegoing pages.In one sense it is an extremely simple factor in the determination of accentuation, in another extremely diiiicult and complex, like so manyproblems ofrhythmic perception resistant to comprehensively objective treatment. But it is an essential consideration in any analysis ofmeter. This discussionshould take place in two stages:first, let us state some of the questions involvedand someof their implications; then, we shall have a look at examples in which preconditioning structure or the lack of it! would appear to be a factor in the expression andeffects ofmeter, remembering that this issueis of recurrent general importance. The basicquestions arethese: With a particular metric structure established, to what extent does it precondition the sense ofaccent in subsequent events and their groupings? Does apparent metric divergence in a given instance-vertical or horizontal-constitute changedmeter or simply counteraccentuation againsta prevailing, persistently manifest,established meter? Moreover, to what extent does apparent metric opposition in one voice yield, in fact, in composite metriceffect, to the establishedmeter of another ? Or, given two meters in concurrent voices, doesone of them, depending in part perhaps on its position in the texture and other differentiating factors, precondition the perceived groupingin the other?

Suppose meter, a the at mensural level, 3ofis well established and may be assumedto have preconditioning force. Might sarabande!rhythms like

JJ

l J J then shift the real bar-line at some point,one in more or

voices, thelonger, agogicallystressed half-note taking precedenceas themetric initiator? Presumably, this dependsin any given context on the stability of other voices and of other temporal areas in the preceding or surrounding preconditioned scheme,and on the applicability of other kinds of accentual forces-that of pitch, for example, as well as factors of punctuation and articulation. But the weighing of evidence in particular, complex situations is a problem of great difficulty and great importance in performance and critical understanding of structure. Moreover, the question of the presumed absence of preconditioning regularity is especialiyimportant in situations rj initial metricdissonance and instabiligf see Exx. 3-9 and 3-20 in this connection!. Useful analogy can bedrawn tothe problemof thesignificance aconsiderably subjective matter!of tonalfluctuation intraditional musictheory; theview ofsyncopation developed hereis ofa kindof incipient fluctuation-noncongruity of unit not really achieved,not really eclipsingthe established unit, not itself becomingreferential.

rhythm and meter 373

Throughout these issues thecrucial significance of interpretive judgment is evident in the fact that the performer can fortify or attenuate preconditioning Wet and its functions by even restrained interventions. These areexamples ofthe kinds of essentialquestions whichinvolvethe issue ofpreconditioning structure. And the issues arestill more complex when cast in other dimensions: what is preconditioning in the particular musical work might be affected by factors of preconditioning in the style, not to mention what is preconditioning in the listeners experience. And what about traditions e.g., dance!, or the possible preconditioning implications of earlier movements in a multimovement

work?

As basisfor continued discussion, weshall make a number of assumptions.

l. It is very doubtful that there areconditions ofpreconditioning meter that can be stated as generalprinciples; questions of preconditioning effect must be considered withrespect to the particular context involved." 2. In many instances there appears to be little or no significance in stated mensural meters aspreconditioners; obvious instances ofthis can be seen inlater stylesof very fluctuant meters, and in early music in which the bar-line is largely a notational convenience. 3. Where preconditioning is an insistent,effective determinant,opposing accentsare commonlyasserted ininteraction with preconditioned meter; we shall understand theseas syncopation. 4. Where preconditioning may be of relevance, weshall assumethat in some degree an opposing meter countermeter, or contrameter! is at least ineipientbv established by confiicting accents,its significance i.e., whether it constitutes syncopation or fluctuation! to be evaluated on the basis of specific conditionsof context ultimately a matter for reasoned judgment. 5. The issue of preconditioning, surely applicable to levels of motive and phrase in much music, is probably of diminishing relevance and complexity at higher levels,where _fewer units over longer timespans areless likely to precondition metric experience. The perception of anacrusis is often significantly, even decisively, governed by preconditioning circumstances the iambic-trochaic, anapesticdactylic issue!. For example, the theme of the slow movementof Schuberts Quartet in D minor Death and the Maiden! is perfectly clear in the function of recurring longer notes asmensural initiatives. But the relation of the half-note to surrounding impulses-the position of the half-note as endaccent or beginning-accent-would lack its functional ambiguity were "One very important factorin that consideration of context isthe extentof unanimity throughout thetexture atpoints ofchange andpoints ofreference see Ex. 3-9and others!.

374 rhythm

and meter

Ex. 3-24. Schubert, StringQuartet No.14 in D minor,D. 810 "Death andthe Maiden"!, second movement. Andante con moto , .---~-2 ' 'll'_ ' _ .Cfill 1_-1'--1 Y

VlnC2 C 1I

*

Vln 2 _ -1 _ __ _` ! Qi.-1l_l1§IfI_!l-*-11 | M2-H-IZ! 1/ Vln.l

there an anticipative approach cf the second phrase!at the beginning Ex. 3-24!; normally, it is interpreted as an end-accent,requiring slight punctuating articulation at the middle of the first bar, itself a preconditioning factor. N oissue couldbe of greater moment to the performer in making interpretive decisions of articulation.

The question of preconditioning arises ina different sense inthe example from Beethoven Ex. 3-25!. The first note it begins the movement! can scarcely beheard asan unequivocal initiative impulse, given the Bbs agogic and intensity values; rather, again dismissingthe implications of the listeners prior familiarity all listening to music that one knows requires an overt submission toits terms!, it is conceivable that the initial note of the upper voice has some effect-in the theme melody-as anticipative, while syncopated against the bass regularity and against increasingly prevalent perpendicularity of bar-line. What is especially intriguing in the applications of this theme

throughout themovement isthe increasing ¢ct cy" preconditioning. Even in the secondbar the grace-note anticipativesupport and the articulation begin to have accord with the notated bar-line, and of coursethe prevailing mensural structure is quickly established sothat subsequentappearances ofthe theme becomeclarified cfi the resolution of metric dissonance! as to

increasingly distinct syncopation in relation toa preconditioned3 0"6 . But it may be that even ultimately a disjunct bar-line is felt in some lingering sense asexpressively counteractiveto relative regularity in the lower voice. The point of the anomalous accentis its expression ofincipient noncongruity. In the matter of fluctuation or syncopation there is a broad range of possibilities circumstances : inwhich presumablyanyone willfeel theprecondi"While I Yit ma seem contradictory, Iit should be notedthat the anticiPative I I IimIpulse canI Ionoccasion have stress and a arent accentual value su error to that of the initiative I II II IP II II Ito which it relates; but this can onl occur in situations in which a basis or anomal is lard-1.e., in II II I I yi I II II II y situations inwhich the anticipative-initiative relationis firmly preconditioned. 52Compare theCooper-Meyer discussion in which stress on the secondnote is regarded asdistinct frommetrically significantaccent The Rhythmic Structure of Music,p. 30!.

rhythm and meter 375 Ex. 3-25. Beethoven, Concerto No. 2,Op. 19,for pianoand orchestra, thirdmove ment. Molto allegro 4' :' |' 27.11. r.¢ _ _ l.'l_--._ lifji ll..§'°:l-1-_IU Luv o 'QI -'H ' I _il

31 '

it If

.

I I/g" vit I1 ammoT.¢|-l-|*-*"' - I-» i| » *T

I1 . *r ''

. .iinrii I1-I _ _Z A°

* Metric dissonance , incipient disjunct bar-line. TStabiIizing factors:continued conformityof lower voice organization;upper voice ornament; uppervoice agogicaccent onc2; uppervoice articulation.

tioning effect of established metric grouping, others in which such effect seems doubtful,and of course manyin which the effectof preconditioning as a determinant of syncopation seems virtually inconceivable. Throughout this spectrum, one is dealing with real or incipient metric fluctuation in some degree,for while syncopation isnot anindependent metric manWstation, as in polymeter or true noncongruity between contiguous groups, it is functional and expressiveas assertinga kind of shadow unit challenging the prevailing metric context.The question of classificationis not idle; it is the question of eject: there is a vital functional and expressive distinctionbetween metric opposition felt against a preconditioned, governing basis with preconditioning meter felt as referential in relation to counteraccentuation! as opposed toone of actual shift of unit size and/or content. Of examples of syncopation, some illustrate incipient vertical noncongruity, and others incipient horizontal noncongruity; with stronger projection and persistence anycould result in true metric shift or fluctuation Terminology here rests uponthe ideathat wherethere issyncopation thereis relative uniformityof prevailing,preconditioned meter. Thus, ina fluctuantcontext, wherethere is frequentchange ofmetric grouping,a consistent frame ofreference i.e.,a preconditioning context! is improbable of cognition. Again, compare theproblem of tonal fluctuation.! And, wherea patterninitially felt as syncopation within a preconditioned context is highly persistent, thepreconditioning metermay ultimatelybe disrupted, not justdisturbed, infavor of the syncopated grouping, which then becomes inits turn the prevailingstructure. In a sense, too, the questioncan beone oflevel ofreference where conflict of contiguous .. 3

unitsconcerned. is for In, example, the common hemiola interposed intoprevalent a menV

__ 2 sural structure, therencipient?! is fluctuation at levels of3V and,Vbut not the at level of unitsVofexcept , the insense of internal ordering involving lower-level groupings.

rhythm and meter 377 Stabri/'ty and Hux; metric progression and recession; compensatory and complementary functions in re/at/'on to other e/ement-structures

To say that a unit of structure is symmetrically ordered is to say that its constituent subunits are at some level possibly not at others! unvarying, or in someother balancedrelation e.g., 3 2 2 3!. To saythat there is fluctuation is to observe a relation between

a unit and those which follow

or are con-

current at a corresponding level. The notion of symmetry is, within a given level, associated with the condition of stability, and asymmetry with fluctuation.

Except in random musical situations, changes in meter like those involving any element! are structurally functional. As with any structural element, there is a vast breadth of possibilities with extremes of metric stabiligf onthe onehand constant,unchanging intervalof initiative accentuation at all levels -in all textural components! and of instability on the other total avoidance of like contiguous or concurrentunits in the metric structure, continued Huctuations ofmeter at all levels!. Control of meter within such extremes is a fundamental factor

in the articulation of

musical structure.

Any given work projects its own norms ofmetric stability or flux, while usually consistentwith, yet at times anomalous in relation to, norms of its style. For most works there is thus a condition of normalcy whose characteristics are at a given level referential, some prevailing unit of meter constituting a basis of variation, and some point in the range of possible states between extreme constancy and extreme fluctuation. In many works, the meter signature or signatures! suggests those conditions of normalcy at the mensural level.

The spacingand frequency,and the quality i.e., degree ordistance! of metric change constitute a rhythm like those of other element-changes,so that it is possibleto speak of a metric rhythm in music-a rhythm conditioned by the qualities and frequency of changes in metric relations e.g., temporal distancesbetween points of metric resolution!. Metric rhythm in this sense like tonal rhythm, etc.! is an expressive determinantin music. As the concept of shaped and shaping! metric fluctuation-that of recessive andprogressive process within the metric structure-is approached, it is vital to keep in mind that any theory of such recessive andprogressive operations must have reference, in applications, to comparable units normalbt initiated bycomparable accents at a given level,the judgment of comparability in these respects animportant part of analysis.There is again no possibility of absolute objective specificationwithin these concerns, but it is clear that any analysisof compared relations and controlled metric change must have reference to units interpreted as comparable, and uniformin level. The potential

378 rhythm

and meter

for shaped rhythms of change in metric relations between levels, while not explored here,could well constitute a provocative further area of concern.!" As to unit size, one aspect by which normalcy is characterized, it should be noted that variation in unit orderingwithin apparent constancy of size wouldnormally denotefluctuation in size at some lowerlevel. Common situations in which meter is functional as afactor of great importance can be listed in the abstract, after which we shall turn our attention to

some exam-

ples of shaping metric change in works of various styles.Note that reference is alwaysto relation to a norm for a given context of level and temporal span. 1. Relative stability, often a factor in the process ofthematic statement except of course wheremetric flux is a thematic attribute!. 2. Metric instability, often a factor in developmental and transitional processes.

3. Fundamental variational changes in metric unit at mensural and intramensural levels,

in such works as theme with

variations or

rondo or

other recapitulative forms. 4. Metric progression, indirection of a! shorter unit and b! increased instability, often significant in a processof mounting intensity; progressive effect is of acceleration.

5. Metric recession, indirections of increased stabilityand longer units note frequencyof temporetardation and fermata!, often a factor in introductory and cadential processes;recessive effectis of deceleration. 6. Interesting is the possible relation of metric indecision to states of relative tension; metric indecision may involve uncertainty or suspension of regular beats or pulses often coincident with dissonance! or a suspension of activity fermata, or indeterminate pause! in an effect of momentary temporal hiatus unmarked, unpunctuated or empty time; measure-free time! unlike the time of unit measurements the time of quanta--of punctuating events!. The factor of ambiguityuncertainty, indecision! of meter cannot besaid Disparities at different levels-e.g.,mensural unitsof fouras opposed to phrase-level units of three, constituteasymmetry ofan important kind whosesignificance ofeffect and function couldwell beexplored; sucha relation,crossing structural levels, isone ofadimension ofdepth in the musicalexperience. The fermata has twobasic functions:it can contribute asthe ultimate broadened unit to recessive process e.g., finalcadence!, andit can intensify thefeeling of instability in fluctuant circumstances by theinterjection ofa unit of ambiguouslength. In the latterfunction it is often complementary todissonance. Whether its effect complements finalityor intensifies instability seems to depend onother element-characteristics to which it is allied as well as on its position in relation to other lmits comprisinga progressive,recessive, or erratically fluctuant series; in the latter circumstance itseems likelythat the fermata is stabilizing ineffect. 'I'hefermata asemphatic, recessive metric broadeningoccurs onlyas a final event.

rhythm and meter 379

clearly to contribute to intensification or resolution. Recessiveor progressive inclination in metrically ambiguous situations dependson how such situations are perceived asto greater or lesserfrequency of accent at a specified level.

Some important progressive andrecessive tendencies of metric structure are summarized in Fig. 3-6; in the summary an effort is made to identify parameters of meter subject to processivemanipulation in the two broad tendencies noted.

Fig. 3-6. A summaryof recessive and progressive tendencies inmetric structure. Features/objectsrecessive of change*

Features/objectsprogressive of change*

l. Larger units at common level 1. Smaller units at common level 2. Relatively symmetrical unit rela- 2. Relatively asymmetrical unit relations tions 3. Increased constancy of unit size 3. Fluctuation in unit size greater greater horizontal congruity! horizontal noncongruity! at at common level common level 4. Increased vertical congruity 4-. Increased vertical noncongruity homometer, resolution! polymeter, dissonance! 5. Restoration of unit norm sta5. Departure from unit norm instability! bility! *Including incipient fluctuation, asymmetry,noncongruity, etc. syncopation!.

Attention is directed to the Final cadence of Schoenbergs secondof five piano pieces, Op. 23, in study of metric structure as complementary to elements ofcadential processalready discussed Ex. l-37, pp. 107-l l!. The recession composite in r/yrthm, one of the cadential processes citedearlier, should here bereemphasized tempo ; changeis a factor within that rhythmic aspect. In the Schoenberg example,the processby which the metric structure is projected in broadening units, beginning with m. 18, can be expressed asin Ex. 3-27b. In this representation, referenceis to the sequential pattern. The initiative event at m. 17, quoted in Ex. 3-27a, is of broad significance-interpreted as the primary initiative impulse for the entire piece, one inrelation towhich allearlier events are broadly anticipative. Inmm. 18-19 see

p. 108!, a mensural F group" is sharply defined; has itanacrustic and agogic 38

l'|*92 5

l"5'_I

"T he symbolJetc.! or E, denotesal or§ J!. Similarly, 9 , ¢h= or s

id ;¢l7=%¢l;

382 rhythm

and meter

Ex. 3-28: continued.

- .=,; $=_.¬ %2!F-.1 lr Q; Q rJ VJ

_

I _4 J/`d g JM 1 1'

J/'wi x1

'

The concluding phrase ofthe Scherzoportion of this movementdeparts in variation of the theme at the typical tonal level of the 3rd above, culminating in a tonic cadence eight measures later.This phrase, which first appears in this form at m. 59, is shown asEx. 3-28b, with three metric levels represented. Ex. 3-zab. 4/5

rf' 1

~

f' 3

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m. 59

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-,

._

__

JJ

do T E*If E . 1 1 T 1 .1

*?

ix

E1 xi

°i

it

The issueis one of agogicas opposed to pitch

accent, andof preconditioningof phrase-levelmeter.!

rhythm and meter 383

,`_/Y

:: "'

lin _

;"'° JT

Wr

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EQ- lt

f

OJ

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II

In the cadential development of this phrase there is considerable fluctuation in grouping as delineated by phrase repetitionsand truncations, as isquickly apparent if one notes that from the point of its inception m. 59! to the conclusion of the Scherzothere are twenty-six measures. The stagesin metrically progressive cadentialapproach are the following: l! repetition of all eight measures; ! repetition of only the last four measures; ! repetition of only the last two measures. Thehorizontal noncongruity at the phrase level is thus immediately visible, with initiative impulses ofincreasing frequency. The pattern is thus one of metric contraction, a progression lending an activating influence of increasing urgency;compensating recessivetendencies haveto be seen inharmonic rhythm and, especially, in predominant movement toward tonal resolution, typically unequivocal. On the other hand, the progressive action of dynamic level is intensifying. All in all, the cadence ismetrically relatively active while emphatic in other parameters; but within the metric process itself;the steadyingfurwtion of jinal metric expansion out of 2! is highly important. Again, one has only to imagine the cadence withoutthis expansion,i.e., with persistent unitsof two. Moreover, the effect of ultimate expansion in the phrase-level metricunit is especially strong in view zy' thepreceding contractive accelerative! succession. The process isrepresented in Ex. 3-28c,d in two distinct modes of graphic illustration.

A composerwhose workhas beenof interest in the direction of particular very overt! solutions to rhythmic-metric problems is Boris Blacher, who has developed a practice of variable meters-the employment of planned sequencesof _measures of varying lengths." Blachers method can be illustrated in brief reference to Ornaments fzir Klavier, Op. 37. The prefatory note identifies the kinds of numerical series SA cadenceof greaterfinality-increased recessivetendencies ofelement-structures in greateraccord-could be imagined asa reversedmetric process. 5°Blachers Piano Concerto No.2 appliesthe practiceof variablemeters extensively.

384 rhythm

and meter

Ex. 3-28c.d. Representations metric of process in the Beethoven Scherzo.

ho C. Acceleration u' I

|

3 IL' | ' A I I e

3

92

e/

92

/ 92

/

Deceleration 92____F___.___/ 92 e Q_ l92__,| "'

, ;e

d. A 1 20 F

4_ ,

F

s_ I £1

rhythm and meter

385

adopted forthepieces inthisset, witha basic unitofP forallthepieces. Thus, the first and secondemploy a simple arithmetical seriesand its retro-

grade;No. 3 employsthe series2 34, 3 45, 4 56, etc., and its retrograde; No. 4 employs a "cyclical"

or "rotational" series 5632,

6 3 2 4 5, etc.!; No. 5 employsan additive series

3 5 8 13 the

56324, Fibonacci

series!;No. 6 applies24 permutationsof the four integers3 45 6

4 56,

3 4 6 5, 3 6 4 5, etc.!; and No. 7 follows the distribution 87 876 8765 ~ ~~ ~~ ~~ ~~ ~ 8 7654 32

65 654 6543 65432

and its retrograde. Brief extracts from three of the pieces,quoted as Ex. 3-29, illustrate theseprocedures. Ex.3-29. Blacher,Ornemente fiir Klevier,Op.37; Nos.3, 5, 6.

AllegroJ -96 No.5

AllegroJ -92

386 rhythm

and meter

Ex. 3-29 continued. -7. I,:» ,

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Processes of active, static, progressive, recessive, and otherwise shaped Huctuant element-changesthus have in Blachers methods a strict systematization at times surelysimplistic!. Blachers accents are,in general,expressed unequivocally: textural, articulative, of pitch, of duration,. etc., and often underscored inmotivic grouping and recurrenceof readily identifiable ideas. An example of minimal difficulty, in which a process of metric progression andrecession canbe seenin a relatively simple context, comesfrom a piano piece of Chopin Ex. 3-30a!. An intensifying progressive process is followed by recessionat the mensural level. The example occurs at the conclusionof an active middle section 6

in theparallel minor, where motivic V units have been prevalent. There is in the quoted excerpta complementaryfunction of dynamic changes,and in .e . .. 4. the recession a rztardando acts expansion in the of larger, V unit as the formal

rhythm and meter

387

Ex.3-30a. Chopin,Nocturnein F, Op. 15, No. 1. Con fuoco

repriseis approached.The metric changes,whetherperceived asa shifting interval of real bar-lineor asincipientfluctuationagainsta preconditioned, referentialnorm, are as summarizedin Ex. 3-30b. Ex.3-30b. Metricprocesses in the Chopinexample. rn.41 49

45 2+2

progression recession

Although the location of initiative accent in the motive might other-

wisebe subjectto debate in view of the agogicsuperiorityof the arrival

388

rhythm and meter

note!, Chopin's articulation ~ ! throughout of the first, higher note tends to clarify the issue; the first, accentednote is also intensified by dissonance. Becauseof the enormousimpact of persistently variable, sometimesunequivocal metric structuresoften asymmetrical and noncongruentto a degree startling for their time, the earlier works of Stravinsky are a landmark in the evolution of increasingmetric flexibility, as are the works of Bartok" and Webern in different ways, all of these stimulative of a range of inquiry, resource,and creative impulse by which rhythm and meter have undergone profound transformations in the earlier twentieth century and up to the present. While any of theseor comparable literatures could be the basis for extended thesis in itself, we are of course limited to discussion of a few relevant

techniques in selected examples, by way of illustration of approaches to metric analysis.' The following referencesare to Stravinsky's CanticumSacrum,subject of inquiry in earlier connections see Ex. 2-13, pp. 225 32!. In a substantial degree,the impact of textual prosody is here, as in certain prior examples,a factor of difficulty, yet of great evendecisive!importance in metric effect: the extent to which accent is projected in textual inflection as opposedto purely musical factors where the two are not in consistentaccord! dependsof course to a great extent on interpretive choice. The metric formulation of the chorus parts is thussubjectto textual inflection asinterpretedia performance and realized in subtle emphases!an area of crucial decision making for the performer. It is apparent that low-level metric structure is one of extreme asymmetry and fluctuation, less pronounced at phrase level, contributing to a generaleffect of great vigor and mobility. In Part V as a whole, large sections recur in a rondo-like manner; metric and other aspectsof rhythmic, e.g., tempo! change from section to section constitutes a higher metrical structure.

level of asym-

Metric noncongruity is of courseat times assertedby very conventional means; the hemiola of the organ refrain of Part V see Ex. 2-13a, p. 226ff., mm. 327 35! is an instanceof this; yet it occursin a context of polymeter of rich polyphonic vitality in which the lowest and highest voices project hemiola at diferenttimesin a composite structure of disjunct bar-line. Such anomalies as the noncoincident hemiolas, the constantly altered and nonconformant metric grouping of the repeated motive of the third voice, and the

82

or ~ but noncongruent meter of the fourth voice, make for a con-

siderablerange of metric effect even within the relatively unassuming,recurrent organ theme. Metric functions in the final cadence of Canticumare chiefly but not entirely! decelerativein complementation to such other element-recessions as pitch-line descent,slowing of tempo, and total dissipationof contrapuntal

rhythm and meter 389

textural interactions in the final events. Example 2-l3a, p. 229, includes the cadence.! In one sense, meteris recessiveat the mensural level,but only slightly so: that is, the final three, punctuating chords are separated by slightly increasingintervals of time. The attack on the final chord isseparated

by lld!from that of the preceding which chord,follows the first of the three by 9Eventhe slight broadening represented in the 9-ll relation is felt as contributive to the more persuasive, functionalrecessions intempo, line,texture, and tonal gfntlzesis.

An analysis of meter at the mensural level, defined on the basis of attacks initiating the descendingmotive of m. 341 and the final three chords, is asshown in Fig. 3-7. The example, which interprets the rest which begins Fig. 3-7. Meter inthe finalcadence ofCanticum, represented at mensurallevel.

ll-9-ll-9 U the adagioas part of the first adagiounit, shows a still fluctuant metric structure. Its vacillation betweenasymmetrical unitsexplains inpart the cadences active quality, fitting in culmination of the movements restlessly fluctuant metric structure."

Beethoven, Symphony No. 8 in F. Op. 93. first movement

A further look at Beethoven is afforded by several excerptsfrom the iirst movement of the eighth symphony,for eachof which unfortunately only brief comment regarding prominent factors-and questions-of metric structure can be given. Let the examples serveas a kind of summary recitation of devices bywhich Beethovenrenders themetric structure expressively variable and off constantly engrossingasymmetries, aswell as functional in its controlled changesin the various processesof statement and climactic development. Example 3-31a poses a question of interpretation comparable to that on p. 329 concerningthe Quartet, Op. 18, No. 1, also in The first twelve measuresof the movement fall easily into a 4 + 4 + 4 intermensural grouping 3-3lb!, although the stress atm. 10 its basis "Interpretation of the quarter-restat the middle ofm. 341as initiative is basedon

considerably%-Q preconditioned units, andfactor theradical of change, tempo which by the restis governed. Inclusion ofthefinal restsas aspecific quantitydefining thefinal mensural unit is of courseproblematic, involvingmore diiiicult questions of preconditioned structure.

390

rhythm and meter

«. 3-31a. Beethoven,SymphonyNo.8 in F, Op.93,first movement.

OF

A «Agogicvalue,but discountingany accentualsignificancein merefact of arrival on 1. !Pitch accent,relative agogicvalue, feeling of downbeatphrasebeginning; but cf. accentualvaluesof impulsesat mm. 4, 6, 8, 10 addeddynamicthrust!, etc., with implications for the interpretationof m.2.

«a 3-31b. A representation of

4

~ phrase-level metricunitsin the openingof the Beethoven

eighthsymphony.

laid in the agogic accentinterpreted as subordinate at phrase level at mm. 2 and 6! creates a provocative intermensural syncopation." A continuation of the thematic element maintains this relative sym2 metry inaneight-measure component ofIs units toand beyond m.20Ex.

3-31c!. At m. 21 see Ex. 3-31d!, the two-measuremotive continues with its establishedequilibrium pitch accent on the higher, initiative note but soon reachesa stasisof pitch level after m. 23! and subsequentlycontracts

from ] to inttclimactic metric progression The initiat.ive higher note has up 'The interpretationof m. 9 asanticipative to a shiftedphraseaccentat m. 10 is a questionof the preconditioningvalidity of the ~ structureof the basicmotive Ex. 3-31a! 4

and thatoftheIs unitsEx. 3-3 lb!with which themovement begins.

62Asyncopatingtextureaccenton the lowerof the two notes,havingimplicationsshort of real fluctuation,occurredat the inceptionof the motive in mm. 13-14; yet the first note assertsinsistentinitiative functionin significantpart becauseof the preconditioningeffectof

the original motive from which itisderived!, the precedence ofIsunitsofwhich m.13is a preconditionedinitiative!, and the important pitch accentof m. 13.

rhythm and meter

to now been in accord with the notated bar-line, while Beethovengives it dynamic accent sf ! at the sametime in confirmation of its accentualsignificance?!. This is no doubt perceived as syncopation in counterpoint to a bar-line preconditioned in thestyleas well asin theworkitself, but the effect is nonethelessan acceleration of implied grouping in incipient fluctuation in 6-5-3

, from m. 23, the point at which the motive becomes

static in pitch-line. The incipient metric asymmetry of the five-unit is a further jarring event in a climactic progressionoccurring only twenty-eight measuresinto the exposition.The entire progressionis followed by a measure of silence,after which the motive now contracted to two impulses! is transposeda 9th below, piano, lacking the disruptive accents. Ex. 3-31d.

rhythm and meter 393

The initial impulse of m. 34, following the silence, maywell be felt as a downbeat, the motive now clearly transformed from trochee to iamb-its first impulse upper note! now clearly anticipative; but it is interesting to think about the experience of this passage in the absence of a conductor signalling every notated downbeat indeed, the sensitive conductorwill not do so! and in the absenceof gratuitous stresses onthe notated downbeats of mm. 34-36. Possibly themetric dissonanceis resolved fully, if temporarily, only with the new theme, and its unequivocal, regular texture accents of accompanying chords. See Ex. 3-3le.!

Ex. 3-31o. Transformationsof motiveand theirmetric implications in the Beethoven. -_ -_ - 92/ Z.-92 I

m, is

-- 92/

*U

P T _?

CO

mea 92|'-B-,; ==:"92|'L&=-

m.28 ,§E== :" J cf.

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92/ -

m.ao m.34

I 92/

_§ m.e1

-If tf

0'

*Relation confirmedby motivic expansion withstronger pitchaccent. TResidual pitch and now dynamic! accentconfused byplacement in relation to notated bar-line. :Dynamic accenton first impulse abandoned. §Dynamic accent now occurringin support of transformedmetric implication.

At another point, intensifying metric contraction occurs in a context of total textural unanimity see m. 64111 inEx. 3-3lf Horizontal noncongruity grows out of a passage of strong accents inconformity with the notated bar-line-a final step in the transformation of the motive as represented in Ex. 3-3le.

At m. 70 a decisive initiative

accent in accord with

that bar-line

2. _. launches In motive a which sequentially is . repeated to form four contracted

units, an extremely vitalizing asymmetry before a further thematic entry again restoresthe bar-line without equivocation. It is significant that in both of thesecontractions mm. 28-32 and 7072! Beethovens textural components are in full accord, an indication of the force with which metric process hereapproaches disruption of the notated

394 rhythm

and meter

Ex. 3-31f. m.64 A'

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I;

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=L'l§1I-== = lin--1- I 92 2

I 92

I 92

d

' v .1 pr we0 'cisnon

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=

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*As to agogic accentand anacrusis. '|'As toinitial anacrusis,pitch accent,and preconditionedbar-line.

bar-line. The style, ofcourse, doesnot permit maintenance ofthe Huctuant tendency: invariably, metric dissonanceis followed by resolution. Unfortunately, the movement hasto be left with these fewsuggestions of areas of potential exploration see also Ex. 3-20, concerning the second movement!. Chopin. Prelude in E. Op. 28. No. 9

The vital theoretical question of the relation between intermensural grouping asmanifest in phraseology cadentialpunctuation and resumption! and grouping as delineated by metric accent is of recurrent importance, especially sincethe latter clearly requires the knowing involvement of the performer while the former is morelikely to be explicit in the phenomenon of cadential punctuation itself. Such an issue of possible noncorrespondence between phraseologyand meter-i.e., of phrase grouping as opposed to accent-to-accent groupingat phraselevel-arises in many works of Chopin, including the piece which is now to be fairly comprehensively treated. The Prelude in E is reproduced in part as Ex. l-18, p. 68, and brief summary extracts given in the following series shouldmake the discussion comprehensible; ideallythe readershould havereference tothe entire piece.

rhythm and meter 395

In instances of formal phraseological! symmetry the problem of organization of higher-level metricunits can of course be elusive and difficult; yet, the question is inescapablein arriving at interpretive decisions. In the Chopin Prelude in E, twelve notated bars fall into three phrases conspicuously felt in cadential punctuation and resumptions ofcommon initiating material at mm. 5 and 9: the entire form is an enlarged period with two antecedents, eachending on the primary dominant harmony but with some striking tonal diversions in the accelerated tonal rhythms of the second antecedent and in the final, consequent phrase again, see Ex. 1-l8!.°° It would be easy to assumeparallel metric structure in which phraselevel meter corresponds tothat of the normal four-beat measure,in which phrase-level impulsefunctions directly reflect thoseof the mensural unit.Such a view would see the last two phrasesas shownin Ex. 3-32a; the quotation is in renotation in

diminution.

Ex. 3-32a. Chopin, Prelude in E.Op. 28,No. 9; last two phrases renotated in diminution. _ lo92 |31 1 'I

f|?|wi¢. ____._.. -_--1_."_ .l||~l; ____ 1. l il-_-_--_-=1:=:?192__"= 11 1 92l:_*l`1f'1'Y'1lY |'

_|

°°

P """'*./7°I`--=-P

i.. l: '

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Some hard questions areimmediately apparent: ! Is the forteinitiative of the first phrase a preconditioning frame of reference in which the piano-to-fortissimo dynamic progressions ofthe secondand third phrases are understood asresistant syncopationsat phrase level? If so, the representation shown in Ex. 3-32a could have validity.! ! Should the intermensural bar-line, on the other hand, be associated with be thought of as prior to! the climactic fortissimo, alsoa point of powerful pitch-texture accent? If so, should the metric structure be thought rj as beginning with an anticipotive unit of three impul.ses? See Ex. 3-32b.! ! Is there, in effect, then, a seriesof phrase-

/-l

level 3 -|- 1 groupings creatingmetric orders counteractively associatedwith Ex. 3-32b. The question of anticipativefunction inphrase beginnings in the Chopin. 4 4 . I i`/

_. _$_

_»

, uf:

_ ,4 ''

Ill All

' etc.

We are concerned here not with metric orderat thehighest level i.e., whetherthe three phrases are ina v 92/ - relation!, but with metricorder internalto phrase-levelunits.

396 rhythm

and meter

the symmetriesof formal phraseology? Doesnot the representation givenin Ex. 3-32b bring into focus somekind of truth about metric structure in the experience ofthis piece?! ! Does phrase-level metricorganization location of phrase-levelaccent! perhaps fluctuate from phrase to phrase? If dynamic stress ismeter-determining accent,associated withinitiative function in the metric organization or with .gyncopation in resistance to a preconditioned organization, such questionsare imperative to an understanding of this piece, whoselow-level regularities are opposedby higher-level, counteractive metric structure. These questions are reflected in the following sketch Ex. 3-32c!, in which various interpretations of metric structure are represented in addressing the basic questionof the location of higher-level accents ultimately,tlw structuraldownbeat!. Although these interpretations embody metric units of largely symmetrical relation, such a relation should of coursenot be construed asa necessaryassumption, aswe haveseen, despite the symmetry of formal phraseology. And if these interpretations are dismissed in favor of that of metric structure in compliance with phraseology

Ex. 3-32c. A representation of the Chopin asto broadmetric structure in whichhigher-level accents are seen asnoncoincident withphrase beginnings. Phrase-level renotated,° meter J=.! 9

ll U",

ll,

ii

_ ,QJ J Q' .Dil al £152 .L_ ablqlhl. gli; t riff;

_,isa -it

or aflnctumt phrase accent? !

1 accent of stress, pitch, maximal textural expanse, extreme tonal distance, etc.

*C-ll groupingas tophrase punctuationinception tocadence! and, on theother hand, phrase-level bar-lineto bar-line.

rhythm and meter 3.97

initiative impulses at phrasebeginnings!, it is impossibleto conceiveon what basis abroad, primary accent might be deduced. Although the particular formulations of thesequestions mayhave the appearance of unorthodoxy, it is easy to see them, with thought, as very familiar questions, and very constant ones in analysis and performance, couched here in terms that direct attention to musical events as qf metric, thereby rhythmic, significance. Thefamiliar, analogous questions might appear in discourseabout music in forms like these: What are the central points in the piece,the primary events towhich succcssionslines of intensity, motion,

directions ofactivity! are aimed? Whatevents inthe pieceare of special moment, of particular centrality of rhythmic function and interpretive projection? Or, what is the point in relation to which all others are, ultimately, subsidiary? Comparable questions are the basis for the following inquiry into metric structure in examples fromWebern. Webern: Three Pieces, Op. 11, for cello and piano, No. 3; Five Pieces, Op. 5, for string quartet, fourth movement

Weberns'meters are generally thought to be more fluctuant and noncongruent than those of the other two of the three great figures of early twentieth-century serialism, Berg and Schoenberg. And it is very possible that Schoenbergs vast influence with respect to the organization of pitch content in post-Romantic music, an influence reaching dominantly across the century, is equalled by Weberns influence in certain of his works especially! in the direction of increasing metric flexibility. Moreover, metric functions of

the kinds to which

we have devoted much

attention in

these

pages-progressive and recessive functional changes, vitalizing noncongruities, motivic meter, and in general the applicationsof rhythmic factors in the functional and expressiveshaping of music-are significantly discernible and important in Weberns music. The third of the cello-piano piecesof Op. ll can be seenas anexample of what is sometimestermed nonmeter, antimeter, suspended meter, or ameter Ex. 3-33a!. The senseof relative suspension of meter isunderstandable inview of what might be describedas verytenuous articulationat a predominantly subdued dynamic level in this very short piece. But, again, no series of differentiated soundevents canfail to assert metricgrouping, with its inevitable implications for structure of relatively stable or fluctuant orders: in these conditions meter may well assume avery unstable and, at times, ambiguous character. Then, it exists in a particular state; it does not ceaseto exist. And it would be argued here that in Webern meter is palpably functional if not immediately apprehensible!; metric structure is'shaped in telling configura-

398 rhythm

and meter

Ex. 3-33a. Webern, Three Pieces for cello andpiano, Op. 11; No.3.

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tions far from the erratic, contextually arbitrary, apparently aimless state in which change cancease tohave functional effect. It is true that in a music of severe dynamicrestraint and minimal eventfulness with embodied silencesof constant relevance! nearly every attack may be felt as initiative of a metric unit at some appreciablelevel, especially in an atmosphere of slow tempo, with consequent considerable distances between articulateevents. Thisis true in a sense comparableto that in which music of great eventfulncss, especially in a texture of constant, complex

rhythm and meter 3.99

interactions, conveysan impression of densemeter ty" continually variegated and competing implications, as doesthe first movement of Weberns Op. 22, cited as Ex. 2-22 and referred to again in the present chapter. Let us consider the example from Op. 11 as to qualities and implications of many of the attacks,as designatedby lowercaseletters in the foregoing quotation. In view of its extreme dynamic and pitch inferiority even considering the limited dynamic potential of the subsequent harmonic!,

pointwhile a!, initiative at the low level of the F unit, is anticipative in relation to the more important event b!, recipient of the directed thrust of the preceding, anticipative wind-up! trill, and strong in its .gf attack, pitch, striking coloration, and in relation to the subsequent intensityrecession and silence.Point b! embraces thefollowing two notes asa mensuralgroup, recessive inrelation to it, these impulses, with subsequent rests,constituting

in the celloitself partEaf2+F! unit. Point g! initiative is a further of 3

unit. Points i! are relatively brief; this factor, along with their relative pitch inferiority, seems to convey the senseof function as anticipative impulses in relation to the agogically stronger and higher pitch of the final note, whose metric unit is of course indefinitein length; they at the sametime form low3

level, subsidiary D units.

a

Point initiatinga j!, unit, final, isthe highest pitchthe in piece, quite possiblythe longestin duration, and of striking coloration-a relatively high harmonic on the cellos highest string-and it is not unthinkable that all precedingevents might plausibly beregarded asits anacrusticpreparation i.e., as having the highest-level function of preparing the final sound!. ll _ _ Point inc!the piano initiates a noncongruent U unit since second the

component, point e! and the note following, is clearly anticipative V-like approach, erescendo! in relation to the accent g of the pianos upper voice." The anacrusticpreparation of the pitch g, just described, andits pitch superiority throughout the piano part, as well as its position as point of highest dynamic intensity in the piano part, lend this event, labeled f!, a feeling of major initiative function and centrality for the entire piano part. With the recessive lowerpitch, dynamic reduction! harmonic interval EE, eb! which follows immediately, and the subsequent silence,point f! is initiative of

another of unit Elfé ! inthe piano cfl size thethe of preceding one!!. A s A number of references to the pianos gof m. 5 mustnot beconfused withthe attack labeled g! in the cello.

400 rhythm

and meter

final unit is initiated point h! by an event of maximal density-number and -compression. Point f! has, with the harmonic interval entering below the attack on g, maximal textural expanse for the entire piece-a factor in its accentual primacy." The final piano event is, broadly, recessive inrelation to the initiative accent

of m. ~5.

Seeing thecello and piano parts as metrically noncongruent, interact-

ing, relatively indepqndent structures, we arrive at thefollowing summary

Ex. 3-33b!, taking ab! 5 as smallest a common denominator." is a It structure of pervasive asymmetry,yet of unmistakable progressive-recessive functions and overall poise, its chief accentual eventsoccurring as balancing rwrential pointsjust Q#center. Thereemerges animpression of noncongruent meter in which a high-level bar-line for the two strata of events is disjunct at an only slightangle nearthe piecescenter-a circumstanceof splendid equilibrium in the structure, with constant asymmetries andfluctuation at the same time.

It is evident that critical interpretive decisions restupon understanding yielded bythe aboveapproaches, orupon somecomparable, justifiableunderstanding. Any interpretation requires, of course, enormous subtlety of projection but firm awareness! in so tenuous and vulnerable a context of restrained articulation. Again, the questions which are the basis of inquiry in the foregoing analysis are very critical ones, as are the conclusions to which they lead: for example, shapeddecline through the b-bb of the cello after the accent of m. 2, or treatment of the pianos f if-G# of mm. 3-4 as anticipative preparation of a leaning toward! m. 5. And the suggested conception of the metric centrality of the cellos f at point g! points to a delicately recessive,withdrawing articulation of the concluding three attacks. That Weberns music isa languageof subtle metric flux and polymetric structure of often elusive definition is apparent in almost any of his works. Compare Stravinsky,whose accentsare in general somuch lessequivocal.! At the same time, Weberns meters, and fluid rhythmic surfaces, are too little understood in real depth; grouping influences are often especially interdependent-motive often very influential, even essential,in conditioning meter in atmospheres of considerable equivocation of accent. In this respect, an occasional work, like the Canons, Op. 16, for soprano and clarinets, is significantly exceptional-not in the common equivalence of motivic with metric unit, but in that works unusual decisiveness of accent, Study ofthe pieces structure from thestandpoint oftextural-spatial shape illuminates one qfits mostpalpably perceptible aspects. The pivotal spatial expansion ofm. 5 includes, of course, thecello f1. The unit taken asdenominator has the valueof courseof avoidingfractional expres-

sions,that so twelve units per notated bar J ! provide basis a accommodating for the in

symbology ofEx. 3-33bdivisions offour, three,and sixwithin the notated measure.

rhythm and meter 401 Ex. 3-33b. Metric proportions in theWebern piece.

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I4 40

+? 32 rr ,¢92 iv *so

evident in the first piece of the set.! Another Webern excerpt is given as Ex. 3-34a; it is a short movement of much intrinsic interest for the study of meter and other structural elements. Example 3-34b is a sketched representation of implications and questions of metric order and ambiguity at the mensural level in its opening bars. There are many devices,with typical understatement ofcoloration and rhythmic nuance, by which events underscorethe notated bar-line and the

J pulse: on-the-beatof attacks mm. 1-2, with the viola attackm. of2, third beat, emphasizedby anacrustic approach; the overall, general steadiness of

articulation theof,alsubject to various divisions; initiative attackthe of violin 2 in m. 4-, part of the clarifying tendency in relation to the relative metric indecision of m. 3; similarly, the initiative attack of the cello in m. 6; in mm. 7-9, the bracing effect of viola Hguration the first violins initiative b in m. 9 is steadying asthe viola rhythm is scattered!; all of these are examplesof metrically clarifying events relating to the notated bar-line and the beat. Moreover, there is the omnipresent device of anticipative impulse at every level, normally preparing attacks on the beat and/or at the notated bar-line. A few of these clarifying anacrustic events are given in Ex. 3-34-c.

402

rhythm and meter

Ex. 3-34a. Webern,FivePieces,Op.5, for string quartet,fourth movement.

SehrlangsamJ=.ca.58! am Steg Vln.l

Vln.lI

Vla.

Vc.

m.4

zogernd

im tempo am Steg- - - - - - - --

rhythm and meter

403

rit. ----

tempo !b,_92

verklingend

_/m'7 zart so rnéglich als IF E ,F ' /I? 1 °/ ~ 1 7' G5 ml :_-=_

-- *

am_Steg-- _....... -__ .... --- ....

-1

RM %l o0 o *| pizz° .' bo o .4 .o I' r . 0 |»;92i111.;ii.111lh.;=1_;i1.; l|`!lY§i-QFHZQZZCF--_F--_ 7 .li l|[ i li* _4--i "J 9 33 I3

-` -1

_ .... -- ------- -_

0

.o

°o

l' r

. 1 = _ _ . 'L |__3_| ;'

_...QT ,_-2-7 8 3__|

K _.ff

I

-O

0

:-

H? --

111.10 pk

J *n

--

tem rit.'

ll .-

1 L_#

tempo

=l M -l '

é|

fl-» _

_ _ __ __ __

,I } am Ste ", am /»Steg `Ili]--{ll§Y*lJ11-111| L ,-92 I pizz. arco

I' _

V ,"

'-

l1

--'-$:::.:=!:f;==:

_ viii:/'

`&'usserst ruhig _ ui

al-co ,-3 'I am li*§ i'i=ECl|'J= If#=¥IT_'£If&= on

Stag e

A am

;r

``

: I 'mf ._.__. i

il

»-92 :_ fliichtig pnzz. '

Steg Pizz_

V HP Copyright 1922, Universal Edition. Used by permission the of publisher. Theodore Presser Company sole representative United States, Canada and Mexako.

rhythm and meter 405

persuasive anticipativefeeling. There is a suggestion oflinking, conjunctive,

functionally ambivalent conclusive-anticipative! tendency one associates even withthe finalstatement of the motive." The general importance of anacrusis inexpression ofmetric structure is thus, with some ambivalences,evident in these motivesand in purely specimen extracts quoted in Ex. 3-34c. Additional examples of anticipative impulse can readily be identified. Let us consider proportional relations in the movement as to two di:tinct modes ofgrouping. Onebasis ofpartitioning is the capacity of the piece to be divided into four, extremely epigrammatic sections on the basis of ritardando zogemd, etc.! and a tempodelineations. Each of the sections so delineated mm. l-2, 3-6, 7-10, ll-13! begins with a rest, and these rests are felt as parts of the sections II, III, IV! which follow rather than asymmetrical extensionsof preceding material in view of ! relatively consistent preconditioning of the notated bar and ! the a tempoin which each of the

rests question in included. is relative The proportions expressed eb in values! are shown in Ex. 3-34d. Ex. 3-34d. Relative proportions among fourtempo-delineated units in theWebern.

I. 10D+ fit. + opening rest2ofJ5 ?! II. 24J!-P fit. III. 24D+ fit. IV. 14

+ 4 of rest + internal rit.

Of the outer sections,IV is slightly longer while both are significantly shorter than the equivalent II and III!, although if the opening rest is included aspart of I When does thepiece begin?!-which seems more problematic than the concluding rests, more easily felt as incorporated on the

These recurrent, unifying motives are thus punctuative! and 92 anticipative /P! , i.e., ,

in feeling, their highly expectant qualities looking aheadwhile other factors

especially tempoalliliations and subsequent rests! seem punctuativeof prior events. It is in this perspective that eventhe linal appearance ofthe motivecan be felt asa kind of curiously anomalous up impulse a quasi-anticipative gesture expecting but unfulfilledby thetic response!;yet, its more precipitateaction fastesttempo, shortestrhythmic values, marked jiuehtig! sets itapart, asnoted earlier,as amore decisive punctuation subsequent to the cadentialbroadening consummated just before.

406 rhythm

and meter

basisprior of structure-we have proportions of I= 12 nb and IV =l8|D in addition to uncertain added quantities occasioned byslowing of tempi!. Thus, flanking the symmetrically related internal units are the shorter I and IV, related in the ratio 2:3, the shorter duration of the first section appropriate to its introductory function, and the conclusivefunction of IV served by its slightly broader length than I! but compromised byits brevity in relation to II and III, its more active pace of tempo change accelerated tempo rhythm!, and its final a tempomotive, a casual, precipitate,punctuative gesture.

The segmentation in the relation 2-4-3 is thus deducible as the numbers of measures inthe four tempo-delineated sections Ex. 3-34e!.°° Ex. 3-34e. A further expression ofrelative proportions numbers of measures intempodelineated sections!.

mmmm 2443 In approaching the issue of broad metric structure, that of accentdelineated grouping, one can plausibly interpret as primary downbeats defining intermensural units the initiative accents of m. 5 the crescendo, extended anacrustic preparation and immediate thrust in complementary outer voices,etc.!, m. 9 expressedin violin 1 only, but dynamically stressed as compared with the c following, and of strong anacrustic preparation!, and m. 12 high pitch, some dynamic emphasis, more modest anacrustic approach, but all voices in concurrence!. All three of these accents affirm the notated bar-line and emerge inthe experienceof the movement aspoints of downbeat focus of more than local significance. Proportions, again of 12 - 12 -9 . 'T .e.,

.. 4-4_3!, can be seen as temporal intervals qfseparatzon of these

important intermensural initiative accents. It seems reasonable, too,to interpret mm. l-4 as of anticipative impulse function at the broadest level: consider, for example, the relative ambiguities of these measurescoming into relative focus following their It is interesting thatthis modeof partitioningagrees with others: thatof articulation by rests eachtotal silence,except forthe anomalous eighth-rest ofm. 12,setting offthe final motive,begins one of thefour sections! and thatof relativelyfixed registrallocus foreach section i.e.,the core of essentialactivity for each isa tightly restricted ambitusof pitch events, differentfor eachsection!. The issue is againgrouping byaccent to accent or bar-line tobar-line! asopposed to other modesof groupingnoted.

408 rhythm

and meter

tonal and harmonic rhythms and relationsof PC content-e.g., those of m. 5 and m. 12, key points in the metric structure!, as well as motivic-thematic relations beyond their ancillary mention in the foregoing study.

Some twentieth-century

problems /h rhythm and meter?

recent developments in seria/ism of durational un/'ts-theory and practice

The severity of changein approachesto and concepts ofrhythm in twentiethcentury music has provokeda breadth of theoretical concern peculiar to the present time, and one hopes thatthis concernwill eventuate in an accelerating rate and range in the study of rhythm, toward better understanding of rhythmic-metric structure and effect in all music. Certainly rhythm in twentieth-century music is often of particular difliculty." Problems of meter in the twentieth century center as with melody, tonality, and other elements! in extremes of fluctuation. The analysis of meter in recent music often reveals ! an increased asymmetryand noncongruity in both horizontal and vertical dimensions; ! an increased fragmentation of metric structure resulting from increased tendenciesof the motivic surfaceto relate to and condition meter and grouping; ! occasional inapplicability of the referential unit of regularly recurrent beat, asin certain contexts ofrandom succession;and in some music! the prescribed, predetermining serialization or other premanipulation! of rhythmic durations and metric grouping, even at all levels. The field of possibletechniques inthe serializationof rhythmic elements is of course endless;any factor by which numerical sets are conceived or derived may yield a basis for serialization of durational quantities. The composers precise techniques in rhythmic often an aspect of total! serialization canbe a very elusive,sometimes virtually indecipherable, problem, since the composerchooses amongan infinite range of possibilities of procedure and permutational principle. Of course, serialization of any ele"Nevertheless, theproblems oftwentieth-century meterare easily and often!overstated. Thereis, forexample, anenormous volume of significanttwentieth-century practice, including someof theimportant literaturescited in foregoing analyses, in which the notated bar-line haspractical importancein representingmensural structure.It is possible thatthe principle ofuncertain significance in notatedmeter isespecially applicable to certainliteratures ofthe presentcentury. Butit is also possible that the reverse issometimes true-that in tradition metric changewas conventionallyeH`ected in devices ofchanging pattemsof initiative accentwithout generallyaltering thestandard ofnotated bar-lineconsistency, but that in many morerecent styles change ofnotated meterhas itselfbecome a convention, more often asignificant clueto realstructure.

rhythm and meter 409

ment has repercussions formeter, since meter is the outcome of variously differentiated element-eventsand rhythmic durations. For example, interval content of the twelve-tone set the IC set! may yield a series ofnumerical values Ex. 3-35! as basisfor a set of durations. EX. 3-35.

|

'

IC: `__4__,

921-1.1 ii! Or! é¢_.4_-;/ 92 ,1

etc.

L

withas ohunit! .DF...D.l5 Or, a durational set is invented conceived or derived from some extrinsic source! as a predetermining concept subject to variation as is a twelve-tone set in classical serialism!, employed as a kind of mode in permutations analogous to those of classical pitch serialism. Messiaen, for example in Mode devaleurs etdinten.sités, oneof Quatre Etudes for piano!,

derives of sets durations simply twelve as even multiples of , then nb ,

then J!. Luigi Nonos Canto Sospeso Ex. l-51! usesseries the 2 35 l 813 the so-called Fibonacciseries, afterits thirteenth-century originator: each integer after the initial consecutive pair the sum of the two preceding!; the series is applied to sets ofnote durations in which it is expressed invarious permutations, for example, its retrograde-13 8 5, etc. Most procedures in rhythmic serialization are based in durations derived directbfom numerical parameters inherent inthe PCset itseU'; some examples of these aredetailed on the following pages. Of great importance in the evolution of serially-orderedrhythm is the composer Olivier Messiaen, notonly becauseof proceduresembraced in his own music but becauseof his influence on a very great number of younger composers. Messiaenuses arhythm series in his music as early as 1940, in his Quatuorpour lafin du temps." "Quartetjbr the End qfTime. See Messiaens own discussion of his rhythmicprocedures in the book, Technique de mon langage musical Paris: A. Leduc, 1944!.

410 rhythm

and meter

Messiaens Ilede jeu2 the second ofthe Four Etudes forpiano! employs sets ofattack qualities !, intensities !, PCs 2!, and durations also 12!. The twelve-tone set is shown in Ex. 3-36a; included is its first permutation, Ex. 3-36a. Set for Messiaen's /lede feu 2 and its first, concurrent permutation, with the principle ofrotational permutation shown.

I3 l2

b 34

l lz:|

56

lw

he Q

76

85

lu ¢

L_' I

94

Ih 78

3 by 10 3

.Q 9 I0

ll 12

'

_h ll 2

a

l2 l

IC

l23456;?89l0lll2 `§'*;'_./_!¢t¢. 64

2l

7/ 928/5929/ 920/392ll/ 922/ which appears as a lower-voice counterpoint at the point of initial application of the various sets, thesimultaneous statementof InterversionPermutation! I and Interversion II, quoted in Ex. 3-36b.

The duration seriessimply is set a of values extending fromcb l to 12 ob _ The permutational applied principle it is to analogous thatto which on the twelve-tone sets lower voice variation is based,as shownin Ex. 3-36c, following. At Interversions I-IIsee Ex.3-36b!, the upper voice has the duration series in the order 6-7-5-8-4-9-3-10-2-l

l-l-12; the

series occurs in the

lower voice in the order 3-9-10-4-2-8-11-5-1-7-12-6. This rotational principle of ordering durational values is as seen in Ex. 3-36c, in which a graphic representation of pitch ordering is given, followed by similar representations ofthe ordering of durations in the upper and lower voices. Durational values are as to .spacings qf attacks i.e., include rests!. There are, in all, ten such permutations punctuated by a recurrent

rhYthm and meter Kx.3-3B PCand durationsets '>n he -3Bc. Rotationalprincipleo ofordering o t Nieseiaea. 12

34

56

78

9 10

11 12

Uppervoicepitches:

Lowervoicepitches:

Uppervoicedurations: 7 5 8 4 9 3 10 2 11 I 12!

Lowervoicedurations:

4510 291t!3254 6 67

89

10

11

12

rhythm and meter 413 Ex. 3-386. Continuation ofrotational principle in Iledo feu2.

1.

a 11.

7

.2 se 3

PITCHES

etc. Z"-,:`.T` 45

67

89

IO I

1 12

1

4 10

3 ll

2 12

I

61

7

47

10

`92

.2 .5 ln.

10

il 91

lv. 2

etc.

59

etc. /"'*` 15

28

12

W Z"" etc.

11 12

96 92g.

31

V

etc. DURATIONS

1. 6

etc. /;92 84

2 11.

93

*

10- 2

ll

l 12

51

7 12

6

10 12

96

3

83

ll

'

.2 ac 0

42

ew.

8 Il

92`_`.7

.5

.S nl.

ll'

2- l

etc. /"" 47

92`:/ lv. 1

I 12

/-"etc, v5 6

29 X.

etc. *A rotation from thecenter ofthe originalrow, butbeginning with6 sincepitch rotation beginswith 7.

414 rhythm

or

and meter

Thisbasic set reacts modifications to theinPC set; for example,

if the PC set is transposedup a minor 2nd, a new sequence oforder numbers is derived, basedon the changed positionsof the twelve PCs; this new sequence oforder numbersthen yieldsa permutation of the original time set, as shown in Ex. 3-37.

Ex. 3-37. Principle ofderived duration set in Boulez Structures see alsoEx. 1-31!. P0

i E.| Q

E]

I2

34

7. #I '

#ZI 56

78

he I 9 10

S-

ll l2

P1

| In 7l

[H . I0 3

etc.

yields dh with aasunit!: 71

10 3

ju dh L45

etc.

The Boulez is thus an example of a totally serialized work in which all elements pitches, registers, durations, dynamics, etc.! are subject to prescription according to a predetermining system, applied in such a way that for a particular pitch there is nevera recurrence of the same intensity, duration, etc. This, in contrast to the Messiaenexamples, isa procedure of constant renewal sought in many works of this kind. Milton Babbitt, in works as early as 1947-48,has employedpervasive serial operations, presenting in 1955 the following general description of procedures for the serialization of nonpitch components. The twelve-tonestructuralization of non-pitch componentscan beunder-

stood onlyin termsof arigorously correct definition ofthe natureof theoperations associatedwith the system. In characterizing the prime set, it is necessary to associatewith each note the ordered number couple-order number, pitch number, measuredfrom the first note as origin-required to defineit completely with regard to the set. Then, as transpositionis revealedto be mere addition of a constant to the pitch number, inversion-in the twelve-tone sense-is revealed to be complementation mod. 12 of the pitch number. In other words, pitch number 4 becomespitch number 8 .... ! Likewise, retrogression iscomplementation of the order number, and retrograde inversion is complementa-

rhythm and meter 415 tion of both order and pitch numbers. Any set of durations-whether the

durations bedefined interms ofattack, pitch,dynamics, orregister-can be, like the pitch set, uniquely pennuted by the operations of addition and complementation, with the modulus most logically determined by a factor or a multiple of the metric unit. Thus, the rhythmic component, for example, can be structured in precisely the same way, by the identical operations, as the pitch component; rhythmic inversion, retrogression, and retrograde inversion are uniquely defined, and combinatoriality, derivation, and related properties are analogously applicable to the durational set."

These operations, and consequent associations of order and pitch numbers, are illustrated as

Ex. 3-38.

Ex. 3-38. Illustration ofpermutations by addition andcomplementation mod. 12 asdesclibed by Babbitt. 1 [Addition

and complementation in a mod. 12

7 0' 2 3 1 5 6 7 9 '° U .gutemzdrawn lines from two numbers

00 11 22

12 23 34

35 4 :s 5»

67 9 1° 'I intersect at the number which is their 6 7 s _ 10 11 0 sum e.g., 8 +8= 4, as indicated!; 7 8 9 ,D H 0 1 numbers whose linesintersect at-whose

3 34 56 I 8 9lo I 0 I 2 sum is-0 e.g., 8 and 4! are complements.

44 56 79

10 11 1 2 3 ]

55

11 0 1 2

67

s 10

34

66789l1l0|1345 77s9101o12:145e 567 99l01l0l2345678

10l0l10l23456789 11110l23456789l0 ~. '

s

0,9 1,11 2,10 9,0

-' 0,0 1,2

0_

lzl

*i _

2,1 9,3

[The set abbreviated!; the first number

0 , ofthepairisthe ordernumber to ll!;

` the

second is the pitch number C = 0,

10,2 11,1 C, = 1, e,c_!_] l [Transpositionz

a constanthere, 3!is

added toeach pitchnumber.]

10,5 11,4

"3NIilton Babbitt, Some Aspects of Twelve-ToneComposition, in The Score and I. M. A. Magazine june 1955!, p. 60. Peter Westcrgaardrefers to these procedures and discusses theirapplications in Ba.bbitts Composition for I2 Instrurnents, in Some Problems Raised bythe Rhythmic Procedures inMilton Babbitts Composition forTwelve Instruments, in PerspectivesNew of Music,IV, l 965!, pp. 109-18.

416 rhythm

and meter

Ex. 3-38 continued.

0 "" 0 0,3 1,12,2 9,0 . ll,l l0,2

' *J

L*

9,0 2,l0

[lnversionz transformation of the series

of pitch numbers into the series of its 10,10 11,11 °°'Pl°°°°] [Retrogradez complementation of order

l numbers plus--i. rotated e., by-a factor l,ll 0,9

[Retrograde inversion: complementation

'» *J

H, 10,10 9,0 2,2

of ll!.]

of pitch numbers of retrograde series

1,1 0,3

order numbers as in retrograde, above!;

or complementation of order numbers of inversion, as in deriving retrograde of basic set.]

Procedures forthe serialization of nonpitch factors arethus established on the basis ofnumerical values paired order and pitch numbers in which C = 0! derived entirely from the row and its conventional alterations. By these procedures,all serial operations permutations of sets, combinatorial applications, etc.! are applicable to the derived set of durations, as to those of all other elements."

Since rhythm and meter are in every sensethe consequencesof differentiations amongthe qualities therefore the functions! of sound eventsas shaped byall parameters one of these duration!,to the extent that durations, pitches, andother elementsare preestablishedin serial operations, sois meter significantly predetermined, or totally predetermined where there is total serialism except when predetermination is precluded within a particular parameter by such practical impediments aslimited ranges of instruments!. Numerical values,however derivedand applied, yield proportional relations among units at all levels; functional interactions of such units are, in many serializations ofrhythm, problematic. Such proceduresthus raisethe question, one that is reiterated in this book, of control ofthe specific qualities ofsound events and theirinterrelations in and "Westergaard SomeRhythmic Problems. . . , p. 115! comments onthe problem of controlof interaction of pitch and rhythmicfactors and the questionof what happens to the traditional differentialrole ofrhythm . . . still operativein Webern, noting p. 114! thatwhile the precompositional construction of pitchand durationsets isrigorously parallel, their compositionaluse ishighly independent in view of the variability of the number of pitches perattack or attacks perpitch. He raises thepoint that in certain textures the rhythmic orderingperceived aasresultof theseoperations may not bethat intended.

rhythm and meter 417

as toparticular contexts and contextual alignments; inserial operations of these kinds, conlluencesof eventsare capable.of total or nearly total predetermination by systemic procedureswhich do not foresee, in any functional sense with which these studieshave been concerned, particular contextual associations- andconsequences. To the extent that serial prescriptionsshaping the various parameters of a work set in motion governing, or significantly conditioning, operations that in no way foresee contextual,functional relations and associations,they must be seen ascomparable tothose ofovertly random operations inconsequent Wet of arbitrary aftiliations of events. However, asnoted in analogous contexts, they are evenmore random in that the influenceof spontaneous,intuitive responsesof the performer, usually signiiicantly applicable to chance operations in music, are definitely precluded to the extent that events are

predetermined. I

The validity of the supposition of blind predetermination of associations of events in total serialism is denied by some composers, while acknowledged by others who consider the consequences ofunpredictability a desideratum to be sought. But even to the extent to which contextual associations may be controlled and foreseen in any given procedure, the question mustarise asto the nature and depth of functional signiticancesof such associationsin the experience of works whose unfolding is prescribed in this sense..

Concluding notes

This study has sought to provide a basis for the consideration and analysis of metric

structure, the

identification of

units at

various levels,

and the

analysis of weak-strong associationsand relations internal to those units; it has sought aswell to illustrate various kinds and applications of structural techniques-functional and expressive-within the parameters of meter and other aspects ofrhythm, seeing meter as the accent-to-accent grouping of events delineated around musical projections of relative strength. We have seen that in dealing with problems of rhythmic analysis, especially inits concernwith meter and other manifestations ofgrouping, one is often led to relatively equivocal conclusions in circumstances of some ambiguity. To some extentthis is due to inconclusiveness anduncertainties of knowledge about perceptual responses tomusical eventswhich result in groupings at different levels of structure; but the equivocal nature of analytically derived information is also to be traced to the fact that rhythmic structure is only rarely, and in situations of least artistic value, unambiguously expressed.Significant ambiguity in metric structure, apart from

418 rhythm

and meter

conditions of gross accentuationand obvious impulse function, is unavoidable, given the extreme range andcomplexity ofrelations amongthe many parameters ofmusical accent.For accentual values areof a scope potentially involving everyelement ofmusical structure; and the evaluation of relative accentual values is not just a matter of discerning and listing those factors, it is a matter of making difficult judgments respecting their relations of effect. It is a question not of simple comeasurement of absolute values, but of evaluating contextual relations dwrrent_hr ez/eg! musical situation. Hence, it is extremely unlikely that a purely objective evaluation of metric structure can be achieved for anything like a comprehensive scopeof real musical situations. Moreover, it is for related reasons that the penetrating study of rhythm engagesquestions ofthe natureand functional-expressive consequences cy" change in all theelements of musical structure. The difficulties of rhythmic-metric analysis do not, however, suggest resort to a passive,irrational insistence on purely subjective criteria, or abdication of the effort to understand rhythmic structure according to demonstrable evidence:plausible evidence,in the score andin the experience, can be adduced in support of interpretations of structure, even when the complexity of real musicis suchthat the evaluation of such evidence-the facts explicit in the notation of music and in general or particular discernible responses thereto-cannot more than rarely lead to perfectly unambiguous conclusions.

We have sought in this study to identify some ofthe objective data and circumstances bywhich rhythmic and metric structures canbe understood, and by which such understanding can illuminate the decisions that have to be made in performance or in critical or stylistic analysis!if interpretation is to be more than mere execution.

Since rhythm is, in a real sense, all of music i.e., since all musical eventfulness is-usually functionally-rhythmic!, a study of rhythm is an appropriate vehicle by which to conclude an exploration into musics structural functions; in rhythmic processes weare reminded powerfully of the interrelations and interactions of element-successions indevelopment and resolution. We see hereagain that only in simplest situations are elementactions in clear convergenceof functional direction; real music bespeaks,in the projection of cooccurring, confluent element-structures, a miraculous and continually absorbing complexity that no musician would deny. That complexity is evident in the breadth and diversity of problems and approaches inmusic theory,and in the elusiveness of understandingof complex structure and of the relations of structure to experiential effect. Many of the observationswe make in analysiswith respect to tonality, harmony, melody, texture, and rhythm and concerning such further dimensions as color, dissonance, dynamic intensity, articulation, density, space, andso on! are citations of fact, but the farther we move beyondsuch

rhythm and meter 4

1.9

facts into the evaluation of interrelations of eventsand their functional-expressive rignyicance, the more difficult, and the more important, the analysisof music and of the musical experience becomes. Nor do we demeanin any way the understanding derived in analysis when we submit demonstrable facts of musical content to varying, plausible interpretations whose relative evaluation is the analysts further, perhaps most exciting, challenge.

NOTES The twelfth part ofSchoenbergs Pima! Lunaire Galgenlied! is astriking exampleof the delineation ofan entiresmall structurein significantpart by controlled progression in tempo asto both its parameters!in a line of acceleration qualified only by moderate cadential recession in the ultimate measure. Complementation of this tempo-structure by dynamic changes, texture, and pitch rise isimmediately apparentand of potent functional eB`ect. In a further example,the third of SchoenbergsPiano Pieces, Op. 19, tempo deceleration complements an explicitdynamic succession of release:

éi *;

'P *PP

@

bRecessive change in rhythmic motion is of coursenormal as a facet of cadential function; thisis strikinglyevident incomposite rhythmic activity in so manyexamples that furtherillustration hardlyseems indicated. But seeEx. 1-49,p. 163,from Weberns Op. 7, No. l, an excellentexample ofactivity-tempo curveapplicable tothe entire piece-progressive, thenrecessive. Note too the parallel fluctuationin pulse-tempo: the lateritardando, and final fermata.Related actionsof otherelement-changes, some of them treated elsewhere in this book, shouldbe considered.The activity curve is instructively seenin arbitrary isolation; that composite up,down! activity curve is as follows: fit. ........ .... 92"'3"92l*"92*92 r_J-ar-3-1 113+

t_

°/ r

* Ill

I1

Ill lllll

!!! !!!

!!! !!!

!!! !!

71

W

4 °Metric and other typesof groupingcan in themselves assume motivic significances. An instanceof this at the level ofthe phrase!may beseen inthe final movement of

Beethovens Symphony No. The 2. initial themegg ! is itself a period of sixmeasure phrases. But what is strikingin this movement isBeethovens employment of six-measure units at crucial points throughout.See, forexample, mm.26 the theme introduced here!,84--89, l39-4-3 and therepetition, 151-56,165-70 or166-7l!, 21027, 282-93, 290-95, 300-l l , 312-21 a 4- + 6 unit!, and other cases. Thus,the character andordering ofgrouping in the abovehaving to do mainly with unit size as opposed to internal ordering! may be functionallymotivic. Or, the five-measure unitof theHaydn themeemployed byBrahms inorchestral variations is motivic in this important sense,and the five-organization is at

420 rhythm

and meter

points inthe variationsa primary and essentialfactor ofthematic relation,as it is in the concludingostinato form. As to all of these factors,regrettably limitedreference isafforded bythe Beethoven example Ex.3-6a!. With specific regardto harmonicrhythm it should bereemphasized thatthe qualitativeaspect, as with other element-rhythms, is conditioned bythe nature and degree qfehange. Even insituations ofrelative stasis, a kindof harmonicchange e.g., a rhythm of harmonic inversion!can of course beinduced, and it can be delicately movingin functionalefl`ect. Forexample, inthe openingphrase ofBrahms Intermezzo inEb, Op. 117, No. l, harmonic rhythm is in one sense inert almost entirely I!, in another sense subtlyactive: the feeling of tightly restrained eventfulness, in varying inverted positions of the prolonged I, is beautifully expressive within- avery narrowspectrum ofmovement. °Element-rhythms can be symbolized and represented by conventionalrhythmic notation as is harmonicrhythm very commonly! or, graphically, asto any number of devices analogous to or extensions ofthose ofEx. 3-6b-e.g., by numbersof events within any given unit of time the smallerthe unit the morespecific therepresentation!. Qualities of events can be symbolized as, for example, p L 1, ._

4 g,

f il

f ll, or by numbers representing any quantifiable factor

texture-space, etc.!, or by chosen symbols to representsuch itemsas tonal-harmonic events involvingstrong factorsof chromaticism,dissonance, distance, and the like. Representation of proportional relationsof event-groups can beshown byany graphic device ofmeasurement-e.g., bylines of`varying lengths.Exercises inanalytical and graphic illustrationof element-rhythms and theirinteractions can be greatlyinstructive. Examples forstudy areof specialinterest whenparticular element-rhythmic functions are concealed beneath asurface ofapparent regularityof too narrowly conceived! rhythmic eventfulness;consider, forexample, ChopinsEtude in EI; minor, Op. 10, No. 6, as to fluctuations inharmonic and,especially, tonalrhythms. The question of where the duple division is initiated is more difficult than it may appear, yetan important one for the performer to resolve.In one sense, thefirst notated beatof m. 29 includedin Ex. l-3! is the initiator: supporting thisview are the preconditioningeffect ofthis pointas mensural downbeat inthe contextualnorm, and theharmonic-rhythmic change; in anothersense, thefollowing beat is initiator consider itsapproach byleap inboth outervoices, and its pitchsuperiority inrelation to the impulse following!. While this situation is cited as equivocal manycomparable analyses raise questions of this kind!it is clear thatthe interpretationarrived atwill radically govern performance andafrct thenature qfthe resultant asymmetry. Ifthe first alternative is regarded as correct m. 32, thirdbeat, isconsidered weak. On theother hand,assump-

tionthe of second alternative render will beginning them. weak, 29of effecting aF

unit in the realmeter commencingwith m. 28, requiringarticulation of the second beat ofm. 29as amensural downbeat, and causingthe entiresequence to end with a strongerimpulse, anda resultantstrong-strong mensural pattern throughthe barline separatingmm. 32 and 33.The questionmight be summed up:Do sequential motivic! and metric unitscoincide? See Ex. 3-IO, amongmany. Afurther examplein which a fairly low-level anacrustic complex cf.Ex. 3-10!has itsown distinctlower-level metricstructure isthe powerful opening gesture of StraussDon juan:the accentit prepares an upper-voiceg#3! is strong inpitch, duration,and noncadential tonal changeof strikingchromatic thrust: .. §¢& 'Z f II { |J '|`}'1hI' l-iIf_f_Ll1l1-ill-.-l|'il:f_$2l|'l'Y'b Q---_-32-__ l._|-Ll'-l--_ , -7' 1_ Q U'_l1_1 Y' 'D li' _ |:v- I' S ' "' ' P* I |

rhythm and meter 421 Only one example canbe treatedin the above discussion, but the entire movement should ideallybe explored in all of theseconnections. Note, in asingle limitedinstance, mensural progressive fluctuation ofgreat intensityat mm. 129-39, adevelopmental area. Theinitial attack at m. 139 isa downbeatof broadstructural importance;it is

approached a process of metric inacceleration the pattern ,in and following the peak of this progression underscored by dynamic and pitch-line 23

changes!isthere recessive a actually -hemiola!, F complemented by retardation

in harmonic rhythm. Do these processes suggest slight,supportive accelerando and trattenuto inperformance? Suchinterpretive questionsare of special relevancein Brahms, asin comparablelater stylesof greaterflexibility. The questionof the accelerative asopposed todecelerative effectof hemiola comes uphere-the questionas towhether itis perceived as threeshorter unitsof two or onelonger unit of six, a matter subject qf course interpretive to control. One important factor wouldseem tobe therelative accentualforces oflow-level impulses initiating

the and second Jthird units: #V F or E E ,again amatter consi subject toperformance controlon the basis ofanalysis offunction. In the Brahms 3

example given Ex. 3-l8a!, the decisively stronger accent initiating F the unit, and

complementary recessions of harmonicrhythm andpitch-line descent,are supportive of broadeningeffect. 'One of the movementscited earlier Ex. 3-9! in this regard, theAndante of Mozarts Symphony No.4-l in C, K. 551 f]upiter!, is alsoa splendidexample ofintensifying metric progression in the passage following original thematicstatement. Beginning at

m. 19, thereprogression is the in pattern of Jbeats!: - 66- 2- 2- 2- 2.'I`he unit

of6 r,as in mm. 19-20, is established by insistent motivic recurrence; the initiative accentual character of its first impulseis a product of strong factorsof pitch-high upper voice,low bassvoice, hence broad texture-space ; big leaps; forte dynamic level; duration; and others. With the metric contractions of such palpableeffect in m. 2351 harmonicrhythm is accelerated. Inconsulting thispassage forsupplementary analytical thought,the readershould consider factors ofrecessive approach to cadence following the progression notedabove, andthe subsequentincreased stabilityand symmetry andcorrespondence to notated bar-line!in accentualpatterns ofthematic statement after m. 28.'I'here isrenewed metricacceleration soon after. Themovement is, all in all, a truly rich resourcefor the study ofmetric structureand process. -'The concludingbars ofthe motetsfirst part project oneof twohighly diversifiedtextures the other is the approachto the m. 54-cadence! whichare focal points for progression of many element-structures of directedintensities. Thistextural diversity is manifestin part in me-tricnoncongruity inthe final bars, whereduple andtriple units establishdivergences throughout the texture and amongcontiguous segments in an atmosphere of extraordinary flux.The factthat the two mostcomplex segments in the work are at immediateapproaches tomajor cadencesone of them the final cadence! is significant andstriking, andit would be amatter ofgreat interestto know the extentto which this is true in works ofjosquin. _]osquins Sanctus from the Mass Faisant Regret: isa further useful subjectof reference: itsvoices influence one anotherand interactexpressively in the extentto which downbeatsare felt. Most important in this sense isthe steadyinginfluence of

422

rhythm and meter the derivedphrase inthe tenor,whose uninterrupted and unperturbedintermensural meter

4l

..

._

o» -sanc-tus! is a foil for the fluctuationsof other voices inonly relatively

restrained complexityas comparedwith the De przyfundis. An evidenceof beautifully expressive noncongruity, for example, occurs in the conflicting tenor and bassus attacks atm. 5H`. The uniformityof the metric structurein the tenor is compromised _. .. .7 _ only at the end,where itsfinal unit is broadenedinto an asymmetncally related o~1n recessive action toward thefinal cadence. *Comparable displacements of the bar-line are of great interest in Liszt, as is evident inthe quotationfrom oneof the Transcendental Etudes, given inEx. l-13, p. 57. 1The Finaleof theHorn Trio is musicof supremerhythmic energyand drivein a context ofrelative texturalsimplicity. Atthe outsetthe rhythmof pitchchange brings agogic

emphasis the half-bar to initiative impulses in the rhythml JJ! J

J , etc.

The bassof the piano carriesthis out in a different way,with the agogic valueof the Eb expressed as anote followedby rest; but the reader will appreciate, ifhe listens carefully tothe bass, that thesame rhythmis thereexpressed in the underlyingpattern of pitch change. Within this apparent regularitythere isregularity atintermensural levels too, as againcan be felt with imaginary conducting of notated measures as impulses forming groupsof two,or four,or eight!,there arestriking syncopations in theopening bars at intramensural levels. One ofthese ismanifest inthe irregularstresses of mm.

3-4, which recur later, in asense superimposxng D units in trochaic rather than

iambic patterns as rhythm to pitch of change: ! atpointsof out joint with the notated, otherwise supported, bar-line.No doubt this is felt assyncopation against apersistent, strongly asserted regularityof mensuralorder. The fourth Etude ofElliott CartersEight Etudes and aFantasy for woodwind quartet is excellent forstudy ofopposite, precipitate, progressive metric process. Its opening bars, for example,should beconsidered as to metricconsequencesevents of as they take place, without regardto preconditionedaccent andassuming noarbitrary bar-linestresses

in performance. Is the92_ motive perceivedvas - or - 92./ In ?! thefirst five measures ofthe Etude,metric acceleration in the series 6- 4 - 3 - 5 - 2 - l takes placein unmistakable intensification, with counteractivechanges indynamic intensity. For further illustration of metric progressionin which tempo, crescendo, and dissonance function complementarily andstatic harmonicrhythm in counteraction! see Debussys lAprés-midi dun Faune, mm.42-46. Immediatelyfollowing thisarea in the work there is metric structureof broadenedbut vertically noncongruent units, and continuedrecessive process in cadentialpreparation. Examples of textural processin Bartok, discussed inChapter 2, will be found in analysis tohave significant correlative features of metricacceleration and deceleration as afundamental expressive and stylistictrait. The opening ofthe StringQuartet No.5 see Ex.2-l l , pp. 218-20! isa caseinpoint, ascan beseen inprogressively contracted intervals ofrecurrence ofthe viola-celloaccent ofm. 5 in the bars following.Another of manyexamples in the same movement is the approachto theimportant hypermetric

accent m. 170 atifégiz-ll '8 4' 8' !,or the to stronger accent

following, or comparable processes involving functional progressive andrecessive metric structures.

rhythm and meter 423 °An interesting application of the principle of rhythmic-metric content asthematic arises inStravinskys Variationsfor orchestra,whose secondm. 23!, fifth m. 47!, and eleventh m. 118! sections area recurrent refrain of contrarhythmic twelvevoice polyphonyin which the factof thematic recurrence ischiefly asto rhythmicmetric content.Thus, whilethere arechanges inviola 4-,and in the double-bass! of minor dimensions, the polyphonyof SectionV is a restatement of therhythmic-metric content ofthe linesof SectionII, etc. Rhythmic-metric contentthus is motivic asa basis forrecapitulative formin theVariations. Themetric structureof theopening six bars isalso thesubject ofrecapitulative procedures; its metricasymmetry isa product of horizontal rather than vertical noncongruity in a context of extreme textural simplicity at the works inception seeEx. 2-4,p. 197!. Discussion ofthese elements of rhythmic-metric recapitulation, and of metric scheme of the twelve-voice polyphony asindicated inits metersignatures, is included inSpies, Noteson Stravinskys Variations, in Perspectives qf NewMusic, IV, l Fall-Winter 1965!, pp.62-74. PSee, for example, theopening ofthe development, where akind ofmetric acceleration

occurs with piled-up elisions of themotive, makingan forintense feelingconof traction complemented by pitch rise andprolonged dissonance; in this process there is strongbar-line conformityyielding at mm. 112-15!to startling ambiguity in ac-

celeration mensural into of .EAt units mm. I4-3-52 there metric is progressi in

relation to which pitch-line descent iscounteractive, butin which intensity isagain increased byunsettling, syncopatedreactive impulses;following m. 164 there is enormous stress on the second beatof everymmsure tom. 180,a processin which,

after 167, m. contradirectional imitations theinbass at ld!are conflicting in assertion qf the notated bar-line. Afurther intensifyingfactor isBeethovens employment of a threenote fragmentof the motive in metric progression in the upper voiceafter m. 180; A I » in _Z "iII:1':i1'f_'Ii'III ct.. 9292| 92 l92

_n

I 92

, formingthe severest unit contraction

ff -if of all just beforethe normal bar-line accentis againdecisively restored at m. l84H`. As to the practiceof extrememetric-motivic contractions, compare theSonata, Op. 53, first movement, mm.23-28.! Such mattersas the proportional relationsof unit segments athigher levels, other instancesof progressivecontraction, andcomplementary actionse.g., dissonance! orcounteractive eventse.g., tonal-harmonicstasis! relatingto these procedures wouldbe importantin continuedstudy. °The matterof correspondence between metric unit andphrase can be explored,too, in others ofthe Preludes-e.g.,the Eb,which is characterized byphraseological groupings of eight notated barsthroughout whilethere isgreat fluctuationin metricstructure expressed inShmiflg points qfphrase-level accent. Conliict betweenthese twofactors, one implying symmetryand the other asymmetry,seems achief expressive point in this piece.! Consider, for example,the lirst eight-measure phrase in renotationin diminution-

4 ---z§1;'T-in 'lQ["l;i'§T1---2_**-Z_ if lm i_l.l]-l-_l-if `-iiili--2--2 IVH-ll?" .92lI1§_l..j--?"..2 ;.l 11121-L.__-"i ff'I ,;¢ 3 *'

TY _-

424 rhythm

and meter

or in relation to the followinggraphic chartingof likely phrase-level and lower-level accents: 92 I __ 'Q | 'Glu

I0 *;

I aI

1'

II

II

II

I-°

The samequestion canbe carriedto a higher level:How do the two fomially synunetrical and symmetrically related thirty-two-measure sectionscompare in internal metricorganization? 'The Preludein F minor, No. 18, isin a different waya veryvaluable basisfor study

of thequestionmetric of structure at thehighest level. Such study a would consider ! the recurrences of Z Q

' at

many levels, ultimately across the whole

expansesupport in the of l at thesecond beat of m.l7; ! lower-level progressive and recessivechanges asevident in such anacrusticgroups e.g.,ci mm. l, 3-5, 7-9, 15, l6, etc., in this regard!; ! progressive approach to m. 17 involving the

succession; !_ 21% complementary of crescendo, pitch roles chrorise, maticism, tonal flux; ! the broad, underlying succession V -V/iv -iv- V - i; and ! background successions involving PCs F-Bb-Al»-C-F andincluding the horizontal expressionof f :i in mm. 9-17, culminating in the m. 17 accent-the

highest pitch, a representation of By no means isthis to say that the worksof Schoenbergand Bergare not of great interest asto rhythmic structures. The reader is referred, forexample, tothe Five Orchestra Pieces, Op. 16,of Schoenberg. The first piece ofthat set,for example,is of constant metricfluctuation, with noncongruities inhorizontal andvertical relations manifest inan exceptional clarity of unit delineationby accentand motivicdefinition. Progressive and recessive tendencies within that structure, andareas ofrelative resolution, are of importanteffect ina pieceof compellinglyactive qualityfrom beginning to end,metric diversityan essential aspect ofthat activity. Such questions as therelations of actual mensuralmeter to the notatedsignatures andbar-lines, higher-level tendencies inmetric-rhythmic shaping,polymeters e.g.,m. 64HI!, and in general the functionaleffects ofcontinuous changes in metric structure, arerewarding bases of inquiry in this piece.

Appfrvmx owe

editorial nofes

Certain editorial and notational practices adoptedin this book require brief explanatory comment. While mostfootnotes appearon the pages ofreference, somedesignated by superscript lowercase letters!appear at the ends of chapters. These are of two types: referencesto supplementary examples within a particular concern, or supplementary comment with respect to cited examples. Uppercase andlowercase lettersdenote, asconventionally, major and minor tonics and tonal systems F,or f ! except when the word minor is used as part of the reference, asin F minor. A system conceivedbimodally may be referred to by the conjunction of both symbols, as in C,c. The symbols M and m are occasionallyused torepresent majorand minor systems ingeneral or M,m to denote the bimodal system asa generic concept!; thus, mziv may stand for the subdominant harmony in the minor mode without respect to any individual system. I use the uppercase roman numeral where reference is to the generic, nonparticular, abstracted harmonic element IV, as a concept apart from any specific tonal identity!; otherwise, uppercaseand lowercase romannumerals denotemajor and minor triads, respectively, with the diminishedtriad representedby the symbol ° as inii°! and the augmented by the plus sign as in III*!. The major or minor structure of the fundamental triad governs the roman numeral usage with regard to 7th-chords, while I normally rely on the qualifying sense of context in distinction between half-diminishedand fully-diminished 7th-chords. I also use uppercaseand lowercaseletters to denote tonicizea' degrees or harmonies: T or t for tonic, D for dominant or D/T where it is necessary to distinguish it from the pitch-class D!, sm/T or SM/t for submediant, etc. Secondary relationsare expressedas with roman numerals V/V, IV/ii! : for example, SD/SD/T for the harmony on Bb where it is in the relation subdominant of subdominant in the systemon C, and where, as distinct from IV/IV, it is tonicized. Often the term /T is omitted as understood. The Arabic figure with caret is used to represent thescale degreeitself

l, Q, etc.!, andalterations inscale degrees may beshown asl, 5, and the ll lv 425

426 editorial

notes

like. Attimes, such symbols as §/5 maydenote particular a scale degree as to a secondary system or scale here, on5! occurringas referentialin some localized sense, as distinctfrom that on the primary l cf ii/V!. Pitch-classes arenamed asuppercase letters.Where specificpitches are cited I use the symbols GG, G, g, g' a 5th above middle C, or c!, g, etc. Enharmonic equivalents are represented as, for example, Al;/Gil. In such expressions asBb,C# the comma standsfor the conjunction and ; in the symbol C-Cll the hyphen stands for the preposition to C to Cil, a succession!. Concepts of pitch-class abbreviated as PC!, pitch-class complex PCC!, and interval-class IC! are explained at appropriate points. Intervals are identified as to class, orare symbolizedas m2,M7, etc. Where a pitch-class complex PCC! expresses more than one discernibly functional harmony simultaneously the symbol I-I-V or, perhaps, I-l-V! is used to denote a kind of hybrid function. The symbol V, or the like, expresses a quasi-functionalharmonic elementhaving appreciable tonal meaning in some demonstrable senseas often in modal contexts, or in music of styles more recent than those of tonal conventions, whereV, for example, may be felt as functional despite modifications of interval content and structure!. A symbol in parentheses, or accompanied by the ? symbol as in AP, N ?, or iv?! often indicates an uncertain, speculative, item of functional identification, or a factor of questionable significance or other uncertainty.

A symbol suchLlgiyas denotes fluctuation a fixed over harmony which is concurrent, or over a pedal implying such fixed harmonic function

against Huctuation. An underlyingbracket orbrace asin ` e '§_"

A '!

embraces asuccession oflocal harmonies essentially prolongingand embellishing one of broader significance. There are outlined in Chapter l various symbols andtechniques forrepresenting pitch structures and affiliations in synoptic sketchesof diverse kinds see pp. 114-l6!; and many devices for graphic and other representations of textural and rhythmic factors and processes are explained asthey occur in Chapters 2 and 3. In the discussion of rhythm, I adopt the terms anacrusticand thetic, adjectival forms of anacrusisand thesis upward-anticipative and downward impulses, respectively!. And throughout the book I employ such words as cqfunctioning, cooccurring, and confluentto describe the interactive relation between discretely identifiable, yet concurrently operative elements in parallel or different ways contributive to structural process. W.B.

APPENDIX Two

translations'

Josquin, Tupauperum refugium from Motet,Magnus estu, Domine Exx. 1-7,pp. 46-47,and 2-16. PP.238-40!. Tu pauperum rqfugium, tulanguorum remedium, spas 6X.l`|ll1lm,_f07litUd0 laborantium, via errantiurn,veritas etvita. Et num:redernptor Domine, ad tesolum confugio, te verumDeum adoro,in te spew, inte canfido. Salas mea, _]e.su Christe, aa§uva me, ne unquam obdormiat inmorte anima mea. Thou refuge of the poor, remedy for the weak, hope of the exiled, support for the heavy-laden, path for the wandering, truth and life. Now, Lord and redeemer, to thee alone I flee; I adore thee, the true God; in thee I hope, in thee I conhde. My salvation, jesus Christ, sustain me, that my soul may never reposein death.

Josquin, Deprofundis clamaviExx. 1-26a,pp. 88-89,2-2, p.190, and3-22, p. 368; Psalm 130, quotedonly inpamtranslation asin theKing James version!. De prqfundisclamavi adte, Domine;Domine, exaudi vocem meam. Fiant aurestuae intendentes in vocemdcprecationis meas. Si iniquitatesobservaveris, Domine, Domirw, qui:rustinebit? Out of the depths have I cried unto thee, O Lord. Lord, hear my voice. Let thine ears be attentive to the voice of my supplications. If thou, Lord, shouldest mark iniquities, O Lord, who shall stand?

Except whereotherwise indicated, translations are by theauthor. Verybrief examples are notincluded, whilethose translations given hereappear inthe orderof their occurrence in the book. NocH`ort ismade tomaintain meteror rhyme in the English renderings. 427

428 trans/ations Gregorian Chant, Veni creator spiritus Ex.1-41 a,p. 121!. Veni Creator Spiritus, Mentes tuorum visita : Imple superna gratia Quae tucreasti pectora. Come, Creator Spirit, Inhabit the hearts of thine own. Pervade with thy supernal grace The souls by thee conceived. Bach, Choralaria, "Draufschliess ichmich in deine Hinde' from Motet.Komm, Jesu, komm Ex. 1-44a,pp. 127-29; poet unknown!. Draqf schliessich michin deineHdnde und sage,Welt, zu guter Nacht! Eilt gleich mein Lebenslazgf zu Ende, ist dochder Geistwohl angebracht. Er soll bei seinem Schripfer schweben, weil Jesusist undbleibt derwahre Wegzum Leben. I yield myself, then, into thy hands, and bid the world good night. Though my lifes course hastens toits end, yet is my spirit well prepared. It would soar to its creator, for jesus is and remains the true way to life. Wolf, "Dasverlassene Méigdlein""The ForsakenMaiden," text by E. Morike; Ex. 1-46a, pp. 139-40!. Fnih, wanndie Hdhnekrahn, eh die Stemlein schwinden, muss :cham Herdestehn, muss Feuer zanden. Sch5n istder Flammen Schein, es springen die Funken ; ich schaue so darcin, in Leid versunken. Pl6tzlich, dakommt esmir, treuloser Knabe, dass ichdak Nachtvon dir getrdumet habe. Trane azgf Trane dann stdrzet hemieder; so kornmtder Tag herano ging er wieder!

translations

429

Early 9when roosters crow3 _ before the stars disappear, I must be at the hearth to kindle the fire The glow of the flames 18beautiful, as the sparks fly about _3 I stare into the fire _3 bowed in sadness Suddenly, it. comes to me, unfaithful lad D that I have dreamed of you in the night. Tear after tear then drops down; and so the day beginsif only it were ended!

Ravel, "Le Martin-P6cheur" from Histoires Nature//es, text by Jules Renard; Ex. 1-47a pp.143-45!. Qa na pas mordu,ce sair,maisje rapporte une rareémotion. Comme je tennis ma perche dc ligne undue, un martin-pécheur est venu :fy poser. Nous navom° pas doiseauplus éclatant. Il .semblaitune grossejleur bleue aubout ¢lun¢ langue tige. Laperclw pliait sous Ie poids. _fene respirais plus, toutjier ¢l¬trepri: pour un arbrc par un martin-pécheur. Et je suis' .sur quil ne s¢.sl pasenvolé de peur, mais quil a cfu quil nefaisait que passer d un¢brandw 5une autrc. Nothing nibbled this evening, yet I take home a rare emotion. As I was holding out my rod, a kingfisher appeared and perched on it. We have no bird more resplendent. He seemeda great blue flower at the end of a long stem. The pole bent under his weight. I stopped breathing, so proud to be taken for a tree by a kinglisher. And I feel certain it was not fear that made him Hy away, but that he believed he was only passing from one branch to another.

430 trans/ations Gesualdo, Or, che ingioia credea viver contentofrom Madrigals, Book 4;text sourceunknown Ex. 2-268.pp. 25a_s9!. Or, chein gioia credea viver contento, M apre la gioia il Jeno, Fuggesi lalmael cor, oimé, vienmeno. Now that, in happiness, I believe to live content, _I oyrends my breast, My soul flees, and my heart, alas, faints away.

Bach, "Denndas Gesetz des Geistes," from Motet,Jesu, meine Freude E x. 2-28.PP. 265-66 text fromRomans 8:2; translationas inthe KingJames version!. Dann dasGesetz des Geistes, der dalebendig machet in Christojesu, hat mich_/ici gemacht von demGesctz der Szindc und desTodes. For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death.

Dallapiccola, Goethe-Lieder No. 1,|n tausend Formen"; Ex.2-30a, pp.281 -82!. In tausendFormen magst du dichverstecken, Doch, Allerliebste, gleich erkerm ich dich; Du mags!mit Zauberschleiem dich bedqcken, Allgegenwdrt ke, gleich efkenn ich dich. You may conceal yourselfin a thousand forms, Yet, most beloved, I know you; You may clothe yourself in magic veils, Omnipresent one, still I know you.

indexes

index of musical examples and citations

Bach, _IohannSebastian 1685-1750! : Motet, _/esu, meine Freude;Denn das Gesetz des Geisler, 264-66,430 Motet, Komm, Jesu, komm; Drauf schliess ich mzbh in deine Hdnde, 12 7-34,428 Partita in C minor; Prelude, 297n Well- Tempered Clavier, I; Prelude in C, 63-67 Bartok, Béla 881-1945!, 388 Divertimento for string orchestra; second movement, l8ln, 234-36, 298n; third movement, 182n Quartet No. 2 for strings; third movement, 147-62, l83n Quartet No. 4 for strings; first movement, 217n Quartet No. 5 for strings; first movement, 217-21, 295n, 42211 Beethoven, Ludwig van 770-1827!, 313,380 Concerto No. 2 for piano and orchestra; third movement, 374-75 Quartet, Op. 18, No. l, for strings; first movement, 329, 389 Sonata, Op. 2, No. 3, for piano; first movement, 2541: Sonata, Op. 13, for piano; second movement, 74, 331, 359 Sonata, Op. 22, for piano; first movement, 328, 420n

Sonata, Op. 53, for piano; first movement, 55-57, 423n Symphony No. 2; first movement, 376; third movement, 380-83 ; fourth movement, 30-34, 134-38 Symphony No. 3; fourth movement, 34-35 Symphony No. 5, 313 Symphony No. 6, 313 Symphony No. 8; first movement, 389-94, 423n; second movement, 364-65, 372 Variations, Op. 120, on a Waltz of Diabelli; Var. XIV, 313-16, 320, 360,420n Berg, Alban 885-1935!, 397, 42411 Concerto for violin and orchestra, 173 Four Pieces, Op. 5, for clarinet and piano; No. 2, 90-93; No. 4, 166-68 Berio, Luciano b. 1925!: Sequenza II,for harp, 38-40, 309-10 Blacher, Boris 903-75!: Concerto No. 2 for piano and orchestra, 38311 Omamentefzlr Klavier,Op. 37; Nos. 3, 5, 6, 363n, 383-86 Boulez, Pierre b. 1925!: Le Marteau sam' Maftre; third movement, 102-3 433

434 index

of musical examples and

Boulez cont.! Structures for two pianos; Part Ia, 97-100, 411, 414 Brahms, Johannes833-97! : Alto Rhapsody,60, 183n Intermezzo, Op. 117, No. 1, for piano, 420n Intermezzo, Op. 118, No. 1, 180-81n, 183n Intermezzo, Op. 118, No. 2, 94 Quintet, Op. lll, for two violins, two violas, and cello; first movement, 299n; second movement, 246-48 Symphony No. 1; first movement, 266-80, 360 Symphony No. 3; first movement, 327 Trio, Op. 4-0, for piano, violin, and horn; third movement, 101-2, 299n, 369-71; fourth movement, 422n Trio, Op. 87, for piano, violin, and cello; first movement, 353-55, 421n Trio, Op. 101, for piano, violin, and cello; third movement, 38-39 Variations, Op. 56A, on a Theme of Haydn; Theme, 328, 351-52, 359, 419-20n

Carter, Elliott b. 1908!: Eight Etudes and a Fantasy for woodwind quartet; No. 2, 204-6; No. 3, 182n; No. 4, 422n Qpartet No. 1 for strings; first movement, 307-9 Chopin, Frédéric 810-49! : Etude, Op. 10, No. 6, 420n Nocturne, Op. 15, No. 1, 386-88 Prelude, Op. 28, No. 9, 68-69, 394-97 Prelude, Op. 28, No. 18, 424n Prelude, Op. 28, No. 19, 35-36, 324-26, 420n Corelli, Arcangelo 653-1713!: Concerto Grosso, Op. 6, No. 3; third movement, 117-19, 213-16, 34549, 362

citations

Couperin, Francois 668-1733!: La Bondissantefrom Ordre 21!, 118-19

Dallapiccola, Luigi b. 1904!: Goethe-Lieder; No. 1, 280-88, 430; No. 2, 299n; No. 6, 299-300n Quaderno Musicale di Annalibera ; Accenti, 104n, l8ln Debussy, Claude 862-1918!, 336 lApr2s-midi dun Faune, 422n Ariettes Oubliées; No. 3, l0mbre des Arbres, 105-7 Prelude No. 2 for piano Book I!, 254-55 Dowland, john 563-1626! : Come, Healy# Sleep from First Book of Airs!, 321-22, 323

Fischer, _Iohann Kaspar Ferdinand c. 1665-174-6!: Ariadne Musica; Preludes in D minor and E phrygian!, 125-27

Gesualdo, Don! Carlo 560-1613!: Madrigal, Moro, moro; first part, 298n Madrigal, Or, che in gioia credeaviver contento; firstpart, 257-59, 430 Gregorian chant, Veni creator spiritus, 120-24, 428

Handel, George Frideric 685-1759!: Suite No. 3 for harpsichord; Prelude, 253-54 Haydn, Franz! joseph 732-1809!, 323n Sonata in D for piano B. & H. No. 37!; third movement, 335 Symphony No. 102; first movement, 94-96, 296-97n, 330-31, 359 Symphony No. 104; fourth movement, 323, 359

_losquin des Prez c. 1450-1521!, 169n, ` 200, 321-22

index of musical examples Mass, Faitant regretz; Sanctus, 421-22n Motet, De prqfundis clamavi, 87-90, 189-90, 241-44, 367-69, 42ln, 427 Motet, Magnus es tu, Domine; Tu paupefum re/izgium, 45-47, 237-40, 297n, 427

Kienek, Ernst b. 1900!: Eight Piano Pieces; No. 4, 119-20, 182n Sachs Vermessene for piano, 291-92 Sestina, 289-91,289n

Lasso, Orlando di c. 1532-94!: Chanson, Bonjour mon coeur, 180n, 200 Liszt, Franz 811-86!: Transcendental Etudes for piano; No. 3, 57-58, l80n, 422n

and citations

435

Symphony No. 41, K. 551 ]upiter!; second movement, 324-26, 372, 373n, 421n

Nono, Luigi b. 1924!: Canto Sospeso, 173, 409

Purcell, Henry c. 1659-95!: Fantasy, Z. 738, for strings, 206-9 Gloria Patri, 233-34, 321

Ravel, Maurice 875-1937! : Histoires Naturelles;No. 4, Le Martin-

Ploheur, 142-47, 429 Trois Poemsde Mallarmi; No. 3, Surgi de laoroupe et du bond,182-83n

schoenberg, Arnold 874-1951!, 177, 397 Concerto for violin and orchestra; first movement, 244-45 Five Pieces, Op. 16, for orchestra; No. 1, 424n

Machaut, Guillaume de c. 1300-1377!: Messe Notre-Dame, 312n Mendelssohn, Felix 809-47! : Sonata No. 2 for cello and piano; third movement, 332-34, 360, 362 Five Pieces, Op. 23,for piano; No. 2, Messiaen, Olivier b. 1908!, 409, 414 107-1 1,232, 339, 379-81 Livre dOrgue; No. 1, Reprires par Piano Piece, Op. 33a, 172 interversion, 311 Pierrot Lunaire;No. 12, Galgenlied, 4l9n Qudtre Etudes for piano; Ile de _ku Six Pieces,.Op. 19, for piano; No. 3, 2, 410-ll; Modes de valeurs et 182n, 297-98n,4l9n dintensités, 409 Three Songs, Op. 48; No. 3, Tot, Quatuor pour la _/indu temps,409 172-76, 211 Milhaud, Darius 892-1974! : Schubert, Franz 797-1828!: Six Sonnets for mixed chorus; No. 3, Quartet in D minor, D. 810, for 186-89 strings; first movement, 74-75 ; second movement,373-74 Mozart, Wolfgang Amadeus 756-91!, 179, 323n Sessions, Roger b. 1896!: Concerto, K. 218, for violin and From My Diary; third movement, 244 orchestra; first movement, 355Strauss, Richard 864-1959!: 59, 362 Don juan, 4201: Fantasy and Fugue, K. 394, for piano; Stravinsky, Igor 882-1971!, 222, 232, Fugue, 77-78 296n, 336,369, 388,400 Sonata, K. 332, for piano; first moveCanticum Saorum; Part III, 296n; ment, 41-45 Part V, 225-32, 388-89

436 index

of musical examples and citations

Stravinsky cont.! Concerto for violin and orchestra; first movement, 254n lHz1¢toire du Soldat; PetitConcert, 365-66 Symphonies of Wind Instruments, l83n Symphony ofPsalms; first movement, 254-56 Symphony in Three Movements; first movement, 222-25, 295-961| Variations for orchestra Aldous Huxley in memoriam!, 196-99, l99n, 423n

Telemann, Georg Philipp 681-1767!: Fantasy No. 4 for violin alone; first movement, 257-64, 298-99n Sonata in E minor for two flutes, 296n

Wagner, Richard 813-83! , 336 Webern, Anton 883-1945!, 250, 336, 388, 4l6n Canons, Op. 16, for soprano andclarinets, 400

Five Pieces,Op. 5, for string quartet; No. 4, 401-8 Five Pieces, Op. 10, for orchestra; No. l, 181-82n Four Pieces, Op. 7, for violin and piano; No. 1, 162-65, 249-50, 41911; No.3, 93, 211-13 Quartet, Op. 22, for piano, clarinet, saxophone, andviolin; first movement, 176, 250-53, 305-7, 399

Six Bagatelles, Op. 9,

for string

quartet; No. 1, 176-77 Three Pieces, Op. ll, for cello and piano; No. 3, 6-7, 397-400 Three Songs, Op. 25; No. 1, Wie bin ich jiohl, 202-4, 295n Variations, Op. 27, for piano; third movement, 312 Wolf, Hugo 860-1903!: Der Mend hat eineschwere Klagerhoben Italienisehes Liederbuch,Vol. I, No. 7!, 85 Selig ihrBlinden Italienisches Liederbuch, Vol. I, No. 5!, 85 Das verlassene Mdgdlein from Gedichte von Mdrilce!,138-42, 428

index of subjects, names, and terms

Acceleration, 9, ll, 71, 76, 78, 86-87, 136-38, 243, 248, 250, 267, 277, 306-10, 313-16, 324, 339, 344, 355, 367n, 371, 376, 378-83, 392-95, 4l9n, 421-23n see also Motion; Progression, recession! Accent, metric initiative impulse, 63n, 66, 90-95, l01n, 123, 126, 134, 149, 154,303, 3l0n, 313, 317-61, 364-77, 379-80, 386, 394-97, 418 see alsoMetric functions! ambiguity of see Ambiguity,in metric structure! of associatedimpulse functions, 342 conjectural factors, 342-44 counter-, 305, 326, 344, 355-59, 372-76, 390-92, 422-23n seealso Metric fluctuation; Metric functions, syncopation! criteria, 63n, 303, 325, 329, 333-49, 358-61, 365-69,376, 379-82,386-

95, 399-400,406, 418,420-21n equivalent to higher-level pulse, 35051, 373 preconditioned see Meter, preconditioned, preconditioning structure!

l

primary, structural downbeat see Metric functions, structural downbeat! of superior values, 339-41, 386 textual, 368-69, 388-89 Ambiguity, l0n in metric structure, 318, 326, 373-79, 380n, 393, 397, 400-406, 417-18, 420n, 423n of pitch, 213 of tonal orientation, 47, 76, 92, 103-4, 136-4-0, 150-51, 154-58, 170, l75n, l8ln Ametric structure, 31811, 397 Anacrusis see Metric functions, anacrusis! Analysis: as interpretation, 61, 100, 179-80, 302-3, 337-38, 344, 348-51, 377, 418-19 symbols, 44, 48-51, 66, 70n, 114-16, 136, 188, 210, 2l3n, 2l5n, 264n, 323n, 328, 334, 350, 352, 379n, 40011, 420n, 425-26 types, 22, 103-14, 178,350 Apel, W., l20n, 310n

1The index omits namesof someauthors andeditors ofworks citedin very minor, passing references. Within items of very broad, recurrentconcern, onlyselected references are given.The symboln isnot usedwhere acitation is to a footnote aswell as text on the same page. A referenceto individual elements concernsany or all of the following:color, density, dissonance, harmony, melody,meter, tempo,texture, andtonality. 437

438 index

of subjects, names, and terms

Asymmetry, 11, 186, 203-4, 305, 406 in metric structure seeMetric Huctuation; Metric relations, of noncongruity! Atonality, 28, 98, 171-72, 176-77, 3l8n

Babbitt, M., 289n, 414-16 Background of structure see Hierarchy, leveled; Levels of structure! Beat, 349, 367, 401,408 Bitonality, 158, 183n, 236

Cone, E., l9n, 172, 225n, 3l9n Convention s! in triadic tonality sec Tonal conventions! Cooper, G. W., 301-2n, 3l9n, 324-25, 336, 361, 374n Counterpoint see Textural relations, contrapuntal!

Deceleration, 9, 107, 136-38, 232, 248, 250, 278, 283, 306-9, 316, 34-4, 367n, 378-89, 419n, 421-22n Deceptive tonal-harmonic succession, 58, 68, 131, 155-59 Density, 11, 86, 104, l82n, l9ln, 209-13, 234, 278-80, 341 Cadential process see Process, forand coloration, 184-85, 210 mal! Chromaticism : -compression, 185, l88n, l9ln, 204, in expansion of tonal system, 41, 205n, 209-13, 244, 290, 29511, 298n, 316,400 60-61, 68, 77, 81, 134-42, 170, 315 and dissonance, 182n, 184, 195, 20915 incipient, 47, 169, l80n of linear auxiliaries, 36, 68, 129, -number, 185, 18811, l9ln, 204, 205n, 140-41, 160, 164-67, 257 209-13, 232, 237, 244-46, 257modal implications, 56n, 70 64, 274-80, 284, 287-88, 296n, 400 in modulation see Tonal fluctuation, chromatic technique! progressive, recessive fluctuations, 103, in pitch successions, 30-37,49, 56-59, l04n, 107-9, 111, 119, 18111, 201, 72, 81, 132-33, 138-47, 157, 168, 211-13, 232-37, 241-48, 257-60, 181n, 183n,257, 275, 348, 42011 267, 275-80, 287, 292-93, 296n, 298n Cognition: of ambiguous states, 10n,375n rhythm of changes, 201-4 of broad structures, 15-16, 297n, Depth of structure, '52, 67, 102, 249n, 361-62 255, 350, 361-62, 378n see also and perception, 2 Hierarchy, leveled; Levels of relative immediacy in certain elementstructure! actions, 5, 189, 293, 299n, 301, Developmental process see Process, 306 formal! Diatonicism see also Tonal system, Color coloration!: as aspectof sonority, 192, 279,298n diatonic!: and multimovement form, 15 in modulation see Tonal fluctuation, progressive, recessiveactions, 11, 86diatonic technique! in pitch successions, 48-51, 56, 71, 87, 107-8, 283, 299n, 313-14, 395, 422n, 424n 79-80, 84, 146 structural functions, 20, 182n, 184-85, Dissonance, within various parameters, 13, 85, 109-11, 209-10, 215-16, 199, 205,211-13, 225, 283, 294n, 340 217n, 234-37, 24-6-48, 278,299n,

index of subjects, names, and terms 43.9 326, 341, 359, 363-66, 372, Forte, A., 1l3n 374-75, 378-79, 381, 388, 393- Function, 23-24 see alsoLinear func94, 42311see alsoResolution; and tion; Process, formal; Progresindividual elements, progressive sion, recession; Tonal function; and underindividual elements! actions! progressive, recessive fluctuations in cadential see Meter, and cadential pitch structures, ll, 63, 86-87, function; Process, formal! complementary and compensatory 103-4, 107-11,117-19, 147,158, 181-82n, 262-64, 275, 295n, 315, counteractive!, 4-9, 63n, 76, 378, 423n 78, 87, 90-91, 95, 109, 180, Dominant, 56, 62-63, 83-84, 96 seealso l82n, 185, 189, 199-205, 20915, 225, 231-36, 241-55, 260Leading-tone relation! forms, 54-55, 138-40 67, 275-80, 286-87, 294, 295n, quasi- seeTonal function, quasi-! 297n, 299n,313-16, 324, 331, 335, Downbeat seeAccent! 367n, 369, 378-83, 386-89, 4078, 419-24n contextual, 17, 23-24, 98n, 292n, Elements of structure, 4-13, 22, 178-79, 302, 338, 340n, 373, 416-17 and isolated or random events, 182n, 185, 199, 204-5, 213, 225, 16-17, 24-25, 416-17 see also 232-36, 248-50, 266-67, 275-80, 283, 286, 294, 297n, 299n, 302, Randomness; Serialism, tota1, and contextual function! 306, 313, 367n, 380-83, 388-89, 418, 420-2ln developmental seeProcess, formal! Enharmonicism, 58, 73-75, 141-42, expository seeProcess, formal! 149, 155, 160-61 introductory see Metric functions, in modulation see Tonal fluctuation, anacrusis; Process,formal! enharmonic technique! multileveled see Hierarchy, leveled; Levels of structure! of succession diatonic, chromatic!, 73-75, 275 I, IV, V-primacy, tonal and linear, Erickson, R., 30511 of, 62-63, 129-34 Eventfulness in music seeTempo! structural, expressive, 3, 4, 12, 23-24, Expository process see Process, for180, 201, 277-80, 293-94, 302, 313, 375-77, 417-19 mal! Extramusical association, 25-26 tonal and linear, 29-37, 61-67, 113, 179 Fermata, 305n, 364 functions of, 137, 264, 378n, 419n 5th succession see Succession, in 5t.hs! F oreground of structure see Hierarchy, leveled; Levels of structure! Form s!, 9, 62, 67, 71, 76, 79, 94, 101-2, 118, 129,136, 158, 171, 176, 178, 185, 199n,217, 231, 233, 236-41, 260, 267, 280-83, 295-99n, 306, 333n, 345,352, 359-61, 378, 388, 395, 42011,423n

Gombosi, O., 3l2n Groupings of structural events,4, 301-3, 316, 320-24, 405-7 see also Rhythm, as grouping! Grout, D., l69n Harmonic distance, 84n, l32n, 341 Harmonic elision, 146 see also Tonal fluctuation, elision! Harmonic motive, 147-48, 150

440 index

of subjects, names, and terms

Harmonic rhythm, 67, 79n, 86, 112, 182n, 248,299n, 313-16,326, 335, 383, 408, 420-23n Harmony see also Linear function; Tonal function!: analysis in later styles, 103-12, 164-65 see also Tonal function, quasi-! progressive, recessiveactions, ll, 84n, 86-87, 103-12, l32n, 164-65 quartal, 160-61 Hemiola see Metric fluctuation, hemiola! Hierarchy, leveled see also Levels of structure!: background, generic content of; 37 and chromaticism, 38 criteria for in linear functions, 61-63, 96, 100-101, 112-13 criteria in nontonal styles, 63, 100, l02n, 112 and dissonance,15 and fluctuation, 14-15, 201 foreground, unique content of, 37, 275-80 and harmonic as opposed to tonal significance, 15, 67-71, 141 in metric structures, 322, 349-62 see also Levelsof structure, in metric structure! and relations among elements, 14-16, 178-79 in textural structure, 255, 267 in tonal and linear functions see Levels of structure! and tonal fluctuation," 14-15, 38, 48, 70-71, 76-79 of tonal system see Tonal system, hierarchic ordering in! Hindemith, P., 14-n, 102n,109n Homorhythm seealso Texturalrelations, homorhythmic!: and chromaticism, 170, 200,298n and tonal conventions, 170 Horizontalization see Linear function, triadic and other tertian horizontalizations! Iamb, 310, 327, 337n,339, 342,373, 393, 422n

Imbrie, A., 319n Imitation, 213, 216-21, 225, 232-34, 237-40, 257, 264, 280-83, 28588, 297-99n,315 in diagonal, multiangular relations, 216, 245, 264, 278-79, 283n, 285-86, 299n harmonic interval, 186, 2l 7n in multiple counterpoint, 221, 280n as opposedto dia1ogue, 2l6n, 267 time interval, 186, 206-9, 216-21, 243, 257, 266, 279, 285-87, 295-96n, 299n, 315 types, 233n,243, 283,285-86, 296-97n, 299-300n Impulse, 305n, 317, 326-34, 349, 360-61 functions seeMetric functions! Intensity sea also Progression, recession!: curve, 4-12, 26, 216-21, 232-34, 243, 255, 257,262, 267, 274, 302, 307, 320, 380, 397 and motion, 5 qualities states! of, 9-11, 84-87, 156-58, 189-90, 216-17, 233-34, 243, 262, 275-78, 286, 367n, 379-89, 421n

Interval classIC!, 193n,285 content in texture, 289, 295n Introductory process see Process, formal! Isorhythm, 306, 312

Katz, A. T., l13n Ki"enek, E., 16, 110n, 289-92 Krueger, T. H., l13n

Leading-tone relation, 122 cancellation ol; 37, 60, 70, 131 and minor mode, 49, 71n secondary see Tonal system,secondary regions of; Tonicization! tonicizing, 36, 40, 46, 54-57, 70-71, 83, 93, 103-4, 148-69, 171, 175n

unfulfilled see Linear function,superseding tonal function!

index of subjects, names, and terms 441 Levels of structure, 5-7, 10, 13-16, 2930, 113, 120-24, 179, 201, 249n, 255, 267, 29712, 316, 320-21, 34962 seealso Hierarchy, leveled! background, intermediate grounds, foreground, 37-40, 113, 24911 cognition, experience of broadest, 15-16, 361-62 as toformal units, 13, 123-24, 130-31, 330-34, 351 in metric structure multileveled metric functions!, 305, 309, 31719, 322-23, 32411, 327-36, 338, 343-73, 375n, 376-83, 386, 38891, 394-400, 404, 406-7, 419-24n in multileveled tonal and linear functions, 29-40, 49, 52-69, 7679, 96, 117-19, 122-34, 138-68 as to temporal spans in meter, 14, 267, 351, 373, 378 in textural structure, 188-89, 193-94, 209, 211, 221, 241-43, 253-55, 257-60, 263-67, 275-76, 278-84, 288, 293, 29611 in tonal systems see Tonal system, hierarchic ordering in! Ligeti, G., 411 Linear function, 29-37, 61-66 chromatic seeChromaticism, of linear auxiliaries! as encirclement, 148-62, 165-66, 171 see alsoLeading-tone relation! multilateral, 30n, l23n, l30n multileveled see Levels of structure, in multileveled tonal and linear functions! neighbor auxiliary, 29, 34, 90-92, 122-25, 129-33, 147-48, 164-65, 171, 231,323 I, IV, V, 62-63, 129-34 passing auxiliary, 29, 34, 90-93, 105, 118, 122-25, 129-33, 141, 146 prolongation, 30-38, 45, 66,85, 87,96, 117-18, 123-34, 138-47, 164-68 rhythmic implications, 63n, 320-21, 340n, 342, 348, 351 see also Metric functions, compared to tonal, linear!

structural essential! as opposed to auxiliary embe1lishinS!» 29-38, 45, 61-66, 85, 87-93, 96, 100-103, 117-18, 122-34, 138-47, 152-53, 158-62, 164-68, 230-31, 323, 351 superseding tonal function in nonfunctional elaboration!, 30, 36, 68n, 103-5, 138-47, 179 and tonal function see Function, tonal and linear! of tonics embellishing tonal system components!, 44-45, 52-53, 5561, 69, 76-79, 134-41, 158, 382 triadic and other tertian horizontalizations, 36-37, 69, 92, 96, 100, 107, 118, 126, 129-33, 138, 14142, 146, 159, 180-81n, 424n Lowinsky, E., 47n

Melody : analysis, 38-40, 87-103, 117-18, 12024, 152-53, 167, 18ln compound line see Texture, compound line! progressive, recessive actions, 11, as-87, 107, 122-24, 162, 167-68, 232-34, 267, 276-78, 283, 29912, 313-15, 324-26, 333-34, 353, 388-89, 4l9n, 421n, 423-241: step relations, 92, 95, 102, 118, 150, 181n, 234, 323 Meter seealso Accent; Metric functions; and relatedentries!: as accent-delineated grouping, 63n, 303, 306, 317-20, 322-26, 338, 344, 355-61, 394-97, 406-7 and cadential function, 63n, 321-24, 329-34, 352-53, 359-61' and disjunct bar-line see Metric relations, polymeter! levels seeLevels of structure, in metric structure! mensural, intramensural, intermensural see Levels of structure, in metric structure!

442 index

of subjects, names, and terms

Meter oont.! motivic, 397, 419-20n, 423n in multimovement forms, 15, 373 and notated bar-line, 324-26, 333-34, 359, 363-75, 377, 392-93, 401, 406, 408n,423-24n and other modes of grouping, 63n, 321-24, 335, 338, 340n, 348, 359-61, 400, 405-8, 420ri, 42324n as parameter of rhythm, 23, 303-6, 317-19 preconditioned, preconditioning structure, 304-5, 324-25, 327, 331-34, 337, 341n,343-4-4, 346n,347, 349, 355, 358n,372-76, 380, 382, 387, 389-9ln, 395-96, 404-5, 420n, 42211 progressive, recessive actions, 11, 232-34, 299n, 307, 321, 324-26, 333, 345, 359, 363, 371, 376-94, 397-400, 421-24-n Metric dissonance and resolution, 326n, 359, 363-67, 372, 374-75, 377, 393-94, 401, 404, 406-7, 4241: Metric fluctuation, 305, 3l0n, 318-19, 321-22, 324-26, 334n, 359-60, 362-400, 404, 408, 422-24n see also Meter, progressive, recessive actions; Metric relations, of noncongruity! hemiola, 371, 375n, 388,42ln incipient, 372n, 374-75, 379, 387, 392 see also Metric functions, syncopation! in opposition to stability, 365-67, 377, 389-94 Metric functions, 318, 326-34, 345-53, 418 anacrusis anticipative impulse!, 126, 266-67, 277, 280, 303, 3l0n, 320, -322-23n, _324-36,337n, 339-42, 345-51, 355, 360, 373-74, 379, 39011, 393-407, 420n, 4247! compared totonal, linear, 63n, 320-21, 330, 340-41n,342, 352-53,359-61

conclusive impulse,320, 323n,32 7-36, 338, 345-49, 355, 359, 404-5 see also Meter, and cadential function! dualities elisions, conjunctions!, 32830, 334, 353, 376, 405 initiative impulse see Accent! multileveled see Levels of structure, in metric structure! reactive impulse, 303, 320, 322n, 327-29, 338, 345-49, 423n of silences,326, 346n, 349, 405-6 structural downbeat primary accent!, 347-50, 379-80, 396-400, 407, 424n syncopation, 276, 303-5, 355-58,

372-76, 379, 387, 390-92, 395-96, 422-231: Metric relations, 362-71 between levels, 378 of congruity symmetry!, 318, 347-48, 362-71, 377-79, 390, 421-24n homometer, 363, 373n, 379 of noncongruity asymmetry!, 318, 322, 326, 333, 336, 347-48, 35960, 362-71,375-93, 397-400,406, 408, 420-24n polymeter, 193n, 274, 363, 365-66, 371, 375, 379, 388, 400, 42ln, 424n preconditioned see Meter, preconditioned, preconditioning structure! in seriesand permutations, 383-86, 409 Metric rhythm, 302, 377 Metric unit, 309, 317-22, 327-30, 335-40, 346-51, 363-64, 373, 376-78, 399, 423n calculation of durations, 363-64 Metric variation, 45, 378 Meyer, L., 19n, 301-2n, 3l9n, 324-25, 336, 336-37n, 361, 374n Middlegrounds of structure see Hierarchy, leveled;Levels ofstructure! Mode: ambivalence of, 121-22, l48n, 1.53, 161, 170, 172, l82n

index of subjects, names, and terms interchangeability of major, minor, 49, 80-81, 134 modal systems see Tonal function, quasi-, in modal practice! Monody, 200 Motion see also Stasis!: and directed change, 4-9, 98, 267, 397, 419n see also Progression, recession! in space, 7-8, 193, 274-80 in time, 7-8, 305-8, 397, 419n Mursell, _]. L., 337n Neighboring linear function see Linear function, neighbor auxiliary! Nelson, R. U., 241n Nonfunctional elaboration see Linear function, superseding tonal function! Organum, 200 Pantonality, 158, 172, 176-77 Parameters of structure, 11, 193, 201-2, 213, 257, 267, 313-16, 318, 418 see alsoElements of structure! Parrish, C., 121n, 260n Passing linear function see Linear function, passing auxiliary! Pentatonic relations, 147 Perception: and cognition, 2' relative immediacy in certain elementactions, 5, 189, 293,299n, 301,306 Performance problems, 3, 18, 292, 299n, 301, 317n, 321, 324-36, 348-49, 364-74, 388, 393-97, 400, 418, 420-21n Perle, G., 20", 289n Pitch-class PC!, 27n Pitch-class-complex PCC!, 27n Plagal action, 70n, 84 in tonic elaboration, 45-46, 125-26, 138

443

Pointillism, 200 Polyphony see Textural relations, polyphonic! Pousseur, H., 6-7, 98-100, 177 Process, formal: cadential, developmental, expository, introductory, 5-6, 61-62, 63n, 76, 107-9, 185, 205-6, 215-16, 232-48, 260, 264, 266-67, 277, 283, 288, 294, 295-99n, 306, 331, 339, 345, 353, 367n, 378-89, 405n, 419n, 422n see alsoForm; Function! Progression, recession,2, 4-13, 17, 23, 26, 63n, 66, 84-87, 92, 104, 107-8, 180, 180-82n, 201, 203, 232-34, 267, 294, 302, 320, 331, 344, 361, 377-89, 399-400, 42ln see alsoAcceleration; Deceleration; Intensity; Resolution; and under individualelements! Projection, 24, 303 Prolongation see Linear function, prolongation! Pulse, 305n, 307-9, 317, 326, 349-51, 358n, 363-64, 366, 378-81-,401

Randomness, 16-19, 23-24, 98-100, 292, 398,408, 417 Resolution, within various parameters, 5-7, 10-13, 26, 84-87, 162, l8ln, 190, 206, 216, 219, 236-37, 244, 267, 275, 283, 326, 352-53, 359, 364-67, 374, 379, 393-94 see also Dissonance; and individual elements, recessive actions! Reti, R., 117n Rhythm seealso Acceleration;Deceleration!: aspects, 267,301-8, 313, 318-24, 363, 379-80, 418, 422n complexities, 302-3, 310n, 318, 33638, 340n, 366-69, 372-73, 388,

400, 408,417-18 of element-changes, 5, 8-9, 20, 23, 136-38, 201-2, 301-2, 306,

444 index

of subjects, names, and terms

Rhythm cont.! of element-changes cont! 313-16, 3771388, 420n, 422n see also under individual elements! as grouping, 304-6, 313-24, 338, 405-7 grouping, factors in, 63n, 320-24, 338, 355, 394-97, 405-7 grouping as accent-delineated see Meter! grouping of associated linearfunctions, 63n, 320-23, 330, 340n, 348, 351 see alsoLinear functions: rhythmic implications! grouping of class-afliliated events,320, 323, 326,406n grouping as to extramusical factors, 321-22, 338, 367-69, 388 grouping in formal units, 322-23, 335, 338, 352-55,383-88, 394-97, 400, 405-6, 420n, 423-24n grouping of tendency-aiiiliated events, 9, 320 see also Progression, recession! iterative neutral!, 317 as pattern, mode, motive, 304-6, 310-13, 318, 369-71, 379-80,

Sequence, 36-37, 66, 93, 101, 117-18, 2l6n, 260, 393 Serialism, 204 combinatoriality, 289, 416 of densities, 289-92 and derived quantities, 289, 408-16 of pitch-class twelve-tone procedures!, 19, 171-77, 280, 288-89, 296n, 29911, 312, 409 of time segments, 291-92,313, 408-17 and tonality, 18-19, 171-79 tota1, and contextual function, 1619, 98-100,408-17 Sessions, R.,16, 329 Slatin, S., ll3n Smither, H., 319n, 340-41n, 363n Sonority, 192, 195, 196n, 199, 210, 236, 255, 267,274-79, 295n, 297-98n Space see Texture, space and spatial relations! Spies, C., l99n, 423n Stasis, 7-8, 11, 86, 92, 94, 204, 221-25, 234, 243, 248, 275, 278, 280, 299-300n, 309,316, 390-92, 42223n seealso Motion! Stretto seeImitation, time interval! Structural downbeat see Metricfunctions, structural downbeat! 386, 397, 400, 424n see also Structure, 9 lamb; Trochee! of pitch or PC! change, 422n Style: and metric preconditioning, 373, 392 proportional notation, 309 relevance of particular elementas tempo, 23, 304-6, 313, 318, 379, 388-89, 405,419n see also Tempo! actions to, 12, 63, 170, 178-79, and unpunctuated time Held, 305n, 200-201, 217n, 221-22n, 305-6, 364, 378 319-21, 359, 362-63, 367, 373, 376-77, 394, 4-22n Rhythmic mode see Rhythm, as patand texture, 200-201, 217n, 221-22n, tern! 243, 293-94 Rhythmic ostinato see Isorhythm! and tonal fluctuation, 59, 171 tonal system of, 40, 47, 53 Succession, 23,84 Sachs, C., 361-62 in 5ths, tonal significance of, 40, 54Salzer, F., 1441, ll3n Schachter, C., l4n, ll3n 57, 83-84, 118, 132-33, 169-70, 180n, 314 Schenker, H., 14n, 113 Symbols of analysis seeAnalysis, symSchoenberg, A., l4n, 47n, 48-49, 52, 289n bols!

index of subjects, names, and terms Symmetry:

formal, 395-96,406 in metric structure see Metric relations, of congruity! in space, 176 of tonal relations, 56-57, 136-37, 156, 176, 18011 Syncopation see Metric functions, syncopation! Syntactic relations, 2, 16-1711 see also Function; Process, formal; Progression, recession!

Tempo: activity-, 305-7, 364, 36711,388-89, 398, 41911 aspects, 305 as parameter of rhythm see Rhythm, as tempo! progr ive, recessive actions, 11, 29711, 299n, 306-10, 379-81, 388-89, 41911 pulse-, 305-9, 316, 380-81, 41911 Tempo rhythm, 302, 388 Tempo-structure, l99n, 307, 36711,388, 405-6, 419n in multimovement forms, 15, 297n Text, problems of, 124, 257, 285n, 29811, 321-22, 367-69, 388 Textural relations, 192-99 chordal, 192, 222, 237-40, 246, 267, 275-77, 297-9811,333 contradirectional, 192-96, 243, 253, 263-64, 274, 278, 283, 286-88, 296-98n, 42371 contraintervallic, 193-96, 26311, 29697n contrapuntal, 192, 213-15, 225, 237, 264, 274-75, 278 contrarhythmic, 193-94, 234, 243, 263-64, 274-75, 27811, 283,293, 296-9711, 300n,423n of doubling, 192, 192-9311,196, 222, 225, 236, 261, 264, 267, 274 hcterodirectional, 193-94, 196

445

heterointervallic, 192, 194-96, 222, 225, 231, 29711 heterophonic seeheterointervallic! heterorhythmic, 193-94, 196-99, 22225, 230-31, 246, 29711 homodirectional, 192-96, 222, 243, 28311, 284, 286, 296-9711, 299300n homointervallic, 192, 194-96, 243, 283-84, 296n, 300n homophonic, 192, 200, 216 homorhythmic, 45, 189, 192-96, 21516, 237, 240, 243, 275, 285, 293, 296-9911, 333, 393 mirror, 192, 280n, 284,29611 monophonic, 185, 191-93, 199, 241, 267, 277, 293-94, 296n, 29911 polyphonic multivoiced!, 192-93, 200, 29711, 366 Textural rhythm, 201-4 Textural variation, l99n, 221, 236, 241, 260-61, 262n, 280n Texture, 23, 184, 191 activation of, 199, 222-32, 275-76, 293, 29611, 29811 classifications and terminology, 19195, 363 see also Textural relations! components real, sounding!, 186-91, 192-9311, 196n, 225, 236,264, 267, 274, 288 compound line, 101, 167-68, 260-61, 278, 298-9911,348 concurrence of lines, 18411 density seeseparate entry! imitation see separate entry! independence, interdependence see Textural relations; Texture, qualitative! levels seeLevels of structure, in textural st1'ucture! line, as component, 192-93n, 267 motivic-thematic, l99n, 202, 254-55, 298n in multimovement forms, 241, 297n

446 index

of subjects, names, and terms

Texture cont.! multiple counterpoint, intersecting diagonal relations see Imitation, in multiple counterpoint! pitch-class displacement transference!, 230-32, 24-6, 296n poly-, 225, 232 progressive, recessive actions, ll, 18590, 196, 199, 204-22, 232-37, 241-54, 257-67, 275-88, 293, 295-300n, 307, 389 qualitative interlinear relations, 185200, 204-9, 213-17, 232-37, 24346, 250, 257, 262-63, 274-75, 278-80, 284-87, 293, 297n, 421n see also Textural relations! qualitative relations, factors parameters! in, 191-95, l96n, 206, 213-17, 232, 234, 262-64, 363 quantitative aspects, 185-89, 204-13, 232, 234-36, 244-48, 262-64, 267, 293, 341 seealso Density! space and spatial relations, 7-8, 16465, 195-96, 199, 204-5, 210-13, 234-36, 244,248-55, 267,2 75-76, 280, 284, 295n, 298n, 316, 333n, 4-00 and style see Style,and texture! symmetrical space, 176, 249-53 and value in art music, 293 voice, as component, 192-93n, 262, 264 Timbre see Color! Tonal conventions, 28n, 40, 62-63, 12934, 148, 169-70, 172 Tonal distance, 11, 44-45, 48-51, 53n, 59-60, 72n, 79-86, 131-32, 141, 204n, 276, 314, 341 criteria of measurement, 80-84,85n and direction, 84, 132n between parallel keys seeMode, interchangeability of major, minor! between relative keys, 53-54, 80-81 Tonal elision see Tonal fluctuation, elision! Tonal fluctuation, 41-47, 55-57, 59, 6779, 130-32, 146, 153-57, 176-77,

183n, 266,297n seealso Tonality, progressive, recessiveactions; Tonicization! chromatic technique, 56-60, 66, 68, 72-75, 141, 146, 170 chronology of reference, as structural factor, 41-45, 85 in Classical development, 34, 42-45, 79, 297n, 382 diatonic technique, 50-51, 56, 71, 74, 77-79, 132, 146 distances of see Tonaldistance! elision, 58, 72 enharmonic technique, 58, 69, 73-75, 155 in modal practice, 45-47, 75 in opposition to stability, 47, 75-79, 140-41, 154-59, 162, 171, l83n, 260, 264, 280, 299n,372n, 375n rate of, as rhythmic factor see Tonal rhythm! in recent styles, 28,75-76, 92, 153-57, 171-77 Tonal function, 29-37, 62-67, 129-33, 278 see also Harmony, progressive, recessive actions; Tonal fluctuation; Tonality; Tonicization; and related entries! and linear function see Function, tonal and linear! multileveled see Levels of structure, in multileveled tonal and linear functions! I, IV, V, 62-63, 129-34 seealso individual entries! primary, secondary see Tonalfluctuation; Tonal system, secondary regions of; Levels of structure, in multileveled tonal and linear functions! quasi-, in modal practice, 28, 45-47, 75, 85n,87-90, 100, 120-26, 16970, 172, 180n quasi-, in recent styles, 28, 39, 75-76, 90-93, 102-5, 111, 119-20, 14668, 171-76, 181-83n, 225, 296n, 299n, 399

index of subjects, names, Tonality, 27-29, 46 see also Tonal system; andrelated entries! conditions oi] 172, 225,231 progressive, recessiveactions, 11, 60, 84-87, 132n, 161, 164-68, 313, 353, 389, 424n see also Tonal fluctuation! quasi-tonal orders see Tonalfunction, quasi-! and serialism, 18-19, 171-77 significance of, 27-28, 178 triadic conventions see Tonalconventions! Tonal range, 41, 204n Tonal regions, secondary see Tonal system, secondaryregions of! Tonal rhythm, 41, 59, 67, 71, 76-79, 134-38, 222, 395, 408, 420n accentual strength of tonicization, 79 defined, 78-79 functional orderings, 136-38, 314

Tonal system, 27-29

and terms

447

parenthetical components of, 41, 56, 105, 118, 146 primary, 34, 38, 41-42, 46, 52-53, 55-59, 67-68, 129-50, 159-68, 173-75 secondary regions components! oi, 33-38, 41-47, 50-53, 55-56, 6671, 118, 125, 129-38, 141, 147, 149n, 173-75 see alsoTonicization! specific particular!, 41-47, 134-36 symbols, 48-51, 134-36 Tonic, 27-29, 40n seealso relatedentries! forms, 49-50, 53-54 quasi- seeTonal function, quasi-! substitute, 131 Tonicization see also Leading-tone relation!: in inliation of harmonic content, 3435, 48, 67-69, 141 see also Tonal system, expanded! secondary see Tonal system,secondary regions of! techniques, 40, 71-75, 164, 166 see also Tonal Huctuation, chromatic technique, diatonic technique, enharmonic technique! Travis, R., 47n Trochee, 310, 327, 337n, 342, 373, 393, 422n

chromatic expansions of see Tonal fluctuation, chromatic technique! component of, 441: diatonic, 27-28, 71n, 87, 147, 162, 1801: embellishing componentsof seeLinear function, of tonics! expanded, 34-35,41-53, 59-61, 67-68, 76-78, 129-62, 170, 172, 204n Voice-leading, 29, 66, 68, 105, 117, 147, generic, 40, 47-51, 72 201 hierarchic ordering in, 13-14, 41-47, as harmonic determinant, 103-7, 112, 52-53, 57, 134-38 seealso Levels 119-20, 160, 164-65, 168, 171, of structure, in multileveled l82n tonal and linear functions; Linear function of tonics! of modal practice seeTonal function, Westergaard, P., 336n, 415-l6n quasi-, in modal practice! Woodrow, H. W., 337n