9780195384277_Steven Rings_Tonality and Transformation

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onality and ransormation


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OXFORD SUDIES IN MUSIC HEORY

Series Editor Richard Cohn

Studies in Music with Text , David Lewin

Music as Discourse: Semiotic Adventures in Romantic Music , Ko Agawu

Playing with Meter: Metric Manipulations in Haydn and Mozart’s Chamber Music for Strings, Danuta Mirka

Songs in Motion: Rhythm and Meter in the German Lied, Yonatan Malin

A Geometry of Music: Harmony and Counterpoint in the Extended CommonPractice, Dmitri ymoczko

In the Process of Becoming: Analytic and Philosophical Perspectives on Form in EarlyNineteenth-Century Music, Janet Schmaleldt

Tonality and Transformation, Steven Rings


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STEVEN RINGS

1


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1Oxord University Press, Inc., publishes works that urtherOxord University’s objective o excellence

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Library o Congress Cataloging-in-Publication Data

Rings, Steven.

onality and transormation / Steven Rings.

p. cm.—(Oxord studies in music theory)

Includes bibliographical reerences and index.

ISBN 978-0-19-538427-7

1. Music theory. 2. onality. 3. Musical intervals and scales. 4. Musical analysis.

I. itle.

M6.R68266 2011

781.2—dc22 2010019212

Publication o this book was supported by the AMS 75 PAYS Publication Endowment Fund o the American

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CONTENTS

Acknowledgments vii

Note to Readers ix

A Note on Orthography xi

Introduction 1

PART I Theory and Methodology

Chapter 1: Intervals, Transormations, and Tonal Analysis 9

Chapter 2: A Tonal GIS 41

Chapter 3: Oriented Networks 101

PART II

Analytical Essays

Chapter 4: Bach, Fugue in E major, Well-Tempered Clavier ,Book II, BWV 878 151

Chapter 5: Mozart, “Un’aura amorosa” rom Così fan tutte 171

Chapter 6: Brahms, Intermezzo in A major, op. 118, no. 2 185

Chapter 7: Brahms, String Quintet in G major, mvt. ii, Adagio 203

Aferword 221

Glossary 223

Works Cited 231

Index 239


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ACKNOWLEDGMENS

Tanks are due rst to my wie Gretchen, or her prodigious patience and grace;though she remains blissully innocent o this book’s contents, she has been a lim-itless source o inspiration. My son Elliott served as a sort o inadvertent qualitycontrol: this volume would have been completed at least a year sooner had he notcome along. I hope my thinking is a year more rened than it would have beenwithout him; my lie is certainly immeasurably richer. Te book is dedicated tothem. I also thank my mother Linda, my ather Dale, and my brother Mike ortheir unwavering love and support.

My Doktorvater Daniel Harrison oversaw the rst iteration o many o theseideas; his spirit imbues this work. David Clampitt introduced me to this styleo music theory and offered trenchant observations on ideas new to this book.My other teachers at various stages in my graduate work—in particular, PatMcCreless, Allen Forte, James Hepokoski, Leon Plantinga, Craig Wright, MichaelCherlin, and David Damschroder—all played crucial roles in shaping my theo-retical and critical outlook. Ian Quinn saved me rom mysel at one very earlystage, pointing out a rather hilarious, exhaustion-induced slip (I had decided orseveral pages that there were only 11 chromatic pitch classes). Ramon Satyendraand Julian Hook both read the manuscript and provided invaluable commentson matters large and small. Henry Klumpenhouwer offered critical perspectiveson the conceptual oundations o Lewinian theory, while Peter Smith and FrankSamarotto lent insight into matters Schenkerian. Dmitri ymoczko has been a

valued sparring partner and riend; this book has beneted greatly rom his criti-cal perspective. Te anonymous readers or Oxord challenged me on severalimportant points and orced me to clariy my thinking. Lucia Marchi answeredquestions on Da Ponte’s verse. My assistant Jonathan De Souza has done yeoman

work as a design consultant, typeace expert, punctuation cop, indexer, and editorextraordinaire (his passion or the Chicago Manual almost exceeds my own). Allo these individuals helped improve this book. Te errors that inevitably remainare mine, not theirs.

Richard Cohn has been a supporter o this work rom the beginning—he hasin many ways opened the theoretical space within which my ideas have unolded.As the editor o this series, he also helped acquire the book or Oxord. I cannotthank him enough. Suzanne Ryan has been a model editor, enthusiastic and

patient in equal measure; her condence and understanding helped me weathera couple rough patches. Norm Hirschy and Madelyn Sutton at Oxord have also

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viii Acknowledgmentsbeen a delight to work with, making the whole process run more smoothly than Icould have hoped. I would also like to thank the Mrs. Giles Whiting Foundationor supporting the year’s research leave necessary to complete the book, and the

American Musicological Society or a generous subvention rom the AMS 75 PAYSPublication Endowment Fund.

My scholarly home during this book’s writing has been the University oChicago. I would like to thank the chairs o the music department during thistime—Robert Kendrick and Martha Feldman, as well as interim chairs AnneRobertson and Larry Zbikowski—or providing unstinting support and helpingto create the ideal intellectual environment or a junior aculty member. My el-low theorists at Chicago—Larry Zbikowski and Tomas Christensen—are the

best proessional and intellectual role models I could hope or; they are also goodriends. My colleagues in historical musicology, ethnomusicology, and composi-tion have at once inspired me with their own achievements and helped to keep mythinking productively unsettled. Te same can be said or the remarkable cohort ograduate students that I have had the pleasure to work with over the last ve years.Finally, I would like to thank Berthold Hoeckner or many hours o stimulatingconversation and or his valued insights into the balance between work and amily.While I cannot imagine a better personal home than the one I described in the

rst paragraph o these acknowledgments, I cannot imagine a better intellectualhome than the one created by my riends and mentors in the University o ChicagoDepartment o Music.


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NOE O READER S

I have sought to make this book both accessible and ormally substantive. It isa tricky balance: one risks disappointing both specialists (who wish there wasmore math) and nonspecialists (who wish there was less). Te latter readersmay indeed wonder why any math is needed at all.1 While I hope that thebook will speak or itsel in this regard, I will answer here that statements intransormational theory gain their musical suggestiveness in signicant partrom the ormal structures that support them. Tose structures situate eachmusical interval or gesture within a richly developed conceptual space; the

algebraic contours o that space contribute in important ways to the charactero the interval or gesture in question. Tis underlying ormal context is some-times operative only “behind the scenes”; at other times it is thematized in theoreground o an analysis. In either case, an awareness o the pertinent under-lying structure adds considerably to the allusiveness o any observation made inthe theory, sensitizing us to the expressive particularity o a given musical rela-tionship. Moreover, the theory’s ormalism can act as a generator o insights:once a basic musical observation has been rendered in transormational terms,the technology can lead the analyst toward new observations. Te ormal pre-cision o the apparatus assures that the new insights will be related to the old,ofen in compelling ways.

I have nevertheless limited the amount o ormalism in the book, and notonly out o ethical concerns or accessibility. Te simplicity o the book’s mathresults rom my conviction that much o the ascination in this style o musictheory resides in the reciprocal interaction that it affords between ormal ideas andmusical experience. One does not need to delve very ar into the math to explorethat interaction. My interest has thus been to employ only as much ormalism as

needed, and to expend somewhat greater effort seeking out the ways in which theresulting technical ideas may be brought to bear on musical experience—modelingor shaping it—in ways that are at once concrete and immediate. For the specialistwishing or more mathematical development, I hope the basic ideas introduced

1. In addition to my comments here, readers may wish to consult John Clough’s eloquent answer to

this question in his review o David Lewin’s Generalized Musical Intervals and Transformations (Clough 1989, 227).

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x Note to Readershere might provide preliminary material or exploration and extension in morespecialized contexts.

I have introduced all ormal ideas in prose, using plain English rather than

ormal denitions, theorems, or proos. I have also included a Glossary thatdenes basic concepts rom transormational theory and abstract algebra—againin plain English. Chapter 1 is an introduction to transormational thought orthose new to it; readers well versed in the approach may wish to skip it and beginwith Chapter 2.2

2. Te book assumes a basic knowledge o tonal and post-tonal theories, including Schenkerian anal-

ysis. For readers new to Schenkerian thought, Cadwallader and Gagné 2010 is a ne introduction.Straus 2005 offers an accessible overview o post-tonal theory.


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A NOE ON ORHO GRAP HY

ransormational theorists ofen write the names o transormations in all capsand/or in italics.3 I have not employed either orthographic convention here. I havecome to nd the all-caps approach visually obtrusive; the all-italics approach ishard to sustain in all cases (or example, when transormation names are wordsrather than single letters). ransormation names in this book are thus simply writ-ten in Roman type, using a combination o upper and lowercase letters, dependingon what seems most readable in a given case. (For example, Chapter 1 introducesa transormation that resolves all elements to C; it is notated ResC.) Tere is onlyone exception to this orthographic rule: the identity element in a group o inter-

vals or transormations is always written as a lowercase italicized e, in keeping withmathematical conventions.

Note names ollow the Acoustical Society o America, with C4 as middleC. Curly brackets { } indicate unordered sets, while parentheses ( ) indicateordered sets. Pitch classes are ofen indicated by integers, using the amiliar C =0 convention. Te letters t and e sometimes substitute or the numbers 10 and 11when indicating pitch classes. Note that Roman e means “pitch class 11,” while ital-icized e means the identity element in a group o intervals or operations. In many

contexts, major and minor triads are indicated using amiliar neo-Riemanniannotation: the note name o the root is ollowed by a plus sign (+) or major or aminus sign (–) or minor: C+ = a C-major triad; E– = an E-minor triad.

3. Te all-caps approach derives rom articial intelligence and early computer programming lan-

guages, both inuential when David Lewin rst ormulated his transormational ideas in the 1970s.

Te italic approach derives rom mathematical orthography, in which variables are typicallyitalicized.

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1

Te research that gave rise to this book began as an effort to connect neo-Rieman-nian theory more ruitully to traditional ideas about tonal music.1 Te volume thathas resulted, as is so ofen the case, is something rather different: an exploration o

the ways in which transormational and GIS technologies may be used to modeldiverse tonal effects and experiences.2 Neo-Riemannian theory makes an appear-ance here and there, but the book is not primarily about harmonic transormations;nor is it limited in scope to chromatic music. Nor, or that matter, does it have muchto say about effi cient voice leading, a major ocus o neo-Riemannian studies and acentral preoccupation o geometrical music theory.3 Instead, the book seeks toreturn to certain undamental ideas rom transormational and GIS theory,exploring their potential to illuminate amiliar aspects o tonal phenomenology.

Te advent o neo-Riemannian theory nevertheless paved the way or theproject in a broader disciplinary sense. Te transormational models o chromaticharmony introduced in inuential studies by David Lewin, Brian Hyer, and RichardCohn gave rise to one o the more striking discursive shifs in music theory inrecent decades, as an algebraic technology once reserved or atonal musics wasapplied to the tonal repertory.4 I suspect that this work captured the imagina-tion o many theorists not only because it ocused on ear-catching progressions,or because it provided persuasive models o analytically challenging music, butalso because o the risson it generated by applying a mathematical metalanguage

to amiliar chromatic passages in Wagner, Schubert, Brahms, and others—musictraditionally modeled by Schenkerian or other tonal methodologies. Tis dis-cursive shit nevertheless seems to have created some conusion, binding

Introduction

1. For an accessible introduction to neo-Riemannian theory, see Cohn 1998. A brie denition is also

provided in the Glossary.

2. On “transormational and GIS technologies,” see Chapter 1.

3. Te major contributions to geometrical music theory have come rom Clifon Callender, Ian Quinn,

and Dmitri ymoczko. See especially ymoczko 2006 and Callender, Quinn, and ymoczko 2008.

4. Te seminal works in neo-Riemannian theory include Lewin 1982 and 1987 (the latter is citedhereafer as GMI; see pp. 175–80); Hyer 1989 and 1995; and Cohn 1996 and 1997.


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2 onality and ransormationthe technology employed to (implicit or explicit) assumptions about tonalityand atonality. Te mathematical apparatus o transormational analysis is stillmost amiliar to theorists rom atonal theory; this can oster the impression that

something in its underlying logic is undamentally nontonal. Such an impressionis heightened by the act that neo-Riemannian approaches have typically sought tomodel chromatic progressions whose tonal status is somehow in doubt. Tis canlead to the view that any application o transormational methods is an (implicit orexplicit) assertion that the passage in question is, in some sense, “not tonal,” or per-haps “not as tonal as we once thought.” O course, the underdenition o “tonality”(and its opposite) is a central issue here. We will return to that issue in a moment.

Neo-Riemannian analysis—with its ocus on local, chromatically striking

passages, ramed by more traditionally diatonic music—also seems to haveled to a view that some works are divvied up into some music that is tonal (orexample, because it is well analyzed by Schenkerian methods) and some that istransormational (because it is well analyzed by neo-Riemannian methods).5 Butthis is to misconstrue the word transormational, treating it as a predicate or acertain kind o music, rather than as a predicate or a certain style o analytical andtheoretical thought. Tat style o thought is moreover extremely capacious: GISand transormational theories simply provide generalized models o musical inter-

vals and musical actions, respectively. Intervals and actions are as undamentalto tonal music as they are to any other kind o music. Tere is nothing abouttransormational theory that makes it atonal in principle.

Nevertheless, to assert that transormational theory may be used to illumi-nate certain specically tonal aspects o tonal music—as I intend to do in this

volume—is to go one step urther. It raises the question o what is meant by theitalicized adjective, and its nominative orm, the rst word in this book’s title.“onality” is at once one o the most amiliar and most elusive terms in music-theoretical discourse.6 It is tied to a set o aural habits and experiences that areso deeply ingrained and seemingly immediate among Western listeners that theconcept is easily naturalized, complicating attempts to pin it down discursively.When theorists have sought to pin it down, they have constructed the concept ina bewildering variety o ways, leading to a “veritable prousion o denitions,” asBrian Hyer has put it.7 Further, the term has acquired a considerable amount oideological reight over its relatively short lie, making it not merely a descriptivelabel or a musical repertory or a set o aural habits, but a concept that has serveda variety o ideological interests.8

5. Such a view is evident, or example, in Samarotto 2003.

6. For the denitive history o the term and its French and German cognates, see Beiche 1992.

7. Hyer 2002, 726. Hibberd 1961; Tomson 1999; Krumhansl 2004, 253–54; and Vos 2000 also offer

valuable accounts o the term’s denitional tensions and its semantic “uzziness” (Vos’s term).

8. See Hyer 2002, 745–50. As Hyer notes (pp. 748–49), the term’s ideological baggage dates back to its

earliest popularization in the writings o Fétis, in which it participates in a conspicuously Orientalist

narrative o race and musical competence. Perhaps the most potent material example o such ideo-

logical entanglements has been presented by Ko Agawu, who argues that tonality has served as a“colonizing orce” in Arica (2003, 8ff; 2010).


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4 onality and ransormation But surely we already have plenty o satisying accounts o tonal experience.

Why add to them with this book? First, and most pragmatically, many questionshave arisen in recent years regarding the relationship between transormational

theories—most notably neo-Riemannian ones—and more traditional tonal ideas.As noted above, my original purpose in beginning this project was to clariy thatrelationship. Tough the book has outgrown its initial neo-Riemannian ocus, Ihope it can nevertheless help to make clear the ways in which transormationalideas can be re-inused with amiliar tonal concepts. o be sure, neo-Riemanniananalysts have purposeully avoided tonal categories in the past, as they have soughtto explain music in which such categories are understood as problematic or eveninappropriate. But many tonal theorists o more traditional bent will likely eel

that tonal concepts, however weakened or problematized, still have at least some relevance to the music typically studied in neo-Riemannian analyses. Te tech-nologies introduced in this book will allow both parties to explore the interac-tion between typical neo-Riemannian harmonic progressions and amiliar tonaleffects. And they will allow transormational analysts to integrate tonal conceptsinto their work i so desired, and, perhaps more important, to exclude them out oa conscious choice, not simply aute de mieux. 12

Second, and more substantively, this book is driven by the conviction that

no musical phenomenon, however amiliar, can be exhausted by a single theo-retical paradigm. As an entailment to this conviction, i we wish to illuminate aphenomenon as brightly as possible, we will do well to bring multiple theoreticalsearchlights to bear on it. Tis commitment to pluralism is very much in keep-ing with the project o transormational theory in general. Te technology o thetheory is designed to encourage pluralism: a given musical phenomenon admits omultiple GIS and transormational perspectives, none o which excludes any othera priori (which is not the same as saying that all o them are equally valuable). Teapproach simply asks the analyst to pursue and extend various musical appercep-tions13 within an algebraic ormal context, with the ull recognition that no singleormal context can lay claim to comprehensiveness—thus making obligatory mul-tiple perspectives on the same music. Given this commitment to pluralism, it isdisappointing, but perhaps not surprising, that antagonisms have arisen betweenpractitioners o transormational methods and those committed to other method-ologies, such as Schenkerian analysis. In this book I hope to show that competitionbetween such divergent modes o analytical engagement is misplaced, that theirmethodologies differ in crucial ways, and that they may in act coexist in analytical

praxis. Tese matters are addressed more thoroughly in section 1.4.onal music, like any richly allusive cultural phenomenon, exceeds any single

interpretive or analytical method. I do not adopt this view out o obeisance to ash-ionable philosophical positions—though such positions could likely be adducedin its support—but instead rom a simple awareness that any analytical approachwill, o necessity, tend to ocus on some aspects o musical experience and neglect

12. C. Lewin 1982–83, 335.13. Lewin preers the word intuitions . On both terms, see section 1.2.3.


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Introduction 5 others.14 Any analytical act will thus leave a surplus—a vast, unruly realm omusical experience that eludes the grasp o the single analytical model. Corners othat vast realm may nevertheless be illuminated via other analytical approaches,

but those approaches will leave their own surpluses. And so on. In short, and inless grandiose terms, there will always be something new to notice in music wecherish—new sonic characteristics to attend to, new ways to ocus our ears onamiliar patterns, new ways to experience sounds as meaningul. Tis is no lesstrue o tonal music, with its rich history o theoretical models. o borrow AlredNorth Whitehead’s memorable phrase, tonal music, like all music, is “patient ointerpretation.”15

Te models I develop here thus offer new ways o thinking about some very

amiliar aural experiences. Te hope is that those experiences may be deamil-iarized in the process, making us acutely alive to them again, and allowing us tosense tonal effects with renewed intensity, and in new ways.16 Surely one o thegreat values o music theory is its potential to reract, alter, and intensiy musicalexperience, in ways both subtle and not so subtle, as new discursive concepts arebrought to bear on the sonic stuff o music. onal music is no different rom anyother music in this regard: it admits o, and rewards, many modes o analyticalengagement.

Part I o the book covers theoretical and methodological ground, while Part IIcontains our analytical essays. Chapter 1 is intended primarily or those newto transormational theory: it includes primers on GIS and transormationalapproaches as well as two model analyses in a traditional transormational style(o passages by Bach and Schubert). Section 1.4 discusses methodological differ-ences between transormational analysis and Schenkerian analysis.

Chapters 2 and 3 present the main theoretical substance o the book. Chapter 2introduces a GIS that models intervals between pitches imbued with special tonalcharacters, or qualia . In addition to surveying the ormal resources o this GIS, thechapter includes several short analytical vignettes on music rom Bach to Mahler.While Chapter 2 is GIS-based, Chapter 3 is transormational in ocus, introducinga special kind o transormational network that can impose an orientation ona given transormational space, directing all o its elements toward one centralelement. Such networks model the ways in which a tonal center can act as a locuso attraction in a musical passage. Te directing o the listener’s attention towarda tonal center, which I call tonal intention, can be conceived as a special kind o

transormational action.Te chapters o Part II present analyses o a Bach ugue, a Mozart aria, a Brahms

intermezzo, and a Brahms quintet movement. Te act that there are two Brahms

14. Tis view in act accords with the methodology o a rather un ashionable school o literary criti-

cism: the Chicago School o R. S. Crane, Elder Olson, Richard McKeon, and others. For a good

overview o this style o pluralism, see Booth 1979.

15. Whitehead 1967, 136.

16. Here I ollow Hepokoski and Darcy (2006, 12), who express a similar desire or deamiliarizationthrough novel modes o analytical reection.


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6 onality and ransormationanalyses is not meant to suggest that these ideas are more applicable to his musicthan to that o others. Tese our chapters aim to demonstrate applications o thetechnologies in Part I to diatonic music that might not be immediately obvious to

the reader as apt or transormational study; it just so happened that two Brahmsmovements t the bill nicely. Te transormational literature abounds in studies ohighly chromatic works, and I expect the reader will nd it obvious how to applythe technologies o Part I to avorite chromatic passages in Schubert, Wagner,Wol, Strauss, and others. I have indeed included analytical vignettes in Part I thatpoint the way toward the application o these ideas to typical neo-Riemannianprogressions.17 Part II, by contrast, stresses the applicability o the present ideasto diatonic idioms that have been largely neglected in the recent transormational

literature. Te essays employ the concepts rom Part I in various ways. While theBach analysis draws primarily on the GIS rom Chapter 2 , the Mozart analysismakes considerable use o the oriented networks rom Chapter 3; both technol-ogies are in evidence in the Brahms analyses. Te Mozart and Brahms analysesurther include Schenkerian components, while the Bach draws on ideas romFuxian ugal pedagogy.

17. See in particular the Liszt example in section 2.8; the analysis o the chromatic Grail motive in

section 2.9.2; and the analysis o Brahms’s op. 119, no. 2, in section 3.7. Te reader is also reerredto my analyses o works by Schubert in Rings 2006, 2007, and 2011a.


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PAR I

Theory and Methodology


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9

CHAPER One

Introduction

Tough transormational theory is by now a amiliar presence on the musicological

landscape, ubiquitous in conerence programs and theoretical journals, it remains aspecialist subdiscipline within a specialist eld, largely the province o initiates. Mostmusic theorists have at least a casual acquaintance with transormational ideas, butonly a handul actively pursue research in the area; or other music scholars (histo-rians, or example), the theory is surely a closed book. As Ramon Satyendra hasnoted, this is due at least in part to the mathematical aspects o the approach, whichhe calls a “language barrier” that has inhibited “broad-based critique and commen-tary” (2004, 99). While that broad-based discussion has yet to emerge, the theory’s

reception among specialists has moved into a new critical phase, with certain o themethod’s oundational assumptions being held up to scrutiny on both technologicaland conceptual grounds—a sign o the theory’s continuing vitality. But such revi-sions also raise a worry: as renements to transormational methodologies becomeever more recherché, the theory threatens to leave behind a host o scholars whonever had a chance to come to terms with it in its most basic guise. Tis would beunortunate, or transormational methods, even in their simplest applications, rep-resent a style o music-theoretic thought o considerable power and richness, andone that is in principle accessible to a wide range o analytically minded musicians.

Tus, while this book is primarily about the application o transormationalideas to tonal phenomena, I hope it can also serve as an accessible general intro-duction to transormational theory. Te present chapter presents an overviewo the theory or the reader new to the approach (or or those who would likea reresher).1 Afer a capsule summary in section 1.1, sections 1.2 and 1.3 serveas primers on the two main branches o transormational thought: generalized

Intervals, ransformations,

and onal Analysis

1. Te discussion here complements the ne introductions to the theory rom Satyendra (2004) andMichael Cherlin (1993).


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10 onality and ransormationintervals and transormational networks. Tese sections introduce what we mightcall “classical” Lewinian intervals and transormations, as ormulated in DavidLewin’s Generalized Musical Intervals and ransormations (hereafer GMI ), the

oundational text in the eld. Tey also survey recent criticisms o and revisionsto Lewin’s ideas. Each section includes a little model analysis o a tonal passage,the rst by Bach, the second by Schubert. Te analyses are meant to display thetheory in action and to demonstrate its effi cacy in illuminating aspects o tonalworks, even beore introducing the new technologies o this book. Te analyses areagain in a rather “classical” transormational idiom, adopting modes o interpre-tation common in the literature. Tis will allow us, in section 1.4, to contrast suchtransormational approaches with Schenkerian analysis.

1.1 ransformational Teory in Nuce

ransormational theory is a branch o systematic music theory that seeks to modelrelational and dynamic aspects o musical experience. Te theory explores themaniold ways in which we as musical actants—listeners, perormers, composers,

interpreters—can experience and construe relationships among a wide range omusical entities (not only pitches). Te ormal apparatus o the theory allows theanalyst to develop, pursue, and extend diverse relational hearings o musical phe-nomena. Te theory articulates into two broad perspectives. One is intervallic,in which the subject “measures” the relationship between two musical objects, asa passive observer. Te other is transormational, in which the subject activelyseeks to recreate a given relationship in his or her hearing, traversing the spacein question through an imaginative gesture.2 Te conceptual difference betweenintervals and transormations is subtle, and some recent theorists have sought todownplay it.3 We will explore such matters in more detail later. For now we cansimply note that the emphasis in both modalities is on the relationships betweenmusical entities, not on the entities as isolated monads. ransormational theorythematizes such relationships and seeks to sensitize the analyst to them.

In both the intervallic and transormational perspectives, musical entities aremembers o sets, while the intervals or transormations that join them are mem-bers o groups or semigroups. We will discuss the italicized terms in the ollowingsection (denitions may also be ound in the Glossary); readers need not worry

about their ormal meaning or the moment. Intervallic structures are modeled viaGeneralized Interval Systems, or GISes, which comprise a set o elements, a groupo intervals, and a unction that maps the ormer into the latter. ransormational

2. As Henry Klumpenhouwer puts it, transormations model “moments o action carried out by and

within the analyst” (2006, 278).

3. See, or example, Hook 2007b, 172–77. Te distinction between the intervallic and transormational

perspectives was o central importance to Lewin, orming part o a general critique o Cartesian views o musical experience, as discussed in section 1.2.2 (see also Klumpenhouwer 2006).


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CHAPER 1 Intervals, ransormations, and onal Analysis 11relationships are modeled by transormational networks: congurations o nodesand arrows, with arrows labeled by transormations (drawn rom some semigroup)and nodes lled with musical entities (drawn rom some set).4 Any GIS statement

may be converted into a transormational statement, a technological conversionthat also implies (in Lewin’s thought) a conceptual conversion rom (passive)intervallic thinking to (active) transormational thinking.5 Te converse, however,is not true: there exist transormational statements that cannot be rendered in GISterms. Te transormational perspective is thus broader than the GIS perspective.For this reason, the term transormational theory is ofen used, as here, to encom-pass both modes o thought.

1.2 Intervals

1.2.1 GISes

GIS statements take the orm int(s, t) = i. Tis is a mathematical expression withormal content, which we will unpack in a moment. I would like rst, however,

simply to note that its arrangement on the page mimics a plain English sentence: itcan be read rom lef to right as a ormal rendering o the statement “Te intervalrom s to t is i.” We can understand GIS technology as an attempt to render explicitthe conceptual structure underlying such everyday statements about musicalintervals.6

Figure 1.1 will help us begin to explore that underlying conceptual structure.Te gure shows our GIS ormula again, now with its various components labeled.Te italicized words indicate mathematical concepts. Here I will present inormaldenitions o these words, offering just enough inormation so that the readerunderstands their overall structure and can begin to appreciate their suggestive-ness—both singly and in combination—as models or intervallic concepts. Moredetailed discussions o each term may be ound in the Glossary.

int(s, t) = i

a function membersof a set

memberof a group

Figure 1.1 A GIS statement with components labeled.

4. Given the theory’s emphasis on relationships over isolated musical elements, the technical emphasis in

the discourse is typically on groups and semigroups, basic concepts rom abstract algebra.

ransormational theory is thus an algebraic music theory. Recent developments in geometrical music

theory—see, or example, Callender, Quinn, and ymoczko 2008—represent a departure rom this

algebraic oundation. Tough such geometrical approaches are sometimes considered subsets o

transormational theory writ large, I will limit the term transormational here to algebraic approaches.

5. Klumpenhouwer (2006) describes the conceptual transition rom intervallic to transormational

thinking as the general theme o GMI. 6. Whether GISes succeed ully in this regard is a question to which we will return.


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12 onality and ransormation As the gure shows, the elements s and t are both members o a mathematical

set . For present purposes, a set may simply be understood as a collection o distinctelements, nite or innite. Te elements are distinct in that none o them occurs

more than once in the set.7 Lewin calls the set that contains s and t the space othe GIS, which he labels S. Te space S may consist o pitches, or pitch classes, orharmonies o a particular kind, or time points, or contrapuntal congurations, ortimbral spectra—and so on. GISes thus extend the idea o interval to a whole hosto musical phenomena, not just pitches; this is one o the senses in which they are“generalized.”

Note that the elements s and t are given in parentheses in the ormula, sepa-rated by a comma. Tis indicates that they orm an ordered pair: (s, t) means “s

then t.” Te ordered pair (s, t) is distinct rom (t, s). GISes thus measure directed intervals—the interval rom s to t, not simply the undirected interval between s andt. For example, measuring in diatonic steps, the interval rom C4 to D4 is differentrom the interval rom D4 to C4: int(C4, D4) = +1, while int(D4, C4) = –1. Tisdiffers rom some everyday uses o the word interval, in which we might say, orexample, “Te interval between C4 and D4 is a diatonic step.” GISes do not modelsuch statements, but instead statements o the orm “Te interval rom C4 to D4 isone diatonic step up (i.e., +1 in diatonic space)” or “Te interval rom D4 to C4 is

one diatonic step down (i.e., –1 in diatonic space).”Te element i to the right o the equals sign is a member o a group . Lewin callsthe group o intervals or a given GIS IVLS. A group is a set (that is, a collectiono distinct elements, nite or innite) plus an additional structuring eature: aninner law or rule o composition that states how any two elements in the set can becombined to yield another element in the set. Lewin calls this inner rule a “binarycomposition,” and we will ollow that usage here.8 Groups underlie a great manyamiliar conceptual structures. For example, take the set o all integers, positive,negative, and zero. As a set, this is simply an innite collection o distinct entities:{… , –3, –2, –1, 0, 1, 2, 3, …}. But once we introduce the concept o addition asour binary composition, the set o integers coheres into a group, which we call“the integers under addition.” Addition, as a binary composition, offers one wayin which we can combine any two integers to yield another integer: given any twointegers x and y, x + y will always yield another integer z . Tis is called the groupproperty o closure: the composition o any two elements under the binary compo-sition always yields another element in the same set.

Groups have three other properties. First, they contain an identity element,

labeled e (or the German word Einheit ). Te composition o e with any othergroup element g yields g itsel. In our group o integers under addition, the iden-tity element is 0: 0 added to any integer x yields x itsel. Further, or every element

7. A set in which elements appear more than once is called a multiset. Multisets have music-theoretical

applications, but we will not explore them in this study.

8. Most mathematicians call Lewin’s “binary composition” a group operation or binary operation .

Lewin, however, somewhat idiosyncratically reserves the word operation or a different ormal con-cept, as we will see, thus making binary composition preerable in this context.


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CHAPER 1 Intervals, ransormations, and onal Analysis 13 g in a group there also exists an element g –1 in the group such that when g and g –1 are combined e is the result. Te element g –1 is called the inverse o g ( g –1 is read“ g -inverse”). In the group o integers under addition, the inverse o any integer x is

–x (e.g., the inverse o 3 is –3, as the two o them added together yield 0, the iden-tity element). Finally, composition within any group is associative . Tat is, giventhree group elements , g, and h, then ( • g ) • h = • ( g • h ).9 o return once againto our example, addition o integers is clearly associative: or any three integersx, y, and z, (x + y ) + z = x + ( y + z ).

Mathematicians study groups primarily or their abstract structure, a structurethat is suggested in its most basic terms by the our conditions outlined above (clo-sure, existence o an identity, existence o inverses, associativity). Te GIS ormula-

tion rests on the idea that intervals, at a very general level, have this same abstractstructure—they are group-like. Tat is, the combination o any two intervals will yieldanother interval (closure). Any musical element lies the identity interval rom itsel(existence o identities). Given an interval i rom s to t, there exists an interval rom tto s that is the “reverse” o i—that is, i–1 (existence o inverses). Finally, we recognizethat intervals combine associatively: given intervals i, j, and k, (i • j) • k = i • (j • k). 10

Note that these abstract, group-like characteristics do not encompass certaincommon ideas about intervals. For instance, there is nothing in the our group

conditions that says anything about direction or distance —two attributes ofenattributed to intervals. Tis is one area in which the GIS concept has recently beencriticized.11 Tough it is tempting to interpret the numbers that we use to labelgroup elements—like the integers +1, –5, and so on in a diatonic or chromaticpitch space—as representative o distances and directions (treating +1 as “onestep up,” and –5 as “ve steps down,” or example), those interpretations are notinherent in the abstract structure o the group. Tat is, the group itsel, qua abstractalgebraic structure, knows nothing o “one step up” or “ve steps down.” Instead,it knows only about the ways in which its elements combine with one anotheraccording to the properties o closure, existence o an identity and inverses, andassociativity. We can conclude two things rom this: (1) GISes are ormally quiteabstract, and may not capture everything we might mean by interval in a givencontext; and (2) not all o the intervals modeled by GISes need to be bound upwith the metaphor o distance.12 While the distance metaphor will likely be quitecomortable or most readers in discussions o pitch intervals, it nevertheless willeel inappropriate in other GIS contexts—or example, when one is measuringintervals between timbral spectra, or between contrapuntal congurations in tri-

ple counterpoint (à la Harrison 1988).13 Indeed, the metaphor o distance will not

9. Te symbol • here is a generic symbol or the binary composition in any group. When the idea o

group composition is understood, such symbols are sometimes eliminated. In that case, our asso-

ciativity notation would look like this: ( g )h = ( gh ).

10. See GMI, 25–26.

11. ymoczko 2008 and 2009.

12. As Rachel Hall puts it, “GISes can express notions about distance, but are not orced to do so”

(2009, 209).13. C. Hall 2009, 208–9.


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14 onality and ransormationalways eel apt in the primary GIS in this book, introduced in Chapter 2.14 TeGIS concept thus abstracts away rom notions o distance, generalizing the ideao interval to relational phenomena in which the distance metaphor might not be

appropriate.We are nevertheless ree to add notions o distance and direction to our inter-

pretations o GIS statements, i so desired. Dmitri ymoczko (2009), Lewin’s maincritic on this ront, has indicated how distance may be reintroduced into a GIS byadding a metric that ormally ranks the distances between all pairs o elementsin the space o the GIS. In practice, this usually amounts to reading numeric GISintervals—like +1, –5, and so on—as indicators o distance and direction, in theusual arithmetic sense (with –5 larger than +1, and proceeding in the opposite

direction). We will not employ ymoczko’s distance metric explicitly in our ormalwork in this study, but we will ofen rely on the idea implicitly, whenever we wishto interpret intervals as representing various distances.15

Groups, or all o their abstraction, nevertheless remain suggestive as a modelor generalized intervals. Tis is because each group has an underlying abstractstructure—or, we might say guratively, a certain “shape.” Tis shape is determinedby the number o elements in the group and the various ways they combine withone another (and with themselves). A group, or example, may be nite or innite.

It may contain certain patterns o smaller groups (called subgroups ) that articulateits structure in various ways. A group may be commutative or noncommutative:two group elements and g commute i • g = g • ; in a noncommutative group thisproperty does not always hold.16 I two groups are isomorphic they have the sameabstract structure. A GIS inherits the particular structural characteristics o itsgroup IVLS. One way to think o this is that a given intervallic statement in Lewin’smodel inhabits a certain conceptual topography—a sort o landscape o intervallicrelationships given shape by the structure o the group IVLS. Different types ointerval may thus inhabit considerably different conceptual topographies, based onthe structure o their respective groups (e.g., whether the groups are nite or in-nite, commutative or noncommutative, articulated into subgroups, and so orth).Tis suggests that the intervallic experiences corresponding to such intervals have

14. Tat GIS, which calculates intervals between qualitative tonal scale degrees, involves a group o

intervals that might better be understood as comprising amiliar intervallic qualities rather than

distances, such as the quality o a minor third, as opposed to that o an augmented second. Te

distance metaphor is especially inapt in connection with certain exotic interval types that we willexplore in sections 2.5 and 2.6.

15. Edward Gollin (2000) explores another model or distances in a GIS, measuring word lengths in

the elements o the intervallic group. In the group o neo-Riemannian operations, or example, the

word PLP, o length 3, is longer than the word RL, o length 2. A given group admits o multiple

distance-based interpretations, based on which group elements are chosen as unitary (words o

length 1) via the ormalism o group presentation, as Gollin demonstrates.

16. wo amiliar noncommutative groups in music theory are the group o transpositions and inver-

sions rom atonal theory, and the group o neo-Riemannian operations. In the ormer group, it is

not generally true that m ollowed by In is the same as In ollowed by m . For example, 3 -then-I2

equals I11 , while I2 -then-3 equals I5 . In the neo-Riemannian group, given operations X and Y, it isnot generally true that XY = YX. For example, PL ≠ LP, RL ≠ LR, PR ≠ RP, and so on.


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CHAPER 1 Intervals, ransormations, and onal Analysis 15certain crucial differences in structure, differences embodied in the structures otheir respective groups. Such differences are ofen interpretively productive—theormalism encourages us to attend to them careully, as we pursue and extend any

given intervallic statement within a particular analytical context.Tus ar in our survey o GIS structure, we have two separate collections that

are as yet entirely independent: the space S o musical elements and the groupIVLS o intervals. We have not yet shown how various intervals in IVLS can beunderstood to span pairs o elements in S. Te lefmost element in the GIS or-mula, int, provides that connection. As indicated in Figure 1.1, int is a unction ormapping (the two words are synonymous or our purposes). A unction rom a set X to a set Y sends each element x in X to some element y in Y . Drawing on amiliar

schoolbook notation, we write (x ) = y to reer to the action o unction sendingelement x to element y . Note how the schoolbook orthography exactly matchesthe layout o our GIS statement: compare (x ) = y and int(s, t) = i. Te element x in the statement (x ) = y is called the argument o the unction, and the element y is the image or value o the argument x under . Te set X o all arguments is calledthe domain o the unction, while the set o all images in Y is called the range .

Te domain or our unction int in a GIS is not simply the space o musicalelements S itsel, but the set o all ordered pairs o elements rom S. Our arguments

are thus not single elements rom S, but ordered pairs o the orm (s, t). We can seethis by comparing again our two statements (x ) = y and int(s, t) = i; the orderedpair (s, t) is “in the role o x ” in our GIS statement, not simply some single elementrom S. Te set o all ordered pairs (s, t) is labeled S × S and is called “S cross S” orthe Cartesian product o S with itsel. Te unction int sends each ordered pair toan element in IVLS. So, ormally speaking, int maps S × S into IVLS.

As an example o how all o this works, let us take the two GIS statements sug-gested above, measuring the interval rom C4 to D4 (and the reverse) in diatonicsteps:

int(C4, D4) = +1int(D4, C4) = - 1

In both GIS statements, the space S consists o the conceptually innite collectiono diatonic “white-note” pitches (NB, not pitch classes). Te group IVLS is the inte-gers under addition, our amiliar group discussed above. Te mapping int sendsevery ordered pair o diatonic pitches to some element in the group o integers. Itsends the ordered pair (C4, D4) to the group element +1 in IVLS, modeling the

statement “Te interval rom C4 to D4 is one diatonic step up.” It then sends theordered pair (D4, C4) to a different element in IVLS, –1, modeling the statement“Te interval rom D4 to C4 is one diatonic step down.”17 Te two intervals, +1 and–1, are inversionally related, indicating that int(C4, D4) ollowed by int(D4, C4)will leave us back where we started, with an overall interval o 0, as intuition dic-tates. Tis relates to a general condition or a GIS, Condition (A): given any threemusical elements r, s, and t in S, int(r, s)int(s, t) = int(r, t). Tat is, the interval rom

17. Te locutions “one diatonic step up” and “one diatonic step down” evoke ideas o distance anddirection, suggesting the pertinence o ymoczko’s distance metric to this particular GIS.


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16 onality and ransormationr to s, plus the interval rom s to t, must equal the interval rom r to t. Tus, in ourexample int(C4, D4)int(D4, C4) = int(C4, C4) = 0. Or int(C4, D4)int(D4, E4) =int(C4, E4) = +2. A second condition, Condition (B), states that, or every musical

element s in S and every interval i in IVLS, there exists exactly one element t in Ssuch that int(s, t) = i.18 Again, a musical context makes the condition clear: let theelement s be the note C4 and the interval i be “one diatonic step up.” Within the seto all diatonic “white-note” pitches, there is o course only one pitch that lies “onediatonic step up” rom C4, that is, D4. Tese two conditions lend a certain logicaltightness to GIS structure, providing only one interval between any two musicalelements within a GIS.19 Tis property is called simple transitivity. As a result o thetwo conditions, in any GIS there will always be exactly as many elements in S as

there are intervals in IVLS. For example, in the GIS corresponding to pitch classesin 12-tone equal temperament, there are 12 elements in S (the 12 pitch classes) and12 intervals in IVLS (the integers mod 12).

We now turn to some philosophical and methodological matters raised byGISes.

1.2.2 GISes and Cartesian Dualism

Te cumbersome structure o GIS statements enacts aspects o Lewin’s critique oCartesian dualism. Note that the main action modeled in a GIS is the action car-ried out by the mapping int. It is int that carries us “across the equals sign” rom thelef-hand side to the right-hand side o the ormula int(s, t) = i. Te active natureo int is especially evident i we use an arrow notation to rewrite the unction. Teschoolbook unction (x ) = y may also be written x y , showing that the unction takes x to y . Similarly, we can rewrite the GIS unction int(s, t) = i as (s, t) int i,showing that int takes (s, t) to i. Tis notation makes visually vivid the act that intis the primary action involved in a GIS statement, capturing the act o pairing twomusical elements with an interval. Te relevant thought process might be verbal-ized thus: “I just heard a C4 and now I hear a D4; the interval rom the ormer tothe latter is one diatonic step up.”

Lewin characterizes this attitude as Cartesian because it is the attitude osomeone passively calculating relationships between entities as points in someexternal space. Te action o passively measuring is embodied by the mapping intitsel. One might think o int as analogous to pulling out some calculating device

and applying it to two musical entities “out there” to discern their intervallic rela-tionship. Te action in question is not one o imaginatively traversing the spacerom C4 to D4 in time, construing and experiencing a musical relationship along

18. Conditions (A) and (B) appear in the ormal denition o a GIS in GMI, 26.

19. o be clear, there is only one interval between two musical entities within a single GIS. As dis-

cussed in section 1.2.5, GIS methodology rests on the idea that there is in act an indeterminate

multiplicity o possible intervals between two musical entities. Tat multiplicity arises not within

a single GIS, but via the multiple potential GIS structures that may embed the two elements inquestion.


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CHAPER 1 Intervals, ransormations, and onal Analysis 17the way. Te GIS ormula is urther like the Cartesian mindset in that it exhibits acertain racturing o experience, a conceptual split between musical elements (thespace S), musical intervals (the group IVLS), and the conceptual action (int) that

relates the two. Te cumbersome nature o the GIS ormalism—with its three com-ponents (S, IVLS, int), all o which need to be coordinated, and with the action intplacing the musical “perceiver” in an explicit subject-object relationship vis-à-visthe music being “perceived”—thus encodes aspects o the Cartesian split betweenres cogitans and res extensa , a amiliar trope in Lewin’s writings.20

We should not conclude rom this that GISes are “bad” and that we shouldnot use them in our analytical and theoretical work. Lewin himsel continuedto nd intervallic thinking ascinating and productive long afer GMI, as a his-

torical phenomenon, a theoretical/ormal problem, and a mode o generatinginsights into musical works.21 In GMI itsel he also observes certain ways inwhich transormational thinking is “impoverished” in comparison to intervallicthinking (GMI, 245–46). In short, despite the act that the GIS ormalism enactsthe Cartesian problematic that Lewin so eloquently criticized, it is still a produc-tive and suggestive technology in many theoretical and analytical contexts.22 GISmodels will play an important role in this book.

1.2.3 Intervallic Apperceptions

Lewin ofen reers to intuitions in his writings about intervals and transorma-tions, but he never says exactly what he means by the word. It will be valuable orus to spend a little time here thinking about the matter, as the questions that itraises bear directly on the relationship between transormational technology andmusical experience.

Tough Lewin gives us no clear denition o what he means by intuitions, wecan iner two crucial characteristics o the term as he uses it in his writings:

(1) His intuitions are culturally conditioned.(2) Tey may be sharpened, extended, or altered through analytical reection.

Lewin states (1) explicitly: “Personally, I am convinced that our intuitions arehighly conditioned by cultural actors” (GMI, 17). By “cultural actors,” Lewin

20. Te relevant philosophical matters are penetratingly treated in Klumpenhouwer 2006 . On the prob-

lematics o the subject-object relationship in passive musical perception , see Lewin’s well-known

phenomenology essay (Lewin 2006, Chapter 4).

21. Lewin published three extensive articles specically on GISes, not transormational systems, afer

GMI : Lewin 1995, 1997, and 2000–2001.

22. My ideas on these matters were claried through conversation with Henry Klumpenhouwer. My

view differs slightly rom Klumpenhouwer’s published comments, in which he states that Lewin

wants us to “replace intervallic thinking with transormational thinking” (2006, 277). I eel that

Lewin’s ethical directive in GMI is not quite this strong—that he wants us not to replace inter-

vallic thinking but to become more aware o its Cartesian bias, and to be sel-conscious about thatbias whenever “thinking intervallically” in some analytical context.


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18 onality and ransormationseems to mean not only differences between various world cultures—thoughhe certainly does mean that—but also historical cultural differences withinthe history o European art music. For example, a sixteenth-century musician

conditioned by ideas about modes, hexachordal mutation, mi-contra-a prohi-bitions, and so orth would have different intuitions about a given musical pas-sage—say in a motet by Palestrina—than would a modern musician conditionedby ideas about keys, diatonic scales, tonal modulation, and so orth.23 Te modernlistener can o course seek to develop hexachordal hearings o the music inquestion, but to that extent—and this leads to characteristic (2)—the listener willbe modiying her or his intuitions (à la Lewin) by analytical intervention. Tegeneral pertinence o characteristic (2) to Lewin’s thought is maniest throughout

his writings, as theoretical structures o various kinds are brought to bear on var-ious musical experiences, sharpening, extending, or altering those experiencesin diverse ways. Indeed, Lewin’s entire analytical project can be understood asa process o digging into musical experience and building it up through analyt-ical reection. Stanley Cavell, paraphrasing Emerson, provides a very suggestivewording that we can borrow or the idea: such work involves a reciprocal “play ointuition and tuition,” or, even more suggestively, it is a project o “providing thetuition or intuition.”24

Lewin’s intuitions are special in the degree to which they reect the inuencenot only o broad cultural and historical conditioning, but also o theoretical con-cepts and other discursive constructions.25 I thus preer to think o such “intui-tions” as apperceptions: perceptions that are inuenced by past experience and mayinvolve present reection.26 Te second clause makes clear that such experiencesare responsive to current analytical contemplation: a GIS or transormationalstatement need not be a report on some prereective experience, but might insteadhelp to shape a new experience (an apperception), or alter an old one, throughanalytical mediation. Te word intuition, by contrast, runs the risk o naturalizingGIS and transormational statements, treating them as unmediated reports onprereective (or at least minimally reective) experience. Tis risk is especially

23 . C. the discussions o hexachordal versus tonal perceptions in Lewin 1993, 48n13 and Lewin

1998b.

24. Cavell 1999, 236 (“a play o intuition and tuition”) and Cavell 2002, section 4 (“providing thetuition or intuition”). For Emerson’s original quote see Emerson 1993, 27.

25. As Henry Klumpenhouwer notes, “In distinction to other uses o the term, Lewin’s intuitions have

some conceptual content” (2006, 278n3).

26. In this book I will understand apperceptions loosely in William James’s sense, as experiences col-

ored by “the previous contents o the mind” (1939, 158). Such apperceptions may involve conscious

reection, or they may not. For example, one’s past experiences with a certain musical idiom will

strongly color one’s current and uture musical experiences with music in that idiom, whether one

has consciously reected on the idiom or not. Apperceptions, thus conceived, are simply current

experiences under the inuence o past experience, and open to present reection. Tis departs

rom certain philosophical understandings, in which conscious reection is a necessary compo-nent o all apperceptions.


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CHAPER 1 Intervals, ransormations, and onal Analysis 19evident when a given statement is made seemingly universal by locutions suchas “when hearing music x, we [NB] have intuition y ”—a rhetorical device thatoccurs with disconcerting requency in Lewin’s writings. By hewing to the word

apperception, I instead hope to make clear that the sorts o experiences explored inthis book are by no means universal, and will be strongly shaped not only by one’scultural background and historical context, but also by the concrete particulars opresent analytical engagement.

1.2.4 GISes: Formal Limitations

As noted above, the abstract nature o GISes allows them to model a wide array ointervallic phenomena via algebraic groups. Yet, despite this abstraction, GISes arenot as general as they might at rst appear, nor are they applicable to all musicalsituations. GISes, or example, cannot model intervals in musical spaces that havea boundary or limit. Consider an example that Lewin himsel raises: S is the spaceo all musical durations measured by some uniorm unit. Tis space has a naturallimit: the shortest duration lasts no time at all—there is no duration shorter thanit. Now imagine that we choose to measure the interval rom duration s to dura-

tion t in this space by subtracting s rom t (IVLS would then be the integers underaddition). For example, i s is 6 units long and t is 4 units long, the interval roms to t is 4–6 = –2. Formally, int(s, t) = int(6, 4) = –2. Now recall the Condition (B)or a GIS: given any element s in S and any i in IVLS, there must exist some t inS such that int(s, t) = i. Let us now set s = 0 and i = –2. Tere exists no t in S suchthat int(0, t) = –2. Such a t would be 2 units shorter than no time at all. As Lewinhimsel notes, this is an absurdity (GMI, 29–30). Tus, the given musical space odurations under addition, though it is musically straightorward, cannot be mod-eled by a GIS. Similar problems arise with any musical space that has a boundarybeyond which no interval can be measured.

Tis relates to a more general limitation. Given Lewin’s denition, any intervalin a GIS must be applicable at all points in the space: i one can proceed the intervali rom s, one must also be able to proceed the interval i rom t, no matter whati, s, and t one selects. Tis limits GISes to only those spaces whose elements all haveuniorm intervallic environments. Te vast majority o amiliar musical spaces do have this property. For example, in the space o chromatic pitches, one can moveup or down rom any pitch by +1 semitone or –1 semitone. By extension, one can

theoretically move up or down rom any pitch by +n semitones or –n semitones,or any integer n . Tis is so even when the result would be too high or too low tohear—the space is still in principle unbounded.27 Similarly, in modular spaces, suchas the space o 12 pitch classes, or the space o seven scale degrees, every elementinhabits an identical intervallic environment. Neo-Riemannian spaces are also

27. o bound it, one would need to assert a specic high pitch beyond which one cannot progress up

by one semitone, and/or a specic low pitch beyond which one cannot progress downward by onesemitone. On theoretically unbounded spaces that can be perceived only in part, see GMI, 27.


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20 onality and ransormation uniorm in this sense: one can apply any neo-Riemannian transormation to anymajor or minor triad. Nevertheless, there do exist spaces that do not have this uni-orm quality, such as the durational space outlined above, or any number o voice-

leading spaces that are better modeled geometrically (as discussed in ymoczko2009). Tus, despite their generalized qualities, GISes are not as broad in scope asthey might initially appear to be: they only apply to uniorm intervallic spaces.28

ymoczko (2009) raises another important criticism o GISes: they do not admito multiple, path-like intervals between two entities. We will return to this importantcriticism in section 2.3, in which I will integrate ymoczko’s path-like conception intothe GIS introduced in that chapter. ymoczko also objects that GISes do not modelentities such as “the interval G4→E♭4” at the opening o Beethoven’s Fifh Symphony.

Instead, a given GIS would model the interval rom G4 to E♭4 as an instance o a moregeneral intervallic type that applies throughout the space: or example, as a manies-tation o the interval “a major third down.” Such an interval could obtain between anyother pair o major-third-related elements in the space, say F4→D♭4, or B6→G6. Moregenerally, unlike ymoczko’s “interval G4→E♭4,” intervals in a GIS are not dened bytheir endpoints, but by the relationship the listener or analyst construes between thoseendpoints, a relationship that is generalizable apart rom the endpoints in question.Te construing o that relationship is modeled by the statement int(s, t) = i, which

produces generalized interval i as output. ymoczko’s ormulation provides a differ-ent and useul perspective, ocusing more attention on the concrete endpoints o aspecic interval (s and t), and less on the ways in which a listener or analyst mightconstrue the relationship between those endpoints as some general interval-type i.But an attractive aspect o GIS theory is lost in the process, to which we now turn.

1.2.5 GIS Apperceptions and Intervallic Multiplicity

GIS technology is responsive to the act that one will be inclined to experience aninterval rom G4 to E♭4 in diverse ways based on the musical context within whichone encounters those pitches. Tere is thus no single “interval rom G4 to E♭4.”Imagine the succession G4→E♭4 in: (1) the opening o Beethoven’s Fifh; (2) a serialwork by Schoenberg; (3) an octatonic passage by Bartók; (4) a pentatonic passageby Debussy (or, or that matter, in a Javanese gamelan perormance in slendrotuning). Tese diverse contexts suggest the pertinence o various GIS appercep-tions or the G4→E♭4 succession. In the Beethoven, the pitch topography is diatonic,

and the GIS might be any one o a number o diatonic GISes (pitch-based or pitch-class-based).29 Te interval in Schoenberg would likely suggest a chromatic GIS,

28. What I have been callinguniorm , ymoczko (2009) calls homogenous and parallelized. Te latter term

means that one can move a given interval rom point to point, applying it anywhere in the space.

29. Te GIS introduced in Chapter 2 would be especially well suited to modeling the interval in

question. It would urther distinguish the G4→E♭4 at the outset o the Fifh rom the same pitch

succession in E♭ major (say, in the primary theme area o the Eroica ), or rom an enharmonically

equivalent succession in E minor (say, in the “new theme” in the Eroica development [e.g., cello,downbeat o m. 285 to that o m. 286]).


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CHAPER 1 Intervals, ransormations, and onal Analysis 21while in Bartók it would evoke an octatonic GIS, and in Debussy (or the gamelanperormance) a pentatonic GIS; any one o these GISes could be pc- or pitch-based.Te various GISes capture the ways in which one’s apperceptions o the G4→E♭4

succession might vary in response to its diverse musical/stylistic contexts.Tis is a rather obvious instance o what we might call “apperceptive multiplicity”

in intervallic experience. Less obvious, perhaps, is GIS theory’s insistence on apper-ceptive multiplicity when conronting a single interval in a single musical passage.Tis suggests that the interval in question can inhabit multiple musical spaces at once.Lewin puts it somewhat more strongly than I would: “we do not really have one intu-ition o something called ‘musical space.’ Instead, we intuit several or many musicalspaces at once” (GMI, 250). Per the discussion in section 1.2.3 above, I would rephrase

this as “we can conceive o a given interval in several different conceptual spaces whenwe are in the act o analytical contemplation. Tose different conceptions can subtlychange our experience o the interval, leading to new musical apperceptions.”30

1.2.6 Vignette: Bach, Cello Suite in G, BWV 1007, Prelude, mm. 1–4

Figure 1.2(a) shows the rst two beats o the Prelude rom Bach’s Cello Suite in G

major, BWV 1007. An arrow labeled i extends rom the cello’s opening G2 to theB3 at the apex o its initial arpeggio. Figures 1.2(b)–(d) model three intervallicconceptions o i, situating it in different musical spaces.

Figure 1.2(b) models i as an ascending tenth . Tis suggests the context shownon the staff: B3 is nine steps up the G-major diatonic scale rom G2. Te gureshows this by placing in parentheses the elements that i “skips over” in the space So the relevant GIS. S in this example comprises the elements o the (conceptuallyinnite) G-major diatonic pitch gamut, and IVLS is our amiliar group o integersunder addition, hereafer notated (ℤ, +).31 Given two diatonic pitches s and t in Gmajor, int(s, t) in this GIS tells us how many steps up the diatonic G-major gamut tis rom s. Te gure thus models the GIS-statement int(G2, B3) = +9.32

30. While my rewording ocuses on listening experiences stimulated by analytical reection, it is not

clear rom his comment what sorts o listening contexts Lewin has in mind. He may indeed have

meant that multiplicity is a act o everyday musical experience: when we hear music in any con-

text, we “intuit” multiple musical spaces at once and thus hear intervallic relationships in maniold

ways—even when we are not in an analytically reective mode. Tis may be true, but I am notsure how one could test the idea, nor do I know what exactly is meant by “intuit” and “intuition”

in Lewin’s passage. Is the listener consciously aware o these maniold “intuitions”? Or are they

perhaps instead a congeries o more or less inchoate sensations that one has when listening, which

can be brought into ocus through analytical reection? I am more comortable with the latter

position, which moves toward my rewording.

31. ℤ is a common label or the set o integers, taken rom the German Zahlen (numerals).

32. Lewin (GMI, 16–17) discusses the discrepancy between the amiliar ordinal intervallic names o

tonal theory (tenths, fhs, thirds, etc.), which indicate number o scale steps spanned between two

pitches, and the intervals in a scalar diatonic GIS, which indicate the number o scale steps up rom

one pitch to another (negative steps up are steps down). Te GIS introduced in Chapter 2 employsthe amiliar ordinal names or intervals between scale degrees.


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22 onality and ransormation

Te GIS o 1.2(b) does not do ull justice to the harmonic character o i. I wesay that i is a tenth , we are likely not thinking primarily about a number o stepsup a scale, but about a privileged harmonic interval. Figure 1.2(c) provides oneharmonic context or i, depicting it as spanning elements in a G-major arpeggio .Our space S no longer consists o all o the elements o the G-major diatonic gamut,but just those pitches belonging to the (conceptually innite) G-major triad, thatis: {. . . G1, B1, D2, G2, B2, D3, G3, B3, D4, G4, . . .}. G2 is adjacent to B2 in this space,as is B2 to D3, and so on.33 In this space, B3 is “our triadic steps up” rom G2: that

is, int(G2, B3) = +4.

34

Note that the D3 in the opening gesture divides i into twosmaller intervals, labeled j and k on the gure; both are “skips” o +2 in the GIS.

1 2 3 4 5

i

i

i

i

j

k ( )

( )

i

1 2 3 4 5

i

jk

? ?

jk

k –1 k –1

=

1 2 3 4 5

j

k

lm

( )

j

k

l

m

=

(a)

(b) (c) (d)

(e)

(f)

Figure 1.2 Bach, Prelude rom the rst suite or solo cello, BWV 1007: (a) themusic or beats one and two, with one interval labeled; (b)–() various GIS per-spectives on that interval.

33. Adjacent pitches here correspond to the “steps” in Fred Lerdahl’s “triadic space” (2001), or to

“steps” in the “chordal scale” o William Rothstein’s imaginary continuo (1991, 296).

34. IVLS is once again (ℤ, +). Note, however, that the integers now represent acoustically larger inter-

vals than did the same group elements in the GIS o Figure 1.2(b). For example, in the GIS o

1.2(b) int(G2, B2) = +2, while in the GIS o 1.2(c), int(G2, B2) = +1. Hook 2007a offers relevant

comments on relating two GISes that have the same abstract group o intervals (like (ℤ, +) here),

though the group elements in the two different GISes may represent intervals o different acousticsize. See also ymoczko 2008 and 2009.


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CHAPER 1 Intervals, ransormations, and onal Analysis 23Tis arpeggio space is highly relevant to the historical and stylistic context o theprelude, which imitates the French lutenists’ style brisé . Te intervals available tothe style brisé lutenist within any given harmony are exactly those o the present

GIS.Figure 1.2(d) invokes a different harmonic space, one o just intervals in which

i is the ratio 5:2. Tis model is suggestive, given the spacing o Bach’s openingarpeggio: the G2–D3–B3 succession corresponds to partials 2, 3, and 5 in the over-tone series o G1. (In a more historical-theoretic vein, we might say that the notesproject elements 2, 3, and 5 o a Zarlinian senario .) Te group IVLS here differs inalgebraic structure rom those in Figures 1.2(b) and (c): it is the positive rationalnumbers under multiplication, not the integers under addition. Tis suggests that

the harmonic interval o 5:2 inhabits a considerably different “conceptual topog-raphy” than do our stepwise (and additive) intervals o +9 and +4 in 1.2(b) and(c). We can sense that difference in topography when we recognize that, in thearpeggio GIS o 1.2(c), the interval rom G2 to D3 is the same as the interval romD3 to B3: that is, both represent an interval o +2. In the just ratio GIS o 1.2(d),however, the intervals are different: int(G2, D3) = 3:2 while int(D3, B3) = 5:3. Tedifference registers the acoustic distinction between a just perect fh and a justmajor sixth. Te resonant, partial-rich open strings o G2 and D3 with which the

arpeggio begins strengthen the relevance o the just-ratio GIS here.35

Figure 1.2(e) shows the articulation o i into its two subintervals, again labeled j andk, as in 1.2(c). While both j and k were “skips” in 1.2(c), in 1.2(d) only k represents a“skip” in the overtone series above G1; j connects two adjacent elements in the series.36 Bach emphasizes the “gapped” interval k, repeating it twice, as k–1 , in the second hal othe bar. Tis calls attention to the “missing” G3, partial 4 in the overtone series (notethe question-marked dotted arrows on the example’s right side). As Figure 1.2() shows,this G3 does eventually arrive in m. 4, at the close o the movement’s opening harmonicprogression. Te G3 bears a considerable tonal accent as a pitch that completes severalprocesses set in motion in the work’s opening measures. Note that the interval romD3 to G3—labeled l in the example—is lled in by step. Tis stepwise motion is the

35. Te cellist can emphasize the partial series by placing a slight agogic accent on the opening G2,

a gesture that makes good musical sense anyway, given the work’s upcoming stream o constant

sixteenth notes. Te partials activated by the G2 are an octave higher than those in Figure 1.2(d),

but they nevertheless still suggest the pertinence o the just-ratio GIS in this resonant opening.

36. I have worded this careully: interval k in 1.2(d) is a skip in the overtone series above G1. It is not,

however, a “skip” in the GIS , in the same sense that j and k are “skips” in the GIS o 1.2(c)—urther

evidence o a shif in conceptual space. In 1.2(c) the space S o the GIS consists o the pitcheso the G-major arpeggio, which are spanned by “steps” (modeled by additive integers). In the

GIS underlying 1.2(d), the space S in act consists o an innitely dense set o pitches, which are

spanned by requency ratios (modeled by multiplicative rational numbers). Te group IVLS in

this GIS consists o all o the positive rational numbers—not just the low-integer ratios explored

in the gure (3:2, 5:3, and so on), but also higher integer ratios like 16:15 (a “major semitone” in

Pythagorean theory), and even enormous integer ratios such as 531,441:524,288 (the acoustically

tiny Pythagorean comma). In conormance with GIS Condition (B), the space S o the GIS thus

includes innitely many pitches in the gap between, say, G2 and D3. (For example, it includes the

pitch residing a Pythagorean comma above G2.) Tis makes clear that the GIS structuring Figure

1.2(d) is not a linearly plotted space o “steps” and “skips” as in 1.2(b) and (c)—it is a space o re-quency ratios, which has a considerably different shape.


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24 onality and ransormationrst concrete maniestation o the scalar GIS-space rom Figure 1.2(b), now explicitlycoordinating that scalar space with an interval rom the harmonic spaces o 1.2(c)–(e).In act, by the end o m. 3, G3 is the only note that has not been heard in the dia-

tonic G-major gamut rom D3 to C4—it has thus been “missing” in both the scalar andharmonic conceptual spaces; its arrival lls a notable gap.

One could invoke other GIS contexts or i as well. One could model i as span-ning the interval rom 1 ̂to 3 ̂in an abstract scale-degree space, or joining root andthird o the tonic harmony (the ideas are related, but not identical). Many otherintervallic contexts or i are possible as well, but not all o them are relevant to theopening bar o Bach’s prelude. For example, one could conceive i to extend up 16semitones in a chromatic pitch gamut. Tis is a somewhat strained understanding

within the context o m. 1, which as yet explicitly invokes no such chromatic divi-sion o pitch space. Such a space is invoked, however, at the work’s climax in mm.37–39, via the cello’s chromatic ascent to G4, the work’s apex. Here it is very easyto hear the interval spanned rom D3 in m. 37 to G4 in m. 39 in terms o steps in achromatic gamut, and to coordinate the steps in this chromatic GIS with those inother diatonic and harmonic GISes relevant to the music in these bars.

Such an analysis could continue, modeling other notable intervallic phe-nomena in the prelude and exploring their interactions. For present purposes, it is

important merely to note the style o the analysis, particularly its ocus on multipleintervallic interpretations o single musical gesture.

1.3 ransformations

1.3.1 Te “ransformational Attitude”

As already noted, the transormational model represents a shif in perspectiverom the GIS view o the passive, outside observer “measuring intervals” to thato an active participant in the musical process. As Lewin puts it in one o his mostrequently quoted passages,

instead o regarding the i-arrow on gure 0.1 [an arrow labeled i extending rom a

point s to a point t] as a measurement o extension between points s and t observedpassively “out there” in a Cartesian res extensa, one can regard the situation actively,

like a singer, player, or composer, thinking: “I am at s; what characteristic transor-mation do I perorm to arrive at t?” (GMI, xxxi)

Lewin elsewhere dubs this the “transormational attitude,” and it has become aamiliar part o the interpretive tradition o transormational theory. It is a subtleand somewhat elusive concept; I will offer my own gloss on the idea and its relevanceto certain acts o tonal hearing in section 3.2.1. For now, the reader may simply con-ceive o transormational arrows as goads to a rst-person experience o various

gestural “actions” in a musical passage, actions that move musical entities or cong-urations along, or that transorm them into other, related entities or congurations.


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CHAPER 1 Intervals, ransormations, and onal Analysis 25 While ormal statements in GIS theory take the orm o int(s, t) = i, ormal

statements in transormational theory are expressed using transormational graphsand networks . A transormational network is a conguration o nodes and arrows

whose nodes contain elements rom some set S o musical elements (analogous tothe set S o elements in a GIS) and whose arrows are labeled with various transor-mations on S. A transormational graph resembles a transormational network inall respects but one: its nodes are empty.

1.3.2 ransformations and Operations

A transormation on S is a unction rom S to S itsel: that is, a mapping thatsends each element in S to some element in S itsel. We have already encoun-tered unctions in the GIS discussion above, with the unction int. Te transor-mations and operations in a transormational graph or network are also unctions,but rather than mapping pairs o elements to intervals (as int does in a GIS), theyact directly on single musical entities, transorming them into each other. Beoreexploring how this works in practice, it will be valuable to distinguish between atransormation and an operation.

Let us dene S as the seven diatonic pitch classes in C major, that is, S = {C,D, E, F, G, A, B}. We now dene a transormation on that we will call “resolve toC,” abbreviated ResC. Tis transormation sends every element in S to the elementC. ResC is indeed a unction rom S to S itsel: it takes as input each element o S,and returns as output an element o S. We can represent it by a mapping table, likethat shown in Figure 1.3(a).37 Figure 1.3(b) shows the mapping table or anothertransormation on S, which we will call Step: it moves each element in S up onediatonic step.

Both o these transormations can be conceived as idealized musical actions.But it is only when we consider the entire mapping table that we get a ull senseo just what these actions are . o see this, consider the act that both transorma-tions have the same effect on the note B: they both map it to C. At this local level,the transormations appear to be indistinguishable. But i we perorm the sameactions elsewhere in the space, their differences emerge. For example, Step maps Dto E, but ResC maps D to C; and Step maps E to F, while ResC maps E to C; and soon. It is only in this broader context that we can see that Step raises pitches by onestep, while ResC resolves notes to C. Tese two actions have the same effect when

applied to B, but the specic kinetics they imply are different—ResC suggests agravitational centering on C, or an action that yields to such gravitation, whileStep suggests a more neutral, uniorm motion o single-step ascent anywhere inthe space.

Step also differs rom ResC in a more ormal way. Every element rom S appearson the right-hand side o the table or Step (Fig. 1.3(b)), while only the element C

37. Note that we could also use the unctional notation rom the discussion o GISes above as areplacement or any one o the arrows in this table: or example, we could write ResC(D) = C.


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26 onality and ransormation

appears on the right-hand side o the table or ResC (Fig. 1.3(a)). While both ResCand Step are transormations, Step is a special kind o transormation that we willcall (afer Lewin) an operation: an operation is a transormation that is one-to-one and onto .38 I a transormation is one-to-one and onto, every element in the setappears once and only once as the “target” or an arrow in the relevant mappingtable—in other words, on the right-hand side o Figure 1.3(b). Operations thushave inverses: one can “undo” any operation simply by reversing the arrows in its

mapping table. Tus, we can dene Step–1 , as shown in Figure 1.3(c); as the tableindicates, Step–1 moves each element in S one diatonic step down. We c

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