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Yale University Department of Music

Parsimony and ExtravaganceAuthor(s): Robert C. CookSource: Journal of Music Theory, Vol. 49, No. 1 (Spring, 2005), pp. 109-140Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/27639392 .

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PARSIMONY AND EXTRAVAGANCE

Robert C. Cook

Introduction

Among the terminological and conceptual contributions made to current music theory by the work gathered under the "neo-Riemannian"heading, perhaps no other has entered the common parlance so thoroughly as "parsimonious voice leading" and its variants. Richard Cohnfirst used the phrase to describe the ability of the consonant triad to generate more of its own kind through (1) the movement of a single pitchclass by step while (2) preserving two common tones. In either one orboth aspects, this ability has been a compositional desideratum for centuries, so much a habit that it is easy?and warranted?to imagine thatthe two aspects are reciprocal descriptions of the same behavior. Indeed,though Cohn presented each aspect as a separate property in his firstpublic presentation on the subject, he then gathered them together as the"P relation" (Cohn 1994). They have since remained essentially togetherunder a single rubric in neo-Riemannian literature.In this article, Iwant to take the opposite approach and see what consonant triad relations look like when we understand stepwise voice motionJournal ofMusic Theory, 49:1DOI 10.1215/00222909-2007-003 ? 2008 byYale University

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and common-tone retention to be strictly separate properties. I wish toask what relations between triads are possible when we insist on twocommon tones, while allowing the moving voice to go where it can and,conversely, what relations between triads are possible when we insist thateach voice move, but only by semitone. I shall call the first property

parsimony and the second property extravagance.In the process, Iwish to probe the interpretive and terminological habits of neo-Riemannian theory and the expression of these habits at themoment when a musical event or series of events encourages intuitions ofcoherence while at the same time this coherence resists ascription to theinfluence of a diatonic tonic. The neo-Riemannian response to this momenthas been to set aside traditional tonal, acoustically based interpretive models for triadic music?those whose antecedents are fundamental-bass,scale-step, and function theories?and to invoke in their place variouslygenerated algebraic transformational models of chord succession.1

Following the practices of transformational theory, these models areusually generalized to identify possible broader families of musical phenomena to which the transformations in the model might belong. Theimpression of unity in a generalized model for triad relations can, however, obscure the differences among the intuitions about chromatic musicthat one is attempting to convey in a transformational model.21 wish toshow that some formal particularization can illuminate such differencesand encourage thinking about the premises of neo-Riemannian theory.

Algebraic models of musical relations and diatonic, tonal, or acousticpremises are not mutually exclusive in the history of harmonic theory, ofcourse. Nora Engebretsen has shown how one may find implicit algebraic groups in the theories of Moritz Hauptmann and Arthur von Oettingen, both of whom sought tomodel diatonic tonal concepts, and thelatter of whom worked explicitly from an acoustic perspective (Engebretsen 2002, 101-17, 168-81). By contrast, Engebretsen shows that CarlFriedrich Weitzmann's work, though presenting a fully chromatic viewof tonal space, obscures the algebraic structures of his teacher Haupt

mann (135-42).3Hugo Riemann explicitly uses algebra to calculate distances betweentonics in the Verwandtschaftstabelle and thus to illustrate how we mightconceive of relations between keys (Riemann 1914-15, 19-24). Theimplicit algebraic features of Riemann's harmonic systems have been well

documented.4 However, Riemann's identification of tonal functions andtransformations of tonal functions with individual triads rather than withconnections between triads has been viewed as an impediment to fulfilling his theories' (algebraic) transformational potential (Lewin 1987, 177;

Hyer 1995, 116 and 128).When Riemann posits a tonic, he deforms theotherwise smooth lattice of the Tonnetz, creating a metaphorical centripetal center (Hyer 1995, 127-28).110



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This is another way of framing the analytical moment to which Ireferred above. When chromatic music seems to break free of this center's gravitational pull, the algebraic coherence of triad relations becomes

more apparent. Hence, it is through group algebra that neo-Riemanniantheorists seek to capture intuitions of coherence inmusic that is generallyagreed not to find accommodation under the roof of a tonal hierarchy ashistorically constructed, implicitly or explicitly algebraic as various historical models might be.Voice leading and tonality are deeply and historically intertwined, too.Indeed, Cohn finds antecedents of his P property in the work of seventeenth-century theorist Charles Masson, in the writings of A. B. Marxand Ottokar Hostinsky, and inArnold Schoenberg's "Law of the Shortest

Way" (Gesetz des n?chsten Weges), under which common-tone preservation and stepwise voice motion are features of tonal coherence and gooddiatonic part-writing (Cohn 1997, 62).5 Still, aside from the interpretiveutility of separating parsimony from extravagance, as I shall show, thereis some historical precedent for distinguishing between these voiceleading properties.

Schoenberg's Law of the Shortest Way rehearses the familiar heuristicthat one should move a voice from chord to chord only if necessary, andthen by step if possible (he says he is recalling the teaching of AntonBruckner). But he goes on to refer to common tones as forming a "harmonic bond" (harmonisches Band), marking common-tone connectionsas having tonal meaning beyond simply good syntax (Schoenberg 1966,41-42). Hauptmann classifies triad successions by the number of com

mon tones that change "harmonic meaning" from chord to chord (Hauptmann 1893, 45-46). Disjunct triads must be "mediated" by chords ofopposite mode with which the two disjunct triads share tones (46-48).6Stepwise voice leading is important toMarx, of course, given his melodicorientation, but common tones between diatonic triads?and the impliedcommon tones between the scales of which the triads themselves aretonics?are bearers of tonal coherence, both within a single key and in

modulations to other keys (Engebretsen 2002,68-79; Kopp 2002,45-51).Broadly speaking, we can consider the privileging of common-tone relations to be a special case of thinking about tones as belonging with or toother tones in general, which leads ultimately to the ways tones belongto the overtone series of a given fundamental.

In contrast to common-tone retention, stepwise voice leading has beenunderstood to provide coherence that draws little or not at all on diatonictonal intuitions. As Engebretsen and Kopp show, Marx explained bothdiatonic and chromatic successions lacking common tones as acceptablewhen smooth voice leading made the connection (Engebretsen 2002,7479; Kopp 2002, 48-51). Cohn shows how Weitzmann first finds eachaugmented triad in the conjunction of nebenverwandten Akkorde,1 and

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(a)

(b) mm. 90-93 94-97 98-99 100-101 102

Example 1.Franck, Piano Quintet inF minor, first movement,mm. 90-102112



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then reconfigures his presentation to show how each augmented triadproduces six consonant triads by "single semitonal displacement" and sixother consonant triads by "double semitonal displacement," thus implying "that the augmented triads bear conceptual priority, with theKl?nge[consonant triads] receding into secondary status" (Cohn 2000, 92-94).

In sum, there is good reason to distinguish between, on the one hand,the various algebraic structures?implicit or explicit?of diatonic harmonictheories and, on the other hand, neo-Riemannian uses of group algebra toformalize ways of interpreting intuitions about chromatic music. Like

wise, there is good reason to distinguish between common-tone retention,which as an aspect of voice-leading practice supports intuitions of diatonic tonal relatedness, and stepwise voice motion, which as an aspect ofthe same practice supports intuitions of fluency and coherence thatmaybe separate from diatonic tonal relatedness.

I. One Extravagant Sequence and One Parsimonious SequenceTwo passages from the first movement of the Piano Quintet inF minor

by C?sar Franck, both of which move sequentially by minor third but todifferent effects, suggest different transformational models. Example lais the score of the later passage (mm. 90-102), which I treat first becauseit serves as the lens through which I see the distinctions between parsimony and extravagance and interpret their effects. It was not until Ithought of this later passage as "extravagant" that I began to hear theearlier passage as a "parsimonious" complement.

This passage is a sequence in which each iteration consists of a rootposition major triad elaborated in neighboring fashion by its own bVlKCohn's "hexatonic pole" (1996, 19).8 Each successive iteration sounds aminor third higher than the preceding, but this transposition is not articulated directly, moving from themajor triad of one iteration to the next.Instead, seventh chords mediate the major triads?a half-diminished seventh followed by a dominant seventh inmm. 93 and 97, and a dominantseventh inmm. 99 and 101, where the sequential unit is abbreviated fromfour bars to two.

Now, any two consonant triads of the same mode with roots a minorthird apart share a common tone. One might reasonably expect Franck atleast tomake use of this link if not emphasize it. Instead, themediatingseventh chords obscure the common tone. One notices instead the extravagance of the voice leading. Nearly every voice moves, and nearly every

motion is by semitone, even into m. 102, where the sequence breaks. (Ifind that the change of register and texture inm. 98 does not disturb mysense of the voice leading.)Example lb sketches what I hear in four voices. Imark those motions

that are not by semitone with asterisks. In each case, the motion is by

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(a)

J^WTf

hy^i s i?ip,jp",.t-j^

pil^m m^^nh> Trn-% ^#

?*^ Jf,?^^ Jf?pVln I;Vln II8-*

J"?j..bfl J"?jta tffi ta wV

MVln I;others *and15"*

^i^> i?!g^gpp ?H=

W^g^^M Hi>5 "h-?h^;

Example 2. Franck,Piano Quintet in F minor, first movement,

mm. 26-37

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whole tone in the tenor and reaches between themajor triad of a four-barsequential unit and the first of themediating seventh chords. (Abbreviation of the sequential unit inmm. 98-101 leaves only motion by semitonein the upper voices and by minor third in the bass.) The motion creates asubtle la-ti-do passing through the minor third from the root of onemajor triad to another. Imark this minor third with brackets in the tenor,and also in the bass, where it comes through in the lowest register.Just before this passage, there is a clear modulation toDt major (mm.80-88), encouraging one to interpret the first iteration of the sequence inrelation to the global F-minor tonic. This interpretation soon fades in relevance, however, leaving only the effects of voice leading and the transpositions between each iteration.

The earlier passage, Example 2a, contains the only use of hexatonicpoles in the piece before the extravagant sequence. The poles, A minorand Dt major, do not sound until mm. 34-37, however; before that, thevoice leading of Example 2a is as parsimonious as the voice leading ofthe other is extravagant. I sketch out my interpretation in Example 2b.The major triad anchoring each of the first two sequential units has twocommon tones with its elaborating minor triad.9 Furthermore, the singlecommon-tone effect of the minor-third transposition is obscured by thepair of common tones between the elaborative minor triad of the firstiteration and its parallel major triad in the second iteration. In this context, I find the sudden intrusion of voice-leading extravagance betweenDt?major and A minor tremendously jarring.This passage is shorter than the other, and I find a place for itmoreeasily in an F-minor tonal context. Inmm. 19-25, just before Example2a opens, El?major has been V of Al? minor (which in turn is III^ of F

minor) and is tonicized inmm. 25-26. Past the point at which Example2a ends, D\? major becomes VI of F minor. Despite the tonal clarity ofExample 2a in relation to Example la, there are several similaritiesbetween the passages, and it is these that draw me to juxtapose the twosequences analytically:

(1)Until I hear the complete, eight-bar second theme later in the piece(mm. 124-31), these sequences are the only two places I hear thecharacteristic sound of hexatonic poles.

(2) In both cases, the interval of sequential motion from one iterationto the next is by minor third. The parsimonious passage descendsby minor third; the extravagant passage ascends.

(3) The structural major triad in the sequential unit of each passagesupports its fifth in the top voice. This support is prolonged ineach passage by an upper neighbor tone harmonized by the minortriad.

(4) The elaborating minor triad in the sequential unit of each passage

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is a submediant to the major triad. In the parsimonious passage,this is a locally diatonic VI; in the extravagant passage, as I notedabove, the submediant harmony is bVlk

(5)The sequences have similar formal roles. The parsimonious passage precedes what eventually shows itself to be a long elaborationof VI-V7-I in F minor (mm. 38-51), leading to the first theme ofthe movement. The extravagant passage precedes an approach toan elaborated dominant pedal inAb major (mm. 115-21), the keyof the second theme.

In sum, many of the same things happen in each sequence, but to different cumulative effects. The first sequence (mm. 26-37; Example 2a)stresses a diatonic relation?relative major and minor?and a chromaticrelationship easily integrated into diatonic interpretation?parallel majorand minor. The second sequence (mm. 90-102; Example la) stresseschromatic relationships; the hexatonic poles are difficult to interpret tonally, as are the seventh-chord mediations between sequential iterations. Iattribute the different effects of the two passages to voice leading?to theparsimonious voice leading of the earlier sequence in contrast to theextravagant voice leading of the later sequence.

II. Formal Aspects of the Parsimonious and Extravagant GroupsI define parsimonious relations to be those that retain two commontones when moving between triads. The motion of the third voice is unre

stricted; it may move by as many steps or leaps as necessary to reachanother consonant triad. As it happens, there are only three such relationspossible, and the three are familiar. Example 3 shows the possibilitiesgiven Db major as a reference. Example 3a preserves the members of the

major thirdDb-F while allowing the third voice tomove through the restof the chromatic collection seeking a note that will make a consonanttriad. Example 3b preserves themembers of the minor third F-Ab as thethird voice moves, and Example

3c preserves the members of the perfectfifth Db-Ab as the third voice moves. The three relations shown in Example 3 are identical to the neo-Riemannian R, L, and P, respectively.I define extravagant relations to be those in which each of the threetriadic voices moves by semitone and only by semitone; no commontones are permitted.10 There are also three such relations possible. Example 4 shows them, again using Db major as a reference. In Example 4a,the top voice, Ab, ascends toA4, while the lower two voices descend fromthemajor thirdDb-F to themajor third C-E. Db major and A minor are,of course, hexatonic poles; hence I label the relation H (Cook 1994). Theremaining extravagant relations are Tj and Tn as shown in Examples 4band 4c, respectively.

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(a)

(b)

?h \. \>- \* u * * b' ^(c)

y? u *i . b? u \**-9?

Example 3. Parsimonious relations between consonant triads

(a) (b) (c)

3gW 1 H"U tt^f ??Example 4. Extravagant relations between consonant triads

The different notations of the parsimonious and extravagant relationsinExamples 3 and 4 reflect my intuitions about the two types of relationand their effects in the Franck passages. On one hand, the notation inExample 3 expresses the sense of near-identity between parsimoniouslyrelated triads, a sense that aids the loosely tonal interpretation of the parsimonious sequence inExample 2. The common tones are literally held incommon by the pianist's right hand. Example 3 shows them the same wayand tests them against all possible third tones, searching for another consonant triad. Example 3 implies a claim like, "This new triad is as muchthe same as the first triad as possible."11 Example 3 illustrates a lack of

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(a)etc. etc.

... Elr ^-G\,+ ^-Bb- - ? ... D+ -+-Gk -+-A+ -- A],- C- V

B+ nxD+B

B+ ,' EHEk

M" G+E+ E

D, C+

Gt+ ~

B\,- B\,+

D^+ DA+ F+

Gk DD+

BH E^+

Ab+C- A

C+

F+

B- G+ E

(b) etc.D>+ ?^D+? E;^+ ^E+?*-?+?*- G \,+^0+? A^ ? A^-? Bk+ ?*-B+ ?*-C+

-H?! 1 ? ? I M ? I MBt- B- C? D\,- D- Eb G? AbFigure 1. Product networks of the (a) parsimonious and (b) extravagant

groups acting on the consonant triads

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concern for how exactly the third voice moves by showing all of the possible notes itmay try against the two sustained notes.The notation inExample 4, on the other hand, transmits an impressionof pervasive motion constrained by set-class. I do not include all possiblesonorities created by semitonal movement of all voices because, unlikeinExample 3, here I am not interested in judging the interaction of someor part of the Dt?major triad with other tones. I am concerned only withthemotion between two triads. Example 4 does not implicitly claim somesort of hierarchical, tonal relation between two triads as in Example 3.Instead, it states, "When Imove like this, the type of sonority [set-class]is consistent." Examples 3 and 4 may not reflect the most concise representations of voice leading between triads, but they do a good job ofcapturing salient aspects of the two sequences.

Distinct formal models of parsimonious and extravagant transformations would seem ideal support for the intuitions depicted inExamples 3and 4 and for the interpretations offered above. Indeed, the two familiesof transformations do belong to different algebraic groups. One may reasonably ask, however, whether the formal differences themselves haveany musical relevance. To show that they do, Iwant to set the distinctiondrawn between parsimonious and extravagant transformations in relief bydetaching them from their heuristic mnemonic labels and instead namingthem consistently according to the ways they transform triads.The parsimonious and extravagant transformations are of the sortJulian Hook calls uniform triadic transformations, orUTTs (Hook 2002).A UTT is named by an ordered triple (?,t+, r), where + indicates modepreservation, - indicates mode reversal, t+ indicates the root-intervaltransposition when acting on a major triad, and t~ indicates the rootinterval transposition when acting on aminor triad (63-64). Where Hook'snomenclature sacrifices the intuitively salient connotations of diverse mne

monic labels, it offers an orderly, analytically transparent means of representing transformations and of grasping their algebraic properties.The parsimonious transformations, as UTTs, are (-, 0, 0) (or P),(-, 9, 3) (orR), and (-, 4, 8) (or L). For each transformation, t++ r = 0;that is, each of the parsimonious transformations transposes major triadsand minor triads by complementary root-intervals mod 12 (Hook 2002,74-75). Any combination of parsimonious UTTs will do likewise.12 Figure la illustrates this non-commutative behavior.13 The figure depicts twocycles alternating parsimonious UTTs (-, 4, 8) with (-, 9, 3) (or L alternating with R). The inner cycle matches D\?minor with its (-, 0, 0) (or P)sibling Dl?major. In both cycles, clockwise is "up" and counterclockwiseis "down." Synchronized motion through the cycles?starting from thesame pair of triads?in the same direction uses opposite alternations oftransformations. Synchronized motion through the cycles in oppositedirections uses the same alternation of transformations.

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The extravagant transformations, as UTTs, are (-, 8, 4), or H;(+, 1, 1), or T?; and (+, 11, 11), orTn. Figure lb illustrates the commutative behavior of extravagant UTTs. No matter the order of UTTs withwhich one moves through the network, (-, 8, 4) followed by (+, 1, 1) or(+,1,1) followed by


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(a)S

^m3

^it

M3

^f I:

m3

f f?fM3

(b)

kifc ?

Example 5. Parsimonious and extravagant UTTs beginning on D\?majorand minor triads

braic group are simple elements, differing only in theways they act on aset of objects (in this case, consonant triads). But because the concernhere iswith the path from my interpretation of themusic to the construction of appropriately evocative transformation groups, I imagine PLP andLPL to be some distance from the center of comprehensibility in theparsimonious realm. Conversely, I imagine H to be the very center ofcomprehensibility in the extravagant realm.17 It is not true, however, thatparsimonious transformations are members of the extravagant group. Forexample, the parsimonious UTT taking Dl? major to B\? minor and Bl?minor toDl?major is (-, 9, 3), or R, but the extravagant UTT in the formercase is (-, 8, 4)(+, 1, 1) =


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Parallel or divergent interpretations are not usually the goal of transformational analyses.19 Transformational studies move, broadly speaking, through three stages of engagement with a composition:

(1) As the analyst listens, certain events?motives, gestures?emergeas articulative of the experience. She names some transformationT (or a few such transformations) that depicts the motion of a particularly salient event or events.

(2) The analyst constructs a formal model of the family of transformations towhich riogically belongs. "If Tis a transformation in thismodel, then T' must also be a transformation in this model," theanalyst might say.(3)The analyst generalizes themodel to the greatest pertinent extentpossible. By "generalize" in the mathematical sense, Imean pursuing algebraic aspects of the model in order to identify possiblebroader families of musical phenomena towhich the transformations in the model might belong. By "generalize" in a morecolloquial sense, I mean seeking to understand the model as a

metaphorical space, or what David Lewin would call a "paradigmthat [is] only sometimes fulfilled in any given piece" (1982-83,335-36). Situating the piece and its transformational paths inthat space enriches and critiques both the interpretation of thepiece and the suitability of the initial interpretive choices. Thiscritical moment creates a cyclic path back to the first stage ofengagement.20

Figure 2a illustrates the process Ihave described, especially its circularpath and the gathering of intuitional strands, as they emerge out of the

musical experience, into a small family of canonical transformations. Thisgathering is a selective activity because, in doing it, the analyst necessarilyexcludes some portions of musical experience. The closure of an algebraicgroup, a construct central to transformational practice, explicitly and for

mally defines both this exclusion and the characteristics of the space or"paradigm" through which the music moves.

By "exclusion" Imean "exclusion from the model," certainly not exclusion from engagement with themusic. Under a given model, the excludedobjects and relations may be of asmuch interest as those included. Indeed,much music, especially chromatic music of the late nineteenth century,sounds rich inmultiple, oddly complementary or mutually contradictoryrelations that can motivate multiple complementary or contradictory models. As Lewin puts it, "[W]e should generally want our analysis to conveythe characteristic multiplicity of the perceptions involved and the characteristic incompatibility of their assertion in-the-same-place at-the-sametime. The rhythm of the dialectic thus engaged will be a significant aspectof our rhythmic response to the music" (1986, 371).122



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(a) Canonicaltransformations

Generalizedspace Formalization

(b) Canonicaltransformations I

Intuitions

Canonicaltransformations II

Formalization I Generalizations I & II,separate or interacting

Formalization II

Figure 2. Illustrations of transformational practicesIn that spirit, the approach in this article follows the paths illustrated by

Figure 2b, dispersing rather than gathering the intuitional strands into apair of canonical families, each of which collects different interpretationsof salient relations, of "characteristic gestures" (Lewin 1987, 159).Whatfollows will accent the critical moment of the third stage of transformational engagement, where two interpretive paths meet, and use it to pickapart the common thread of triad relations in neo-Riemannian thinking.

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III. Transformational InterpretationsLet us now take a second look at the sequences, but from a transformational perspective. Example 6 reproduces the sketches from Examples lband 2b. I have added arrows and labels for the transformations. Example6a is a sketch of the earlier, parsimonious sequence, and Example 6b is asketch of the later, extravagant sequence. Each is labeled with transforma

tions from the appropriate group. I mean for these figures to serve asguides to the transformational networks atwhich we shall look below.

Figures 3a and 3b give network interpretations of each passage. Figure3a conveys a sense of the parsimonious connection between the two iterations of the sequence through the vertical arrangement of the parallels Cminor and C major (as if they have some phenomenological identity). Oneeffect of this interpretive arrangement is to suggest that the relationshipbetween Ebmajor and C major is not simply a transpositional one, but onethat also entails the tonal relationship between Eb major and its submediant C minor, which in turn is parsimoniously inflected by the parallelmove toC major. The "long way" around the right triangle formed by the

(b) mm. 90-93

Example 6. Interpretations of Franck, Piano Quintet in F minor,mm. 26-37 and 90-102, labeled with (a) parsimonious and(b) extravagant transformations

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(a)D^+

LPL/ PLP

C

H H H HA- C- El,- Gk

Figure 3. Networks interpreting Examples 6a and 6b

triad vertices conjures an image of the edge not taken, an image seen interms of the two edges that are taken. Figure 3a also emphasizes the starkcontrast of the disrupted third iteration by depicting the relation betweenA minor and Db major as a sudden, far-reaching PLP gesture.In contrast, Figure 3b transmits the impression that the toggling H pairsof each iteration in the extravagant sequence carry none of the relationalinterdependence of the transformations in the parsimonious sequence.There are no diagonal paths to illustrate the sort of relationship heard inthe parsimonious sequence.

Now, if one imagines the latent transpositions by minor third inFigure3a, then one might just as easily imagine a shearing of the network to cre125



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(a) T9 (b) T9

A CT3 T3 GkT3

Figure 4. Reshaped networks interpreting Examples 6a and 6b

ate Figure 4a. (For clarity, Ihave temporarily disconnected the PLP arrow;itwill reappear below with a new name.) This rendition of my interpretation of the parsimonious sequence is less representative of my listeningexperience. The parallel relationship between C minor and C major, whichis important for Figure 3a, is questioned because it renders the network

malformed.21 However, the arrangement of Figure 4a facilitates comparison with the extravagant sequence.Figures 4b and 4c render each sequence as a product network combin

ing two or more vertical transformations?parsimonious Rs in the firstcase, extravagant Hs in the second?with horizontal T3s or T9s. If onewere to add an H arrow (equivalent to PLP and LPL) descending fromthe A-minor triad inFigure 4a to aDb-major triad, then the two networkscould belong to a single larger product network. I have done exactly thisto produce Figure 5 a.

Figure 5a is a larger metaphorical space inwhich I imagine the twosequences tracing parallel paths that weave by T3 or T9 among the conso

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nant triads. I have isolated each relevant portion of the larger network inFigures 5b and 5c. I described above how we may hear the two sequencesas similar. But in order to transmit this sense of similarity through transformational networks, I have had to deform and abbreviate the networkfor the parsimonious sequence (compare Figure 3a with Figure 5b). Thisreshaping draws attention away from my tonal impressions of thepassage?the way I hear C minor and A minor in terms of their Eb-majorand C-major relatives (the R transformations), the way I then hear Cmajor as inflected by C minor (the P transformation), and the way I hearthe overall motion of the passage articulating a progression from Abminor to F minor (fflb-I) through the agencies of Ab minor's dominantand F minor's submediant. Accommodating similarities between the twopassages requires removing the P arrow and trading some of the parsimonious sequence's individual character for a way to show its interactionwith the extravagant sequence.

H

E+

Fjt+

(c)

H

Gk E|r

E+ Cfrf B^+ G+

Figure 5. Product networks including portions interpretingExamples 6a and 6b

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IV. Utter ExtravaganceThe extravagant interpretation has come to overshadow the parsimo

nious one, though not because the extravagant interpretation is better.The generality of the extravagant model encourages accommodation of

less extravagant uses rather than the reverse. The extravagant model isgeneralized in the sense that it makes no reference to tonal intuitions.Indeed, itmodels the behavior of consonant triads as collections of pitchclasses in the twelve-note equal-tempered system, rather than as sonorities having potential tonal references. I can demonstrate the extent towhich this epistemological shift takes place by reformulating and generalizing the concept of voice-leading extravagance to account not just forconsonant triads but for the seventh chords Franck uses in the extravagantsequence and, indeed, for voice leading between any sonority that is aminimal perturbation of an equal division of the octave.

First, another look at Example la is in order. As noted in part I, themajor triads that anchor each iteration of the extravagant sequence aremediated by seventh chords: in m. 93, bts. 3-4, and m. 97, bts. 3-4, by ahalf-diminished seventh chord and a dominant seventh chord (spelled asan augmented sixth chord), and inm. 99, bt. 4, and m. 101, bt. 4, bydominant seventh chords alone. Cohn notes that these members of setclass [0, 2, 5, 8] are "minimal perturbations of a symmetrical division ofthe octave," namely, the fully diminished seventh chord (1996, 39 n. 40).The consonant triads (set-class [0, 3, 7]) are also minimal perturbationsof a symmetrical division of the octave, namely, the augmented triad.These mediating seventh chords are related entirely by semitonal voice

motion, just as the hexatonic-polar triads in the passage are. We shouldbe able to generalize the extravagant group to account for these seventhchords as well as the consonant triads.

Imagine that there is some passage of music inwhich the composerhas perturbed an augmented triad such that first one note moves down bya semitone and returns to its original position, then the same note movesup by a semitone and returns to its original position. Next, the secondnote in the triad follows the same routine. Finally, the third note in thetriad does the same. We can name a function m to model the motions inthis passage. The result of m is a set of six consonant triads. Table 1shows the four possibilities.22The definition of m has to do with augmented triads only because weimagined it acting on them. In principle, what m does has to do only withequal divisions of the octave, not with any properties, cardinality or otherwise, of equal divisions themselves. We could apply m to fully diminished seventh chords; Table 2 shows the three sets that result from doingso.23 In principle, we could apply

m to whole-tone collections, yieldingMystic chords; or to tritones, yielding perfect fourths and fifths.24128



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Table 1. Set m for each augmented triadFunction Resultm( {C, E, Ab} ) {E+, Db-, Ab+, F-, C+, A-}m({Db, F, A}) {F+, D-, A+, Ft-, Db+, Bb-}m({D, Gb, Bb}) {Gb+,Eb-, Bb+, G-, D+, Bb-}m({Eb, G, B}) {G+, E-, B+, Ab-, Eb+, C-}

Table 2. Set m for each fully diminished seventh chordFunction Resultm({C, Eb, Gb, A}) {B7, Eb07, D7, Ft07, F7, A07, Ab7, C07}m({Db, E, G, Bb}) {C7, E07, Eb7, G07, Gb7, Bb07, A7, Db07}m({D, F, Ab, B}) {Db7, F07, E7, Ab07, G7, B07, Bb7, D07}

As we apply m to each equal division, each minimal perturbationsends one pitch class of the given equal division into a semitonally adjacent equal division. For example (see Table 1), to reach Db minor from{C, E, Ab}, we move C toDb, which is in {Db, F, A}. In fact, each of theminor triads in the set obtained from m({C, E, Ab}) involves a motionfrom one pitch class in {C, E, Ab} to one pitch class in {Db, F,A}. Likewise, the major triads in the same set involve motions from one pitchclass in {C, E, Ab} to one pitch class in {Eb,G, B}. In each case, the othertwo pitch classes of {C, E, Ab} are held as common tones.We can refinem, therefore, to account for the equal division of origin and the equaldivision into which a voice moves through minimal perturbation.Let X be an equal division of the octave of cardinality n with the pitchclasses xm in an arbitrary but fixed order (x0,.xn-\)- Next, let F bean equal division of the octave of cardinality n ordered such that eachpitch class ym is a semitone higher or lower than each pitch class xm.Thendefine M(X, Y) to be a function thatminimally perturbs the equal divisionX into the equal division Y by moving xm to ym, records the result, and

returns ym to xm. The function then does the same with xm+x, and so forth,0 ^ m ^ n - 1. The cardinality of the resulting set of sonorities is n.Tables 3 and 4 list the resulting sets for augmented triads and diminishedseventh chords, respectively. In each case, I have taken Tto be one semitone higher than X, but one could do the reverse. In any case, the contentsof the sets of three consonant triads or four [0, 2, 5, 8] tetrachords wouldbe identical to those in the table.

Now, the behavior of triads under the extravagant transformation Hbears an interesting resemblance to the behavior of sonorities under thefunction M. Example 7 shows the triads of the first phrase of the extravagant sequence, Cttmajor and A minor, in the middle bar of the example.129



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Table 3. Sets M(X, Y) for the augmented triadsX YResult

{C,E, Ab} {Eb,G,B) {E+,Ab+,C+}{C,E, Ab} {Db,F,A} {Db-,F-,A-}{Db,F,A} {C,E,Ab} {F+,A+,Db+}{Db,F,A} {D,Gb,Bb} {D-,Ft-,Bb-}{D,Gb,Bb} {Db,F,A} {Gb+,Bb+,D+}{D,Gb,Bb} {Eb,G,B} {Eb-,G-,B-}{Eb,G,B} {D,Gb,Bb} {G+,B+,Eb+}{Eb,G,B} {CE,Ab} {E-,Ab-,C-}

Table 4. SetsM{X, Y) for the fully diminished seventh chordsX YResult

{C,Eb, Gb,A} {D, F,Ab, B} {B7,D7, F7,Ab7}{C,Eb, Gb, A} {Db,E, G, Bb} {Eb07, t07,A07, C07}{Db,E, G, Bb} {C,Eb, Gb, A} {C7,Eb7,Gb7,A7}{Db,E, G, Bb} {D,F,Ab, B} {E07,G07,Bb07,Db07}{D, F,Ab, B} {Db,E, G, Bb} {Db7,E7, G7, Bb7}{D, F,Ab, B} {C,Eb, Gb, A} {F07, b07,B07,D07}

The outer bars show the two augmented triads involved. The open noteheads aremembers of {C, E, Gtt}, and the closed note heads aremembersof {Ct, F,A}. Cttmajor is an element of the setM({Ct, F,A}, {C, E, Gt}),while A minor is an element ofM({C, E, G?}, {C#, F, A}). The musicalnotation shows that H moves each pitch class from one augmented triadto the other as the voices shift from one consonant triad to another. Thus,H flips the arguments ofM, toggling adjacent pitch classes between thetwo augmented triads.

The only other transformations under which each voice of a consonanttriadmoves by exactly one semitone are the transpositions T} and Tn, theother extravagant transformations. One can see how Tj relates to M bycomparing E major in the first result set of Table 3 with F major in thethird set. The pitch classes of the arguments toM in the third row are eachone semitone higher than the pitch classes of the arguments toM in thefirst row.

Though we defined H as a characteristic transformation between triads, this brief exploration of M suggests that we can generalize H toapply in all cases of minimally perturbed equal divisions of the octave.Take X and Y to be semitonally adjacent equal divisions of the octave ofthe same cardinality in an arbitrary but fixed ordering as above. Let S bea sonority inM(X, Y), and let T be a sonority inM(Y, X) such that eachpitch class sm in S is related by semitone to each pitch class tm n T.130



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Note that a given pitch class s or tmay be amember of X or itmay beamember of Y because S and T are minimal perturbations of the equaldivisions of the octave X and Y.Then define H' as the operation thatmoves from S to T by taking each

sm to tm. Put more casually, H' acts on any chord that is a member of a setM(X, Y), flipping each pitch class in the chord that is an element of X intoT and each pitch class that is an element of y intoX, effectively producingamember of the setM(Y,X). This is exactly what happens between thehexatonic poles Cttmajor and A minor inExample 7.With H\ the reformulated model of extravagant voice leading drops

already tenuous connections to a strictly triadic model and thus to any apriori privilege the consonant triad and its traditional roles have in organizing musical motion. From the standpoint of atonal theory, H' is simplya complementation operation on subsets of any equal division of theoctave. H' generates the complete hexatonic collection [0, 1, 4, 5, 8, 9]from one of its [0, 3, 7] subsets, a complete octatonic collection [0, 1, 3,4, 6, 7, 9, 10] from one of its [0, 2, 5, 8] subsets, and so on.

We need not define any transformations T' because in common usage,transposition operates in the same way on any pitch-class set. Clearly,though, we can describe the transpositions in terms ofM. Let W, X, Y, andZ be equal divisions of cardinality n, let W and X be ordered such thateach pitch class xm in X is one semitone higher than each pitch class wminW, and let Y and Z be ordered such that each pitch class zm inZ is onesemitone higher than each pitch class ym in Y, 0 ^ m ^ n - 1 in all cases.Finally, let t equal the interval of transposition from W to Y and X to Z.Then T, of any sonority that is an element ofM(W, X) is a sonority that isan element ofM(Y,Z).From this revised view of voice-leading extravagance, it is possible tointerpret triads and seventh chords in Franck's extravagant sequence usingthe same group of transformations. The left-hand network in Figure 6depicts the triadic portion of the phrases; the right-hand network depictsthemediating seventh chords. (The reader may wish tomove slowly here,comparing Figure 6 with Example la when necessary.) Brackets aroundA07 and C07 indicate portions of the network that remain unused when thesequential phrase contracts to two bars inmm. 98-101. The final A7 of the

l ," i ,* '


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y

T3y

F*7T3

yA?

triads and seventh chordsFigure 6. Networks interpreting Example 6b using H' and T3

seventh-chord network appears inmm. 102^4- (not shown inExample la),though the sequential pattern of mm. 90-97 leads one to expect it earlier.Ifwe imagine triads and seventh chords inhabiting amusical space dividedinto regions by equal divisions of the octave, then the triads and seventhchords of the sequence fulfill the same product network.I am not offering M as amodel for use in the practical study of music.

Among the issues I have not considered are the precise definition of ageneralized equal division of the octave and what, exactly, a completespace defined byM would look like. Such a space would certainly be sobroad as to be unwieldy. Virtually no relation would be excluded, makingcritical distinctions next to impossible. I find it absurd to think meaningfully about themusic inExample 2a as somehow moving through a spaceinwhich set-classes [0, 3,7], [0,4, 8], [0, 2,5, 8], [0, 3,6,9], and all otherminimal perturbations and their associated equal divisions of the octavereside equally. Rather, M is an exercise in generalization that attemptssome excavation of the thought behind the notion of extravagant voiceleading. It turns the discussion here toward that moment when we consider just how the voice leading in a succession of sonorities, a passage,or awork is primarily tonal in effect or primarily algebraic in effect.

132

Q+ -?? A- Df7H'T3 T3

H'

E+ H' Ff7Ta T3 x3

H'

G+ -?- EkH'

B^+ -?- GkH'

A07 H'

C07 H'



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V. A Critical MomentThe phenomenological order in which the networks of Figures 3-5

arrange the interpretations of part I corresponds nicely?and suggestively?with the process of judgments, questionings, and reinterpretings throughwhich I go as I listen to Franck's quintet. I cross from tonal, even diatonicinterpretation of the parsimonious sequence?and minor-third transposition with it?to an algebraic voice-leading interpretation of the extravagant sequence, and then find Imight retrospectively hear the parsimonious sequence in the same terms as the extravagant one but, as part IIIshows, with some reservations about the extent of the extravagance.

I suggested above that, in one way or another, such a crossing andmoment of reinterpretation is crucial in neo-Riemannian analysis ofchromatic music. We may observe traces of the moment at many placesin the literature, of which Iwill mention three, two by Hyer and one byCohn.

Near the beginning of his dissertation onWagner's Tristan und Isolde,Brian Hyer proposes a reading of functional harmony as a semiotic system. D[ominant] and S[ubdominant] are read as signifiers of T[onic], thesignified. He writes, "The [three triads] . . .form a system in which thereare but two signs, the dominant and the subdominant... .The system hasthree elements, the tonic, dominant, and subdominant triads, none of

which, however, are signs in themselves." The tonic is not a sign. Thisseems difficult to accept until one recalls that it takesmore than the simplestatement of a harmony to assert it as tonic. More specifically, "[i]t is therelation between the dominant seventh and the tonic that actually hassignificance, however, not the dominant seventh itself" (Hyer 1989,26-28).

Hyer writes only of the traditional Riemannian tonal functions, butalready from a transformational perspective that drives his comprehensivedevelopment of a group-theoretical model for harmonic relations later inthe project. Hyer's semiotic approach allows him to disconnect the triadsthemselves from their tonal functions or the production of tonalmeaning.Function andmeaning are carried instead by the relations between triads.The result is an interpretive stance that is both extremely flexible andhighly sensitive. For example, one does not say, "This chord is a dominantbecause its root [implied, inferred, or present] is 5 and it contains both 7and 4." Rather, one says, "The relation between this chord and another[implied, inferred, or present] signifies 'dominantness' in some way or

complex of ways."It is a rather small conceptual step to extend this perspective to relations between triads other than fifth relations, and even to do away withtonic altogether. Hyer's algebraization of triad relations provides a foun

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dation from which to assert that "tonal coherence does not require a pieceto elaborate a single prolonged tonic, but rather that we regard relationsbetween harmonies as being tonal" (Hyer 1995, 130)."Tonal" here does not mean "diatonic," but it does mean that there issome heuristic quality of repose. In the Schlafakkorden fromWagner'sDie Walk?re, this quality adheres toE major:It is the transformational process that relates (E, +) with (Ab, +) and(C, +), then, that prolongs (E, +)'s tonal significance as tonic in these

measures. The realization that LP applied three times in succession to an(E, +) tonic produces an equivalent (E, +) tonic thus engages the notionof "closure," both musical and algebraic: (E, +)(LP)3 = (E, +). (115)

More generally, I interpret this quality as "I know where I am," a sensethat one frequently does not have in chromatic music if one is looking fora diatonic tonal center.

Interestingly, Hyer asserts such coherence in the Schlafakkorden whileexplicitly attributing that quality to aspects of triad relations other thanvoice leading: "[T]he tonal structure of the Schlafakkorden lies not in thethree recurring triads, nor in the melodic processes connecting them, butin the transformational relations that bind them together, relations thatgain an intrinsic intelligibility from the algebraic structure of the group"(115; emphasis added).In Cohn's opening gambit to "Maximally Smooth Cycles," traditionalmeans of interpreting triad relations founder on the shoals of amajor-thirdprogression (1996, 9-11).25 Cohn then notes that writings by Hyer andLewin, "[i]n assuming a priori status of consonant triads, leave unaddressed the reasons that late nineteenth-century composers continued tofavour triads as harmonic objects" (12). Cohn then outlines the objectivesof his essay, which include treating the elements of triads?the notes andthe intervals between the notes?as elements of pitch-class sets, and thusonly incidentally as unitary tonal objects. He writes, "[T]his article adoptsthe convention of invoking acoustic roots for their mnemonic value," but

use of these labels should not be interpreted as implying generative statuson the part of the named pitch-class. On the contrary, the component pcs[pitch classes] should in all cases be considered as equally weighted. Per

haps more importantly, the analysis presented here suggests that thevocabulary chosen to describe triad relations situates the analyst to oneside of this crossing. (13)There is deep consonance between Hyer's and Cohn's projects in the

main, despite their different locutions. They do represent, however, twodifferent threads of thinking that have met in the last twenty years in thestudy of chromatic music from the late nineteenth century. The strandthatHyer's work represents is one that grows out of familiar, but perhaps134



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not recently critically enough examined, ideas of tonality and chord function, but seeks to generalize these ideas to account for a wide range ofcomplex harmonic procedures heard in Romantic music. Cohn's work,on the other hand, represents the application of tools conceived for posttonal music.

This is not to say thatHyer clings to the intuitions of diatonic tonality,or that Cohn uncritically applies atonal pitch-class theory to an oftentonal repertoire. Rather, each takes a different transformational approachtomusic inwhich harmony and tonal process on the one hand, and thetwelve-note, equal-tempered chromatic scale and associated motivic

developmenton the other, are both important compositional palettes. The

interpretations presented here suggest that the vocabulary chosen todescribe triad relations situates the analyst to one side of a conceptualboundary, and she or he iswise to think clearly about the implications ofthe model ormodels employed.

On one hand, when we name transformations with symbols derivedfrom such words as "parallel," "Leittonwechsel," or "relative," we implicitly refer to a broad, historically nuanced collection of tonal intuitionsabout triads and relations among them, even while generalizing interpretations of these relations to describe more satisfyingly the coherence,closure, or progressions we sense in their passing (Hyer 1995, 115). Onthe other hand, when we derive our symbols from such words as "hexatonic pole," "transposition," or "minimal perturbation," we refer to intuitions about triads as pitch-class sets, and as subsets of some larger pitchclass set?the hexatonic collection [0, 1,4, 5, 8, 9] or the equal-temperedaggregate (Cohn 1996, 12-13).Iwould like to recall part of the third stage of transformational engagement. In part III wrote:

By "generalize" in the mathematical sense, I mean pursuing algebraicaspects of the model in order to identify possible broader families of musical phenomena to which the transformations in the model might belong.

By "generalize" in amore colloquial sense, Imean seeking to understandthe model as a metaphorical space, or what David Lewin would call a"paradigm that [is] only sometimes fulfilled in any given piece" (1982-83,335-36). Situating the piece and its transformational paths in that spaceenriches and critiques both the interpretation of the piece and the suitability of the initial interpretive choices. This critical moment creates acyclic path back to the first stage of engagement.Two diverging circular paths began when I separated parsimony and

extravagance as a way to interpret differences between the two sequentialpassages from Franck's quintet. Part II gave formal expression to thesedifferences in the non-commutativity of the parsimonious group in

contrast to the commutativity of the extravagant group. The consonant triad

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is privileged under parsimony (Cohn 1996 and 1997), but under the generalized model of commutative extravagance in part IV, that status diminishes. The formalization thus underscores historical precedents linking

maximal common-tone retention with diatonic coherence and maximalsemitonal voice leading with chromatic relations.With this formal perspective, interpretations of Franck's music in Figure 3 led to the networks of Figure 4 and to situating them in themoregeneral space of Figure 5. Taking into account the trajectory from Figure3 to Figure 5, we gained critical insight to aspects of neo-Riemanniananalytical practice. We might say that themoment chromatic music becomes

more than or other than tonal?or the moment our interpretations of themusic are likely to do so?is the moment when motion between triadstraces commutative paths through themetaphorical space triads inhabit.The change of motion leads to a change in interpretation of the objects orlocations in the space: these may be triads, or these may be all sonoritiesthat areminimal perturbations of equal divisions of the octave. The privilege consonant triads continue to enjoy is no more than convention.

Thus, the circle of Figure 2, inscribed by the parsimonious and extravagant aspects of Cohn's P relation as they arced away from each other inthe first paragraphs of this article, is closed again. We should considerthe properties together, but we have a clearer sense of just where and howin their union we experience the interface of traditionally diatonic andincreasingly tonally non-committal voice leading.

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NOTESThis article began as an invited paper delivered at the Symposium on Neo-Riemannian Theory, the State University of New York at Buffalo, July 2001. Part IVdraws on Cook 2001. An earlier version of the present text was read at the University ofWashington School of Music inApril 2005. Thanks to John Rahn, Brandon

Derfler, Peter Shelley, and Jason Yust for their insight and suggestions for improvements. Thanks also to Lawrence Fritts for requesting a small but important andwarranted bit of formal clarification.

1. For a history of neo-Riemannian theory and a sketch of its transformational character, see Cohn 1998.

2. Hook (2002, 60, 89-93) calls attention to this issue in work that both generalizesand particularizes triad relations.3.Weitzmann's treatise on the augmented triad is an exception, as Cohn 2000 shows.

For a comprehensive survey of the issues sketched in this and following paragraphs, see Engebretsen 2002.

4. See Klumpenhouwer 1994; Hyer 1995, 110-11; Mooney 1996, 210-68; Gollin2000, 210-40; and Engebretsen 2002, 209-51.

5. For a detailed treatment of Hostinsky's ideas, which encompass acoustic, common-tone, and stepwise voice-leading relations as determinants of triad proxim

ity, see Engebretsen 2002, 181-96.6. For a thorough study of Hauptmann's common-tone theory, see Engebretsen 2002,

79-117.7. Nebenverwandten Akkorde, "neighbor-related" or "adjacent" chords, may be under

stood as a major triad and its minor subdominant or a minor triad and its majordominant?Db major and Gb minor, for example. Cohn says the two interpretations of the pair are of "equal strength" for Weitzmann (2000, 92).

8. Cohn discusses the use of hexatonic poles in this very passage later in the samearticle (26-28).

9. Iwill refer to mm. 26-37 as a sequence because I expect a third iteration, beginning inA major, atm. 34. As will become clear below, this experience of disruptedexpectation, through the agency of the hexatonic poles, helps to shape my transformational interpretation of the two passages.

10. Obviously, I have difficulty escaping my habit of hearing a whole tone as a "step,"but it is only

astep

in reference to a particulardiatonic collection. On the semi

tonal and whole-tone adjacencies in diatonic and chromatic contexts and the statusof whole-tone-related notes in chromatic music, see Proctor 1978, 143.

11. One might reasonably read "same" tomean "identical, but for a contrapuntal transformation," such as a root-position triad transformed through 5-6 motion againstthe bass. This is a common way to imagine the Relative relation in a major key.

The tonal implications are clear: a submediant thus expressed is part of a composing-out of the tonic Stufe. The connection of tonal intuitions about Relative and

other relations to neo-Riemannian practices is addressed in Part V of this article.12. Imagine two parsimonious transformations S = (-, s, s~{) and T= (-, t,r1). Trans

forming a triad by ST would be equivalent to transforming the same triad by (+, s+ r1, s~[ + t) (Hook 2002, 61 and 68-69). ST is clearly the same sort of UTT asthe parsimonious transformations because (s + rl) + (s~{ +t) = 0; that is, the trans

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position levels of ST for major and minor triads are complementary. Similar resultsobtain if the signs of S and T are opposite or if they are both positive.13. The dualistic character of these transformations has been reflected in neo-Rie

mannian literature, especially by the ascription of different qualities to progressions by thirds, depending on the mode of initial triad or key. A recent example:though Kopp (2002, 183-85) takes issue with Lewin's (1992, 49-52) networks ofprogressions by third inWagner's Ring, both Kopp and Lewin interpret a progression beginning on a minor triad as figuratively opposite in direction to a progression beginning on amajor triad.

14. Imagine two extravagant transformations U = (-, u, u~x) and V = {+, v, v) (V isequivalent to the transposition Tv). Transforming a triad by UV would be equivalent to transforming the same triad by (-, u + v, u~l + v). Clearly, UV is equivalentto VU; the transposition by root-interval v will have the same effect applied beforeor after the root intervals u or ux.

15. The extravagant UTTs generate the commutative group of UTTs 7^(1, 8) (Hook2002, 84-88). The parsimonious group is isomorphic toD[2, the dihedral group oforder 24. The extravagant group is isomorphic to Z2 X Z12, the direct product ofthe groups under addition of integers modulo 2 and modulo 12.

16. Lewin (1993, 25-30) first defined contextual inversion. They have been studied inthe neo-Riemannian literature by Clough (1998) and Kochavi (1998).

17. Gollin 2000 (4-11) and Engebretsen 2002 (15-29) address the conceptual differences between apparently fundamental single transformations and compound transformations through "group presentations," which characterize groups in terms oftheir generating transformations. See also Kopp 2002 (142-64).

18. Cohn 2004 explores the technical and figurative roles of H in chromatic music.19. Hook (2002, 121 n. 24) cites a few papers that illustrate plural modeling, but his

own discussion of different group-algebraic models for triad progressions (91-92)would be unnecessary if the practice were habitual.

20. A similar view of transformational practice is offered inKlumpenhouwer 2000 (157).21. To see how Figure 4a ismalformed, replace the Eb-major triad in the upper left of

the network with C minor. The resulting rearrangement of triads in the remainingcorners preserves the horizontal and vertical transformations, but the arrow fromlower left to upper right would extend from Eb major to A minor, which is not P.I thank Julian Hook for calling attention to this fact and for agreeing that myinterpretive point makes this "bad graph" a good idea.

22. The resulting triads inhabit what Cohn calls "Weitzmann regions" after the nineteenth-century theorist Carl Friedrich Weitzmann (2000, 89-103). See particularly Cohn's Example 4 (93), which presents the material of Table 2 inWeitz

mann's manner. Cohn's "SSD," or "single semitonal displacement" property (94?98,101), is identical tom, though Cohn does not pursue the notion of minimal pertur

bation as I do here.23. Boretz (1972) uses a similar approach to treat fully diminished seventh chords as

conceptually prior to dominant-seventh and "Tristan" chords.24. Callender (1998, 222-23) gives an example of Mystic chords as minimal pertur

bations of whole-tone collections.25. Cohn's example is E major-G major-Ab major-E major. Riemann uses the same

cycle, beginning on C major, to assert that tonal function may be expressed without IV and V (Musik-Lexicon, "Tonalit?t").

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