Neo-Riemannian operations Cohn 1997.pdf

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Yale University Department of Music

Neo-Riemannian Operations, Parsimonious Trichords, and Their "Tonnetz" RepresentationsAuthor(s): Richard CohnReviewed work(s):Source: Journal of Music Theory, Vol. 41, No. 1 (Spring, 1997), pp. 1-66Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843761 .

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NEO-RIEMANNIAN OPERATIONS,PARSIMONIOUS TRICHORDS, AND

THEIR TONNETZREPRESENTATIONS

Richard ohn

1. The Over-Determined TriadIn workpublished n the 1980s, David Lewinproposed o model rela-tions between triads'using operationsadaptedfrom the writingsof theturn-of-the-centuryheorist Hugo Riemann.2Subsequent work alongneo-Riemannian ines has focused on three operationsthat maximize

pitch-classintersectionbetween pairsof distincttriads:P (for Parallel),which relates triads that share a common fifth; L (for Leading-toneexchange),which relates triadsthatshare a common minorthird;andR(forRelative),whichrelatestriads hatsharea commonmajor hird.3Fig-ure 1illustrates hethreeoperations,which I shallreferto collectively asthe PLR family, as they act on a C minortriad,mapping t to threedif-ferentmajortriads.(Throughout his paper,+ and - are used to denotemajorand minortriads,respectively.)Singly applied,each PLR-familyoperation nvertsa triad(major minor).Doubly applied,each PLR-familyoperationmapsa triadto its identity; .e., each is an involution.A striking feature of PLR-family operations is their parsimoniousvoice-leading.4To a degree, parsimonyis inherentto the PLR family,whose definingfeature s doublecommon-toneretention.What s notin-herent is the incrementalmotion of the thirdvoice, which proceedsby

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L P RC- - Ab+ C- - C+ C- Eb+

Figure1:ThePLR-Family f C-MinorA . - . I

Figure2: ThePLR-Family f {0, 1,51semitone in the case of P andL, andby whole tone in thecase of R. Thisfeature s not withoutsignificanceto the developmentof a musicalcul-ture whereconjunctvoice-leadingin general,andsemitonalvoice-lead-ing in particular,are enduringnorms throughan impressiverange ofchronologicaleras and musical styles.5The parsimonyof PLR-familyvoice-leadingis so engrained n theproceduralknowledgeof a musiciantrained n theEuropean lassical tradition hat it hardlyseems to warrantnotice, much less scrutiny. t is scrutinizedhere with the aim of demon-strating hat,from a certainpointof view, the feature s fortuitous.Inorder o cultivate hispointof view, imaginea musicalculturewhereset-class 3-4 (015) was theprivilegedharmony o the extentthatthe triadprevailsin Europeanmusic c. 1500-1900. Forany memberof set-class3-4, therearethreeothermemberswithwhich it shares wo commonpcs.Figure 2 leads the primeform of 3-4 to its threedouble-common-tonerelatedpeers.Note the magnitudeof the moving voice: in two cases byminorthird,in the thirdcase by tritone. Such a culturewould be inca-pableof achievingthe degreeof voice-leading parsimonycharacteristicof triadicmusic, particularly s it developedin the nineteenthcentury. tcan be easily verified,and will be demonstrated hortly,thatreplicationof the exercise using any other mod-12 trichord-classyields similarlyunparsimonious esults.6Itmaycome as no surprise hat,amongtrichord-classes, onsonant ri-adsarespecial.Theiruniqueacousticpropertiesarewell established,andindeed are fundamental o standardapproachesto triadic music. Thepotentialof consonant triadsto engage in parsimoniousvoice-leading,however, is unrelatedto those acoustic properties.This potential is,rather,a functionof theirgroup-theoretic ropertiesas equallytemperedentities modulo-12.To demonstrate his claim, the definitionof PLR-familyoperations snow generalized,initially to the prime forms of set-classes defined byT,/T,Iequivalence,subsequently o all trichords.Althoughour primary2



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attentionwill be focused on trichords n the usual chromaticsystem oftwelvepitch-classes, he definition s opento application o othersystemsas well, forreasons that will emergeas the expositionunfolds.DEE (1). c is a positive integerrepresentinghe cardinalityof a chro-maticsystem.DEE (2). Q is a mod-c trichord{0, x, x + y} such that0 < x < y 5 c -(x + y).

The conditioninsuresthatQ is theprimeformof its trichord-class.DEE (3) Iuis the inversionthat maps pitch-classes v and u to eachother.7

The threePLR-familyoperations an now be definedon aprime-formri-chordQ = {0, x, x + y} as follows:DEE (4a) P = I,+y I DEF.(4b) L = I I DEF.(4c) R = IFigure 3a (p. 4) demonstrates he mappingof abstractpitch-classeswhen P,L, and R act on the abstract richordalprimeformQ. Figures3band 3c realizeQ as the two trichordsexploredin Figures 1 and 2. Eachoperation swaps two of the pitch-classes in Q. The remainingpc ismapped outside of Q, and this mapping is perceived as the "movingvoice." We now define a set of variables,a, /, and4, which expressthemagnitudeof thoseexternalmappingsas mod-ctranspositional alues.

x----y,hence: x + y-y, hence: 0--2x+ y, hence:DEE (5a)9 = y - x DEE (5b)l = -2y - x DEE (5c) , = 2x + y

Observethatp + / + 00, since (y - x) + (-2y - x) + (2x + y) = (2x - 2x)+ (2y - 2y) = 0.The valuesof , 1,and,zarenow linked to the structureof trichordQ.First, following Bacon (1917), Chrisman(1971), and others since, wedefinea step-interval eries as follows:DEF (6). The step-interval series for a normal-orderrichord{i,j,k}is the orderedset , modulo c.

Via Def. (2), Q = {0, x, x+ y } is in primeform,which presupposesnor-mal order.Thusthe step-interval eriesof Q is .3



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P ( Q ) (Q L ( Q ) (Q R ( Q ) (Qx+y-> x+y--y x+y-xx --4y x --4O x- x + yO---)x+y 0---x 0--2x + v

Figure :PLR-FamilyMappings(a)forQ= {0,x, x +y}P(Q) = I o(Q) L(Q)= I (Q) R(Q) = I' (Q)7 -- 0 7- 8 7 --433 - 4 3--4 0 3--470--4 7 0 - 3 0--4 10

Figure :PLR-FamilyMappings(b)forC-Minor; = 3,y =4, Q = {0,3,7}P(Q) = I

o(Q) L(Q) = I (Q) R(Q) = I; (Q)

5--4 0 5--4 8 5-- 11---44 1--40 1--~ 50--4 5 0--41 0-- 6

Figure :PLR-FamilyMappings(c) x = 1,y = 4, Q = {0, 1,5)Thefollowing theoremstatesthatp, 1,andz are eachequivalent o thedifferencesbetween a distinctpairof step-intervals.THEOREM1. Fora prime-form richordQ = {0, x, x + y} with step-intervals,1.1)p is the differencebetween the firstand second step interval,y - x. Proof:y - x = p via Def. (5a).1.2)1 is the differencebetween the second andthirdstep interval,

- (x+y) - y. Proof:- (x+y) - y = - 2y - x = / via Def. (5b).1.3) ' is the difference between the third and first step interval,x - (-(x+y)). Proof:x - (- (x+y)) = 2x + y = ' via Def. (5c).Theorem1facilitatescalculationof thevaluesofp.,1,and foranytri-chordalprime form in a chromaticsystem of any size. The results of

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these calculations orthemod-12trichordal rimeformsaregiveninFig-ure 4 (p. 6). Conjunct ntervalsare enclosed in boxes.The figuredemon-strateswhat was asserted above: that 037 is unique among trichordalprimeforms (modulo 12) in its preservationof conjunctmelodic inter-vals when anyof the threePLR-familyoperations s executed.8The generalization rom primeform to other class members s easilycarriedout. Calculate heTnorTnIof the trichord n relation o theprimeform of its class, reassign 0 to the pitch-class that had been formerlyassignedtheintegern, andreassignthe integers n ascentor descentfromthe new pitch-class 0, dependingon whetherthe trichord s Tnor TnI-related o the primeform. Thenapplythe operationsas in definition(4).If Tn-related o the primeform,the values of p, 1,and, are the same aswhen the operationacts on the primeform;if TnI-related, he valuesareinverted. Thisfollows from the involutionalnatureof PLR-familyoper-ations.) In either case, the magnitudesare invariant.Consequently, heuniquecharacteristicnotedabove for trichordalprimeform {037} gen-eralizes to all membersof its set-class.To summarizeour findingsso far: (1) Among mod-12 trichords, heconsonant riadalone is susceptible o parsimoniousvoice-leadingunderthe threePLR-familyoperations; 2) This circumstance s a function ofthe trichord's tep-interval izes, whichareanaspectof its internal truc-ture;(3) the optimalvoice-leading propertiesof triadsthereforestand nincidentalrelationto theiroptimalacousticproperties.In a word:the triad s over-determined.The fortuitousrelation of the consonant triad'svoice-leading parsi-monyto its acousticgenerability s asprofound o thedevelopmentof theEuropeanmusicaltraditionas othersorts of over-determinationhatwerefirst broughtto light at Princeton in the 1960s (Babbitt 1965, Gamer1967, Boretz 1970):of the chromaticdivision of the octave into twelveparts, 12 being at once the smallest abundant nteger and the smallestintegern suchthat3n approximates ome powerof 2; of theproximityofthe perfect fifth's 2 geometricdivision of the octave (the source of itsacoustic power) to its 7 arithmetic division of the octave (a fractionwhose irreducibility, arefor its divisor,is necessaryfor the deep-scalepropertyof diatoniccollections, a circumstancewhich in turnleads tothe gradedcommon-tonedistributionof the set of diatoniccollectionsunder ransposition,andhence to the systemof key signatures).Equallyremarkable s the extent to which the triad'sacoustic propertieshavemaskedrecognitionof its group-theoreticpotential.9Our sensibilities,born of incessantexposureto a musical tradition hathabitually mple-mentstheacousticpropertiesof triads,as well as to a music-theoretic ra-dition that habituallymodels this habitualimplementation,have beentrained o resistby defaultanyeffort to regard hetriadas anythingotherthan acoustic in essence. Like the stock figure of the Cold War spy

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primeform step-intervals p 1 r(0, x, x + y) y -x -2y - x 2x + y

0,1,2) 0 9 3

{0,1,3} 7 4

{0,1,4} 5 5

{0,1,51 3 3 6{0,1,6} 4 W 7{0,2,4 0 6 6

{0,2,5} W 4 7{0,2,6} E 8

{0,2,7) 3 0 9

{0,3,6) 0 3 9

{0,3,7.} 1--F01{0,4,8} 0 0 0

Figure4: Values of p, 1,and r for trichordalprimeforms



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F ---- C - G D

A E H Fs

Cs Gs Ds B

Figure : TwoTonnetze(a)fromEuler1926 1739)

thriller, he dazzlingbeautyof the triad has blinded us to its substantialintellectualresources.2.1. The Over-Determined Tonnetz

The two-dimensional matrix known as the table of consonant (ortonal)relations(Verwandtschaftsverhdltnistabelle)r harmonicnetwork(Tonnetz,Tongewebe)has long presentedmusic theorists with a usefulgraphicinstrument or representing riadicprogressions.Although thematrix originated in response to the acoustic propertiesof triads, itresponds n equalmeasure o theirgroup-theoretic roperties.The over-determinationof the triad is thus encoded in the over-determination fthe Tonnetz.The Tonnetzwas initiallyconceived to reconcile the first two distinct(non-complementary) ub-octave intervalsgeneratedfrom a resonatingbody.'"LeonhardEuler(1926 [1739]) situatedustlytunedversionsof thetwelve pitchclasses on a bounded4x3 matrixwhose axes aregeneratedby acoustically pure fifths (3:2) and major thirds (5:4) (Figure 5a)."Arthurvon Oettingen(1866, 15) invertedEuler'smatrixabout the hori-zontalaxis, andprojected t onto aninfiniteplane,as shownin Figure5b(p. 8). (The slashes representsyntonic comma adjustments,and resultfromOettingen'ssensitivityto thejust-intonationaldistinctionsmaskedbynotationaland etter-name quivalence.)Thisversionof thepitch-classtablewasappropriated yRiemann,becamewidelydisseminated hroughhis writings,andhas been passed down by generationsof Germanhar-monictheorists eadingall the way up to thepresentday(see Imig 1970,Harrison1994).

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2 c g d a e h fis cis gis dis a1 as es b f c g d a e h f0 fes ces ges des as es b f c g

- 1 deses asas eses bb fes ces ges des as es-2 bbb feses ceses geses deses asas eses bb fes ces g

Figure5: Two Tonnetze(b) fromOettingen1866



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Thepositionof majorandminor riadsonthe matrixwas firstobservedby Euler(1926 [1773], 585), who notedthattheycould be represented nhis matrixby the conjunctionof two perpendicular ine-segments.Oet-tingen,less concerned han Eulerabouttheacousticallyresidualstatusofthe minorthird,suggested addinga hypotenuseto close the structure oa righttriangle(1866, 17).This move bringsPLR-familyrelations o theforefront, incetriadsso relatedarerepresentedby triangles hatshareanedge, and are thereby maximallyproximate.One might infer fromthiscircumstance hatPLR-familyoperationswouldbe the vehicle of choicefor navigating riadicprogressionson the Tonnetz,but this has not beenthe case historically.The developmentof Oettingen'stable as a "game-board"for mappingprogressions among triads was insteadguided byconvictionsabout the acoustic,tonallycentricstatus of consonanttriadsand their relations to each other.This resultedin the subordinationofPLR-familyrelations o theTonic/Subdominant/DominantTSD) "func-tional" rameworkdevelopedby Riemann n the 1890s,a frameworkhathas continued to dominate NorthernEuropeanharmonictheory eversince. 2The mappingof PLR-familyoperationsindependentlyof the TSDframeworkwas firstproposed by Lewin (1987, 175-180) andhas beendeveloped by Brian Hyer (1989, 1995), whose work demonstrates heheuristicvalueof chartingPLR-familyoperationson the Tonnetzwithoutnecessaryrecourse to assumptionsabout tonal centers,TSD-functionalrelations,ortheacousticpropertiesof triads.Theemphasison PLR-fam-ily operations n the work of Lewin and Hyer is apparentlymotivatedempiricallyby the desire to model characteristicallyate-Romanticpro-gressionsin a manner hat s faithfulto themusicalqualities hattheyareperceived o project.Thefocus on voice-leadingparsimonycultivated nSection 1abovesuggestsa complementarydeductive-rationalistmotiva-tion for liberatingthe triad,PLR-familyoperations,and their Tonnetzrepresentationromtheiracousticorigins.It is this motivation hatdirectsthe remainingprogram or this paper.Inthematerial hatfollows immediately,a versionof the Tonnetz f Oet-tingenand Riemann s situatedwithina genusandspecies of two-dimen-sional matrices.The definingcharacteristic f the genus is thatpairsoftrichordsrepresentedby adjacenttrianglesare related by PLR-familyoperations,as broadlydefined in Section I above (cf. Def. (4)). Thedefiningcharacteristic f the species is thatpairsof trichords epresentedby adjacenttriangles featureparsimoniousvoice-leading. Section 2.4refines the conception of the Oettingen/Riemannmatrix, and of thespecies that it represents,by acknowledgingthe toroidalgeometrythatunderlies them when theircontentsand relationsare interpretedn thecontext of equaltemperament.Thefocus on the TonnetzhroughoutSec-tion 2 servesas a largestructural pbeat o Section3, the musical coreof

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-2x + 2y -x + 2y 2y x + 2y 2x + 2y

-2x+ y -x +y y- x +y- 2x + y

I/1/2x -x 0O x 2x-2x-y -x-y -y x-y 2x-y

Tx-2y(Q)2x2y-2x- 2y -x- 2y -2y x - 2y-2x - 2y

Figure : TheAbstract onnetz

the paper,whichuses PLR-familyoperations o navigatethe Tonnetz,nits variousversions at variousdegreesalongtheabstraction/specificationcontinuum.2.2. The Generic Tonnetz

Our nvestigationof the Tonnetz f harmonic heory nitiallysituates tas a member of an infinite class of two-dimensional matrices whosegenericform is presented n Figure6. The primaryaxes of Figure6 aregeneratedby the generic intervalsx and y, in the sense that each rowincrements rom left to rightby the value of x, andeach column incre-ments frombottom to top by the value of y. The figureshould be inter-pretedas projecting ts structurebeyondits boundaries.The elements ofFigure6 represent eal numbersas they increment o infinity,andshouldnot be interpretedn thecontext of theclosed modular ystemsthat werethe focus of ourpreviouswork.Figure6 is neithermore nor less thantheCartesiancoordinateplaneof analyticgeometry.Intermsof Def. (2), theprimaryaxes of Figure6 areinterpreted s thesmallest two step-intervals f a primeformtrichordQ = {0, x, x+y} withstep-intervals. Theremaining tep-intervals the inverseofthe sum of the two smallerstep-intervals,and hence generatesthe diag-10



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onal which runs from northeast o southwest, n the sense thateach suchdiagonaldecrements,slopingsouthwestward, y x+y.Figure6 represents richordQ as a darklybordered riangleat its cen-ter.Eachvertexof thetrianglerepresentsapitchorpitch-class,13ndeachedge a dyadicsubset,of Q. Any geometrictranslationof this triangle-that is, any trianglewhose hypotenusesubtends northwestof the rightangle-represents apitchorpitch-classtransposition f Q. (Anexample,labeledTx-2y(Q),s providedby the isolatedtriangle n the southeastcor-ner of Figure6.) Furthermore, ny geometricinversionof the Q triangleabouta secondarydiagonal-that is, anytrianglewhose hypotenusesub-tends southeast of the right angle-represents a pitch or pitch-classinversionof Q. Figure6 indicatesthree such inverted riangles,all shar-ing an edge with the centraltriangle.These threetrianglesrepresent hePLR-familyof Q (cf. Figure3a). The unique edge thatthe centraltrian-gle shares with each of its adjacent rianglesrepresents he unique dyadthat trichordQ shares witheach memberof its PLR-family.Figure7 replicatesthe core of Figure6 and adds threearrows,eachlabeled with one of thevoice-leadingvariables romDef. (5). Each arrowrepresents he magnitudeof the moving voice when Q is subject to aPLR-familyoperation:

*p labelsx-y whenP takes {0, x, x + y} to {0, y, x + y} across theirsharedhypotenuse;*/ labels x + y - -y when L takes 10, x, x + y} to 10, x, - y} acrosstheir sharedhorizontaledge;*4 labels0 --- 2x + y when R takes {0, x, x + y) to {2x + y, x, x + y)across theirsharedverticaledge.When the same operationsare enacted on other members of trichord-class Q, theirgeometricorientationon Figure7 is invariant. n all cases,P-related riadssharea hypotenuse,andp executes a pawn-capture longthe main diagonal;L-relatedtriads sharea horizontaledge, andI exe-cutes a knight'smove, a displacementby two rows andone column;R-related riadssharea verticaledge, and4 executes a knight'smove, a dis-placementby twocolumnsandone row.Themagnitudesofp, 4, and4 arelikewise invariant, lthoughwhen theobjectof themapping s atriadTnI-elated o theprimeform,thedirectionof the arrowreverses,and theval-ues of p, 1,and4 invert.

The unboundedgrid inferable from Figure6 is applicableto pitchesand intervals n a varietyof ways. If x andy areassignedto acousticallypure intervals(as in Euler,etc.), or to intervalsin pitch-space,then thestructuremplicitlyprojects ntoaninfiniteplane.Therealizationsof theFigure6 gridthatwill hold ourfocus aregeneratedby equally temperedintervals n some modularsystem,wherethe modularcongruencerepre-11



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y x+y 2x+y

-x 2x

-y x-y 2x-y

Figure :Tonnetz epresentationfPLR-Family perationsnQ= (0, x, x + y}

sents octave equivalence. In such interpretations,both x and y axesbecome cyclic rather hanlinear,and the plane inferredfrom Figure6thereforeprojectsinto itself as a torus.14These cyclic featureswill bestudied n some detailin Section2.4 below.2.3. The Parsimonious Tonnetz

In Figure 7, PLR-family operationsare only associated with voice-leading parsimony n the restrictedsense that each operationpreservestwo common tones. The degreeof parsimonyassociated with the thirdvoice dependson the magnitudesof p, / and4, which in turndependonthe valuesof x andy, as yet unassigned.Furthermore,he interpretationof thatmagnitudeas representingan interval-class n a modularpitch-class system dependson the impositionof a congruence, .e. a specificvalueforc. The first sectionof thispaperestablished hat,wherec = 12,the threePLR-familyoperationsareparsimoniousonly when x = 3 andy = 4, i.e whenthetrichord-classs theconsonant riad, n whichcase thegeneric Tonnetz s realized as an equal-tempered ersion of the Oettin-gen/RiemannTonnetz f Figure5b. Since this is the case of historicalandanalytical nterest, t will soon be subject o detailedscrutiny.First,how-ever,it will be instructive o considera structure f intermediate bstrac-tion;a "middleground" onnetzhat"composesout"Figure6 in a partic-ularway,at the same time as it positionsthe propertiesof the Oettingen/RiemannTonnetzn a generalcontext.What values of c, x, andy will lead to optimallyparsimoniousvoice-leading when PLR-family operationsare executed?Intuitively, he de-gree of parsimony s optimalwhenthe magnitudes i.e. absolutevalues)of the voice-leading intervalsp, I and 4 are as small as possible, butgreater hanzero.(Thelastcondition nsures hatvoice-leading"motion"12



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is perceptibleas such; cf. note 6.) Ideal parsimony,then, would beachievedwhena = +1, = + 1,and4 = ? 1.Butthis combination s impos-sible. As observed n Section 1,9 + / +4 = 0, and so each variable s theinverse of the sum of the othertwo. Consequently, , 1,and4 must in thiscase have identicaldirectionsas well as magnitudes,andso a = == -.ViaDefs. (5b) and(5c), ifl/= 4 then- 2y - x = 2x + y, andso 3x = - 3y, hencex = - y, which implies thaty - x is even. Thusp is even (via Def. (5a)),and so # + 1,contrary o what was stipulated.The next recourse s to increment he magnitudeof one of the voice-leading variables.In principle,any of the three variablescan be incre-mented,butQ is in primeform (cf. Def. 2) only if p = 1,1 = 1, 4 = -2.These areexactly the values for the familiarcase of the consonanttriadmodulo12.Oncewe cease to assume a chromatic ystemof twelvepitch-classes, as we didin section 1,what othercombinations ead to theseval-ues ofp, 1,and4? Thisproblem s easily solved usingTheorem1, whichlinked the voice-leadingintervals o step-intervaldifferences.If the firststepinterval s x, and = 1,then the secondstepinterval s x + 1,viaThe-orem 1.1.Further, ince/ = 1,thenthethirdstepinterval s x + 2, viaThe-orem 1.2. Thethreestep intervalsof a parsimonious richord, hen,mustformthe ascendingconsecutive series.15The sum of these threestep intervals s 3x + 3, distributed s 3(x + 1).Since c, the numberof pitch-classes n the system,is the sumof the step-intervals, t follows thatparsimonious richordsareonly available f c isanintegralmultipleof 3. Ineach suchsystem,there s a single step-inter-val series of the form thatrepresentsa parsimonious ri-chord-classwhose primeform is {0, x, 2x + 1}. The genericversion ofsuch a trichordwill be representedusing the variableQ':

DEE (7). Q'= {0, x, 2x + 1}, modulo3x + 3Figure8 (p. 14)presentsa genericTonnetzorparsimonious richords,which will be referred o as the parsimonious Tonnetzfor short. Theaxes of Figure8 aregeneratedby x andx + 1, the smalleststep-intervalsin Q'. A modularcongruenceof 3x + 3 is imposedon the figure,so thatthe first term of each expressionis representedas a non-negativevalue.Thetrichordalprime-formQ' = {0, x, 2x + 1} is portrayed longwith itsPLRfamilyat thecenterof Figure8 . The arrowsdepictthe following:*p labels the TImotion x - x + 1whenP takes {0, x, 2x + 1) to {0,x + 1, 2x + 1} across theirsharedhypotenuse;*/ labels the TImotion2x + 1- 2x + 2 when L takes {0, x, 2x + 1}to {0, x, 2x + 2} across theirsharedhorizontaledge;

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x + 3 2x+ 3 0 x 2x 3x

2 x+2 2x+2 3x+2 x- 1 2x- 1

2x+4 1 x+1 2x+1- 3x+1 x-2

x + 3 2x+ 3 0 7 x 2x 3x

2 x+2 2x+2 3x+2 x- 1 2x- 1

2x+ 4 1 x + 1 2x+ 1 3x+ 1 x- 2

x +3 2x+ 3 0 x 2x 3x

Figure : TheParsimoniousonnetz*, labels the T2motion0 -- 3x + 1 -2 whenR takes {0, x, 2x + 1}to {3x + 1, x, 2x + 1})acrosstheirsharedverticaledge.Figure 8 is a powerful representationof parsimonious trichordalmotion. No matterwhatvalue is assignedto x, any local trianglewith asouthwest/northeastypotenuserepresentsaparsimoniousrichord.Con-versely,allparsimoniousrichordsarerepresentablehrougha realizationof Figure8. Toinvestigate he scope of this power,we now explorethree

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2 x + 3

2 x+2 2x+2 3x+2 x-1I 2x- 1

2x+4 1 x+1 2x+1 3x+ 1 x-2

x + 3 2x + 3 0 - x 2x 3x

1 '2 x+2 2x+2 3x+2 x- 2x-2x+4 1 x+ 1 2x+ 1 3x+ 1 x-2

x+3 2x+3 0 x 2x 3x[6 9] 03 [6 9

Figure : ThreeRealizationsf theParsimoniousonnetz.(a)x = 3, y = 4, c = 12,Q'= {0,3, 7} modulo2(OettingenRiemann onnetz)such realizations,for x = 3, 5, and 7 respectively.In each of the threematrices hatcomprise Figure9, the abstract xpressionsof Figure8 areretainedalong with their realizationsso that derivationscan be easilytraced.Figure9a is a versionof the Oettingen/RiemannTonnetz. t partiallyrotatesFigure5b, replacing pitch-classnames with integers.The seriesof majorthirds s retainedon the y axis, but the seriesof minor thirds sdisplaced romthenorthwest/southeast iagonalof Figure5b to thex axisof Figure9a, therebyshiftingthe series of perfectfifths fromthe x axis

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2 x + 3

2 x+2 2x+2 3x+2 x-1 2x- 1

2x+4 1 x+ 1 2x+ 1 3x+ 1 x-214 6 11 16p r-x+3 2x+3 0 -% x 2x 3x

813 0 5,151 '2 x+2 2x+2 3x++2 x-1 2x- 1w W 12 1 W

2x+4 1 x+ 1 2x+ 1 3x+ 1 x-2

x+3 2x+3 0 x 2x 3x13 ] [5 1] [5

Figure : ThreeRealizationsf theParsimoniousonnetz.(b)x = 5, y = 6, c = 18,Q'= (0, 5,11)modulo8

to the southwest/northeast iagonal.This particularversion of the Ton-netzactuallyantedatesOettingen: t wasfirst ntroducedbyCarlFriedrichWeitzmannn 1853,although n abounded orm(as inFig. 5a), andusingstaff-notatedpitchesrather hanintegers.'6Thetriangular omplexat thecenterof Figure9aportrayshe trichordal rime orm,{0, 3, 7 } = Cminor,togetherwith its PLRfamily.The arrowsrepresent he following:16



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*p labelstheTIsemitonalmotion3- 4 = Eb-- E when P takes {0,3, 7} = C minor to 10, 4, 7} = C major across their sharedhypotenuse;*1 labelstheTIsemitonalmotion7 - 8 = G- Abwhen L takes {0,3, 7) = C minorto {0, 3, 8) = Abmajoracross their sharedhorizon-tal edge;*4 labels theTl0o T-2whole-stepmotion0 - 10 = C-- Bb whenRtakes{0, 3, 7) = C minor o {10, 3, 7) = Ebmajoracrosstheirsharedverticaledge.In Figure9b, the parsimonious richord s Q' = {0, 5, 11) in a modulo18 ("third-tone")ystem, with step-intervals. The triangular

complex at the center of Figure9b portrays10, 5, 11} togetherwith thethreetrichords hatcompriseits PLRfamily.The arrowsportray he fol-lowing:p*labels theTImotion 5 --- 6 when P takes 10, 5, 11) to 10, 6, 11)acrosstheir sharedhypotenuse;*l labels theT1motion 11 -- 12when Ltakes {0, 5, 11} to {0, 5, 12acrosstheir sharedhorizontaledge;*,z abels theT16 T-2motion0-- 16 when R takes {0, 5, 11 to {16,5, 11} acrosstheirsharedverticaledge.

In Figure9c (p. 18), the parsimonious richord s Q' = 10, 7, 15} in amodulo 24 ("quarter-tone")ystem,withstep-intervals. Thetri-angularcomplex at the center of Figure9c portrays10, 7, 15} togetherwith the threetrichords hatformits PLRfamily.The arrowsportray hefollowing:p*labels theT1motion 7 -- 8 when P takes 10, 7, 15) to 10, 8, 15)across theirsharedhypotenuse;* labels theT, motion 15- 16when L takes {0, 7, 15} to {0, 7, 16)across theirsharedhorizontaledge;* labels theT22= T-2motion0 - 22 whenR takes10,7, 15} to {22,7, 15) acrosstheirsharedverticaledge.

2.4. The Toroidal TonnetzBeforenavigating he Tonnetze,we need to confront heir imitationasa representation f pitch-classrelations n equal temperament.nFigures8 and9, pitch-classesoccurin multiplelocations,obscuring heirequiv-alence.Alternatively epresenting achpitch-classata single location,asin Figure5a, has the equally perniciousconsequenceof obscuringadja-

17



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2 x + 3

2 x+2 2x+2 3x+2 x- 1 2x- 1

2x+4 1 x+ 1 2x+ 1 3x+ 1 x-2Fx181 1 x8 15 22 F5x + 3 2x + 3 0 - x 2x 3xFl 1 7 1

2 x+2 2x + 2 3x + 2 x-1 2x- 1S 91 16 236 [132x+4 1 x+ 1 2x+ 1 3x+ 1 x-2

x+10 1W+5]3

Figure : ThreeRealizationsf the Parsimoniousonnetz.(c)x = 7, y = 8,c = 24,Q'= {0,7, 15) modulo4

cency relationships, ausingaxes to float off the edge of the two-dimen-sional surfaceonly to reappearon the opposite edge. These obscuritiesresultartificially romthe mismatchbetween thecyclical natureof pitch-class spaceandtheflat surfaceof theprintedpage.A toruspresentsageo-metric figure more appropriateo representing he cyclic propertiesofequal-tempered itch-class.Although hetorus s eschewed here becauseit is difficult o renderandinterpret nthetwo-dimensional urfaceof thepage, its underlyingstatusneeds to be sufficientlyacknowledgedbeforethe Tonnetz an be navigatedwithfull comprehension.18



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At issue aboveall is the cyclic periodicitiesof the axes, which fluctu-ate according o the generating ntervalsand the cardinalityof the chro-matic system.The axes relevant o Tonnetznavigationarethe threethatare generatedby step-intervalsof the parsimonious richord.These in-clude not only the primaryx and y axes, but also the -(x + y) axis thatgenerates the southwest/northeastdiagonal. Assigning an abbreviatedvariable o this step-intervalwill simplifyour treatment f the axis-cyclegeneratedby it.DEE (8). z = c - (x + y).All threestep-interval-generatedxes are circularizedby equal tem-

perament,and thuswill be referred o as axis/cycles.Ourstudyof theperiodicityof axis/cycles will be aidedby putting ntoplay a variableq*, representingheperiodicityof step-intervalq moduloC.

DEF.(9). q = where gcd(c,q), the greatest common divi-gcd(c,q) 'sor of c andq, is the largest nteger such that-c and arepositiveintegers. J JThe asterisk s attachable o any variablethatrepresentsa step-inter-val;hence x* is the periodicityof x moduloc, andso forth.Thereare two generalcases.17 If q andc areco-prime,thengcd(c,q) =1, in which case q* = c. The q cycle then exhauststhe chromaticsystem,andall cycles along the q axis areidentical; n effect, thereis a single qaxis-cycle. A familiarexample is presentedby the z axis of Figure 9a(p. 15), wherec = 12 andz = 5. gcd(12,5) = 1, and so z* = c = 12. (The12-periodicitycannotbe directlyverified on the attenuatedrepresenta-

tion of the z-axis given in Figure9a, but mustbe inducedby extendingthe boundariesof the figure.)All z axes in Figure9a thushave a period-icityof 12,andexhaustthe 12pitch-classesvia the"circleof fifths."Thusthere is only a single distinctz axis-cycle.In the second generalcase, gcd(c,q) > 1, in which case q* < c. The qaxis does not exhaust the chromaticsystem, but instead runs throughsome propersubsetof its pcs. The pcs moduloc are thenpartitionedntogcd(c,q) = ~ co-cycles (providedthat 1 is the greatestcommon divi-sorof the threestep-intervalsn Q, as is indeed true for all cases relevantto this study).An exampleis presentedby the z axis of Figure9c, wherec = 24 and z = 9. gcd(24,9) = 3, andso z* = c = 8: each z cyclegcd(c,z)includeseight of the 24 pcs in the chromaticsystem. (Again this claimmustbe inducedfromthefigure.Theorderedcontentof the z axis begin-

19



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ning at the southwestcornerof 9c is as follows: .)There aregcd(24,9) = 3 distinct z axis-cycles, which partitionthe 24 pitch-classes.Retreatingby one level of abstraction, onsidernow thecyclic featuresof Figure8 (p. 14), wherec = 3x + 3, andthe threestep intervals are. c = 3x + 3 = 3 (x+1) = 3y, and so y = . Since yevenly divides c, it follows that gcd(c,y) = y, and so y* =- = 3. ThisYexplains why there areexactlythree distinctelementsin each column ofthe matrices n Figures8 and9. Thesignificantpointhere is that hetripleperiodicityassociatedwith the secondstep-intervals proper o the under-lying structure f Figure8, andthus the inclusion of an octave-trisectinginterval,acousticallyequivalent o a temperedmajor hird, s common toallparsimoniousrichords.As for the number f distinctcolumns, herearegcd(c,y)= y. That s: there s one y-axis co-cycle foreachdegreeof sepa-rationbetween the second and thirdpitch-class n the primeform of Q'.In contrast,the remainingstep-intervalsare divisors of c only underlimited conditions. c is co-primewith x (the firststep-interval)unless xis a multipleof one of c's divisors,3 or x + 1. x cannotbe a multipleof x+ 1;thus c and x areco-primeunless x is an integralmultipleof 3, i.e.,there s some positive integern such that 3n = x. If so, then c = 3(3n) + 3= 9n + 3, in which case c is congruent o 3 modulo9. The smallestexam-ples of suchsystemsare c = {12 (N.B.), 21, 30}.Of the systems portrayedn Figure9, only Figure9a (p. 15), where c= 12,meets thiscondition,andconsequently t is only here thatthex axispartitions ts pcs into co-cycles rather hanexhaustingthem in a single12cycle. In this case, x = 3, gcd(c,x) = 3, and x* 4. Each x axisthus contains our distinctpcs, andtherearex = 3 distinctx axes. (Instan-dard erms,of course,whatwe havehereis thepartitionof the aggregateinto threediminishedseventhchords.)By contrast, n Figures9b and9c,neitherc = 18 nor c = 24 arecongruent o 3 modulo 9. Consequently,andx areco-prime,and so there is only a single x axis thatexhauststhesystemof 18 or 24 pcs.A similarsituationholds fortherelationship f c to thethirdstep-inter-val. x + 2 cannot be a multipleof x + 1, andso, in parallelwith the caseof the x axis just described,c andx + 2 areco-primeunless x + 2 is anintegralmultipleof 3, i.e., thereis some positive integern such that 3n -2 = x. In suchcases, c = 3(3n - 2) + 3 = 9n - 3. Thatis, c - 6 modulo 9.The smallestsuch systemsarec = 16, 15, 24). Of the systemsportrayedin Figure9, only Figure9c, where c = 24, meets thiscondition,and con-sequentlyit is only here that the z axis partitions ts pcs into co-cyclesrather hanexhaustingthemin a single cycle. By contrast, n Figures9aand9b, neitherc = 12 norc = 18 arecongruent o 6 modulo9, and so c20



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and z areco-prime,and thereis only a single z axis which exhauststhesystemof 12 or 18pcs.A practicaldemonstration f the features discussed in this section isprovided in the pioneering microtonaltreatise of Alois Hiba (1927),which systematicallyexploresthe generativepowersof intervals n bothquarter-tonec = 24) and third-tone c = 18) systems (where "tone" staken n the sense of "wholetone").In his discussion of the quarter-tonesystem, Hiba notes that the "neutral hird,"equivalentto 7/24 of anoctave, generatesall 24 pitch-classes.In contrast,Hiba writes thattheinterval"aquarter-tone igherthan a major hird[i.e. 9/24 of an octave]leads only as far as an octachord[Achtklang].This octachordis com-posed of the symmetricpartitionof a majorsixth.... Twotranspositionsof the octachordupwardby a quarter-tone se theremaining16 tonesofthequarter-tone-scale"1927, 166). Hiba's three octachordsareequiva-lent to the threedistinct z-axis co-cycles discussed above in associationwith Figure9c. In a subsequentchapter,Hiba approaches he third-tonesystem from a similarperspective,observingthat "the successive seriesof 5/3 steps forms a unified collection of 18 tones in the third-tonesys-tem,andindeedin a morebroadlyexpandeddistribution crossa spanoffive octaves. The successive series of 18 seven-third ones forms a col-lection spreadacross seven octaves"(202-203). H1iba's ggregate-com-pletingNacheinanderfolgenareequivalent o the x and z axes of Figure9b. Theperspectivecultivated hroughouthis sectionprovidesa system-atic foundation or Hdiba's bservations.Figure 10 (p. 22) summarizes he workof this sectionby providingatable for calculatingthe cyclic periodicitiesfor the chromaticsystemsthatcanhostparsimonious richords,as exemplified n the threematricesof Figure9. The significantpoint to be carriedout of this exposition isthatthey axis, generatedby the secondstep-interval,has a constantperi-odicity, and the numberof y co-cycles varies with the size of the chro-matic system. Conversely,the x and z axes, generatedby the first andthirdstep-intervals espectively,havea variableperiodicity,but the num-ber of x andz co-cycles is constantto withinthe modulo-3congruenceof c. The relevanceof these findings,andparticularly f the special sta-tus of the y axis, to progressionsbased on PLR-familyoperationswillbecome apparentn Section 3.

3.1. PLR-family CompoundsWeare now in apositionto navigate hetoroidalTonnetz,n all itsvar-ious manifestations,using PLR-family operationsas our vehicle. Themaximal common-tone retention inherent to these operationsinsuresthat thecruise will be smooth,particularlywhenthe Tonnetzs parsimo-nious.Ourexplorationwill follow a systematicprogram, ocusingon tri-

21



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(a) (b) (c)x axis y axis

c mod-9 periodicity # co-cycles periodicity # co-cycles(x*) (c/x*) (y*) (c/y*)0 c 1 3 c33 c 3 3 c3 36 c 1 3 c3

Figure 10:Cyclic Periodicitiesof StepIntervalso



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chordalprogressions,orchains,generatedby therecursiveapplicationofa patternof PLR-familyoperations.'8Whereappropriate, hainswill beviewed as segmentsof cycles. The basic cycle-classes formedby gener-ated PLR-familychains are few in number,and theirrelation to PLR-family operations s roughlyanalogousto the role of the chain of fifths(Sechterkette)n diatonicprogressions: hey constitutenormativeproto-types againstwhichparticular rogressions, n all theirvarietyandcom-plexity, maybe gauged.The ultimategoal of this investigation s the pragmaticone of explor-ing parsimoniousvoice-leading amongconsonanttriadsin a 12-pc sys-tem. Consistentwith the framework stablished n Section 2, this famil-iar phenomenon is situated as a particularmanifestation of a moregeneral one: the behavior of parsimonious richords n any pitch-classsystemthat s suitablysizedto host them.Somereadersmaybe frustratedby this strategy,since it defersan encounterwith music in systems thatwe care about, instead inviting contemplationof hypotheticalmusicalsystems whose sounds we may havedifficultyimagining. Nowadays,ofcourse,the pragmatic alloutof such a study, n the form of "microtonaluniverses," s readily available to composers, analysts, and listeners.From hisviewpoint,theresearchpresented nthispaperreflectsthedeepinfluence of GeraldBalzano's classic study (Balzano 1980). Like Bal-zano,my motivationsare notonly compositional.Theystemas well froman intuition,perhapsa credo,thatinsightsinto thepropertiesand behav-ior of individual nstancesare furnishedby studyingthe propertiesandbehaviorsof generalphenomenawhichtheyrepresent.As inhabitants fa planetthatsustains ife, the value of exploringotherplanets,solarsys-tems, or galaxies for theirlife-sustainingproperties,or lack thereof,po-tentially transcends he conceivable materialbenefits,extendingto theself-knowledgethatemerges from the differentiating ontextfurnishedby the Other.19We begin with some notations,definitions,and observations nvokedthroughout herest of the paper:3.1.1. Notation of Compound Operations. A compoundPLR-familyoperation s denotedas an orderedset of individualPLR-familyopera-tions, enclosed in angled brackets.The operationsapply in the order inwhichtheyappear n the set, fromleft to right.Forexample,in the com-poundoperation,R is applied,P is appliedto the productof R,andL is appliedto the productof R-then-P.3.1.2. T/I Equivalences of Compound Operations. All compoundoperationsareequivalent o eithertranspositionsor inversions,depend-ing on the cardinalityof the orderedset. Compoundoperationsof oddcardinalityare inversions,those of even cardinality ranspositions.Thisfollows from the inversionalstatus of PLR-familyoperations,together

23



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with the groupstructureof inversionand transposition see, e.g., Rahn1980, 52).3.1.3. Generators. A PLR-familycompoundis generated if it canbe partitioned nto two or more identical orderedsubsets.The orderedsubset, singly iterated,constitutes the generator of the compound.Thecompoundcanbe expressedas Hn, where H is thegeneratorandn countsits iterations n thecompound.A generatorof cardinality#H is classifiedas #H-nary(hence binary, ternary, etc.). For example, the compound s binary-generated, ince it can be partitionedas ,andexpressedas 3.3.1.4. T/I Equivalences of Generators. Thecardinality f a generatedcompoundHn is #H - n. If either#H or n are even , thenHn is a transpo-sition. In orderfor Hn to be an inversion,#H and n mustboth be odd.This follows from the observationsmadein 3.1.2 togetherwith the mul-tiplicativepropertiesof odd and even integers.3.1.5. Cycles. H generatesa cycle if, operatingon some trichordQ,there is some integer q* > 0 such thatHq* Q) = Q. The smallest suchvalue q* is the operational periodicity of H. It will also be useful onoccasion to count the trichordsthat result from an H-generatedcycle.That number, he trichordal periodicity of the cycle, is equivalentto#H -q*.

3.1.6. Tonnetz Representations of Cycles. Generatorsof odd cardi-nalityareinvolutions: heyretreat o theirpointof originon the Tonnetzaftertwo iterations.This is restated ormallyas Theorem2 in the appen-dix, wherea proofis offered.Generators f evencardinality,by contrast,aredevolutions: heymoveperpetually wayfrom theirpointof originonthe Tonnetz.Cycles aregeneratedonly whena modularcongruencegov-erns the Tonnetz,n whichcase thegeneratorhas the sameperiodicityasthe transpositional peration o which it is equivalent(cf. 3.1.4). Theseperiodicitiescan be computedby firstexpressingthe PLR-familygener-atoras a transposition perationT,, and thendetermining heperiodicityof n in relation o the size of the chromaticsystem.For this second step,we will rely on the workcarriedout in Section 2.4 and summarized nFigure10.3.2. Binary Generators

Because the unarygenerators,

, and areinvolutions, heprogressions hatthey generateareinsufficientlyvariedto serveas com-pelling musicalresources.Thus ourexplorationbeginswith binarygen-erators hatpairdistinctPLR-familyoperations.Thereare six such gen-erators,which group nto threeretrograde-relatedairs:

(1) and; (2) and; (3) and.24



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Ithasalreadybeendetermined cf. 3.1.2) thateachbinaryoperation sequivalent o some T,. Eachtranspositional aluen associatedwitha bi-naryoperation s equivalent o a non-zerodirected ntervalwithinthetri-chordthatis the objectof operation.To see why this is so, considerthateach individualPLR-familyoperationaltersone pitch-class n Q, andsoeach binary operationalters two pitch-classes in Q. It follows thattheproductof a binaryoperationsharesat least one commonpc withQ. Thecommon-tone heorem ortranspositionRahn1980, 108)dictatesthat fQr~T,(Q)> 0, then there s some {q1,q2) E Qsuch thatq2- q1= n. Fromthis it follows that each binaryPLR-family operationtransposesQ bysomeinterval nternal o it. Thetranspositional aluesassociatedwiththesix binary operationsare exactly the three step-intervalsand their in-verses,?x, ?y, and?(x + y).Figure 11 (p. 26) matches directed ntervals o theirassociatedbinaryoperations.The six directedintervals of Q are listed as transpositionalvalues at (a). Theirbinaryoperationequivalentsare shown at (b). Thetranspositional alues areimplementedon Qat(c). Thebinaryoperationsareimplementedat (d), wherethey arerepresentedon the abstractTon-netzas arrows eadingoutof Q. Eachoperation ransposesQby oneposi-tion alongthe axis representing he step-intervalwith which it is associ-ated.(Forexample,theassociationbetweenTxand is confirmedbythe matrixposition of (Q), one step rightwardof Q along the xaxis.) Consequently,the vertices of each resultanttriangle at (d) areexactly the pitch-classes resultingfrom the associated transpositionat(c). (Forexample, {x, 2x, 2x + y} appearsboth as the set of verticesof(Q)at (d) and as the result of Tx(Q) at (c).)Note that inversely related step-intervalsare associated with retro-grade-related perations.Forexample, the association of Tx with is complementedby an association of Tx with .Moregenerally:

THEOREM . Foran orderedPLR-familyoperation etHand ts retro-grade Ret(H), if H = T,, then Ret(H) = T,.A proofis given in theAppendix.Moving one step forward nto the "middleground," e now examinethese relations as they apply to the abstractparsimonioustrichordalprimeformQ' = {0, x, 2x + 1 , withdirected ntervals?x, ?(x + 1), and+(x + 2). Ingeneral,these intervalsrepresent ix distinctvalues,withthe

lone exception that, if x = 1 and c = 6, then x + 2 = - (x + 2). This excep-tion aside, the six binaryPLR-familyoperationsproducesix distinct tri-chordswhenimplementedon Q'.Figure 12 (p. 27) translatesFigure 11 into termsspecific to parsimo-nioustrichords.Thetranspositions t(a) aregiven in threeforms:as pos-itive and negative generic step-intervals,as positive and negative step-25



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(a) (b) (c)Tx {O,x,x+ y} -{-y,x-Y,x)

Tx+y {O,x, x + y} {x + y, 2x + y, 2x + 2y)T x-y {0, x, x + y


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(a) (b) (c)Tx {10, , 2x + )-- x, 2x, 3x + I)


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x+4 2x + 4 1 x + 1 2x+ 1

3 x + 3 2x + 3 0 x

x+4 2x+4 I x + I -- 2x + I 3 x+3 2x + 3 0 - x

Figure 13:BinaryChainson the Parsimonio



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intervalsof the generic parsimonious richordQ', and as positivevaluesmodulo 3x + 3. It is these lattervalues thatgenerate he mappingsat (c).Figure 12(d) transfers he label/arrownetworkfrom Figure 11(d) ontotheparsimoniousTonnetzcf. Figure8). As withFigure11,a comparisonof the trianglevertices at (d) with the resultsof the transpositionalmap-pings at (c) confirmsthe associationsof binaryPLR-familyoperation opc-transposition ssertedat (a) and(b).3.3. Binary Chains and Cycles

Having explored the transpositionalbehavior of binary operationssingly iterated,we now studythe chains generated hrough heir recur-sive application.Figure13appliesthe six generators o Q'= {0, x, 2x+1),as represented n theparsimoniousTonnetz f Figures8 and12.Pursuantto Theorem3, retrograde-relatedeneratorsproceedinverselyout of Q'.Consequently,any generatorand its retrograde ombine to forma singlechain. For the sake of procedural conomy,it will be useful to provideaunified label for each chain:invoking alphabeticalprecedence,the threebinary-generatedhains will be referred o as ,,and .Each of the three chains representedon Figure 13 threadsa spaceboundedby two parallelandadjacentaxes. The chain,whose gen-erator s associated with the x interval, hreadsa pairof x axes andthusproceeds horizontally; he chain, associated with the y interval,threadsapairof y axes and thusproceedsvertically;andthechain,associated with the x + y = z interval,threadsa pairof z axes andthusproceedsdiagonally.These affiliationsbetweenbinarychains,stepinter-vals, andTonnetzdirectionsareconstant o all realizationsof thegenericTonnetz.Inwhatsense arethe chainsof Figure13cyclic?Withthechain,thecyclicity is readilyapparent: oth3,at thetopof thefigure,and3,at its bottom,areequivalent o Q', indicating hattheoperationalperiodicityof thechain is 3. (Note thatthisperiodicity s proper othe parsimoniousTonnetz;werethe chainstracedon the genericTonnetzof Figures6 and 11, no suchcyclicity would be evident.)Withthe othertwo chains, no such equivalencesare apparent,and so these chains arenot cyclic in the abstract.Once a specific integervalue is assignedto x,however, he and chainsbecome cyclic. Forexample,if x =3, then(Q)= {3, x + 4, 2x + 4} at the figure'sleft edge isequivalent o (Q)= {x, 2x + 1, 3x + 1 just rightof center:both tri-chords are equivalent o {3, 7, 10). This indicates thatthe chainhas anoperationalperiodicityof 4. This periodicitydoes not hold, how-ever, if x is assigneda differentvalue. It is proper o the Oettingen/Rie-mannTonnetz,butnot to the parsimoniousTonnetz f Figure 13, where,

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(a) (b) (c) (d)operational chain-class trichordal pitch-classModuloc periodicity cardinality periodicity cardinalityper co-cycle

c 6

c - 3 c 3 2c 2cmodulo 3 3 3c0, 6 c 1 2c cmodulo 9c0, 3 c 1 2c cmodulo9c-6 c 3 2c 2cmodulo9 3 3 3

Figure14:PeriodicitiesndCardinalitiesf Binary-Generatedyclesunlike , neither the nor the chain exhibits constantperiodicity.Althoughvariable, heoperationalperiodicitiesof andforspecific values of x can nonetheless be easily determined.To do so, weonly need coaston themomentumof ourlaborfrom Section 2.4. Becausebinary operationsare equivalentto step-interval ranspositions,as dis-cussed in Section3.2, the two necessarilyhaveidenticalperiodicities.Wehavealreadyseen this for thecase of the chain,whose operationalperiodicityof 3 derives fromthe tripleperiodicityof its associatedstep-intervaly (cf. Figure10(c),p. 22). Theperiodicities or and likewise transferfrom that of their associated step-intervals,x and z,respectively.Figure10(b)gives the value of x*, theperiodicityof x mod-ulo c, and these values apply directly to the periodicityof : if c- {0,6 } modulo9, then the operationalperiodicityof is c; if c -

3modulo 9, the periodicityis -. Figure 10(d) gives the value of z*, theperiodicityof z moduloc, andthese values apply directlyto the period-icity of : if c = {0,3) modulo9, then the periodicityof is c;if c -6 modulo9, the periodicity s C. These periodicitiesaresumma-rizedin column(a) of Figure 14.30



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Othersignificantfeatures of binary-generated ycles are directly in-ferrable rom the foregoing.Cardinality of cycle-classes. When one of the chains on Figure 13(p. 28) is transposed,axes and thus trichordsmaybe exchanged,buttheclass membership f the chain(,,or)is preserved. t isthusimportanto make anontologicaldistinctionbetweenspecificcyclesand the classes thatthey represent.This distinction eads us to askhowmanydistinctcycles belong to each class, i.e. whatis the cardinalityofthatclass. To answer hisquestion,we begin by observing hateachcycleengages two pitch-classaxes, andeach axis participatesn two cycles. Itfollows thatthecardinalityof eachbinary-generated ycle-class is equiv-alent to the quantityof distinctpitch-classaxes of thatgenerator's tep-intervalssociate, sgiven nFigure10(p.22).Thisquantityquals-,qwherec measures he chromaticsystemas in Def. (1), and q* is theperi-odicity of the binarygeneratoras at Figure 14, column (a). This infor-mationis summarizedn Figure 14, column(b).Trichordal periodicity. The trichordalperiodicity of a generatedcycle equalstheproductof operationalperiodicity imesgenerator ardi-nality (cf. 3.1.5). In the case of binarygenerators, richordalperiodicitydoubles operationalperiodicity.All cycles thus engage six tri-chords,butthe trichordalperiodicity s variable or the and cycles. This information s summarized n Figure14,column(c) foreachcycle-class.Pitch-class engagement. ConsiderabinaryoperatorHassociatedwithstep-intervalq, whereq* represents he periodicityof q andhence of H.

*If q and c areco-prime,q* = c (cf. 2.4). Thenbothaxes threadedbyanH-generated haininclude theentirepc-aggregate. t follows thatan H-generatedchain engages all c pitch-classesof the chromaticsystem.* If, however,q and c share a common divisor greaterthan 1, thenq* < c, and thec pitch-classespartition ntoco-cycles (cf. 2.4). Thetwo q axes threadedby an H-generatedchain thushavedistinctpc-content. Since each of these axes has a periodicityof q*, it followsthat anH-generated hainengages2q* pitch-classes.Thus an cycle always engages a hexachord.(The primeform of this hexa-chord is {0, 1, x + 1, x + 2, 2x + 2, 2x + 3), with step-intervals.) Dependingon themodulo9 congruenceof c, theandcycles eitherengage the entirepitch-classsystem,or of that system. (In the lattercase, the primeform of the setengagedby the cycle can be representedas {


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(a) (b) (c) Tx T3 {0, 3, 7)-----} {3, 6, 10) C--8,11, 3} C ------8 Ab -

T x+y T7 {0, 3, 7 ----- {7, 10, 21 C------- G- T -x-y T5 {0, 3, 7)- f{5, 8, 0) C- ----- F-

(d)

1 4 10

(Q') C-Minor (Q')= A-Minor i = Eb-Minor9 0O3 6

A6:(Q') (Q')= F-Minor = Ab-Minor5 8 11 2

Figure 15:BinaryGeneratorsandConsonantTriadsmodulo 12



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In general,the smallerthe periodicityof the step intervalassociatedwith an operation, he more economical is the pitch-classdesign of thecycle generatedfrom that operation.The significance of this circum-stancefor nineteenth-century armonywill become clearin Section3.4.3.4 Binary Chains and Cycles in Modulo 12

In this section, the propertiesof parsimonious richordsestablishedabove are studied as they applyto the objectof historicalandanalyticalinterest, he consonant riad n a systemof 12pitch-classes.Webegin bystudyingthe six binaryoperations,singly iterated,as we did in associa-tion withFigure12(p. 27). Figure15providesa realizationof Figure12,with x = 3. This assignment mmediatelybringsthe familiarobjectandsystem into view. Q' = {0, x, 2x + 1} is realized as the consonanttriad{0, 3, 7} = C minor,the step intervals as , andthe chromatic ystem3x + 3 as the 12-pc system.TheparsimoniousTon-netzbecomes an equal-tempered ersion of the one discoveredby Eulerand Oettingenand propagatedby Riemann(cf. Figure 9a). The triadsproducedby the six binary operations ndicatedon Figure 15 constitutea complete inventoryof the minortriads that share a pitch-classwith Cminor.

To explore the recursive propertiesof these operations, Figure 16(p. 34) transfers he threebinary-generatedhains of Figure 13 onto theOettingen/RiemannTonnetz. Thesechains are somewhatre-positioned,in partfor visual clarity.)Each of these threeclasses of triadicprogres-sion is familiarfrommusic of the nineteenthcentury,and each has dis-tinctiveproperties.The groupstructure f chainshas been studiedby Hyer(1989,1995), and some of their special qualitieswere exploredfromdifferentperspectives n an earlierpaperof mine (Cohn 1996, where I refer to achain of this type as a "maximallysmoothcycle") and in Lewin (1996,who classifies it as a "CohnCycle").The particular chain thread-ing the vertical axes of Figure 16 is the one thatis traversed n a down-warddirection n a passagefromBrahms'sDoubleConcerto hat s mod-elled in Example 1 andthatI havealreadystudied n some detail (Cohn1996, 13-15). The matrixrepresentation llustratesseveral significantfeaturesof cycles identified n my earlierpaper:

Set-class Non-Exhaustion. generatesa cyclic progressionof 6triads,a sub-groupof the 24 triads.Limited pc engagement. Both engaged columns have a cyclic peri-odicityof 3, so thatan cycle engagesonly 6 pitch-classes.Thepc-set unites two adjacentIC-4 cycles (representedby the two y-axis columns)into a hexatonic set belongingto Forte-class6-20.2033



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1 4 .........-------7 10 1 4 7

9 0 3Brahms 6 9 0 3Ab+5 8 11 2 5 8 111 4 7 10 4 79 0 3 6 9 0 3

A+ Ab+5 8 11 2 5 8 11

Bb-1 ---------4 ---- 7 --------10 1------- 4--------7.10b+ CES Eb- F#-,A------ 0 6---------------0 3B+Schubert

5 8 11 2 5 8 111 4 -------7 10 1 4 79 0 --------" 3 6 9 0 3

Figure 16:BinaryChains on the Oettingen Rie



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tI I... , I I IExample1:Brahms,Concerto or ViolinandCello,Op. 102,FirstMovement,mm.270-78

oi mII

O0 ) OExample2: Schubert,Overture o Die Zauberharfe,peningAndante

Multiple distinct cycles. The numberof distinctcycles is equiv-alent to the numberof distincty axes: four. In my earlierpaper,Ireferto these fourcycles as hexatonic systems, in reference o theirlimited pc-content.In the currentcontext they are more appropri-ately referred o as sub-systems.

The particular cycle threading he horizontalaxes of Figure16is the one that s traversed ight-wardinreference o thespatial ayoutofFigure 16) in Example2, which models theAndante ntroductionof theovertureto Schubert'soperaDie Zauberharfe 1820).21 The three fea-turesproper o cycles areanalogousto those identifiedabove for cycles, and this analogy is reflectedin the paralleldescriptionofthembelow:Set-class Non-Exhaustion. generatesa cyclic progressionof 8triads,a sub-groupof the 24 triads.Limited pc engagement. Both engagedrows have a cyclic periodic-ity of 4, so thata cycle engages 8 pitch-classes.The pc-setunitestwo adjacent C-3cycles (represented y the two x-axis rows)intoan octatonic set belongingto Forte-class8-28.Multiple distinct cycles. The number of distinct cycles isequivalent o the numberof distinctx-axes: three.These cycles canbe referred o as octatonic sub-systems (cf. Lerdahl1994, 132-33).Any cycle-segmentis a (properor improper) ubset of one ofthe threecycles.

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The situationwith the cycle, whichthreads hediagonalaxes ofFigure16, is entirelydifferent.Thethreefeaturesnoted for the andcycles have no analoguehere.Set-class Exhaustion. The compoundoperation generatestheentiregroupof 24 triads.PC-Aggregate Completion. Both engaged z-axis diagonals have acyclic periodicityof c = 12,andan chainengagesall 12pitch-classes;One Single Cycle. There is a single cycle, of whichanycycle-segmentis a subset.The completecycle is too long to sustaincompositional nterestunder normalconditions. This is broughtout by a thirty-four-measurepassagewhich I have writtenabouton two earlieroccasions(Cohn 1991and 1992) from the second movementof Beethoven'sNinth Symphony(mm. 143-76) . From a startingpointat C-major n the northeast ornerof Figure 16, the passageobsessively whirlsthrough9, raversingeighteentriadsbeforehaltingat the nineteenth ink, theA-majortriad nthe southwestcorner.Evena squallof suchvelocityand force cannotpro-pel itself about the entirecycle.Nonetheless, hecycle has amorevenerable egacythan he otherbinary-generated ycles. Singly iterated, transposesby perfectfifth, the transpositionalvalue that preservesmaximumpc-intersectionbetween diatoniccollections (as well as the interval hatsportsacousticprivilege). The -cycle was initially recognized in thorough-bassmethodsfrom the late seventeenthcenturyas a gaugeof modulatorydis-tance(Lester1992,215). In 1827,JohannBernhardLogierrecommendedthatyoung musicians earn to play the entirecycle at the pianoforte,"asit forms the groundwork n which maybe constructedan almostinfinitenumberof passages and variations."Logier's assessment of the cycle'sproperties s pertinent o the approachadoptedhere:Weperceive, romthebeginning o theend,notonlya beautiful ymme-tryandregularity ervadinghewhole,butalso a double unionof inter-vals-two of them always remainingundisturbed....he whole pro-gressionforminga chain of harmonyunequalledn any of ourformerexercises.22

As these passages suggest, the entirecycle is less a model of a surfacestructure o be traversed n a single gesturethan a compositionalspacewhich, like a city, becomesentirelyknownthroughan exertionof mem-ory across a set of partialexplorations.In a more recent formulationdirectlyrelated to the approachdeveloped in this paper,Lewin (1987,180) notes that the 24 triadsaregenerableby a single operation,MED,36



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(a) (b) (c) (d)c = 12 operational chain-class trichordal pitch-classperiodicity cardinality periodicity cardinality

3 modulo-9 perco-cycle 3 4 6 6

4 3 8 8

12 1 24 12Figure17:Binary-GeneratederiodicitiesndCardinalities odulo12

which mapsa triad o itsdiatonic submediant.Thepowersof MEDorga-nize the 24 triads into a simply transitivegroup,a GeneralizedIntervalSystem isomorphicwith the cycle. In this sense, the cyclecan be seen as a source for all triadicprogressions.The andcycles emerge, among other routines, as patternedsub-groupsof thecycle.23Figure 17, modelledafterFigure 14, summarizes hepropertiesof thebinary-generated ycles modulo 12. Surveyingthe three cycles as anensemble, whatis most striking s the affinitybetweenthe (hexa-tonic) and (octatonic)cycles, whose periodicitesandcardinalitiesarenearlyidentical,and the anomalousstatusof the cycle by thesame standard.These affiliationsare unexpected n the generalcontextprovidedby Section3.3, whichemphasized heaffinitiesof the and cycles, on the basis of theirvariableperiodicities,andthe anom-alousstatusof the cycle, on the basis of its uniquelyconstantperi-odicity.The sectionthat now follows will suggestthat the strongaffinitybetween the and cycles inmodulo12is accidentalrather haninherent.Inno otherchromaticsystemis the affinityso strong.

3.5. Hexatonic and Octatonic AnaloguesWebegin by investigating herelativelysimplecase of cycles insystems otherthanmodulo 12. Ineach pitch-classsystemthathostspar-simonioustrichords,cycles have a constantoperationalperiodicityof 3, trichordalperiodicityof 6, andpitch-classcardinalityof 6 (cf. Fig-ure 14, p. 30). Thus cycles are inherentlyhexatonic.The variable

37



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II I I I/06

G - Thirdones Dritteltdne)

G# .- .--E- . G GE.

C . E F

H1HC- CC

GGGOGOA ' , FGCk. D G , C ,E GFGO..

C CE,C,

E, GO"C#", F+, A+,H SDG+, B+ D, F , AB+ BD . .' .AAOF A...AB+ DA . . F.. AG+A

,E.GOAGB F

H1H

-Dl D rG+ D$ FO " ". FOA$ " G+kB+k Al#B+k

Figure 18: The Hyper-Hexatonic-Analogue ystem f



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propertys thecardinalityof the cycle-class, i.e. the numberof dis-tinct hexatonicsub-systems.The 2c parsimonious richords n a systemof c pitch-classespartition nto c= cfco-cycles. This yields four6 3hexatonicsub-systemsfor c = 12, six sub-systemsfor c = 18, eight sub-systemsfor c = 24, and so forth.In my earlierpaper,which studied the relationshipamong the fourhexatonicsub-systemsin the mod-12 system, I observed(Cohn 1996,23-25) thateachsub-systemsharespitch-classeswith two othersub-sys-tems,andis pitch-classcomplementaryo theremainingsub-system.Onthisbasis,the ensembleof sub-systems s shaped nto a four-element ys-tem at a higherlevel. I proposeda two-tierdesign, where both the six-element sub-systemsandthe four-element"hyper-system"hatembedsthem areGeneralizedIntervalSystems(GIS) (Lewin 1987). Our currentwork suggests that the same two-tiered GIS design applies to cycles in all chromaticsystems where they occur. The individualsub-systems,whichI shall call hexatonic analogues, constituteGIS's whoseelements are the six trichordsof an cycle, andthus whose size isconstant to all chromatic systems. The higher-level hyper-hexatonicanalogue system, organizedby pc-intersectionamongthe sub-systems,constitutesa GIS whose size of - sub-systemsvarieswiththechromaticcardinality.Figure 18 demonstrates he hyper-hexatonic-analogueystem for c =18, using Haiba'sDritteltonsystempitch-class designations, which areinventoriedat the top of the figure.(Haibauses +for 1/6 sharp,$for 1/3sharp,# for 2/3 sharp, and # for 5/6 sharp.)The 18 pitch-classes arearrangednto 6 T6-cyclesat the center of the figure.Each suchcycle tri-sects the octave, and is thus acousticallyequivalent o a temperedaug-mentedtriadmodulo 12.Thecycles arepaired ntosix overlappingovalswhich portray he six hexatonic-analoguepc sets. Each set is connectedby an arrow o the hexatonicanalogue sub-systemof trichords or whichit furnishesa pitch-classsource.Eachsuch sub-system s ancyclemodulo 18, and each is connecteddirectly,by shared"augmentedriad,"to two neighboring ystems.A uniquefeatureof thischromatic ystemisthat the hexatonicsub-systemsandthe hyper-systemshare a cardinalityof 6. This isomorphism,which arises from the equalityof the variablecto the constant6 when c = 18, presentssome interestingcompositionalpossibilities which I shall not explorein the current ontext.

The (octatonic)sub-systemsof modulo 12generalizein a quitedifferentway.Whereasproper sub-systemsappear n all suitablysized chromaticsystems,propersub-systemsappearonly in thosesystemswherec = 3 modulo 9. Even in thesesystems,thesub-sys-tems generalizedifferently han the . Wehave seen that sub-systems havea constantsize but a variablequantity(cf. Figure 14).The39



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ECq E G GEI,7CE G\

F C C# B E4G//[ / CI

D E C CGGABF# D4A D qD / G q D# E

- - G - I GE G GF F G#A# B4 A B

E G#

F G B GC#\ / DF C A.4 B Dj G

Fq \C q B/ DDF AlAA Dq/F4 D, --D_

D Ft\F Fjd

Figure19:TheHyper-Octatonic-AnalogueSystemfor c = 24



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situationwith the sub-systemsis converse.The quantityof sub-systems is constant: he 2c parsimonious richordsof each systemparti-tion into three sub-systems.The size of each sub-system svariable,engaging -~trichords.Consequently,generalized sub-

systemsare not"octatonic"n the same sense thatgeneralizedsub-systemsare hexatonic: heyhave apc-cardinality f 8 only in case c = 12.The most salient propertyof the mod-12 octatonic that generalizes tomoduloc is thestep-interval attern. Inthis sense theterm"l:2-analogue" basedon Lendvai1971) wouldbetteraccommodate hebasis of the analogythan "octatonicanalogue." nonetheless retain thelatter erm n order o stressthepositive aspectsof themeta-analogywith"hexatonicanalogue."Although, as we have seen, the cycle yields no propersub-systems modulo 12, it does yield proper sub-systemsin chromaticsys-tems congruent o 6 modulo9, such as c = 15, 24, and so forth.In thesecases, the sub-systemsarestructuredn thesameway as the sub-systems, n the sensethatthereare threesub-systems,each witha tri-chordalperiodicityand pitch-class cardinalityof - . Because of thissimilarstructuring,t makes sense to applythe term octatonic analogueto the as well as the sub-systems.Figure 19 demonstrates he hyper-octatonic-analogueystem for c =24, using Haiba'sVierteltonsystem itch-classdesignations,inventoriedin ascendingorderatthe topof the figure.The 24 pitch-classesareparti-tioned into three T3-cyclesof cardinality8 at the center of the figure,which arepairedby the ovals intooctatonic-analogue ollections of car-dinality 16. These collections furnishthe pc content for the octatonic-analoguesub-systemsof triads,equivalent o the -cycles, modulo24.c = 24, then, is an example of a system whose 2c parsimonious ri-chords partitioninto both hexatonic- and octatonic-analoguesub-sys-tems. The two sub-system classes are equivalentto two of the threebinary-generatedycles, and(in thiscase) .Thethirdbinary-generatedcycle, in this case, generatesthe entireset-class of 48parsimonious richords,andthusits rolein thec = 24 systemis analogousto that described n Section3.4 forthe -cyclein thec = 12 system.Figure20 (p. 42), which is cut fromthe sametemplateas Figures 14and 17, summarizes he periodicitiesfor a mod-24 quarter-tone ystem.

The affinitybetweenthe and cycles of modulo 12 hasbeenreplacedby anaffinitybetween and.Bothfeatureattenuatedperiodicitiesandnon-exhaustionof thetrichord lass andthepitch-classaggregate.But the degreeof affinityis greatlyweakened,as the period-icities andcardinalitiesdiverge.Thiscircumstanceunderlines hespecialnatureof modulo 12suggestedattheend of Section 3.4. In no otherchro-41



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(a) (b) (c) (d)c = 24 operational chain-class trichordal pitch-classperiodicity cardinality periodicity cardinality6 modulo-9 perco-cycle

3 8 6 6

24 1 48 24

8 3 16 16

Figure 0:Binary-GeneratederiodicitiesndCardinalities odulo 4matic system does the variableoperationalperiodicityf so closely ap-proximate he constantperiodicity3. (Bothc = 9 and c = 6 aredeficient:c = 9 producesno octatonic-analogue ub-systemsbecause9 is congru-ent to 0 modulo9; althoughc = 6 producesoctatonicanaloguesub-sys-tems under, its two (!) hexatonicsub-systemsare not pitch-classdistinct.)I imaginethe hexatonicandoctatonicsub-systemsas objectsin space,one fixed andone transient.As the transientobject momentarilypassesby the fixedone, theirrelationshipbecomesrecorded, rozenin time likethe objects in a photograph.An observerwho knows the objects onlythrough he recording s in no positionto understand hattheir associa-tion is anythingotherthanpermanent nd intrinsic.Ourstudyof thisphe-nomenonin the contextof generalizedchromaticsystemsunfreezes themoment,allowingus to see the accidentalnatureof therelationship,andtherebyenhancesourappreciation f the over-determined ualitiesof theinteger12.This concludes the inventoryof binaryoperationsand the trichordalprogressions hattheygenerate.Wenowturnour attention o ternary en-erators,which are of a differentnaturebecausetheyare inversionsratherthantranspositions.3.6. Ternary Generators and LPR Loops

We begin ourstudyof ternarygeneratorsby demonstratinghat,oncecertainrulesof reductionareinvoked, heirnumbersareconstrained.Thereductionprocessis facilitatedby the conceptof generator form. Gen-42



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erators that have identical order-positionequivalences will be said toshare an abstract orm. Forms are designatedby assigning the abstracttokens A, B, and C to the three distinct PLR-familyoperationsin theorder that they appear in the generator.Accordingly andarebothof the form.Because two successive iterationsof a single operation"undo"eachother,ternarygeneratorsof the form reduceto unarygenerators( = ), and accordinglyhave no independent nterest.Nei-ther do generatorsof the form , since reiterationuxtaposesthetwo A's, triggeringa chain of involutionsthat unravelsthe entirestruc-ture: 2 = = = = To.Conse-quently, ndependent ernarygeneratorsarelimited to the form.The six realizationsof this form correspond o the six orderingsof theunordered et {L,P,R . These six orderingsare equivalentunderretro-gradeand rotation.We will view them as members of a single class ofternarychain whose canonicallabel, invoking alphabeticprecedence, s.Toillustrate he involutionalnatureof ternarygenerators cf. Theorem2), Figure21 (p. 44) models a cycle of six triadson the Oettingen/Rie-mannTonnetz.SelectingF minor as a startingpoint,we move clockwisethrougha semi- cycle, engagingR, P,andL in thatorder,andstoppingatE major.Continuing lockwise fromthispoint,thesamethreeoperationsareengaged in the same fixed order,closing the cycle back to F minor.Thecompoundoperationrepresents2.Regardlessof starting riador direction, he sameset of triads s traversed,demonstratinghe equiv-alence of the six orderingsof {L, P, R}. I will refer to such structuresgenericallyas LPR loops.Twoexamplesof LPR oopsfromnineteenth-centurypera,bothusingthespecifictriads ncluded n Figure21, arepresented n Examples3 and4 (p.45). Example3 models the successionof triads n thecantabilesec-tion of "Ahsi, ben mio,"fromAct IIIof Verdi'sII Trovatore.Beginningin F minor,the firstquatrain loses by tonicizing its relativemajor.Thearia'ssecondquatrain"mutates"oAbminor,prolongsFbmajor hrough-out its consequentphrase,repeatsthe text of the consequentphraseovera Dbminortriad,andreachesa fermataover a V7 of Db.The finalqua-trainof the cantabile then prolongsDb major.The six consonanttriadstraversean LPRloop, althoughthe loop does not close with a return othe initial F minor.

Example4 is the EngelmotivfromWagner'sParsifal, in a transposi-tion thatoccurs in Amfortas'sPrayerfromAct III.The passage beginsandends on Db major,androtatesclockwise through hetriadsof Figure21, omittingAbminorandDb minor.As the analysisbeneathExample4indicates,the omissions areaccountedfor by compoundoperations hat"elideacross" he omittedtriads.2443



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10 1 4 7 10

R P6 9 0 3 6

Ab+F - A -

2 L 5 8 11 2D b + F6 +

10 r 4 7 10

6 9 0 3 6

Figure 1:LPR-looproundAb/G#A significant eatureof LPRloops is thattheirsix triadssharea singlepitch-class,located at the center of the matrixrepresentation. n Figure

21, Ab/G#plays thatrole. Themembersof a loop furnisha completeros-ter of the triads hatinclude the sharedpitch-class.25t follows that therearetwelve distinctLPRloops (one per pc), thateach triadparticipatesnthreesuchloops, and thatthe threeinterlocking oops in which thattriadparticipates urnish a complete roster of the triads with which it sharesone or morepitch-classes.Figure22 (p. 46) illustrates or the case of Fminor.The figureis in the formof an interlockedset of "honeycombs."The circle at the centerof each loop encloses the pitch-classcommon toall the triadsin thatloop. The entire structureof the twelve interlockedloops can be easily inferredby projectingoutward romFigure22.26Becauseof their common-toneproperties,LPRloops furnishan idealprogressionfor consonantlysupportinga single melodic pitch with di-verse harmonywhile maximizing voice-leading parsimony.Such pro-gressions, with their implicationof inner action or turmoil beneath a44



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Ah! si ben mio... Mapur,se... ...le vittime.., fraquegli... in ciel precederti......avrbpiuiorte ... resti fra le vittime... ...trafitto... ...pensierverrb...

Ii Ii vI v I1

jetzt in g6tt - li-chem Glanz den Er - 16 - ser

(La.L>


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10 1 4 7 10C +

A- C-

10 1 4 7 10

6 9 0 3 63 6

Figure 2:InterlockingPR-Loops

3.7 Quaternary GeneratorsLiketernarygenerators,quaternary enerators educeto a single formonce operations hatinvolute(eitherdirectlyor across the boundariesofsuccessive iterations) are eliminated and rotational equivalence isinvoked. Forms with two sets of duplicate operationsare eliminatedeitherbecause the duplicatesarejuxtaposedand thus undo each other( = = Toand = To),orbecause the gener-ator is itself binary-generated = 2).Thus, all indepen-dent quaternary enerators nclude all threedistinct operations,with asingle operationrepresented wice. The two iterationsof the duplicateoperationcan be neitheradjacent as in ) normaximally sep-arated as in ,whosereiteration auses),and

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(a) (b) (c) (d) (e) (f)L T T-(x+y) T(2y+x) Ti

P TyT-x Ty x TpR Tx+yTx T2x+y Tr

Figure 3:Transpositionalquivalencesf Quaternaryeneratorsaccordinglymust be separatedby two orderpositions. All quaternarygeneratorshatfit thisdescriptionareequivalent o viarotation.The six generators hatrealizegroup ntothreerotation-relatedpairs,as follows:

(1)nd;2)nd;3)nd.A significantpropertyof anyquaternary enerator s that the transpo-sition that it enacts is equivalentto the voice-leading intervalproducedby the movingvoice when a single iterationof theduplicateoperation sexecuted.Forexample,an operationwith duplicateR, such as ,is equivalent o T4 Thus a quaternary operation unfolds in the largethe transpositional operation that its duplicate operation expressesin the small.Figure23 demonstrates his "exfoliation"property.The threeopera-tions,listed at(a),arerepresented s duplicatesby aquaternary perationat (b). These areeach partitioned nto a pairof binaryoperationsat (c),and convertedto t

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