Yale University Department of Music
Neo-Riemannian Operations, Parsimonious Trichords, and Their "Tonnetz" Representations Author(s): Richard Cohn Source: Journal of Music Theory, Vol. 41, No. 1 (Spring, 1997), pp. 1-66 Published by: Duke University Press on behalf of the Yale University Department of Music Stable URL: http://www.jstor.org/stable/843761 Accessed: 24/11/2008 03:08
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http://www.jstor.org NEO-RIEMANNIAN OPERATIONS,
PARSIMONIOUS TRICHORDS, AND
THEIR TONNETZ REPRESENTATIONS
RichardCohn
1. The Over-Determined Triad
In work publishedin the 1980s, David Lewin proposedto model rela- tions between triads'using operationsadapted from the writings of the turn-of-the-centurytheorist Hugo Riemann.2Subsequent work along neo-Riemannianlines has focused on three operations that maximize pitch-class intersectionbetween pairs of distinct triads:P (for Parallel), which relates triads that share a common fifth; L (for Leading-tone exchange), which relates triadsthat share a common minor third;and R (for Relative),which relatestriads that share a common majorthird.3 Fig- ure 1 illustratesthe threeoperations, which I shall referto collectively as the PLR family, as they act on a C minor triad,mapping it to three dif- ferent major triads. (Throughoutthis paper, + and - are used to denote major and minor triads, respectively.)Singly applied, each PLR-family operationinverts a triad (major-- minor). Doubly applied, each PLR- family operationmaps a triadto its identity;i.e., each is an involution. A striking feature of PLR-family operations is their parsimonious voice-leading.4To a degree, parsimony is inherent to the PLR family, whose definingfeature is double common-toneretention. What is not in- herent is the incrementalmotion of the third voice, which proceeds by
1 p R Qb'. []ll b[] R T 0o |' 11 o0 o0 o0 0 0& 0 006^ .0 b?y C- - Ab+ C- - C+ C- Eb+
Figure 1: The PLR-Familyof C-Minor
?J ee-Oe -*
Figure2: The PLR-Familyof {0, 1, 5} semitone in the case of P and L, and by whole tone in the case of R. This featureis not without significance to the developmentof a musical cul- ture where conjunctvoice-leading in general, and semitonal voice-lead- ing in particular,are enduring norms through an impressive range of chronological eras and musical styles.5 The parsimony of PLR-family voice-leading is so engrainedin the proceduralknowledge of a musician trainedin the Europeanclassical traditionthat it hardlyseems to warrant notice, much less scrutiny.It is scrutinizedhere with the aim of demon- stratingthat, from a certainpoint of view, the featureis fortuitous. In orderto cultivatethis point of view, imagine a musical culturewhere set-class 3-4 (015) was the privilegedharmony to the extent thatthe triad prevails in Europeanmusic c. 1500-1900. For any memberof set-class 3-4, thereare threeother members with which it sharestwo common pcs. Figure 2 leads the prime form of 3-4 to its three double-common-tone relatedpeers. Note the magnitudeof the moving voice: in two cases by minor third, in the third case by tritone. Such a culture would be inca- pable of achieving the degree of voice-leading parsimonycharacteristic of triadicmusic, particularlyas it developed in the nineteenthcentury. It can be easily verified, and will be demonstratedshortly, that replication of the exercise using any other mod-12 trichord-classyields similarly unparsimoniousresults.6 It may come as no surprisethat, among trichord-classes,consonant tri- ads are special. Theirunique acoustic propertiesare well established,and indeed are fundamentalto standardapproaches to triadic music. The potential of consonant triads to engage in parsimoniousvoice-leading, however, is unrelated to those acoustic properties. This potential is, rather,a function of their group-theoreticproperties as equally tempered entities modulo-12. To demonstratethis claim, the definitionof PLR-familyoperations is now generalized, initially to the prime forms of set-classes defined by TI/TnIequivalence, subsequentlyto all trichords.Although our primary
2 attentionwill be focused on trichordsin the usual chromaticsystem of twelve pitch-classes,the definitionis open to applicationto othersystems as well, for reasons that will emerge as the exposition unfolds.
DEE (1). c is a positive integerrepresenting the cardinalityof a chro- matic system.
DEE (2). Q is a mod-c trichord{0, x, x + y) such that 0 < x < y < c - (x + y).
The condition insuresthat Q is the prime form of its trichord-class.
DEF. (3) Iu is the inversion that maps pitch-classes v and u to each other.7
The threePLR-family operations can now be definedon a prime-formtri- chord Q = (0, x, x + y} as follows:
DEF. (4a) P = I+y DEF.(4b) L = Ix DEF.(4c) R = IX+y
Figure 3a (p. 4) demonstratesthe mapping of abstractpitch-classes when P, L, and R act on the abstracttrichordal prime form Q. Figures 3b and 3c realize Q as the two trichordsexplored in Figures 1 and 2. Each operation swaps two of the pitch-classes in Q. The remaining pc is mapped outside of Q, and this mapping is perceived as the "moving voice." We now define a set of variables,p, /, and ,, which express the magnitudeof those externalmappings as mod-c transpositionalvalues.
x- y, hence:h0--c -2x + y, hence: DEE (5a) p = y - x DEE (5b) l = -2y - x DEE (5c) , = 2x + y
Observethat p + I + = 0, since (y - x) + (-2y - x) + (2x + y) = (2x - 2x) + (2y - 2y) = 0. The values of , l, and , are now linked to the structureof trichordQ. First, following Bacon (1917), Chrisman(1971), and others since, we define a step-intervalseries as follows:
DEF (6). The step-interval series for a normal-ordertrichord {i,j,k} is the orderedset , modulo c.
Via Def. (2), Q = {0, x, x+ y} is in prime form, which presupposesnor- mal order.Thus the step-intervalseries of Q is .
3 P(Q) = I +y(Q) L(Q) = I (Q) R(Q) = Ix.y (Q)
x + y -4 x x ->x + y 0 -* 2x + Y
Figure3: PLR-FamilyMappings (a) forQ = {0, x, x + y}
P(Q)= 17(Q) L(Q)= I 3(Q) R(Q)= 17(Q)
7 ->0 7-8 7->3 3 - 4 3 ->0 3-> 7 0- 7 0- 3 0 -10
Figure 3: PLR-FamilyMappings (b) for C-Minor;x = 3, y = 4, Q = {0, 3, 7}
P(Q) = I 5 (Q) L(Q) = I o(Q) R(Q) = I 5 (Q)
5 --->0 5 -> 8 5 -1 1->4 1 -0 1 5 0-->5 0-> 1 0-6
Figure3: PLR-FamilyMappings (c)x= ,y=4,Q ={0, 1,5}
The following theoremstates thatp, I, and, are each equivalentto the differencesbetween a distinct pair of step-intervals.
THEOREM1. For a prime-formtrichord Q = {0, x, x + y with step- intervals , 1.1) p is the difference between the first and second step interval, y - x. Proof: y - x = p via Def. (5a). 1.2) l is the difference between the second and third step 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).
Theorem 1 facilitatescalculation of the values of , l, and4 for any tri- chordal prime form in a chromatic system of any size. The results of
4 these calculationsfor the mod-12 trichordalprime forms are given in Fig- ure 4 (p. 6). Conjunctintervals are enclosed in boxes. The figuredemon- strates what was asserted above: that 037 is unique among trichordal prime forms (modulo 12) in its preservationof conjunct melodic inter- vals when any of the three PLR-familyoperations is executed.8 The generalizationfrom prime form to other class members is easily carriedout. Calculatethe Tnor TnIof the trichordin relationto the prime form of its class, reassign 0 to the pitch-class that had been formerly assigned the integern, and reassignthe integersin ascent or descent from the new pitch-class 0, depending on whether the trichordis Tnor TnI- relatedto the prime form. Then apply the operationsas in definition(4). If Tn-relatedto the prime form, the values of 2, l, and', are the same as when the operationacts on the prime form; if TnI-related,the values are inverted.(This follows from the involutionalnature of PLR-familyoper- ations.) In either case, the magnitudesare invariant.Consequently, the unique characteristicnoted above for trichordalprime form {037} gen- eralizes to all membersof its set-class. To summarizeour findings so far: (1) Among mod-12 trichords,the consonanttriad alone is susceptibleto parsimoniousvoice-leading under the three PLR-familyoperations; (2) This circumstanceis a function of the trichord'sstep-interval sizes, which are an aspect of its internalstruc- ture;(3) the optimal voice-leading propertiesof triadstherefore stand in incidentalrelation to their optimal acoustic properties. In a word:the triadis over-determined. The fortuitous relation of the consonant triad's voice-leading parsi- mony to its acousticgenerability is as profoundto the developmentof the Europeanmusical traditionas othersorts of over-determinationthat were first brought to light at Princeton in the 1960s (Babbitt 1965, Gamer 1967, Boretz 1970): of the chromaticdivision of the octave into twelve parts, 12 being at once the smallest abundantinteger and the smallest integern such that 3 approximatessome power of 2; of the proximityof the perfect fifth's - geometric division of the octave (the source of its I acoustic power) to its arithmetic division of the octave (a fraction whose irreducibility,rare for its divisor, is necessary for the deep-scale propertyof diatonic collections, a circumstancewhich in turn leads to the graded common-tone distributionof the set of diatonic collections undertransposition, and hence to the system of key signatures).Equally remarkableis the extent to which the triad's acoustic propertieshave masked recognition of its group-theoreticpotential.9 Our sensibilities, born of incessant exposure to a musical traditionthat habituallyimple- ments the acousticproperties of triads,as well as to a music-theoretictra- dition that habitually models this habitual implementation,have been trainedto resist by defaultany effort to regardthe triadas anythingother than acoustic in essence. Like the stock figure of the Cold War spy 5 prime form step-intervals p I r (0,x, x+ y) y -x -2y -x 2x +y
(0,1,2) <1,1,10> 0 9 3
(0,1,3) < 1,2,9> FW 7 4
{0,1,4) < 1,3,8> [_15 5
{0,1,5) < 1,4,7> 3 3 6
(0,1,6) < 1,5,6> 4 FiW
(0,2,41 <2,2,8> 0 6 6
10,2,5) <2,3,7> WI 4 7
(0,2,61 <2,4,6> R 8
(0,2,7) <2,5,5> 3 0 9
{0,3,6} <3,3,6> 0 3 9
(0,3,7). <3,4,5> ww1
10,4,81 <4,4,4> 0 0 0
Figure 4: Values of p, 1, and r for trichordalprime forms F C G D
A E H Fs
Cs Gs Ds B
Figure5: Two Tonnetze (a) fromEuler 1926 (1739)
thriller,the dazzling beauty of the triad has blinded us to its substantial intellectualresources.
2.1. The Over-Determined Tonnetz
The two-dimensional matrix known as the table of consonant (or tonal) relations(Verwandtschaftsverhiltnistabelle) or harmonicnetwork (Tonnetz,Tongewebe) has long presentedmusic theorists with a useful graphic instrumentfor representingtriadic progressions.Although the matrix originated in response to the acoustic properties of triads, it respondsin equal measureto their group-theoreticproperties. The over- determinationof the triad is thus encoded in the over-determinationof the Tonnetz. The Tonnetzwas initially conceived to reconcile the first two distinct (non-complementary)sub-octave intervalsgenerated from a resonating body.10Leonhard Euler (1926 [1739]) situatedjustly tunedversions of the twelve pitch classes on a bounded4x3 matrixwhose axes are generated by acoustically pure fifths (3:2) and major thirds (5:4) (Figure 5a)." Arthurvon Oettingen (1866, 15) invertedEuler's matrixabout the hori- zontal axis, and projectedit onto an infiniteplane, as shown in Figure5b (p. 8). (The slashes representsyntonic comma adjustments,and result from Oettingen'ssensitivity to the just-intonationaldistinctions masked by notationaland letter-nameequivalence.) This versionof the pitch-class table was appropriatedby Riemann,became widely disseminatedthrough his writings, and has been passed down by generationsof Germanhar- monic theoristsleading all the way up to the presentday (see Imig 1970, Harrison1994).
7 m -- 2 c g d a e h fis cis gis dis ais eis his fisis cisis gisis disis
1 as es b f c g d a e h fis cis gis dis ais eis his
0 fes ces ges des as es b f c g d a e h fis cis gis
-1 deses asas eses bb fes ces ges des as es b f c g d a e
-2 bbb feses ceses geses deses asas eses bb fes ces ges des as es b f c
Figure 5: Two Tonnetze (b) from Oettingen 1866 The position of majorand minortriads on the matrixwas firstobserved by Euler(1926 [1773], 585), who noted thatthey could be representedon his matrixby the conjunctionof two perpendicularline-segments. Oet- tingen, less concernedthan Euler about the acousticallyresidual status of the minor third, suggested adding a hypotenuseto close the structureto a right triangle(1866, 17). This move brings PLR-familyrelations to the forefront,since triadsso relatedare representedby trianglesthat share an edge, and are thereby maximally proximate.One might infer from this circumstancethat PLR-familyoperations would be the vehicle of choice for navigatingtriadic progressions on the Tonnetz,but this has not been the case historically.The developmentof Oettingen'stable as a "game- board"for mapping progressions among triads was instead guided by convictions about the acoustic, tonally centric statusof consonanttriads and their relations to each other. This resulted in the subordinationof PLR-familyrelations to the Tonic/Subdominant/Dominant(TSD) "func- tional"framework developed by Riemannin the 1890s, a frameworkthat has continued to dominate Northern European harmonic theory ever since.12 The mapping of PLR-family operations independentlyof the TSD frameworkwas first proposed by Lewin (1987, 175-180) and has been developed by Brian Hyer (1989, 1995), whose work demonstratesthe heuristicvalue of chartingPLR-family operations on the Tonnetzwithout necessary recourse to assumptionsabout tonal centers, TSD-functional relations,or the acoustic propertiesof triads.The emphasison PLR-fam- ily operations in the work of Lewin and Hyer is apparentlymotivated empiricallyby the desire to model characteristicallylate-Romantic pro- gressions in a mannerthat is faithfulto the musical qualitiesthat they are perceivedto project.The focus on voice-leading parsimonycultivated in Section 1 above suggests a complementarydeductive-rationalist motiva- tion for liberating the triad, PLR-family operations, and their Tonnetz representationfrom their acoustic origins. It is this motivationthat directs the remainingprogram for this paper. In the materialthat follows immediately,a version of the Tonnetzof Oet- tingen and Riemannis situatedwithin a genus and species of two-dimen- sional matrices.The defining characteristicof the genus is that pairs of trichordsrepresented by adjacent triangles are related by PLR-family operations, as broadly defined in Section 1 above (cf. Def. (4)). The definingcharacteristic of the species is thatpairs of trichordsrepresented by adjacent triangles feature parsimonious voice-leading. Section 2.4 refines the conception of the Oettingen/Riemannmatrix, and of the species that it represents,by acknowledgingthe toroidal geometry that underlies them when their contents and relations are interpretedin the context of equal temperament.The focus on the Tonnetzthroughout Sec- tion 2 serves as a large structuralupbeat to Section 3, the musical core of
9 -2x + 2y -x + 2y 2y x + 2y 2x + 2y
-2x +y, -x + y y x+y 2x + y
-2x -x 0 x 2x
-2x- y -x- y -y x- y 2x- y
/T-2y(Q) -2x - 2y -x - 2y -2y x - 2y - 2x- 2y
Figure6: TheAbstract Tonnetz the paper,which uses PLR-familyoperations to navigate the Tonnetz,in its variousversions at variousdegrees along the abstraction/specification continuum.
2.2. The Generic Tonnetz
Ourinvestigation of the Tonnetzof harmonictheory initially situatesit as a member of an infinite class of two-dimensional matrices whose generic form is presentedin Figure 6. The primaryaxes of Figure 6 are generated by the generic intervals x and y, in the sense that each row incrementsfrom left to right by the value of x, and each column incre- ments from bottom to top by the value of y. The figure should be inter- pretedas projectingits structurebeyond its boundaries.The elements of Figure 6 representreal numbersas they incrementto infinity,and should not be interpretedin the context of the closed modularsystems that were the focus of our previous work. Figure6 is neithermore nor less thanthe Cartesiancoordinate plane of analytic geometry. In termsof Def. (2), the primaryaxes of Figure6 are interpretedas the smallest two step-intervalsof a prime form trichordQ = {0, x, x+y } with step-intervals. The remainingstep-interval is the inverseof the sum of the two smaller step-intervals,and hence generatesthe diag- 10 onal which runs from northeastto southwest,in the sense that each such diagonal decrements,sloping southwestward,by x+y. Figure 6 representstrichord Q as a darklybordered triangle at its cen- ter.Each vertexof the trianglerepresents a pitch or pitch-class,13 andeach edge a dyadic subset, of Q. Any geometric translationof this triangle- that is, any triangle whose hypotenuse subtends northwestof the right angle-represents a pitch or pitch-classtransposition of Q. (An example, labeled TX2y(Q), is providedby the isolated trianglein the southeastcor- ner of Figure 6.) Furthermore,any geometric inversionof the Q triangle abouta secondarydiagonal-that is, any trianglewhose hypotenusesub- tends southeast of the right angle-represents a pitch or pitch-class inversionof Q. Figure 6 indicates three such invertedtriangles, all shar- ing an edge with the centraltriangle. These three trianglesrepresent the PLR-familyof Q (cf. Figure 3a). The unique edge that the centraltrian- gle shares with each of its adjacenttriangles represents the unique dyad that trichordQ shares with each memberof its PLR-family. Figure 7 replicates the core of Figure 6 and adds three arrows, each labeled with one of the voice-leading variablesfrom Def. (5). Each arrow representsthe magnitude of the moving voice when Q is subject to a PLR-familyoperation:
* labels x-y when P takes 0, x, x + y} to 0, y, x + y across their sharedhypotenuse; * labels x + y - -y when L takes (0, x, x + y} to {0, x, - y} across their sharedhorizontal edge; *?,labels 0 - 2x + y when R takes {0, x, x + y} to {2x + y, x, x + y} across their sharedvertical edge.
When the same operations are enacted on other members of trichord- class Q, their geometric orientationon Figure 7 is invariant.In all cases, P-relatedtriads share a hypotenuse,and a executes a pawn-capturealong the main diagonal; L-relatedtriads share a horizontaledge, and I exe- cutes a knight's move, a displacementby two rows and one column; R- relatedtriads share a verticaledge, and, executes a knight'smove, a dis- placementby two columns and one row.The magnitudesof p, I, and, are likewise invariant,although when the object of the mappingis a triadTJI- relatedto the primeform, the directionof the arrowreverses, and the val- ues of a, I, and 4 invert. The unboundedgrid inferable from Figure 6 is applicable to pitches and intervalsin a varietyof ways. If x and y are assigned to acoustically pure intervals (as in Euler, etc.), or to intervalsin pitch-space, then the structureimplicitly projectsinto an infiniteplane. The realizationsof the Figure 6 grid that will hold our focus are generatedby equally tempered intervalsin some modularsystem, where the modularcongruence repre- 11 2x
x-y 2x-y
Figure7: TonnetzRepresentation of PLR-FamilyOperations on Q = {0, x, x + y sents octave equivalence. In such interpretations,both x and y axes become cyclic ratherthan linear, and the plane inferredfrom Figure 6 thereforeprojects into itself as a torus.14 These cyclic features will be studiedin some detail in Section 2.4 below.
2.3. The Parsimonious Tonnetz
In Figure 7, PLR-family operations are only associated with voice- leading parsimonyin the restrictedsense that each operationpreserves two common tones. The degree of parsimonyassociated with the third voice depends on the magnitudes of ?, I and 4, which in turn depend on the values of x and y, as yet unassigned.Furthermore, the interpretation of that magnitudeas representingan interval-classin a modularpitch- class system depends on the imposition of a congruence, i.e. a specific value for c. The first section of this paperestablished that, where c = 12, the three PLR-familyoperations are parsimoniousonly when x = 3 and y = 4, i.e when the trichord-classis the consonanttriad, in which case the generic Tonnetzis realized as an equal-temperedversion of the Oettin- gen/RiemannTonnetz of Figure5b. Since this is the case of historicaland analyticalinterest, it will soon be subjectto detailed scrutiny.First, how- ever, it will be instructiveto consider a structureof intermediateabstrac- tion; a "middleground"Tonnetz that "composes out" Figure 6 in a partic- ular way, at the same time as it positions the propertiesof the Oettingen/ RiemannTonnetz in a general context. What values of c, x, and y will lead to optimally parsimoniousvoice- leading when PLR-family operations are executed? Intuitively,the de- gree of parsimonyis optimal when the magnitudes(i.e. absolute values) of the voice-leading intervals z, I and 4 are as small as possible, but greaterthan zero. (The last conditioninsures that voice-leading "motion"
12 is perceptible as such; cf. note 6.) Ideal parsimony, then, would be achieved when p = ? 1, = ? 1, and = ? 1. But this combination is impos- sible. As observedin Section 1, p + l +- = 0, and so each variableis the inverseof the sum of the othertwo. Consequently,?, l, ande must in this case have identicaldirections as well as magnitudes,and so z = l = . Via Defs. (5b) and (5c), if I = 4 then - 2y - x = 2x + y, and so 3x = - 3y, hence x = - y, which implies that y - x is even. Thus p is even (via Def. (5a)), and so p ? ? 1, contraryto what was stipulated. The next recourse is to incrementthe magnitudeof one of the voice- leading variables. In principle, any of the three variablescan be incre- mented, but Q is in prime form (cf. Def. 2) only if p = 1, I = 1, 4 = -2. These are exactly the values for the familiarcase of the consonanttriad modulo 12. Once we cease to assume a chromaticsystem of twelve pitch- classes, as we did in section 1, what othercombinations lead to these val- ues of , 1, and 4? This problemis easily solved using Theorem 1, which linked the voice-leading intervalsto step-intervaldifferences. If the first step intervalis x, and = 1, then the second step intervalis x + 1, via The- orem 1.1. Further,since I = 1, then the thirdstep intervalis x + 2, via The- orem 1.2. The three step intervalsof a parsimonioustrichord, then, must 5 form the ascendingconsecutive series . The sum of these three step intervalsis 3x + 3, distributedas 3(x + 1). Since c, the numberof pitch-classes in the system, is the sum of the step- intervals,it follows that parsimonioustrichords are only availableif c is an integralmultiple of 3. In each such system, thereis a single step-inter- val series of the form thatrepresents a parsimonioustri- chord-classwhose prime form is (0, x, 2x + 1 }. The generic version of such a trichordwill be representedusing the variableQ':
DEE (7). Q' = {0, x, 2x + 1 }, modulo 3x + 3
Figure 8 (p. 14) presentsa generic Tonnetzfor parsimonioustrichords, which will be referredto as the parsimonious Tonnetz for short. The axes of Figure 8 are generatedby x and x + 1, the smallest step-intervals in Q'. A modularcongruence of 3x + 3 is imposed on the figure, so that the first term of each expression is representedas a non-negativevalue. The trichordalprime-form Q' = I0, x, 2x + 1 } is portrayedalong with its PLR family at the center of Figure 8 . The arrowsdepict the following:
* labels the T1motion x - x + 1 when P takes (0, x, 2x + 1 } to {0, x + 1, 2x + 1 } across their sharedhypotenuse; * labels the T1motion 2x + 1 - 2x + 2 when L takes {0, x, 2x + 1} to {0, x, 2x + 2} across their sharedhorizontal edge;
13 x+3 2x + 3 0 x 2x 3x
2 x+2 2x + 2 3x + 2 x- 2x-
2x+4 1 x + I 2x+ I 3x+ l x-2 /',"P I /
I I x+3 2x + 3 0 2x 3x
I
2 x+2 2x + 2 3x + 2 x- l 2x- 1
2x +4 1 x+ 1 2x+ 1 3x+ 1 x-2
x+3 2x + 3 0 x 2x 3x
Figure 8: The ParsimoniousTonnetz
*? labels the T_2motion 0 - 3x + 1 - -2 when R takes 10, x, 2x + 1 to {3x + 1, x, 2x + 1 across their shared vertical edge.
Figure 8 is a powerful representation of parsimonious trichordal motion. No matter what value is assigned to x, any local triangle with a southwest/northeast hypotenuse represents a parsimonious trichord. Con- versely, all parsimonious trichords are representable through a realization of Figure 8. To investigate the scope of this power, we now explore three
14 x+3 2x + 3 0 x 2x 3x F6
2 x+2 2x+2 3x+2 x-I 2x- 1 w w E B5
2x +4 I x+ 1 2x+ 1 3x+ 1 x-2 a E r .%
2x+3 0 / ', x 2x 3x [x1 , Ex-'' E+ 1
x+2 2x +2/ 3x + 2 x- I 2x- I 811] 11 F5]
2x +4 I x+ 2x+ 3x+ I x- 2 10 B4
x+3 2x+3 0 x 2x 3x D FB9 Fo [_] [H 0
Figure9: ThreeRealizations of theParsimonious Tonnetz. (a) x = 3, y = 4, c = 12,Q' = {0, 3, 7} modulo12 (Oettingen/ Riemann Tonnetz) such realizations, for x = 3, 5, and 7 respectively.In each of the three matricesthat comprise Figure 9, the abstractexpressions of Figure 8 are retained along with their realizations so that derivationscan be easily traced. Figure 9a is a version of the Oettingen/RiemannTonnetz. It partially rotates Figure 5b, replacing pitch-class names with integers. The series of majorthirds is retainedon the y axis, but the series of minor thirdsis displacedfrom the northwest/southeastdiagonal of Figure5b to the x axis of Figure 9a, thereby shifting the series of perfect fifths from the x axis
15 x+3 2x+3 0 x 2x 3x 281 213 OF 3 x 10 215
2 x+2 2x+2 3x+2 x-1 2x - 1 m m H z]3
2x +4 I x+ 1 2x+ I 3x+ 1 x-2 (14 W BL. , 16 El
x+3 2x + 3 0 x 2x 3x 5 10 15 x1 2+' - ,
2 x+2 2x +2/ 3x+2 x- 1 2x- 1 m a7 g [17 [4 El
2x +4 1 x+ 1 2x+ 1 3x+ 1 x-2 m14 Li [] 11 16 3]
x+3 2x + 3 0 x 2x 3x [i [5] 10 15
Figure9: ThreeRealizations of theParsimonious Tonnetz. (b) x = 5, y = 6, c = 18, Q'= {0, 5,11} modulo 18
to the southwest/northeastdiagonal. This particularversion of the Ton- netzactually antedates Oettingen: it was firstintroduced by CarlFriedrich Weitzmannin 1853, althoughin a boundedform (as in Fig. 5a), andusing staff-notatedpitches ratherthan integers.16The triangularcomplex at the centerof Figure9a portraysthe trichordalprime form, {0, 3, 7 = C minor, togetherwith its PLR family. The arrowsrepresent the following:
16 - *p labels the T, semitonal motion 3 - 4 = Eb E when P takes {0, 3, 7} = C minor to {0, 4, 7} = C major across their shared hypotenuse; * l labels the Ti semitonal motion 7 - 8 = G - Al when L takes (0, 3, 7 = C minorto {0, 3, 8} = Ab majoracross their sharedhorizon- tal edge; *a labels the T0lo T_2whole-step motion 0 - 10 = C - B when R takes {0, 3, 7} = C minorto { 10, 3, 7} = El majoracross theirshared verticaledge.
In Figure 9b, the parsimonioustrichord is Q' = {0, 5, 11} in a modulo 18 ("third-tone")system, with step-intervals<5, 6, 7>. The triangular complex at the center of Figure 9b portrays10, 5, 11 } togetherwith the three trichordsthat comprise its PLR family. The arrowsportray the fol- lowing:
* labels the T, motion 5 - 6 when P takes {0, 5, 11} to (0, 6, 11} across their sharedhypotenuse; *?labels the T1motion 11 -12 when L takes {0,5, 11} to {0, 5, 12} across their sharedhorizontal edge; *-zlabels the T16= T_2motion 0- 16 when R takes {0,5, 11 to 16, 5, 11 across their sharedvertical edge.
In Figure 9c (p. 18), the parsimonioustrichord is Q' = {0, 7, 15} in a modulo 24 ("quarter-tone")system, with step-intervals<7, 8, 9>. The tri- angularcomplex at the center of Figure 9c portrays {0, 7, 15} together with the threetrichords that form its PLR family. The arrowsportray the following:
* labels the T1motion 7 - 8 when P takes (0, 7, 15} to {0, 8, 15} across their sharedhypotenuse; * labels the Tl motion 15 16 when L takes {0, 7, 15) to {0, 7, 16} across their sharedhorizontal edge; * labels the T22r T-2motion 0- 22 when R takes {0, 7, 15} to {22, 7, 15) across their sharedvertical edge.
2.4. The Toroidal Tonnetz
Before navigatingthe Tonnetze,we need to confronttheir limitation as a representationof pitch-classrelations in equal temperament.In Figures 8 and 9, pitch-classes occur in multiple locations, obscuringtheir equiv- alence. Alternativelyrepresenting each pitch-classat a single location, as in Figure 5a, has the equally perniciousconsequence of obscuringadja-
17 x+3 2x+3 0 x 2x 3x 1101 17] 14] [21]
2 x +2 2x + 2 3x + 2 x- 1 2x- 1 2 1 6 E23] 131
2x +4 1 x+ 1 2x+ 1 3x + x-2
x+3- 2x+3 0 x 2x 3x E[10 [21F717 0 / 7 E14 E2
2 x+2 2x+2 3x+2 x- 1 2x- 1 [ 16 26 [13]
2x +4 x+ 1 2x+ 1 3x+ 1 x-2 [18] E [15 1221 E
x+3 2x+3 0 x 2x 3x E10 17 E [14] 21
Figure9: ThreeRealizations of theParsimonious Tonnetz. (c) x = 7, y = 8, c = 24, Q'= {0, 7, 151 modulo24
cency relationships,causing axes to float off the edge of the two-dimen- sional surface only to reappearon the opposite edge. These obscurities resultartificially from the mismatchbetween the cyclical natureof pitch- class space and the flat surfaceof the printedpage. A toruspresents a geo- metric figure more appropriateto representingthe cyclic propertiesof equal-temperedpitch-class. Although the torusis eschewed here because it is difficultto renderand interpreton the two-dimensionalsurface of the page, its underlyingstatus needs to be sufficientlyacknowledged before the Tonnetzcan be navigatedwith full comprehension. 18 At issue above all is the cyclic periodicitiesof the axes, which fluctu- ate accordingto the generatingintervals and the cardinalityof the chro- matic system. The axes relevantto Tonnetznavigation are the three that are generatedby step-intervalsof the parsimonioustrichord. These in- clude not only the primaryx and y axes, but also the -(x + y) axis that generates the southwest/northeastdiagonal. Assigning an abbreviated variableto this step-intervalwill simplify our treatmentof the axis-cycle generatedby it.
DEF. (8). z = c - (x + y).
All three step-interval-generatedaxes are circularizedby equal tem- perament,and thus will be referredto as axis/cycles. Ourstudy of the periodicityof axis/cycles will be aidedby puttinginto play a variableq*, representingthe periodicityof step-intervalq modulo c.
DEF. (9). = where the greatest common divi- q* gcd(c,q)cc ' gcd(c,q), sor of c and q, is the largest integerj such that c and 9 are positive integers. J J
The asterisk is attachableto any variablethat representsa step-inter- val; hence x* is the periodicityof x modulo c, and so forth. There are two generalcases."7 If q and c are co-prime, then gcd(c,q) = 1, in which case q* = c. The q cycle then exhauststhe chromaticsystem, and all cycles along the q axis are identical;in effect, there is a single q axis-cycle. A familiar example is presentedby the z axis of Figure 9a (p. 15), where c = 12 and z = 5. gcd(12,5) = 1, and so z* = c = 12. (The 12-periodicitycannot be directly verified on the attenuatedrepresenta- tion of the z-axis given in Figure 9a, but must be induced by extending the boundariesof the figure.)All z axes in Figure 9a thus have a period- icity of 12, and exhaustthe 12 pitch-classesvia the "circleof fifths."Thus there is only a single distinct z axis-cycle. In the second general case, gcd(c,q) > 1, in which case q* < c. The q axis does not exhaust the chromatic system, but instead runs through some propersubset of its pcs. The pcs modulo c are then partitionedinto gcd(c,q) = c co-cycles (providedthat 1 is the greatest common divi- sor of the three step-intervalsin Q, as is indeed truefor all cases relevant to this study).An example is presentedby the z axis of Figure 9c, where c = 24 and z = 9. gcd(24,9) = 3, and so z* = c = 8: each z cycle gcd(c,z) includes eight of the 24 pcs in the chromaticsystem. (Again this claim must be inducedfrom the figure.The orderedcontent of the z axis begin-
19 ning at the southwestcorer of 9c is as follows: <10, 1, 16, 7, 22, 13, 4, 19, 10>.) There are gcd(24,9) = 3 distinct z axis-cycles, which partition the 24 pitch-classes. Retreatingby one level of abstraction,consider now the cyclic features of Figure 8 (p. 14), where c = 3x + 3, and the three step intervals are . c = 3x + 3 = 3 (x+l) = 3y, and so y = c. Since y divides it follows that = and so = c = 3. This evenly c, gcd(c,y) y, y* Y explains why there are exactly three distinctelements in each column of the matricesin Figures8 and 9. The significantpoint here is thatthe triple periodicityassociated with the second step-intervalis properto the under- lying structureof Figure 8, and thus the inclusion of an octave-trisecting interval,acoustically equivalent to a temperedmajor third, is common to all parsimonioustrichords. As for the numberof distinctcolumns, there are gcd(c,y) = y. That is: thereis one y-axis co-cycle for each degree of sepa- rationbetween the second and thirdpitch-class in the prime form of Q'. In contrast,the remaining step-intervalsare divisors of c only under limited conditions. c is co-prime with x (the first step-interval)unless x is a multipleof one of c's divisors, 3 or x + 1. x cannotbe a multipleof x + 1; thus c and x are co-prime unless x is an integralmultiple of 3, i.e., thereis some positive integern such that 3n = x. If so, then c = 3(3n) + 3 = 9n + 3, in which case c is congruentto 3 modulo 9. The smallest exam- ples of such systems are c = { 12 (N.B.), 21, 30}. Of the systems portrayedin Figure 9, only Figure 9a (p. 15), where c = 12, meets this condition,and consequentlyit is only here thatthe x axis partitionsits pcs into co-cycles ratherthan exhausting them in a single 12 cycle. In this case, x = 3, gcd(c,x) = 3, and x* = = 4. Each x axis thus containsfour distinctpcs, and thereare x = 3 distinctx axes. (In stan- dardterms, of course, what we have here is the partitionof the aggregate into three diminishedseventh chords.) By contrast,in Figures9b and 9c, neitherc = 18 nor c = 24 are congruentto 3 modulo 9. Consequently,c and x are co-prime, and so there is only a single x axis that exhausts the system of 18 or 24 pcs. A similarsituation holds for the relationshipof c to the thirdstep-inter- val. x + 2 cannot be a multiple of x + 1, and so, in parallelwith the case of the x axis just described,c and x + 2 are co-prime unless x + 2 is an integralmultiple of 3, i.e., there is some positive integer n such that 3n - 2 = x. In such cases, c = 3(3n - 2) + 3 = 9n - 3. That is, c _ 6 modulo 9. The smallest such systems are c = {6, 15, 24}. Of the systems portrayed in Figure 9, only Figure 9c, where c = 24, meets this condition, and con- sequently it is only here that the z axis partitionsits pcs into co-cycles ratherthan exhaustingthem in a single cycle. By contrast,in Figures 9a and 9b, neitherc = 12 nor c = 18 are congruentto 6 modulo 9, and so c
20 and z are co-prime, and there is only a single z axis which exhausts the system of 12 or 18 pcs. A practicaldemonstration of the features discussed in this section is provided in the pioneering microtonal treatise of Alois Haba (1927), which systematicallyexplores the generativepowers of intervalsin both quarter-tone(c = 24) and third-tone(c = 18) systems (where "tone"is taken in the sense of "whole tone"). In his discussion of the quarter-tone system, Haba notes that the "neutralthird," equivalent to 7/24 of an octave, generates all 24 pitch-classes. In contrast,Haba writes that the interval"a quarter-tonehigher than a majorthird [i.e. 9/24 of an octave] leads only as far as an octachord [Achtklang].This octachord is com- posed of the symmetricpartition of a majorsixth.... Two transpositions of the octachordupward by a quarter-toneuse the remaining 16 tones of the quarter-tone-scale"(1927, 166). Haba'sthree octachordsare equiva- lent to the three distinct z-axis co-cycles discussed above in association with Figure 9c. In a subsequentchapter, Haba approachesthe third-tone system from a similarperspective, observing that "the successive series of 5/3 steps forms a unified collection of 18 tones in the third-tonesys- tem, and indeed in a more broadlyexpanded distribution across a span of five octaves. The successive series of 18 seven-thirdtones forms a col- lection spreadacross seven octaves" (202-203). Haiba'saggregate-com- pleting Nacheinanderfolgenare equivalentto the x and z axes of Figure 9b. The perspectivecultivated throughout this section providesa system- atic foundationfor Haiba'sobservations. Figure 10 (p. 22) summarizesthe work of this section by providinga table for calculating the cyclic periodicities for the chromatic systems thatcan host parsimonioustrichords, as exemplifiedin the threematrices of Figure 9. The significantpoint to be carriedout of this exposition is thatthe y axis, generatedby the second step-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 and thirdstep-intervals respectively, have a variableperiodicity, but the num- ber of x and z co-cycles is constant to within the modulo-3 congruence of c. The relevanceof these findings, and particularlyof the special sta- tus of the y axis, to progressionsbased on PLR-family operationswill become apparentin Section 3.
3.1. PLR-family Compounds
We are now in a position to navigatethe toroidalTonnetz, in all its var- ious manifestations,using PLR-family operations as our vehicle. The maximal common-tone retention inherent to these operations insures that the cruise will be smooth, particularlywhen the Tonnetzis parsimo- nious. Ourexploration will follow a systematicprogram, focusing on tri- 21 (a) (b) (c) (d) (e) x axis y axis z axis c mod-9 periodicity # co-cycles periodicity # co-cycles periodicity # co-cycles example (x*) (c/x*) (y*) (c/y*) (z*) (c/z*)
0 c 1 3 c c 1 c=18 3 (Fig. 9b)
3 c 3 3 c c 1 c=12 3 3 (Fig. 9a)
6 c 1 3 c c 3 c=24 3 3 (Fig. 9c)
Figure 10: Cyclic Periodicitiesof Step Intervalsof ParsimoniousTrichords chordalprogressions, or chains, generatedby the recursiveapplication of a patternof PLR-familyoperations.'8 Where appropriate,chains will be viewed as segments of cycles. The basic cycle-classes formed by gener- ated PLR-family chains are few in number,and their relation to PLR- family operationsis roughly analogous to the role of the chain of fifths (Sechterkette)in diatonic progressions:they constitutenormative proto- types against which particularprogressions, in all their varietyand com- plexity, may be gauged. The ultimategoal of this investigationis the pragmaticone of explor- ing parsimoniousvoice-leading among consonant triads in a 12-pc sys- tem. Consistentwith the frameworkestablished in Section 2, this famil- iar phenomenon is situated as a particularmanifestation of a more general one: the behavior of parsimonioustrichords in any pitch-class system thatis suitablysized to host them. Some readersmay be frustrated by this strategy,since it defers an encounterwith music in systems that we care about, instead inviting contemplationof hypothetical musical systems whose sounds we may have difficultyimagining. Nowadays, of course, the pragmaticfallout of such a study, in the form of "microtonal universes,"is readily available to composers, analysts, and listeners. Fromthis viewpoint,the researchpresented in this paperreflects the deep influence of Gerald Balzano's classic study (Balzano 1980). Like Bal- zano, my motivationsare not only compositional.They stem as well from an intuition,perhaps a credo, that insights into the propertiesand behav- ior of individualinstances are furnishedby studying the propertiesand behaviorsof general phenomenawhich they represent.As inhabitantsof a planetthat sustainslife, the value of exploring other planets, solar sys- tems, or galaxies for their life-sustainingproperties, or lack thereof, po- tentially transcendsthe conceivable material benefits, extending to the self-knowledge that emerges from the differentiatingcontext furnished by the Other.19 We begin with some notations, definitions, and observationsinvoked throughoutthe rest of the paper: 3.1.1. Notation of Compound Operations. A compoundPLR-family operationis denoted as an orderedset of individualPLR-family opera- tions, enclosed in angled brackets.The operationsapply in the order in which they appearin the set, from left to right. For example, in the com- pound operation, R is applied, P is applied to the productof R, and L is applied to the productof R-then-P. 3.1.2. T/I Equivalences of Compound Operations. All compound operationsare equivalentto either transpositionsor inversions,depend- ing on the cardinalityof the ordered set. Compoundoperations of odd cardinalityare inversions,those of even cardinalitytranspositions. This follows from the inversionalstatus of PLR-family operations,together
23 with the group structureof inversion and transposition(see, e.g., Rahn 1980, 52). 3.1.3. Generators. A PLR-family compound is generated if it can be partitionedinto two or more identical ordered subsets. The ordered subset, singly iterated,constitutes the generator of the compound.The compoundcan be expressedas Hn, where H is the generatorand n counts its iterationsin the compound.A generatorof cardinality#H is classified as #H-nary (hence binary, ternary, etc.). For example, the compound is binary-generated,since it can be partitionedas < >,and expressedas 3. 3.1.4. T/I Equivalences of Generators. The cardinalityof a generated compoundHn is #H ?n. If either#H or n are even , then H" is a transpo- sition. In order for Hn to be an inversion,#H and n must both be odd. This follows from the observationsmade in 3.1.2 togetherwith the mul- tiplicativeproperties of odd and even integers. 3.1.5. Cycles. H generates a cycle if, operatingon some trichordQ, there is some integer q* > 0 such that Hq* (Q) = Q. The smallest such value q* is the operational periodicity of H. It will also be useful on occasion to count the trichordsthat result from an H-generatedcycle. That number,the trichordal periodicity of the cycle, is equivalent to #H q*. 3.1.6. Tonnetz Representations of Cycles. Generatorsof odd cardi- nality are involutions:they retreatto their point of origin on the Tonnetz aftertwo iterations.This is restatedformally as Theorem2 in the appen- dix, where a proof is offered. Generatorsof even cardinality,by contrast, are devolutions:they move perpetuallyaway from theirpoint of origin on the Tonnetz.Cycles are generatedonly when a modularcongruence gov- erns the Tonnetz,in which case the generatorhas the same periodicityas the transpositionaloperation to which it is equivalent(cf. 3.1.4). These periodicitiescan be computedby firstexpressing the PLR-familygener- atoras a transpositionoperation T,, and then determiningthe periodicity of n in relationto the size of the chromaticsystem. For this second step, we will rely on the work carriedout in Section 2.4 and summarizedin Figure 10.
3.2. Binary Generators
Because the unarygenerators , , and are involutions,the progressionsthat they generateare insufficientlyvaried to serve as com- pelling musical resources.Thus our explorationbegins with binarygen- eratorsthat pair distinct PLR-familyoperations. There are six such gen- erators,which group into three retrograde-relatedpairs:
(1) and
THEOREM3. Foran orderedPLR-family operation set H andits retro- grade Ret(H), if H = Tp,then Ret(H) = Tp.
A proof is given in the Appendix. Moving one step forwardinto the "middleground,"we now examine these relations as they apply to the abstract parsimonious trichordal prime form Q' = {0, x, 2x + 1}, with directedintervals ?x, ?(x + 1), and +(x + 2). In general,these intervalsrepresent six distinctvalues, with the lone exception that, if x = 1 and c = 6, then x + 2 = - (x + 2). This excep- tion aside, the six binaryPLR-family operations produce six distinct tri- chords when implementedon Q'. Figure 12 (p. 27) translatesFigure 11 into terms specific to parsimo- nious trichords.The transpositionsat (a) are given in threeforms: as pos- itive and negative generic step-intervals,as positive and negative step- 25 (a) (b) (c) Tx
T-x {O,x,x+y} >{-x, ,y}
Ty 0, x +y} > ,x +y, xx+2y}
T_y (O,x, x+y} v >(-y,x-y,x}
Tx+y {0, x, x + y} >{x + y, 2x + y, 2x + 2y}
T-x-y (O,x,x +y}y>{-x-y, -y, O
(d)
-x + 2y 2y x + 2y 2x + 2y
Figure 11: AbstractTranspositional Equivalences for Binary Generators (a) (b) (c)
x> Tx
< T2x+3> T-x T2x+3 {0, x,2x2 1 +> {2x+3,0,x+ 1
Ty Tx+1 {0, x, 2x+ 1} >{x+ , 2x+ 1, 3x +2}
T-y T-(x+l) T2x+2 {0,x,2x+} )>{2x+2,3x+2, x}
Tx+y T-(x+2) T2x+l {0, x, 2x+l >{2x+ 1,3x+ 1,x -1}
T-x-y Tx+2 {0,x, 2x+1 >x+ 2,2x+2, 0
(d)
x+2 2x+2 3x + 2 x- 1
1
x- 1
Figure 12: Binary Generatorsand ParsimoniousTrichords x+4 2x + 4 1 x + 1 2x+ 1 3x + I x- 2 2x- 2
< >3 ~ x+4 2x + 4 1 x + 1 2x+ 1 3x+ 1 x- 2 2x- 2
