Yale University Department of Music
Neo-Riemannian Operations, Parsimonious Trichords, and Their "Tonnetz" Representations Author(s): Richard Cohn Reviewed work(s): 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: 18/03/2013 12:53 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp
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NEO-RIEMANNIAN OPERATIONS, PARSIMONIOUS TRICHORDS, AND THEIR TONNETZREPRESENTATIONS
Richard Cohn
1. The Over-Determined Triad In work publishedin the 1980s, David Lewin proposedto model relations between triads'using operationsadaptedfrom 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 sharea common minorthird;and R (for Relative),which relatestriadsthatsharea commonmajorthird.3Figure 1 illustratesthe threeoperations,which I shall referto collectively as the PLR family, as they act on a C minortriad,mappingit to three different majortriads. (Throughoutthis paper,+ and - are used to denote majorand minor triads,respectively.)Singly applied, each PLR-family operationinvertsa triad(majorminor).Doubly applied,each PLRfamily 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, parsimonyis inherentto the PLR family, whose definingfeatureis double common-toneretention.Whatis not inherent is the incrementalmotion of the third voice, which proceeds by 1
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L C-
-
P C-
Ab+
-
R C+
C-
Eb+
Figure 1: The PLR-Familyof C-Minor A .
-
.
I
Figure2: The PLR-Familyof {0, 1, 51 semitone in the case of P and L, and by whole tone in the case of R. This featureis not without significanceto the developmentof a musical culture where conjunctvoice-leading in general, and semitonalvoice-leading in particular,are enduring norms through an impressive range of chronological eras and musical styles.5 The parsimonyof PLR-family voice-leading is so engrainedin the proceduralknowledgeof a musician trainedin the Europeanclassical traditionthatit hardlyseems to warrant notice, much less scrutiny.It is scrutinizedhere with the aim of demonstratingthat,from a certainpoint of view, the featureis fortuitous. In orderto cultivatethis pointof view, imaginea musicalculturewhere set-class 3-4 (015) was the privilegedharmonyto the extent thatthe triad prevailsin Europeanmusic c. 1500-1900. For any memberof set-class 3-4, therearethreeothermemberswith which it sharestwo commonpcs. 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 thirdcase by tritone. Such a culture would be incapable 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,amongtrichord-classes,consonanttriads arespecial. Theiruniqueacousticpropertiesarewell established,and indeed are fundamentalto standardapproachesto triadic music. The potential of consonant triads to engage in parsimoniousvoice-leading, however, is unrelated to those acoustic properties. This potential is, rather,a functionof theirgroup-theoreticpropertiesas equally tempered entities modulo-12. To demonstratethis claim, the definitionof PLR-familyoperationsis now generalized, initially to the prime forms of set-classes defined by T,/T,Iequivalence, subsequentlyto all trichords.Although our primary 2
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attentionwill be focused on trichordsin the usual chromaticsystem of twelve pitch-classes,the definitionis open to applicationto othersystems as well, for reasonsthat will emerge as the exposition unfolds. DEE (1). c is a positive integerrepresentingthe cardinalityof a chromatic system. DEE (2). Q is a mod-c trichord{0, x, x + y} such that0 < x < y 5 c (x + y). The condition insuresthat Q is the prime form of its trichord-class. DEE (3) Iu is the inversion that maps pitch-classes v and u to each other.7 The threePLR-familyoperationscan now be definedon a prime-formtrichord Q = {0, x, x + y} as follows: DEE (4a) P = I,+y
I
DEF. (4b) L = I I
DEF. (4c) R =
I
Figure 3a (p. 4) demonstratesthe mapping of abstractpitch-classes when P, L, and R act on the abstracttrichordalprimeform Q. Figures3b 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,a, /, and 4, which express the magnitudeof those externalmappingsas mod-c transpositionalvalues. x----y,hence: DEE (5a) 9 = y - x
x + y-y, hence: DEE (5b)l = -2y - x
+ y, hence: 0--2x(5c) , = 2x + y DEE
Observethatp + / + = 00, since (y - x) + (-2y - x) + (2x + y) = (2x - 2x) + (2y - 2y) = 0. The values of , 1,and are now linked to the structureof trichordQ. ,z Chrisman(1971), and others since, we First, following Bacon (1917), 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 presupposesnormal order.Thus the step-intervalseries of Q is . 3
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=
P(Q)
I
sy
(Q)
L(Q)
=
x+y-> x --4y
x+y--y x --4 O
O---)x+y
0---x
I
(Q)
R(Q)
=
(Q)
Ix
x+y-x x +y x+v
0--2x
Figure3: PLR-Family Mappings (a) forQ = {0, x, x + y} P(Q) = I o(Q)
L(Q) = I (Q)
R(Q) = I' (Q)
7 -- 0
7- 8
7 --43
3- 4 0--4 7
3--4 0 0- 3
3--47 0--4 10
Figure3: PLR-Family Mappings (b) forC-Minor;x = 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 1--40 0--41
5-- 1 1--~ 5 0-- 6
1---44 0--4 5
Figure3: PLR-Family Mappings (c) x = 1, y = 4, Q = {0, 1, 5) The following theoremstates thatp, 1,and z are each equivalentto the differencesbetween a distinctpairof step-intervals. THEOREM1. For a prime-formtrichordQ = {0, x, x + y } with stepintervals, 1.1) p is the difference between the first and second step interval, Proof:y - x = p via Def. (5a). y - x. 1.2) 1 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 facilitatescalculationof the values ofp., 1,and for any trichordal prime form in a chromatic system of any size. The results of '
4
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these calculationsfor the mod-12 trichordalprimeformsaregiven in Figure 4 (p. 6). Conjunctintervalsare enclosed in boxes. The figuredemonstrates what was asserted above: that 037 is unique among trichordal prime forms (modulo 12) in its preservationof conjunctmelodic intervals when any of the three PLR-familyoperationsis executed.8 The generalizationfrom prime form to other class membersis easily carriedout. Calculatethe Tnor TnIof the trichordin relationto the prime form of its class, reassign 0 to the pitch-class that had been formerly assignedthe integern, andreassignthe integersin ascentor descentfrom the new pitch-class 0, depending on whether the trichordis Tnor TnIrelatedto the prime form. Then apply the operationsas in definition(4). If Tn-relatedto the prime form, the values of p, 1,and, are the same as when the operationacts on the primeform;if TnI-related,the values are inverted.(This follows from the involutionalnatureof PLR-familyoperations.) In either case, the magnitudesare invariant.Consequently,the uniquecharacteristicnoted above for trichordalprime form {037} generalizes to all membersof its set-class. To summarizeour findings so far: (1) Among mod-12 trichords,the consonanttriadalone is susceptibleto parsimoniousvoice-leadingunder the three PLR-familyoperations;(2) This circumstanceis a functionof the trichord'sstep-intervalsizes, which are an aspect of its internalstructure;(3) the optimalvoice-leading propertiesof triadsthereforestandin incidentalrelationto their optimalacoustic properties. In a word:the triadis over-determined. The fortuitousrelation of the consonant triad's voice-leading parsimony to its acousticgenerabilityis as profoundto the developmentof the thatwere Europeanmusicaltraditionas othersortsof over-determination 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 n integern such that3 approximatessome powerof 2; of the proximityof the perfect fifth's 2 geometric division of the octave (the source of its acoustic power) to its 7 arithmetic division of the octave (a fraction whose irreducibility,rarefor 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.9Our sensibilities, born of incessantexposure to a musical traditionthat habituallyimplementsthe acousticpropertiesof triads,as well as to a music-theoretictradition 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
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prime form (0, x, x + y)
step-intervals
p y -x
1
r
-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
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7
{0,2,6}
E
{0,2,7)
3
0
9
{0,3,6)
0
3
9
{0,3,7.}
--F01
1
{0,4,8}
0
0
0
8
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
Figure5: TwoTonnetze (a) fromEuler1926(1739)
thriller,the dazzling beauty of the triadhas blinded us to its substantial intellectualresources. 2.1. The Over-Determined Tonnetz The two-dimensional matrix known as the table of consonant (or or harmonicnetwork tonal) relations(Verwandtschaftsverhdltnistabelle) has music theorists with a useful (Tonnetz,Tongewebe) long presented graphic instrumentfor representingtriadic progressions.Although the matrix originated in response to the acoustic properties of triads, it respondsin equal measureto their group-theoreticproperties.The overdeterminationof 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 intervalsgeneratedfrom a resonating body.'"LeonhardEuler(1926 [1739]) situatedjustly tunedversionsof 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 matrixaboutthe horizontal 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-intonationaldistinctionsmasked by notationalandletter-nameequivalence.)This versionof the pitch-class tablewas appropriatedby Riemann,becamewidely disseminatedthrough his writings, and has been passed down by generationsof Germanharmonic theoristsleading all the way up to the presentday (see Imig 1970, Harrison1994). 7
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2
c
g
d
a
e
h
fis
cis
gis
dis
ai
1
as
es
b
f
c
g
d
a
e
h
fi
0
fes
ces
ges
des
as
es
b
f
c
g
d
eses
bb
fes
ces
ges
des
as
es
b
eses
bb
fes
ces
ge
-1 -2
deses asas
bbb feses ceses geses deses asas
Figure 5: Two Tonnetze (b) from Oettingen 1866
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The positionof majorandminortriadson the matrixwas firstobserved by Euler(1926 [1773], 585), who notedthatthey could be representedon his matrixby the conjunctionof two perpendicularline-segments. Oettingen,less concernedthanEuleraboutthe acousticallyresidualstatusof the minorthird,suggested adding a hypotenuseto close the structureto a righttriangle(1866, 17). This move bringsPLR-familyrelationsto the forefront,since triadsso relatedarerepresentedby trianglesthatsharean edge, and are therebymaximally proximate.One might infer from this circumstancethatPLR-familyoperationswould be the vehicle of choice for navigatingtriadicprogressionson the Tonnetz,but this has not been the case historically.The developmentof Oettingen'stable as a "gameboard"for mapping progressions among triads was instead guided by convictions aboutthe acoustic, tonally centric statusof consonanttriads and their relations to each other. This resulted in the subordinationof PLR-familyrelationsto the Tonic/Subdominant/Dominant (TSD) "functional"frameworkdevelopedby Riemannin the 1890s, a frameworkthat has continued to dominate Northern Europeanharmonic theory ever since.12 The mapping of PLR-family operations independentlyof the TSD frameworkwas first proposedby Lewin (1987, 175-180) and has been developed by Brian Hyer (1989, 1995), whose work demonstratesthe heuristicvalue of chartingPLR-familyoperationson the Tonnetzwithout necessary recourse to assumptionsabout tonal centers, TSD-functional relations,or the acousticpropertiesof triads.The emphasison PLR-family operationsin the work of Lewin and Hyer is apparentlymotivated empiricallyby the desire to model characteristicallylate-Romanticprogressions in a mannerthatis faithfulto the musical qualitiesthatthey are perceivedto project.The focus on voice-leadingparsimonycultivatedin Section 1 above suggests a complementarydeductive-rationalistmotivation for liberatingthe triad, PLR-family operations,and their Tonnetz representationfrom theiracoustic origins. It is this motivationthat directsthe remainingprogramfor this paper. In the materialthatfollows immediately,a versionof the Tonnetzof Oettingen and Riemannis situatedwithina genus and species of two-dimensional matrices.The defining characteristicof the genus is that pairs of trichordsrepresentedby adjacent triangles are related by PLR-family operations, as broadly defined in Section I above (cf. Def. (4)). The definingcharacteristicof the species is thatpairsof 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 contextof equal temperament.The focus on the TonnetzthroughoutSection 2 serves as a large structuralupbeatto Section 3, the musicalcore of 9
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-2x + 2y
-x + 2y
2y
x + 2y
2x + 2y
-2x + y
-x +y
y-
x + y-
2x + y
-2x
-x
0O
-2x-y
-x-y
-2x - 2y
-x - 2y
I/1/ x
2x
-y
x-y
2x-y
-2y
Tx-2y (Q)2x 2y x - 2y-2x -- 2y
Figure6: TheAbstractTonnetz the paper,which uses PLR-familyoperationsto navigatethe Tonnetz,in its variousversionsat variousdegreesalong the abstraction/specification continuum. 2.2. The Generic Tonnetz Ourinvestigationof the Tonnetzof harmonictheoryinitially 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 generatedby the generic intervals x and y, in the sense that each row incrementsfrom left to right by the value of x, and each column increments from bottom to top by the value of y. The figure should be interpretedas projectingits structurebeyond its boundaries.The elements of Figure6 representreal numbersas they incrementto infinity,and should not be interpretedin the context of the closed modularsystems thatwere the focus of our previouswork.Figure6 is neithermore nor less thanthe Cartesiancoordinateplane of analytic geometry. In termsof Def. (2), the primaryaxes of Figure6 areinterpretedas the smallesttwo step-intervalsof a primeform trichordQ = {0, x, x+y } with step-intervals. The remainingstep-intervalis the inverseof the sum of the two smaller step-intervals,and hence generatesthe diag10
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onal which runsfrom northeastto southwest,in the sense thateach such diagonaldecrements,sloping southwestward,by x+y. Figure6 representstrichordQ as a darklyborderedtriangleat its center.Eachvertexof the trianglerepresentsa pitchor pitch-class,13andeach edge a dyadic subset, of Q. Any geometrictranslationof this trianglethat is, any triangle whose hypotenuse subtendsnorthwestof the right angle-represents a pitch or pitch-classtranspositionof Q. (An example, labeled Tx-2y(Q),is providedby the isolated trianglein the southeastcorner of Figure6.) Furthermore,any geometricinversionof the Q triangle abouta secondarydiagonal-that is, any trianglewhose hypotenusesubtends southeast of the right angle-represents a pitch or pitch-class inversionof Q. Figure 6 indicatesthree such invertedtriangles,all sharing an edge with the centraltriangle.These threetrianglesrepresentthe PLR-familyof Q (cf. Figure 3a). The uniqueedge that the centraltriangle shares with each of its adjacenttrianglesrepresentsthe unique dyad thattrichordQ shareswith each memberof its PLR-family. Figure 7 replicates the core of Figure 6 and adds three arrows,each labeled with one of the voice-leadingvariablesfromDef. (5). Eacharrow representsthe magnitudeof the moving voice when Q is subject to a PLR-familyoperation: *p 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 10, x, x + y} to 10, x, - y} across their sharedhorizontaledge; *4 labels 0 --- 2x + y when R takes {0, x, x + y) to {2x + y, x, x + y) across their sharedverticaledge. When the same operations are enacted on other members of trichordclass Q, theirgeometricorientationon Figure 7 is invariant.In all cases, P-relatedtriadssharea hypotenuse,andp executes a pawn-capturealong the main diagonal; L-relatedtriads share a horizontaledge, and I executes a knight's move, a displacementby two rows and one column; Rrelatedtriadssharea verticaledge, and4 executes a knight'smove, a displacementby two columns andone row.The magnitudesofp, 4, and4 are likewise invariant,althoughwhen the objectof the mappingis a triad TnIrelatedto the primeform,the directionof the arrowreverses,and the values of p, 1, and4 invert. The unboundedgrid inferablefrom Figure 6 is applicableto 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 Figure6 grid that will hold our focus are generatedby equally tempered intervalsin some modularsystem, where the modularcongruencerepre11
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y
x+y
2x+y
x
2x
x-y
2x-y
-
-y
of Figure7: TonnetzRepresentation on Q = (0, x, x + y} Operations PLR-Family 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 operationsare only associated with voiceleading parsimonyin the restrictedsense that each operationpreserves two common tones. The degree of parsimonyassociated with the third voice dependson the magnitudesof p, / and 4, which in turndependon the values of x and y, as yet unassigned.Furthermore,the interpretation of that magnitudeas representingan interval-classin a modularpitchclass system depends on the imposition of a congruence,i.e. a specific value for c. The firstsection of this paperestablishedthat, where c = 12, the three PLR-familyoperationsare 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 Oettingen/RiemannTonnetzof Figure5b. Since this is the case of historicaland analyticalinterest,it will soon be subjectto detailedscrutiny.First,however, it will be instructiveto considera structureof intermediateabstraction; a "middleground"Tonnetzthat"composesout"Figure6 in a particularway, at the same time as it positions the propertiesof the Oettingen/ RiemannTonnetzin a generalcontext. What values of c, x, and y will lead to optimallyparsimoniousvoiceleading when PLR-familyoperationsare executed? Intuitively,the degree of parsimonyis optimal when the magnitudes(i.e. absolutevalues) of the voice-leading intervalsp, I and 4 are as small as possible, but greaterthanzero. (The last conditioninsuresthatvoice-leading"motion" 12
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is perceptible as such; cf. note 6.) Ideal parsimony,then, would be achievedwhen a = + 1,/ = + 1, and4 = ? 1. But this combinationis impossible. As observedin Section 1,9 + / + 4 = 0, and so each variableis the inverseof the sum of the othertwo. Consequently,p, 1,and4 mustin this case have identicaldirectionsas well as magnitudes,and so a = == -.Via Defs. (5b) and (5c), ifl/= 4 then - 2y - x = 2x + y, and so 3x = - 3y, hence x = - y, which implies that y - x is even. Thusp is even (via Def. (5a)), and so # +I1, contraryto what was stipulated. The next recourseis to incrementthe magnitudeof one of the voiceleading variables.In principle, any of the three variablescan be incremented, but Q is in prime form (cf. Def. 2) only if p = 1, 1 = 1, 4 = -2. These are exactly the values for the familiarcase of the consonanttriad modulo 12. Once we cease to assumea chromaticsystem of twelve pitchclasses, as we did in section 1, whatothercombinationslead to these values of p, 1,and4? 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 Theorem 1.1. Further,since / = 1, thenthe thirdstep intervalis x + 2, via Theorem 1.2. The three step intervalsof a parsimonioustrichord,then, must form the ascendingconsecutive series .15 The sum of these three step intervalsis 3x + 3, distributedas 3(x + 1). Since c, the numberof pitch-classesin the system, is the sum of the stepintervals,it follows thatparsimonioustrichordsare only availableif c is an integralmultipleof 3. In each such system, thereis a single step-interval series of the form thatrepresentsa parsimonioustrichord-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 Figure8 (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 modularcongruenceof 3x + 3 is imposed on the figure, so that the first term of each expression is representedas a non-negativevalue. The trichordalprime-formQ' = {0, x, 2x + 1} is portrayedalong with its PLR family at the center of Figure 8 . The arrowsdepict the following: *p labels the motion x - x + 1 when P takes {0, x, 2x + 1) to {0, TI x + 1, 2x + 1} across their sharedhypotenuse; * labels the motion 2x + 1 - 2x + 2 when L takes {0, x, 2x + 1 / TI } to {0, x, 2x + 2} across their sharedhorizontaledge; 13
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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
2x+1-
3x+1
x-2
x+3
2x + 3
0
x
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3x
2
x+2
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x- 1
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2x + 4
1
x+1
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x- 2
x +3
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0
x
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3x
x+1•
7
Tonnetz Figure8: TheParsimonious *, labels the T2motion 0 -- 3x + 1 -2 when R takes {0, x, 2x + 1} to {3x + 1, x, 2x + 1})across their sharedverticaledge. Figure 8 is a powerful representationof parsimonious trichordal motion. No matterwhat value is assigned to x, any local trianglewith a southwest/northeasthypotenuserepresentsa parsimonioustrichord.Conversely,all parsimonioustrichordsarerepresentablethrougha realization of Figure 8. To investigatethe scope of this power,we now explore three 14
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2x+3
x+3
3x
2x
0
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x+1
2x+1
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x-2
x+3
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0
x
2x
3x
2
-
1 '
2 2x+4
[6 x+3
x+2 2x+2 3x+2 x1
9]
2x+3
x+ 1
03 0
2x+ 1
3x+ 1
x
[6 2x
1 2xx-2
9 3x
of theParsimonious Tonnetz. Figure9: ThreeRealizations (a)x = 3, y = 4, c = 12,Q'= {0, 3, 7} modulo 12 / RiemannTonnetz) (Oettingen such realizations,for x = 3, 5, and 7 respectively.In each of the three matricesthatcompriseFigure 9, the abstractexpressionsof 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 rotatesFigure 5b, replacingpitch-class names with integers.The series of majorthirdsis retainedon the y axis, but the series of minorthirdsis displacedfromthe northwest/southeastdiagonalof Figure5b to the x axis of Figure 9a, thereby shifting the series of perfect fifths from the x axis 15
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2x+3
x+3
x+2
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of theParsimonious Tonnetz. Figure9: ThreeRealizations = = = = modulo 18 x c (0, 5, 6, 18, Q' 1) (b) y 5,1
to the southwest/northeastdiagonal. This particularversion of the TonnetzactuallyantedatesOettingen:it was firstintroducedby CarlFriedrich Weitzmannin 1853, althoughin a boundedform(as in Fig. 5a), andusing staff-notatedpitches ratherthanintegers.'6The triangularcomplex at the centerof Figure9a portraysthe trichordalprimeform, {0, 3, 7 } = C minor, togetherwith its PLR family.The arrowsrepresentthe following: 16
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*p labels the TIsemitonalmotion 3- 4 = Eb-- E when P takes {0, 3, 7} = C minor to 10, 4, 7} = C major across their shared hypotenuse; * labels the semitonalmotion 7 - 8 = G - Abwhen L takes {0, 1 TI 3, 7) = C minorto {0, 3, 8) = Ab majoracrosstheir sharedhorizontal edge; * 4 labels the Tl0o T-2whole-step motion 0 - 10 = C-- B b when R takes {0, 3, 7) = C minorto { 10, 3, 7) = Eb majoracrosstheirshared verticaledge. In Figure9b, the parsimonioustrichordis Q' = {0, 5, 11) in a modulo 18 ("third-tone")system, with step-intervals. The triangular complex at the center of Figure 9b portrays10, 5, 11} togetherwith the threetrichordsthatcomprise its PLR family.The arrowsportraythe following: 6 when P takes 10, 5, 11) to 10, 6, 11) p* z labels the TImotion 5 --across their sharedhypotenuse; *l labels the T1motion 11-- 12 when L takes {0, 5, 11} to {0, 5, 12 across their sharedhorizontaledge; *,zlabels the T16= T-2motion 0-- 16 when R takes {0, 5, 11) to { 16, 5, 11} across their sharedverticaledge. In Figure 9c (p. 18), the parsimonioustrichordis Q' = 10, 7, 15} in a modulo24 ("quarter-tone") system, with step-intervals. The triangularcomplex at the center of Figure 9c portrays10, 7, 15 } together with the threetrichordsthatform its PLR family.The arrowsportraythe following: p* z labels the T1motion 7 -- 8 when P takes 10, 7, 15) to 10, 8, 15) across their sharedhypotenuse; *l labels the T, motion 15 - 16 when L takes {0, 7, 15} to {0, 7, 16) across their sharedhorizontaledge; * labels the T22= T-2motion 0 - 22 when R takes 10, 7, 15 to {22, } • 7, 15) across their sharedverticaledge. 2.4. The Toroidal Tonnetz Before navigatingthe Tonnetze,we need to confronttheirlimitationas a representationof pitch-classrelationsin equaltemperament.In Figures 8 and 9, pitch-classesoccur in multiplelocations, obscuringtheirequivalence.Alternativelyrepresentingeach pitch-classat a single location,as in Figure 5a, has the equally perniciousconsequenceof obscuringadja17
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of theParsimonious Tonnetz. Figure9: ThreeRealizations 24 (c) x = 7, y = 8, c = 24, Q'= {0, 7, 15) modulo
cency relationships,causing axes to float off the edge of the two-dimensional surface only to reappearon the opposite edge. These obscurities resultartificiallyfromthe mismatchbetween the cyclical natureof pitchclass space andthe flat surfaceof the printedpage. A toruspresentsa geometric figure more appropriateto representingthe cyclic propertiesof equal-temperedpitch-class.Althoughthe torusis eschewed here because it is difficultto renderandinterpreton the two-dimensionalsurfaceof the page, its underlyingstatusneeds to be sufficientlyacknowledgedbefore the Tonnetzcan be navigatedwith full comprehension. 18
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At issue above all is the cyclic periodicitiesof the axes, which fluctuate accordingto the generatingintervalsand the cardinalityof the chromatic system. The axes relevantto Tonnetznavigationare the threethat are generatedby step-intervalsof the parsimonioustrichord.These include 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. DEE (8). z = c - (x + y). All three step-interval-generatedaxes are circularizedby equal temperament,and thus will be referredto as axis/cycles. Ourstudyof the periodicityof axis/cycles will be aidedby puttinginto play a variableq*, representingthe periodicityof step-intervalq modulo C.
DEF. (9). q = gcd(c,q) ' where gcd(c,q), the greatestcommon divisor of c and q, is the largestintegerj such that -c and are positive integers. J J The asteriskis attachableto any variablethat representsa step-interval; hence x* is the periodicityof x modulo c, and so forth. Thereare two generalcases.17 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 attenuatedrepresentation 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 periodicity of 12, andexhaustthe 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) = ~ co-cycles (providedthat 1 is the greatestcommon divisor of the three step-intervalsin Q, as is indeed truefor all cases relevant to this study).An example is presentedby the z axis of Figure9c, 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 mustbe inducedfromthe figure.The orderedcontentof the z axis begin19
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ning at the southwestcornerof 9c is as follows: .) There are gcd(24,9) = 3 distinct z axis-cycles, which partition the 24 pitch-classes. Retreatingby one level of abstraction,considernow the cyclic features of Figure 8 (p. 14), where c = 3x + 3, and the three step intervals are . c = 3x + 3 = 3 (x+1) = 3y, and so y = . Since y evenly divides c, it follows that gcd(c,y) = y, and so y* =- = 3. This Y explains why there are exactly three distinctelementsin each column of the matricesin Figures8 and 9. The significantpointhere is thatthe triple periodicityassociatedwith the second step-intervalis properto the underlying structureof Figure 8, and thus the inclusionof an octave-trisecting interval,acousticallyequivalentto a temperedmajorthird,is common to all parsimonioustrichords.As for the numberof distinctcolumns,thereare gcd(c,y) = y. Thatis: thereis one y-axis co-cycle for each degreeof separationbetween the second and thirdpitch-classin the prime form of Q'. In contrast,the remainingstep-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 modulo9. The smallestexamples of such systems are c = { 12 (N.B.), 21, 30}. Of the systems portrayedin Figure 9, only Figure 9a (p. 15), where c meets this condition,andconsequentlyit is only here thatthe x axis 12, = partitionsits pcs into co-cycles ratherthan exhaustingthem in a single cycle. In this case, x = 3, gcd(c,x) = 3, and x* 12 4. Each x axis thuscontainsfourdistinctpcs, andtherearex = 3 distinctx axes. (In standardterms,of course, what we have here is the partitionof the aggregate into threediminishedseventhchords.)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 exhauststhe system of 18 or 24 pcs. A similarsituationholds for the relationshipof c to the thirdstep-interval. x + 2 cannotbe a multipleof 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 integralmultipleof 3, i.e., there is some positive integern 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 = 16, 15, 24). Of the systems portrayed in Figure9, only Figure9c, where c = 24, meets this condition,and consequently it is only here that the z axis partitionsits pcs into co-cycles ratherthanexhaustingthem 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
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and z are co-prime, and there is only a single z axis which exhauststhe system of 12 or 18 pcs. A practicaldemonstrationof the featuresdiscussed in this section is provided in the pioneering microtonal treatise of Alois Hiba (1927), which systematicallyexplores the generativepowers of intervalsin both quarter-tone(c = 24) and third-tone(c = 18) systems (where "tone"is takenin the sense of "wholetone").In his discussion of the quarter-tone system, Hiba notes that the "neutralthird,"equivalent to 7/24 of an octave, generates all 24 pitch-classes. In contrast,Hiba writes that the interval"a quarter-tonehigherthan a majorthird [i.e. 9/24 of an octave] leads only as far as an octachord [Achtklang].This octachordis composed of the symmetricpartitionof a majorsixth.... Two transpositions of the octachordupwardby a quarter-toneuse the remaining16 tones of the quarter-tone-scale" (1927, 166). Hiba's threeoctachordsare equivalent to the three distinct z-axis co-cycles discussed above in association with Figure9c. In a subsequentchapter,Hiba 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-tonesystem, and indeed in a morebroadlyexpandeddistributionacross a spanof five octaves. The successive series of 18 seven-thirdtones forms a collection spreadacross seven octaves"(202-203). H1iba'saggregate-completing Nacheinanderfolgenare equivalentto the x and z axes of Figure 9b. The perspectivecultivatedthroughoutthis section providesa systematic foundationfor Hdiba'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 constantperiodicity, and the numberof y co-cycles varies with the size of the chromatic system. Conversely,the x and z axes, generatedby the first and thirdstep-intervalsrespectively,have a variableperiodicity,but the number of x and z co-cycles is constantto within the modulo-3 congruence of c. The relevanceof these findings, and particularlyof the special status of the y axis, to progressionsbased on PLR-familyoperationswill become apparentin Section 3. 3.1. PLR-family Compounds We are now in a position to navigatethe toroidalTonnetz,in all its various manifestations,using PLR-family operations as our vehicle. The maximal common-tone retention inherent to these operations insures thatthe cruise will be smooth, particularlywhen the Tonnetzis parsimonious. Ourexplorationwill follow a systematicprogram,focusing on tri21
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(a)
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Figure 10: Cyclic Periodicitiesof Step Intervalsof
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chordalprogressions,or chains, generatedby the recursiveapplicationof a patternof PLR-familyoperations.'8Where appropriate,chains will be viewed as segments of cycles. The basic cycle-classes formedby generated PLR-familychains are few in number,and their relation to PLRfamily operationsis roughly analogous to the role of the chain of fifths (Sechterkette)in diatonicprogressions:they constitutenormativeprototypes againstwhich particularprogressions,in all theirvarietyand complexity, may be gauged. The ultimategoal of this investigationis the pragmaticone of exploring parsimoniousvoice-leading among consonanttriadsin a 12-pc system. Consistentwith the frameworkestablishedin Section 2, this familiar phenomenon is situated as a particularmanifestation of a more general one: the behavior of parsimonioustrichordsin 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 hypotheticalmusical 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 researchpresentedin this paperreflectsthe deep influence of Gerald Balzano's classic study (Balzano 1980). Like Balzano, my motivationsarenot only compositional.They stem as well from an intuition,perhapsa credo, that insights into the propertiesand behavior of individualinstances are furnishedby studying the propertiesand behaviorsof generalphenomenawhich they represent.As inhabitantsof a planetthat sustainslife, the value of exploringotherplanets, solar systems, or galaxies for their life-sustainingproperties,or lack thereof,potentially transcendsthe conceivable materialbenefits, 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-familyoperations, enclosed in angled brackets.The operationsapply in the order in which they appearin the set, from left to right.For example, in the compound operation, R is applied, P is applied to the productof R, and L is appliedto the productof R-then-P. 3.1.2. T/I Equivalences of Compound Operations. All compound operationsare equivalentto either transpositionsor inversions,depending on the cardinalityof the orderedset. Compoundoperationsof odd cardinalityare inversions,those of even cardinalitytranspositions.This follows from the inversionalstatus of PLR-family operations,together 23
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with the group structureof inversionand 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 orderedsubsets. The ordered subset, singly iterated,constitutesthe generator of the compound.The compoundcan be expressedas Hn, whereH is the generatorandn 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 Hn is a transposition. In orderfor Hn to be an inversion,#H and n must both be odd. This follows from the observationsmade in 3.1.2 togetherwith the multiplicativepropertiesof odd and even integers. 3.1.5. Cycles. H generatesa 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 equivalentto #H - q*. 3.1.6. Tonnetz Representations of Cycles. Generatorsof odd cardinality are involutions:they retreatto their point of origin on the Tonnetz aftertwo iterations.This is restatedformallyas Theorem2 in the appendix, where a proof is offered. Generatorsof even cardinality,by contrast, aredevolutions:they move perpetuallyaway fromtheirpointof origin on the Tonnetz.Cycles are generatedonly when a modularcongruencegoverns the Tonnetz,in which case the generatorhas the same periodicityas the transpositionaloperationto which it is equivalent(cf. 3.1.4). These periodicitiescan be computedby firstexpressingthe PLR-familygeneratoras a transpositionoperationT,, 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 progressionsthatthey generateare insufficientlyvariedto serve as compelling musical resources.Thus our explorationbegins with binarygeneratorsthat pair distinctPLR-familyoperations.Thereare six such generators,which groupinto threeretrograde-related pairs: (1) and ; (2) and ; (3) and . 24
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It has alreadybeen determined(cf. 3.1.2) thateach binaryoperationis equivalentto some T,. Each transpositionalvalue n associatedwith a binaryoperationis equivalentto a non-zerodirectedintervalwithinthe trichordthat is the object of operation.To see why this is so, considerthat each individualPLR-familyoperationaltersone pitch-classin Q, and so each binary operationalters two pitch-classes in Q. It follows that the productof a binaryoperationsharesat least one commonpc with Q. The common-tonetheoremfor transposition(Rahn 1980, 108) dictatesthatif Qr~T,(Q) > 0, thenthereis some {q1,q2) E Qsuch thatq2- q1 = n. From this it follows that each binary PLR-family operationtransposesQ by some intervalinternalto it. The transpositionalvalues associatedwith the six binary operations are exactly the three step-intervalsand their inverses, ?x, ?y, and ?(x + y). Figure 11 (p. 26) matchesdirectedintervalsto their associatedbinary operations.The six directed intervalsof Q are listed as transpositional values at (a). Their binary operationequivalentsare shown at (b). The transpositionalvalues areimplementedon Q at (c). The binaryoperations are implementedat (d), where they are representedon the abstractTonnetzas arrowsleadingout of Q. EachoperationtransposesQ by one position along the axis representingthe step-intervalwith which it is associated. (Forexample,the associationbetween Txand is confirmedby the matrix position of (Q), one step rightwardof Q along the x axis.) Consequently,the vertices of each resultanttriangle at (d) are exactly the pitch-classes resulting from the associated transpositionat (c). (For example, {x, 2x, 2x + y } appearsboth as the set of vertices of (Q) at (d) and as the resultof Tx(Q) at (c).) Note that inversely related step-intervalsare associated with retrograde-relatedoperations.For example, the association of Tx with is complementedby an associationof Tx with . More generally: THEOREM3. Foran orderedPLR-familyoperationset H andits retrograde Ret(H), if H = T,, then Ret(H) = T,.
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 intervalsrepresentsix distinctvalues, with the
lone exception that, if x = 1 and c = 6, then x + 2 = - (x + 2). This excep-
tion aside, the six binaryPLR-familyoperationsproducesix distinct trichordswhen implementedon Q'. Figure 12 (p. 27) translatesFigure 11 into terms specific to parsimonious trichords.The transpositionsat (a) aregiven in threeforms:as positive and negative generic step-intervals,as positive and negative step25
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(a)
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Tx
{x + y, 2x + y, 2x + 2y) {O,x, x + y}
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