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(major-- minor).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 thirdvoice, which proceeds by
1
p
T
Q
b'
.
0o
0&
C- -
[]ll
|'
0
Ab+
11
R
b[]
0o
0o
0o
.0
C- -
C+
C-
006^
R
0
b?y
Eb+
Figure1: The PLR-Familyof C-Minor
?J
O
eee-
-*
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 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-leadingis 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 are special.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
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 reasonsthatwill 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 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-familyoperationscan now be definedon a prime-formtrichord 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 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,p, /, and ,, which express the
magnitudeof those externalmappingsas mod-c transpositionalvalues.
xhence:
y,
h0--c
DEE (5a) p = y - x
DEE (5b) l = -2y - x
-2x + y, hence:
DEE (5c) , = 2x + y
Observethatp + 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 <j-i, k-j, i-k>, modulo c.
Via Def. (2), Q = {0, x, x+ y} is in prime form, which presupposesnormal order.Thus the step-intervalseries of Q is <x, y, - (x + y)>.
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-Family
Mappings
(a) forQ = {0, x, x + y}
P(Q)= 17(Q)
L(Q)= I3(Q)
R(Q)= 17(Q)
7 ->0
3- 4
7-8
3 ->0
7->3
3-> 7
0-
0-
7
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
0-->5
0-> 1
0-6
5
Figure3: PLR-Family
Mappings
(c)x= ,y=4,Q ={0, 1,5}
The following theoremstates thatp, I, and, are each equivalentto the
differencesbetween a distinctpair of step-intervals.
THEOREM1. For a prime-formtrichordQ = {0, x, x + y with step-
intervals <x, y, - (x + y)>,
1.1) p is the difference between the first and second step interval,
Proof: y - x = p via Def. (5a).
y - x.
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).
Theorem1 facilitatescalculationof the values of , l, and4 for any trichordal prime form in a chromatic system of any size. The results of
4
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 conjunct melodic 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 2, l, 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
unique characteristicnoted 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 function of
the trichord'sstep-intervalsizes, which are an aspectof its internalstructure;(3) the optimalvoice-leadingpropertiesof triadsthereforestandin
incidentalrelationto theiroptimalacoustic 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
Europeanmusicaltraditionas othersortsof over-determinationthatwere
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 that3 approximatessome powerof 2; of the proximityof
the perfect fifth's - geometric division of the octave (the source of its
acoustic power) to its I 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 incessant exposure 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
I
r
<x, y, -(x+ y)>
p
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
{0,1,5)
< 1,4,7>
3
(0,1,6)
< 1,5,6>
4
(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
(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
prime form
(0,x, x+ y)
step-intervals
5
3
6
FiW
8
Figure4: Values of p, 1, and r for trichordalprimeforms
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(Verwandtschaftsverhiltnistabelle)
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.10LeonhardEuler(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 matrixabout the 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
m
--
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
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 European harmonic 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 musicalqualitiesthatthey 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 directs the 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 1 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
context of equal temperament.The focus on the TonnetzthroughoutSection 2 serves as a large structuralupbeatto 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- y
-x- y
-y
x- y
-2x - 2y
-x - 2y
-2y
x - 2y -
2x
2x- y
/T-2y(Q)
2x- 2y
Figure6: TheAbstractTonnetz
the paper,which uses PLR-familyoperationsto navigatethe Tonnetz,in
its variousversions at variousdegrees along 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 analyticgeometry.
In termsof Def. (2), the primaryaxes of Figure6 are interpretedas the
smallesttwo step-intervalsof a primeform trichordQ = {0, x, x+y } with
step-intervals<x, y, -(x+y)>. The remainingstep-intervalis the inverseof
the sum of the two smaller step-intervals,and hence generatesthe diag10
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,13 andeach
edge a dyadic subset, of Q. Any geometric translationof this trianglethat is, any triangle whose hypotenuse subtendsnorthwestof the right
angle-represents a pitch or pitch-classtranspositionof Q. (An example,
labeled TX2y
(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 indicates three such invertedtriangles,all sharing an edge with the centraltriangle.These three trianglesrepresentthe
PLR-familyof Q (cf. Figure 3a). The unique edge that the centraltriangle shareswith 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:
* 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 sharedhorizontaledge;
*?,labels 0 - 2x + y when R takes {0, x, x + y} to {2x + y, x, x + y}
across their sharedverticaledge.
When the same operationsare enacted on other members of trichordclass Q, theirgeometricorientationon Figure7 is invariant.In all cases,
P-relatedtriadssharea hypotenuse,anda 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, and, executes a knight'smove, a displacementby two columns andone row.The magnitudesof p, I, and, are
likewise invariant,althoughwhen the objectof the mappingis a triadTJIrelatedto the primeform,the directionof the arrowreverses,and the values of a, I, 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
2x
x-y
2x-y
of
Figure7: TonnetzRepresentation
on Q = {0, x, x + y
PLR-Family
Operations
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.14These 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 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 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 intervals z, I and 4 are as small as possible, but
greaterthanzero. (The last conditioninsuresthatvoice-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 impossible. 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 identicaldirectionsas 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 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, I = 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 , 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, thenthe second step intervalis x + 1, via Theorem 1.1. Further,since I = 1, thenthe thirdstep intervalis x + 2, via Theorem 1.2. The three step intervalsof a parsimonioustrichord,then, must
form the ascendingconsecutive series <x, x + 1, x + 2>. 5
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 <x, x + 1, x + 2> 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' = I0, x, 2x + 1} is portrayedalong with its
PLR family at the centerof 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 sharedhorizontaledge;
13
x+3
2x + 3
0
x
2
x+2
2x + 2
3x + 2
1
x+ I
2x+ I
/',"P I
2x+4
I
x+3
2x + 3
0
I
2x
x-
3x
2x-
3x+ l
x-2
2x
3x
/
I
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
2
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. Conversely, 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
2x
x
0
3x
F6
2
x+2
w
I
2x +4
E
3x+2
w
E
x+ 1
2x+ 1
a
2x+3
0 /
E+
Ex-''
1
x+2
2x +2/
811]
I
10
2x- 1
B5
3x+ 1
x-2
2x
3x
x- I
2x- I
r .%
',
x
[x1
2x +4
x-I
2x+2
x+
,
3x + 2
11
2x+
F5]
3x+ I
x- 2
2x
3x
B4
0
x+3
2x+3
D
FB9 Fo
x
[_]
[H
0
of theParsimonious
Tonnetz.
Figure9: ThreeRealizations
12
(a) x = 3, y = 4, c = 12,Q'= {0, 3, 7} modulo
/ 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
2x+3
x+3
2x+2
x+2
2
m
m
x+ 1
x1
x+2
2
m
2x +4
a7
1
m14
Li
2x + 3
x+3
[i
x-1
3x+ 1
2x+ I
,
2x - 1
16
x-2
El
x
2x
3x
5
10
15
3x+2
x- 1
2x- 1
[17
[4
x+ 1
2x+ 1
3x+ 1
[]
11
16
3]
x
2x
3x
10
15
0
2x + 3
x+3
215
z]3
W BL.
(14
10
x
3x+2
H
I
2x +4
3
3x
2x
x
0
OF
281 213
2+' -
,
2x +2/
g
0
[5]
El
x-2
of theParsimonious
Tonnetz.
Figure9: ThreeRealizations
(b) x = 5, y = 6, c = 18, Q'= {0, 5,11}
modulo 18
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.16The triangularcomplex at the
centerof Figure9a portraysthe trichordalprimeform, {0, 3, 7 = C minor,
togetherwith its PLR family.The arrowsrepresentthe 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 theirsharedhorizontal 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 majoracrosstheirshared
verticaledge.
In Figure9b, the parsimonioustrichordis 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
threetrichordsthatcomprise its PLR family.The arrowsportraythe following:
* 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 sharedhorizontaledge;
16 when R takes {0,5, 11 to 16,
*-zlabels the T16= T_2motion 05, 11 across their sharedverticaledge.
In Figure 9c (p. 18), the parsimonioustrichordis Q' = {0, 7, 15} in a
modulo24 ("quarter-tone")
system, with step-intervals<7, 8, 9>. The triangularcomplex at the center of Figure 9c portrays{0, 7, 15} together
with the threetrichordsthatform its PLR family.The arrowsportraythe
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 sharedhorizontaledge;
* labels the T22r T-2motion 022 when R takes {0, 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 multiple locations, obscuringtheirequivalence.Alternativelyrepresentingeach pitch-classat a single location,as
in Figure 5a, has the equally perniciousconsequence of obscuringadja17
x+3
1101
2
2x+3
2x + 2
1
2
2x +4
2
1
0
0
2x +4
E10
2x+3
17
x- 1
/
2x- 1
131
3x +
x-2
3x
x
2x
7
E14
2
E
3x+2
x- 1
2x- 1
16
26
x+ 1
2x+ 1
3x+ 1
x-2
[15
1221
E
E
[18]
x+3
2x+2
2x+ 1
3x
[21]
E
23]
x+ 1
717
F
[21
[
3x + 2
6
2x+3
x+2
2x
14]
17]
x +2
x+3
E[10
x
0
0
E
x
[13]
2x
[14]
3x
21
of theParsimonious
Tonnetz.
Figure9: ThreeRealizations
24
(c) x = 7, y = 8, c = 24, Q'= {0, 7, 151 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 renderand interpreton the two-dimensionalsurfaceof the
page, its underlyingstatusneeds to be sufficientlyacknowledgedbefore
the Tonnetzcan be navigatedwith full comprehension.
18
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 three that
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.
DEF. (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* = cc
where gcd(c,q), the greatestcommon divigcd(c,q) '
sor of c and q, is the largestintegerj such that c and 9 are positive
J
J
integers.
The asteriskis attachableto any variablethat representsa step-interval; hence x* is the periodicityof x modulo c, and so forth.
There are two generalcases."7If 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) = c co-cycles (providedthat 1 is the greatestcommon divisor of the threestep-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
must be inducedfrom the figure.The orderedcontentof the z axis begin19
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,considernow the cyclic features
of Figure 8 (p. 14), where c = 3x + 3, and the three step intervals<x, y,
z> are <x, x+1, x+2>. c = 3x + 3 = 3 (x+l) = 3y, and so y = c. Since y
evenly divides c, it follows that gcd(c,y) = y, and so y* = Yc = 3. This
explains why there are exactly threedistinctelements in 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 that3n = 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 Figure9, only Figure9a (p. 15), where c
= 12, meets this condition,andconsequentlyit is only here thatthe x axis
partitionsits pcs into co-cycles ratherthan exhaustingthem in a single
12 =
4. Each x axis
cycle. In this case, x = 3, gcd(c,x) = 3, and x* =
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 thatexhauststhe
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., thereis 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 = {6, 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
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 Haba (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 "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-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). Haba'sthreeoctachordsare equivalent to the three distinct z-axis co-cycles discussed above in association
with Figure9c. 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-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). Haiba'saggregate-completing Nacheinanderfolgenare equivalentto the x and z axes of Figure
9b. The perspectivecultivatedthroughoutthis section providesa systematic foundationfor Haiba'sobservations.
Figure 10 (p. 22) summarizesthe work of this section by providinga
table for calculating the cyclic periodicities for the chromaticsystems
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 operationsas 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
(a)
c mod-9
(b)
(c)
x axis
y axis
periodicity
# co-cycles
periodicity
# co-cycles
(x*)
(c/x*)
(y*)
(c/y*)
0
c
1
3
c
3
3
c
3
3
3
c
3
6
c
1
3
c
3
Figure 10: Cyclic Periodicitiesof Step Intervalsof
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
systemthatis 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, thatinsights 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 orderin
which they appearin the set, from left to right.For example, in the compound operation<RPL>, 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
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
<PRPRPR>is binary-generated,since it can be partitionedas <<PR>
<PR><PR>>,and expressedas <PR>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 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 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 equivalentto
#H q*.
3.1.6. Tonnetz Representations of Cycles. Generatorsof odd cardinality are involutions:they retreatto theirpoint 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 originon
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<L>, <P>, and <R> are involutions,the
progressionsthatthey generateare insufficientlyvariedto serve as compelling musical resources.Thus our explorationbegins with binarygeneratorsthat pair distinctPLR-familyoperations.There are six such generators,which groupinto threeretrograde-related
pairs:
(1) <PR> and <RP>; (2) <LP> and <PL>; (3) <LR> and <RL>.
24
It has alreadybeen determined(cf. 3.1.2) thateach binaryoperationis
equivalentto some Tn.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 common pc with Q. The
common-tonetheoremfor transposition(Rahn 1980, 108) dictatesthatif
Q n T,(Q) > 0, thenthereis some {ql, q2} E Q such thatq2- q, = 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 T,and <RP> is confirmedby
the matrix position of <RP>(Q), one step rightwardof Q along the x
axis.) Consequently,the vertices of each resultant triangle 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
<RP>(Q) at (d) and as the resultof T,(Q) at (c).)
Note that inversely related step-intervalsare associated with retrograde-relatedoperations.For example, the association of Txwith <RP>
is complementedby an associationof Tx with <PR>. More generally:
THEOREM3. Foran orderedPLR-familyoperationset H andits retrogradeRet(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 intervalsrepresentsix distinctvalues, with the
lone exception that,if x = 1 and c = 6, then x + 2 = - (x + 2). This exception aside, the six binaryPLR-familyoperationsproducesix distincttrichords when 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
(a)
(b)
(c)
Tx
<RP>
{O,x,x+y}
T-x
<PR>
{O,x,x+y}
Ty
<PL>
T_y
<LP>
(O,x, x+y}
Tx+y
<RL>
{0, x, x + y}
T-x-y <LR>
0,
<Tx >
>{x, 2x, 2x+y
>{-x, ,y}
x +y}
>
<T-v >
v
<Tx+y >
,x +y, xx+2y}
>(-y,x-y,x}
>{x+ y, 2x + y, 2x + 2y}
-y, O
(O,x,x +y}
y>{-x-y,
(d)
-x + 2y
2y
x + 2y
2x + 2y
Figure 11: AbstractTranspositionalEquivalences
for BinaryGenerators
(a)
Tx
T-x
(c)
(b)
T2x+3
<RP>
{0, x,2x+ 1
<PR>
2
{0, x,2x
<PL>
{0, x, 2x+ 1}
x>
){x, 2x, 3x+ 1
<T2x+3>
1
Ty
Tx+1
T-y
T-(x+l)
T2x+2
<LP>
{0,x,2x+}
Tx+y
T-(x+2)
T2x+l
<RL>
{0, x, 2x+l
T-x-y
Tx+2
<LR>
{0,x, 2x+1
+>
{2x+3,0,x+
>{x+
<T2x+2 >
<T2x+1>
<Tx+2 >
1
, 2x+ 1, 3x +2}
)>{2x+2,3x+2,
x}
>{2x+ 1,3x+ 1,x -1}
>x+ 2,2x+2,
0
(d)
x+2
2x+2
3x + 2
x- 1
1
x- 1
Figure 12: BinaryGeneratorsand ParsimoniousTrichords
x+4
2x + 4
3
x+3
2x+ 1
x +1
1
2x+3
<
0
>3
x
<PLP
x+2
2
2x + 5
2x + 4
x+4
<RP
<PRP
<PRPR <PRPR
>
PRP
RP>
3
<PR>
3
x +3
<PL>~
3x + 2
2x + 2
<PLP>
<PL>
2x+ 1
x +1
1
<
<R>
<P>
<PRP>
<PR>
<PR>
0
2x + 3
<
x
<L>>
2x + 5
2
x +2
<LRL
x+4
3
2x + 4
<LRL
RL>
><LR>
x+3
"1
2x + 3
<LR>
2x + 2
<LP>
3x + 2
<LPL>
~<L LP>
2x + 1
x +1
<LPL
PL>
<
x
0
Figure 13: BinaryChains on the Parsimonio
intervalsof the generic parsimonioustrichordQ', and as positive values
modulo 3x + 3. It is these lattervalues thatgeneratethe mappingsat (c).
Figure 12(d) transfersthe label/arrownetwork from Figure 1l(d) onto
the parsimoniousTonnetz(cf. Figure8). As with Figure 11, a comparison
of the trianglevertices at (d) with the results of the transpositionalmappings at (c) confirmsthe associationsof binaryPLR-familyoperationto
pc-transpositionassertedat (a) and (b).
3.3. Binary Chains and Cycles
Having explored the transpositionalbehavior of binary operations
singly iterated,we now study the chains generatedthroughtheir recursive application.Figure 13 appliesthe six generatorsto Q'= {0, x, 2x+1 ),
as representedon the parsimoniousTonnetzof Figures8 and 12. Pursuant
to Theorem3, retrograde-related
generatorsproceedinverselyout of Q'.
Consequently,any generatorand its retrogradecombine to form a single
chain. For the sake of proceduraleconomy, it will be useful to providea
unifiedlabel for each chain: invokingalphabeticalprecedence,the three
binary-generatedchains will be referredto as <LP>, <LR>, and <PR>.
Each of the three chains representedon Figure 13 threads a space
boundedby two paralleland adjacentaxes. The <PR> chain, whose generatoris associated with the x interval,threadsa pair of x axes and thus
proceeds horizontally;the <LP> chain, associated with the y interval,
threadsa pairof y axes andthusproceedsvertically;andthe <LR> chain,
associated with the x + y = z interval,threadsa pair of z axes and thus
proceedsdiagonally.These affiliationsbetweenbinarychains, step intervals, and Tonnetzdirectionsare constantto all realizationsof the generic
Tonnetz.
In whatsense are the chains of Figure 13 cyclic? Withthe <LP> chain,
the cyclicity is readilyapparent:both <PL>3,at the top of the figure,and
<LP>3,at its bottom,areequivalentto Q', indicatingthatthe operational
periodicityof the <LP> chain is 3. (Note thatthis periodicityis properto
the parsimoniousTonnetz;were the chains tracedon the generic Tonnetz
of Figures6 and 11, no such cyclicity would be evident.)Withthe other
two chains, no such equivalences are apparent,and so these chains are
not cyclic in the abstract.Once a specific integer value is assigned to x,
however,the <LR> and <PR> chains become cyclic. For example, if x =
3, then <PRPRPRP>(Q)= {3, x + 4, 2x + 4} at the figure's left edge is
equivalentto <R>(Q) = {x, 2x + 1, 3x + 1 } just rightof center:both trichords are equivalentto {3, 7, 10}. This indicates that the <PR> chain
has an operationalperiodicityof 4. This periodicitydoes not hold, however, if x is assigned a differentvalue. It is properto the Oettingen/Riemann Tonnetz.but not to the parsimoniousTonnetzof Figure 13, where,
29
(a)
(b)
(c)
(d)
operational chain-class trichordal pitch-class
periodicity cardinality periodicity cardinality
perco-cycle
Modulo
<LP><LP>
c
r
3
modulo9
3
c
6
6
2c
2c
3
3
3
c
3
3
<PR>
c
1
2c
c
c
1
2c
c
c =6
c
3
2c
2c
modulo 9
3
3
3
c0,
6
modulo9
c=0,3
modulo9
<LR>
andCardinalities
of Binary-Generated
Figure14:Periodicities
Cycles
unlike <LP>, neither the <PR> nor the <LR> chain exhibits constant
periodicity.
Althoughvariable,the operationalperiodicitiesof <PR> and<LR> for
specific values of x can nonetheless be easily determined.To do so, we
only need coast on the momentumof ourlaborfromSection 2.4. Because
binary operationsare equivalent to step-intervaltranspositions,as discussed in Section 3.2, the two necessarilyhave identicalperiodicities.We
have alreadyseen this for the case of the <LP> chain, whose operational
periodicityof 3 derives from the triple periodicityof its associatedstepintervaly (cf. Figure 10(c), p. 22). The periodicitiesfor <PR> and <LR>
likewise transferfrom that of their associated step-intervals,x and z,
respectively.Figure 10(b) gives the value of x*, the periodicityof x modulo c, and these values apply directly to the periodicity of <PR>: if c
- {0,6} modulo 9, then the operationalperiodicityof <PR> is c; if c = 3
modulo 9, the periodicity is ?. Figure 10(d) gives the value of z*, the
3.
periodicityof z modulo c, and these values apply directly to the periodicity of <LR>: if c - {0,3 modulo 9, then the periodicityof <PR> is c;
if c = 6 modulo 9, the periodicityis -. These periodicitiesare summarized in column (a) of Figure 14.
30
Other significantfeatures of binary-generatedcycles are directly inferrablefrom the foregoing.
Cardinality of cycle-classes. When one of the chains on Figure 13
(p. 28) is transposed,axes and thus trichordsmay be exchanged,but the
class membershipof the chain (<LP>, <PR>, or <LR>) is preserved.It is
thus importantto makean ontologicaldistinctionbetween specific cycles
and the classes that they represent.This distinctionleads us to ask how
many distinct cycles belong to each class, i.e. what is the cardinalityof
thatclass. To answerthis question,we begin by observingthateach cycle
engages two pitch-classaxes, and each axis participatesin two cycles. It
follows thatthe cardinalityof each binary-generatedcycle-class is equivalent to the quantityof distinct pitch-class axes of that generator'sstepintervalassociate, as given in Figure 10 (p. 22). This quantityequals -c,
wherec measuresthe chromaticsystem as in Def. (1), and q* is the penodicity of the binary generatoras at Figure 14, column (a). This informationis summarizedin Figure 14, column (b).
Trichordal periodicity. The trichordalperiodicity of a generated
cycle equalsthe productof operationalperiodicitytimes generatorcardinality (cf. 3.1.5). In the case of binarygenerators,trichordalperiodicity
doubles operationalperiodicity.All <LP> cycles thus engage six trichords,but the trichordalperiodicityis variablefor the <PR> and <LR>
cycles. This informationis summarizedin Figure 14, column (c) for each
cycle-class.
Pitch-class engagement. Considera binaryoperatorH associatedwith
step-intervalq, where q* representsthe periodicityof q and hence of H.
* If q and c are co-prime,q* = c (cf. 2.4). Then both axes threadedby
an H-generatedchain includethe entirepc-aggregate.It follows that
an H-generatedchain engages all c pitch-classes of the chromatic
system.
* If, however, q and c share a common divisor greaterthan 1, then
q* < c, and the c pitch-classespartitioninto co-cycles (cf. 2.4). The
two q axes threadedby an H-generatedchain thus have distinctpccontent. Since each of these axes has a periodicityof q*, it follows
thatan H-generatedchain engages 2q* pitch-classes.Thus an <LP>
cycle always engages a hexachord.(The prime form of this hexachord is {0, 1, x + 1, x + 2, 2x + 2, 2x + 3}, with step-intervals
<1, x, 1, x, 1, x>.) Dependingon the modulo 9 congruenceof c, the
<LR> and <PR> cycles eitherengage the entirepitch-class system,
or 2 of that system. (In the latter case, the prime form of the set
engaged by the cycle can be representedas {<<3n,3n + 1 } for n
0 to c with step intervals<1, 2 1, 2...>.) This informationis summarizedin Figure 14, column (d).
31
(a)
(b)
(c)
Tx
T3
<RP>
{0,3, 7}
T-x
T9
<PR>
{0, 3, 7}
Ty
T4
<PL>
{0,3, 7}
T_y
T8
<LP>
{0,3,7}
Tx+y
T7
<RL>
{0,3, 7}
T-x-y
T5
<LR>
{0,3, 7}
<T3>
<T9>
<T4 >
<T8 >
<T7>
>{3,6,10}
C-
>{9, 0, 4}
C-
>{4,7, 11}
C-
>{8, 11,3}
C-
>{7, 10, 2}
C-
>{5, 8, 0}
C-
<T3>
<T9>
<T4>
> Eb-
>A -
>E-
<T8 >
<T5>
>Ab<T7>
<T5>
>G-
>F-
(d)
5
8
11
2
Figure 15: Binary Generators and Consonant Triads modulo 12
In general, the smaller the periodicity of the step intervalassociated
with an operation,the more economical is the pitch-class design of the
cycle generated from that operation.The significance of this circumstance for nineteenth-centuryharmonywill become clear in Section 3.4.
3.4 Binary Chains and Cycles in Modulo 12
In this section, the propertiesof parsimonioustrichordsestablished
above are studiedas they apply to the object of historicaland analytical
interest,the consonanttriadin a system of 12 pitch-classes.We begin by
studyingthe six binaryoperations,singly iterated,as we did in association with Figure 12 (p. 27). Figure 15 providesa realizationof Figure 12,
with x = 3. This assignmentimmediatelybrings the familiarobject and
system into view. Q' = {0, x, 2x + 1 is realized as the consonanttriad
{0, 3, 7) = C minor,the step intervals<x, x + 1, x+ 2> as <3, 4, 5>, and
the chromaticsystem 3x + 3 as the 12-pc system. The parsimoniousTonnetz becomes an equal-temperedversion of the one discoveredby Euler
and Oettingen and propagatedby Riemann (cf. Figure 9a). The triads
producedby the six binaryoperationsindicatedon Figure 15 constitute
a complete inventoryof the minortriadsthat share a pitch-class with C
minor.
To explore the recursive properties of these operations, Figure 16
(p. 34) transfersthe threebinary-generatedchains of Figure 13 onto the
Oettingen/RiemannTonnetz.(These chains are somewhatre-positioned,
in partfor visual clarity.)Each of these three classes of triadicprogression is familiarfrom music of the nineteenthcentury,and each has distinctiveproperties.
The group structureof <LP> chains has been studiedby Hyer (1989,
1995), and some of their special qualities were explored from different
perspectivesin an earlierpaperof mine (Cohn 1996, where I refer to a
chain of this type as a "maximallysmooth cycle") and in Lewin (1996,
who classifies it as a "CohnCycle"). The particular<LP> chain threading the vertical axes of Figure 16 is the one that is traversedin a downwarddirectionin a passage from Brahms'sDouble Concertothatis modelled in Example 1 and that I have alreadystudiedin some detail (Cohn
1996, 13-15). The matrix representationillustrates several significant
featuresof <LP> cycles identifiedin my earlierpaper:
Set-class Non-Exhaustion. <LP> generatesa cyclic progressionof 6
triads,a sub-groupof the 24 triads.
Limited pc engagement. Both engaged columns have a cyclic periodicity of 3, so thatan <LP> cycle engages only 6 pitch-classes.The
pc-set unites two adjacentIC-4 cycles (representedby the two yaxis columns) into a hexatonicset belonging to Forte-class6-20.20
33
1
4 .........7
9
Ab+/
/Ab-
10
11
2
5
8
11
10
1
4
7
1
4
|
7l
9
0
3
6
5
8
|
11
2
9 0:-
.......56
9
-
I<
X
...
-------...... 3.
....
Eb-
8
1
4
A+
9
11
| Db+/
1 .--
4 ---------7
j
1 XGb+
C-
6-----------0
B+/
X+^^ |
5
8
/Bb- I
7 --------10
A
9 0 3 6 90Eb+
0
3
/
| Ab+/
5
^
E+/
1 - 4------
7
9
8
Ab+/
4
3 Brahms 6
5
IE+/
1
| Eb+
3
Schubert
11
2
5
8
11
7
10
1
4
7
6
9
0
3
---
0 --------3
Figure 16: BinaryChains on the Oettingen/ Rie
(Fy ,|J
J
r #r
Lml
J
rf
7
7
J
1>
,!J
/
r
J
J
r
Example1: Brahms,Concertofor Violin andCello,
Op. 102, FirstMovement,mm. 270-78
6 66
J
hh
#d
^
i
6
1
Example2: Schubert,Overtureto Die Zauberharfe,openingAndante
Multiple distinct cycles. The numberof distinct<LP> cycles is equivalent to the numberof distinct y axes: four. In my earlierpaper,I
referto these four cycles as hexatonic systems, in referenceto their
limited pc-content. In the currentcontext they are more appropriately referredto as sub-systems.
The particular<PR> cycle threadingthe horizontalaxes of Figure 16
is the one thatis traversedright-ward(in referenceto the spatiallayout of
Figure 16) in Example 2, which models the Andanteintroductionof the
overtureto Schubert'sopera Die Zauberharfe(1820).21The three features properto <PR> cycles are analogous to those identifiedabove for
<LP> cycles, and this analogy is reflected in the parallel descriptionof
them below:
Set-class Non-Exhaustion. <PR> generatesa cyclic progressionof 8
triads,a sub-groupof the 24 triads.
Limited pc engagement. Both engaged rows have a cyclic periodicity of 4, so that a <PR> cycle engages 8 pitch-classes. The pc-set
unitestwo adjacentIC-3 cycles (representedby the two x-axis rows)
into an octatonic set belonging to Forte-class8-28.
Multiple distinct cycles. The number of distinct <PR> cycles is
equivalentto the numberof distinctx-axes: three.These cycles can
be referredto as octatonic sub-systems (cf. Lerdahl1994, 132-33).
Any <PR> cycle-segment is a (properor improper)subset of one of
the three<PR> cycles.
35
The situationwith the <LR> cycle, which threadsthe diagonalaxes of
Figure 16, is entirelydifferent.The threefeaturesnotedfor the <LP> and
<PR> cycles have no analoguehere.
Set-class Exhaustion. The compoundoperation<LR> generatesthe
entire groupof 24 triads.
PC-Aggregate Completion. Both engaged z-axis diagonals have a
cyclic periodicityof c = 12, and an <LR> chainengages all 12 pitchclasses;
One Single Cycle. There is a single <LR> cycle, of which any <LR>
cycle-segment is a subset.
The complete <LR> cycle is too long to sustaincompositionalinterest
under normal conditions. This is broughtout by a thirty-four-measure
passage which I have writtenabouton two earlieroccasions (Cohn 1991
and 1992) from the second movementof Beethoven's Ninth Symphony
(mm. 143-76) . From a startingpoint at C-majorin the northeastcorner
of Figure 16, the passage obsessively whirls through<RL>9,traversing
eighteen triadsbefore halting at the nineteenthlink, the A-majortriadin
the southwestcorer. Even a squallof such velocity andforce cannotpropel itself aboutthe entire cycle.
Nonetheless,the <LR>cycle has a morevenerablelegacy thanthe other
binary-generatedcycles. Singly iterated, <LR> transposes by perfect
fifth, the transpositionalvalue that preserves maximum pc-intersection
between diatonic collections (as well as the intervalthat sports acoustic
privilege). The <LR>-cycle was initially recognized in thorough-bass
methodsfrom the late seventeenthcenturyas a gauge of modulatorydistance(Lester1992, 215). In 1827, JohannBernhardLogierrecommended
that young musicianslearnto play the entire cycle at the pianoforte,"as
it forms the groundworkon which may be constructedan almost infinite
numberof passages and variations."Logier's assessment of the cycle's
propertiesis pertinentto the approachadoptedhere:
We perceive,fromthe beginningto the end, not only a beautifulsymmetry andregularitypervadingthe whole, butalso a double unionof intervals-two of them always remainingundisturbed....the whole progressionforminga chain of harmonyunequalledin any of our former
exercises.22
As these passages suggest, the entire cycle is less a model of a surface
structureto be traversedin a single gesture than a compositional space
which, like a city, becomes entirelyknown throughan exertionof memory across a set of partial explorations.In a more recent formulation
directly related to the approachdeveloped in this paper, Lewin (1987,
180) notes that the 24 triads are generableby a single operation,MED,
36
= 12
- 3 modulo-9
(c)
(d)
(a)
(b)
operational chain-class trichordal pitch-class
periodicity cardinality periodicity cardinality
per co-cycle
<LP>
3
4
6
6
<PR>
4
3
8
8
<LR>
12
1
24
12
andCardinalities
Periodicities
Modulo12
Figure17:Binary-Generated
which maps a triadto its diatonicsubmediant.The powersof MED organize the 24 triadsinto a simply transitivegroup, a GeneralizedInterval
System isomorphicwith the <LR> cycle. In this sense, the <LR> cycle
can be seen as a source for all triadicprogressions.The <PR> and <LP>
cycles emerge, among other routines, as patternedsub-groups of the
<LR> cycle.23
Figure 17, modelled afterFigure 14, summarizesthe propertiesof the
binary-generatedcycles modulo 12. Surveying the three cycles as an
ensemble, what is most strikingis the affinitybetween the <LP> (hexatonic) and <PR> (octatonic)cycles, whose periodicitesand cardinalities
are nearly identical, and the anomalousstatusof the <LR> cycle by the
same standard.These affiliationsare unexpectedin the general context
providedby Section 3.3, which emphasizedthe affinitiesof the <PR> and
<LR> cycles, on the basis of their variableperiodicities,and the anomalous statusof the <LP> cycle, on the basis of its uniquelyconstantperiodicity.The section thatnow follows will suggest thatthe strongaffinity
betweenthe <LP> and<PR> cycles in modulo 12 is accidentalratherthan
inherent.In no otherchromaticsystem is the affinityso strong.
3.5. Hexatonic and Octatonic Analogues
We begin by investigatingthe relativelysimple case of <LP> cycles in
systems otherthanmodulo 12. In each pitch-classsystem thathosts parsimonioustrichords,<LP> cycles have a constantoperationalperiodicity
of 3, trichordalperiodicityof 6, and pitch-classcardinalityof 6 (cf. Figure 14, p. 30). Thus <LP> cycles are inherentlyhexatonic.The variable
37
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propertyis the cardinalityof the <LP> cycle-class, i.e. the numberof distinct hexatonic sub-systems.The 2c parsimonioustrichordsin a system
of c pitch-classespartitioninto = c <LP> co-cycles. This yields four
6 3
hexatonic sub-systemsfor c = 12, six sub-systemsfor c = 18, eight subsystems for c = 24, and so forth.
In my earlier paper, which studied the relationshipamong the four
hexatonic sub-systems in the mod-12 system, I observed (Cohn 1996,
23-25) thateach sub-systemsharespitch-classeswith two othersub-systems, and is pitch-classcomplementaryto the remainingsub-system.On
this basis, the ensemble of sub-systemsis shapedinto a four-elementsystem at a higher level. I proposed a two-tier design, where both the sixelement sub-systems and the four-element"hyper-system"that embeds
them are GeneralizedIntervalSystems (GIS) (Lewin 1987). Ourcurrent
work suggests that the same two-tiered GIS design applies to <LP>
cycles in all chromaticsystems where they occur. The individual subsystems, which I shall call hexatonic analogues, constituteGIS's whose
elements are the six trichordsof an <LP> cycle, and thus whose size is
constant to all chromatic systems. The higher-level hyper-hexatonic
analogue system, organizedby pc-intersectionamong the sub-systems,
constitutesa GIS whose size of c sub-systemsvaries with the chromatic
cardinality.
Figure 18 demonstratesthe hyper-hexatonic-analoguesystem for c =
18, using Haba's Dritteltonsystempitch-class designations, which are
inventoriedat the top of the figure. (Haba uses +for 1/6 sharp,t for 1/3
sharp, t for 2/3 sharp, and # for 5/6 sharp.)The 18 pitch-classes are
arrangedinto 6 T6-cyclesat the center of the figure.Each such cycle trisects the octave, and is thus acoustically equivalentto a temperedaugmentedtriadmodulo 12. The cycles arepairedinto six overlappingovals
which portraythe six hexatonic-analoguepc sets. Each set is connected
by an arrowto the hexatonicanaloguesub-systemof trichordsfor which
it furnishesa pitch-class source. Each such sub-systemis an <LP> cycle
modulo 18, and each is connecteddirectly,by shared"augmentedtriad,"
to two neighboringsystems.A uniquefeatureof this chromaticsystem is
that the hexatonic sub-systemsand the hyper-systemshare a cardinality
of 6. This isomorphism,which arises from the equality of the variablec
to the constant6 when c = 18, presents some interestingcompositional
possibilities which I shall not explore in the currentcontext.
The <PR> (octatonic)sub-systemsof modulo 12 generalizein a quite
differentway. Whereasproper<LP> sub-systems appearin all suitably
sized chromaticsystems, proper<PR> sub-systemsappearonly in those
systems where c - 3 modulo9. Even in these systems, the <PR> sub-systems generalizedifferentlythanthe <LP>. We have seen that<LP> subsystems have a constantsize but a variablequantity(cf. Figure 14). The
39
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situationwith the <PR> sub-systems is converse. The quantityof subsystems is constant:the 2c parsimonioustrichordsof each system partition into three<PR> sub-systems.The size of each <PR> sub-systemis
trichords.Consequently,generalized <PR> subvariable, engaging 2c
3
systems arenot "octatonic"in the same sense thatgeneralized<LP> subsystems arehexatonic:they have a pc-cardinalityof 8 only in case c = 12.
The most salient propertyof the mod-12 octatonic that generalizes to
moduloc is the step-intervalpattern<1, 2, 1, 2... >. In this sense the term
"1:2-analogue"(based on Lendvai 1971) would betteraccommodatethe
basis of the analogy than "octatonicanalogue."I nonetheless retainthe
lattertermin orderto stressthe positive aspectsof the meta-analogywith
"hexatonicanalogue."
Although, as we have seen, the <LR> cycle yields no proper subsystems modulo 12, it does yield propersub-systems in chromaticsystems congruentto 6 modulo 9, such as c = 15, 24, and so forth. In these
cases, the <LR> sub-systemsare structuredin the same way as the <PR>
sub-systems,in the sense thattherearethreesub-systems,each with a trichordal periodicity and pitch-class cardinalityof 2c. Because of this
3
similarstructuring,it makes sense to apply the termoctatonic analogue
to the <LR> as well as the <PR> sub-systems.
Figure 19 demonstratesthe hyper-octatonic-analoguesystem for c =
24, using Haba's Vierteltonsystempitch-class designations,inventoried
in ascendingorderat the top of the figure.The 24 pitch-classesare partitioned into three T3-cyclesof cardinality8 at the center of the figure,
which are pairedby the ovals into octatonic-analoguecollections of cardinality 16. These collections furnish the pc content for the octatonicanalogue sub-systems of triads,equivalentto the <LR>-cycles, modulo
24.
c = 24, then, is an example of a system whose 2c parsimonioustrichords partitioninto both hexatonic- and octatonic-analoguesub-systems. The two sub-system classes are equivalent to two of the three
binary-generatedcycles, <LP> and (in this case) <LR>.The thirdbinarygeneratedcycle, <PR> in this case, generatesthe entire set-class of 48
parsimonioustrichords,andthus its role in the c = 24 system is analogous
to thatdescribedin Section 3.4 for the <LR>-cycle in the c = 12 system.
Figure 20 (p. 42), which is cut from the same templateas Figures 14
and 17, summarizesthe periodicitiesfor a mod-24 quarter-tonesystem.
The affinitybetween the <LP> and <PR> cycles of modulo 12 has been
replacedby an affinitybetween <LP> and <LR>. Both featureattenuated
periodicitiesand non-exhaustionof the trichordclass and the pitch-class
aggregate.But the degree of affinityis greatly weakened,as the periodicities and cardinalitiesdiverge.This circumstanceunderlinesthe special
natureof modulo 12 suggestedat the end of Section 3.4. In no otherchro41
= 24
6 modulo-9
(d)
(b)
(c)
(a)
operational chain-class trichordal pitch-class
periodicity cardinality periodicity cardinality
per co-cycle
<LP>
3
8
6
6
<PR>
24
1
48
24
<LR>
8
3
16
16
Modulo24
Periodicities
andCardinalities
Figure20: Binary-Generated
matic system does the variableoperationalperiodicity- so closely approximatethe constantperiodicity3. (Both c = 9 and c = 6 are deficient:
c = 9 producesno octatonic-analoguesub-systemsbecause 9 is congruent to 0 modulo 9; althoughc = 6 producesoctatonic analogue sub-systems under<LR>, its two (!) hexatonic sub-systems are not pitch-class
distinct.)
I imagine the hexatonicand octatonicsub-systemsas objects in space,
one fixed and one transient.As the transientobject momentarilypasses
by the fixed one, theirrelationshipbecomes recorded,frozen in time like
the objects in a photograph.An observer who knows the objects only
throughthe recordingis in no position to understandthat their association is anythingotherthanpermanentand intrinsic.Ourstudyof this phenomenon in the context of generalizedchromaticsystems unfreezes the
moment,allowing us to see the accidentalnatureof the relationship,and
therebyenhancesourappreciationof the over-determinedqualitiesof the
integer 12.
This concludes the inventoryof binary operationsand the trichordal
progressionsthatthey generate.We now turnour attentionto ternarygenerators,which are of a differentnaturebecause they are inversionsrather
thantranspositions.
3.6. Ternary Generators and LPR Loops
We begin our study of ternarygeneratorsby demonstratingthat, once
certainrulesof reductionareinvoked,theirnumbersareconstrained.The
reductionprocess is facilitatedby the concept of generator form. Gen42
erators that have identical order-positionequivalences will be said to
share an abstractform. Forms are designatedby assigning the abstract
tokens A, B, and C to the three distinct PLR-family operationsin the
order that they appear in the generator.Accordingly <LPLR> and
<RPRL>are both of the form <ABAC>.
Because two successive iterationsof a single operation"undo"each
other,ternarygeneratorsof the form <ABB> reduceto unarygenerators
(<A(BB)> = <A>), and accordinglyhave no independentinterest.Neither do generatorsof the form <ABA>, since reiterationjuxtaposes the
two A's, triggeringa chain of involutionsthat unravelsthe entire structure: <ABA>2 = <AB(AA)BA> = <A(BB)A> = <AA> = To. Consequently,independentternarygeneratorsare limited to the form <ABC>.
The six realizationsof this form correspondto the six orderingsof the
unorderedset {L,P,R}. These six orderingsare equivalentunder retrograde and rotation.We will view them as membersof a single class of
ternarychain whose canonical label, invoking alphabeticprecedence,is
<LPR>.
To illustratethe involutionalnatureof ternarygenerators(cf. Theorem
2), Figure 21 (p. 44) models a cycle of six triadson the Oettingen/Riemann Tonnetz.Selecting F minoras a startingpoint, we move clockwise
througha semi- cycle, engaging R, P, and L in thatorder,and stoppingat
E major.Continuingclockwise fromthis point, the same threeoperations
are engaged in the same fixed order,closing the cycle back to F minor.
The compoundoperationrepresents<RPL>2.Regardlessof startingtriad
or direction,the same set of triadsis traversed,demonstratingthe equivalence of the six orderingsof {L, P, R}. I will refer to such structures
genericallyas LPR loops.
Two examplesof LPRloops fromnineteenth-centuryopera,bothusing
the specific triadsincludedin Figure21, arepresentedin Examples3 and
4 (p. 45). Example3 models the succession of triadsin the cantabilesection of "Ahsi, ben mio," from Act III of Verdi'sII Trovatore.Beginning
in F minor,the first quatraincloses by tonicizing its relative major.The
aria'ssecondquatrain"mutates"to Abminor,prolongsFb majorthroughout its consequentphrase,repeatsthe text of the consequentphraseover
a Db minor triad,and reaches a fermataover a V7 of D. The final quatrain of the cantabilethen prolongs Db major.The six consonanttriads
traversean LPR loop, althoughthe loop does not close with a returnto
the initial F minor.
Example 4 is the Engelmotivfrom Wagner'sParsifal, in a transposition that occurs in Amfortas's Prayerfrom Act III. The passage begins
and ends on Db major,and rotatesclockwise throughthe triadsof Figure
21, omittingAb minorand Db minor.As the analysis beneathExample4
indicates,the omissions are accountedfor by compoundoperationsthat
"elide across"the omittedtriads.24
43
10
1
6
9
4
R
/0
6
2
Ab+
F-
_^/
5
10
P
/
L
7
/
/
Ab-
8
2
L
Db+
Fb+
/
/
Df10
4
(1
p
6
9
~
7
10
3
6
R
0
Figure21:LPR-looparoundAb/G#
A significantfeatureof LPR loops is thattheirsix triadssharea single
pitch-class, located at the center of the matrixrepresentation.In Figure
21, Ab/G#plays thatrole. The membersof a loop furnisha complete roster of the triadsthat include the sharedpitch-class.25It follows that there
are twelve distinctLPR loops (one per pc), thateach triadparticipatesin
three such loops, and thatthe threeinterlockingloops in which thattriad
participatesfurnisha complete rosterof the triads with which it shares
one or more pitch-classes. Figure 22 (p. 46) illustratesfor the case of F
minor.The figure is in the form of an interlockedset of "honeycombs."
The circle at the centerof each loop encloses the pitch-classcommon to
all the triadsin that loop. The entire structureof the twelve interlocked
loops can be easily inferredby projectingoutwardfrom Figure 22.26
Because of theircommon-toneproperties,LPR loops furnishan ideal
progressionfor consonantly supportinga single melodic pitch with diverse harmonywhile maximizing voice-leading parsimony.Such progressions, with their implication of inner action or turmoil beneath a
44
...in ciel precederti...
fra quegli...
Ma pur,se..
...le vittime...
Ah! si ben mio...
...trafitto...
...avropiu forte. ...restifra le vittime...
...penslerverra...
I
Example 3: Verdi, II Trovatore, Act III: Ah si b'en mio
r
Ihf
b bf^ T|I)fT^I
in
jetzt_
@
TIQ0
-
gott
Glanz
#Xil
den Er
r n
I
-
16 -
ser
I;
9
#<L
<#R
L
L
li-chem
R
P>
<RP
<PL>
<RP>
Example 4: Wagner, Parsifal, Act III: Engelmotiv
placidandharmonioussurface,were well suitedto symbolizenineteenthcentury notions about the relationshipof the inner and outer worlds.
Examples include the openings of Liszt's "II Penseroso" (Annees de
Pelerinage, Deuxieme Annee, composed 1839), and of the Monk's Cho-
rus fromVerdi'sDon Carlos.
All of the LPR-loop propertiesidentifiedhere for mod-12 consonant
triads also hold, mutatis mutandis,for parsimonioustrichordsin other
systems, as can be seen by transferringthe cyclic design of Figure 21
onto the backgroundgrid first introducedin Figure 8 (p. 14). Furthermore, analogous propertieshold for any trichordin any system, minus
the parsimoniousvoice-leading, as can be seen by transferringthe same
cyclic design on the backgroundgrid of Figure6 (p. 10).27This transferability results from the odd cardinality of ternary generators, which
causes them to involute,in contrastto the devolutionarynature(cf. 3.1.6)
of binarygenerators,and of the quaternarygeneratorsthatwe now study.
45
10
10
6
9
2
)Ab+
F+
a/
/B
6
11
2
/
58)
B+
3
)
/
Db+
/
E+
B6-
/
C#-
/
/
10
1
4
7
10
6
9
0
3
6
Figure22:Interlocking
LPR-Loops
3.7 Quaternary Generators
Like ternarygenerators,quaternarygeneratorsreduceto a single form
once operationsthat involute (eitherdirectlyor across the boundariesof
successive iterations) are eliminated and rotational equivalence is
invoked. Forms with two sets of duplicate operations are eliminated
either because the duplicates are juxtaposed and thus undo each other
(<A(BB)A> = <AA> = Toand <(AA)(BB)> = To),or because the generator is itself binary-generated(<ABAB> = <AB>2). Thus, all independent quaternarygeneratorsinclude all three distinct operations,with a
single operationrepresentedtwice. The two iterationsof the duplicate
operationcan be neitheradjacent(as in <(AA)BC>) nor maximally separated(as in <ABCA>, whose reiterationcauses <ABC(AA)BCA>), and
46
(a)
(b)
(c)
(d)
(e)
(f)
Ty T(x+y)
T-(2y+x)
Ti
L
<LPLR>
P
<PLPR> <PL><PR>
TyT
Tyx
Tp
R
<RLRP> <RL><RP>
Tx+yTx
T2x+y
Tr
<LP><LR>
of Quaternary
Generators
Figure23:Transpositional
Equivalences
accordingly must be separatedby two order positions. All quaternary
generatorsthatfit this descriptionareequivalentto <ABAC> via rotation.
The six generatorsthatrealize <ABAC> groupinto threerotation-related
pairs,as follows:
and<LRLP>;
and<PRPL>;
and<RPRL>.
(1)<LPLR>
(2)<PLPR>
(3)<RLRP>
A significantpropertyof any quaterary generatoris thatthe transposition that it enacts is equivalentto the voice-leading intervalproduced
by the moving voice when a single iterationof the duplicateoperationis
executed. For example, an operationwith duplicateR, such as <RPRL>,
is equivalentto T, . Thus a quaternary operation unfolds in the large
the transpositional operation that its duplicate operation expresses
in the small.
Figure 23 demonstratesthis "exfoliation"property.The three operations, listed at (a), arerepresentedas duplicatesby a quaternaryoperation
at (b). These are each partitionedinto a pair of binaryoperationsat (c),
and convertedto their associated T, values at (d) (cf. Figure 11, p. 26).
The T,values at (d) are composed at (e) by summingsub-scripts,and the
productis expressedin termsof a voice-leadingintervalat (f), following
the equivalencesestablishedat Def. (5) in Section 1. A comparisonof (f)
with (a) demonstratesthe correspondenceof the transpositionalvalue
with the duplicateoperation.Note thatreplacingan operationat (b) with
its rotation(e.g. <LRLP> for <LPLR>) does not affect the result, since
transpositionoperatorscommuteandthus <LP><LR>= <LR><LP>(cf.
Kopp 1995, 272-73).
Figure24a uses the the abstractTonnetz(cf. Figure6) to portraythe six
quaternaryoperationsas they act on the generic prime-formtrichordQ =
47
Generators
on theAbstractTonnetz
Figure24a:Quaternary
{0, x, x + y} with step-intervals<x, y, -(x+y)>. Each of the three paths
leading out of Q implements the first operationof the quaternaryset,
traversingone of the three edges. The paths then bifurcate, reuniting
at the point that the operationis completed. For example, <PLPR> and
<PRPL> both start along the northwest path out of Q, traversingthe
hypotenuse to P(Q). <PLPR> turns right and proceeds counter-clockwise, while <PRPL>forks left and proceedsclockwise. Both operations
terminateat Tp(Q) = Ty.x(Q)at the northwestcornerof the figure.
Figure24b transfersthe same design onto the Oettingen/RiemannTon-
netz, where p = 1, / = 1, and , = -2 . These voice-leading intervals are
echoed as transpositionalvalues:
*Tothe south of Q', the duplicateL operations<LPLR>and <LRLP>
transpose Q' = C minor to T/(Q) = Ti(Q) = C# minor;
48
9
on theOettingen
Generators
/ RiemannTonnetz
Figure24b:Quaternary
*Tothe northwestof Q', duplicateP operations<PLPR>and<PRPL>
transposeQ' = C minorto Tp(Q) = T1(Q)= C# minor;
*To the east of Q', duplicate R operations <RLRP> and <RPRL>
transpose Q' = C minor to T,(Q') = T2(Q) = B, minor.
Figure25 (p. 50) presentsthe six operationsof Figure24b in a format
that facilitates examination of individual voices. Rotationally related
pairs occupy the same row. Pairs of generatorsare connected by arrows
which indicate a particularlyintimatevoice-leading relationship:paired
generatorshave identicalC-voices, and each G-voice is T4-relatedto the
Eb voice of its partner.One surprisingaspect of this pair-wisepartition
of the generatorsis thatit is not identicalto the pair-wisepartitionon the
basis of rotationalequivalence (i.e., sharedduplicate).What principles
underliethese observationsis still an open question, as are the potential
compositional and analytic applications.Readers may enjoy exploring
49
<LPLR>
G
Ab
Ab
Ab
Ab,-
G
A
Eb
Eb
E
Fb
Fb -
Eb
E
C
Cb
Cb
Db
C
C
-c
T4
TO
To
<PLPR>
G
G
G
Gt
G<*
G
G
Eb
E
E
E
E
El
E
C
B
B
CO
- C
G
<RLRP>
G G
F
T4
El
Eb
D
D
Db
C
Bb
Bb
Bb
Bb <
C
-C
G
El
E
> C
B
T
Figure 25: Voice-leading Affinities among Quatern
the aural kinship between these paired progressions, and between the
chromaticsequences thatthey generate.28
Havingexploredthe transpositionalbehaviorof quaternaryoperations
singly iterated,we now study the progressionsgeneratedby their recursive application.Complete quaternary-generated
cycles of triads are so
lengthy as to be of negligible musical value and are impracticalto represent in a compact graphic space. Consequently,I will focus here on
cycle-segments, or chains, which present one way to conceptualizethe
stepwise chromaticsequencesfavoredby nineteenth-centurycomposers.
Figure26 (p. 52) superimposesone chain of each rotation-classonto the
generic Tonnetz.
* <PLPR>, which representsa Tp sequence, proceeds from 5:00 to
11:00, moving along the main diagonal in the same directionas its
hypotenuse-traversingduplicate,P;
* <LPLR>,which presentsa T1sequence, proceedsfrom 1:00 to 7:00,
approximatelyin the same verticaldirectionas its duplicate,L;
* <RLRP>,which presentsa T, sequence,proceedsfrom 8:00 to 2:00,
approximatelyin the same horizontaldirectionas its duplicate,R.
It is characteristicof actual compositional settings that extended sequences are heardnot in termsof the continuousflow implied by Figure
26, but ratheras a series of terraces.Each terraceis individuallyunified,
but the junction between them characteristicallymoves "acrossthe barline,"inauguratinga new iterationof the sequenced segment at a metrically markedposition. Potentiallyany of the four operationsin a quaternarygeneratorcan assume the transitional(or external)role, leaving the
remainingthree operationsto assume the role of coherentlyarticulating
each terracedregion.
A significant feature of quaternarygeneratorsis that these terraced
regions tend to be coherentin termsof the binaryand ternarychains discussed in Sections 3.4 and 3.6. Figure 27 (pp. 54-57) illustrates this
point, transferringa representativechain from Figure 26, <PLPR>,onto
the surfaceof the Oettingen/RiemannTonnetzin fourdifferentways. The
four representationshave identical content, moving through the same
series of triads,but their graphicinflections suggest four differentways
of partitioningthe chain into terraces.Each Tonnetzportraitis accompanied by a schematic notationalrealizationin orderto reinforce some of
the observedqualities.
* Partition(a) (p. 54): <... (PLP) R (PLP) R (PLP)...>. Each set of
four triads joined by <PLP> constitutes a terrace unified by the
<LP>-cycleorhexatonicconstituencyof its triads.R plays the role of
effecting the "modulation"between neighboringhexatonicregions.
51
<LPLR>
- 2y
-x - 2y
-2y
x - 2y
2x - 2y
/ <PLPR>
3x - 2y
Chainson theAbstractTonnetz
Figure26:Quaternary
The structureis graphicallyportrayedas a series of unifiedupward
(y-axis) motions, interruptedby a transitionalleft-wardjog.
* Partition (b) (p. 55): <... P (LPR) P (LPR) P (LP... >. Each set of four
triadsjoined by <LPR>constitutesa terraceabstractlyunifiedby the
LPR-loopconstituencyof its triads.In concrete musical terms, this
unity is insuredby the invariantpitch-class inherentto LPR-loops.
The P operationexternalto <LPR> plays the role of "modulating"
between adjacentLPR-loops.The modulationis markedby the dis-
52
placementof the invariantpitch-class that defines the region. (The
voice bearingthis pc is highlightedin the notationalrealizationby
registral placement.) The structureis graphically portrayedas a
series of arcs veering northeastof the invariantpc, interruptedby a
diagonaljog.
* Partition(c) (p. 56): <... P) L (PRP) L (PRP) L (P... >. Sets of four
triadsjoined by <PRP> constitute terraces unified by the <PR>cycle or octatonic constituency of the triads. L plays the role of
"modulating"between adjacentoctatonic regions. The structureis
graphicallyportrayedas a series of leftward(x-axis) motions, interruptedby a transitionalupwardjog.
*Partition(d) (p. 57): <... PL) P (RPL) P (RPL) P...> An alternative
set of LPR-loop segments, with the pitch-classes unifying the
regionsresidingin a differentvoice thanat partition(b). In this case,
the loop is articulatedvia <RPL>.The Tonnetzportrayseach region
as an arc veering southwestof the invariantpc, which againis highlighted in the sopranovoice of the notationalrealization.
The generaldesign of Figure27 holds for the otherquaternarygenerators, but with a single degree of attenuation:one of the four partitions
articulatesa series of segmentsof the <LR> chain. Since the 24 triadsare
united into a single <LR> chain, the terraces do not define harmonic
regions in any obvious way. The design of Figure 27 also transfersto
other parsimonioustrichordsin microtonal systems. For example, the
quaternarygenerator<LPLR>mod-24 leads to a coherentset of terraces
for all four partitions,two of which articulateLPR-loopregions unified
by pitch-classinvariance,the othertwo of which are articulatedby hexatonic-analogue<LP> cycles articulatedby <LPL>, and octatonic-analogue <LR> cycles articulatedby <LRL>.
This concludes our explorationof generatedPLR-familychains in the
abstract,and as they apply to consonanttriads in the familiarcase. Although there are a limited numberof independentgeneratorsfor even
cardinalitieslargerthan four, my preliminaryinvestigationsuggests that
their explorationyields diminishingreturnsboth theoreticallyand analytically.
4. Some Open Questions
Although our tour of compound PLR-family operations has been
densely packed, it has hardlyexhaustedthe terrain.I conclude by suggesting several questions and topics that may rewardfurtherinvestigation. My hope is that some of the investigatorsmight be "microtonal"
composers attractedto set-class consistency and smooth voice-leading,
and analysts seeking to interpretthe triadicallybased repertoryof late
53
4
1
1
4
9
0
5
8
I
n
ir
QV
-
I1i
LI HI I
-
I V
1
Ii
I4P vP
7
10
1
6
9
2
5
11
I
41
I"
i
i
I
! M
I
I
t,-
I
I
I
I
-
h
Figure 27 (a)-(d): Four Portraitsof a <PLPR>Chain
on the Oettingen/RiemannTonnetz.
Figure 27 (a)
I
i
I
i
I
4
1
1
4
9
0
5
8
7
10
1
3
6
9
11
2
5
11
2
5
I
r#- -
-
'i
!'
"
^# -
'
- - --
Fi h 27 b r r r 'IF I rr
Figure27 (b)
1
A
_l
4
1
4
9
0
5
8
I- I I I,I
I
TI_
#
#
7
11
I I
W
I
I
*I
i
I
I
I
10
1
6
9
2
5
10
1
2
5
I
1J 1J ..J I ..J iJ,i
W
;._
Figure27 (c)
I
I:
I
I
I
mJ
., -
-
I
1
4
7
10
1
9
0
3
6
9
5
8
11
2
5
1
4
7
10
1
9
0
3
6
9
5
8
11
2
5
19
0
3
Figure 27 (d)
a9
-I 6
Romanticism(andperhapsalso the fledgling microtonalrepertoryof our
own century).The following comments startdown some theoreticaland
historicalpaths,but pass by points of analyticinterestas well.
(1) Like the work presented here, Balzano (1980) adopts a grouptheoreticapproachto pitch-relationsnormallydiscussed in an acoustic
framework,uses a versionof the Tonnetzto representthese relations,and
generalizespropertiesof c = 12 to otherequal-temperedsystems.The triadic featuresthatBalzanogeneralizesdifferfromthose presentedherehis generalizedtriadshave step-intervals<x, x + 1, x2 - x - 1> in a chromatic system of size x2 + x-so that his generalizedtriads are not the
parsimoniousones that have focused our attentionhere, nor do they inhabitthe same set of chromaticsystems. It seems worthwhileto explore
the relationshipbetween Balzano's generalizationand mine, and what is
gained and lost by each in relationto the other.
(2) An alternativedefinitionof "parsimony"might open up the category of parsimonioustrichords.On the accountadoptedabove, a trichord
is parsimoniousif its voice-leading is non-zerobut minimal underL, P,
and R. One alternativeavenue that seems promisingis to classify a trichordas parsimoniousif its voice-leadingis non-zerobut minimalunder
two of the three PLR-familyoperations.This would allow not only P =
1, l = 1, 7 = -2 with step-intervals<x, x + 1, x + 2>, but also p = 0, l = 1,
z = -1 with step-intervals<x, x, x + 1>, and = 1, = 0, a = -1 with stepintervals<x, x + 1 , x + 1>. On this account,all chromaticsystems possess one parsimonious trichord-class.29Unlike the parsimonious trichord-classes considered in this paper,the newly yielded trichordsare
inversionallysymmetric,and maximally even in the sense of Clough &
Douthett1991. The new groupincludes,amongothers,the set of diatonic
(024) triadsin a mod-7 diatonic system, and the set of consonant(025)
trichordsin a mod-8 octatonic system. Furthermore,invoking a distinction made by Lewin (1996), the formergroup are of the antithesistype,
whereas this new group is of the generatortype. One advantageof this
expandedinterpretationof parsimonyis thatit suggests a way to collapse
Lewin's distinction,at least in the case of trichords.
(3) The crooks in the arrowsof Figure 11 and relatedfigures suggest
that compound operationsproceed in multiple stages through a set of
intermediateterms,ratherthandirectlyto theirtargettriad.I includedthe
crooks to facilitate the tracing of compound operations,but pedagogy
should not be confused with ontology. The ontological problemis most
generallyframedin termsof a path/goalduality. Given some triadicprogression <C+, Ab+>, whose most economical PLR-family analysis is
<PL>, is the progressionto be intuitedas a single motion C + -'PLAb+,or
as a pairof GestaltsC + -PC - --Ab+ whose mediantermis elided out?
Whatrelationholds between the two interpretations,and what is the status of C-?
58
Along the same lines, how can we choose between two equally economical analyses of a single progression,for example,f- <LP>E + and
f_ <RPL>. E +? Furthermore,does economy always overrideother "preference rules" in the assignmentof a PLR analysis to a triadic progression?Again to groundthe questionby example:is <PL> always andonly
the appropriateanalysisfor <C+, Ab+>, or aretherecircumstancesunder
which some other descriptivelyaccurateanalysis would be considered
more appropriate,for example, <LP>2, <RRPL>, <PRRL>, <PLRR>,
or <RL>4?The problemis a generalone inherentto the interpretationof
relationson a two-dimensionalmatrixsuch as a mapor a chessboard,and
it shows up in nineteenth-centuryTonnetzanalyses (cf. note 24 above)
andin otherneo-Riemanniantransformationalaccountsof triadicmotion
(e.g. Lewin 1992, Hyer 1995, Mooney 1996).
(4) The final topic is a historicalone. It is apparentthat a group-theoretic orientationtoward musical materials and their relations did not
arrivefully formedin the twentiethcentury,but ratheremergedfromelements that had been long present, if not fully articulatedor mobilized.
Whatrole do the over-determinedtriadand the over-determinedTonnetz
play in this emergence?
The responsivenessof the Tonnetz,designed to model acoustic relations, to a group-theoreticmodel potentiallyfurnishesa lever for prying
apartthe acousticfromthe group-theoreticaspectsof triadicprogressions,
and for exploringthe cohabitationof a nascent and tacit group-theoretic
perspective with an explicitly acoustic one in nineteenth-centuryharmonictheory.To whatextentdid nineteenth-centurytheoristsunwittingly
smuggle an implicitlygroup-theoreticorientationbeneaththe cover of an
apparatuswhose essentially acoustic nature they never doubted for a
moment? Clues can be found not only in the writings of Tonnetznavigators, but also in the way that common-tone preservationand incremental voice-leading is treatedby nineteenth-centurytheorists such as
Reicha, Fetis, Marx, Hauptmann,Weitzmann,Helmholtz,Tchaikovsky,
and Hostinsky.This investigationmight give new meaning to the timeworn adage that "the seeds of the tonal system's destructionwere sewn
from within."What is suggested is that the sower is none otherthan the
structuremost emblematicof thatsystem, the triaditself. The association
of the triadwith divinityand perfectionas initiallyconceived by Lippius,
and carrieddown throughSchenkerand beyond, suggests an allegorical
interpretationwhich is best carriedout by scholarstrainedin theology and
the historyof ideas.
59
APPENDIX
The proofs for Theorems2 and 3 use the formulasfor compositionof
transpositionand inversionprovidedin Rahn (1980, 52), translatedinto
an ordered-setformatas follows:
(Fl) <Ta,Tb>= Ta+b;
(F3) <Ta,TbI>= Tb-al;
(F2) <TaI,Tb>= Ta+bI;
(F4) <TaI, TbI>= Tba
To use these formulas,we will need to translateLewin's conventionfor
labeling inversions(see Def. (3)) into the index numberconventionused
by Rahn.The translationproceeds as follows:
(F6) TnI= I?.
(F5) = Tu+vI;
*
*
*
THEOREM2. Proof that, if #H is odd, H2 = To.
(1) H = <Hi,H2 ... Hn>, where n is odd.
H1is an inversion,by virtueof being a PLR-familyoperation.The cardinality of <H2,.. . Hn> is even, and hence <H2,... Hn> is a transposition
(cf. Section 3.1.2). Thus
(2) H = <a,Tp>.
A second iterationof H likewise can be analyzedas an inversionfollowed
by a transposition,and so
(3) H2= d<,Tp,;,Tq>.
Althoughthe inversionaloperationsare identicalin PLRterms,the intervening transpositioncauses the second inversionto invertaroundan axis
to thatof the firstinversion.Hence
Tp-related
(4) c = a + p; d =b + p.
The transpositionaloperationsare likewise identical in PLR terms, and
thus necessarilytransposeby the same magnitude,but not necessarilyin
the same direction.Since one inversionoperationintervenesbetween Tp
and Tq,the trichordsubject to Tqis inversionallyrelated to the trichord
subjectto Tt,,and so q = -p. Thus:
60
(5) H2 -
Tp>.
(7) <Ta+bI,Tp> = Ta+b+pI
By translationvia F5
F2
(8) H2 = <Ta+b+p+lII4=,
= T2p+a+bI
(9) Ib+P
By substitutionto (5)
By translationvia F5
(6) Ib= Ta+bI
T->
(10) <Ta+b+pI,T2p+a+bI>=
(II)
F4
T(2p+a+b)-(a+b+p)
=TP
By substitutionto (8)
Fl
H2 = <Tp,Ip
(12) H2 = T0
QED
*t
*i
*c
THEOREM3. Proofthat,if orderedset H = Tp,and Ret(H)is the retrogradeof H, then Ret(H) = Tp.
(1) If H = T,, then H can be partitionedinto
3.1.2).
H transpositions(cf.
(2) And so H = <Ta,Tb,..., T,,Tn>, where a + b +... m + n = p.
(3) Each transpositionTqof H is comprised of two inversionoperations; hence Tq= <Tj'I,ITjl>.
(4) Thus H = <<TaJIT,4I>,<Tb,TbI>,...,
(5) It follows from (3) that q =
(6) And so H = < Ta"-d,Tb"-b',
<T'I,TmI>,<TJ'IT,4">>
- q', via F4.
Tm-m',Tn"-n'>.
(7) Since (4) analyzes H to its atomic elements, it follows that
Ret(H) = <<?TnI,TI>,<Tm4l
J>,..., <Tb4I,TbJ>,<Ta4I,TJI>>
=
Via
F4,
(8)
Ret(H) <T',_,,,,Tm_m,,",...
Tb'-b",Ta'-a">
(9) It follows from (5) that q' - q" = -q.
(10) Thus Ret(H) = < T,TnT, ... Tb,Ta>
(I 1) It follows from (2) that -n - m...-b - a=-p.
(12) Then Ret(H) = Tp-,via Fl
QED
61
NOTES
The workpresentedherehas benefittedfromsustainedconversationsandcorrespondencewith DavidClampitt,JohnClough,JackDouthett,DavidLewin,and
MichaelSiciliano.AdrianChildsprovideda helpfulreadingof the finaldraft.
1. Throughout
this presentation,
"triad"restrictivelyrefersto the class of harmonies
thatis morespecificallydenotedby theterms"consonanttriad,""harmonic
triad,"
"Klang,"or "memberof Forte-class3-11."
2. Lewin 1982, 1987.The firststagesin the developmentof neo-Riemannian
theory
aretracedin Kopp1995,254-269.
3. Lewin 1992;Hyer 1989, 1995.The relationof thesetermsto those usedby Riemanncan be causefor confusion.Riemann's"Parallelklang"
is equivalentto the
Relativemajoror minorin currentEnglishusage;whatwe call a parallelrelation,
Riemanntermsa "Variant."
4. Theterm"parsimony"
is usedin this contextin Hostinsky1879, 106.In Schoenberg'swritings,the sameprincipleis referredto as the "lawof the shortestway."
See Schoenberg1983 (1911), 39. Fora formulationof this "law"datingfromthe
late seventeenthcentury,see Masson1967(1694), 47.
5. See Dahlhaus1990(1968),73-74, 86-87, 241. Beginningwith(atlatest)theeighteenth century,the normativestatus of common-toneretentionand stepwise
motionis notonly statisticalbutcognitive:oneconceivesof themas occuringeven
whenthe actualleadingof the "voices"violatesthem,e.g. wheninstantiations
of
the commonor step-relatedpitch-classesarerealizedin differentregisters.
6. Thereis a single exception:the 3-12 [048] class. In a certainsense, the exercise
doesnotapplyto thisclass,sincethosemembersof 3-12 thatsharecommontones
are not distinctfromeach other.In a differentsense, we can view C,E,G) as
holding two tones-it mattersnot which two-in common with its inversion
aroundC (or aroundany other"even"pc), while the thirdvoice "progresses"
by
theintervalof zerosemitones.Herethevoice-leadingis parsimonious
indeed.And
no moneyis spentby the dead.
7. Definition(3) is fromLewin(1987, 51). Fora moreflexibledefinitionof Q, see
note 13 below.
8. Otheraspectsof Figure4 areintriguingandsuggestive,althoughnot pertinentto
the currentproject.Comparethe followingtriosof trichord-classes
for theirstepintervaldifferences:{012, 027, and036}; {013,016, and025 . Notealsothemod3 congruenceof the step-intervaldifferencesof each trichord.For six of the trithe three step-interval
chord-classes,includingthe five that are TnI-invariant,
differencesarecongruentto 0, modulo3.
9. Butsee Clough& Douthett1991andAgmon 1991,bothof whichidentifyspecial
properties of the triad as an object in modulo-7 diatonic space.
10. For a valuablerecenthistoryof the Tonnetz,see Mooney 1996. See also Vogel
1993(1975).
11.Euler1926(1739), 319, 349. The Tonnetzwas alreadyimplied,althoughnot laid
out in Euler'sgeometricallycompactform, in Rameau'sNouveauSystemede
musiquetheoriqueof 1726.See Popovic1992, 119, 127.
A more remarkableharbingerof Euler'smatrixis impliedby the pitch-class
namesusedin ancientChinesemusic.A set of 65 bells fromtheZeng stateof the
MarquisYi, datingfrom433 B.C. butonly unearthedby archaeologistsin 1978,
62
revealsa systemof twelvepitch-classesperoctave.Thenamesandtonalfunctions
of pitches,whicharetransmitted
throughinscriptionson eachbell, indicateoctave
equivalence.The twelve pitch-classesare namedby prefix-suffixcombinations
whichconstitutea 4x3 Cartesianproductisomorphicwith Euler'smatrix.There
are four prefixes, gong = C, zhi = G, shang = D, and yu = A, which are identical to
fourof the pentatonicdegreesyllablesstill in use in China.The threesuffixesare
0 = no transposition,jue = transpose upward by major third, and zeng = transpose
upwardby two majorthirds.Thefollowingtable,whichtransposesEuler'smatrix
(fromFigure5a) upwardby perfectfifth,andthenplacesZengsyllablesalongside
the corresponding
Europeanpitch-classnames,clarifiesthe isomorphism:
C gong
E gongjue
Gt gongzeng
zhi
D
shang
A
zhijue
ES zhizeng
Ft
shangjue
Ct yujue
F yuzeng
G
B
Bb shangzeng
yu
Forinformationon the Zeng bells, see Falkenhausen1993,especiallyChapter8,
andChen, 1994.
12. This claimrequiressome elaborationin light of the recentdissertationsof Kopp
(1995) andMooney(1996). In the contextof Riemann'stheoryof functions,the
PLR-family relations (where P = Riemann's Variantand R = Riemann's Parallel)
modifyone of the threefunctions.KopparguesthatRiemann'sseparatelydeveloped systemof musicalsyntax,which conceivesof triadicrelationsin termsof
Schritteand Wechsel,is not subordinated
to the functionalframework,butrather
co-exists with it in a relationshipof mutualautonomy.The syntacticsystem
assigns differentlabels to PLR-familyrelationsand conceives of them more
dynamicallythandoes the systemof functions.AlthoughRiemannwas not much
concernedwithTonnetzrepresentations
of thesyntacticoperations,Mooneynonethelessarguesthatthe systemof Schritteand Wechselis deeplyintertwinedwith
thegeometryof theTable(see 236-268, esp. 266.)YetRiemannneverbestoweda
privilegedpositionto PLR-familyoperationsin the contextof his syntacticsystem, partlybecausehe increasinglyinterpretedthe objectsin the tableas tonics,
andso triangular
ceasedto be relevant.
representations
13. "Pitchor pitch-class"becausewe havenot yet committedthe matrixto a modularcongruence.To makeFigure6 susceptibleto a pitch-spaceinterpretation,
Def.
(2) wouldneedto be mademoreflexibleas follows:"Qis a trichord{0, x, x + y}
suchthat0 < x < y."
14. Tomy knowledge,theearliestrecognitionof thetoroidalnatureof anequallytemperedversionof the Tonnetzis Lubin1974.I thankJudithSchwartzfor recognizing the relevanceof this workto my research.
15.JohnCloughand David Clampittwere the firstto enumerateparsimonioustrichordson the basis of theirstep-intervalproperties.Theirinsightswerecommunicatedto me by Cloughin a letterdatedJune25, 1993.
16. Weitzmann1853, 23. An unboundedversionof Figure9a, implicitlyprojecting
into a torus,was presentedin Balzano 1980, 72. The x and y axes of Balzano's
matrixareswappedin relationto Figure9a; otherwisethe structuresareidentical.
17.Forbackground,see Morris1987,.132.
63
18.This materialbuildson Hyer'sexplorationsof the algebraandgroupstructureof
in "TonalIntuitions"and"Reimag(in)ing
Rieneo-Riemannian
transformations,
mann."
19.Thisdictumis axiomaticfor theoristsof personality,culture,andgeo-politics.Its
venerablerootsare suggestedby the followingancientallegory:"Thebearwent
overthemountainto see whathe couldsee. Theothersideof themountainwas all
thathe couldsee."
20. Cohn1996cites a numberof studiesof 6-20; to thislist shouldbe addedthemore
recenttreatmentin vandenToorn1995: 123-142.
21. AlthoughcomposedforDie Zauberharfe,
thismusicwas initiallypublishedas the
"Rosamunde
underwhichtitle it is morefamiliar.
Overture,"
22. Logier1976(1827),p. 38. I amgratefulto ScottBurnhamforrecognizingthe significanceof Logier'streatiseto my research.
23. MichaelSicilianopresentsa compellingargumentfor the sourcestatusof the
<LR>cycle in unpublishedworkwhichis currentlyin preparation
as a dissertation.
24. See Rothstein1991for an indicationof the long reachof elisions in music-theoreticwritings.Sourcesparticularly
germaneto invocationof elisionsin thecurrent
contextincludeHauptmann1991(1853), 57, 160-161; Oettingen1866, 145-48;
andHostinsky1879, 103-05.
25. Popovic1992refersto the set of pitch-classesthatsharetriadicmembershipwith
The triadsof an LPR-loopare grouped
some centralpc as its "neighborhood."
togetherin relationto thecentralpc (althoughnotnecessarilyarrangedcyclically)
by theoristswho view triadsas assembliesof acousticconsonances,ratherthanas
directlygeneratedentities.See Helmholtz1954(1862, 1877),212 andHostinsky
1879,67.
26. The interlockingLPR loops correspondwith resultsof experimentsreportedin
Krumhansland Kessler 1982 and Krumhansl1990, Chapter2. Krumhansland
Kessleraskedsubjectsto ratethe goodnessof fit betweena givenpitch-classand
a givenkey on the basisof acousticinput,andusedthatdatato profilesimilarity
relationsamongthe 24 keys.Althoughtheprofileis bestportrayedin fourdimenof thatfigure-a rectangleimplyinga
sions, theirtwo-dimensional"flattening"
torus-is a rotatedversionof Figure22.
27. Similarstructures
areexploredformod-12trichordsin Bennighof1987andLewin
1996.
28. Itis of historicalinterestthattheP-duplicate
in Figure25 leadto Ct minor,
operations
buttheL-duplicates
leadto Dbminor.Thisenharmonic
distinctionvestigiallyreflects
anacousticone.RecallthatP-duplicateandL-duplicate
operationsleadto different
theorists(andsome
Tonnetzlocationsin Figure24. For manynineteenth-century
modemones as well, e.g. Vogel1993(1975))the notationalandlocationaldistinctionswouldreflectan acousticdistinctionof threesyntoniccommas.
29. These were firstenumeratedby CloughandClampittin the same documentreferredto in note 15 above.
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