Leah Frederick | Indiana University | lnfreder@indiana.edu Society for Music Theory | Arlington, VA | 11.3.2017

GENERIC (MOD-7) VOICE-LEADING SPACES

ABSTRACTIn the burgeoning field of geometric music theory, scholars have explored ways of spatially representing voice leadings between chords. The

OPTIC spaces provide a way to examine all “classes” of n-note chords formed under various types of equivalence: octave, permutational, transpositional, inversional, and cardinality. Although it is possible to map diatonic progressions in these spaces, they often appear irregular since the spaces are constructed with the fundamental unit of a mod-12 semitone, rather than a mod-7 diatonic step. Outside of geometric music theory, the properties of diatonic structure have been studied more broadly: Clough has established a framework for describing diatonic structure analogous to that of Forte’s set theory; Hook provides a more generalized, “generic,” version of this work to describe any seven-note scale. This paper employs these theories in order to explore the fundamental difference between mod-12 and mod-7 spaces: that is, whether the spaces are fundamentally discrete or continuous.

After reviewing the construction of these voice-leading spaces, this paper will present the mod-7 OPTIC-, OPTI-, OPT-, and OP-spaces of 2- and 3-note chords. Although these spaces are fundamentally discrete, they can be imagined as lattice points within a continuous space. This construction reveals that the chromatic (mod-12) and generic (mod-7) voice-leading lattices both derive from the same topological space. In fact, although the discrete versions of these lattices appear to be quite different, the topological space underlying each of these graphs depends solely on the number of notes in the chords and the particular OPTIC relations applied.

Figure 1. Continuous, 3-note, OP-Space (Mod-12) (Tymoczko 2011, fig. 3.8.2)

Figure 2. Discrete, 3-note, OP-Space (Mod-7) (Tymoczko 2011, fig. 7.5.5)

1. Chromatic (mod-12) / Generic (mod-7)2. Continuous/ Discrete3. OPTIC Equivalence Relations Applied4. Number of Notes per Chord

Figure 5. Generic Pitch Space [GPITCH] (Hook, forthcoming, fig. 1.5)

Figure 4. Pitch Space [PITCH]/Continuous Pitch Space [CPITCH] (Hook, forthcoming, fig. 1.1)

Figure 6. Definitions and examples of the OPTIC Relations in mod-7 space

Figure 3. Properties of Voice-Leading Spaces

O P T I C Permutational Transpositional Inversional Cardinality

relates points whose notes appear in a different order

relates points whose notes differ by the same level of generic transposition

relates points whose pitches are related by inversion about generic C4

relates points that differ only by the appearance of consecutive doublings

Octave

relates points whose pitches are equivalent mod 7

(C2, E3, G4)~O(C2, E6, G1) (!14, !5, 4)~O(!14, 16, !17)

(C2, E3, G4)~P(G4, C2, E3) (!14, !5, 4)~P(4, !14, !5)

(C2, E3, G4)~T(G2, B3, D5) (!14, !5, 4)~T(!10, !1, 8)

(C2, E3, G4)~I(C6, A4, F3) (!14, !5, 4)~I(14, 5, !4)

(C2, E3, G4, E3)~C(C2, C2, E3, G3, E3) (!14, !5, 4, !5)~C(!14, !14, !5, 4, !5)

!

Figure 7. Discrete, 2-note, OPTI-Space (Mod-7)

Figure 8. Discrete, 2-note, OP-Space (Mod-7)

Figure 9. Discrete, 3-note, OPTI-Space (Mod-7)

Figure 10. Discrete, 3-note, OPT-Space (Mod-7)

00 01 02 03

GC

FB

EA

DGC

F

BE

BD

AD

AB

GA

FG

EF

DE

CDBC

CC

AC

BB

GB

AAFA

GG

EG

FF

DF EE

CE

DD

000 001 002 003

011 012 013 014

022 023 024

014033

012 013 014

024

000 001 002 003

Figure 11. Continuous, 2-note, OP-Space (Mod-12) (after Tymoczko 2011, fig. 3.3.1)

Figure 13. Discrete, 2-note, OP-Space (Mod-7) embedded in the Möbius strip

Figure 12. Continuous, 2-note, OP-Space (Mod-7) (after Tymoczko 2011, fig. 4.1.4b)

DECD EF [FG]

DDCC EE FF

BE CF DG

BD CE DF

FA GB AC

EA FB GC

GG AA BB

FG GA AB

[EA]

EG

AD

[BD]

BC

[CC]

GG A!A! AA BBB!B!

F#G GA! A!A B!BAB! BC

F#A! GA A!B! B!CAB

FA F#B! GB AC#A!C

EB! FB CF# DA!C#G

BE! CE C#F E!GDF#

CD C#E! DE EF#E!F

C#C# DD E!E! FFEE

FA! F#A GB! ACA!B B!C#

EA FB! F#B A!C#GC AD

B!E! BE CF DGC#F# E!A!

BD CE! C#E E!F#DF EG

CC# C#D DE! EFE!E FF#

F#F#

FG

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E!A

B!D

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CC

CC

BC#

B!D

E!A

EA!

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F#F#

GG AA BB [CC]

GA AB BC

FB GC AD

CE DF EG

DD EE FF

FA GB AC [BD]

BE CF DG [EA]

CD DE EF [FG]

FG

EA

BD

CC

Figure 14. Cross Section of Continuous, 3-note, OP-space (Mod-12) (after Tymoczko 2011, fig. 3.8.6)

Figure 15. Cross Section of Continuous, 3-note, OP-space (Mod-7)

Figure 16. Continuous, 3-note, OP-space (Mod-12) (after Hook, forthcoming, fig. 9.9)

Figure 17. Continuous, 3-note, OP-space (Mod-7)

CCC

BCD

BBE

GAA

ADD

EEF

ACE

FGC

FABDEG

DFF GGB

CCC

BCC#

BBDB!C#C#

B!CD

AC#D B!BE!

ACE!

C#E!A!

DDA!

E!E!F#

EEE DFF F#F#C GGB! G#G#G#

E!EF C#FF# F#GB GA!A

F#A!B!

C#EG

DEF#

DE!G CEG#

FGC

FA!B F#AA

FAB!

ABE

B!B!E

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