generic (mod-7) voice-leading spaces...
TRANSCRIPT
Leah Frederick | Indiana University | [email protected] 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
EA!
E!A
B!D
BC#
CC
CC
BC#
B!D
E!A
EA!
FG
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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