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

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1 Leah Frederick Indiana University Society for Music Theory Arlington, VA GENERIC (MOD-7) VOICE-LEADING SPACES ABSTRACT In 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 ) Figure 2. Discrete, 3-note, OP-Space (Mod-7) (Tymoczko 2011, fig )

2 Figure 3. Properties of Voice-Leading Spaces 1. Chromatic (mod-12) / Generic (mod-7) 2. Continuous/ Discrete 3. OPTIC Equivalence Relations Applied 4. Number of Notes per Chord Figure 4. Pitch Space [PITCH]/Continuous Pitch Space [CPITCH] (Hook, forthcoming, fig. 1.1) Figure 5. Generic Pitch Space [GPITCH] (Hook, forthcoming, fig. 1.5) Figure 6. Definitions and examples of the OPTIC Relations in mod-7 space O P T I C Octave Permutational Transpositional Inversional Cardinality relates points whose pitches are equivalent mod 7 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 (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)!

3 FG EA Figure 7. Discrete, 2-note, OPTI-Space (Mod-7) Figure 9. Discrete, 3-note, OPTI-Space (Mod-7) Figure 8. Discrete, 2-note, OP-Space (Mod-7) BB BC CD DD BD Figure 10. Discrete, 3-note, OPT-Space (Mod-7) AC AD BE CE DE AB GC CF GB FB DG DF EE GA FA EG EF AA GG FF

4 Figure 12. Continuous, 2-note, OP-Space (Mod-7) (after Tymoczko 2011, fig b) DD EE FF Figure 11. Continuous, 2-note, OP-Space (Mod-12) (after Tymoczko 2011, fig ) BC# B!D C#C# DD E!E! EE FF # C#D DE! E!E EF CD C#E! DE E!F EF# BD CE! C#E DF E!F# BE! CE C#F DF# E!G FF# EG F#F# FG EA! BD EA FG CD DE EF [FG] CE DF EG BE CF DG [EA] FB GC AD FA GB AC [BD] GA AB BC GG AA BB [] B!E! BE CF C#F# DG E!A! E!A EB! FB CF# C#G DA! EA FB! F#B GC A!C# AD E!A Figure 13. Discrete, 2-note, OP-Space (Mod-7) embedded in the Möbius strip EA! FG F#F# FA F#B! GB A!C AC# FA! F#A GB! A!B AC F#A! GA A!B! AB B!C F#G GA! A!A AB! B!B GG A!A! AA B!B! BB B!C# BC B!D BC# DD EE FF CD DE EF [FG] BD CE DF BE CF DG EA FB GC FA GB AC EG AD [EA] [BD] FG GA AB BC GG AA BB []

5 Figure 14. Cross Section of Continuous, 3-note, OP-space (Mod-12) (after Tymoczko 2011, fig ) C Figure 15. Cross Section of Continuous, 3-note, OP-space (Mod-7) C B!C#C# B# B!CD BBD ADD BCD BBE AC#D B!BE! ACE DDA! ACE! B!B!E DEG FAB E!E!F# DE!G C#E!A! C#EG CEG# ABE FA!B FAB! F#AA EEF DFF FGC GGB GAA DEF# FGC F#A!B! E!EF C#FF# F#GB GA!A EEE DFF F#F#C GGB! G#G#G# Figure 16. Continuous, 3-note, OP-space (Mod-12) (after Hook, forthcoming, fig. 9.9) Figure 17. Continuous, 3-note, OP-space (Mod-7)

6 BIBLIOGRAPHY Callender, Clifton Continuous Transformations. Music Theory Online 10 (3) callender.pdf Callender, Clifton, Ian Quinn, and Dmitri Tymoczko Generalized Voice-Leading Spaces. Science 320: Clough, John Aspects of Diatonic Sets. Journal of Music Theory 23: Clough, John, and Gerald Myerson Variety and Multiplicity in Diatonic Systems. Journal of Music Theory 29: Cohn, Richard A Tetrahedral Graph of Tetrachordal Voice- Leading Space. Music Theory Online 9 (4) _frames.html Douthett, Jack, and Peter Steinbach Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition. Journal of Music Theory 42: Forte, Allen The Structure of Atonal Music. New Haven, CT: Yale University Press. Hook, Julian. Forthcoming. Exploring Musical Spaces. New York: Oxford University Press. Roeder, John A Geometric Representation of Pitch-Class Series. Perspectives of New Music 25: Straus, Joseph N Uniformity, Balance, and Smoothness in Atonal Voice Leading. Music Theory Spectrum 25: Voice Leading in Set-Class Space. Journal of Music Theory 49: Tymoczko, Dmitri Scale Networks and Debussy. Journal of Music Theory 48 (2): The Geometry of Musical Chords. Science 313: Three Conceptions of Musical Distance. In Mathematics and Computation in Music, Proceedings of the Second International Conference of the Society for Mathematics and Computation in Music and John Clough International Conference, New Haven, CT, June 2009, edited by Elaine Chew, Adrian Childs, and Ching-Hua Chuan, pp Berlin: Springer A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford University Press The Generalized Tonnetz. Journal of Music Theory 56: Lewin, David. [1987] Generalized Musical Intervals and Transformations. New York: Oxford University Press. Original publication, New Haven, CT: Yale University Press.