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Science 18 April 2008:
Vol. 320. no. 5874, pp. 346 - 348
DOI: 10.1126/science.1153021
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Reports

Generalized Voice-Leading Spaces

Clifton Callender,1 Ian Quinn,2 Dmitri Tymoczko3*

Western musicians traditionally classify pitch sequences by disregarding the effects of five musical transformations: octave shift, permutation, transposition, inversion, and cardinality change. We model this process mathematically, showing that it produces 32 equivalence relations on chords, 243 equivalence relations on chord sequences, and 32 families of geometrical quotient spaces, in which both chords and chord sequences are represented. This model reveals connections between music-theoretical concepts, yields new analytical tools, unifies existing geometrical representations, and suggests a way to understand similarity between chord types.

1 College of Music, Florida State University, Tallahassee, FL 32306, USA.
2 Music Department, Yale University, New Haven, CT 06520, USA.
3 Music Department, Princeton University, Princeton, NJ 08544, USA.

* To whom correspondence should be addressed. E-mail: dmitri{at}princeton.edu

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THIS ARTICLE HAS BEEN CITED BY OTHER ARTICLES:
Trading codes for errors.
J.-P. Eckmann (2008)
PNAS 105, 8165-8166
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Continuous Harmonic Spaces.
C. Callender (2007)
Journal of Music Theory 51, 277-332
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Science. ISSN 0036-8075 (print), 1095-9203 (online)