Generalized Voice-Leading Spaces

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 eq...

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Bibliographic Details
Published in:Science (American Association for the Advancement of Science) 2008-04, Vol.320 (5874), p.346-348
Main Authors: Callender, Clifton, Quinn, Ian, Tymoczko, Dmitri
Format: Article
Language:English
Subjects:
Equivalence relation
Exact sciences and technology
Geometry
Major chords
Mathematical methods in physics
Mathematical models
Mathematical sequences
Music theory
Musical chords
Musicology
Octaves
Optics
Other topics in mathematical methods in physics
Physics
Quotients
Sound pitch
Theory
Voice leading
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Summary: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.
ISSN:0036-8075
1095-9203
DOI:10.1126/science.1153021