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Generalized voice-leading spaces
Clifton Callender1, Ian Quinn, Dmitri Tymoczko
1College of Music, Florida State University, Tallahassee, FL 32306, USA.
Science (New York, N.Y.)
|April 19, 2008
Summary
This study introduces a mathematical model for understanding musical pitch sequences, revealing 32 equivalence relations for chords and 243 for sequences. The model offers new analytical tools and insights into chord similarity.
Area of Science:
- Music Theory
- Mathematical Modeling
- Computational Musicology
Background:
- Traditional Western music theory classifies pitch sequences while ignoring transformations like octave shifts and transpositions.
- Existing geometrical representations of musical structures can be complex and fragmented.
Purpose of the Study:
- To develop a mathematical model that accounts for five key musical transformations (octave shift, permutation, transposition, inversion, cardinality change).
- To reveal underlying mathematical structures and equivalences within chords and chord sequences.
- To provide new analytical tools for music theory and a unified framework for geometrical representations.
Main Methods:
- Mathematical modeling of musical transformations.
- Derivation of equivalence relations for chords and chord sequences.
- Construction of geometrical quotient spaces for musical elements.
Main Results:
- Identification of 32 distinct equivalence relations for chords.
- Identification of 243 distinct equivalence relations for chord sequences.
- Development of 32 families of geometrical quotient spaces representing chords and sequences.
Conclusions:
- The mathematical model provides a unified framework for understanding musical structures.
- The model reveals novel connections between music-theoretical concepts.
- It offers new analytical tools and a method for assessing similarity between chord types.
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