Constrained voice-leading spaces

Abstract

Dmitri Tymoczko describes the voice-leading space of N-note chords as the orbifold ${\mathbb{T}}^N/S_N$, the N-torus modulo the Nth symmetric group action, “an N-dimensional prism whose simplicial faces are glued together with a twist, and whose remaining boundaries act like mirrors” (2011. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Oxford: Oxford University Press). This quotient space ${\mathbb{T}}^N/S_N$ is produced from the space of all ordered sequences of N pitch classes ${\mathbb{T}}^N$ by identifying each sequence with all its reorderings, indicating that we consider a chord unchanged under any permutation of its voices. Here instead we consider a polyphonic setting in which not all voices are free to move independently. Such constraints describe “power chords” in rock (bare fifths or fourths played on guitar) and can also be found in the classical repertoire. We present chord spaces describing excerpts from Bartók and Stravinsky.

Keywords:

• voice-leading space
• orbifold
• linear constraint
• power chord
• Bartók
• Stravinsky

AMS Classification codes:

• 00A65
• 57R18

Acknowledgments

The authors are grateful for many helpful suggestions from Dmitri Tymoczko, Thomas Fiore, Clifton Callender, and the anonymous reviewer.

Disclosure statement

No potential conflict of interest was reported by the authors.