Constrained voice-leading spaces

David Bulger*, Richard Cohn

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    Abstract

    Dmitri Tymoczko describes the voice-leading space of N-note chords as the orbifold T1N/SN , 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 T1N/SN is produced from the space of all ordered sequences of N pitch classes TN 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.

    Original languageEnglish
    Pages (from-to)1-17
    Number of pages17
    JournalJournal of Mathematics and Music
    Volume10
    Issue number1
    DOIs
    Publication statusPublished - 2 Jan 2016

    Keywords

    • voice-leading space
    • orbifold
    • linear constraint;power chord
    • Bartok
    • Stravinsky

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