Constrained voice-leading spaces

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