Composing (with) automorphisms in the colored Cube Dance: an interactive tool for musical chord transformation

Abstract

The `colored Cube Dance' is an extension of Douthett's and Steinbach's Cube Dance graph, related to a monoid of binary relations defined on the set of major, minor, and augmented triads. This contribution explores the automorphism group of this monoid action, as a way to transform chord progressions. We show that this automorphism group is of order 7776 and is isomorphic to $({{\mathbb{Z}}_3}^4⋊D_8)⋊(D_6\times {\mathbb{Z}}_2)$. The size and complexity of this group makes it unwieldy: we therefore provide an interactive tool via a web interface based on common HTML/Javascript frameworks for students, musicians, and composers to explore these automorphisms, showing the potential of these technologies for math/music outreach activities.

Accepted to the Mathematics and Computation in Music 2022 (MCM 2022) conference. Final version will appear in the proceedings of MCM 2022, in the Springer LNCS series