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

TL;DR

This paper investigates the automorphism group of a musical graph structure called the colored Cube Dance, revealing its complex symmetry properties and providing an interactive web tool for musical exploration and education.

Contribution

It characterizes the automorphism group of the colored Cube Dance monoid and develops an interactive web tool for practical musical and educational use.

Findings

Automorphism group order is 7776.

Group isomorphic to a semi-direct product involving Z3^4, D8, D6, and Z2.

Web-based tool enables exploration of chord transformations.

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 $({\mathbb{Z}_3}^4 \rtimes D_8) \rtimes (D_6 \times \mathbb{Z}_2)$. 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.