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arXiv · 2204.03022

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

Alexandre Popoff·Corentin Guichaoua·Moreno Andreatta
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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.

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Alexandre Popoff, Corentin Guichaoua, Moreno Andreatta. 2022-04-06. Composing (with) automorphisms in the colored Cube Dance: an interactive tool for musical chord transformation. https://arxiv.org/abs/2204.03022

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