Musical Systems with $\mathbb{Z}_n$ -- Cayley Graphs

Abstract

We apply geometric group theory to study and interpret known concepts from Western music. We show that chords, the circle of fifths, scales and certain aspects of the first species of counterpoint are encoded in the Cayley graph of the group $\mathbb{Z}_{12}$, generated by $3$ and $4$. Using $\mathbb{Z}_{12}$ as a model, we extend the above music concepts to a particular class of groups $\mathbb{Z}_{n}$, which displays geometric and algebraic features similar to $\mathbb{Z}_{12}$. We identify a weaker form of counterpoint which, in particular leads to Fux's dichotomy in $\mathbb{Z}_{12}$, and to consonant sets in $\mathbb{Z}_n$. Using Maple software, we implement these new constructions and show how to experiment with them musically.