Musical Systems with Zₙ -- Cayley Graphs

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 Z₁₂, generated by $3$ and $4$. Using Z₁₂ as a model, we extend the above music concepts to a particular class of groups Zₙ, which displays geometric and algebraic features similar to Z₁₂. We identify a weaker form of counterpoint which, in particular leads to Fux's dichotomy in Z₁₂, and to consonant sets in Zₙ. Using Maple software, we implement these new constructions and show how to experiment with them musically.