Vol. 8 - 2024

MusMat • Brazilian Journal of Music and Mathematics

Papers

Musical Systems with Zn - Cayley Graphs

Gabriel Picioroaga, Olivia Roberts

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 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.

Keywords: Cayley graph, finitely generated group, chords, scales, circle of fifths, first species counterpoint.

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