Musical Systems with ℤ_n -- Cayley Graphs

2024/02/09 by Picioroaga, Gabriel, Roberts, Olivia
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.06833

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 ℤ_n, 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 ℤ_n. Using Maple software, we implement these new constructions and show how to experiment with them musically.

Related

• Some combinatorial aspects of the musical chords
• The Generalized Tonnetz
• Groupoids and Wreath Products of Musical Transformations: a Categorical Approach from poly-Klumpenhouwer Networks
• Transformations of Triads and Seventh Chords: Group Extensions and Duality
• A group-theoretical classification of three-tone and four-tone harmonic chords