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Musical Systems with $\mathbb{Z}_n$ -- 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 $\mathbb{Z}_{12}$, 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.

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 , generated by and . Using as a model, we extend the above music concepts to a particular class of groups , which displays geometric and algebraic features similar to . We identify a weaker form of counterpoint which, in particular leads to Fux's dichotomy in , and to consonant sets in . Using Maple software, we implement these new constructions and show how to experiment with them musically.
Paper Structure (11 sections, 10 theorems, 7 equations, 5 figures)

This paper contains 11 sections, 10 theorems, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Terminology and Notation
  3. Groups, Generators and Cayley Graphs
  4. Understanding Western Music with $\mathbb{Z}_{12}$
  5. Chords
  6. The Circle of Fifths
  7. Major and Minor Scales
  8. Counterpoint
  9. Sound and Tuning
  10. Musical Systems in $\mathbb{Z}_n$
  11. Conclusion

Key Result

Theorem 3.13

Let $G$ be a group, $e$ its identity element, and $f\in\text{Aut}(G)$. Suppose $S\subset G$ is a symmetric, generating set for $G$. The following are equivalent: i) $f(S)=S$. ii) $f$ is an isometry on the unoriented Cayley graph of $G$ with respect to $S$.

Figures (5)

  • Figure 1: $\mathbb{Z}_{6}= \langle 2, 3\rangle$ oriented Cayley graph
  • Figure 2: $\mathbb{Z}_{12}= \langle 3, 4\rangle$ oriented Cayley graph
  • Figure 3: The Circle of Fifths in Western Classical Music
  • Figure 4: The Circle of Fifths for $\mathbb{Z}_{6}=\langle 2,3\rangle$
  • Figure 5: The Circle of Fifths for $\mathbb{Z}_{10}=\langle 2,5\rangle$

Theorems & Definitions (62)

  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • Definition 3.5
  • Example 3.6
  • Definition 3.7
  • Remark 3.8
  • Definition 3.9
  • Definition 3.10
  • ...and 52 more