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A Theory of Scales and Orbit Covers

Drew Flieder

Abstract

This paper develops a formal theory of musical scales and their harmonic coverings and introduces orbit covers: coverings obtained by translating a fixed subset across a scale via a group action. Orbit covers generalize familiar constructions, such as the covering of the diatonic scale by tertian triads, and are motivated by the search for a generalized harmonic framework extending common-practice tonality. We model modes as group structures associated with pitch-class sets and scales as torsors, introducing scale covers and, in particular, orbit covers. To each orbit cover we associate a nerve complex encoding its intersection structure and associated topological invariants. We classify triadic orbit covers of heptatonic scales up to affine symmetry and nerve isomorphism. These results support a broader theory of harmonic organization with analytical and compositional applications.

A Theory of Scales and Orbit Covers

Abstract

This paper develops a formal theory of musical scales and their harmonic coverings and introduces orbit covers: coverings obtained by translating a fixed subset across a scale via a group action. Orbit covers generalize familiar constructions, such as the covering of the diatonic scale by tertian triads, and are motivated by the search for a generalized harmonic framework extending common-practice tonality. We model modes as group structures associated with pitch-class sets and scales as torsors, introducing scale covers and, in particular, orbit covers. To each orbit cover we associate a nerve complex encoding its intersection structure and associated topological invariants. We classify triadic orbit covers of heptatonic scales up to affine symmetry and nerve isomorphism. These results support a broader theory of harmonic organization with analytical and compositional applications.

Paper Structure

This paper contains 12 sections, 8 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Modes and Scales
  3. Modes
  4. Scales
  5. Scale Covers and Their Classification
  6. Scale Covers and Morphisms
  7. Orbit Covers and Primitive Covers
  8. Classification of Orbit Covers
  9. Topology of Scale Covers
  10. Nerve Complexes and Orbit Covers
  11. Affine Classification of Nerves
  12. Outlook and Future Prospects

Key Result

Proposition 3.1

Let $\sigma, \sigma' \in \Sigma(n,k)$. Then $\mathrm{T}_p(\sigma(x)) = \sigma'(x)$ for some $p \in \mathbb{Z}_n$ if and only if $\sigma'$ is a rotation of $\sigma$, i.e., there exists $\mathrm{R}_i \in \mathbf{R}\mathbb{Z}^k$ such that $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Comparison of harmonic structure in diatonic and exotic triadic orbit covers. The transformation preserves orbit structure and local common-tone relations.

Theorems & Definitions (24)

  • Definition 2.1: Mode
  • Example 2.1: Modes of a Major Scale
  • Definition 2.2: Mode Homomorphism
  • Definition 2.3: Torsor
  • Definition 2.4: Scale
  • Definition 2.5: Scale Homomorphism
  • Definition 3.1: Scale Cover
  • Definition 3.2: Orbit Cover
  • Example 3.1: Diatonic Triadic Orbit Cover
  • Proposition 3.1
  • ...and 14 more