A Theory of Scales and Orbit Covers

TL;DR

This paper introduces a formal framework for musical scales and harmonic coverings using group actions, orbit covers, and topological invariants to analyze and classify harmonic structures.

Contribution

It develops a novel mathematical theory of scales and orbit covers, extending harmonic analysis with topological and group-theoretic tools.

Findings

Classified triadic orbit covers of heptatonic scales up to symmetry.

Introduced a nerve complex to encode intersection structures of orbit covers.

Supported a broader harmonic organization theory with analytical applications.

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.