A Theory of Scales and Orbit Covers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910098901696512 |
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| author | Flieder, Drew |
| author_facet | Flieder, Drew |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02373 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Theory of Scales and Orbit Covers Flieder, Drew General Mathematics 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. |
| title | A Theory of Scales and Orbit Covers |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2604.02373 |