| _version_ | 1866902017504444416 |
|---|---|
| author | Lesperance, Joel Michael |
| author_facet | Lesperance, Joel Michael |
| contents | <p><span>This paper presents a spherical geometric framework that integrates two–three tonal relations, two–three–five relational structures, and the circle of fifths within a single non-linear container. Linear and circular tonal representations are treated as valid planar projections of a higher-order volumetric geometry. The spherical model demonstrates how intervallic ratios, cyclic ordering, and harmonic closure coexist without hierarchy when expressed as boundary-contained relationships. The framework offers a unified geometric description applicable to tonal organization, harmonic modeling, and relational systems that require simultaneous containment of sequence, cycle, and symmetry.</span></p> <p><span>spherical geometry, tonal geometry, harmonic structure, two three relation, two three five relation, circle of fifths, musical topology, harmonic cycles, tonal symmetry, geometric music theory, boundary systems, volumetric modeling, rotational symmetry, great circles, tonal containment, relational geometry, harmonic integration, cyclic structures, intervallic ratios, tonal organization, non-linear systems, geometric projection, symmetry groups, harmonic space, tonal mapping, music and geometry, mathematical music theory, pitch space, tonal coherence, structural harmony, harmonic closure, rotational equivalence, geometric integration, tonal relations, cyclic order, proportional systems, surface topology, musical structure, harmonic modeling, spatial music theory, boundary geometry, tonal systems, volumetric relations, cyclic harmony, tonal symmetry space</span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18489927 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Spherical Geometric Integration of Two–Three–Five Tonal Relations and the Circle of Fifths Lesperance, Joel Michael spherical geometry, tonal geometry, harmonic structure, two three relation, two three five relation, circle of fifths, musical topology, harmonic cycles, tonal symmetry, geometric music theory, boundary systems, volumetric modeling, rotational symmetry, great circles, tonal containment, relational geometry, harmonic integration, cyclic structures, intervallic ratios, tonal organization, non-linear systems, geometric projection, symmetry groups, harmonic space, tonal mapping, music and geometry, mathematical music theory, pitch space, tonal coherence, structural harmony, harmonic closure, rotational equivalence, geometric integration, tonal relations, cyclic order, proportional systems, surface topology, musical structure, harmonic modeling, spatial music theory, boundary geometry, tonal systems, volumetric relations, cyclic harmony, tonal symmetry space <p><span>This paper presents a spherical geometric framework that integrates two–three tonal relations, two–three–five relational structures, and the circle of fifths within a single non-linear container. Linear and circular tonal representations are treated as valid planar projections of a higher-order volumetric geometry. The spherical model demonstrates how intervallic ratios, cyclic ordering, and harmonic closure coexist without hierarchy when expressed as boundary-contained relationships. The framework offers a unified geometric description applicable to tonal organization, harmonic modeling, and relational systems that require simultaneous containment of sequence, cycle, and symmetry.</span></p> <p><span>spherical geometry, tonal geometry, harmonic structure, two three relation, two three five relation, circle of fifths, musical topology, harmonic cycles, tonal symmetry, geometric music theory, boundary systems, volumetric modeling, rotational symmetry, great circles, tonal containment, relational geometry, harmonic integration, cyclic structures, intervallic ratios, tonal organization, non-linear systems, geometric projection, symmetry groups, harmonic space, tonal mapping, music and geometry, mathematical music theory, pitch space, tonal coherence, structural harmony, harmonic closure, rotational equivalence, geometric integration, tonal relations, cyclic order, proportional systems, surface topology, musical structure, harmonic modeling, spatial music theory, boundary geometry, tonal systems, volumetric relations, cyclic harmony, tonal symmetry space</span></p> |
| title | A Spherical Geometric Integration of Two–Three–Five Tonal Relations and the Circle of Fifths |
| topic | spherical geometry, tonal geometry, harmonic structure, two three relation, two three five relation, circle of fifths, musical topology, harmonic cycles, tonal symmetry, geometric music theory, boundary systems, volumetric modeling, rotational symmetry, great circles, tonal containment, relational geometry, harmonic integration, cyclic structures, intervallic ratios, tonal organization, non-linear systems, geometric projection, symmetry groups, harmonic space, tonal mapping, music and geometry, mathematical music theory, pitch space, tonal coherence, structural harmony, harmonic closure, rotational equivalence, geometric integration, tonal relations, cyclic order, proportional systems, surface topology, musical structure, harmonic modeling, spatial music theory, boundary geometry, tonal systems, volumetric relations, cyclic harmony, tonal symmetry space |
| url | https://doi.org/10.5281/zenodo.18489927 |