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IJMSTA - Vol. 8 - Issue 1 - January 2026
ISSN 2612-2146
Pages: 14
Fractal Structures, Prime Number Cycles, and Geometric Models of Jazz Voice-Leading: A Novel Framework for Harmonic Analysis Based on Seventh Chords
Authors: Biagio Russo
Categories: Journal
Abstract - Jazz harmony exhibits complex patterns of transformation that extend beyond traditional functional tonality, particularly in the use of seventh chords, chromatic motion, and non-cadential resolutions. This study proposes a formal analytical framework for modeling jazz voice leading among seventh chords using prime-number interval constraints within mod-12 pitch-class space. By restricting harmonic transfor-mations to prime intervals (2, 3, 5, and 7 semitones), the model generates non-redundant and non-periodic harmonic trajectories that minimize trivial symmetries while preserving parsimonious voice leading.
These transformations are represented through a set of geometric and topological models, including a two-dimensional Tonnetz-like graph adapted for seventh chords, a three-dimensional orbifold-inspired em-bedding, and a Euclidean triangle describing tonic, subdominant, and dominant functional regions in jazz harmony. Structural properties of the resulting harmonic networks are examined using entropy-based measures, tonal distance functions derived from tonal pitch space theory, and voice-leading cost metrics.
The analysis reveals recurrent patterns of structural self-similarity across multiple harmonic levels, sup-porting the interpretation of jazz harmonic motion as a scale-invariant and hierarchically organized system. Case studies drawn from works by Debussy, Bill Evans, and Messiaen further illustrate how non-functional harmonic resolutions can be understood as expressive pivots within this framework rather than as deviations from tonal norms. The proposed model offers applications in harmonic analysis, algorithmic composition, and computational music cognition, providing a mathematically grounded approach to the study of complex tonal systems.
Keywords: Jazz Harmony, Seventh Chords, Voice-
Prime Number Theory, Fractal Music, Structures Transformational Theory, Mod-12 Tonal Spaces, Orbifold Embeddings, Algorithmic Composition, Cognitive Music Theory
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