Granular 12-Tone Harmony via Mixed-Radix Balanced Gray Codes: Interval Sets and the BalaGray Solver

Chromatic pitch clusters such as {C, C♯, D} produce harsh roughness. Conventional twelve-tone techniques distribute pitch classes evenly in melody but do not prevent chromatic clusters in harmony. We partition the twelve-tone space into interval sets—contiguous mixed-radix spans that generate complete chord vocabularies free of chromatic clusters—and traverse those chords via Gray codes, so that successive chords differ by exactly one pitch. We introduce a heuristic crawler, BalaGray, to find optimally balanced, minimal-span Gray cycles in arbitrary mixed-radix spaces, and show that the scarcity of such cycles justifies a dedicated solver. We provide open-source code together with exhaustive tables of prime-form interval sets and their balanced Gray sequences.