Scale Atlas: A Set-Theoretic Framework for Harmony, Enumeration, and Musical Exploration

Scale Atlas presents a set-theoretic and computational framework for the systematic description, enumeration, and exploration of musical scale space in equal temperament.

The work treats scales as structured subsets of pitch-class space and develops a formal vocabulary for comparing them by inclusion, intersection, difference, symmetric distance, interval cycles, canonical normalization, and transposition. In 12-tone equal temperament, this leads to a complete catalogue of the 351 non-empty playable cyclic scale classes, organized as canonical interval structures.

Beyond enumeration, the manuscript introduces a harmonic affordance model that interprets a scale as a local field of support, tension, stability, and possible transformation. This model is connected to Scale Atlas, a software environment for scale generation, cent-based visualization, ranked transformation suggestions, export, and fretboard mapping.

The aim of the work is to make scale space more explicit, navigable, and musically usable for analysis, composition, improvisation, and further computational exploration.