Tonal parsimony in chord-sequence analysis: combining modulation cost and tonal vocabulary
2026-06-02 • Sound
SoundArtificial Intelligence
AI summaryⓘ
The authors look at how to assign musical keys or tonal centers to sequences of chords, which helps in music analysis and creation. They compare different methods that focus on either minimizing key changes or using fewer different keys overall. They introduce a new combined method that balances these goals and provide exact ways to solve the problem efficiently since there are only 24 possible keys (major and minor). Their method results in fewer keys and key changes in many cases, improving how well the assigned keys fit actual jazz music. This makes detailed harmonic analysis more manageable for complex music like jazz standards.
local tonalitychord sequencesdynamic programmingtonal modulationtonal vocabularyharmonic analysisjazz substitutionchord-scale theorymajor/minor keyslexicographic optimization
Authors
François Pachet
Abstract
We study the assignment of local tonalities to chord sequences, a task useful for harmonic analysis, composition, and jazz-oriented improvisation. Standard dynamic-programming approaches minimize modulations but can introduce unnecessarily many tonal centers. We compare this transition-only objective with pure minimum-vocabulary analysis and with tonal parsimony, which minimizes lexicographically the number of modulations and then the number of distinct tonalities. Although this joint objective is combinatorially hard in general, we give exact algorithms exploiting the fixed 24-tonality major/minor universe. On 31,032 LMD Chords sequences, tonal parsimony preserves the transition optimum while reducing tonal vocabulary in 55.8% of cases. With weighted jazz-substitution closure, it lowers mean tonalities from 3.802 to 3.206 and modulations from 16.728 to 12.141. On 1,555 annotated jazz standards, it improves compatible chord-scale agreement to 95.6%, supporting tractable professional-scale harmonic analysis.