Tonal parsimony in chord-sequence analysis: combining modulation cost and tonal vocabulary

2026-06-02

AI Summary

This study addresses the challenge in traditional tonal assignment methods for chord sequences, which often fail to simultaneously minimize both the number of modulations and the diversity of tonalities, leading to either tonal redundancy or excessive modulation. To resolve this, the paper introduces the principle of “tonal parsimony” and formulates the joint optimization of these two objectives as a lexicographic minimization problem—a first in the field. Within the 24 major–minor key system, the authors develop an exact dynamic programming algorithm that integrates weighted jazz substitution chord closures with tonal space modeling. Evaluated on 31,032 LMD sequences, the method reduces the number of tonalities in 55.8% of cases, lowering the average tonality count from 3.802 to 3.206 and the average modulation count from 16.728 to 12.141. On a corpus of 1,555 jazz standards, it achieves a chord–scale compatibility rate of 95.6%.

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.

Problem

• tonal parsimony
• chord-sequence analysis
• modulation cost
• tonal vocabulary
• harmonic analysis

Innovation

• tonal parsimony
• chord-sequence analysis
• modulation cost
• tonal vocabulary
• dynamic programming