Tonal parsimony in chord-sequence analysis: combining modulation cost and tonal vocabulary
和弦序列分析中的调性简约性:结合调制代价与调性词汇
François Pachet
AI总结 提出调性简约性方法,通过字典序最小化调制次数和不同调性数量,结合动态规划与固定24调性空间,在和弦序列分析中减少调性词汇并保持调制最优。
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- 20 pages, 1 figure
我们研究将局部调性分配给和弦序列,这一任务对和声分析、作曲和爵士即兴演奏很有用。标准的动态规划方法最小化调制,但可能引入不必要多的调性中心。我们将这种仅转移目标与纯最小词汇分析以及调性简约性进行比较,后者按字典序最小化调制次数,然后最小化不同调性的数量。尽管这个联合目标通常组合困难,但我们利用固定的24调性大调/小调宇宙给出了精确算法。在31,032个LMD和弦序列上,调性简约性在55.8%的情况下保持了转移最优,同时减少了调性词汇。在加权爵士替换闭包下,它将平均调性数从3.802降至3.206,调制次数从16.728降至12.141。在1,555个带注释的爵士标准曲上,它将兼容和弦-音阶一致性提高到95.6%,支持可处理的专业级和声分析。
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.