A General Approach to Enharmonicism

Euler's Gradus Suavitatis (1739) assigns a dissonance value to a musical interval p/q by the formula G(p/q) = 1 + Ω^(p) + Ω^(q), where Ω^(n) = \sum_i e_i(p_i - 1) sums the weighted prime exponents of n. We propose the simpler asymmetric formula f(p/q) = p + Ω^(q), which treats numerator and denominator differently and performs comparably on standard consonance data. We also show that, under a model in which harmonics are integer-indexed and counted uniformly up to a fixed truncation level, Gradus is equivalent to a weighted harmonic coincidence count with weights w(n) = Ω^(n), connecting it to Galileo's earlier pulse-coincidence model (1638). The formula naturally generates a coprime integer triangle T(n,k) = n + Ω^(k), whose rightmost diagonal gives the two-stage dissonance of the superparticular (consecutive-harmonic) intervals. The formula f admits a simple two-stage interpretation in terms of harmonic context and partial recognition, which we offer as a speculative perceptual hypothesis.

Since antiquity, musical scales have been explained by harmony rather than melody. This view relies on the mathematically designed scales of a few traditions, and was never directly tested. Testing it requires cross-cultural data and a method that judges theories by what they get wrong as well as right. We provide both, modelling scale evolution across 1,314 scales from 96 countries. A Melody model explains the near-universal preference for step-sizes of 1-3 semitones, and matches independent data from melodies, singing, and psychoacoustics. Harmony does far less: it explains the music-theoretic scales, but in those measured from performance it adds only a weak bias towards fourths, fifths, and octaves. Harmony's importance has been overstated, likely due to the historical focus on music-theoretic rather than measured scales. Melody is the primary driver of global scale structure; harmonic constraints are less impactful and mainly reflect musicological theory over musical performance.

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.