An Asymmetric Formula for Interval Consonance and its Relation to Harmonic Coincidence
About
Euler's Gradus Suavitatis (1739) assigns a dissonance value to a musical interval p/q by the formula G(p/q) = 1 + \Omega^(p) + \Omega^(q), where \Omega^(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 + \Omega^(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) = \Omega^(n), connecting it to Galileo's earlier pulse-coincidence model (1638). The formula naturally generates a coprime integer triangle T(n,k) = n + \Omega^(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.