An Asymmetric Formula for Interval Consonance and its Relation to Harmonic Coincidence

2026-06-15

AI Summary

This study addresses the quantitative characterization of musical interval consonance and its intrinsic relationship with harmonic coincidence. The authors propose an asymmetric consonance measure \( f(p/q) = p + \Omega(q) \), which treats the numerator and denominator of a frequency ratio distinctly, where \( \Omega(q) \) denotes the sum of the exponents in the prime factorization of \( q \). Grounded in a harmonic truncation model, this formula reinterprets Euler’s Gradus Suavitatis as a weighted count of coinciding harmonics and establishes a theoretical link to Galileo’s pulse coincidence model. Furthermore, an integer triangle \( T(n,k) \) is constructed to capture the two-stage consonance behavior of superparticular intervals. Experimental results demonstrate that the proposed measure achieves performance comparable to Gradus on standard consonance datasets while offering a more perceptually intuitive interpretation, and for the first time provides a precise mathematical formulation of the connection between consonance and harmonic coincidence.

Abstract

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.

Problem

• interval consonance
• dissonance measure
• harmonic coincidence
• asymmetric formula
• Gradus Suavitatis

Innovation

• asymmetric consonance formula
• harmonic coincidence
• Gradus Suavitatis
• prime exponent sum
• superparticular intervals