An Asymmetric Formula for Interval Consonance and its Relation to Harmonic Coincidence
2026-06-15 • Sound
Sound
AI summaryⓘ
The authors study a way to measure how dissonant musical intervals sound, based on Euler’s 1739 formula that uses prime factors of the interval's numbers. They suggest a simpler formula that treats the numerator and denominator differently but works just as well on known sound tests. They also connect Euler’s original method to an older model by Galileo that counts overlapping sound pulses. Their new formula creates a special number pattern and offers an idea about how we might perceive sounds in two steps.
Euler's Gradus Suavitatismusical intervaldissonanceprime factorizationharmonicssuperparticular intervalsharmonic coincidenceGalileo's pulse-coincidence modelcoprime numbersperceptual hypothesis
Authors
David De Roure
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.