An Asymmetric Formula for Interval Consonance and its Relation to Harmonic Coincidence (opens in new tab)

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 inte...