An Asymmetric Formula for Interval Consonance and its Relation to Harmonic Coincidence
Authors: David De Roure
Organizations: Department of Engineering Science, University of Oxford, Oxford, UK. · Technical Director, Centre for Practice & Research in Science & Music (PRiSM), Royal Northern College of Music, Manchester, UK.
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.
We propose a general approach to enharmonicism within syntactic music theory; that is, we formalize several abstract structures of enharmonicism without any acoustic considerations. The structures of concern to this paper are constructed from diatonics, chromatics, and enharmonics, and we seek to present a general enharmonic theory that builds upon Hook's enharmonic theory published in "Enharmonic Systems: A Theory of Key Signatures, Enharmonic Equivalence and Diatonicism" (2007). We also propose that the two criteria of reflection and maximal evenness can be used to determine whether an arbitrary enharmonic system is musically practical. Moreover, we argue that preferential treatment of the standard enharmonic system (SES) can be motivated from an abstract mathematical perspective divorced from pitch consideration and conventional acoustic constructions of SES.
We propose a continuous measure of tonal ambiguity that extends the established concept of uniqueness. While uniqueness is widely regarded as necessary for tonality, it cannot (i) discriminate among sets that possess it, (ii) capture hierarchical organization in modes of limited transposition, or (iii) account for temporal unfolding. To address these limitations, we introduce a companion measure, grounded in information theory, that quantifies tonal ambiguity on a continuous scale. The measure applies across pitch-class sets and tuning systems, expanding analytic coverage of tonal relationships and offering a practical tool for theory and analysis.
We present vega-mir, an open-source Python library that bundles nine information-theoretic and statistical metrics for the analysis of symbolic music corpora behind a small, tested, citable API, and demonstrates two of them at corpus scale in case studies not addressed by the upstream Cygnus paper. Of the nine metrics, three (Shannon entropy, Kullback-Leibler divergence, Zipfian fits) were deployed in the companion Cygnus arXiv preprint; two (network analysis on chord-transition graphs and spectral analysis of rubato curves) are deployed in full case studies here; the four remaining (multi-dimensional Gini, chi-squared stationarity, Higuchi fractal dimension, interval distribution) are validated against analytic anchors and exercised as sanity checks on a bundled 8-composer dataset. The two case studies yield two main observations. First, on the fourteen MAESTRO composers with N >= 10 pieces, the PageRank value of the gravity-centre node correlates with the marginal Kullback-Leibler distance at rho = 0.61 (Spearman, composer-level jackknife N = 14); the categorical gravity-centre identity takes five distinct values across the corpus but is not itself correlated with marginal KL (rho = 0.13, p = 0.21). Second, on the 247-piece Bach multi-master corpus (Schiff, Gould, Richter), Gould holds the highest periodicity ratio of the three performers, not the lowest, inverting the cliché that low scalar rubato reads as "metronomic": Gould's rubato is small in amplitude but structured in time, with a median dominant period of 66 beats against Schiff's 102 and Richter's 104.