Consonance in music -- the Pythagorean approach revisited

TL;DR

This paper provides a mathematically principled reexamination of musical consonance by modeling tones as pure and complex signals and quantifying consonance with a cosine-similarity measure. The authors show that the local maxima of the consonance function along frequency ratios $r=\frac{n}{m}$ align with classical Pythagorean intervals and that the number of maxima grows with harmonic depth, connecting abstract ratios to psychoacoustic curves such as Plomp--Levelt. The work delivers both numerical evidence (recovering known consonance curves) and analytic results (explicitly characterizing maxima and their dependence on depth and harmonic amplitudes), offering a purely analytic explanation for consonance that does not rely on physiological assumptions. The findings illuminate how the richness of timbre shapes the set of perceptually consonant intervals and provide a bridge between historical music theory and modern signal-based consonance concepts with potential implications for timbre design and music analysis.

Abstract

The Pythagorean school attributed consonance in music to simplicity of frequency ratios between musical tones. In the last two centuries, the consonance curves developed by Helmholtz, Plompt and Levelt shifted focus to psycho-acoustic considerations in perceiving consonances. The appearance of peaks of these curves at the ratios considered by the Pythagorean school, and which were a consequence of an attempt to understand the world by nice mathematical proportions, remained a curiosity. This paper addresses this curiosity, by describing a mathematical model of musical sound, along with a mathematical definition of consonance. First, we define pure, complex and mixed tones as mathematical models of musical sound. By a sequence of numerical experiments and analytic calculations, we show that continuous cosine similarity, abbreviated as cosim, applied to these models quantifies the elusive concept of consonance as a frequency ratio which gives a local maximum of the cosim function. We prove that these maxima occur at the ratios considered as consonant in classical music theory. Moreover, we provide a simple explanation why the number of musical intervals considered as consonant by musicians is finite, but has been increasing over the centuries. Specifically, our formulas show that the number of consonant intervals changes with the depth of the tone (the number of harmonics present).

Figures (7)

• Figure 1: (a) Left: Graph of a pure tone. (b) Right: Cosine similarity of a pure tones $f$ and $rf$ for $r$ ranging between one and two.
• Figure 2: The consonance of samples of complex tones $f$ and $rf$ of depth $N$ ranging from three to six (compare to Figure \ref{['fig:Continuous_Cosine_Similarity']} presenting the continuous counterparts in place of finite samples).
• Figure 3: The Plomp--Levelt consonance diagram, taken from Plomp65
• Figure 4: Continuous cosine similarity of complex tones with frequencies $f$ and $rf$ for depth $N$ ranging from three to six. Note the similarity to Figure \ref{['fig:Cosine_Similarity']} containing the sampled version of the curves.
• Figure 5: The function $AC$ for $f=440$ Hz and (a) $n=3, m=5$ and (b) $n=5, m=3$.

Theorems & Definitions (9)

• Example 1
• Lemma 1
• proof
• Corollary 1
• proof
• Theorem 1
• Lemma 2
• proof
• Theorem 2