Tonal Consonance and Critical Bandwidth

TLDR

The paper reviews theories that explain tonal consonance as arising from tone intervals with frequency ratios of small integers. The study aims to determine whether critical bandwidth influences musical chords by analyzing interval distributions in Bach and Dvořák compositions. The authors performed psychophysical tests of simple‑tone interval judgments across frequencies and interval widths, and analyzed chord interval distributions from Bach and Dvořák pieces. The results indicate that the consonant–dissonant transition range corresponds to the critical bandwidth, with consonance occurring for frequency differences exceeding this bandwidth and maximum dissonance at about a quarter of it; these findings also suggest that critical bandwidth shapes the density of partials in musical chords.

Abstract

Firstly, theories are reviewed on the explanation of tonal consonance as the singular nature of tone intervals with frequency ratios corresponding with small integer numbers. An evaluation of these explanations in the light of some experimental studies supports the hypothesis, as promoted by von Helmholtz, that the difference between consonant and dissonant intervals is related to beats of adjacent partials. This relation was studied more fully by experiments in which subjects had to judge simple-tone intervals as a function of test frequency and interval width. The results may be considered as a modification of von Helmholtz's conception and indicate that, as a function of frequency, the transition range between consonant and dissonant intervals is related to critical bandwidth. Simple-tone intervals are evaluated as consonant for frequency differences exceeding this bandwidth. whereas the most dissonant intervals correspond with frequency differences of about a quarter of this bandwidth. On the base of these results, some properties of consonant intervals consisting of complex tones are explained. To answer the question whether critical bandwidth also plays a rôle in music, the chords of two compositions (parts of a trio sonata of J. S. Bach and of a string quartet of A. Dvořák) were analyzed by computing interval distributions as a function of frequency and number of harmonics taken into account. The results strongly suggest that, indeed, critical bandwidth plays an important rôle in music: for a number of harmonics representative for musical instruments, the “density” of simultaneous partials alters as a function of frequency in the same way as critical bandwidth does.