A biological rationale for musical consonance

Abstract

The basis of musical consonance has been debated for centuries without resolution. Three interpretations have been considered: (i) that consonance derives from the mathematical simplicity of small integer ratios; (ii) that consonance derives from the physical absence of interference between harmonic spectra; and (iii) that consonance derives from the advantages of recognizing biological vocalization and human vocalization in particular. Whereas the mathematical and physical explanations are at odds with the evidence that has now accumulated, biology provides a plausible explanation for this central issue in music and audition.

Keywords: audition; biology; consonance; music; vocalization.

Fig. 1.

The relative consonance of musical intervals. (A) The equal tempered chromatic scale used in modern Western and much other music around the world. Each interval is defined by the ratio between a tone’s fundamental frequency and that of the lowest (tonic) tone in the scale. In equal temperament, small adjustments to these ratios ensure that every pair of adjacent tones is separated by 100 cents (a logarithmic measure of frequency). (B) The relative consonance assigned by listeners to each of the 12 chromatic intervals played as two-tone chords. The filled black circles and dashed line show the median rank for each interval; colored circles represent data from ref. ; open circles from ref. ; crosses from ref. ; open squares from ref. . These data were collected between 1898 and 2012, in Germany, Austria, the United Kingdom, the United States, Japan, and Singapore.

Fig. 2.

The harmonic series generated by a vibrating string. (A) Diagram of the first 10 vibrational modes of a string stretched between fixed points. (B) A spectrogram of the frequencies produced by a vibrating string with a fundamental frequency of 100 Hz. Each of the dark horizontal lines is generated by one of the vibrational modes in A. The first mode gives rise to the fundamental at 100 Hz, the second mode to the component at 200 Hz, the third mode to the component at 300 Hz, etc. These component vibrations are called harmonics, overtones, or partials and their frequencies are integer multiples of the fundamental frequency. Note that many ratios between harmonic frequencies correspond to ratios used to define musical intervals (cf. Fig. 1A).

Fig. 3.

Auditory roughness as the basis of dissonance and its absence as the basis of consonance. (A) Acoustic waveforms produced by middle C, C#, and their combination (a minor second) played on an organ. Small differences in the harmonic frequencies that comprise these tones give rise to an alternating pattern of constructive and destructive interference perceived as auditory roughness. (B) Waveforms produced by middle C, G, and their combination (a perfect fifth). Helmholtz argued that the relative lack of auditory roughness in this interval is responsible for its consonance.

Fig. 4.

Auditory roughness versus perceived consonance. Roughness scores plotted against mean consonance ratings for the 12 possible two-tone chords (dyads), 66 three-tone chords (triads), and 220 four-tone chords (tetrads) that can be formed using the intervals of the chromatic scale. The labeled chords highlight an example of a rough chord (the tetrad comprised of a major second, major third, and perfect fifth) that is perceived as more consonant than a chord with less roughness (the major-seventh dyad). The chord stimuli were comprised of synthesized piano tones with fundamental frequencies tuned according to the ratios in Fig. 1A and adjusted to maintain a mean frequency of 262 Hz (middle C). Roughness scores were calculated algorithmically as described in ref. . Consonance ratings were made by 15 music students at the Yong Siew Toh Conservatory of Music in Singapore using a seven-point scale that ranged from “quite dissonant” to “quite consonant.” (Data from ref. .)

Fig. 5.

Musical intervals in voiced speech. The graph shows the statistical prominence of physical energy at different frequency ratios in voiced speech. Each labeled peak corresponds to one of the musical intervals in Fig. 1A. The data were produced by averaging the spectra of thousands of voiced speech segments, each normalized with respect to the amplitude and frequency of its most powerful spectral peak. This method makes no assumptions about the structure of speech sounds or how the auditory system processes them; the distribution simply reflects ratios emphasized by the interaction of vocal fold vibration with the resonance properties of the vocal tract. (Data from ref. .)

Fig. 6.

The conformity of widely used pentatonic and heptatonic scales to a uniform harmonic series. (A) Schematic representations of the spectra of a relatively consonant (perfect fifth) and a relatively dissonant (minor second) two-tone chord. Blue bars represent the harmonic series of the lower tone, red bars indicate the harmonic series of the upper tone, and blue/red bars indicate coincident harmonics. Gray bars indicate the harmonic series defined by the greatest common divisor of the red and blue tones. The metric used by Gill and Purves (37) defined harmonic conformity as the percentage of harmonics in the gray series that are actually filled in by chord spectra (for another metric, see “harmonic entropy” in ref. 4). (B) When ranked according to harmonic conformity, the top ten pentatonic and heptatonic scales out of >40 million examined corresponded to popular scales used in different musical traditions. (Data from ref. .)