PERSPECTIVE PERSPECTIVE A biological rationale for musical consonance Daniel L. Bowlinga,1 and Dale Purvesb,1 aDepartment of Cognitive Biology, University of Vienna, 1090 Vienna, Austria; and bDuke Institute for Brain Sciences, Duke University, Durham, NC 27708 Edited by Solomon H. Snyder, Johns Hopkins University School of Medicine, Baltimore, MD, and approved June 25, 2015 (received for review March 25, 2015) 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 accumu- lated, biology provides a plausible explanation for this central issue in music and audition. consonance | biology | music | audition | vocalization Why we humans hear some tone combina- perfect fifth (3:2), and the perfect fourth revolution in the 17th century, which in- tions as relatively attractive (consonance) (4:3), ratios that all had spiritual and cos- troduced a physical understanding of musi- and others as less attractive (dissonance) has mological significance in Pythagorean phi- cal tones. The science of sound attracted been debated for over 2,000 years (1–4). losophy (9, 10). many scholars of that era, including Vincenzo These perceptual differences form the basis The mathematical range of Pythagorean and Galileo Galilei, Renee Descartes, and of melody when tones are played sequen- consonance was extended in the Renaissance later Daniel Bernoulli and Leonard Euler. tially and harmony when tones are played by the Italian music theorist and composer Vincenzo Galilei, who had studied under simultaneously. Geoseffo Zarlino. Zarlino expanded the Py- Zarlino, undermined theories based on Musicians and theorists have long consid- thagorean “tetrakys” to include the numbers mathematical simplicity by demonstrating ered consonance and dissonance to depend 5 and 6, thus accommodating the major third the presence of more complex ratios in on the fundamental frequency ratio between (5:4), minor third (6:5), and major sixth (5:3), numerologically “pure” scales (e.g., 32:27 tones (regularly repeating sound signals that which had become increasingly popular in between the second and fourth tones of we perceive as having pitch). A number of thepolyphonicmusicoftheLateMiddle Zarlino’s justly tuned major scale). He also studies have asked listeners to rank the Ages (2). Echoing the Pythagoreans, Zarlino’s showed that simple ratios do not account for relative consonance of a standard set of two- rationale was based on the numerological consonance when their terms express the tone chords whose fundamental frequency significance of 6, which is the first integer that relative weights of hammers or volumes ratios range from 1:1 to 2:1 (Fig. 1A)(5–8). equalsthesumofallofthenumbersofwhich enclosed in bells, as proponents of mathe- The results show broad agreement about it is a multiple (1 + 2 + 3 = 1 × 2x3= matical theories had assumed. Galileo con- which chords are more consonant and which 6). Additional reasons included the natural firmed his father’s work, and went on to less (Fig. 1B). But why two-tone combina- world as it was then understood (six “planets” show that, properly conceived, the terms of tions with different frequency ratios are dif- in the sky), and Christian theology (the world consonant ratios express the frequencies at ferently appealing to listeners has never was created in 6 days) (11). According to one which objects vibrate (10). been settled. music historian, Zarlino sought to create “a At about the same time, interest grew in Inwhatfollows,wediscussthemajorways divinely ordained natural sphere within the sounds produced by vibrating objects that that mathematicians, physicists, musicians, which the musician could operate freely” do not correspond to the pitch of the fun- psychologists, and philosophers have sought (ref. 10, p. 103). damental frequency. Among the early con- to rationalize consonance and dissonance, Although Pythagorean beliefs have long tributors to this further issue was the French concluding that the most promising frame- been derided as numerological mysticism, the theologian and music theorist Marin work for understanding consonance is evo- coincidence of numerical simplicity and Mersenne, who correctly concluded that the lutionary biology. pleasing perceptual effect continues to in- pitches of these “overtones” corresponded to fluence music theory and concepts of con- specific musical intervals above the funda- Consonance Based on Mathematical sonance even today (12, 13). The idea that mental (10). A physical basis for overtones Simplicity tone combinations are pleasing because they was provided shortly thereafter by the Most accounts of consonance begin with the are simple, however, begs the question of French mathematician and physicist Joseph interpretation of Greek mathematician and why simple is pleasing. And theories of Sauveur, who showed that the overtones of a philosopher Pythagoras in the sixth century consonance based on mathematical simplic- BCE. According to legend, Pythagoras showed ity have no better answer today than did Author contributions: D.L.B. and D.P. analyzed data; and D.L.B. and that tones generated by plucked strings Pythagoras. D.P. wrote the paper. whose lengths were related by small integer The authors declare no conflict of interest. ratios were pleasing. In light of this observa- Consonance Based on Physics This article is a PNAS Direct Submission. tion, the Pythagoreans limited permissible Enthusiasm for mathematical explanations 1To whom correspondence may be addressed. Email: purves@ tone combinations to the octave (2:1), the of consonance waned during the scientific neuro.duke.edu or [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1505768112 PNAS | September 8, 2015 | vol. 112 | no. 36 | 11155–11160 Downloaded by guest on September 29, 2021 of two-tone chords and showed that the combinations perceived as relatively consonant indeed exhibited little or no roughness, whereas those perceived as dissonant had relatively more. Helmholtz concluded that auditory roughness is the “true and suffi- cient cause of consonance and dissonance in music” (ref. 16, p. 227). Studies based on more definitive physio- logical and psychophysical data in the 20th century generally supported Helmholtz’s interpretation. Georg von Békésy’smap- ping of physical vibrations along the basilar membrane in response to sine tones made it possible to compare responses of the inner Fig. 1. The relative consonance of musical intervals. (A) The equal tempered chromatic scale used in modern ear with the results of psychoacoustical Western and much other music around the world. Each interval is defined by the ratio between a tone’s fundamental studies (17, 18). This comparison gave rise frequency and that of the lowest (tonic) tone in the scale. In equal temperament, small adjustments to these ratios to the idea of “critical bands,” regions ∼1mm ensure that every pair of adjacent tones is separated by 100 cents (a logarithmic measure of frequency). (B) The in length along the basilar membrane within 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. 5; open which the inner ear integrates frequency in- circles from ref. 6; crosses from ref. 7; open squares from ref. 8. These data were collected between 1898 and 2012, in formation (19–21). Greenwood (18) related Germany, Austria, the United Kingdom, the United States, Japan, and Singapore. critical bands to auditory roughness by comparing estimates of their bandwidth to the psychophysics of roughness perception plucked string arise from different vibrational that “in nature we find not only beauty but (22). The result suggested that tones falling modes and that their frequencies are neces- ugliness... proof that anything is natural within the same critical band are per- sarily integer multiples of the fundamental ” doesnotsufficetojustifyitesthetically (ref. ceived as rough whereas tones falling in dif- (Fig. 2). By the early 19th century, further 16, p. 232). He proposed instead that con- ferent critical bands are not (see also ref. 23). contributions from Bernoulli, Euler, Jean le “ ” sonance arises from the absence of jarring A link was thus forged between Helmholtz’s Rond d’Alembert, and Joseph Fourier had amplitude fluctuations that can be heard in conception of dissonance and modern provided a complete description of this some tone combinations but not others. The sensory physiology, and the phrase “sensory “harmonic series,” which arises not only from “ ” basis for this auditory roughness is the dissonance” was coined to describe this strings, but also from air columns and other physical interaction of sound waves with synthesis (24–26). The fact that perceived physical systems (14). similar frequencies, whose combination gives roughness tracked physical interactions on The ratios among harmonic overtones also rise to alternating periods of constructive and the basilar membrane was taken as support drew the