What is permitted? The nature of music

3. Music as psychology and physiology

During the nineteenth century the sciences were both a resource for those who wanted to make the study of musical systems more rigorous, and a foil for the Romantic cult of solitary genius. Fourier showed in 1822 that any arbitrarily-shaped wave form can be represented as the superposition of a number of simple harmonic (sinusoidal) curves. Two decades later the German physicist Ohm demonstrated that this applied to sound waves as well: complex sounds composed of many frequencies could be understood as an appropriate set of simple frequencies, amplitudes, and phases added together. Various machines for generating sounds with definite combinations of frequencies were also invented in the early decades of the century. Scientists established that humans can hear tones ranging from roughly 20 Hz at the low end to around 20,000 Hz, and also showed that the threshold of audibility varies with frequency.

Late in the nineteenth century music theorists aspired to decipher the code of music in their theories of harmony and meter. (The first chair of musicology was established at Vienna in 1870, but imitators followed only gradually.) This vision of scientific study of music encompassed more than music aesthetics and practical music theory; it also extended to acoustics, and to the audiophysiology and psychology of perception. Not least, it would benefit from the study of music history. One of the aims of this science of music, wrote Hugo Riemann in 1913, would be to move interest away from "the life stories of the great masters towards the development of tonal forms and stylistic features." Appeals to musical "genius" would be rendered unnecessary as scholars approached ever closer to the underlying principles of music through scientific means. As you might expect, this nonetheless led to tension between what harmony as natural phenomenon might permit in theory, and what composition had permitted as historical practice.

Music theory may have aspired to scientific status, but this was not simply evidence of a generalized positivistic spirit. Physicists in particular made contributions to music theory from 1850 onward that directly shaped the terms of debate. The most influential of these studies was made by the physiologist and physicist Hermann von Helmholtz in his On the Sensations of Tone of 1863. Helmholtz sought to ground "the entire theory of musical harmony on natural scientific principles." He was anything but hostile to aesthetics per se—he simply found it more plausible than we perhaps do today that science could serve as the foundation for aesthetic culture. The assumption he shared with music theorists—an assumption widely held since Descartes’ day—was that the laws of nature should also be the laws of music.

In classical compositional practice the major triad: 

 

and the minor triad: 

 

are essentially equivalent, occupying symmetrical aesthetic positions. Helmholtz declared somewhat regretfully that the upper harmonics (or overtones: the multiples of a fundamental frequency which are also produced in most musical instruments; the lowest-frequency whole wave that fits a given violin string when plucked will be accompanied by fainter waves whose lengths are 1/2, 1/4, 1/8, 1/16 the length of the original, and whose frequencies are thus 2, 4, 8, 16 times higher) did not match up mathematically and empirically quite as neatly for minor harmonies as for major harmonies, and thus they had to be regarded as "less consistent." This pronouncement had greater significance than you might think, once you remember that an empirical psychology of music began to develop at around the same time as the discipline of psychology itself, spurred on by Helmholtz's work. The hope was that it would bridge the gap between physical and physiological acoustics on the one hand, and musicology and aesthetics on the other. So when the authority of science suggested that the acoustical inferiority of the minor harmonies did not justify their aesthetic symmetry with the major harmonies, several late-nineteenth century musicologists took the challenge very seriously, and they struggled mightily to come up with "empirical" counterdemonstrations. There are earlier examples of attempts to construct musical theories justifying this duality, but it is only in the wake of Helmholtz’s reluctant challenge that scholars try to identify its basis in Nature. (Helmholtz himself found ample room for "artistic invention," provided one did not insist that the natural function of the ear directly dictated the construction of harmonies.)