The definition of music as “sound” wrongly suggests that music, like all natural phenomena, adheres to the laws of nature. In this case, the laws would be the acoustical patterns of sound such as the (harmonic) relationships in the structure of the dominant tones, which determine the timbre. This is an idea that has preoccupied primarily the mathematically oriented music scientists, from Pythagoras to Hermann von Helmholtz.
The first, and oldest, of these scientists, Pythagoras, observed, for example, that “beautiful” consonant intervals consist of simple frequency relationships (such as 2:3 or 3:4). Several centuries later, Galileo Galilei wrote that complex frequency relationships only “tormented” the eardrum.
But, for all their wisdom, Pythagoras, Galilei, and like-minded thinkers got it wrong. In music, the “beautiful,” so-called “whole-number” frequency relationships rarely occur—in fact, only when a composer dictates them. The composer often even has to have special instruments built to achieve them, as American composer Harry Partch did in the twentieth century.
Contemporary pianos are tuned in such a way that the sounds produced only approximate all those beautiful “natural” relationships. The tones of the instrument do not have simple whole number ratios, as in 2:3 or 3:4. Instead, they are tuned so that every octave is divided into twelve equal parts (a compromise to facilitate changes of key). The tones exist, therefore, not as whole number ratios of each other, but as multiples of 12√2 (1:1.05946).
According to Galilei, each and every one of these frequency relationships are “a torment” to the ear. But modern listeners experience them very differently. They don’t particularly care how an instrument is tuned, otherwise many a concertgoer would walk out of a piano recital because the piano sounded out of tune. It seems that our ears adapt quickly to “dissonant” frequencies. One might even conclude that whether a piano is “in tune” or “out of tune” is entirely irrelevant to our appreciation of music.
[fragment from Honing, 2011.]