Notes repeat themselves, higher or lower, at the interval we now call an octave. Double or halve the speed at which a string vibrates and the sound, in some sense which is as undeniable as it is gratuitous, remains the same. And between notes in simple ratios, most of all the interval we call the fifth, there is a sweetness sweeter and more dizzying than wine.
Between the octave and the fifth, the world almost seems made for us. This appearance is deceiving. The world is not just unfair, but rigged. Chances are you know what it is to pick up part A, and part B, never having doubted they went together, only to find that they don't quite fit. The world is like that. Between the octave and the fifth there is a small but shattering discrepancy we call the Pythagorean comma.
The comma of Pythagoras is as bad as the flaming sword. It means that music, even music, must always be compromised, whether by a diet of a few safe notes, or an intricate microtonal dissection of the octave, or a distortion of the fifth.
This distortion (the Western approach) goes by the name of temperament. Since the Middle Ages the West has known and used several exquisite systems of temperament for particular purposes, but in the last century they gave way to a single system brutal in its simplicity. Equal temperament deals with the Pythagorean comma the way the senators dealt with Romulus, when they caught him in a sudden fog, hacked him to pieces and, walking away with the pieces hidden under their togas, called it apotheosis.
(Are the jitters of the West, its frantic days and restless nights, the symptoms of our addiction to this uneasy music, the Pythagorean comma working its way deeper and deeper under our skins?)
Of all things with value, music is the purest, the most abstract. If even music must compromise, what hope is there for anything else? None at all; but do not take it too hard. Consider poor Pythagoras, twice betrayed, once by music, once by math. Traumatic as Gödel, Turing, Russell, and Tarski were for us, how much worse was it for him, the philosopher who thought number was truth and music was beauty, only to find that numbers could be irrational and music sheltered wolves.
The last century was not, as it boasted, the moment when thought ran up against the limits of certainty and perfectibility. From the very beginning, the whole arc from faith to doubt, from certainty to anxiety, has always been with us in Pythagoras and his comma.
