“…let’s construct a virtual keyboard. The first step is to gather all the parts: wood for the keys and frame; quills to pluck the strings; and, of course, the strings themselves – in many different lengths. Each length will vibrate at its own particular frequency; the longer the string, the lower its tone.
We’ll begin by selecting a string of medium length and installing it for the key we call do. (Today, there is a standard length for do; but we are constructing our instrument during the Middle Ages).
To find the correct string lengths for the rest of the keyboard’s notes we’ll apply the proportions we learned from Pythagoras. For sol, a perfect fifth above, we will select a shorter string to form the ratio 3:2 with do. Our sol string (assuming, again, that tension and materials are consistent) will thus be two-thirds the length of the string for do.
We can select the string for fa by using the formula for a perfect fourth above do. The proportion between the string length for do and the one for fa will be 4:3.
Next, we’ll find the proper string for re calculating a perfect fourth below sol. Once again, the proportion between those two strings will be 4:3, with the lower tone getting the longer string.
So far, all has gone smoothly. As we continue, searching next for the correct string length of la, however, difficulties abound. There is more than one option. For example, la lies a perfect fifth above re, and these two string lengths should form the proportion 3:2. However, la could also be a major third above fa, so we might form this tone by putting those strings in the proportion 5:4. The problem is, the two solutions to finding la actually yield different tones. Which one is the real la?
The question is crucial. The interval between fa and la created by means of Pythagorean tuning – that is, through a series of perfect fifths (3:2) – will be slightly wider than the “pure” third (5:4) the human ear finds so rich and rewarding; as a result, its sound is far less pleasant – edgy, even acidic. But the la that is a pure, lovely major third above fa is unacceptably ugly when serving in harmony as the fifth of re.
The difference between these two versions of la is called a comma (in this case, it is known as the comma of Didymus, named after the ancient Greek who first described it, or the synotic comma). In the grammar of languages, a comma provides a pause – the space of a hair’s breadth within a sentence. In music, the term also describes a tiny gap, the difference between tones that are so close they share the same name but have been calculated through different proportions. Just as a misplaced comma in the written word has the power to stop a sentence dead in its tracks, and errant musical comma can bring a lyrical cadence to a screeching halt. Indeed, in music, even a numerically infintesimal gap can seem the sonic equivalent of a harrowing abyss.
Thirds are not the only ‘problem’ interval for keyboards. As we have seen, Pythagoras at his monochord had realized millenia ago the imperfection in his chain of perfect fifths – that in order for the twelve pitches generated through the proportion 3:2 to complete a path from do to do, the circle has somehow to be adjusted, ’rounded off.’ That is, one ‘link’ in this chain must be made shorter (by a gap known as the Pythagorean comma) so that the top note of the series can form a perfect octave with its lower namesake. But the act of artificially forcing the fifths into a seamless circle dooms that shortened link to an unsavory role.”
– Stuart Isacoff’s Temperament