The self-described "grumpy old coot" who writes the blog Right Wing Nation has recently put up a generally admirable post ("On An Entirely Different Note", 2/19/2009) on the relations among music, mathematics, physics. Unfortunately, his account of the Pythagorean comma is psychoacoustically, historically and mathematically wrong.
He gets off to a good start by introducing the Circle of Fifths, showing that 12 fifths bring you back to the name nominal pitch class:
Tune a starting note, say middle C (or if you’re tuning a piano, A-440). The human ear can hear, and tune, a perfect fifth, so we would next tune G, because G is a fifth above C. We would then tune a fifth up from G (D), then a fifth above D (A), and so forth, until it led us finally back to our starting note, C (and that brings us back to do!) It looks like this (the acoustic space between each adjacent pair of tones is a fifth):
But then he goes off the rails:
Acoustically, what the human ear hears as a perfect fifth is just slightly more than a fifth (about 2 cents, and there are 100 cents in a semi-tone). But as we tune from fifth to fifth, that tiny difference adds up, one after another, so when we get back to C, it is WAY out of tune with our starting C.
No, no, no. The problem is that a perfect fifth is a ratio of 3/2, while an octave is a ratio of 2/1; and (3/2)12 — the result of 12 successive intervals of a fifth– is not the same as 27 — the result of 7 successive intervals of an octave.
The results are close — 27 is 128, while (3/2)12 is around 129.7463. Another way to see the relationship is to note that
(3/2)12/27 = 312/219 = 531441/524288
But they're not the same, and the difference is what is known as the "Pythagorean Comma". This is just mathematics, and has nothing to do with the properties of the human auditory system.
As discussed here a couple of years ago ("Pavarotti and the crack to chaos", 9/9/2007), the Pythagoreans saw this difference as one of three crucial flaws in the fundamental mathematical fabric of reality that challenged their belief in its coherence. So when the G.O.C. writes that "the ear is imperfect, so we have the Pythagorean Comma, a serious problem identified by Pythagoras that was not solved until the 18th Century", he's wrong. And it's historically as well as physically and mathematically important that this issue is not merely a matter of some imperfection in the human sensory apparatus.
I shouldn't be writing this, I should be working on the announcements for the new journal that I'm helping to start, or one of the papers that I owe, or the problem set that I need to create. But as Randall Munroe (or his cartoon character, anyhow) put it, "What do you want me to do? LEAVE? Then they'll keep being wrong!"
And I know that the G.O.C. appreciates mathematical rigor and historical accuracy.