It’s over a year since I wrote the first part of this. I was trying to show how there was something very strange going on in the way we’re generally told that music works; something that when I found about it became a powerful metaphor for hidden complexity, fakery and the acceptability of approximations.
Where had we got to? That using various perfect mathematical ratios to divide a vibrating string we could build up a series of musical notes. Playing some of these together produces pleasing harmonies, because of the way the waveforms behave together.
Pythagoras got onto this pretty quickly. You can read a thorough account of his theory here. Cutting to the chase, he described a bunch of different notes using nice whole number ratios. We can order these notes into something we’d call a scale. C, D, E, F, G, A, B, C – that sort of thing.
The great thing about an ordering like this is that you can keep on building… Scales can start on different notes, you can introduce half-way steps between the notes to help make it all work out. C# becomes essential to help make the key of A major work properly, and so on.
I used to think it was amazing how this maths all worked out: how there could be 12 perfectly spaced semitones on a piano keyboard that could make up any scale or key you wanted.
And then I realised that it didn’t work out. Almost exactly, but not quite. And the fact that the maths so nearly works out perfectly is, for me, the most fascinating part of all this. It means that music theory can get away with a bit of cheating and approximation.
How so? Well, take the Pythagorean scale at face value. Each time you get to the fifth note in a scale (so that would be G in the scale beginning with C) you have a “perfect fifth” with a harmonic relationship 3:2 times the frequency of the C you started with.
Take the fifth (G), and build another perfect fifth on that (D). Go from D to A, and so on… eventually, after working through all the 12 semitones in an octave (see what I meant in the previous post about “oct-” being a bit of a red herring?) you get back to the C you started with. It all neatly works.
But it doesn’t. And it can’t. Here’s why.
Go up 7 Cs (each an octave higher than its predecessor). You end up with a pretty high note of course, but because you’ve doubled the frequency each time it’s fairly obvious you are 2^7 times higher in frequency than your starting point. That’s 128 times the frequency.
Now, do it the other way – going from perfect fifth to perfect fifth. You have to go through a few more cycles because you’re not jumping as much as an octave each time. In fact, you need to go through 12 cycles, which sort of makes sense as you’ve covered all the 12 semitones in turn as starting points. The eventual frequency you reach is (3/2)^12. Got a calculator? Well, it’s awfully close to 128. In my humble view, weirdly, bizarrely close. It comes out at about 129.746. That’s only 1.36% out from the C you would have arrived at from just doubling each time.
You’ve gone up 7 octaves and you’re only just over 1% out from where those neat harmonic intervals would have taken you? Doesn’t that strike you as rather interesting?
This tiny difference is called the Pythagorean Comma. It has to exist because the Fundamental Theorem of Arithmetic says you can only make up a number by multiplying one set of smaller prime numbers together. So you couldn’t get to 128 by multiplying anything other than 2 by itself 7 times… So those neat Pythagorean ratios were always going to run into trouble at some point.
So what? The big ‘what’ is that because the Comma (= ‘gap’) is so small, it only takes a bit of tweaking to the Pythagorean ratios to get 12 intervals that DO fit exactly into a repeating pattern. You just have to make some of the notes very slightly lower, and some slightly higher. But eerily, the adjustments are so small that one hardly ever notices.
That’s how a piano could be built. A modern (post c17th or so) piano usually has these adjustments (called temperament) made to its tuning so that each interval is exactly the same. The same goes for most instruments where the notes are ‘fixed’ in its construction (like a guitar, with fixed frets, but unlike a violin where the player has discretion about where precisely the strings are stopped with the fingers).
Because of this mathematical curiosity – that tiny Pythagorean Comma – a totally flexible muscial system can be built. So we can have jazz, infinitely complex harmony, tunes that modulate smoothly from key to key, and far greater flexibility than if we were stuck with Pythagoras’ precise spaces between the notes.
The trade-off is that most things we hear as harmonies in music actually aren’t. Very, very slightly, the whole structure of music involves ‘cheating’ to make it work.
And I think that’s a fantastic story-behind-a-story. Tiny compromises, and acceptance of approximation, produce a far more beautiful (if slightly ‘wrong’) end product than could rigid adherence to the rules.
Filed under: Uncategorized, Approximation, complexity, harmony, Musical theory, pythagorean comma