Equal Temperament
The centuries-old struggle to play in tune, by Jan Swafford:
There have been some 150 tuning systems put forth over the centuries, none of them pure. There is no perfection, only varying tastes in corruption. If you want your fifths nicely in tune, the thirds can’t be; if you want pure thirds, you have to put up with impure fifths. And no scale on a keyboard, not even good old C major, can be perfectly in tune.
Why the mess? As the article points out, if you tune up mathematically perfect fifths, the result is not what is expected — things quickly sound painfully out of tune. So, the art of tuning is approximating a mathematical relationship in the most pleasing way possible. The modern solution? Equal temperament:
Here the poison is distributed equally through the system: The distance between each interval is mathematically the same, so each interval is equally in, and slightly out of, tune. Nothing is perfect; nothing is terrible.
So, here’s the part I didn’t quite understand. I thought we couldn’t tune things using mathematically identical intervals, right? I mean, I just reiterated the article’s example about perfect fifths — if you tune them up through a few octaves, the resulting notes sound really flat!
BUT, the key (har) here is this: — every note shares a little of the off-key-ness because every note is the same distance apart from its immediate neighbor. You aren’t trying to make sure every eighth key on the keyboard is exactly a mathematical fifth, you’re trying to make sure that every key on the keyboard has “exactly the same frequency ratio as its neighbor”. NOW I get it!
Equal temperament is why you can transpose things on the piano, and they don’t sound terrible in that new key. It also allowed the development of jazz.