I’ve composed upwards of a dozen pieces using microtonal equal temperaments — dividing the octave into 17 steps, or 19, or 20, or 31. What I like about these tunings, beyond the fact that you can modulate freely, is that the set of intervals is small enough that you can wrap your brains around it. In a given equal temperament, an interval of, say, seven small-steps always exactly the same sound, no matter what note you start on.
But there’s a price to be paid for this easy-to-grasp uniformity: Even the best, most desirable intervals don’t sound very good.
The two most important intervals are the perfect 5th and the major 3rd. Perhaps this is not surprising; we all know that a major triad sounds stable. The stability arises because the ratios in this chord are found at the very start of the harmonic series. The 5th has a ratio of 3:2 and the major 3rd a ratio of 5:4. At least, in just intonation that’s the case. But it’s not the case in equal temperament.
Our conventional 12-note equal temperament has a very good perfect 5th. It’s within 2 cents (a cent being 1/100 of a semitone) of the ideal 3:2 ratio. The other intervals in 12ET, however, are poor approximations. Our major 3rd is 14 cents sharp when compared to an ideal 5:4 ratio — a difference that is extremely audible, once you know how to listen for it. This out-of-tuneness gives our scale its characteristic edgy sound.
In case you want to whip out your calculator, a 3:2 ratio is about 701.95 cents. A 5:4 ratio is 386 cents and some change.
Several other equal temperaments have better major 3rds, but you have to go clear out Read the rest of this entry »