Temper, Temper

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 to 41 equal steps per octave to get a perfect 5th that’s better than the one we all know and love. 19-tone equal has both major 3rds and perfect 5th that are about 7 cents flat. As a result, its major triad sounds more in tune than the one we’re accustomed to, but it’s a bit dark or sour.

The major 3rd in 31-equal is extremely good (about 1 cent sharp), but its perfect 5th is about 5 cents flat. The major triad in this scale is audibly better than the one in 19-equal, but the 5th is still a bit edgy. And that 41-note scale with the wonderful 5th? Its major 3rd is 6 cents flat. Besides, 41 notes per octave is just too many notes. When you come right down to it, asking the listener to keep track of all those intervals is just being rude.

Having spent a little time playing chords in all of these scales, I’m starting to be able to hear the not-quite-goodness of those “stable” intervals. They’re starting to bug me.

The trouble with just intonation is just the opposite. The intervals sound wonderful, because they’re perfect ratios, but the number of possible notes in the scale is infinite. No matter what scale you choose, if it has enough pitch classes to be interesting, the number of interval classes is going to be much larger than would be the case with an equal temperament containing the same number of pitches per octave. Modulation is still possible in just intonation, but (assuming the total number of pitch classes you use is finite) each key center has its own unique sound, because one or more of the notes in the scale will have unique relations to the tonic — relations not found in scales based on other tonics. If that makes any sense.

I’m starting to wonder how a 19-note scale might be tempered in an unequal way, so as to have mostly better 3rds and 5ths, but with a “wolf 5th” somewhere. I’m not sure how to do the math on that, but it shouldn’t be too difficult.

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