This is surely of little interest to anyone but me and maybe three other tuning freaks in the world, none of whom even knows that my blog exists. I’m just chronicling it. It’s like talking into a Dictaphone, I suppose.
In 31-note equal temperament, the intervals corresponding to our familiar whole-steps are five small-steps wide. Five doesn’t divide neatly by two, so we can’t do symmetrical half-steps. The “half-steps” between E and F and between B and C are three small-steps wide.
If we construct a scale using intervals that are four small-steps wide, it doesn’t sound like anything you’ve ever heard before. This scale (or mode, if you prefer) has good minor 3rds, because minor 3rds are eight small-steps wide. Its next characteristic interval is the semi-diminished 4th, which at 12 small-steps is one small-step narrower than the perfect 4th. For triads, this mode has a first-inversion major triad in which the root is one small-step low, and a second-inversion minor triad in which the fifth is one small-step high. It’s pretty ugly all the way around. But interesting.
If we want the mode to repeat at the octave, we have to knock off a small-step somewhere. If we make the first step up from the tonic three small-steps instead of four, that better-sounding half-step gives the mode a somewhat Phrygian flavor.
We can bring the Phrygian quality a little more into focus by shifting a few of the steps in the mode up by one small-step each. In this new mode, we’ll alternate intervals of three small-steps with with intervals of five. That is, we’ll alternate diatonic half-steps and whole-steps (00-03-08-11-16-19-24-27). The result sounds very like a diminished scale in 12ET — the scale B-C-D-Eb-F-F#-G#-A — except for two things. First, we have to come up with an extra small-step somewhere along the line, so we’ll park it at the top end, making a slightly wide leading tone; and second, we’re going to start with the equivalent of B, making this a Locrian mode. The tonic has no 5th above it.
The two modes I’ve just described are closely related: Five of the eight notes are the same. If you play them for a while, alternating between them, you start to be able to hear that the mode with the slightly higher 3rd, 5th, and 7th steps sounds brighter and more consonant. It sounds, in a word, major. The mode with the lowered steps has a darker, more mysterious flavor.
As a final fillip, while lowering those three steps to make a minorish sound, you can also lower the tonic by a small-step. In 12ET, lowering the tonic causes havoc, but in 31ET the small-steps are close enough together, and the intervals in this scale so unsettling, that lowering the tonic doesn’t seem to put us in a new key. We’re just in a weird version of the original key.
This is some of the outlandish stuff you can get into with 31-note equal temperament.