Last night I started musing about how one might develop a coherent theory of harmony (that is, scales and chords) for a tuning with 31 equal-tempered steps per octave. This tuning, which we can call 31et or 31edo for short, has some very nice properties, but describing those properties in terms of conventional music theory quickly leads into a morass of confusing terminology.
Our terms for intervals, for example (second, third, fourth, and, for that matter, octave), are firmly rooted in the diatonic tradition — the white keys on the piano, in other words. But 31edo provides some scale modes that are not related to the diatonic modes.
If you hang around online, as I do, with people who do microtonal music, you’ll start hearing a lot of very specialized terminology. Things like “porcupine” and “MOS.” Might other theorists have come up with ideas that I could borrow to describe scales and chords in 31edo? This morning I jaunted over to the xenharmonic wiki to find out.
It soon became apparent that microtonal theory is an intellectual playground, or if you prefer a smoking parlor, in which everybody gets to roll their own. And sometimes the tobacco crumbs leak out the sides. Here, lightly edited, is an attempt to describe MOS, which stands for “moment of symmetry”:
“A Moment of Symmetry is a scale that consists of (1) a generator (of any size, for example a 3/2 or a fifth in 12 equal temperament) which is repeatedly superimposed but reduced within the (2) Interval of Equivalence (of any size, for example most commonly an octave), often called a period, (3) where each scale degree or scale unit will be represented by no more than two sizes and two sizes only (Large = L and small = s).”
The first part of that is clear enough. We’re going to choose an interval of some size and stack it on top of itself, periodically folding it back down so that it stays within the octave (or tritave, or whatever). But what exactly does item (3) in the definition tell us? How can a scale degree be represented by a size? How, indeed, can a single scale degree be represented by two sizes, as that sentence seems to say?
I suspect this writer is trying to tell us that we’re going to end up with two sizes of steps in our scale. If we look at a conventional pentatonic scale on a piano keyboard, for instance, we’ll find that some of the intervals in it are major seconds, and some are minor thirds.
Really, we have two problems here. The first is, there are hundreds of people developing their own concepts and terminology, most of which apply only to their own music. The second is, they don’t always explain their concepts very clearly.
Can all of this intellectual ferment be distilled down into anything useful? I have my doubts.