The Theory of 31

The theory of harmony in 12-note equal temperament is pretty interesting, and not terribly complicated. (I’ve written a book about it, in fact — Chords & Harmony is available from Hal Leonard Publishing.) Being able to talk about how chords are built out of intervals is useful, because it lets us describe what we’re hearing.

So as I put together little musical sketches in 31-note equal temperament, I’m starting to think about the theory. You can check out a few of these sketches on my channel at SoundCloud, by the way.

In 31ET, there are four “black keys” between each “whole-step” pair of white keys, and two “black keys” each between E and F and between B and C. Constructing a physical keyboard would be difficult, and if we did it we wouldn’t want all of those keys to be black. We could color them purple, aqua, citrus, and salmon, I suppose. But for theoretical purposes, talking about “G purple” would be silly. We need to find a terminology.

I’m calling the smallest intervals chroma-steps. The distance from C to D or from D to E is five chroma-steps. The distance from E to F or B to C is three chroma-steps.

We might want to call the first note above C “C high.” This would be followed, in order, by “C sharp,” “D flat,” “D low,” and finally D. The note immediately above a white key is “high,” while the key immediately below a white key is “low.” The note two steps above a white key is that key plus “sharp,” while the note two steps below a white key is that key plus “flat.”

We could conveniently abbreviate these: C, C/, C#, Db, D\, D.

One oddity of this system of nomenclature is worth pointing out. C# is below Db — but E# is above Fb. E# is the same note as F\ (F low), while Fb is the same note as E/ (E high). This may seem obtuse, but it preserves the correct enharmonic spellings: If we’re in the key of C#, the major 3rd of a C# major chord will be E#, just as we’d expect. And the 3rd of a Db major triad (one chroma-step higher than C#) will be F natural, again just as we would expect.

I haven’t yet started thinking about the nomenclature for the many varieties of triads and seventh-chords, let alone suspensions and stacked voicings. Maybe the triad C-E\-G is a low major triad. But what about C-E\-G\? That’s even lower.

The names of the intervals are a thorny enough problem. We can preserve the old terminology of modes and say that there is no such thing as a major fourth or minor fourth, only perfect, augmented, and diminished fourths — but we have to add high fourths and low fourths to the list. But that’s potentially confusing. If the interval C-F/ is a high fourth, then so is the interval C\-F. Maybe we should say they’re big and small rather than high and low.

The reason to toy with these ideas is simple: I’m making the sounds already. If I have some terminology, it will be easier to think about what I’m doing. Translating the physical layout of a 12-note MIDI keyboard onto a 31-note scale is challenging enough. If I’m going to play a chord made of stacked intervals 12 chroma-steps wide (octaves, on the MIDI keyboard), it will be useful to say “these are narrow fourths.”

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