9-EDO: Three Augmented Triads in Perfect Symmetry

I've recently acquired a basic synth setup with a view to exploring some non-standard tunings. This is something I've messed with in the past and used for "colour" but never really got deeply into, but that's about to change.

Recently I've been working with 9-EDO. "EDO" stands for "Equal Divisions of the Octave", so this tuning's "chromatic scale" has nine evenly-spaced notes instead of the usual twelve. The semitones are a third larger than those in our standard 12-EDO.

Here's me noodling a bit in 9-EDO; sorry for the sloppy keyboard playing and also for the Flash plugin. I can't promise better piano chops in the future but I'll certainly try to find a better audio hosting solution soon. Probably I should just stick these things on YouTube.

The notes 0, 3 and 6 together form an augmented triad. It's not surprising that these three notes should be shared with 12-EDO: after all, 9 and 12 share a common factor of 3. These are the three notes you get from that common factor.

We can think of 9-EDO as being constructed by interlacing three augmented triads that are spaced equally: 0-3-6, 1-4-7 and 2-5-8. These are the basic structure of the tuning as I see it. (Of course, 12-EDO has four such augmented triads instead of three; we don't usually think of them as being fundamental to it but then 12 has more factors than 9, giving us more ways to slice it.)

Here's the picture I've been using to visualize it; each coloured triangle is of course an augmented triad:

To play it on the guitar's high E string, play the augmented triad E-G#-C. Then divide the region between E and G# into three equal parts by playing the F a third sharp and the G a third flat. The same pattern -- "one third sharp, one third flat" -- repeats for the notes immediately above G# and then C. But I'm interested in chords so I'm using the keyboard for now.

Of course, you can play these three triads as triads and that's fine, but it gets boring very quickly. You'll soon find you want to mix up the notes. Let's see which different three- and four-note chords we can make in that way -- I'll count two chords as "different" if they're inversions of each other but not if they're transpositions of each other. We will of course find 9 copies of each of these chords in the scale (by design, none of them is symmetrical).

There are three distinct ways to combine one note from each augmented chord into a new triad -- here they are in descending order of dissonance as I perceive it (all of them are pretty dissonant, though):


and six ways to combine one note from one augmented chord with two from another:


Taken together this means we have nine "basic triad types" to learn, in terms of sounds at least as importantly as in terms of practical playing.

The thirteen four-note chords are easy to list. One augmented triad plus a note from one of the others yields two possibilities:


Taking two notes from one augmented triad and two from another gives another three:


Taking one note from each augmented triad then adding another one yields a more surprising eight possibilities:


This certainly suggests a very rich harmonic palette; Just imagine if this tuning was all you'd ever known, and these twenty three chords (22 listed here, plus the augmented triad) were the bread and butter of your music...

Aside from these chords we obviously have pentatonics (the complements of the 4-note chords) and some hexatonic scales, both of which are worth exploring. Of the hexatonics the most obvious are the two you get by combining two of the augmented triads; I've been getting a lot of value out of these.


There are lots more.