Some Chords from 10 EDO

10 EDO divides the octave into ten equal parts instead of the usual twelve. We may as well number these notes 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This is a pretty tuning that doesn't take too much getting used to. Let's create some harmony out of it.


On a 12-EDO keyboard I'm currently playing 0-1-2-3-4 on C-C#-D-C#-E and then 5-6-7-8-9 on F#-G-G#-A-#A#. This leaves F and B unused. I'm trying just setting these to the halfway point between the two adjacent notes. These make for pretty embellishments on many of the other harmonies, although I won't discuss them here. Here's the Scala file:


I do most of this kind of work in Pianoteq but I find acoustic piano sounds are rarely suited to tunings like this. Try electric piano or plucked string sounds, which tend (I think) to have fewer high harmonics. Generally you want to pick your sounds carefully when working with far-out tunings as the harmonic structure of the timbre itself can make or break the harmony you get out of the tuning.

On another note, the following analysis really suffers from two problems. The first is that all of our language for talking about musical structures comes from 12-EDO and its historical relatives; I speak in terms of "semitones", "major thirds" and so on but these terms don't mean much here. Inventing new terms seems counterproductive. I'm not sure tuning theory offers any solution to this; if it does I'd be pleased to learn about it.

The other is that playing tunings that are a long way off 12-EDO, and especially those that don't have 12 notes, on a standard keyboard is pretty miserable. I'm considering solving this by buying a different controller but I consider that a bit of a nuclear option -- I'm not a gear-head and anyway if I want to communicate these ideas I don't want to be tied to a proprietary piece of kit. Again, I don't have an answer for this right now.

Whole-Tone Chords

The obvious thing to do with any n-EDO where n is even is to choose every alternate note; in 12-EDO this yields an interlocked pair of whole-tone scales. In 10-EDO we also get whole-tone scales, but the whole tones are fatter and the scales are pentatonic: 0-2-4-6-8 and 1-3-5-7-9.

This scale contains only two three-note chords: 0-2-4, the cluster of adjacent notes, and 0-2-6, the "whole tone triad". These latter have a sound halfway between a major triad and a stack of fourths; overall this three-note harmony is a simplistic, bell-like, even folkish language. But really these two possibilities sound very much, to my ears, like inversions of each other; they're not, but their sounds are so similar they could be. And of course there really is only one four-note chord and one five-note chord.

Overall, then, whole-tone harmony is 10EDO seems rather limited to me. It has a very distinctive sound but really seems to contain only one sound; this can be an ingredient in our harmonic language but it can't really do that on its own unless you're looking for a very floaty, suspended quality.

Half-Whole Chords

Now, how about a "half-whole" language? Our first try seems to be a failure: 0-1-3-4-6-7-9 won't repeat at the octave (there is a semitone before 0 and after it) because 10 is not divisible by 3. But it's not a failure, because it sounds really interesting for reasons I can't put my finger on. It contains four notes from each of the two whole-tone scales, making it an exact halfway point between them.

We can harmonize this in "thirds", i.e. in intervals of three semitones: 0-3-6-9. Continuing the pattern takes us outside this half-whole scale to cover the whole of 10-EDO, producing a sort of "mother chord"; the next four notes are 2-5-8-1. In terms of the keyboard mapping I'm using this is a Cm7 chord fingering (the lower four notes) and a DΔ fingering (the upper four notes). In fact the fingerings for the four possible transpositions of this chord correspond to the fingerings for Cm7, C#m7, DΔ and EbΔ in 12-TET.

Following the same logic, a harmonized version of this scale produces these fingerings (other choices are possible):

  • Cm7 fingering: 0-3-6-9 (Intervals: mT, mT, mT, s) -- TYPE I
  • C#° fingering: 1-4-6-9 (Intervals: mT, t, mT, t) -- TYPE IIa
  • D#7 fingering: 3-6-9-1 (Intervals: mT, mT, mT, s) -- TYPE I
  • E° fingering: 4-6-9-1 (Intervals: t, mT, t, mT) -- TYPE IIa
  • Am7b5 fingering: 8-0-3-6 (Intervals: t, mT, mT, t) -- TYPE IIb
  • Bb° fingering: 9-1-4-6 (Intervals: t, mT, t, mT) -- TYPE IIa

I've categorized them by interval mixture. All contain "minor thirds" (three-step intervals). The TYPE I chord contains three of these, plus of necessity a single semitone (since 3x3 = 9 and 9+1=10). The TYPE II chord contains two minor thirds and two tones, but these can be arranged either so that they alternate (TYPE IIa) or so that like intervals are consecutive (TYPE IIb).

These chords seem to me to be interesting, quite distinctive and quite different from the pentatonic "whole-tone" harmony. However, getting them in your ears is not so easy and the standard keyboard layout doesn't help much.

Special mention should go to the three-note chord 0-4-7. This structure can be obtained as a TYPE IIb with the two tones merged into a "major third" or a TYPE I with the semitone merged into one of its "minor third" neighbours. In my keyboard mapping 0-4-7 is fingered like a 12-EDO augmented triad; in fact in my mapping all augmented triad fingerings produce transpositions of this chord. I'm inclined to call this 10-EDO's "major chord".

I'm planning to use 10-EDO on my next album so I'll be spending more time with it. Getting to the end of this post, though, I feel as if the two issues I mentioned above are really starting to bite: my language for talking about it needs to change, as does my physical relationship with it via the traditional keyboard. We'll have to see how that goes.

But the next thing I want to look at is tunings that blend notes from 12-EDO and 10-EDO. There are a lot of possibilities...