30-EDO Ideas

Dividing the octave into 30 notes is an interesting proposition in part because 30 has the factors 2, 3 and 5, which means it contains a few smaller EDOs. In particular, it has three copies of 10-EDO that sit inside it like the three augmented triads (3-EDO) in standard tuning, and I've been working a lot with 10-EDO lately. But 30 notes is a lot to deal with, and far too many for me to play on a single keyboard. So in this post I muse about some ways to split 30-EDO up into more manageable parcels.

These multiple-of-5 tunings tend not to please the folks looking for "perfect" fourths, fifths, thirds or the like. I'm not saying you can't play something like tonal harmony with them, but as far as I know people don't because that would be a really unnatural choice. I'm not interested in that though; I'm looking for far-out sounds and these tunings certainly provide those.

Naming the Notes

Each "step" in the scale from one note to the next is 40 cents, so five steps make a standard tone. The notes C, D, E, F#, G# and A# from standard tuning are all there; this is one of the Whole Tone scales. The other six notes are not there, though; this is not surprising, since 30 is divisible by 6 but not by 12.

So it's natural for someone raised with 12-EDO to identify the 5-step "whole tone" as the basic interval of this tuning, and to call the corresponding notes by their familiar names: C (0c), D (200c), E (400c), F# (600c), G# (800c) and A# (1000c). However, the presence of sharp or flat signs complicates things. I'd prefer to label these notes more neutrally. Let's instead use letters from the end of the alphabet: U (0c), V (200c), W (400c), X (600c), Y (800c) and Z (1000c).

The in-between notes could then be named as follows. +80c is "sharp" and -80c is "flat". Note that this means that, say, X# (680c) is not the same as Yb (720c). We can then indicate smaller divergences as "raised" (+40c) and "lowered" (-40c), using the half-sharp and half-flat symbols from quarter tone music: X𝄲 is 640c (+40c, "X raised") and X𝄳 is 560c (-40c, "X lowered").

Well, I'm not in love with these names. They'll prove somewhat useful in the last section of this post but the letter names in particular seem a bit silly. I can't imagine them catching on.

As Two Copies of 15-EDO

30-EDO can be thought of as two copies of 15-EDO offset by 80c; the offset could be managed in a similar way or using a footswitch. You can't, of course, map a complete copy of 15-EDO to an octave of a 12-EDO keyboard but you can if you drop three notes; there are probably a lot of ways to do this (the calculation is less straightforward than first appears -- ¹⁵C₃ = 455 but there are many effectively equivalent cases that need to be factored out and as usual transpositional symmetry makes that non-trivial).

Another, I think more viable approach arises from noting that 15-EDO is three copies of 5-EDO. This suggests taking any two of the three copies of 5-EDO and mapping them the same way as we do for 10-EDO (one copy on the black keys, the other on the white keys with E=F and B=C). There are three ways to do this. The most obvious way would be with three copies of a softsynth that you can switch between. Again, with three keyboard no switching is necessary.

Let's call these 5p, 5q and 5r. We build the 10-note tunings 10pq, 10pr and 10qr. Here are their cent values, with duplicated pitches at 5 (F) and 11 (B) to map them onto a 12-note keyboard:

These are not really three different sounds. 10qr can be thought of as (a) 10pq shifted up by 80c or (b) 10pr shifted down by 80c, so they contain exactly the same intervals. But any pair of them covers all of 30-EDO, which could work well across a pair of keyboards which will then match on the white notes but differ on the black notes or vice versa. Then if Keyboard 1 is 10pr and Keyboard 2 is 10pq, playing the black keys on both gives 10qr. After some experimentation this seems to be quite practical and these 10-note scales are actually very flavourful.

To climb up from here to 30-EDO would require four keyboard controllers; more than I have available and more than I think is practical for most of us (although pipe organs run to 4 manuals pretty often). One could try to build an alternative MIDI keyboard such as Hanson or Porcupine that can cover 15-EDO in one controller; this can be done by rearranging the keys of a standard keyboard and then mapping them to the right scale notes: https://www.youtube.com/watch?v=MxUhkcMixm4. But this involves some financial cost and some potentially hazardous DIY. I might give it a go in the summer. Maybe there's even a case for a double manual Porcupine keyboard that's specifically designed to handle 30-EDO (I've had some success with 24-EDO done this way, using two stacked-up keyboards tuned 50c apart, and I've seen others do it too).

As Three Copies of 10-EDO

Similarly, with three keyboards talking to three synths tuned in 10-EDO and shifted 40c apart, you have full access to 30-EDO. The 10-EDO mapping I use is similar to those given above:

This is already quite familiar to me but that could be a disadvantage and three keyboards is a lot to navigate so I might not go down this route at first.

As Five Copies of 6-EDO

30-EDO can also be decomposed into five copies of 6-EDO, the ordinary Whole Tone scale we find in 12-EDO. You can play any two of these on the 12-note keyboard, of course. Let's devise the following shorthand: 6-EDO (C Whole Tone in 12 EDO), 6-EDO+40, 6-EDO+80, 6-EDO+120 and 6-EDO+160. If we label them p, q, r, s and t we get the 12-note tunings 12pq, 12pr, 12ps and 12pt. They're pretty easy to understand:

• 12pq consists of U, V, W, X, Y, Z and their raised counterparts.
• 12pr consists of U, V, W, X, Y, Z and their sharp counterparts.
• 12ps consists of U, V, W, X, Y, Z and their flat counterparts.
• 12pt consists of U, V, W, X, Y, Z and their lowered counterparts.

remembering that U, V, W, X, Y, Z are just the familiar C, D, E, F#, G#, A# and the operations of raising, sharping, flatting and lowering are as described above.

Any pair of these can be mapped to a standard keyboard; the most natural way would be to assign them so that the standard Whole Tone scale fingerings are preserved. Two keyboards with the same mapping suitably offset (e.g. 12pr and 12pr+40) can cover 24 of the 30 notes in 30-EDO. A third keyboard, limited to a single whole-tone scale, would fill in the rest.

That should give me about a year's worth of stuff to tinker with. I still have a 10-EDO-centric album in the works, but progress has been very slow due to work commitments. But I might also have another Minimum Labyrinth collaboration coming up in the spring that needs some music, and some of these tunings could well feature in that.