Third-Tone Tunings

After a bit of exploration into quarter-tones I thought I'd add to my collection of EDO-based 12-note tunings by exploring third-tones, which split the gap between (say) C and D into three equal parts rather than the usual two. The EDO made entirely of this interval is 18-EDO, which we can think of as three equally-spaced whole-tone scales.

Ferrucio Busoni experimented with these sounds but as far as I know never composed for them; in his time it would have been prohibitively hard to get satisfactory acoustic instruments capable of producing them reliably. But note that third-tones are, at least in theory, less difficult to hear than their smaller relatives the quarter-tones, and with modern electronic instruments we can produce them with ease. Well, with two caveats: the instrument must support re-tuning (which most worth their salt do these days) and the notes must be playable using some practical interface. The latter is the main problem, since most of us don't have fancy controllers or don't want to invest the time in developing a skill on a proprietary instrument that will probably go out of production before long, hence the attraction of 12-note subsets of the tuning.

I had a hard time finding any music in third-tones, partly because internet search results are full of very interesting results about Chinese languages. But apparently Xenakis called for them from the clarinets in "Antikhthon", which is a good enough excuse to throw in a video link so you've got something heavy to play in the background:

As an aside: I don't really guitar these days but I know lots of readers of this blog do. On guitar 18EDO isn't hard to pull off, at least for melodic purposes. Using a slide or similar, you just have to stop seeing every alternate fret and fill in the gap with two notes instead of one. Low down on the fretboard this is pretty easy. It helps if you tune the guitar so that all the strings are separated by some number of whole tones -- for example, the tuning E-Ab-C-G#-C-E will do it. And of course visualization is easier if you pull out half the frets, but that's a bit more radical. Technically both notes should be a tiny bit sharper than they would be if you divided the distance exactly equally; practicing with guide notes played by a computer helps nail the intonation, but wobbly tuning that sounds good is better than perfect tuning that doesn't.

(Thinking about this kind of thing makes me really want a lap steel, which I could play despite my busted elbow, but I've yet to get round to it; maybe something for a future project.)

If, on the other hand, we want to play third-tones on an ordinary black-and-white keyboard then we'll need to choose 12 notes from the available 18. There are 476 ways to do this, which isn't very helpful, so instead we'll take a hint from Wyshnegradsky and look at just the semi-regular options.

These start by dividing the 18 notes into a number of equal parts, then subdividing those in the same way for each part. The factors of 18 are 2, 3, 6 and 9. Of these we have to reject 9 because it isn't a factor of 12, but the others are OK. There are 10 such tunings based on the tritone, 3 based on the augmented triad and -- as usual -- just one based on the whole-tone scale.

Of the augmented ones, two contain a pair of 12-EDO augmented triads and another one offset by a third-tone; the other is the opposite, containing one 12-EDO augmented triad and two offset by a third-tone. The tritone-based ones are more of a mixed bag.

So there we go: another 14 interesting tunings to play around with. You can see them individually here or download the whole project and get my whole collection of Scala files in one batch -- look for the "12_from_18EDO" folder under "scala".