Semi-Regular Quarter-Tone Scales

In Manual of Quarter-Tone Harmony, Wyshnegradsky describes a "semi-regular" scale as a scale that divides the octave into equal parts, then divides each of those in the same way. Quarter tones are very practical on an unmodified guitar (using a slide), at least for melodies, and they can be coaxed out of many other instruments too so this may be a bit more friendly than the 30-EDO stuff I've been playing with over the last year. In this post we explore the 12-note semi-regular scales that form nicely symmetrical ways to impose a subset of 24-EDO onto a standard keyboard.

Scala files for all the tunings derived in this post can be found here -- look for the "12_from_24EDO" folder under "scala".

Wyshnegradsky's approach is similar to the one found in Slonimsky and it's also in Hansons' Harmonic Materials of Modern Music -- dividing the available notes into equal parts, then subdividing those again according to a rule. This makes some sense if your interests are primarily atonal; you want a gamut of notes that have symmetry and balance rather than any particular gravitational direction. Something like a Miro mobile: not a solid hierarchy but all the notes suspended in space in perfect equilibrium.

We need some browser-friendly notation for quarter-tones. I'll stick with my usual convention of using the numbers 1-7 for the notes of the major scales and # and b for sharp and flat modifications of them. Since that means we have, say, b3 for the flat third (i.e. minor third) it's not much of a stretch to write d3 for the half-flat third (halfway between 2 and b3). The half-sharp symbol doesn't have an easy-to-type equivalent but it's closest to a lowercase "t" so I'll use that; t3 is halfway between 3 and 4. One way to write out the notes in 24-EDO using this notation is:

1   t1   b2   d2   2   t2   b3   d3   3   d4   4   t4   b5   d5   5   t5   b6   d6   6   t6   b7   d7   7   t7

We start by noticing that the factors of 24 are 2, 3, 4, 6, 8 and 12. However, if we're to make a 12-note scale we can't use the division into 8 parts because 8 is not a factor of 12, and we must have the same number of notes in each part. Furthermore, 12 is not useful since that's just going to give us the chromatic scale.

So here is what we have:

  • 2 equal parts containing 6 notes each (2 x 6 = 12)
  • 3 equal parts containing 4 notes each (3 x 4 = 12)
  • 4 equal parts containing 3 notes each (4 x 3 = 12)
  • 6 equal parts containing 2 notes each (6 x 2 = 12)

The first of these offers 80 possibilities; this is pleasing but too many to be immediately helpful, so we focus on the others. It might be worth coming back to these with finer analytical tools at some point, but as you'll see there's plenty more to play with.

Dividing 24-EDO into three equal parts yields segments that start on (say) C, E and G#, the notes of an augmented triad (the additional quarter-tones don't change that). Each part contains eight notes, but the first is fixed so we have seven notes from which to choose the remaining 3 notes. On paper there are 35 ways to do this but in practice modality trims the number down to 10:

['C', 'Ct', 'C#', 'Dd', 'E', 'Et', 'F', 'Ft', 'G#', 'Ad', 'A', 'At']
['C', 'Ct', 'C#', 'D', 'E', 'Et', 'F', 'F#', 'G#', 'Ad', 'A', 'A#']
['C', 'Ct', 'C#', 'Dt', 'E', 'Et', 'F', 'Gd', 'G#', 'Ad', 'A', 'Bd']
['C', 'Ct', 'C#', 'D#', 'E', 'Et', 'F', 'G', 'G#', 'Ad', 'A', 'B']
['C', 'Ct', 'Dd', 'D', 'E', 'Et', 'Ft', 'F#', 'G#', 'Ad', 'At', 'A#']
['C', 'Ct', 'Dd', 'Dt', 'E', 'Et', 'Ft', 'Gd', 'G#', 'Ad', 'At', 'Bd']
['C', 'Ct', 'Dd', 'D#', 'E', 'Et', 'Ft', 'G', 'G#', 'Ad', 'At', 'B']
['C', 'Ct', 'D', 'Dt', 'E', 'Et', 'F#', 'Gd', 'G#', 'Ad', 'A#', 'Bd']
['C', 'Ct', 'D', 'D#', 'E', 'Et', 'F#', 'G', 'G#', 'Ad', 'A#', 'B']
['C', 'C#', 'D', 'D#', 'E', 'F', 'F#', 'G', 'G#', 'A', 'A#', 'B']

The last of these is just the chromatic scale, so that's not of much interest. The others might be, though; all of them contain a mixture of notes that are in 12EDO and others that aren't, which is where the juice is in these tunings.

You might think that dividing into 4 parts and choosing 3 from each is the same, or at least offers the same number of possibilities, but that's not true; there are far fewer ways to do it. The division into four parts of course follows the notes of a diminished seventh chord, and we only get to choose two notes in between those. There are just four ways to do it, one of which is again the chromatic scale:

1   :   ['C', 'Ct', 'C#', 'D#', 'Ed', 'E', 'F#', 'Gd', 'G', 'A', 'At', 'A#']
2   :   ['C', 'Ct', 'Dd', 'D#', 'Ed', 'Et', 'F#', 'Gd', 'Gt', 'A', 'At', 'Bd']
3   :   ['C', 'Ct', 'D', 'D#', 'Ed', 'F', 'F#', 'Gd', 'G#', 'A', 'At', 'B']
4   :   ['C', 'C#', 'D', 'D#', 'E', 'F', 'F#', 'G', 'G#', 'A', 'A#', 'B']

Again, on the face of it there's no reason to reject any of the others; all might be interesting.

The last option involves dividing 24-EDO using the Whole Tone scale and filling in the gaps. There are only two ways to do this: one creates the chromatic scale as usual; the other is a sort of hyper-augmented scale that might be well worth investigating:

1   :   ['C', 'Ct', 'D', 'Dt', 'E', 'Et', 'F#', 'Gd', 'G#', 'Ad', 'A#', 'Bd']

Many of these tunings have a foot in conventional 20th century harmony, especially the Scriabin school, with those whole-tone and octatonic scales dominating; this gives us a point of refence, at least. And what sounds they are! Queasy, spacetime-continuum-bending intervals and modulations that seem to pass through other dimensions; weird, alien, outer-space stuff that in the 1920s would have seemed like the background music to an H G Wells novel. Or -- better -- an H G Wells movie, a medium for the future rather than the past:

This is about all I got out of my reading of Wyshnegradsky's book. It's short and quite concerned with quarter-tone extensions of tonal harmony, which is only tangentially interesting to me. But it's also just a bit thin because (like the posts I do here) it's theory-before-practice: it's an attempt to lay out some things that might turn out to be interesting. But quarter-tone music never took off as a musical practice, so those things didn't get worked out and expanded. Maybe in the future that will happen; you have people like Adam Neely and Joseph Collier starting to popularize microtones in tonal contexts now so who knows?

My own sense of alternative tunings is that they're like parallel universes you can step into, like a dream or an acid trip: you put your hands down in familiar places but what they touch is transformed and liable to behave unexpectedly. Once I've found some interesting tunings I'd rather not build a great edifice of theory on top of them; I like the mystery.

If you like this approach too, I can highly recommend listening to the pioneers (and perhaps only practitioners) of quarter-tone music as a way to get your ears tuned up to what these tunings can do. I might put together some listening suggestions in a separate post.