I’ve written a little keyboard work (for a retuned electronic keyboard, playable by human hands) that I’m proud of for reasons with which the reader has no reason to sympathize. One is that I’ve finally, after years of trying, broken past the barrier of the 11th harmonic to base a piece on the 13th harmonic and its resultant intervals. This will seem a small achievement to some microtonalists, many of whom run wild with 43rd and 79th harmonics and 53- and 72-tone scales, but I have always found myself unable to compose merely theoretically, without internalizing and being able to hear, almost more in my heart than in my head, the materials I’m using. Thus my approach to microtonality has always been slow and gradual, and I’ve had a devil of a time getting the 13th harmonic into my system. The other reason is that the scale is the simplest I’ve ever come up with (simplicity being an artistic virtue, if not inherently the best or most necessary virtue, and having been considered one for many hundreds of years, no matter how fervently the complexity mavens try to rationalize it out of existence). The scale, defined as ratios to a fundamental (this way of discussing pitch is explained at my just intonation page if you’re interested), comprises nothing more than all possible ratios among whole numbers 1 through 13:
13/12, 13/11, 13/10, 13/9, 13/8, 13/7 (13/6, 13/5, and so on, are merely octaves of those already mentioned)
12/11, 12/7 (12/10 is the same as 6/5, 12/9 = 4/3, and so on)
11/10, 11/9, 11/8, 11/7, 11/6
10/9, 10/7 (10/8 = 5/4, 10/6 = 5/3)
9/8, 9/7, 9/5
8/7, 8/5
7/6, 7/5, 7/4
6/5
5/4, 5/3
4/3
3/2
1/1
It’s 29 pitches in all, all with fairly simple relationships to the tonic, because of which the whole piece takes place over a rhythmicized tonic drone. I figured out that I could make different scales within this network by taking all notes expressible by the form 13/X, or 11/X, or X/7, and the scales with the smallest numbers would be closest to simple tonality, while the larger-numbered scales will have a much more oblique relationship. Thus, by wandering through the 29 pitches on these different scales, the piece goes “in and out of focus,†sometimes comically random-sounding, sometimes purely and simply in tune, with every gradation in-between – and all with a tremendous economy of means. I’ve put it up for you to hear it here. The duration is just under five minutes, the title: Triskaidekaphonia. More detailed information about the tuning and compositional strategy is here. Only a trifle, perhaps, but it provides yet another bit of proof of the miraculous nature of the whole number series.