Superstition Be Damned

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


5/4, 5/3




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.


  1. says

    Nice work! Also I highly enjoyed the explanations on scale structure. This seems like quite fertile material.

    I’m curious about how you think of your melodies. Is there some scale-related structuring underlying the melodic and rhythmic shapes themselves? I have the impression that all the chromas you have are used in very diverse ways, so that it seems that the notes can have many sorts of function; sometimes, the great gamut seems to be used for extra passing notes, sometimes for more stable harmonic scale effects, sometimes for “licks” where it’s quite more active and I lose the harmonic sense a little. Or is it rather based on a free melody within a ‘scale progression chart’? If you can expand on that, I would be very interested.

  2. Jean Lawton says

    What an incandescently beautiful piece. By the way, there’s a name for your scale — it’s a Farey Series. Not that it makes any difference, since as we all know maths have no meaningful connection with music, just as science proves perenially powerless to explain it. (Someone else seems to have said the same thing: My last science class was in 11th grade in 1971 – I think the periodic table was up to aluminum – and from hearing too many lousy pieces of new music based on scientific models, I’ve developed a possibly unfortunate bias that science has nothing to offer art. Gann, Kyle, 31 October, 2003) In any case, Warren Burt has done a splendid Farey Series just intonation piece in his Portrait In Sines Of Erv Wilson. Quite different from yours, albeit worth checking out.

    I do rather wish more chaps would give us insight into the internal construction of their compositions, as you have done here. Nowadays, in the 21st century, older musical forms like the fugue having become passe, we poor listeners find ourselves confronted by a bewildering cornucopia of new musical forms. Often, one-of-a-kind musical wiwaxids which seem to have crawled straight out of the compositional equivalent of the Burgess Shale.

    They make perfect sense in retrospect; but, in so many cases (viz., Basinksi’s The Disintegration Loops or Dodge’s The Earth’s Magnetic Field) it helps ever so much to have the composer along as tour guide so the listener daren’t get lost. Best of all, the web’s ideally suited for these sorts of divers excursions through compositional minutia — as your ever so helpful link to your explanation illustrates. The web can combine graphics with mp3 clips. Infinitely superior, I should think, to the customary muso journal article.

    You remarked: 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.

    Bravo! Well said…as always. As Blaise Pascal averred, The heart has its reasons of which reason knows nothing. One of the most contemptibly infantile conceits of the so-called “modern” era remains the delusion that all of human life takes place above the neck, whereas the reverse remains true. The most important parts of human experience usually occur from the nape of the neck down, not to say below the belt.

  3. says

    The number 13 certainly has some interesting properties. It is a Fibonacci number. The sum of its digits is 4, the sum of its square, 169, is 16, which is 4 squared. Also, the square of 31 is 961, the reverse of 169.

    All this info is from Bryan Bunch’s excellent “field guide” to numbers, The Kingdom of Infinite Number.

    But wait, there’s more:
    13 is prime. And for some reason it is associated with bad luck. Bunch associates 13 with the apostles — Judas would have been the 13th. However, 13 was considered lucky by the Maya.

    Still, the number-theoretic properties of primes are always fascinating. Have you tried 17?

  4. says

    To address Samuel’s questions: I wrote out all the scales on sheets of paper, with the 13-based scale first, then the 11-based, then 9, 7, 5, 3, and 2. Then I just composed freely, starting with the least intelligible scale (13) and moving to the most intelligible (2), back and forth as I wanted the music to go in and out of focus,

    Jean, thanks for all the lovely comments.