by Wendy Carlos

Almost all historical work on multiple divisions of the octave in tuning theory has focused on whole number integer divisions. That assures us that after the particular number of notes of a particular division is added together, we arrive at a note exactly one octave (frequency ratio 2:1) away from the pitch at which we started. The so-called equally tempered scale is one of these divisions, as are all the important tunings based on 19, 31, and 53 equal steps. This notion has been around for so long that it almost sounds impertinent to suggest there might be a useful alternative which has been systematically ignored.
Notice that each of these historical divisions is symmetrically laid-out: you will find the prime ratio of the perfect fifth, 3/2, and also the perfect fourth, 4/3. But once you have 2/1 (the perfect octave) and 3/2, the ratio of 4/3 follows directly. It's not prime like the other two ratios, but embedded in their combination. Similarly in the past you find the major third, 5/4, but also its inversion, the minor sixth, 8/5. And both 6/5 and 5/3 appear.
Since each of the redundant interval pairs is symmetric with respect to the octave, the result is a kind of "over-representation" of this interval. But the octave is a ratio most common to the "strategies" of many instruments, including newer synthesizer architectures. Look at their 16', 8', 4' octaving borrowed from the pipe organ. Most timbres/instrument voices include a similar designation of transpositions up or down by octaves. We have octave possibilities all over the place.
So why not, as an experiment, investigate divisions which are not integer based, but allow fractional parts? That will lose all octave symmetry, but if we handle the octaving later, we might be able to find some really interesting equal-step specimens. Several years ago I wrote a computer program to perform a precise deep-search investigation into this kind of Asymmetric Division, based on the target ratios of: 3/2, 5/4, 6/5, 7/4, and 11/8. Here's what it discovered.
Between 10-40 equal steps per octave only three divisions exist which are amazingly more consonant than any other values around the, like lush tropical islands scattered in a great ocean of uniform chaos. I call them Alpha ('alpha'), Beta ('beta'), and Gamma ('gamma'). These happy discoveries occur at:

  • 'alpha' = 78.0 cents/step = 15.385 steps/octave,
  • 'beta' = 63.8 cents/step = 18.809 steps/octave,
  • 'gamma' = 35.1 cents/step = 34.188 steps/octave.

If you try to play through a one octave scale of Alpha, you'd find there are 4 steps to the minor third, 5 steps to the major third, and 9 steps to the perfect (no kidding) fifth, but, or course, no octave. The closest "attempt" at this is an awful 1170 cent version, which sounds awfully flat. Yet the next step to 1248 cents is even further away, and hopelessly sharp, except for timbres like those in a gamelan ensemble. But that's the trade-off we've requested, and there's no free lunch! Try some harmonies and you'll find they're amazingly pure. The melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before. This is a scale well worth exploring.
Beta is very like Alpha in its harmonies, but with 5 steps to the minor third, 6 to the major third, and 11 to the perfect fifth, melodic motions are different, rather more diatonic in effect than Alpha. That's not so surprising, since this scale is very close in its intervals to the 19 2 Symmetric division, which theorists from Yasser on have praised as a good direction to take eventually as a new diatonic alternative for Western music. But Beta sounds even better than 19-step Equal, which is troubled by a fairly flat major third of less than 379 cents, which sounds rather anemic to our ears, brought up as we are in a very sharp major third world of E.T. Melodically it's quite impossible to hear much difference between Beta and 19-tone Equal. So Beta is suited for more standard types of music which might benefit from the nearly perfect harmonies. Beta also lacks the excellent harmonic seventh chords which can be found in Alpha by using the inversion of 7/4, i.e., 8/7, a fact which I first had overlooked when I first discovered Alpha, and a big reason why Alpha is one of my favorite alternative tunings.
You can manage on the standard keyboard design, sort of, to try experimenting with both Alpha and Beta, by retuning two physical octaves for each acoustic octave. This trick also is an easy way to get octaves back in, if the pure octaves are located each physical two octaves apart on a standard keyboard controller. Other kinds of controllers, like wind controllers, could cope with the problem in much the same way. It then gives a means for notating what keys to play, which is important. Just use standard notation for the physical notes, not the sounds (I have no idea how to notate the sounds yet...).
But Gamma really requires a "Multiphonic" Generalized Keyboard, like most >24 divisions, as it simply has, like the joke in the film, Amadeus, "too many notes." Note that Gamma (9 steps - 11 steps - 20 steps) is also slightly smoother than Alpha or Beta, having no palpable difference from Just tuning in harmonies, which is saying a lot. You really have to go further, up to 53-step E.T., to find another nearly perfect equal division, yet Gamma is noticeably freer of beats than even that venerable tuning. Why was it overlooked for so long? You guessed it, it's not symmetrical about the octave, and so was excluded a priori from everybody's search. Gamma's scale is yet a "third flavor," sort of intermediate to 'alpha' and 'beta', although a melodic diatonic scale is easily available. I have searched but can find no previous description of 'alpha', 'beta' or 'gamma' nor their Asymmetric scale-family in any of the literature.
Alpha has a musically interesting property not found in Western music: it splits the minor third exactly in half (also into quarters). This is what initially led me to look for it, and I merely called it my "split minor 3rd scale of 78-cents-steps." Beta, like the Symmetric 19 division, does the same thing to the perfect fourth. This whole formal discovery came a few weeks after I had completed the album, Beauty in the Beast, which is wholly in new tunings and timbres. The title cut from the album contains an extended study of some 'beta', but is mostly in 'alpha'. I expect to work more with both in the near future, and eventually (with the right hardware) with Gamma as well. Any curious souls out there are invited to try their own hand, too. these are not just theoretical speculations we're talking about here. The sound and the music that results is what counts, and the territory is virgin and ripe with gorgeous possibilities. Happy harvesting.

Text ©1989-96 Wendy Carlos

up(Top of the Page)

Back Back to the Wendy Carlos Home Page

Wendy Carlos, Pitch article
© 1997-2008 Serendip LLC. No images, text, graphics or design
may be reproduced without permission. All Rights Reserved.