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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
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ASYMMETRIC DIVISIONS OF THE OCTAVE
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