Non-octave scales; monkeys banging on ke

Although I believe octave equivalence is universally perceived, I think a
lot of interesting music can be made without octaves. The Bohlen-Pierce
scale (13 notes per 3:1 ("tritave") or 8.2 notes/octave), whether or not
"tritave" intervals acquire an "equivalence" quality, is a fine tuning to
explore such music, as Randy Winchester's music proves. The consonant
intervals within one tritave here are 3:5 (6 steps), 3:7 (10 steps), and 5:7
(4 steps), and supposedly their "inversions", 9:5 (7 steps), 9:7 (3 steps),
and 15:7 (9 steps), plus the 3:1 itself (13 steps). The maximum error here
is 6.5 cents, for the 5:3 and 9:5 approximations.

What if we wanted to extend this idea to 11-limit intervals? The first
really good solution is 39 notes per tritave, or 24.6 notes per octave: a
triplication of the Bohlen-Pierce scale. The largest error remains 6.5
cents, but we gain the 11:9 (7 steps), 11:7 (16 steps), and 11:5 (28 steps)
and their tritave inversions and extensions.

How about 13-limit intervals? The Triple-Bohlen-Pierce scale still works,
without increasing the maximum error! 13:11 (6 steps), 13:9 (13 steps), 13:7
(22 steps), and 13:5 (34 steps) are the new intervals here. That gives us 1
5-limit, 2 7-limit, 3 11-limit, and 4 13-limit intervals, plus their
inversions, plus the tritave iteslf, for 21 consonant intervals out of 39.
These are 54% odds that two notes selected randomly by a monkey will be
consonant. The odds for the Pierce scale itself are the same,
7/13=21/39=54%.

In the world of octave scales, the odds of finding a consonance in 12-note
tuning are slightly better, 7/12=58%, but out-of-tuneness should weigh
against it. Otherwise, we could even allow a 9-note tuning and get 78% odds!
Maybe that's why a lot of composers have been successful with 9tet (not to
suggest that they're monkeys, of course). Other octave scales have worse
odds: if all 7-limit intervals and their octave inversions are allowed, the
odds are 48% in 27 and 42% in 31. If all 9-limit intervals are allowed, the
odds are 46% in 41 and 41% in 46. A 58-note tuning gives the monkey 50% odds
of hitting an 11-limit interval or its inversion, dropping to 40% for 72
notes. With 87 notes to play with, the monkey would be within the 13-limit
47% of the time, dropping to 44% in a 94-note equal temperament.

Phew. All the tuning-limit combinations considered in the last paragraph are
both consistent and unique including octave inversions, which seems fair
since the Pierce and triple-Pierce scales satisfy the corresponding
properties for tritaves. I read them off columns 5 and 7 of my table in TD
781 #2, which is mostly read off Manuel's new table. Anyway, I'm going to go
home and map some square waves (which have no 2nd, 4th, 6th, . . .
harmonics) to Pierce, and triple-Pierce scales, and monkey around.

Other possibilities to explore would be including 2 but omitting 3, etc.
Just as square waves are a natural timbre for Pierce scales, so a waveform
that steps between three different levels (-1 0 1 -1 0 1 etc.) would be
appropriate for a tuning that omits ratios with the number 3. In fact,
quantizing a sawtooth wave to n levels eliminates all harmonics that are
multiples of n times the fundamental frequency. We have considered above the
ideal monkey tunings only for the special cases n=2 and n>24.


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