Penultimate Amazing
Join Date: Jul 2006
Posts: 13,001
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I might as well post the results of a few days of enjoyable work.
Using the "waffle iron" in LMSO, I made a 13-limit JI rectangle, with a long chain of 3/2's or near-3/2's. I trimmed out unnecessary ratios until I had 43 pitches.
1/1, 256/243, 16/15, 13/12, 12/11, 11/10,10/9, 9/8, 8/7, 7/6, 32/27, 6/5, 11/9,16/13, 5/4, 81/64, 9/7, 13/10, 4/3, 11/8, 18/13, (7/5 or 45/32), (64/45 or 10/7), 13/9, 16/11, 3/2, 20/13, 14/9, 11/7, 8/5, 13/8, 18/11, 5/3, 27/16, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 24/13, 15/8, 243/128
Then, using various means including my ear, the Xenharmonic Wiki, LMSO, and my knowledge of theory, I quantized these ratios to a series of equal-tempered approximations.
These were chosen to be fairly accurate, with a 3/2 approximation (a "5th") that is not far from JI ( 701.95319 cents.)
I like a 5th that is between 701 and 703, with 700 cents being acceptable.
Below 77edo, I can't really get enough accuracy for 13-limit.
As the number of the EDO increases, it has less and less "personality" as an EDO, and more accuracy and flexability. One uses subsets of these EDOs, because they have too many pitches to be practical to use all at once.
The higher-numbered EDOs are where JI and equal-tempered thinking meet. Somewhere above 400 it becomes sort of ridiculous to call it an EDO, and it starts having little advantage over simply thinking in cents.
The best EDOs below 77 for JI are 53 and 46, imo.
Here are the best of the best: A combination of subjective and objective judgements. They each have something that they do best.
The honor roll:
77,87,94,99,
130,135,159,164,166,171,183
200,207,224,253,270,
306,311,323,378
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