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Tempered versions of Carl's 12-note JI scales

🔗Gene W Smith <genewardsmith@juno.com>

7/31/2002 11:34:24 PM

I took these five scales and looked for what new 11-limit intervals would
appear if we allowed commas of less than 8 cents. This puts us in a range
well covered by the 72-et, but none of the resulting scales needed the
full power of this, or even of miracle; they all were covered by one or
another planar temperament. Except for the case of "lester", which is in
the {225/224, 441/440} temperament which doesn't improve from the 72-et
values, this meant some improvement in the tuning was possible. Whether
it is worthwhile is another question.

# lumma in {385/384, 441/440} temperament, 873-et version
l873 := [873, 1383, 2026, 2449, 3019];

lum:=[0, 37, 170, 230, 280, 400, 450, 510, 643, 680, 740, 813];
lumd := [37, 133, 60, 50, 120, 50, 60, 133, 37, 60, 73, 60];

42 ingervals, 58 triads--the least harmony, but the best tuning

# {225/224, 385/384} 858-et version
l858 := [858, 1359, 1990, 2408, 2967];

# prism

prs := [0, 83, 144, 191, 274, 357, 418, 501, 584, 631, 692, 775];
prsd := [83, 61, 47, 83, 83, 61, 83, 83, 47, 61, 83, 83];

49 intervals, 86 triads--the champ. It's also fairly regular.

# stelhex

ste := [0, 61, 191, 227, 274, 335, 418, 501, 584, 645, 692, 728];
sted := [61, 130, 36, 47, 61, 83, 83, 83, 61, 47, 36, 130];

46 intervals, 72 triads

# class

cla := [0, 61, 108, 227, 274, 335, 418, 501, 548, 645, 692, 775];
clad := [61, 47, 119, 47, 61, 83, 83, 47, 97, 47, 83, 83];

47 intervals, 80 triads

# {225/224, 441/440} in 72-et version

# lester

les := [0, 5, 12, 16, 23, 30, 35, 42, 46, 53, 58, 65];
lesd := [5, 7, 4, 7, 7, 5, 7, 4, 7, 5, 7, 7];

46 intervals, 71 triads. Considering this is the least in tune, something
of an also-ran.