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Schismic[17] and Miracle[20]

🔗Gene Ward Smith <gwsmith@svpal.org>

3/23/2003 1:22:55 PM

We can make the same comment about these as I made for Orwell[13];
that the augmented triads, from the perspective of the corresponding
septimal temperament, are [1,5/4,14/9] 225/224-tempered asses. Note
however that 5-limit schismic is a microtemperament, and 7-limit
schismic is not nearly as well in tune.

Looking at this, I find myself completely unconvinced by George's
17-revolution alternative history. Going from meantone to 12-equal is
one thing, but I don't think you could ever get people used to
schismic[17] used for 5-limit harmony to move to 17-equal. What do the
17-revolutionaries think of schismic[17]?

Schismic[17]
rms error 0.162 cents

32805/32768
[1, 135/128, 16/15, 9/8, 32/27, 5/4, 81/64, 4/3, 45/32, 64/45, 3/2,
128/81, 8/5, 27/16, 16/9, 15/8, 243/128]
[5/4, 15/8, 45/32, 135/128, 128/81, 32/27, 16/9, 4/3, 1, 3/2, 9/8, 27/
16, 81/64, 243/128, 64/45, 16/15, 8/5]

Circle of fifths
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 25/16]

Miracle[20]
rms error = 1.981 cents

34171875/33554432
[1, 16/15, 1125/1024, 256/225, 75/64, 4096/3375, 5/4, 675/512, 4/3,
45/32, 64/45, 3/2, 1024/675, 8/5, 3375/2048, 128/75, 225/128,
2048/1125, 15/8, 2025/1024]
[1125/1024, 75/64, 5/4, 4/3, 64/45, 1024/675, 3375/2048, 225/128,
15/8, 1, 16/15, 256/225, 4096/3375, 675/512, 45/32, 3/2, 8/5, 128/75,
2048/1125, 2025/1024]

Generator = 16/15
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]