Ennealimmal: Difference between revisions
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=== Intervals === | === Intervals === | ||
=== Derivation of 1\9 period === | === Derivation of 1\9 period === | ||
We want to show that (1) (27/25)^3 is equated to 63/50 and (2) (63/50)^3 is equated to 2/1. | |||
(1) is easy: | |||
(27/25)^3 ~= 27/25 * 7/6 = 9/25 * 7/2 = 63/50. | |||
For (2): | |||
(63/50)^2 = (49/40 * 9/35 * 4/1)^2 | |||
~= 3/2 * 81/(49*25) * 16/1 | |||
= 3/2 * 27/25 * 3/7 * 1/7 * 16/1 | |||
= 3/2 * 27/25 * 6/7 * 1/2 * 1/7 * 16/1 | |||
~= 3/2 * 25/27 * 1/7 * 8/1 | |||
= 25/9 * 4/7 | |||
= 100/63, the 2/1 complement of 63/50. | |||
{{Navbox regtemp}} | {{Navbox regtemp}} | ||
{{Cat|Temperaments}} | {{Cat|Temperaments}} | ||
Revision as of 17:21, 7 April 2026
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Ennealimmal (from "ennea-" = 9 and "large limma" = 27/25), 72 & 99, is a 7-limit rank-2 microtemperament that tempers out the two smallest 7-limit superparticular ratios:
- 2401/2400 = S49, the difference between the 49/40 neutral third and its 3/2-complement 60/49
- 4375/4374, the difference between (27/25)2 and 7/6
This implies a period of 1\9 (representing 27/25) and a generator of 49/40. The generator can also be taken to be 5/3.
Theory
Intervals
Derivation of 1\9 period
We want to show that (1) (27/25)^3 is equated to 63/50 and (2) (63/50)^3 is equated to 2/1.
(1) is easy:
(27/25)^3 ~= 27/25 * 7/6 = 9/25 * 7/2 = 63/50.
For (2):
(63/50)^2 = (49/40 * 9/35 * 4/1)^2 ~= 3/2 * 81/(49*25) * 16/1 = 3/2 * 27/25 * 3/7 * 1/7 * 16/1 = 3/2 * 27/25 * 6/7 * 1/2 * 1/7 * 16/1 ~= 3/2 * 25/27 * 1/7 * 8/1 = 25/9 * 4/7 = 100/63, the 2/1 complement of 63/50.
