Ennealimmal

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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 = S25/S27, 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

Derivation of 1\9 period

This can be derived by showing 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 * 36/35)^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.