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, or the difference between 49/48 and 50/49
• 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

Intervals

Tuning shown is pure-2/1, pure-3/2 tuning.

# gens	0		+1		+2		+3		+4
# periods	Cents*	JI	Cents*	JI	Cents*	JI	Cents*	JI	Cents*	JI
-4 mod 9	666.667	72/49	1017.644	9/5	168.622		519.599	27/20	870.577
-3 mod 9	800.000	100/63	1150.978	35/18	301.955	25/21	652.933	35/24	1003.910	25/14
-2 mod 9	933.333	12/7	84.311	21/20	435.288	9/7	786.266	63/40	1137.243	27/14
-1 mod 9	1066.667	50/27	217.644	(17/15)	568.622	25/18	919.599		70.577	25/24
0 mod 9	0	1/1	350.978	49/40	701.955	3/2	1052.933	90/49	203.910	9/8
1 mod 9	133.333	27/25	484.311		835.288	81/50	1186.266	125/63	337.243	(17/14)
2 mod 9	266.667	7/6	617.644	10/7	968.622	7/4	119.599	15/14	470.577	21/16
3 mod 9	400.000	63/50	750.978	54/35	1101.955	189/100	252.933		603.910	(17/12)
4 mod 9	533.333	49/36	884.311	5/3	35.288	49/48, 50/49	386.266	5/4	737.243	64/49

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.

Extensions

Ennealimmal readily extends to prime 17 by tempering out S50, equating 49/48~50/49 to 51/50.

The most natural extension to prime 11 is a weak extension Hemiennealimmal which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can be extended to prime 17 via the 2.3.7.11.17 comma 2058/2057 which ends up being a weak extension of the above 2.3.5.7.17 extension.

There is no canonical 13 extension; however, 270 is an excellent rank-1 Ennealimmal tuning in the 13-limit.

Praxis

On isomorphic keyboards

To set up Ennealimmal, set one axis to be 1\9 and a different axis to be the ~21/20.

Patent vals

Main article: Ennealimmal/Patent vals