Superkleismic

Superkleismic is a regular temperament defined in the 7-limit such that three 6/5 generators reach 7/4 (tempering out S5/S6 = 875/864, the keema) and such that three 8/7 intervals reach 3/2 (tempering out S7/S8 = 1029/1024, the gamelisma), making it a member of the gamelismic clan and a keemic temperament. It extends extremely easily to the 11-limit as well, by tempering out S10 = 100/99 (as well as 385/384 and 441/440) so that two generators reach 16/11, which serves to extend the structure of orgone in the 2.7.11 subgroup. Since in superkleismic, the interval 21/20 stands for half 10/9 = 20/19 × 19/18, we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out S19 = 361/360 and S20 = 400/399. Superkleismic can also be defined in the 13-limit, where two generators are identified with 13/9 alongside 16/11, tempering out 144/143 and 325/324.

The minor-third generator of superkleismic is ~6.3 cents sharp of pure 6/5, even wider than the kleismic minor third (~317 cents), and from this it derives its name. The two mappings unite at 15edo. While not as simple or accurate as kleismic in the 5-limit, it comes into its own as a 7- and 11-limit temperament, approximating both simply and accurately in good tunings. Discarding the harmonics 3 and 5 and concentrating purely on that subgroup gets you orgone. 41edo is a good tuning for superkleismic, with a minor-third generator of 11\41, and mosses of 11 (4L 7s), 15 (11L 4s), or 26 notes (15L 11s) are available.

See Gamelismic clan #Superkleismic for more technical data.

Interval chain

In the following table, odd harmonics and subharmonics 1–21 are bolded.

#	Cents*	Approximate 11-limit add-19 ratios	13-limit extension
0	0.0	1/1
1	322.0	6/5
2	644.0	16/11, 36/25	13/9, 19/13
3	966.0	7/4, 33/19	26/15
4	88.0	20/19, 19/18, 21/20, 22/21
5	410.0	14/11, 19/15, 24/19
6	732.0	32/21, 38/25	20/13
7	1053.9	11/6	24/13
8	175.9	10/9, 11/10, 21/19
9	497.9	4/3, 33/25
10	819.9	8/5
11	1141.9	35/18, 48/25, 64/33	52/27
12	263.9	7/6, 22/19
13	585.9	7/5
14	907.9	32/19, 42/25, 56/33	22/13
15	29.9	49/48, 55/54, 56/55, 64/63	40/39
16	351.9	11/9	16/13
17	673.9	22/15, 28/19, 40/27
18	995.9	16/9, 44/25
19	117.9	16/15	14/13
20	439.9	32/25
21	761.8	14/9
22	1083.8	28/15
23	205.8	28/25	44/39
24	527.8	49/36
25	849.8	44/27	64/39
26	1171.8	49/25, 88/45, 160/81	128/65

* in 13-limit CWE tuning

Tunings

Tuning spectrum

Edo Generators	Eigenmonzo (unchanged-interval)*	Generator (¢)	Comments
	6/5	315.641		Untempered tuning
4\15		320.000	Lower bound of 7- through 21-odd-limit diamond monotone
	22/21	320.134
	11/10	320.626
	21/20	321.117	1/4-keema
	11/6	321.338
	22/15	321.356
	8/5	321.369	5-odd-limit minimax, 1/10-shibboleth comma
15\56		321.429
	32/21	321.537
	16/15	321.670	2/19-shibboleth comma
	11/9	321.713
	7/5	321.732	7- and 11-odd-limit minimax
	28/15	321.844
11\41		321.951	Upper bound of 15- through 21-odd-limit diamond monotone
	4/3	322.005	9- and 15-odd-limit minimax, 1/9-shibboleth comma
	14/9	322.139
	7/6	322.239
18\67		322.388	67ch val
	10/9	322.800	1/8-shibboleth comma
	7/4	322.942	1/3-keema
7\26		323.077	Upper bound of 7-, 9-, and 11-odd-limit diamond monotone
	11/7	323.502
	11/8	324.341

* besides the octave