Superkleismic

Superkleismic temperament is temperament for the 7, 11, and 13 prime limits. It is a member of shibboleth family, gamelismic clan, keemic temperaments, and octagar temperaments. The minor-third generator of superkleismic is ~6.3 cents sharp of 6/5, even wider than 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 it's own as a 7&11 limit temperament, approximating both simply and accurately in good tunings. Discarding the 3&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 MOS of 11, 15, or 26 notes are available.

Temperament data

Superkleismic temperament (15 & 26)

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 245/242

Mapping: [⟨1 4 5 2 4 8], ⟨0 -9 -10 3 -2 -16]]

POTE generator:

• 7-limit: ~6/5 = 321.93010
• 11-limit: ~6/5 = 321.84656
• 13-limit: ~6/5 = 321.99387

TOP generators:

• 7-limit: ~2 = 1200.76801, ~6/5 = 322.13613
• 11-limit: ~2 = 1200.17605, ~6/5 = 321.89378
• 13-limit: ~2 = 1200.03800, ~6/5 = 322.00406

Diamond monotone ranges:

• 5-odd-limit: ~6/5 = [315.78947, 327.27273] (5\19 to 3\11)
• 7, 9, 11, and 13-odd-limit: ~6/5 = [320.00000, 323.07692] (4\15 to 7\26)
• 15-odd-limit: ~6/5 = 321.95122 (11\41)

Diamond tradeoff ranges:

• 5-odd-limit: ~6/5 = [315.64129, 322.00500]
• 7 and 9-odd-limit: ~6/5 = [315.64129, 322.94197]
• 11, 13, and 15-odd-limit: ~6/5 = [315.64129, 324.34103]

Diamond monotone and tradeoff ranges:

• 5-odd-limit: ~6/5 = [315.78947, 322.00500]
• 7 and 9-odd-limit: ~6/5 = [320.00000, 322.94197]
• 11 and 13-odd-limit: ~6/5 = [320.00000, 323.07692]
• 15-odd-limit: ~6/5 = 321.95122

Optimal ET sequences:

• 7-limit: 11c, 15, 26, 41
• 11-limit: 11c, 15, 26, 41, 179cde, 220cde, 261ccdee
• 13-limit: 11cf, 15, 26, 41

Badness:

• 7-limit: 0.047932
• 11-limit: 0.025659
• 13-limit: 0.021478

Interval chain

Number of minor third	Cents value*	Approximate Ratios
0	0.00	1/1
1	321.99	6/5
2	643.99	13/9, 16/11
3	965.98	7/4
4	87.98	21/20, 22/21
5	409.97	14/11
6	731.96	20/13
7	1053.96	11/6, 24/13
8	175.95	10/9, 11/10
9	497.94	4/3
10	819.94	8/5
11	1141.93
12	263.93	7/6
13	585.92	7/5
14	907.91	22/13
15	29.91
16	351.90	11/9, 16/13
17	673.90	22/15
18	995.89	16/9
19	117.88	14/13, 16/15
20	439.88
21	761.87	14/9
22	1083.87	28/15

* in 13-limit POTE tuning

Tuning spectrum

Gencom: [2 6/5; 100/99 105/104 144/143 245/242]

Gencom mapping: [⟨1 4 5 2 4 8], ⟨0 -9 -10 3 -2 -16]]

eigenmonzo (unchanged-interval)	minor third (¢)	comments
6/5	315.641
18/13	317.420
15/13	318.309
11/10	320.626
12/11	321.338
15/11	321.356
5/4	321.369	5-odd-limit minimax
16/15	321.670
11/9	321.713
7/5	321.732	7 and 11-odd-limit minimax
15/14	321.844
4/3	322.005	9 and 15-odd-limit minimax
9/7	322.139
13/11	322.199	13-odd-limit minimax
7/6	322.239
16/13	322.467
14/13	322.542
10/9	322.800
8/7	322.942
13/12	323.061
14/11	323.502
13/10	324.298
11/8	324.341

Scales

• Shibboleth7 - 4L 3s scale
• Shibboleth11 - 4L 7s scale
• Shibboleth15 - 11L 4s scale