Superkleismic

From Xenharmonic Wiki
Jump to navigation Jump to search

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 GPV sequences:

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