Superkleismic
Superkleismic is a regular temperament defined in the 7-, 11-, and 13-limit. It is a member of shibboleth family as well as of the gamelismic clan. The minor-third generator of superkleismic is ~6.3 cents sharp of 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, 15, or 26 notes are available.
See Shibboleth family #Superkleismic for more technical data.
Interval chain
In the following table, odd harmonics 1–21 are bolded.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 322.0 | 6/5 |
| 2 | 644.0 | 13/9, 16/11 |
| 3 | 966.0 | 7/4 |
| 4 | 88.0 | 21/20, 22/21 |
| 5 | 410.0 | 14/11 |
| 6 | 732.0 | 20/13, 32/21 |
| 7 | 1053.9 | 11/6, 24/13 |
| 8 | 175.9 | 10/9, 11/10 |
| 9 | 497.9 | 4/3 |
| 10 | 819.9 | 8/5 |
| 11 | 1141.9 | 35/18, 48/25, 52/27, 64/33 |
| 12 | 263.9 | 7/6 |
| 13 | 585.9 | 7/5 |
| 14 | 907.9 | 22/13 |
| 15 | 29.9 | 40/39, 49/48, 56/55, 64/63 |
| 16 | 351.9 | 11/9, 16/13 |
| 17 | 673.9 | 22/15, 40/27 |
| 18 | 995.9 | 16/9 |
| 19 | 117.9 | 14/13, 16/15 |
| 20 | 439.9 | 32/25, 35/27 |
| 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, 128/65, 160/81 |
* in 13-limit CWE tuning
Tunings
Tuning spectrum
| Edo Generators |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 5/3 | 315.641 | ||
| 13/9 | 317.420 | ||
| 15/13 | 318.309 | ||
| 4\15 | 320.000 | Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone | |
| 21/11 | 320.134 | ||
| 11/10 | 320.626 | ||
| 21/20 | 321.117 | ||
| 11/6 | 321.338 | ||
| 15/11 | 321.356 | ||
| 5/4 | 321.369 | 5-odd-limit minimax | |
| 15\56 | 321.429 | 56f val | |
| 21/16 | 321.537 | ||
| 15/8 | 321.670 | ||
| 11/9 | 321.713 | ||
| 7/5 | 321.732 | 7- and 11-odd-limit minimax | |
| 15/14 | 321.844 | ||
| 11\41 | 321.951 | 15-odd-limit diamond monotone (singleton) | |
| 3/2 | 322.005 | 9- and 15-odd-limit minimax | |
| 9/7 | 322.139 | ||
| 13/11 | 322.199 | 13-odd-limit minimax | |
| 7/6 | 322.239 | ||
| 18\67 | 322.388 | 67c val | |
| 13/8 | 322.467 | ||
| 13/7 | 322.542 | ||
| 9/5 | 322.800 | ||
| 7/4 | 322.942 | ||
| 21/13 | 323.025 | ||
| 13/12 | 323.061 | ||
| 7\26 | 323.077 | Upper bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone | |
| 11/7 | 323.502 | ||
| 13/10 | 324.298 | ||
| 11/8 | 324.341 |
* besides the octave