Superkleismic: Difference between revisions
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'''Superkleismic''' is a [[regular temperament]] defined in the [[7-limit]] such that three [[6/5]] generators reach [[7/4]] (tempering out {{S|5/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 temperaments|keemic temperament]]; its [[5-limit]] comma is [[1953125/1889568]], the shibboleth comma. 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 | '''Superkleismic''' is a [[regular temperament]] defined in the [[7-limit]] such that three [[6/5]] generators reach [[7/4]] (tempering out {{S|5/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 temperaments|keemic temperament]]; its [[5-limit]] comma is [[1953125/1889568]], the shibboleth comma. It [[extension|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 also serves to extend the structure of [[orgone]] in the 2.7.11 subgroup. Furthermore, 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]], and extended to 17 to reach the full [[19-limit]], based on the equivalence (8/7)<sup>2</sup> ~ [[17/13]] (natural in slendric) and tempering out [[273/272]] and [[833/832]] (in addition to [[120/119]] and [[170/169]]). | ||
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 [[mos]]ses of 11 ([[4L 7s]]), 15 ([[11L 4s]]), or 26 notes ([[15L 11s]]) are available. | 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 [[mos]]ses of 11 ([[4L 7s]]), 15 ([[11L 4s]]), or 26 notes ([[15L 11s]]) are available. | ||
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|- | |- | ||
! Edo<br>Generators | ! Edo<br>Generators | ||
! [[Eigenmonzo| | ! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]* | ||
! Generator (¢) | ! Generator (¢) | ||
! Comments | ! Comments | ||
Latest revision as of 06:03, 6 August 2025
| Shibboleth; superkleismic |
100/99, 385/384, 441/440 (11-limit);
100/99, 133/132, 190/189, 385/384 (L11.19)
((L11.19) 21-odd limit) 8.85 ¢
((L11.19) 21-odd limit) 56 notes
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; its 5-limit comma is 1953125/1889568, the shibboleth comma. 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 also serves to extend the structure of orgone in the 2.7.11 subgroup. Furthermore, 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, and extended to 17 to reach the full 19-limit, based on the equivalence (8/7)2 ~ 17/13 (natural in slendric) and tempering out 273/272 and 833/832 (in addition to 120/119 and 170/169).
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 | Full 19-limit extension |
|---|---|---|---|
| 0 | 0.0 | 1/1 | |
| 1 | 321.8 | 6/5 | |
| 2 | 643.6 | 16/11, 36/25 | 13/9, 19/13 |
| 3 | 965.4 | 7/4, 33/19 | 26/15, 30/17 |
| 4 | 87.3 | 20/19, 19/18, 21/20, 22/21 | 18/17 |
| 5 | 409.1 | 14/11, 19/15, 24/19 | 34/27 |
| 6 | 730.9 | 32/21, 38/25 | 20/13, 26/17 |
| 7 | 1052.7 | 11/6 | 24/13 |
| 8 | 174.5 | 10/9, 11/10, 21/19 | 19/17 |
| 9 | 496.3 | 4/3, 33/25 | |
| 10 | 818.2 | 8/5 | |
| 11 | 1140.0 | 35/18, 48/25, 64/33 | 52/27 |
| 12 | 261.8 | 7/6, 22/19 | 20/17 |
| 13 | 583.6 | 7/5 | 24/17 |
| 14 | 905.4 | 32/19, 42/25, 56/33 | 22/13 |
| 15 | 27.2 | 49/48, 55/54, 56/55, 64/63 | 40/39 |
| 16 | 349.1 | 11/9 | 16/13 |
| 17 | 670.9 | 22/15, 28/19, 40/27 | |
| 18 | 992.7 | 16/9, 44/25 | |
| 19 | 114.5 | 16/15 | 14/13 |
| 20 | 436.3 | 32/25 | 22/17 |
| 21 | 768.1 | 14/9 | 80/51 |
| 22 | 1080.0 | 28/15 | 32/17 |
| 23 | 201.8 | 28/25 | 44/39 |
| 24 | 523.6 | 49/36 | |
| 25 | 845.4 | 44/27 | 28/17, 64/39 |
| 26 | 1167.2 | 49/25, 88/45, 160/81 | 128/65 |
* in L11.19 CWE tuning
Tunings
Tuning spectrum
| Edo Generators |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 6/5 | 315.641 | Untempered tuning | |
| 4\15 | 320.000 | Lower bound of 7- through (L11.19) 21-odd-limit diamond monotone | |
| 22/21 | 320.134 | ||
| 11/10 | 320.626 | ||
| 24/19 | 320.888 | ||
| 21/20 | 321.117 | 1/4-keema | |
| 19\71 | 321.127 | ||
| 22/19 | 321.150 | ||
| 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 | ||
| 32/19 | 321.606 | ||
| 26\97 | 321.649 | ||
| 21/19 | 321.658 | ||
| 16/15 | 321.670 | 2/19-shibboleth comma | |
| 11/9 | 321.713 | ||
| 7/5 | 321.732 | 7- and 11- through (L11.19) 21-odd-limit minimax | |
| 37\138 | 321.739 | 138e val | |
| 28/19 | 321.842 | ||
| 28/15 | 321.844 | ||
| 19/15 | 321.849 | ||
| 11\41 | 321.951 | Upper bound of (L11.19) 15- through 21-odd-limit diamond monotone | |
| 4/3 | 322.005 | 9-odd-limit minimax, 1/9-shibboleth comma | |
| 14/9 | 322.139 | ||
| 20/19 | 322.200 | ||
| 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 | |
| 19/18 | 323.401 | ||
| 14/11 | 323.502 | ||
| 16/11 | 324.341 |
* besides the octave