Superkleismic: Difference between revisions

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updated intro, split 13-limit intervals off since 13-limit superkleismic is considerably higher-damage than 7-limit and 11-limit
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'''Superkleismic''' is a [[regular temperament]] defined in the [[7-limit|7-]], [[11-limit|11-]], and [[13-limit]]. It is a member of [[shibboleth family]] as well as of the [[gamelismic clan]]. It also [[extension|extends]] [[orgone]]. The minor-third generator of superkleismic is ~6.3 cents sharp of [[6/5]], even wider than the [[Kleismic family|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, 15, or 26 notes are available.
'''Superkleismic''' is a [[regular temperament]] defined in the [[7-limit]] such that three [[6/5]] generators reach [[7/4]] (tempering out [[square superparticular|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 temperaments|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 [[extension|extend]] the structure of [[orgone]] in the 2.7.11 subgroup. 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 [[mos]]ses of 11 ([[4L 7s]]), 15 ([[11L 4s]]), or 26 notes ([[15L 11s]]) are available.


See [[Shibboleth family #Superkleismic]] for more technical data.  
See [[Shibboleth family #Superkleismic]] for more technical data.  
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! #
! #
! Cents*
! Cents*
! Approximate ratios
! Approximate 11-limit ratios
! 13-limit extension
|-
|-
| 0
| 0
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|
|-
|-
| 1
| 1
| 322.0
| 322.0
| 6/5
| 6/5
|
|-
|-
| 2
| 2
| 644.0
| 644.0
| 13/9, '''16/11'''
| '''16/11'''
| 13/9
|-
|-
| 3
| 3
| 966.0
| 966.0
| '''7/4'''
| '''7/4'''
|
|-
|-
| 4
| 4
| 88.0
| 88.0
| 21/20, 22/21
| 21/20, 22/21
|
|-
|-
| 5
| 5
| 410.0
| 410.0
| 14/11
| 14/11
|
|-
|-
| 6
| 6
| 732.0
| 732.0
| 20/13, '''32/21'''
| '''32/21'''
| 20/13
|-
|-
| 7
| 7
| 1053.9
| 1053.9
| 11/6, 24/13
| 11/6
| 24/13
|-
|-
| 8
| 8
| 175.9
| 175.9
| 10/9, 11/10
| 10/9, 11/10
|
|-
|-
| 9
| 9
| 497.9
| 497.9
| '''4/3'''
| '''4/3'''
|
|-
|-
| 10
| 10
| 819.9
| 819.9
| '''8/5'''
| '''8/5'''
|
|-
|-
| 11
| 11
| 1141.9
| 1141.9
| 35/18, 48/25, 52/27, 64/33
| 35/18, 48/25, 64/33
| 52/27
|-
|-
| 12
| 12
| 263.9
| 263.9
| 7/6
| 7/6
|
|-
|-
| 13
| 13
| 585.9
| 585.9
| 7/5
| 7/5
|
|-
|-
| 14
| 14
| 907.9
| 907.9
|
| 22/13
| 22/13
|-
|-
| 15
| 15
| 29.9
| 29.9
| 40/39, 49/48, 56/55, 64/63
| 49/48, 56/55, 64/63
| 40/39
|-
|-
| 16
| 16
| 351.9
| 351.9
| 11/9, '''16/13'''
| 11/9
| '''16/13'''
|-
|-
| 17
| 17
| 673.9
| 673.9
| 22/15, 40/27
| 22/15, 40/27
|
|-
|-
| 18
| 18
| 995.9
| 995.9
| '''16/9'''
| '''16/9'''
|
|-
|-
| 19
| 19
| 117.9
| 117.9
| 14/13, '''16/15'''
| '''16/15'''
| 14/13
|-
|-
| 20
| 20
| 439.9
| 439.9
| 32/25, 35/27
| 32/25, 35/27
|
|-
|-
| 21
| 21
| 761.8
| 761.8
| 14/9
| 14/9
|
|-
|-
| 22
| 22
| 1083.8
| 1083.8
| 28/15
| 28/15
|
|-
|-
| 23
| 23
| 205.8
| 205.8
| 28/25, 44/39
| 28/25
| 44/39
|-
|-
| 24
| 24
| 527.8
| 527.8
| 49/36
| 49/36
|
|-
|-
| 25
| 25
| 849.8
| 849.8
| 44/27, 64/39
| 44/27
| 64/39
|-
|-
| 26
| 26
| 1171.8
| 1171.8
| 49/25, 88/45, 128/65, 160/81
| 49/25, 88/45, 160/81
| 128/65
|}
|}
<nowiki>*</nowiki> in 13-limit CWE tuning
<nowiki>*</nowiki> in 13-limit CWE tuning

Revision as of 15:00, 6 May 2025

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. 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 Shibboleth family #Superkleismic for more technical data.

Interval chain

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

# Cents* Approximate 11-limit ratios 13-limit extension
0 0.0 1/1
1 322.0 6/5
2 644.0 16/11 13/9
3 966.0 7/4
4 88.0 21/20, 22/21
5 410.0 14/11
6 732.0 32/21 20/13
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, 64/33 52/27
12 263.9 7/6
13 585.9 7/5
14 907.9 22/13
15 29.9 49/48, 56/55, 64/63 40/39
16 351.9 11/9 16/13
17 673.9 22/15, 40/27
18 995.9 16/9
19 117.9 16/15 14/13
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, 160/81 128/65

* 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

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