Superkleismic: Difference between revisions

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{{Infobox regtemp
| Title = Shibboleth; superkleismic
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.19
| Comma basis = [[875/864]], [[1029/1024]] (7-limit); <br> [[100/99]], [[385/384]], [[441/440]] (11-limit); <br> [[100/99]], [[133/132]], [[190/189]], [[385/384]] (L11.19)
| Edo join 1 = 15 | Edo join 2 = 26
| Generator = 6/5 | Generator tuning = 321.8 | Optimization method = CTE
| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[11L 4s]], [[15L 11s]]
| Mapping = 1; -9 -10 3 -2 -14
| Pergen = (P8, ccP4/9)
| Odd limit 1 = 7 | Mistuning 1 = 6.09 | Complexity 1 = 41
| Odd limit 2 = (L11.19) 21 | Mistuning 2 = 8.85 | Complexity 2 = 56
}}
'''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. 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]].
'''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. 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]].


Line 20: Line 33:
|-
|-
| 1
| 1
| 322.0
| 321.8
| 6/5
| 6/5
|
|
|-
|-
| 2
| 2
| 644.0
| 643.6
| '''16/11''', 36/25
| '''16/11''', 36/25
| 13/9, 19/13
| 13/9, 19/13
|-
|-
| 3
| 3
| 966.0
| 965.4
| '''7/4''', 33/19
| '''7/4''', 33/19
| 26/15
| 26/15
|-
|-
| 4
| 4
| 88.0
| 87.3
| 20/19, 19/18, 21/20, 22/21
| 20/19, 19/18, 21/20, 22/21
|
|
|-
|-
| 5
| 5
| 410.0
| 409.1
| 14/11, 19/15, 24/19
| 14/11, 19/15, 24/19
|
|
|-
|-
| 6
| 6
| 732.0
| 730.9
| '''32/21''', 38/25
| '''32/21''', 38/25
| 20/13
| 20/13
|-
|-
| 7
| 7
| 1053.9
| 1052.7
| 11/6
| 11/6
| 24/13
| 24/13
|-
|-
| 8
| 8
| 175.9
| 174.5
| 10/9, 11/10, 21/19
| 10/9, 11/10, 21/19
|
|
|-
|-
| 9
| 9
| 497.9
| 496.3
| '''4/3''', 33/25
| '''4/3''', 33/25
|
|
|-
|-
| 10
| 10
| 819.9
| 818.2
| '''8/5'''
| '''8/5'''
|
|
|-
|-
| 11
| 11
| 1141.9
| 1140.0
| 35/18, 48/25, 64/33
| 35/18, 48/25, 64/33
| 52/27
| 52/27
|-
|-
| 12
| 12
| 263.9
| 261.8
| 7/6, 22/19
| 7/6, 22/19
|
|
|-
|-
| 13
| 13
| 585.9
| 583.6
| 7/5
| 7/5
|
|
|-
|-
| 14
| 14
| 907.9
| 905.4
| '''32/19''', 42/25, 56/33
| '''32/19''', 42/25, 56/33
| 22/13
| 22/13
|-
|-
| 15
| 15
| 29.9
| 27.2
| 49/48, 55/54, 56/55, 64/63
| 49/48, 55/54, 56/55, 64/63
| 40/39
| 40/39
|-
|-
| 16
| 16
| 351.9
| 349.1
| 11/9
| 11/9
| '''16/13'''
| '''16/13'''
|-
|-
| 17
| 17
| 673.9
| 670.9
| 22/15, 28/19, 40/27
| 22/15, 28/19, 40/27
|
|
|-
|-
| 18
| 18
| 995.9
| 992.7
| '''16/9''', 44/25
| '''16/9''', 44/25
|
|
|-
|-
| 19
| 19
| 117.9
| 114.5
| '''16/15'''
| '''16/15'''
| 14/13
| 14/13
|-
|-
| 20
| 20
| 439.9
| 436.3
| 32/25
| 32/25
|
|
|-
|-
| 21
| 21
| 761.8
| 768.1
| 14/9
| 14/9
|
|
|-
|-
| 22
| 22
| 1083.8
| 1080.0
| 28/15
| 28/15
|
|
|-
|-
| 23
| 23
| 205.8
| 201.8
| 28/25
| 28/25
| 44/39
| 44/39
|-
|-
| 24
| 24
| 527.8
| 523.6
| 49/36
| 49/36
|
|
|-
|-
| 25
| 25
| 849.8
| 845.4
| 44/27
| 44/27
| 64/39
| 64/39
|-
|-
| 26
| 26
| 1171.8
| 1167.2
| 49/25, 88/45, 160/81
| 49/25, 88/45, 160/81
| 128/65
| 128/65
|}
|}
<nowiki>*</nowiki> in 13-limit CWE tuning
<nowiki>*</nowiki> in L11.19 CWE tuning


== Tunings ==
== Tunings ==

Revision as of 20:05, 6 May 2025

Shibboleth; superkleismic
Subgroups 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.19
Comma basis 875/864, 1029/1024 (7-limit);
100/99, 385/384, 441/440 (11-limit);
100/99, 133/132, 190/189, 385/384 (L11.19)
Reduced mapping ⟨1; -9 -10 3 -2 -14]
Edo join 15 & 26
Generator (CTE) ~6/5 = 321.8 ¢
MOS scales 3L 1s, 4L 3s, 4L 7s, 11L 4s, 15L 11s
Ploidacot beta-enneacot
Pergen (P8, ccP4/9)
Minimax error (7-odd limit) 6.09 ¢;
((L11.19) 21-odd limit) 8.85 ¢
Target scale size (7-odd limit) 41 notes;
((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. 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. 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.

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 13-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
4 87.3 20/19, 19/18, 21/20, 22/21
5 409.1 14/11, 19/15, 24/19
6 730.9 32/21, 38/25 20/13
7 1052.7 11/6 24/13
8 174.5 10/9, 11/10, 21/19
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
13 583.6 7/5
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
21 768.1 14/9
22 1080.0 28/15
23 201.8 28/25 44/39
24 523.6 49/36
25 845.4 44/27 64/39
26 1167.2 49/25, 88/45, 160/81 128/65

* in L11.19 CWE tuning

Tunings

Tuning spectrum

Edo
Generators 		Eigenmonzo
(unchanged-interval)* 		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

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