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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 [[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]].
{{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 = 5/3 | Generator tuning = 878.2 | 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 {{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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! Cents*
! Cents*
! Approximate 11-limit add-19 ratios
! Approximate 11-limit add-19 ratios
! 13-limit extension
! Full 19-limit extension
|-
|-
| 0
| 0
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|-
|-
| 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, 30/17
|-
|-
| 4
| 4
| 88.0
| 87.3
| 20/19, 19/18, 21/20, 22/21
| 20/19, 19/18, 21/20, 22/21
|
| 18/17
|-
|-
| 5
| 5
| 410.0
| 409.1
| 14/11, 19/15, 24/19
| 14/11, 19/15, 24/19
|
| 34/27
|-
|-
| 6
| 6
| 732.0
| 730.9
| '''32/21''', 38/25
| '''32/21''', 38/25
| 20/13
| 20/13, 26/17
|-
|-
| 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
|
| 19/17
|-
|-
| 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
|
| 20/17
|-
|-
| 13
| 13
| 585.9
| 583.6
| 7/5
| 7/5
|
| 24/17
|-
|-
| 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
|
| 22/17
|-
|-
| 21
| 21
| 761.8
| 768.1
| 14/9
| 14/9
|
| 80/51
|-
|-
| 22
| 22
| 1083.8
| 1080.0
| 28/15
| 28/15
|
| '''32/17'''
|-
|-
| 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
| 28/17, 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 ==
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|-
|-
! Edo<br>Generators
! Edo<br>Generators
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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