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'''Superkleismic temperament''' is temperament for the 7, 11, and 13 [[Harmonic limit|prime limits]]. It is a member of [[shibboleth family]], [[gamelismic clan]], [[keemic temperaments]], and [[octagar temperaments]]. The minor-third generator of superkleismic is ~6.3 cents sharp of [[6/5]], even wider than [[Kleismic family|kleismic]] minor third (~317 cents), and from this it derives its name. [[41edo]] is a good tuning for superkleismic, with a minor-third generator of 11\41, and MOS of 11, 15, or 26 notes are available.
{{Infobox regtemp
| Title = 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]], [[245/242]], [[385/384]] (11-limit); <br>[[100/99]], [[133/132]], [[190/189]], [[385/384]] (L11.19)
| Mapping = 1; 9 10 -3 2 14
| Edo join 1 = 15 | Edo join 2 = 26
| Generators = 5/3 | Generators tuning = 878.2 | Optimization method = CWE
| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[11L 4s]], [[15L 11s]]
| Pergen = (P8, ccP4/9)
| Odd limit 1 = 7 | Mistuning 1 = 6.09 | Complexity 1 = 15
| Odd limit 2 = 2.3.5.7.11.19 21 | Mistuning 2 = 8.85 | Complexity 2 = 26
}}
'''Superkleismic''' is a [[regular temperament]] defined in the [[7-limit]] such that three [[6/5]] generators reach [[7/4]] (tempering out [[875/864]] ([[S-expression|S5/S6]]), the keema) and such that three [[8/7]] intervals reach [[3/2]] (tempering out [[1029/1024]] ([[S-expression|S7/S8]]), 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 [[100/99]] ({{S|10}}) so that two generators reach [[16/11]], which also serves to extend the structure of [[orgone]] in the 2.7.11 subgroup. This implies [[385/384]] and [[441/440]] are tempered out as well, making it a subtemperament of [[portent]]. Furthermore, since in superkleismic, the interval [[21/20]] stands for half [[10/9]] = ([[19/18]])⋅([[20/19]]), we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out [[361/360]] ({{S|19}}) and [[400/399]] ({{S|20}}). 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]].


== Temperament data ==
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.
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Superkleismic temperament (15 &amp; 26)'''</div>
<div class="mw-collapsible-content">
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 100/99, 105/104, 144/143, 245/242
See [[Gamelismic clan #Superkleismic]] for more technical data.


[[Mapping]]: [{{val| 1 4 5 2 4 8 }}, {{val| 0 -9 -10 3 -2 -16 }}]
== Interval chain ==
 
In the following table, odd harmonics and subharmonics 1–21 are '''bolded'''.  
[[POTE generator]]:
* 7-limit: ~6/5 = 321.93010
* 11-limit: ~6/5 = 321.84656
* 13-limit: ~6/5 = 321.99387
 
[[TOP tuning|TOP generator]]s:
* 7-limit: ~2 = 1200.76801, ~6/5 = 322.13613
* 11-limit: ~2 = 1200.17605, ~6/5 = 321.89378
* 13-limit: ~2 = 1200.03800, ~6/5 = 322.00406
 
[[Diamond monotone]] ranges:
* 5-odd-limit: ~6/5 = [315.78947, 327.27273] (5\19 to 3\11)
* 7, 9, 11, and 13-odd-limit: ~6/5 = [320.00000, 323.07692] (4\15 to 7\26)
* 15-odd-limit: ~6/5 = 321.95122 (11\41)
 
[[Diamond tradeoff]] ranges:
* 5-odd-limit: ~6/5 = [315.64129, 322.00500]
* 7 and 9-odd-limit: ~6/5 = [315.64129, 322.94197]
* 11, 13, and 15-odd-limit: ~6/5 = [315.64129, 324.34103]
 
Diamond monotone and tradeoff ranges:
* 5-odd-limit: ~6/5 = [315.78947, 322.00500]
* 7 and 9-odd-limit: ~6/5 = [320.00000, 322.94197]
* 11 and 13-odd-limit: ~6/5 = [320.00000, 323.07692]
* 15-odd-limit: ~6/5 = 321.95122
 
[[Optimal GPV sequence]]s:
* 7-limit: {{Vals| 11c, 15, 26, 41 }}
* 11-limit: {{Vals| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}
* 13-limit: {{Vals| 11cf, 15, 26, 41 }}


[[Badness]]:
* 7-limit: 0.047932
* 11-limit: 0.025659
* 13-limit: 0.021478
</div></div>
== Interval chain ==
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! Number of <br>minor third
! rowspan="2" | #
! Cents <br>value*
! rowspan="2" | Cents*
! Approximate Ratios
! colspan="2" | Approximate ratios
|-
! 11-limit add-19
! Full 19-limit extension
|-
|-
| 0
| 0
| 0.00
| 0.0
| 1/1
| '''1/1'''
|
|-
|-
| 1
| 1
| 321.99
| 321.8
| 6/5
| 6/5
|
|-
|-
| 2
| 2
| 643.99
| 643.6
| 13/9, 16/11
| '''16/11''', 36/25
| 13/9, 19/13
|-
|-
| 3
| 3
| 965.98
| 965.4
| 7/4
| '''7/4''', 33/19
| 26/15, 30/17
|-
|-
| 4
| 4
| 87.98
| 87.3
| 21/20, 22/21
| 20/19, 19/18, 21/20, 22/21
| 18/17
|-
|-
| 5
| 5
| 409.97
| 409.1
| 14/11
| 14/11, 19/15, 24/19
| 34/27
|-
|-
| 6
| 6
| 731.96
| 730.9
| 20/13
| '''32/21''', 38/25
| 20/13, 26/17
|-
|-
| 7
| 7
| 1053.96
| 1052.7
| 11/6, 24/13
| 11/6
| 24/13
|-
|-
| 8
| 8
| 175.95
| 174.5
| 10/9, 11/10
| 10/9, 11/10, 21/19
| 19/17
|-
|-
| 9
| 9
| 497.94
| 496.3
| 4/3
| '''4/3''', 33/25
|
|-
|-
| 10
| 10
| 819.94
| 818.2
| 8/5
| '''8/5'''
|
|-
|-
| 11
| 11
| 1141.93
| 1140.0
|  
| 35/18, 48/25, 64/33
| 52/27
|-
|-
| 12
| 12
| 263.93
| 261.8
| 7/6
| 7/6, 22/19
| 20/17
|-
|-
| 13
| 13
| 585.92
| 583.6
| 7/5
| 7/5
| 24/17
|-
|-
| 14
| 14
| 907.91
| 905.4
| '''32/19''', 42/25, 56/33
| 22/13
| 22/13
|-
|-
| 15
| 15
| 29.91
| 27.2
|  
| 49/48, 55/54, 56/55, 64/63
| 40/39
|-
|-
| 16
| 16
| 351.90
| 349.1
| 11/9, 16/13
| 11/9
| '''16/13'''
|-
|-
| 17
| 17
| 673.90
| 670.9
| 22/15
| 22/15, 28/19, 40/27
|
|-
|-
| 18
| 18
| 995.89
| 992.7
| 16/9
| '''16/9''', 44/25
|
|-
|-
| 19
| 19
| 117.88
| 114.5
| 14/13, 16/15
| '''16/15'''
| 14/13
|-
|-
| 20
| 20
| 439.88
| 436.3
|  
| 32/25
| 22/17
|-
|-
| 21
| 21
| 761.87
| 768.1
| 14/9
| 14/9
| 80/51
|-
|-
| 22
| 22
| 1083.87
| 1080.0
| 28/15
| 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
|}
|}
<nowiki>*</nowiki> in 13-limit POTE tuning
<nowiki>*</nowiki> In 11-limit add-19 [[CWE]] tuning, octave reduced


== Tuning spectrum ==
== Tunings ==
Gencom: [2 6/5; 100/99 105/104 144/143 245/242]
=== Norm-based tunings ===
 
{| class="wikitable mw-collapsible mw-collapsed"
Gencom mapping: [{{val| 1 4 5 2 4 8 }}, {{val| 0 -9 -10 3 -2 -16 }}]
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~5/3 = 878.2017{{c}}
| CWE: ~5/3 = 878.1077{{c}}
| POTE: ~5/3 = 878.0699{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~5/3 = 878.1854{{c}}
| CWE: ~5/3 = 878.1606{{c}}
| POTE: ~5/3 = 878.1534{{c}}
|}


{| class="wikitable center-all"
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
|-
! [[eigenmonzo|eigenmonzo<br>(unchanged interval)]]
! Edo<br>generators
! minor third<br>(¢)
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! comments
! Generator (¢)
! Comments
|-
|-
| 6/5
|  
| [[6/5]]
| 315.641
| 315.641
| Untempered tuning
|-
| '''[[15edo|4\15]]'''
|  
|  
| '''320.000'''
| '''Lower bound of 7- through (L11.19) 21-odd-limit diamond monotone'''
|-
|-
| 18/13
|  
| 317.420
| [[22/21]]
| 320.134
|  
|  
|-
|-
| 15/13
|  
| 318.309
| [[11/10]]
| 320.626
|  
|  
|-
|-
| 11/10
|  
| 320.626
| [[24/19]]
| 320.888
|
|-
|
| [[21/20]]
| 321.117
| 1/4-keema
|-
| [[71edo|19\71]]
|
| 321.127
|
|-
|
| [[22/19]]
| 321.150
|  
|  
|-
|-
| 12/11
|  
| [[11/6]]
| 321.338
| 321.338
|  
|  
|-
|-
| 15/11
|  
| [[22/15]]
| 321.356
| 321.356
|  
|  
|-
|-
| 5/4
|  
| [[8/5]]
| 321.369
| 321.369
| 5-odd-limit minimax
| 5-odd-limit minimax, 1/10-shibboleth comma
|-
|-
| 16/15
| [[56edo|15\56]]
|
| 321.429
|
|-
|
| [[32/21]]
| 321.537
|
|-
|
| [[32/19]]
| 321.606
|
|-
| [[97edo|26\97]]
|
| 321.649
|
|-
|
| [[21/19]]
| 321.658
|
|-
|
| [[16/15]]
| 321.670
| 321.670
| 2/19-shibboleth comma
|-
|  
|  
|-
| [[11/9]]
| 11/9
| 321.713
| 321.713
|  
|  
|-
|-
| 7/5
|  
| [[7/5]]
| 321.732
| 321.732
| 7 and 11-odd-limit minimax
| 7- and 11- through (L11.19) 21-odd-limit minimax
|-
| [[138edo|37\138]]
|
| 321.739
| 138e val
|-
|
| [[28/19]]
| 321.842
|
|-
|-
| 15/14
|  
| [[28/15]]
| 321.844
| 321.844
|  
|  
|-
|-
| 4/3
|  
| [[19/15]]
| 321.849
|
|-
| '''[[41edo|11\41]]'''
|
| '''321.951'''
| '''Upper bound of (L11.19) 15- through 21-odd-limit diamond monotone'''
|-
|
| [[4/3]]
| 322.005
| 322.005
| 9 and 15-odd-limit minimax
| 9-odd-limit minimax, 1/9-shibboleth comma
|-
|-
| 9/7
|  
| [[14/9]]
| 322.139
| 322.139
|  
|  
|-
|-
| 13/11
|  
| 322.199
| [[20/19]]
| 13-odd-limit minimax
| 322.200
|  
|-
|-
| 7/6
|  
| [[7/6]]
| 322.239
| 322.239
|  
|  
|-
|-
| 16/13
| [[67edo|18\67]]
| 322.467
|  
|  
| 322.388
| 67ch val
|-
|-
| 14/13
| 322.542
|  
|  
| [[10/9]]
| 322.800
| 1/8-shibboleth comma
|-
|-
| 10/9
| 322.800
|  
|  
| [[7/4]]
| 322.942
| 1/3-keema
|-
|-
| 8/7
| '''[[26edo|7\26]]'''
| 322.942
|  
|  
| '''323.077'''
| '''Upper bound of 7-, 9-, and 11-odd-limit diamond monotone'''
|-
|-
| 13/12
|  
| 323.061
| [[19/18]]
| 323.401
|  
|  
|-
|-
| 14/11
|  
| [[14/11]]
| 323.502
| 323.502
|  
|  
|-
|-
| 13/10
| 324.298
|  
|  
|-
| [[16/11]]
| 11/8
| 324.341
| 324.341
|  
|  
|}
|}
<nowiki>*</nowiki> besides the octave


== Scales ==
[[Category:Superkleismic| ]] <!-- main article -->
* [[Shibboleth7]] - [[4L 3s]] scale
[[Category:Rank-2 temperaments]]
* [[Shibboleth11]] - [[4L 7s]] scale
* [[Shibboleth15]] - [[11L 4s]] scale
 
[[Category:Shibboleth family]]
[[Category:Gamelismic clan]]
[[Category:Gamelismic clan]]
{{IoT}}
[[Category:Keemic temperaments]]
[[Category:Octagar temperaments]]
Retrieved from "https://en.xen.wiki/w/Superkleismic"