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{{Infobox regtemp
{{Infobox regtemp
| Title = Shibboleth; superkleismic
| Title = Superkleismic
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.19
| 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)
| 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
| Edo join 1 = 15 | Edo join 2 = 26
| Generator = 5/3 | Generator tuning = 878.2 | Optimization method = CTE
| Generators = 5/3 | Generators tuning = 878.2 | Optimization method = CWE
| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[11L 4s]], [[15L 11s]]
| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[11L 4s]], [[15L 11s]]
| Mapping = 1; 9 10 -3 2 14
| Pergen = (P8, ccP4/9)
| Pergen = (P8, ccP4/9)
| Odd limit 1 = 7 | Mistuning 1 = 6.09 | Complexity 1 = 41
| Odd limit 1 = 7 | Mistuning 1 = 6.09 | Complexity 1 = 15
| Odd limit 2 = (L11.19) 21 | Mistuning 2 = 8.85 | Complexity 2 = 56
| 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]].
'''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.


See [[Gamelismic clan #Superkleismic]] for more technical data.  
See [[Gamelismic clan #Superkleismic]] for more technical data.


== Interval chain ==
== Interval chain ==
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{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! #
! rowspan="2" | #
! Cents*
! rowspan="2" | Cents*
! Approximate 11-limit add-19 ratios
! colspan="2" | Approximate ratios
|-
! 11-limit add-19
! Full 19-limit extension
! Full 19-limit extension
|-
|-
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| 128/65
| 128/65
|}
|}
<nowiki>*</nowiki> in L11.19 [[CWE]] tuning
<nowiki>*</nowiki> In 11-limit add-19 [[CWE]] tuning, octave reduced


== Tunings ==
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ 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}}
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>Generators
! Edo<br>generators
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
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[[Category:Superkleismic| ]] <!-- main article -->
[[Category:Superkleismic| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Shibboleth family]]
[[Category:Gamelismic clan]]
[[Category:Gamelismic clan]]
[[Category:Keemic temperaments]]
[[Category:Keemic temperaments]]
[[Category:Octagar temperaments]]
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