Superkleismic: Difference between revisions

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| 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. 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 [[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]].

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.

== Interval chain ==

@@ Line 11: / Line 11: @@
 | 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. 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 [[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]].
 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.
 == Interval chain ==