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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 [[S-expression|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]]; its [[5-limit]] comma is [[1953125/1889568]], the shibboleth comma. 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 {{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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| 965.4
| 965.4
| '''7/4''', 33/19
| '''7/4''', 33/19
| 26/15
| 26/15, 30/17
|-
|-
| 4
| 4
| 87.3
| 87.3
| 20/19, 19/18, 21/20, 22/21
| 20/19, 19/18, 21/20, 22/21
|
| 18/17
|-
|-
| 5
| 5
| 409.1
| 409.1
| 14/11, 19/15, 24/19
| 14/11, 19/15, 24/19
|
| 34/27
|-
|-
| 6
| 6
| 730.9
| 730.9
| '''32/21''', 38/25
| '''32/21''', 38/25
| 20/13
| 20/13, 26/17
|-
|-
| 7
| 7
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| 174.5
| 174.5
| 10/9, 11/10, 21/19
| 10/9, 11/10, 21/19
|
| 19/17
|-
|-
| 9
| 9
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| 261.8
| 261.8
| 7/6, 22/19
| 7/6, 22/19
|
| 20/17
|-
|-
| 13
| 13
| 583.6
| 583.6
| 7/5
| 7/5
|
| 24/17
|-
|-
| 14
| 14
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| 436.3
| 436.3
| 32/25
| 32/25
|
| 22/17
|-
|-
| 21
| 21
| 768.1
| 768.1
| 14/9
| 14/9
|
| 80/51
|-
|-
| 22
| 22
| 1080.0
| 1080.0
| 28/15
| 28/15
|
| '''32/17'''
|-
|-
| 23
| 23
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| 845.4
| 845.4
| 44/27
| 44/27
| 64/39
| 28/17, 64/39
|-
|-
| 26
| 26
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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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