Superkleismic: Difference between revisions

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{{Infobox regtemp

| Title = ~~Shibboleth; superkleismic~~

| 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); [[100/99]], [[245/242]], [[385/384]] (11-limit); [[100/99]], [[133/132]], [[190/189]], [[385/384]] (L11.19)

| Comma basis = [[875/864]], [[1029/1024]] (7-limit); [[100/99]], [[245/242]], [[385/384]] (11-limit); [[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

| ~~Generator~~ = 5/3 | ~~Generator~~ tuning = 878.2 | Optimization method = CWE

| Generators = 5/3 | Generators tuning = 878.2 | Optimization method = CWE

| 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 = 15

| Odd limit 2 = ~~(L11~~.19) 21 | Mistuning 2 = 8.85 | Complexity 2 = 26

| 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)2 ~ [[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)2 ~ [[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 ==

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{| 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

|-

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| 128/65

|}

<nowiki>*</nowiki> ~~in L11.~~19 [[CWE]] tuning

<nowiki>*</nowiki> In 11-limit add-19 [[CWE]] tuning, octave reduced

== 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 ===

{| class="wikitable center-all left-4"

|-

! Edo ~~Generators~~

! Edo generators

! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*

! Generator (¢)

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[[Category:Superkleismic| ]]

[[Category:Rank-2 temperaments]]

~~[[Category:Shibboleth family]]~~

[[Category:Gamelismic clan]]

[[Category:Keemic temperaments]]

[[Category:Octagar temperaments]]

@@ Line 1: / Line 1: @@
 {{Infobox regtemp
-| Title = Shibboleth; superkleismic
+| 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)
+| 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
-| Generator = 5/3 | Generator tuning = 878.2 | Optimization method = CWE
+| Generators = 5/3 | Generators tuning = 878.2 | Optimization method = CWE
 | 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 = 15
-| Odd limit 2 = (L11.19) 21 | Mistuning 2 = 8.85 | Complexity 2 = 26
+| 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 21: / Line 21: @@
 {| 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
 |-
@@ Line 161: / Line 163: @@
 | 128/65
 |}
-<nowiki>*</nowiki> in L11.19 [[CWE]] tuning
+<nowiki>*</nowiki> In 11-limit add-19 [[CWE]] tuning, octave reduced
 == 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 ===
 {| class="wikitable center-all left-4"
 |-
-! Edo<br>Generators
+! Edo<br>generators
 ! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
 ! Generator (¢)
@@ Line 351: / Line 385: @@
 [[Category:Superkleismic| ]] <!-- main article -->
 [[Category:Rank-2 temperaments]]
-[[Category:Shibboleth family]]
 [[Category:Gamelismic clan]]
 [[Category:Keemic temperaments]]
+[[Category:Octagar temperaments]]