Kleismic rank three family: Difference between revisions

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The '''kleismic rank-3 family''' of temperaments tempers out the kleisma, [[15625/15552]]. If nothing else is tempered out in the 7-limit, we have a [[planar temperament]]. With an 11-limit comma we get an 11-limit temperament, and so forth.

The '''kleismic rank-3 family''' of temperaments [[tempering out|tempers out]] the kleisma, [[15625/15552]]. If nothing else is tempered out in the 7-limit, we have a [[planar temperament]]. With an 11-limit comma we get an 11-limit temperament, and so forth.

== Rank-3 kleismic ==

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* [[7-odd-limit|7-]] and [[9-odd-limit]]

: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1 5/6 0 0 | 0 0 0 1 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.3.7

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7

{{Optimal ET sequence|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }}

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* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1 5/6 0 0 0 }}, {{monzo| 3 1/12 0 1/2 -1/2 }}, {{monzo| 3 1/12 0 -1/2 1/2 }}]

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.3.11/7

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7

{{Optimal ET sequence|legend=1| 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee }}

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[[Badness]]: 0.798 × 10<sup>-3</sup>

=== 13-limit (~~aka~~ sumatra) ===

=== 13-limit (sumatra) ===

Subgroup: 2.3.5.7.11.13

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* 13- and 15-odd-limit

: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 4/3 0 0 -2/9 -2/9 4/9 }}, {{monzo| 19/9 0 0 -5/27 -5/27 10/27 }}, {{monzo| 28/9 0 0 13/27 -14/27 1/27 }}, {{monzo| 28/9 0 0 -14/27 13/27 1/27 }}, {{monzo| 28/9 0 0 -14/27 -14/27 28/27 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.11/7.13/7

: unchanged-interval (eigenmonzo) basis: 2.11/7.13/7

{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }}

Badness: 0.680 × 10<sup>-3</sup>

Complexity spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11

Scales: [[enlil19_13]]

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* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| -18/13 0 6/13 -6/13 12/13 }}, {{monzo| -2/13 0 5/13 -5/13 10/13 }}, {{monzo| -2/13 0 -8/13 8/13 10/13 }}, {{monzo| 0 0 0 0 1 }}]

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7/5.11

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11

{{Optimal ET sequence|legend=1| 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424 }}

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* 13- and 15-odd-limit

: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 2/3 0 -2/3 0 0 2/3 }}, {{monzo| 14/9 0 -5/9 0 0 5/9 }}, {{monzo| 0 0 0 1 0 0 }}, {{monzo| 13/9 0 -4/9 1/2 0 4/9 }}, {{monzo| 14/9 0 -14/9 0 0 14/9 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.7.13/5

: unchanged-interval (eigenmonzo) basis: 2.7.13/5

{{Optimal ET sequence|legend=1| 34d, 49f, 72, 87, 159, 193, 280 }}

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[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Kleismic rank-3 family| ]]

[[Category:Rank 3]]

@@ Line 1: / Line 1: @@
-The '''kleismic rank-3 family''' of temperaments tempers out the kleisma, [[15625/15552]]. If nothing else is tempered out in the 7-limit, we have a [[planar temperament]]. With an 11-limit comma we get an 11-limit temperament, and so forth.
+{{Technical data page}}
+The '''kleismic rank-3 family''' of temperaments [[tempering out|tempers out]] the kleisma, [[15625/15552]]. If nothing else is tempered out in the 7-limit, we have a [[planar temperament]]. With an 11-limit comma we get an 11-limit temperament, and so forth.
 == Rank-3 kleismic ==
@@ Line 15: / Line 16: @@
 * [[7-odd-limit|7-]] and [[9-odd-limit]]
 : {{monzo list| 1 0 0 0 | 0 1 0 0 | 1 5/6 0 0 | 0 0 0 1 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3.7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7
 {{Optimal ET sequence|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }}
@@ Line 46: / Line 47: @@
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1 5/6 0 0 0 }}, {{monzo| 3 1/12 0 1/2 -1/2 }}, {{monzo| 3 1/12 0 -1/2 1/2 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3.11/7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7
 {{Optimal ET sequence|legend=1| 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee }}
@@ Line 52: / Line 53: @@
 [[Badness]]: 0.798 × 10<sup>-3</sup>
-=== 13-limit (aka sumatra) ===
+=== 13-limit (sumatra) ===
 Subgroup: 2.3.5.7.11.13
@@ Line 64: / Line 65: @@
 * 13- and 15-odd-limit
 : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 4/3 0 0 -2/9 -2/9 4/9 }}, {{monzo| 19/9 0 0 -5/27 -5/27 10/27 }}, {{monzo| 28/9 0 0 13/27 -14/27 1/27 }}, {{monzo| 28/9 0 0 -14/27 13/27 1/27 }}, {{monzo| 28/9 0 0 -14/27 -14/27 28/27 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11/7.13/7
+: unchanged-interval (eigenmonzo) basis: 2.11/7.13/7
 {{Optimal ET sequence|legend=1| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }}
 Badness: 0.680 × 10<sup>-3</sup>
+Complexity spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11
 Scales: [[enlil19_13]]
@@ Line 112: / Line 115: @@
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| -18/13 0 6/13 -6/13 12/13 }}, {{monzo| -2/13 0 5/13 -5/13 10/13 }}, {{monzo| -2/13 0 -8/13 8/13 10/13 }}, {{monzo| 0 0 0 0 1 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5.11
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11
 {{Optimal ET sequence|legend=1| 15, 23de, 34d, 49, 72, 159, 193, 265, 352, 424 }}
@@ Line 130: / Line 133: @@
 * 13- and 15-odd-limit
 : [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 2/3 0 -2/3 0 0 2/3 }}, {{monzo| 14/9 0 -5/9 0 0 5/9 }}, {{monzo| 0 0 0 1 0 0 }}, {{monzo| 13/9 0 -4/9 1/2 0 4/9 }}, {{monzo| 14/9 0 -14/9 0 0 14/9 }}]
-: eigenmonzo (unchanged-interval) basis: 2.7.13/5
+: unchanged-interval (eigenmonzo) basis: 2.7.13/5
 {{Optimal ET sequence|legend=1| 34d, 49f, 72, 87, 159, 193, 280 }}
@@ Line 137: / Line 140: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Kleismic rank-3 family| ]] <!-- main article -->
 [[Category:Rank 3]]