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Kleismic rank three family: Difference between revisions

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__FORCETOC__
{{Technical data page}}
These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit 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.


7 and 9 limit minimax
== Rank-3 kleismic ==
[[Subgroup]]: 2.3.5.7


[|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>]
[[Comma list]]: 15625/15552


[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3
{{Mapping|legend=1| 1 0 1 0 | 0 6 5 0 | 0 0 0 1 }}


Map: [<1 0 1 0|, <0 6 5 0|, <0 0 0 1|]
: mapping generators: ~2, ~6/5, ~7


[[generator|Generators]]: 2, 6/5, 7
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 317.0586, ~7/4 = 968.8259


[[EDO|Edos]]: [[15edo|15]], [[19edo|19]], [[23edo|23]], [[34edo|34]], [[53edo|53]], [[68edo|68]], [[72edo|72]], [[87edo|87]], [[121edo|121]], [[125edo|125]], [[140edo|140]], [[212edo|212]], 333, 405
[[Minimax tuning]]:  
* [[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|unchanged-interval (eigenmonzo) basis]]: 2.3.7


=Enlil=
{{Optimal ET sequence|legend=1| 15, 19, 34d, 49, 53, 72, 140, 212, 617c, 829c }}
[[Comma|Commas]]: 385/384, 6250/6237


[[Minimax_tuning|Minimax tuning]]
[[Badness]]: 0.363 × 10<sup>-3</sup>


[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1 5/6 0 0 0&gt;,
=== 2.3.5.7.13 subgroup ===
|3 1/12 0 1/2 -1/2&gt;, |3 1/12 0 -1/2 1/2&gt;]
Subgroup: 2.3.5.7.13


[[Eigenmonzo|Eigenmonzos]]: 2, 4/3, 14/11
Comma list: 325/324, 625/624


Map: [&lt;1 0 1 0 6|, &lt;0 6 5 0 1|, &lt;0 0 0 1 -1|]
Sval mapping: {{mapping| 1 0 1 0 0 | 0 6 5 0 14 | 0 0 0 1 0 }}


[[generator|Generators]]: 2, 6/5, 7
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1110, ~7/4 = 968.8259


EDOs: 53, 68, 72, 140, 159, 212, 284
{{Optimal ET sequence|legend=1| 15, 19, 34d, 49f, 53, 72, 121, 140, 333, 352f, 492f }}


Badness: 0.000798
Badness: 0.290 × 10<sup>-3</sup>


==13-limit==
== Enlil ==
[[Comma|Commas]]: 325/324, 385/384, 625/624
[[Subgroup]]: 2.3.5.7.11


13 and 15 limit minimax tuning
[[Comma list]]: 385/384, 6250/6237


[|1 0 0 0 0 0&gt;, |4/3 0 0 -2/9 -2/9 4/9&gt;,
{{Mapping|legend=1| 1 0 1 0 6 | 0 6 5 0 1 | 0 0 0 1 -1 }}
|19/9 0 0 -5/27 -5/27 10/27&gt;,
|28/9 0 0 13/27 -14/27 1/27&gt;,
|28/9 0 0 -14/27 13/27 1/27&gt;,
|28/9 0 0 -14/27 -14/27 28/27&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 13/11, 14/11
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 317.0667, ~7/4 = 967.6041


Map: [&lt;1 0 1 0 6 0|, &lt;0 6 5 0 1 14|, &lt;0 0 0 1 -1 0|]
[[Minimax tuning]]:
* [[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|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7


[[generator|Generators]]: 2, 6/5, 7
{{Optimal ET sequence|legend=1| 15, 19, 34, 38, 53, 68, 72, 140, 159, 212, 284, 496ce, 583cde, 655cde, 867ccdee, 939ccddee }}


[[EDO|Edos]]: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[53edo|53]], [[57edo|57]], [[68edo|68]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[212edo|212]], [[284edo|284]], [[299edo|299]]
[[Badness]]: 0.798 × 10<sup>-3</sup>


Badness: 0.000680
=== 13-limit (sumatra) ===
Subgroup: 2.3.5.7.11.13


Scales: [[enlil19_13|enlil19_13]]
Comma list: 325/324, 385/384, 625/624


=Enki=
Mapping: {{mapping| 1 0 1 0 6 0 | 0 6 5 0 1 14 | 0 0 0 1 -1 0 }}
[[Comma|Commas]]: 3025/3024, 4375/4356


[[Minimax_tuning|Minimax tuning]]
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1156, ~7/4 = 967.6235


[|1 0 0 0 0&gt;, |-18/13 0 6/13 -6/13 12/13&gt;,  
Minimax tuning:
|-2/13 0 5/13 -5/13 10/13&gt;,  
* 13- and 15-odd-limit
|-2/13 0 -8/13 8/13 10/13&gt;, |0 0 0 0 1&gt;]
: [{{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 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7.13/7


[[Eigenmonzo|Eigenmonzos]]: 2, 11/8, 7/5
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 511cdef }}


Map: [&lt;1 0 1 0 1|, &lt;0 6 5 0 4|, &lt;0 0 0 2 1|]
Badness: 0.680 × 10<sup>-3</sup>


[[generator|Generators]]: 2, 6/5, 66/25
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


[[EDO|Edos]]: [[15edo|15]], [[49edo|49]], [[57edo|57]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], [[231|231]], 265, 280, 337, 352, 439
Scales: [[enlil19_13]]


Badness: 0.00109
== Isimud ==
[[Subgroup]]: 2.3.5.7.11


==13-limit==
[[Comma list]]: 4125/4096, 8019/8000
[[Comma|Commas]]: 325/324, 364/363, 625/624


13 and 15 limit
{{Mapping|legend=1| 1 0 1 0 9 | 0 6 5 0 -21 | 0 0 0 1 0 }}


[|1 0 0 0 0 0&gt;, |2/3 0 -2/3 0 0 2/3&gt;,
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 316.7584, ~7/4 = 968.8259
|14/9 0 -5/9 0 0 5/9&gt;, |0 0 0 1 0 0&gt;,  
|13/9 0 -4/9 1/2 0 4/9&gt;,
|14/9 0 -14/9 0 0 14/9&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 8/7 13/10
{{Optimal ET sequence|legend=1| 19, 34e, 53, 72, 197e, 269ce, 341ce, 375cdee, 447cdee, 519bccddee }}


Map: [&lt;1 0 1 0 1 0|, &lt;0 6 5 0 4 14|, &lt;0 0 0 2 1 0|]
[[Badness]]: 3.690 × 10<sup>-3</sup>


[[generator|Generators]]: 2, 6/5, 66/25
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[EDO|Edos]]: [[15edo|15]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], 280, 439
Comma list: 325/324, 625/624, 1287/1280


Badness: 0.000874
Mapping: {{mapping| 1 0 1 0 9 0 | 0 6 5 0 -21 14 | 0 0 0 1 0 0 }}


=Isimud=
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 316.8446, ~7/4 = 968.8259
Commas: 4125/4096, 8019/8000


Map: [&lt;1 0 1 0 9|, &lt;0 6 5 0 -21|, &lt;0 0 0 1 0|]
{{Optimal ET sequence|legend=1| 19, 34e, 53, 72, 125f, 178def, 197ef, 231eef, 250deff, 303cdeeff }}


EDOs: 19, 53, 72, 197e, 269ce, 341ce, 375cde, 447cde, 519bcde
Badness: 2.273 × 10<sup>-3</sup>


Badness: 0.003690
== Enki ==
[[Subgroup]]: 2.3.5.7.11


==13-limit==
[[Comma list]]: 3025/3024, 4375/4356
Commas: 325/324, 625/624, 1287/1280


Map: [&lt;1 0 1 0 9 0|, &lt;0 6 5 0 -21 14|, &lt;0 0 0 1 0 0|]
{{Mapping|legend=1| 1 0 1 0 1 | 0 6 5 0 4 | 0 0 0 2 1 }}


EDOs: 19, 53, 72, 178def, 197ef, 231ef, 250def, 303cdef
: mapping generators: ~2, ~6/5, ~66/25


Badness: 0.002273
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 317.0396, ~33/25 = 484.2358


[[Category:Theory]]
[[Minimax tuning]]:
[[Category:Temperament family]]
* [[11-odd-limit]]
[[Category:Planar temperament]]
: [{{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 }}]
[[Category:Kleismic]]
: [[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 }}
 
[[Badness]]: 1.09 × 10<sup>-3</sup>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 625/624
 
Mapping: {{mapping| 1 0 1 0 1 0 | 0 6 5 0 4 14 | 0 0 0 2 1 0 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1387, ~33/25 = 483.3828
 
Minimax tuning:
* 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 }}]
: unchanged-interval (eigenmonzo) basis: 2.7.13/5
 
{{Optimal ET sequence|legend=1| 34d, 49f, 72, 87, 159, 193, 280 }}
 
Badness: 0.874 × 10<sup>-3</sup>
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Kleismic rank-3 family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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