Kleismic rank three family: Difference between revisions

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__FORCETOC__
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.
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.


= Kleismic =
7 and 9 limit minimax
7 and 9 limit minimax


[|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>]
[|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>]


[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3
[[Eigenmonzo]]s: 2, 7/6, 4/3


Map: [<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|]


[[generator|Generators]]: 2, 6/5, 7
[[Generator]]s: 2, 6/5, 7


[[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
[[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


=Enlil=
= Enlil =
[[Comma|Commas]]: 385/384, 6250/6237
[[Comma|Commas]]: 385/384, 6250/6237


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Map: [<1 0 1 0 6|, <0 6 5 0 1|, <0 0 0 1 -1|]
Map: [<1 0 1 0 6|, <0 6 5 0 1|, <0 0 0 1 -1|]


[[generator|Generators]]: 2, 6/5, 7
[[Generator]]s: 2, 6/5, 7


EDOs: 53, 68, 72, 140, 159, 212, 284
EDOs: 53, 68, 72, 140, 159, 212, 284
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==13-limit==
==13-limit==
[[Comma|Commas]]: 325/324, 385/384, 625/624
[[Comma]]s: 325/324, 385/384, 625/624


13 and 15 limit minimax tuning
13 and 15 limit minimax tuning
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|28/9 0 0 -14/27 -14/27 28/27>]
|28/9 0 0 -14/27 -14/27 28/27>]


[[Eigenmonzo|Eigenmonzos]]: 2, 13/11, 14/11
[[Eigenmonzo]]s: 2, 13/11, 14/11


Map: [<1 0 1 0 6 0|, <0 6 5 0 1 14|, <0 0 0 1 -1 0|]
Map: [<1 0 1 0 6 0|, <0 6 5 0 1 14|, <0 0 0 1 -1 0|]


[[generator|Generators]]: 2, 6/5, 7
[[Generator]]s: 2, 6/5, 7


[[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]]
[[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]]
Line 53: Line 53:
Badness: 0.000680
Badness: 0.000680


Scales: [[enlil19_13|enlil19_13]]
Scales: [[enlil19_13]]


=Enki=
= Enki =
[[Comma|Commas]]: 3025/3024, 4375/4356
[[Comma]]s: 3025/3024, 4375/4356


[[Minimax_tuning|Minimax tuning]]
[[Minimax tuning]]


[|1 0 0 0 0>, |-18/13 0 6/13 -6/13 12/13>,  
[|1 0 0 0 0>, |-18/13 0 6/13 -6/13 12/13>,  
Line 64: Line 64:
|-2/13 0 -8/13 8/13 10/13>, |0 0 0 0 1>]
|-2/13 0 -8/13 8/13 10/13>, |0 0 0 0 1>]


[[Eigenmonzo|Eigenmonzos]]: 2, 11/8, 7/5
[[Eigenmonzo]]s: 2, 11/8, 7/5


Map: [<1 0 1 0 1|, <0 6 5 0 4|, <0 0 0 2 1|]
Map: [<1 0 1 0 1|, <0 6 5 0 4|, <0 0 0 2 1|]


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


[[EDO|Edos]]: [[15edo|15]], [[49edo|49]], [[57edo|57]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], [[231|231]], 265, 280, 337, 352, 439  
[[EDO|Edos]]: [[15edo|15]], [[49edo|49]], [[57edo|57]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], [[231|231]], 265, 280, 337, 352, 439  
Line 74: Line 74:
Badness: 0.00109
Badness: 0.00109


==13-limit==
== 13-limit ==
[[Comma|Commas]]: 325/324, 364/363, 625/624
[[Comma]]s: 325/324, 364/363, 625/624


13 and 15 limit
13 and 15 limit
Line 84: Line 84:
|14/9 0 -14/9 0 0 14/9>]
|14/9 0 -14/9 0 0 14/9>]


[[Eigenmonzo|Eigenmonzos]]: 2, 8/7 13/10
[[Eigenmonzo]]s: 2, 8/7 13/10


Map: [<1 0 1 0 1 0|, <0 6 5 0 4 14|, <0 0 0 2 1 0|]
Map: [<1 0 1 0 1 0|, <0 6 5 0 4 14|, <0 0 0 2 1 0|]


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


[[EDO|Edos]]: [[15edo|15]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], 280, 439
[[EDO|Edos]]: [[15edo|15]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], 280, 439
Line 94: Line 94:
Badness: 0.000874
Badness: 0.000874


=Isimud=
= Isimud =
Commas: 4125/4096, 8019/8000
Commas: 4125/4096, 8019/8000


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Badness: 0.003690
Badness: 0.003690


==13-limit==
== 13-limit ==
Commas: 325/324, 625/624, 1287/1280
Commas: 325/324, 625/624, 1287/1280


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[[Category:Theory]]
[[Category:Theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Planar temperament]]
[[Category:Kleismic]]
[[Category:Kleismic]]
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 05:53, 6 January 2021

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.

Kleismic

7 and 9 limit minimax

[|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>]

Eigenmonzos: 2, 7/6, 4/3

Map: [<1 0 1 0|, <0 6 5 0|, <0 0 0 1|]

Generators: 2, 6/5, 7

Edos: 15, 19, 23, 34, 53, 68, 72, 87, 121, 125, 140, 212, 333, 405

Enlil

Commas: 385/384, 6250/6237

Minimax tuning

[|1 0 0 0 0>, |0 1 0 0 0>, |1 5/6 0 0 0>, |3 1/12 0 1/2 -1/2>, |3 1/12 0 -1/2 1/2>]

Eigenmonzos: 2, 4/3, 14/11

Map: [<1 0 1 0 6|, <0 6 5 0 1|, <0 0 0 1 -1|]

Generators: 2, 6/5, 7

EDOs: 53, 68, 72, 140, 159, 212, 284

Badness: 0.000798

13-limit

Commas: 325/324, 385/384, 625/624

13 and 15 limit minimax tuning

[|1 0 0 0 0 0>, |4/3 0 0 -2/9 -2/9 4/9>, |19/9 0 0 -5/27 -5/27 10/27>, |28/9 0 0 13/27 -14/27 1/27>, |28/9 0 0 -14/27 13/27 1/27>, |28/9 0 0 -14/27 -14/27 28/27>]

Eigenmonzos: 2, 13/11, 14/11

Map: [<1 0 1 0 6 0|, <0 6 5 0 1 14|, <0 0 0 1 -1 0|]

Generators: 2, 6/5, 7

Edos: 15, 19, 34, 53, 57, 68, 72, 87, 159, 212, 284, 299

Badness: 0.000680

Scales: enlil19_13

Enki

Commas: 3025/3024, 4375/4356

Minimax tuning

[|1 0 0 0 0>, |-18/13 0 6/13 -6/13 12/13>, |-2/13 0 5/13 -5/13 10/13>, |-2/13 0 -8/13 8/13 10/13>, |0 0 0 0 1>]

Eigenmonzos: 2, 11/8, 7/5

Map: [<1 0 1 0 1|, <0 6 5 0 4|, <0 0 0 2 1|]

Generators: 2, 6/5, 66/25

Edos: 15, 49, 57, 72, 87, 159, 193, 231, 265, 280, 337, 352, 439

Badness: 0.00109

13-limit

Commas: 325/324, 364/363, 625/624

13 and 15 limit

[|1 0 0 0 0 0>, |2/3 0 -2/3 0 0 2/3>, |14/9 0 -5/9 0 0 5/9>, |0 0 0 1 0 0>, |13/9 0 -4/9 1/2 0 4/9>, |14/9 0 -14/9 0 0 14/9>]

Eigenmonzos: 2, 8/7 13/10

Map: [<1 0 1 0 1 0|, <0 6 5 0 4 14|, <0 0 0 2 1 0|]

Generators: 2, 6/5, 66/25

Edos: 15, 72, 87, 159, 193, 280, 439

Badness: 0.000874

Isimud

Commas: 4125/4096, 8019/8000

Map: [<1 0 1 0 9|, <0 6 5 0 -21|, <0 0 0 1 0|]

EDOs: 19, 53, 72, 197e, 269ce, 341ce, 375cde, 447cde, 519bcde

Badness: 0.003690

13-limit

Commas: 325/324, 625/624, 1287/1280

Map: [<1 0 1 0 9 0|, <0 6 5 0 -21 14|, <0 0 0 1 0 0|]

EDOs: 19, 53, 72, 178def, 197ef, 231ef, 250def, 303cdef

Badness: 0.002273

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