Kleismic rank three family: Difference between revisions

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= Kleismic =
= Kleismic =
7 and 9 limit minimax


[|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>]
Subgroup: 2.3.5.7


[[Eigenmonzo]]s: 2, 7/6, 4/3
Comma list: 15625/15552


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


[[Generator]]s: 2, 6/5, 7
Mapping generators: ~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
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]
: [|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>]
: [[Eigenmonzo]]s: 2, 7/6, 4/3
 
{{Val list|legend=1| 15, 19, 23, 34, 53, 68, 72, 87, 121, 125, 140, 212, 333, 405 }}


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


[[Minimax_tuning|Minimax tuning]]
Subgroup: 2.3.5.7.11


[|1 0 0 0 0>, |0 1 0 0 0>, |1 5/6 0 0 0>,  
[[Comma list]]: 385/384, 6250/6237
|3 1/12 0 1/2 -1/2>, |3 1/12 0 -1/2 1/2>]


[[Eigenmonzo|Eigenmonzos]]: 2, 4/3, 14/11
Mapping: [<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|]
Mapping generators: ~2, ~6/5, ~7


[[Generator]]s: 2, 6/5, 7
[[Minimax tuning]]:
* [[11-odd-limit]]
: [|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>]
: [[Eigenmonzo]]s: 2, 4/3, 14/11


EDOs: 53, 68, 72, 140, 159, 212, 284
{{Val list|legend=1| 53, 68, 72, 140, 159, 212, 284 }}


Badness: 0.000798
Badness: 0.000798


==13-limit==
== 13-limit ==
[[Comma]]s: 325/324, 385/384, 625/624


13 and 15 limit minimax tuning
Subgroup: 2.3.5.7.11.13


[|1 0 0 0 0 0>, |4/3 0 0 -2/9 -2/9 4/9>,
Comma list: 325/324, 385/384, 625/624
|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>]


[[Eigenmonzo]]s: 2, 13/11, 14/11
Mapping: [<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|]
Mapping generators: 2, 6/5, 7


[[Generator]]s: 2, 6/5, 7
Minimax tuning
* 13- and 15-odd-limit
: [|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>]
: [[Eigenmonzo]]s: 2, 13/11, 14/11


[[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]]
Vals: {{Val list| 15, 19, 34, 53, 57, 68, 72, 87, 159, 212, 284, 299 }}


Badness: 0.000680
Badness: 0.000680
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= Enki =
= Enki =
[[Comma]]s: 3025/3024, 4375/4356


[[Minimax tuning]]
Subgroup: 2.3.5.7.11


[|1 0 0 0 0>, |-18/13 0 6/13 -6/13 12/13>,
[[Comma list]]:  3025/3024, 4375/4356
|-2/13 0 5/13 -5/13 10/13>,  
|-2/13 0 -8/13 8/13 10/13>, |0 0 0 0 1>]


[[Eigenmonzo]]s: 2, 11/8, 7/5
Mapping: [<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|]
Mapping generators: 2, 6/5, 66/25


[[Generator]]s: 2, 6/5, 66/25
[[Minimax tuning]]:
* [[11-odd-limit]]
: [|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>]
: [[Eigenmonzo]]s: 2, 11/8, 7/5


[[EDO|Edos]]: [[15edo|15]], [[49edo|49]], [[57edo|57]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], [[231|231]], 265, 280, 337, 352, 439  
{{Val list|legend=1| 15, 49, 57, 72, 87, 159, 193, 231, 265, 280, 337, 352, 439 }}


Badness: 0.00109
Badness: 0.00109


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


13 and 15 limit
Subgroup: 2.3.5.7.11.13


[|1 0 0 0 0 0>, |2/3 0 -2/3 0 0 2/3>,  
Comma list: 325/324, 364/363, 625/624
|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>]


[[Eigenmonzo]]s: 2, 8/7 13/10
Mapping: [<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|]
Mapping generators: 2, 6/5, 66/25


[[Generator]]s: 2, 6/5, 66/25
Minimax tuning:
* 13- and 15-odd-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


[[EDO|Edos]]: [[15edo|15]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[193edo|193]], 280, 439
Vals: {{Val list| 15, 72, 87, 159, 193, 280, 439 }}


Badness: 0.000874
Badness: 0.000874


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


Map: [<1 0 1 0 9|, <0 6 5 0 -21|, <0 0 0 1 0|]
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 4125/4096, 8019/8000
 
[[Mapping]]: [<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
{{Val list|legend=1| 19, 53, 72, 197e, 269ce, 341ce, 375cde, 447cde, 519bcde }}


Badness: 0.003690
[[Badness]]: 0.003690


== 13-limit ==
== 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|]
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 625/624, 1287/1280
 
Mapping: [<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
Vals: {{Val list| 19, 53, 72, 178def, 197ef, 231ef, 250def, 303cdef }}


Badness: 0.002273
Badness: 0.002273


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Kleismic]]
[[Category:Kleismic]]
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 11:40, 23 April 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

Subgroup: 2.3.5.7

Comma list: 15625/15552

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

Mapping generators: ~2, ~6/5, ~7

Minimax tuning:

[|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>]
Eigenmonzos: 2, 7/6, 4/3

Template:Val list

Enlil

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237

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

Mapping generators: ~2, ~6/5, ~7

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

Template:Val list

Badness: 0.000798

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624

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

Mapping generators: 2, 6/5, 7

Minimax tuning

  • 13- and 15-odd-limit
[|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

Vals: Template:Val list

Badness: 0.000680

Scales: enlil19_13

Enki

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4356

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

Mapping generators: 2, 6/5, 66/25

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

Template:Val list

Badness: 0.00109

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624

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

Mapping generators: 2, 6/5, 66/25

Minimax tuning:

  • 13- and 15-odd-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

Vals: Template:Val list

Badness: 0.000874

Isimud

Subgroup: 2.3.5.7.11

Comma list: 4125/4096, 8019/8000

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

Template:Val list

Badness: 0.003690

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1287/1280

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

Vals: Template:Val list

Badness: 0.002273

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