Sengic family: Difference between revisions

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'''Sengic family''' is a collection of the [[planar temperament]]s tempering out the senga comma, [[686/675]].  
'''Sengic family''' is a collection of the [[planar temperament]]s tempering out the senga comma, [[686/675]].  


== Sengic ==
[[Comma]]: 686/675
[[Comma]]: 686/675


[[Map]]: {{lbrack}} {{val| 1 0 2 1 }}, {{val| 0 1 0 1 }}, {{val| 0 0 3 2 }} {{rbrack}}  
[[Mapping]]: {{lbrack}} {{val| 1 0 2 1 }}, {{val| 0 1 0 1 }}, {{val| 0 0 3 2 }} {{rbrack}}  


[[POTE tuning|POTE generators]]: 2, ~3/2 = 704.154, ~15/14 = 129.824
[[POTE tuning|POTE generators]]: 2, ~3/2 = 704.154, ~15/14 = 129.824


[[EDO]]s: {{Vals| 9, 10, 18, 19, 27, 46 }}
[[Optimal GPV sequence]]: {{Vals| 9, 10, 18, 19, 27, 46 }}


[[Badness]]: 0.000320
[[Badness]]: 0.000320


[[Projection pair]]s: 5 3375/686 7 675/98 to 2.3.7/5
[[Projection pair]]s: ~5 = 3375/686 ~7 = 675/98 to 2.3.7/5


== Demeter ==
== Demeter ==
Commas: 441/440, 686/675
Comma list: 441/440, 686/675


Map: {{lbrack}} {{val| 1 0 2 1 -3 }}, {{val| 0 1 0 1 4 }}, {{val| 0 0 3 2 1 }} {{rbrack}}
Mapping: {{lbrack}} {{val| 1 0 2 1 -3 }}, {{val| 0 1 0 1 4 }}, {{val| 0 0 3 2 1 }} {{rbrack}}


POTE generators: 2, ~3/2 = 705.518, ~15/14 = 130.039
POTE generators: 2, ~3/2 = 705.518, ~15/14 = 130.039


EDOs: {{Vals| 10, 17c, 19e, 27e, 29, 46 }}
Optimal GPV sequence: {{Vals| 10, 17c, 19e, 27e, 29, 46 }}


=== 13-limit  ===
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11
Commas: 91/90, 169/168, 352/351


Map: {{lbrack}} {{val| 1 0 2 1 -3 2 }}, {{val| 0 1 0 1 4 1 }}, {{val| 0 0 3 2 1 1 }} {{rbrack}}
=== 13-limit ===
Comma list: 91/90, 169/168, 352/351
 
Mapping: {{lbrack}} {{val| 1 0 2 1 -3 2 }}, {{val| 0 1 0 1 4 1 }}, {{val| 0 0 3 2 1 1 }} {{rbrack}}


POTE generators: 2, ~3/2 = 705.113, ~13/12 = 129.673
POTE generators: 2, ~3/2 = 705.113, ~13/12 = 129.673


EDOs: {{Vals| 10, 17c, 19e, 27e, 29, 46 }}
Optimal GPV sequence: {{Vals| 10, 17c, 19e, 27e, 29, 46 }}


=== 17-limit  ===
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11
Commas: 91/90, 136/135, 154/153, 169/168


Map: {{lbrack}} {{val| 1 0 2 1 -3 2 -1 }}, {{val| 0 1 0 1 4 1 3 }}, {{val| 0 0 3 2 1 1 3 }} {{rbrack}}
=== 17-limit ===
Comma list: 91/90, 136/135, 154/153, 169/168
 
Mapping: {{lbrack}} {{val| 1 0 2 1 -3 2 -1 }}, {{val| 0 1 0 1 4 1 3 }}, {{val| 0 0 3 2 1 1 3 }} {{rbrack}}


POTE generators: 2, ~3/2 = 705.147, ~13/12 = 129.700
POTE generators: 2, ~3/2 = 705.147, ~13/12 = 129.700


EDOs: {{Vals| 10, 17cg, 19eg, 27eg, 29g, 46 }}
Optimal GPV sequence: {{Vals| 10, 17cg, 19eg, 27eg, 29g, 46 }}
 
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11


== Krypton ==
== Krypton ==
Commas: 56/55, 540/539
Comma list: 56/55, 540/539


Map: {{lbrack}} {{val| 1 0 2 1 2 }}, {{val| 0 1 0 1 1 }}, {{val| 0 0 3 2 -1 }} {{rbrack}}
Mapping: {{lbrack}} {{val| 1 0 2 1 2 }}, {{val| 0 1 0 1 1 }}, {{val| 0 0 3 2 -1 }} {{rbrack}}


POTE generators: 2, ~3/2 = 705.978, ~12/11 = 132.544
POTE generators: 2, ~3/2 = 705.978, ~12/11 = 132.544


EDOs: {{Vals| 8d, 9, 10, 17c, 19, 27e, 36 }}
Optimal GPV sequence: {{Vals| 8d, 9, 10, 17c, 19, 27e, 36 }}


Badness: 0.000856
Badness: 0.000856


Projection pairs: 5 6912/1331 7 854/121 to 2.3.11
Projection pairs: ~5 = 6912/1331 ~7 = 854/121 to 2.3.11


=== 13-limit ===
=== 13-limit ===
Commas: 56/55, 78/77, 91/90
Comma list: 56/55, 78/77, 91/90


Map: {{lbrack}} {{val| 1 0 2 1 2 2 }}, {{val| 0 1 0 1 1 1 }}, {{val| 0 0 3 2 -1 1 }} {{rbrack}}
Mapping: {{lbrack}} {{val| 1 0 2 1 2 2 }}, {{val| 0 1 0 1 1 1 }}, {{val| 0 0 3 2 -1 1 }} {{rbrack}}


POTE generators: 2, ~3/2 = 706.029, ~12/11 = 132.428
POTE generators: 2, ~3/2 = 706.029, ~12/11 = 132.428


EDOs: {{Vals| 8d, 9, 10, 17c, 19, 27e, 36 }}
Optimal GPV sequence: {{Vals| 8d, 9, 10, 17c, 19, 27e, 36 }}


Badness: 0.000727
Badness: 0.000727


Projection pairs: 5 6912/1331 7 854/121 13 144/11 to 2.3.11
Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11


[[Category:Theory]]
[[Category:Theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Sengic]]
[[Category:Sengic]]

Revision as of 01:40, 5 February 2022

Sengic family is a collection of the planar temperaments tempering out the senga comma, 686/675.

Sengic

Comma: 686/675

Mapping: [ ⟨1 0 2 1], ⟨0 1 0 1], ⟨0 0 3 2] ]

POTE generators: 2, ~3/2 = 704.154, ~15/14 = 129.824

Optimal GPV sequence: 9, 10, 18, 19, 27, 46

Badness: 0.000320

Projection pairs: ~5 = 3375/686 ~7 = 675/98 to 2.3.7/5

Demeter

Comma list: 441/440, 686/675

Mapping: [ ⟨1 0 2 1 -3], ⟨0 1 0 1 4], ⟨0 0 3 2 1] ]

POTE generators: 2, ~3/2 = 705.518, ~15/14 = 130.039

Optimal GPV sequence: 10, 17c, 19e, 27e, 29, 46

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11

13-limit

Comma list: 91/90, 169/168, 352/351

Mapping: [ ⟨1 0 2 1 -3 2], ⟨0 1 0 1 4 1], ⟨0 0 3 2 1 1] ]

POTE generators: 2, ~3/2 = 705.113, ~13/12 = 129.673

Optimal GPV sequence: 10, 17c, 19e, 27e, 29, 46

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11

17-limit

Comma list: 91/90, 136/135, 154/153, 169/168

Mapping: [ ⟨1 0 2 1 -3 2 -1], ⟨0 1 0 1 4 1 3], ⟨0 0 3 2 1 1 3] ]

POTE generators: 2, ~3/2 = 705.147, ~13/12 = 129.700

Optimal GPV sequence: 10, 17cg, 19eg, 27eg, 29g, 46

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11

Krypton

Comma list: 56/55, 540/539

Mapping: [ ⟨1 0 2 1 2], ⟨0 1 0 1 1], ⟨0 0 3 2 -1] ]

POTE generators: 2, ~3/2 = 705.978, ~12/11 = 132.544

Optimal GPV sequence: 8d, 9, 10, 17c, 19, 27e, 36

Badness: 0.000856

Projection pairs: ~5 = 6912/1331 ~7 = 854/121 to 2.3.11

13-limit

Comma list: 56/55, 78/77, 91/90

Mapping: [ ⟨1 0 2 1 2 2], ⟨0 1 0 1 1 1], ⟨0 0 3 2 -1 1] ]

POTE generators: 2, ~3/2 = 706.029, ~12/11 = 132.428

Optimal GPV sequence: 8d, 9, 10, 17c, 19, 27e, 36

Badness: 0.000727

Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11

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