Sengic family: Difference between revisions

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== Sengic ==
== Sengic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[686/675]]
[[Comma list]]: [[686/675]]


[[Mapping]]: [{{val| 1 0 2 1 }}, {{val| 0 1 0 1 }}, {{val| 0 0 3 2 }}]
{{Mapping|legend=1| 1 0 2 1 | 0 1 0 1 | 0 0 3 2 }}


Mapping generators: ~2, ~3, ~15/14
: mapping generators: ~2, ~3, ~15/14


[[POTE generator]]s: ~3/2 = 704.154, ~15/14 = 129.824
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.154, ~15/14 = 129.824


{{Optimal ET sequence|legend=1| 9, 10, 18, 19, 27, 46 }}
{{Optimal ET sequence|legend=1| 9, 10, 18, 19, 27, 46 }}
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== Demeter ==
== Demeter ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 441/440, 686/675
[[Comma list]]: 441/440, 686/675


[[Mapping]]: [{{val| 1 0 2 1 -3 }}, {{val| 0 1 0 1 4 }}, {{val| 0 0 3 2 1 }}]
{{Mapping|legend=1| 1 0 2 1 -3 | 0 1 0 1 4 | 0 0 3 2 1 }}


[[POTE generator]]s: ~3/2 = 705.518, ~15/14 = 130.039
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 705.518, ~15/14 = 130.039


{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 29, 46 }}
{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 29, 46 }}
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Comma list: 91/90, 169/168, 352/351
Comma list: 91/90, 169/168, 352/351


Mapping: [{{val| 1 0 2 1 -3 2 }}, {{val| 0 1 0 1 4 1 }}, {{val| 0 0 3 2 1 1 }}]
Mapping: {{mapping| 1 0 2 1 -3 2 | 0 1 0 1 4 1 | 0 0 3 2 1 1 }}


POTE generators: ~3/2 = 705.113, ~14/13 = 129.673
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.113, ~14/13 = 129.673


{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 29, 46 }}
{{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 29, 46 }}
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Comma list: 91/90, 136/135, 154/153, 169/168
Comma list: 91/90, 136/135, 154/153, 169/168


Mapping: [{{val| 1 0 2 1 -3 2 -1 }}, {{val| 0 1 0 1 4 1 3 }}, {{val| 0 0 3 2 1 1 3 }}]
Mapping: {{mapping| 1 0 2 1 -3 2 -1 | 0 1 0 1 4 1 3 | 0 0 3 2 1 1 3 }}


POTE generators: ~3/2 = 705.147, ~14/13 = 129.700
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.147, ~14/13 = 129.700


{{Optimal ET sequence|legend=1| 10, 17cg, 19eg, 27eg, 29g, 46 }}
{{Optimal ET sequence|legend=1| 10, 17cg, 19eg, 27eg, 29g, 46 }}
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== Krypton ==
== Krypton ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 56/55, 540/539
[[Comma list]]: 56/55, 540/539


[[Mapping]]: [{{val| 1 0 2 1 2 }}, {{val| 0 1 0 1 1 }}, {{val| 0 0 3 2 -1 }}]
{{Mapping|legend=1| 1 0 2 1 2 | 0 1 0 1 1 | 0 0 3 2 -1 }}


[[POTE generator]]s: ~3/2 = 705.978, ~12/11 = 132.544
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 705.978, ~12/11 = 132.544


{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 36 }}
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 36 }}
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Comma list: 56/55, 78/77, 91/90
Comma list: 56/55, 78/77, 91/90


Mapping: [{{val| 1 0 2 1 2 2 }}, {{val| 0 1 0 1 1 1 }}, {{val| 0 0 3 2 -1 1 }}]
Mapping: {{mapping| 1 0 2 1 2 2 | 0 1 0 1 1 1 | 0 0 3 2 -1 1 }}


POTE generators: ~3/2 = 706.029, ~14/13 = 132.428
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 706.029, ~14/13 = 132.428


{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 36 }}
{{Optimal ET sequence|legend=1| 8d, 9, 10, 17c, 19, 27e, 36 }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Sengic family| ]] <!-- main article -->
[[Category:Sengic family| ]] <!-- main article -->
[[Category:Sengic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 13:59, 24 August 2023

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

Temperament discussed elsewhere include sensigh (→ Sensamagic family). Considered below are demeter and krypton.

Sengic

Subgroup: 2.3.5.7

Comma list: 686/675

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

mapping generators: ~2, ~3, ~15/14

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.154, ~15/14 = 129.824

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

Badness: 0.320 × 10-3

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

Demeter

Subgroup: 2.3.5.7.11

Comma list: 441/440, 686/675

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.518, ~15/14 = 130.039

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

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

13-limit

Subgroup: 2.3.5.7.11.13

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.113, ~14/13 = 129.673

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

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

17-limit

Subgroup: 2.3.5.7.11.13.17

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.147, ~14/13 = 129.700

Optimal ET 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

Subgroup: 2.3.5.7.11

Comma list: 56/55, 540/539

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.978, ~12/11 = 132.544

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

Badness: 0.856 × 10-3

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

13-limit

Subgroup: 2.3.5.7.11.13

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 706.029, ~14/13 = 132.428

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

Badness: 0.727 × 10-3

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

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