Sengic family: Difference between revisions
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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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, | {{Optimal ET sequence|legend=1| 9, 10, 17c, 19, 27, 46 }} | ||
[[Badness]]: 0.320 × 10<sup>-3</sup> | [[Badness]]: 0.320 × 10<sup>-3</sup> | ||
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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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, | {{Optimal ET sequence|legend=1| 10, 17c, 19e, 27e, 46, 102, 148 }} | ||
[[Badness]]: 1.32 × 10<sup>-3</sup> | |||
[[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11 | [[Projection pair]]s: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~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, 102, 148f }} | ||
Badness: 0.977 × 10<sup>-3</sup> | |||
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11 | Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11 | ||
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Optimal tuning (POTE): ~2 = 1\1, ~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, 102, 148f }} | ||
Badness: 0.830 × 10<sup>-3</sup> | |||
Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11 | Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11 | ||
Revision as of 14:07, 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
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, 17c, 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, 46, 102, 148
Badness: 1.32 × 10-3
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, 102, 148f
Badness: 0.977 × 10-3
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, 102, 148f
Badness: 0.830 × 10-3
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