Sengic family
The sengic family of rank-3 temperaments tempers out the senga a.k.a. sengic comma, 686/675.
Temperament discussed elsewhere include sensigh (→ Sensamagic family). Considered below are demeter and krypton.
Sengic
Sengic is naturally a 2.3.5.7.13 subgroup temperament due to the identity 686/675 = (91/90)(196/195) and 91/90 = (169/168)(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by Keenan Pepper in 2011[1].
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 (CTE): ~2 = 1\1, ~3/2 = 703.7873, ~15/14 = 129.6451
Optimal ET sequence: 8d, 9, 10, 17c, 19, 27, 46
Badness: 0.320 × 10-3
Projection pairs: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5
2.3.5.7.13 subgroup
Subgroup: 2.3.5.7.13
Comma list: 91/90, 169/168
Sval mapping: [⟨1 0 2 1 2], ⟨0 1 0 1 1], ⟨0 0 3 2 1]]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 704.5918, ~14/13 = 129.7585
Optimal ET sequence: 8d, 9, 10, 17c, 19, 27, 46, 111df, 121df
Badness: 0.320 × 10-3
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