Sensamagic family
The sensamagic family of rank-3 temperaments tempers out 245/243. For a list of rank-2 temperaments, see Sensamagic clan.
Sensamagic
- Main article: Sensamagic
Subgroup: 2.3.5.7
Comma list: 245/243
Mapping: [⟨1 0 0 0], ⟨0 1 1 2], ⟨0 0 2 -1]]
- mapping generators: ~2, ~3, ~9/7
Mapping to lattice: [⟨0 1 1 2], ⟨0 0 2 -1]]
Lattice basis:
- 3/2 length = 0.9644, 9/7 length = 1.0807
- Angle (3/2, 9/7) = 86.5288°
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.7424, ~9/7 = 440.9020
- [[1 0 0 0⟩, [0 0 1/5 2/5⟩, [0 0 1 0⟩, [0 0 0 1⟩]
- eigenmonzo (unchanged-interval) basis: 2.5.7
- [[1 0 0 0⟩, [0 1 0 0⟩, [0 5/3 2/3 -2/3⟩, [0 5/3 -1/3 1/3⟩]
- eigenmonzo (unchanged-interval) basis: 2.3.7/5
Optimal ET sequence: 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283
Badness: 0.129 × 10-3
Projection pair: 5 243/49 to 2.3.7
2.3.7 subgroup
- 12: 729/686, 64/63
- 17: 64/63, 19683/19208
- 19: 49/48, 177147/175616
- 22: 64/63, 537824/531441
- 24: 64/63, 15059072/14348907
Overview to extensions
Temperaments discussed elsewhere include supernatural (→ Keemic family). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.
Undecimal sensamagic
- Main article: Sensamagic
Subgroup: 2.3.5.7.11
Comma list: 245/243, 385/384
Mapping: [⟨1 0 0 0 7], ⟨0 1 1 2 -2], ⟨0 0 2 -1 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.8004, ~9/7 = 440.9178
- [[1 0 0 0 0⟩, [21/13 6/13 -1/13 1/13 -3/13⟩, [35/13 10/13 7/13 -7/13 -5/13⟩, [35/13 10/13 -6/13 6/13 -5/13⟩, [42/13 -14/13 -2/13 2/13 7/13⟩]
- eigenmonzo (unchanged-interval) basis: 2.7/5.11/9
Optimal ET sequence: 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd
Badness: 0.722 × 10-3
Projection pairs: 5 243/49 11 896/81 to 2.3.7
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 245/243, 352/351, 364/363
Mapping: [⟨1 0 0 0 7 12], ⟨0 1 1 2 -2 -5], ⟨0 0 2 -1 -1 -1]]
Optimal ET sequence: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd
Badness: 1.20 × 10-3
Sensawer
Subgroup: 2.3.5.7.11
Comma list: 245/243, 441/440
Mapping: [⟨1 0 0 0 -3], ⟨0 1 1 2 5], ⟨0 0 2 -1 -4]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.1900, ~9/7 = 441.1359
Optimal ET sequence: 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd
Badness: 0.796 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 245/243, 352/351
Mapping: [⟨1 0 0 0 -3 2], ⟨0 1 1 2 5 2], ⟨0 0 2 -1 -4 -4]]
Optimal ET sequence: 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd
Badness: 0.928 × 10-3
Octarod
Subgroup: 2.3.5.7.11
Comma list: 100/99, 245/243
Mapping: [⟨1 0 0 0 2], ⟨0 1 1 2 0], ⟨0 0 2 -1 4]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.0464, ~9/7 = 439.5050
Optimal ET sequence: 8d, 14c, 19, 22, 27e, 41, 104, 131e
Badness: 0.581 × 10-3
Scales: octarod1, octarod2, octarod3, octarod4, octarod5
Shrusus
Subgroup: 2.3.5.7.11
Comma list: 176/175, 245/243
Mapping: [⟨1 0 0 0 -4], ⟨0 1 1 2 4], ⟨0 0 2 -1 3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 706.3702, ~9/7 = 442.1147
Optimal ET sequence: 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc
Badness: 0.877 × 10-3
Shrusic
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 176/175, 245/243
Mapping: [⟨1 0 0 0 -4 1], ⟨0 1 1 2 4 1], ⟨0 0 2 -1 3 3]]
Optimal ET sequence: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf
Badness: 1.125 × 10-3
Bisector
Subgroup: 2.3.5.7.11
Comma list: 121/120, 245/243
Mapping: [⟨2 0 0 0 3], ⟨0 1 1 2 1], ⟨0 0 2 -1 1]]
- mapping generators: ~77/54, ~3, ~9/7
Optimal tuning (POTE): ~77/54 = 1\2, ~3/2 = 703.0884, ~9/7 = 441.1060
Optimal ET sequence: 22, 46, 68, 82e, 106de, 114, 128e
Badness: 1.089 × 10-3
Sensigh
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 126/125, 169/168
Mapping: [⟨1 6 8 11 0 10], ⟨0 -7 -9 -13 0 -10], ⟨0 0 0 0 1 0]]
- mapping generators: ~2, ~9/7, ~11
Optimal ET sequence: 19, 27, 46, 111df, 157df
Badness: 0.939 × 10-3
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 126/125, 154/153, 169/168
Mapping: [⟨1 6 8 11 0 10 0], ⟨0 -7 -9 -13 0 -10 1], ⟨0 0 0 0 1 0 1]]