Misty family
The misty family tempers out the misty comma, 67108864/66430125 = [26 -12 -3⟩, equating the Pythagorean comma with a stack of three schismas, making it a member of the schismic-Pythagorean equivalence continuum. It features a third-octave period, which is represented by 512/405 and acts as the interval separating ~128/81 from ~5/4.
The second comma of the normal comma list defines which 7-limit family member we are looking at.
- Adding 3136/3125 (hemimean) leads to septimal misty.
- Adding 225/224 (marvel) leads to fog.
Misty
- Main article: Misty
Subgroup: 2.3.5
Comma list: 67108864/66430125
Mapping: [⟨3 0 26], ⟨0 1 -4]]
- mapping generators: ~512/405, ~3
Optimal ET sequence: 12, 51, 63, 75, 87, 99, 285, 384, 483c
Badness: 0.106540
Septimal misty
- Main article: Misty
Subgroup: 2.3.5.7
Comma list: 3136/3125, 5120/5103
Mapping: [⟨3 0 26 56], ⟨0 1 -4 -10]]
- mapping generators: ~63/50, ~3
Wedgie: ⟨⟨3 -12 -30 -26 -56 -36]]
Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448
Optimal ET sequence: 12, …, 87, 99
Badness: 0.036802
2.3.5.7.17 subgroup
The S-expression-based comma list for this extension is {(S8/S9,) S15/S17, S16, S18/S20(, S50)}, and for the 2.3.5.7.17.19 subgroup extension, {(S8/S9,) S16, S18, S20, S17/S19(, S50)}.
Subgroup: 2.3.5.7.17
Comma list: 256/255, 1701/1700, 2025/2023
Sval mapping: [⟨3 0 26 56 -2], ⟨0 1 -4 -10 3]]
- sval mapping generators: ~34/27, ~3
Optimal tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.0963
Optimal ET sequence: 12, 87, 99, 309cdg, 408cdgg, 507cdggg
Badness: 0.0186
2.3.5.7.17.19 subgroup
Subgroup: 2.3.5.7.17.19
Comma list: 256/255, 324/323, 400/399, 476/475
Sval mapping: [⟨3 0 26 56 -2 8], ⟨0 1 -4 -10 3 1]]
- sval mapping generators: ~24/19, ~3
Optimal tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.0778
Optimal ET sequence: 12, 87, 99, 210gh, 309cdghh
Badness: 0.0132
Undecimal misty
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 4375/4356
Mapping: [⟨3 0 26 56 77], ⟨0 1 -4 -10 -14]]
Optimal tunings:
- CTE: ~44/35 = 1\3, ~3/2 = 703.3127
- POTE: ~44/35 = 1\3, ~3/2 = 703.210
Optimal ET sequence: 12, …, 87, 99e, 186e
Badness: 0.035932
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 364/363, 625/624
Mapping: [⟨3 0 26 56 77 92], ⟨0 1 -4 -10 -14 -17]]
Optimal tunings:
- CTE: ~44/35 = 1\3, ~3/2 = 703.3859
- POTE: ~44/35 = 1\3, ~3/2 = 703.303
Optimal ET sequence: 12f, …, 87, 186ef
Badness: 0.027820
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594
Mapping: [⟨3 0 26 56 77 92 -2], ⟨0 1 -4 -10 -14 -17 3]]
Optimal tunings:
- CTE: ~34/27 = 1\3, ~3/2 = 703.3686
- POTE: ~34/27 = 1\3, ~3/2 = 703.233
Optimal ET sequence: 12f, 87, 99ef, 186efg
Badness: 0.023908
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
Mapping: [⟨3 0 26 56 77 92 -2 8], ⟨0 1 -4 -10 -14 -17 3 1]]
Optimal tunings:
- CTE: ~24/19 = 1\3, ~3/2 = 703.3626
- POTE: ~34/27 = 1\3, ~3/2 = 703.253
Optimal ET sequence: 12f, 87, 99ef, 186efgh
Badness: 0.019878
Mystic
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 441/440, 896/891, 1001/1000
Mapping: [⟨3 0 26 56 77 -46], ⟨0 1 -4 -10 -14 12]]
Optimal tunings:
- CTE: ~44/35 = 1\3, ~3/2 = 703.3285
- POTE: ~44/35 = 1\3, ~3/2 = 703.311
Optimal ET sequence: 12, 87, 186e, 273e
Badness: 0.036872
Semimist
Subgroup: 2.3.5.7.11
Comma list: 3136/3125, 5120/5103, 9801/9800
Mapping: [⟨6 0 52 112 173], ⟨0 1 -4 -10 -16]]
- mapping generators: ~55/49, ~3
Optimal tunings:
- CTE: ~55/49 = 1\6, ~3/2 = 703.0864
- POTE: ~55/49 = 1\6, ~3/2 = 702.9764
Optimal ET sequence: 12, 174e, 186, 198, 408cde, 606bcde
Badness: 0.068509
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 847/845, 1716/1715, 3136/3125
Mapping: [⟨6 0 52 112 173 203], ⟨0 1 -4 -10 -16 -19]]
Optimal tunings:
- CTE: ~55/49 = 1\6, ~3/2 = 703.1045
- POTE: ~55/49 = 1\6, ~3/2 = 703.0087
Optimal ET sequence: 12f, 174ef, 186, 198
Badness: 0.044073
Hemimist
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 3136/3125, 5120/5103
Mapping: [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]]
- mapping generators: ~63/50, ~400/231
Optimal tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.5721
Optimal ET sequence: 24d, 87, 111, 198
Badness: 0.047439
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 676/675, 847/845, 3025/3024
Mapping: [⟨3 0 26 56 8 23], ⟨0 2 -8 -20 1 -5]]
Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 951.5810
Optimal ET sequence: 24d, 87, 111, 198
Badness: 0.022923
Fog
The fog temperament (12 & 63) tempers out the marvel comma, 225/224.
Subgroup: 2.3.5.7
Comma list: 225/224, 156250/151263
Mapping: [⟨3 0 26 37], ⟨0 1 -4 -6]]
Wedgie: ⟨⟨3 -12 -18 -26 -37 -8]]
- mapping generators: ~80/63, ~3
Optimal tunings:
- CTE: ~80/63 = 1\3, ~3/2 = 704.3694
- POTE: ~80/63 = 1\3, ~3/2 = 704.118
Optimal ET sequence: 12, 39c, 51, 63, 75
Badness: 0.086866
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 2420/2401
Mapping: [⟨3 0 26 37 58], ⟨0 1 -4 -6 -10]]
- mapping generators: ~44/35, ~3
Optimal tunings:
- CTE: ~44/35 = 1\3, ~3/2 = 704.6253
- POTE: ~44/35 = 1\3, ~3/2 = 704.411
Optimal ET sequence: 12, 39ce, 51, 63, 138de
Badness: 0.051348