Misty family
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The misty family tempers out the misty comma (monzo: [26 -12 -3⟩, ratio: 67108864/66430125).
Misty
Misty features a third-octave period, which represents the ptolemaic diminished fourth (512/405). Four perfect fourths (i.e. a minor sixth, ~128/81) minus a period give the interval class of 5. Misty has a ploidacot of triploid monocot, and equates the Pythagorean comma with a stack of three schismas, making it a member of the schismic–Pythagorean equivalence continuum with n = 3.
Subgroup: 2.3.5
Comma list: 67108864/66430125
Mapping: [⟨3 0 26], ⟨0 1 -4]]
- mapping generators: ~512/405, ~3
- WE: ~512/405 = 399.8595 ¢, ~3/2 = 702.8645 ¢
- error map: ⟨-0.421 +0.488 +0.261]
- CWE: ~512/405 = 400.0000 ¢, ~3/2 = 703.1489 ¢
- error map: ⟨0.000 +1.194 +1.091]
Optimal ET sequence: 12, 51, 63, 75, 87, 99, 285, 384, 483c, 867bc
Badness (Sintel): 2.50
Overview to extensions
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.
Septimal misty
7-limit
Subgroup: 2.3.5.7
Comma list: 3136/3125, 5120/5103
Mapping: [⟨3 0 26 56], ⟨0 1 -4 -10]]
- WE: ~63/50 = 399.8579 ¢, ~3/2 = 702.7714 ¢
- error map: ⟨-0.426 +0.390 +0.611 -0.236]
- CWE: ~63/50 = 400.0000 ¢, ~3/2 = 703.0551 ¢
- error map: ⟨0.000 +1.100 +1.466 +0.623]
Optimal ET sequence: 12, …, 87, 99
Badness (Sintel): 0.931
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
Subgroup-val mapping: [⟨3 0 26 56 -2], ⟨0 1 -4 -10 3]]
- mapping generators: ~34/27, ~3
Optimal tunings:
- WE: ~34/27 = 399.8023 ¢, ~3/2 = 702.6087 ¢
- CWE: ~34/27 = 400.0000 ¢, ~3/2 = 702.9837 ¢
Optimal ET sequence: 12, 87, 99, 309cdg, 408cdgg, 507cdggg
Badness (Sintel): 0.688
2.3.5.7.17.19 subgroup
Subgroup: 2.3.5.7.17.19
Comma list: 256/255, 324/323, 400/399, 476/475
Subgroup-val mapping: [⟨3 0 26 56 -2 8], ⟨0 1 -4 -10 3 1]]
Optimal tunings:
- WE: ~24/19 = 399.7556 ¢, ~3/2 = 702.4862 ¢
- CWE: ~24/19 = 400.0000 ¢, ~3/2 = 702.9418 ¢
Optimal ET sequence: 12, 87, 99, 210gh, 309cdghh
Badness (Sintel): 0.637
Undecimal misty
Subgroup: 2.3.5.7.11
Comma list: 441/440, 896/891, 3136/3125
Mapping: [⟨3 0 26 56 77], ⟨0 1 -4 -10 -14]]
Optimal tunings:
- WE: ~44/35 = 399.8727 ¢, ~3/2 = 702.9867 ¢
- CWE: ~44/35 = 400.0000 ¢, ~3/2 = 703.2385 ¢
Optimal ET sequence: 12, …, 87, 99e, 186e
Badness (Sintel): 1.19
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:
- WE: ~44/35 = 399.8888 ¢, ~3/2 = 703.1080 ¢
- CWE: ~44/35 = 400.0000 ¢, ~3/2 = 703.3258 ¢
Optimal ET sequence: 12f, …, 87, 186ef
Badness (Sintel): 1.15
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:
- WE: ~34/27 = 399.8046 ¢, ~3/2 = 702.8894 ¢
- CWE: ~34/27 = 400.0000 ¢, ~3/2 = 703.2597 ¢
Optimal ET sequence: 12f, 87, 99ef, 186efg
Badness (Sintel): 1.22
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:
- WE: ~24/19 = 399.7530 ¢, ~3/2 = 702.7601 ¢
- CWE: ~24/19 = 400.0000 ¢, ~3/2 = 703.2216 ¢
Optimal ET sequence: 12f, 87, 99ef, 186efgh
Badness (Sintel): 1.21
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:
- WE: ~44/35 = 399.9543 ¢, ~3/2 = 702.2310 ¢
- CWE: ~44/35 = 400.0000 ¢, ~3/2 = 703.3140 ¢
Optimal ET sequence: 12, 87, 186e, 273e
Badness (Sintel): 1.52
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:
- WE: ~55/49 = 199.9260 ¢, ~3/2 = 702.7162 ¢
- CWE: ~55/49 = 200.0000 ¢, ~3/2 = 703.0062 ¢
Optimal ET sequence: 12, 174e, 186, 198, 408cde, 606bcde
Badness (Sintel): 2.26
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:
- WE: ~55/49 = 199.9293 ¢, ~3/2 = 702.7602 ¢
- CWE: ~55/49 = 200.0000 ¢, ~3/2 = 703.0343 ¢
Optimal ET sequence: 12f, 174ef, 186, 198
Badness (Sintel): 1.82
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 tunings:
- WE: ~63/50 = 399.8922 ¢, ~400/231 = 951.2704 ¢
- CWE: ~63/50 = 400.0000 ¢, ~400/231 = 951.5363 ¢
Optimal ET sequence: 24d, 87, 111, 198
Badness (Sintel): 1.57
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 tunings:
- WE: ~63/50 = 399.8818 ¢, ~26/15 = 951.2442 ¢
- CWE: ~63/50 = 400.0000 ¢, ~26/15 = 951.5358 ¢
Optimal ET sequence: 24d, 87, 111, 198
Badness (Sintel): 0.947
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]]
- WE: ~80/63 = 399.7918 ¢, ~3/2 = 703.7511 ¢
- error map: ⟨-0.624 +1.172 -4.232 +4.712]
- CWE: ~80/63 = 400.0000 ¢, ~3/2 = 704.1865 ¢
- error map: ⟨0.000 +2.232 -3.060 +6.055]
Optimal ET sequence: 12, 39c, 51, 63, 75
Badness (Sintel): 2.20
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]]
Optimal tunings:
- WE: ~44/35 = 399.8078 ¢, ~3/2 = 704.0728 ¢
- CWE: ~44/35 = 400.0000 ¢, ~3/2 = 704.4697 ¢
Optimal ET sequence: 12, 39ce, 51, 63, 138de
Badness (Sintel): 1.70