Misty family: Difference between revisions
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{{Multival|legend=1| 3 -12 -30 -26 -56 -36 }} | {{Multival|legend=1| 3 -12 -30 -26 -56 -36 }} | ||
[[Optimal tuning]] ([[ | [[Optimal tuning]] ([[CTE]]): ~63/50 = 1\3, ~3 = 1903.145 | ||
{{Optimal ET sequence|legend=1| 12, 87, 99 }} | {{Optimal ET sequence|legend=1| 12, 87, 99 }} | ||
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[[Badness]]: 0.036802 | [[Badness]]: 0.036802 | ||
=== | === 2.3.5.7.17 subgroup === | ||
[[Subgroup]]: 2.3.5.7.17 | |||
[[S-expression]]-based [[comma list]]: {([[5120/5103|S8/S9]],) [[2025/2023|S15/S17]], [[256/255|S16]], [[1701/1700|S18/S20]](, [[2500/2499|S50]])} | |||
[[Comma list]]: 3136/3125, 5120/5103, 256/255 | |||
{{Mapping|legend=2| 3 0 26 56 -2 | 0 1 -4 -10 3 }} | |||
[[Optimal tuning]] ([[CTE]]): ~63/50 = 1\3, ~3 = 1903.096 | |||
[[Patent val]] [[EDO]]s: 12, 87, 99, 111 | |||
==== 2.3.5.7.17.19 subgroup ==== | |||
[[Subgroup]]: 2.3.5.7.17.19 | |||
[[S-expression]]-based [[comma list]]: {([[5120/5103|S8/S9]],) [[256/255|S16]], [[324/323|S18]], [[400/399|S20]], [[1445/1444|S17/S19]](, [[2500/2499|S50]])} | |||
[[Comma list]]: 3136/3125, 5120/5103, 256/255, 324/323 | |||
{{Mapping|legend=2| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }} | |||
[[Optimal tuning]] ([[CTE]]): ~24/19 = 1\3, ~3 = 1903.078 | |||
[[Patent val]] [[EDO]]s: 12, 87, 99, 111 | |||
=== Undecimal misty === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Revision as of 19:44, 12 February 2024
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
Subgroup: 2.3.5
Comma list: 67108864/66430125
Mapping: [⟨3 0 26], ⟨0 1 -4]]
Optimal tuning (POTE): ~512/405 = 1\3, ~3/2 = 703.111
Optimal ET sequence: 12, 51, 63, 75, 87, 99, 285, 384
Badness: 0.106540
Septimal misty
Subgroup: 2.3.5.7
Comma list: 3136/3125, 5120/5103
Mapping: [⟨3 0 26 56], ⟨0 1 -4 -10]]
Wedgie: ⟨⟨ 3 -12 -30 -26 -56 -36 ]]
Optimal tuning (CTE): ~63/50 = 1\3, ~3 = 1903.145
Optimal ET sequence: 12, 87, 99
Badness: 0.036802
2.3.5.7.17 subgroup
Subgroup: 2.3.5.7.17
S-expression-based comma list: {(S8/S9,) S15/S17, S16, S18/S20(, S50)}
Comma list: 3136/3125, 5120/5103, 256/255
Subgroup-val mapping: [⟨3 0 26 56 -2], ⟨0 1 -4 -10 3]]
Optimal tuning (CTE): ~63/50 = 1\3, ~3 = 1903.096
Patent val EDOs: 12, 87, 99, 111
2.3.5.7.17.19 subgroup
Subgroup: 2.3.5.7.17.19
S-expression-based comma list: {(S8/S9,) S16, S18, S20, S17/S19(, S50)}
Comma list: 3136/3125, 5120/5103, 256/255, 324/323
Subgroup-val mapping: [⟨3 0 26 56 -2 8], ⟨0 1 -4 -10 3 1]]
Optimal tuning (CTE): ~24/19 = 1\3, ~3 = 1903.078
Patent val EDOs: 12, 87, 99, 111
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 tuning (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 tuning (POTE): ~44/35 = 1\3, ~3/2 = 703.303
Optimal ET sequence: 12f, 87, 99ef, 186ef, 273ef
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 tuning (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 tuning (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 tuning (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 tuning (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 tuning (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 = 951.527
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 = 951.525
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 5 6 7], ⟨0 -1 4 6]]
Wedgie: ⟨⟨ 3 -12 -18 -26 -37 -8 ]]
Optimal tuning (POTE): ~80/63 = 1\3, ~3/2 = 704.118
Optimal ET sequence: 12, 51, 63, 75
Badness: 0.086866
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 2420/2401
Mapping: [⟨3 5 6 7 8], ⟨0 -1 4 6 10]]
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 704.411
Optimal ET sequence: 12, 51, 63, 138de
Badness: 0.051348