Misty family: Difference between revisions

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{{Main| Misty }}
{{Main| Misty }}


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 67108864/66430125
[[Comma list]]: 67108864/66430125


[[Mapping]]: [{{val| 3 0 26 }}, {{val| 0 1 -4 }}]
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}


[[POTE generator]]: ~3/2 = 703.111
[[Optimal tuning]] ([[POTE]]): ~512/405 = 1\3, ~3/2 = 703.111


{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384 }}
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384 }}
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{{Main| Misty }}
{{Main| Misty }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 5120/5103
[[Comma list]]: 3136/3125, 5120/5103


[[Mapping]]: [{{val| 3 0 26 56 }}, {{val| 0 1 -4 -10 }}]
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


{{Multival|legend=1| 3 -12 -30 -26 -56 -36 }}
{{Multival|legend=1| 3 -12 -30 -26 -56 -36 }}


[[POTE generator]]: ~3/2 = 703.021
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~3/2 = 703.021


{{Optimal ET sequence|legend=1| 12, 87, 99 }}
{{Optimal ET sequence|legend=1| 12, 87, 99 }}
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Comma list: 441/440, 3136/3125, 4375/4356
Comma list: 441/440, 3136/3125, 4375/4356


Mapping: [{{val| 3 0 26 56 77 }}, {{val| 0 1 -4 -10 -14 }}]
Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}


POTE generator: ~3/2 = 703.210
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 703.210


{{Optimal ET sequence|legend=1| 12, 87, 99e, 186e }}
{{Optimal ET sequence|legend=1| 12, 87, 99e, 186e }}
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Comma list: 196/195, 352/351, 364/363, 625/624
Comma list: 196/195, 352/351, 364/363, 625/624


Mapping: [{{val| 3 0 26 56 77 92 }}, {{val| 0 1 -4 -10 -14 -17 }}]
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}


POTE generator: ~3/2 = 703.303
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 703.303


{{Optimal ET sequence|legend=1| 12f, 87, 99ef, 186ef, 273ef }}
{{Optimal ET sequence|legend=1| 12f, 87, 99ef, 186ef, 273ef }}
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Comma list: 196/195, 352/351, 364/363, 375/374, 595/594
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594


Mapping: [{{val| 3 0 26 56 77 92 -2 }}, {{val| 0 1 -4 -10 -14 -17 3 }}]
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}


POTE generator: ~3/2 = 703.233
Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 703.233


{{Optimal ET sequence|legend=1| 12f, 87, 99ef, 186efg }}
{{Optimal ET sequence|legend=1| 12f, 87, 99ef, 186efg }}
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Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594


Mapping: [{{val| 3 0 26 56 77 92 -2 8 }}, {{val| 0 1 -4 -10 -14 -17 3 1 }}]
Mapping: {{mapping| 3 0 26 56 77 92 -2 8 | 0 1 -4 -10 -14 -17 3 1 }}


POTE generator: ~3/2 = 703.253
Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 703.253


{{Optimal ET sequence|legend=1| 12f, 87, 99ef, 186efgh }}
{{Optimal ET sequence|legend=1| 12f, 87, 99ef, 186efgh }}
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Comma list: 325/324, 441/440, 896/891, 1001/1000
Comma list: 325/324, 441/440, 896/891, 1001/1000


Mapping: [{{val| 3 0 26 56 77 -46 }}, {{val| 0 1 -4 -10 -14 12 }}]
Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}


POTE generator: ~3/2 = 703.311
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 703.311


{{Optimal ET sequence|legend=1| 12, 87, 186e, 273e }}
{{Optimal ET sequence|legend=1| 12, 87, 186e, 273e }}
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Comma list: 3136/3125, 5120/5103, 9801/9800
Comma list: 3136/3125, 5120/5103, 9801/9800


Mapping: [{{val| 6 0 52 112 173 }}, {{val| 0 1 -4 -10 -16 }}]
Mapping: {{mapping| 6 0 52 112 173 | 0 1 -4 -10 -16 }}


Mapping generators: ~55/49, ~3
: mapping generators: ~55/49, ~3


POTE generator: ~3/2 = 702.9764
Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 702.9764


{{Optimal ET sequence|legend=1| 12, 174e, 186, 198, 408cde, 606bcde }}
{{Optimal ET sequence|legend=1| 12, 174e, 186, 198, 408cde, 606bcde }}
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Comma list: 352/351, 847/845, 1716/1715, 3136/3125
Comma list: 352/351, 847/845, 1716/1715, 3136/3125


Mapping: [{{val| 6 0 52 112 173 203 }}, {{val| 0 1 -4 -10 -16 -19 }}]
Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}


POTE generator: ~3/2 = 703.0087
Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 703.0087


{{Optimal ET sequence|legend=1| 12f, 174ef, 186, 198 }}
{{Optimal ET sequence|legend=1| 12f, 174ef, 186, 198 }}
Line 135: Line 135:
Comma list: 3025/3024, 3136/3125, 5120/5103
Comma list: 3025/3024, 3136/3125, 5120/5103


Mapping: [{{val| 3 0 26 56 8 }}, {{val| 0 2 -8 -20 1 }}]
Mapping: {{mapping| 3 0 26 56 8 | 0 2 -8 -20 1 }}


Mapping generators: ~63/50, ~693/400
: mapping generators: ~63/50, ~400/231


POTE generator: ~693/400 = 951.527
Optimal tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.527


{{Optimal ET sequence|legend=1| 24d, 87, 111, 198 }}
{{Optimal ET sequence|legend=1| 24d, 87, 111, 198 }}
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Comma list: 352/351, 676/675, 847/845, 3025/3024
Comma list: 352/351, 676/675, 847/845, 3025/3024


Mapping: [{{val| 3 0 26 56 8 23 }}, {{val| 0 2 -8 -20 1 -5 }}]
Mapping: {{mapping| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}


POTE generator: ~26/15 = 951.525
Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 951.525


{{Optimal ET sequence|legend=1| 24d, 87, 111, 198 }}
{{Optimal ET sequence|legend=1| 24d, 87, 111, 198 }}
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== Fog ==
== Fog ==
The ''fog'' temperament (12&63) tempers out the marvel comma, [[225/224]].
The fog temperament (12 & 63) tempers out the marvel comma, [[225/224]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 156250/151263
[[Comma list]]: 225/224, 156250/151263


[[Mapping]]: [{{val|3 5 6 7}}, {{val|0 -1 4 6}}]
{{Mapping|legend=1| 3 5 6 7 | 0 -1 4 6 }}


{{Multival|legend=1| 3 -12 -18 -26 -37 -8 }}
{{Multival|legend=1| 3 -12 -18 -26 -37 -8 }}


[[POTE generator]]: ~3/2 = 704.118
[[Optimal tuning]] ([[POTE]]): ~80/63 = 1\3, ~3/2 = 704.118


{{Optimal ET sequence|legend=1| 12, 51, 63, 75 }}
{{Optimal ET sequence|legend=1| 12, 51, 63, 75 }}
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Comma list: 100/99, 225/224, 2420/2401
Comma list: 100/99, 225/224, 2420/2401


Mapping: [{{val|3 5 6 7 8}}, {{val|0 -1 4 6 10}}]
Mapping: {{mapping| 3 5 6 7 8 | 0 -1 4 6 10 }}


POTE generator: ~3/2 = 704.411
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 704.411


{{Optimal ET sequence|legend=1| 12, 51, 63, 138de }}
{{Optimal ET sequence|legend=1| 12, 51, 63, 138de }}

Revision as of 08:39, 12 September 2023

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 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]]

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

Main article: 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 (POTE): ~63/50 = 1\3, ~3/2 = 703.021

Optimal ET sequence: 12, 87, 99

Badness: 0.036802

11-limit

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

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