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Misty family: Difference between revisions

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The '''misty family''' tempers out the [[misty comma]], 67108864/66430125 = {{monzo| 26 -12 -3 }}.
{{Technical data page}}
The '''misty family''' tempers out the [[misty comma]], 67108864/66430125 = {{monzo| 26 -12 -3 }}, equating the [[Pythagorean comma]] with a stack of three [[schisma]]s, 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 lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
Line 8: Line 9:
{{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
: mapping generators: ~512/405, ~3


{{Val list|legend=1| 12, 51, 63, 75, 87, 99, 285, 384 }}
[[Optimal tuning]]s:
* [[CTE]]: ~512/405 = 1\3, ~3/2 = 703.2481
* [[POTE]]: ~512/405 = 1\3, ~3/2 = 703.111
 
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c }}


[[Badness]]: 0.106540
[[Badness]]: 0.106540
Line 23: Line 28:
{{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 }}
: mapping generators: ~63/50, ~3


[[POTE generator]]: ~3/2 = 703.021
Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448


{{Val list|legend=1| 12, 87, 99 }}
{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}


[[Badness]]: 0.036802
[[Badness]]: 0.036802


=== 11-limit ===
=== 2.3.5.7.17 subgroup ===
The [[S-expression]]-based comma list for this extension is {([[5120/5103|S8/S9]],) [[2025/2023|S15/S17]], [[256/255|S16]], [[1701/1700|S18/S20]](, [[2500/2499|S50]])}, and for the 2.3.5.7.17.19 subgroup extension, {([[5120/5103|S8/S9]],) [[256/255|S16]], [[324/323|S18]], [[400/399|S20]], [[1445/1444|S17/S19]](, [[2500/2499|S50]])}.
 
Subgroup: 2.3.5.7.17
 
Comma list: 256/255, 1701/1700, 2025/2023
 
Sval mapping: {{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: {{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: {{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: {{Optimal ET sequence| 12, 87, 99, 210gh, 309cdghh }}
 
Badness: 0.0132
 
=== Undecimal misty ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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 tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3127
* POTE: ~44/35 = 1\3, ~3/2 = 703.210


Vals: {{Val list| 12, 87, 99e, 186e }}
Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}


Badness: 0.035932
Badness: 0.035932
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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 tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3859
* POTE: ~44/35 = 1\3, ~3/2 = 703.303


Vals: {{Val list| 12f, 87, 99ef, 186ef, 273ef }}
Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}


Badness: 0.027820
Badness: 0.027820
Line 68: Line 109:
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 tunings:
* CTE: ~34/27 = 1\3, ~3/2 = 703.3686
* POTE: ~34/27 = 1\3, ~3/2 = 703.233


Vals: {{Val list| 12f, 87, 99ef, 186efg }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efg }}


Badness: 0.023908
Badness: 0.023908
Line 81: Line 124:
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 tunings:
* CTE: ~24/19 = 1\3, ~3/2 = 703.3626
* POTE: ~34/27 = 1\3, ~3/2 = 703.253


Vals: {{Val list| 12f, 87, 99ef, 186efgh }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efgh }}


Badness: 0.019878
Badness: 0.019878
Line 94: Line 139:
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 tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3285
* POTE: ~44/35 = 1\3, ~3/2 = 703.311


Vals: {{Val list| 12, 87, 186e, 273e }}
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 186e, 273e }}


Badness: 0.036872
Badness: 0.036872
Line 107: Line 154:
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 tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 703.0864
* POTE: ~55/49 = 1\6, ~3/2 = 702.9764


Vals: {{Val list| 12, 174e, 186, 198, 408cde, 606bcde }}
Optimal ET sequence: {{Optimal ET sequence| 12, 174e, 186, 198, 408cde, 606bcde }}


Badness: 0.0685
Badness: 0.068509


==== 13-limit ====
==== 13-limit ====
Line 122: Line 171:
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 tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 703.1045
* POTE: ~55/49 = 1\6, ~3/2 = 703.0087


Vals: {{Val list| 12f, 174ef, 186, 198 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 174ef, 186, 198 }}


Badness: 0.0441
Badness: 0.044073


=== Hemimist ===
=== Hemimist ===
Line 135: Line 186:
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 = 951.5721


Vals: {{Val list| 24d, 87, 111, 198 }}
Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}


Badness: 0.047439
Badness: 0.047439
Line 150: Line 201:
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 = 951.5810


Vals: {{Val list| 24d, 87, 111, 198 }}
Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}


Badness: 0.022923
Badness: 0.022923


== 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 0 26 37 | 0 1 -4 -6 }}


{{Multival|legend=1| 3 -12 -18 -26 -37 -8 }}
: mapping generators: ~80/63, ~3


[[POTE generator]]: ~3/2 = 704.118
Optimal tunings:
* CTE: ~80/63 = 1\3, ~3/2 = 704.3694
* POTE: ~80/63 = 1\3, ~3/2 = 704.118


{{Val list|legend=1| 12, 51, 63, 75 }}
{{Optimal ET sequence|legend=1| 12, 39c, 51, 63, 75 }}


[[Badness]]: 0.086866
[[Badness]]: 0.086866
Line 180: Line 233:
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 0 26 37 58 | 0 1 -4 -6 -10 }}
 
: mapping generators: ~44/35, ~3


POTE generator: ~3/2 = 704.411
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 704.6253
* POTE: ~44/35 = 1\3, ~3/2 = 704.411


Vals: {{Val list| 12, 51, 63, 138de }}
Optimal ET sequence: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}


Badness: 0.051348
Badness: 0.051348


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Misty family| ]] <!-- main article -->
[[Category:Misty family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Misty_family"