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Misty family: Difference between revisions

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m Marking this. In 17-limit a comma is missing. Edo list is also inconvincing
 
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__FORCETOC__
{{Technical data page}}
The misty family tempers out the [[misty comma]], 67108864/66430125 = |26 -12 -3>.
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]].  


=Misty=
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
Comma: 67108864/66430125 = |26 -12 -3>
* Adding 3136/3125 (hemimean) leads to septimal misty.
* Adding 225/224 (marvel) leads to fog.


POTE generator: ~3/2 = 703.111
== Misty ==
{{Main| Misty }}


Map: [<3 0 26|, <0 1 -4|]
[[Subgroup]]: 2.3.5


EDOs: 12, 51, 63, 75, 87, 99, 285, 384
[[Comma list]]: 67108864/66430125


Badness: 0.1065
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}


==7-limit==
: mapping generators: ~512/405, ~3
{{see also|Hemimean clan #Misty}}


Commas: 3136/3125, 5120/5103
[[Optimal tuning]]s:
* [[CTE]]: ~512/405 = 1\3, ~3/2 = 703.2481
* [[POTE]]: ~512/405 = 1\3, ~3/2 = 703.111


[[POTE_tuning|POTE generator]]: 703.021
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c }}


Map: [<3 0 26 56|, <0 1 -4 -10|]
[[Badness]]: 0.106540


Wedgie: <<3 -12 -30 -26 -56 -36||
== Septimal misty ==
{{Main| Misty }}


EDOs: [[12edo|12]], [[87edo|87]], [[99edo|99]]
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 3136/3125, 5120/5103
Commas: 441/440, 3136/3125, 4375/4356


[[POTE_tuning|POTE generator]]: 703.210
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


Map: [<3 0 26 56 77|, <0 1 -4 -10 -14|]
: mapping generators: ~63/50, ~3


EDOs: 12, 87, 99, 186
Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448


==13-limit==
{{Optimal ET sequence|legend=1| 12, …, 87, 99 }}
Commas: 196/195, 352/351, 364/363, 635/624


[[POTE_tuning|POTE generator]]: 703.303
[[Badness]]: 0.036802


Map: [<3 0 26 56 77 92|, <0 1 -4 -10 -14 -17|]
=== 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]])}.


EDOs: 12, 87, 273
Subgroup: 2.3.5.7.17


=Mystic=
Comma list: 256/255, 1701/1700, 2025/2023
Commas: 325/324, 441/440, 896/891, 1001/1000


[[POTE_tuning|POTE generator]]: 703.311
Sval mapping: {{mapping| 3 0 26 56 -2 | 0 1 -4 -10 3 }}


Map: [<3 0 26 56 77 -46|, <0 1 -4 -10 -14 12|]
: sval mapping generators: ~34/27, ~3


EDOs: 12, 87, 273
Optimal tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.0963


==17-limit==
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
Commas: 196/195, 352/351, 364/363, 635/624


[[POTE_tuning|POTE generator]]: 703.233
Badness: 0.0186


Map: [<3 0 26 56 77 92 -2|, <0 1 -4 -10 -14 -17 3|]
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19


EDOs: 12, 87, 186
Comma list: 256/255, 324/323, 400/399, 476/475


==19-limit==
Sval mapping: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}
Commas: 154/153, 196/195, 245/243, 273/272, 286/285, 325/324


[[POTE_tuning|POTE generator]]: 703.253
: sval mapping generators: ~24/19, ~3


Map: [<3 0 26 56 77 92 -2 8|, <0 1 -4 -10 -14 -17 3 1|]
Optimal tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.0778


EDOs: 12, 87, 186
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 210gh, 309cdghh }}


=Hemimist=
Badness: 0.0132
Commas: 3136/3125, 5120/5103, 3025/3024


POTE generator: ~11/8 = 551.527
=== Undecimal misty ===
Subgroup: 2.3.5.7.11


Map: [<3 0 26 56 8|, <0 2 -8 -20 1|]
Comma list: 441/440, 3136/3125, 4375/4356


EDOs: 15, 24, 87, 111, 198
Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}


Badness: 0.0474
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3127
* POTE: ~44/35 = 1\3, ~3/2 = 703.210


==13-limit==
Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}
Commas: 352/351, 676/675, 847/845, 3025/3024


POTE generator: ~11/8 = 551.525
Badness: 0.035932


Map: [<3 0 26 56 8 23|, <0 2 -8 -20 1 -5|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 15, 24, 87, 111, 198
Comma list: 196/195, 352/351, 364/363, 625/624


Badness: 0.0229
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}


[[Category:Theory]]
Optimal tunings:
[[Category:Temperament family]]
* CTE: ~44/35 = 1\3, ~3/2 = 703.3859
[[Category:Misty]]
* POTE: ~44/35 = 1\3, ~3/2 = 703.303


[[Category:Todo:review]]
Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}
[[Category:Todo:improve layout]]
 
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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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|legend=1| 3 0 26 37 | 0 1 -4 -6 }}
 
: 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|legend=1| 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: {{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: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}
 
Badness: 0.051348
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Misty family| ]] <!-- main article -->
[[Category:Rank 2]]
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