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Misty family: Difference between revisions

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Move the large chunk of subgroup extensions to the bottom
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{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}
{{Mapping|legend=1| 3 0 26 | 0 1 -4 }}
: mapping generators: ~512/405, ~3
: mapping generators: ~512/405, ~3


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~512/405 = 1\3, ~3/2 = 703.2481
* [[CTE]]: ~512/405 = 400.0000{{c}}, ~3/2 = 703.2481{{c}}
* [[POTE]]: ~512/405 = 1\3, ~3/2 = 703.111
* [[POTE]]: ~512/405 = 400.000{{c}}, ~3/2 = 703.111{{c}}


{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c }}
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c }}


[[Badness]]: 0.106540
[[Badness]] (Smith): 0.106540


=== Overview to extensions ===
=== Overview to extensions ===
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{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}
{{Mapping|legend=1| 3 0 26 56 | 0 1 -4 -10 }}


: mapping generators: ~63/50, ~3
[[Optimal tuning]] ([[CTE]]): ~63/50 = 400.0000{{c}}, ~3/2 = 703.1448{{c}}
 
Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448  


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


[[Badness]]: 0.036802
[[Badness]] (Smith): 0.036802


=== Undecimal misty ===
=== Undecimal misty ===
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Optimal tunings:
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3127
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3127{{c}}
* POTE: ~44/35 = 1\3, ~3/2 = 703.210
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.210{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}
{{Optimal ET sequence|legend=0| 12, …, 87, 99e, 186e }}


Badness: 0.035932
Badness (Smith): 0.035932


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3859
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3859{{c}}
* POTE: ~44/35 = 1\3, ~3/2 = 703.303
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.303{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}
{{Optimal ET sequence|legend=0| 12f, …, 87, 186ef }}


Badness: 0.027820
Badness (Smith): 0.027820


===== 17-limit =====
===== 17-limit =====
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Optimal tunings:
Optimal tunings:
* CTE: ~34/27 = 1\3, ~3/2 = 703.3686
* CTE: ~34/27 = 400.0000{{c}}, ~3/2 = 703.3686{{c}}
* POTE: ~34/27 = 1\3, ~3/2 = 703.233
* POTE: ~34/27 = 400.000{{c}}, ~3/2 = 703.233{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efg }}
{{Optimal ET sequence|legend=0| 12f, 87, 99ef, 186efg }}


Badness: 0.023908
Badness (Smith): 0.023908


===== 19-limit =====
===== 19-limit =====
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Optimal tunings:
Optimal tunings:
* CTE: ~24/19 = 1\3, ~3/2 = 703.3626
* CTE: ~24/19 = 400.0000{{c}}, ~3/2 = 703.3626{{c}}
* POTE: ~34/27 = 1\3, ~3/2 = 703.253
* POTE: ~34/27 = 400.000{{c}}, ~3/2 = 703.253{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efgh }}
{{Optimal ET sequence|legend=0| 12f, 87, 99ef, 186efgh }}


Badness: 0.019878
Badness (Smith): 0.019878


==== Mystic ====
==== Mystic ====
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Optimal tunings:
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3285
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3285{{c}}
* POTE: ~44/35 = 1\3, ~3/2 = 703.311
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 703.311{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 87, 186e, 273e }}
{{Optimal ET sequence|legend=0| 12, 87, 186e, 273e }}


Badness: 0.036872
Badness (Smith): 0.036872


=== Semimist ===
=== Semimist ===
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Mapping: {{mapping| 6 0 52 112 173 | 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


Optimal tunings:
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 703.0864
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0864{{c}}
* POTE: ~55/49 = 1\6, ~3/2 = 702.9764
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.9764{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 174e, 186, 198, 408cde, 606bcde }}
{{Optimal ET sequence|legend=0| 12, 174e, 186, 198, 408cde, 606bcde }}


Badness: 0.068509
Badness (Smith): 0.068509


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 703.1045
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.1045{{c}}
* POTE: ~55/49 = 1\6, ~3/2 = 703.0087
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 703.0087{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12f, 174ef, 186, 198 }}
{{Optimal ET sequence|legend=0| 12f, 174ef, 186, 198 }}


Badness: 0.044073
Badness (Smith): 0.044073


=== Hemimist ===
=== Hemimist ===
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Mapping: {{mapping| 3 0 26 56 8 | 0 2 -8 -20 1 }}
Mapping: {{mapping| 3 0 26 56 8 | 0 2 -8 -20 1 }}
: mapping generators: ~63/50, ~400/231
: mapping generators: ~63/50, ~400/231


Optimal tuning (POTE): ~63/50 = 1\3, ~400/231 = 951.5721
Optimal tuning (POTE): ~63/50 = 400.0000{{c}}, ~400/231 = 951.5721{{c}}


Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}
{{Optimal ET sequence|legend=0| 24d, 87, 111, 198 }}


Badness: 0.047439
Badness (Smith): 0.047439


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 3 0 26 56 8 23 | 0 2 -8 -20 1 -5 }}
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 tuning (POTE): ~63/50 = 400.0000{{c}}, ~26/15 = 951.5810{{c}}


Optimal ET sequence: {{Optimal ET sequence| 24d, 87, 111, 198 }}
{{Optimal ET sequence|legend=0| 24d, 87, 111, 198 }}


Badness: 0.022923
Badness (Smith): 0.022923


== Fog ==
== Fog ==
The fog temperament (12 & 63) tempers out the marvel comma, [[225/224]].
The fog temperament ({{nowrap| 12 & 63 }}) tempers out the marvel comma, [[225/224]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 3 0 26 37 | 0 1 -4 -6 }}
{{Mapping|legend=1| 3 0 26 37 | 0 1 -4 -6 }}
: mapping generators: ~80/63, ~3
: mapping generators: ~80/63, ~3


Optimal tunings:
Optimal tunings:
* CTE: ~80/63 = 1\3, ~3/2 = 704.3694
* CTE: ~80/63 = 400.0000{{c}}, ~3/2 = 704.3694{{c}}
* POTE: ~80/63 = 1\3, ~3/2 = 704.118
* POTE: ~80/63 = 400.000{{c}}, ~3/2 = 704.118{{c}}


{{Optimal ET sequence|legend=1| 12, 39c, 51, 63, 75 }}
{{Optimal ET sequence|legend=1| 12, 39c, 51, 63, 75 }}
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Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}
Mapping: {{mapping| 3 0 26 37 58 | 0 1 -4 -6 -10 }}
: mapping generators: ~44/35, ~3


Optimal tunings:
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 704.6253
* CTE: ~44/35 = 400.0000{{c}}, ~3/2 = 704.6253{{c}}
* POTE: ~44/35 = 1\3, ~3/2 = 704.411
* POTE: ~44/35 = 400.000{{c}}, ~3/2 = 704.411{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}
{{Optimal ET sequence|legend=0| 12, 39ce, 51, 63, 138de }}


Badness: 0.051348
Badness (Smith): 0.051348


== Subgroup extensions ==
== Subgroup extensions ==
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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]])}.  
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
[[Subgroup]]: 2.3.5.7.17


Comma list: 256/255, 1701/1700, 2025/2023
[[Comma list]]: 256/255, 1701/1700, 2025/2023


Subgroup-val mapping: {{mapping| 3 0 26 56 -2 | 0 1 -4 -10 3 }}
{{Mapping|legend=2| 3 0 26 56 -2 | 0 1 -4 -10 3 }}
: mapping generators: ~34/27, ~3
: mapping generators: ~34/27, ~3


Optimal tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.0963
[[Optimal tuning]] ([[CTE]]): ~34/27 = 400.0000{{c}}, ~3/2 = 703.0963{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}
{{Optimal ET sequence|legend=1| 12, 87, 99, 309cdg, 408cdgg, 507cdggg }}


Badness: 0.0186
[[Badness]] (Smith): 0.0186


==== 2.3.5.7.17.19 subgroup ====
==== 2.3.5.7.17.19 subgroup ====
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Subgroup-val mapping: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}
Subgroup-val mapping: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}


Optimal tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.0778  
Optimal tuning (CTE): ~24/19 = 400.0000{{c}}, ~3/2 = 703.0778{{c}}


Optimal ET sequence: {{Optimal ET sequence| 12, 87, 99, 210gh, 309cdghh }}
{{Optimal ET sequence|legend=0| 12, 87, 99, 210gh, 309cdghh }}


Badness: 0.0132
Badness (Smith): 0.0132


[[Category:Temperament families]]
[[Category:Temperament families]]
[[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"