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Misty family: Difference between revisions

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Normalization; POTE -> CTE; review optimal ET sequences
 
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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]].  
{{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.
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: mapping generators: ~512/405, ~3
: mapping generators: ~512/405, ~3


[[Optimal tuning]] ([[CTE]]): ~512/405 = 1\3, ~3/2 = 703.2481
[[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 }}
{{Optimal ET sequence|legend=1| 12, 51, 63, 75, 87, 99, 285, 384, 483c }}
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: mapping generators: ~63/50, ~3
: mapping generators: ~63/50, ~3


{{Multival|legend=1| 3 -12 -30 -26 -56 -36 }}
Optimal tuning (CTE): ~63/50 = 1\3, ~3/2 = 703.1448  
 
[[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 }}
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Comma list: 256/255, 1701/1700, 2025/2023
Comma list: 256/255, 1701/1700, 2025/2023


Sval mapping: {{mapping| 3 0 ​26 ​56 -2 | 0 ​1 -4 -10​ 3​ }}
Sval mapping: {{mapping| 3 0 26 56 -2 | 0 1 -4 -10 3 }}


: sval mapping generators: ~34/27, ~3
: sval mapping generators: ~34/27, ~3
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Comma list: 256/255, 324/323, 400/399, 476/475
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: {{mapping| 3 0 26 56 -2 8 | 0 1 -4 -10 3 1 }}


: sval mapping generators: ~24/19, ~3
: sval mapping generators: ~24/19, ~3
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Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}
Mapping: {{mapping| 3 0 26 56 77 | 0 1 -4 -10 -14 }}


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


Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}
Optimal ET sequence: {{Optimal ET sequence| 12, …, 87, 99e, 186e }}
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Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}
Mapping: {{mapping| 3 0 26 56 77 92 | 0 1 -4 -10 -14 -17 }}


Optimal tuning (CTE): ~44/35 = 1\3, ~3/2 = 703.3859
Optimal tunings:
* CTE: ~44/35 = 1\3, ~3/2 = 703.3859
* POTE: ~44/35 = 1\3, ~3/2 = 703.303


Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}
Optimal ET sequence: {{Optimal ET sequence| 12f, …, 87, 186ef }}
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Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}
Mapping: {{mapping| 3 0 26 56 77 92 -2 | 0 1 -4 -10 -14 -17 3 }}


Optimal tuning (CTE): ~34/27 = 1\3, ~3/2 = 703.3686
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efg }}
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Mapping: {{mapping| 3 0 26 56 77 92 -2 8 | 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 }}


Optimal tuning (CTE): ~24/19 = 1\3, ~3/2 = 703.3626
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 87, 99ef, 186efgh }}
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Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}
Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}


Optimal tuning (CTE): ~44/35 = 1\3, ~3/2 = 703.3285
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12, 87, 186e, 273e }}
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: mapping generators: ~55/49, ~3
: mapping generators: ~55/49, ~3


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 703.0864
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12, 174e, 186, 198, 408cde, 606bcde }}
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Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}
Mapping: {{mapping| 6 0 52 112 173 203 | 0 1 -4 -10 -16 -19 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 703.1045
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 174ef, 186, 198 }}
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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 }}
{{Multival|legend=1| 3 -12 -18 -26 -37 -8 }}


: mapping generators: ~80/63, ~3
: mapping generators: ~80/63, ~3


[[Optimal tuning]] ([[CTE]]): ~80/63 = 1\3, ~3/2 = 704.3694
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 }}
{{Optimal ET sequence|legend=1| 12, 39c, 51, 63, 75 }}
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: mapping generators: ~44/35, ~3
: mapping generators: ~44/35, ~3


Optimal tuning (CTE): ~44/35 = 1\3, ~3/2 = 704.6253
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 }}
Optimal ET sequence: {{Optimal ET sequence| 12, 39ce, 51, 63, 138de }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[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"