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Porwell temperaments: Difference between revisions

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This family of temperaments tempers out the ''porwell comma'', {{monzo| 11 1 -3 -2 }} = [[6144/6125]], and includes hendecatonic, hemischis, twothirdtonic, nessafof, septisuperfourth, whoops, and polypyth.  
{{Technical data page}}
This is a collection of [[regular temperament|temperaments]] that [[tempering out|tempers out]] the porwell comma, {{monzo| 11 1 -3 -2 }} ([[6144/6125]]), and includes hendecatonic, hemischis, twothirdtonic, nessafof, septisuperfourth, whoops, and polypyth.  


Discussed elsewhere are:  
Discussed elsewhere are:  
* ''[[Hexadecimal]]'' (+36/35) → [[Pelogic family #Armodue|Pelogic family]]
* ''[[Armodue]]'' (+36/35) → [[Pelogic family #Armodue|Pelogic family]]
* [[Porcupine]] (+64/63), also in: [[Porcupine family #Porcupine|Porcupine family]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Porcupine|Porcupine family]]
* [[Mohajira]] (+81/80), also in: [[Meantone family #Mohajira|Meantone family]]
* [[Mohajira]] (+81/80) → [[Meantone family #Mohajira|Meantone family]]
* [[Valentine]] (+126/125), also in: [[Starling temperaments #Valentine|Starling temperaments]]
* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]]
* [[Orwell]] (+225/224), also in: [[Semicomma family #Orwell|Semicomma family]]
* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]
* [[Shrutar]] (+245/243), also in: [[Diaschismic family #Shrutar|Diaschismic family]]
* [[Shrutar]] (+245/243) → [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]]
* ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]]
* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* [[Amity]] (+4375/4374 or 5120/5103), also in: [[Amity family #Septimal amity|Amity family]] and [[Ragismic microtemperaments #Amity|Ragismic microtemperaments]]
* [[Amity]] (+4375/4374 or 5120/5103) → [[Amity family #Septimal amity|Amity family]]
* ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]]
* ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]]
* ''[[Grendel]]'' (+16875/16807) → [[Mirkwai clan #Grendel|Mirkwai clan]]
* ''[[Grendel]]'' (+16875/16807) → [[Mirkwai clan #Grendel|Mirkwai clan]]
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* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Trident]]'' (+14348907/14336000) → [[Tricot family #Trident|Tricot family]]
* ''[[Alphatrident]]'' (+14348907/14336000) → [[Alphatricot family #Alphatrident|Alphatricot family]]
* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]
* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]
* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]
* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]
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== Hendecatonic ==
== Hendecatonic ==
{{see also|11th-octave temperaments}}
The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]].
The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]].


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{{Mapping|legend=1| 11 0 43 -4 | 0 1 -1 2 }}
{{Mapping|legend=1| 11 0 43 -4 | 0 1 -1 2 }}


: mapping generators: ~16/15, ~3
: Mapping generators: ~16/15, ~3
 
{{Multival|legend=1| 11 -11 22 -43 4 82 }}


[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\11, ~3/2 = 703.054
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\11, ~3/2 = 703.054
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Comma list: 121/120, 176/175, 10976/10935
Comma list: 121/120, 176/175, 10976/10935


Mapping: {{mapping| 11 0 43 -4 38 | 0 1 -1 2 0 }}
{{Mapping|legend=1| 11 0 43 -4 38 | 0 1 -1 2 0 }}


Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.636
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.636


{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 176e, 275e }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176e, 275e }}


Badness: 0.046088
Badness: 0.046088
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Comma list: 121/120, 176/175, 351/350, 4459/4455
Comma list: 121/120, 176/175, 351/350, 4459/4455


Mapping: {{mapping| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}
{{Mapping|legend=1| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}


Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.291
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.291


{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 176e }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176e }}


Badness: 0.040099
Badness: 0.040099
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Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023
Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023


Mapping: {{mapping| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}
{{Mapping|legend=1| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}


Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301


{{Optimal ET sequence|legend=1| 22, 55, 77, 99, 176eg }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176eg }}


Badness: 0.029054
Badness: 0.029054
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Comma list: 540/539, 896/891, 4375/4356
Comma list: 540/539, 896/891, 4375/4356


Mapping: {{mapping| 11 0 43 -4 73 | 0 1 -1 2 -2 }}
{{Mapping|legend=1| 11 0 43 -4 73 | 0 1 -1 2 -2 }}


Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.686
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.686


{{Optimal ET sequence|legend=1| 22, 77e, 99e, 121, 220e }}
{{Optimal ET sequence|legend=0| 22, 77e, 99e, 121, 220e }}


Badness: 0.038042
Badness: 0.038042
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Comma list: 352/351, 364/363, 540/539, 625/624
Comma list: 352/351, 364/363, 540/539, 625/624


Mapping: {{mapping| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}
{{Mapping|legend=1| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}


Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.888
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.888


{{Optimal ET sequence|legend=1| 22, 77eff, 99ef, 121, 341bdeeff }}
{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 341bdeeff }}


Badness: 0.036112
Badness: 0.036112
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Comma list: 256/255, 352/351, 364/363, 375/374, 540/539
Comma list: 256/255, 352/351, 364/363, 375/374, 540/539


Mapping: {{mapping| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}
{{Mapping|legend=1| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}


Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.877
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.877


{{Optimal ET sequence|legend=1| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}
{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}


Badness: 0.022590
Badness: 0.022590
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Comma list: 3388/3375, 6144/6125, 9801/9800
Comma list: 3388/3375, 6144/6125, 9801/9800


Mapping: {{mapping| 22 0 86 -8 111 | 0 1 -1 2 -1 }}
{{Mapping|legend=1| 22 0 86 -8 111 | 0 1 -1 2 -1 }}


: mapping generators: ~33/32, ~3
: Mapping generators: ~33/32, ~3


Optimal tuning (POTE): ~33/32 = 1\22, ~3/2 = 702.914
Optimal tuning (POTE): ~33/32 = 1\22, ~3/2 = 702.914


{{Optimal ET sequence|legend=1| 22, 154, 176, 198 }}
{{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}


Badness: 0.057725
Badness: 0.057725
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{{Mapping|legend=1| 1 3 2 4 | 0 -13 3 -11 }}
{{Mapping|legend=1| 1 3 2 4 | 0 -13 3 -11 }}


: mapping generators: ~2, ~15/14
: Mapping generators: ~2, ~15/14
 
{{Multival|legend=1| 13 -3 11 -35 -19 34 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 130.401
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 130.401
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{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}
{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}


: mapping generators: ~2, ~8/7
: Mapping generators: ~2, ~8/7
 
{{Multival|legend=1| 19 7 -1 -33 -55 -22 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 226.4834
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 226.4834
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== Nessafof ==
== Nessafof ==
: ''For the 5-limit version, see [[High-badness temperaments #Nessafof]].''  
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments#Nessafof]].''  


Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's short post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101089.html Yahoo! Tuning Group | ''Some more unclassified temperaments'']</ref>, nessafof adds the [[landscape comma]] and has a third-octave period. The name actually refers to the fact that it has a neutral-second generator, and that a semi-augmented fourth, stacked 5 times, makes 5/1<ref name="petr's long post"/>.  
Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's short post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101089.html Yahoo! Tuning Group | ''Some more unclassified temperaments'']</ref>, nessafof adds the [[landscape comma]] and has a third-octave period. The name actually refers to the fact that it has a neutral-second generator, and that a semi-augmented fourth, stacked 5 times, makes 5/1<ref name="petr's long post"/>.  
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{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}
{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}


: mapping generators: ~63/50, ~35/32
: Mapping generators: ~63/50, ~35/32
 
{{Multival|legend=1| 21 15 -12 -25 -78 -70 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~35/32 = 157.480
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~35/32 = 157.480
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{{Mapping|legend=1| 1 6 -7 19 | 0 -9 19 -33 }}
{{Mapping|legend=1| 1 6 -7 19 | 0 -9 19 -33 }}


: mapping generators: ~2, ~45/32
: Mapping generators: ~2, ~45/32
 
{{Multival|legend=1| 9 -19 33 -51 27 130 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 588.782
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 588.782
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{{Mapping|legend=1| 1 17 14 -7 | 0 -33 -25 21 }}
{{Mapping|legend=1| 1 17 14 -7 | 0 -33 -25 21 }}


: mapping generators: ~2, ~441/320
: Mapping generators: ~2, ~441/320
 
{{Multival|legend=1| 33 25 -21 -37 -126 -119 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~441/320 = 560.519
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~441/320 = 560.519
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{{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}
{{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}


: mapping generators: ~2, ~3
: Mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.174
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.174
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{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}
{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}


: mapping generators: ~1323/1280, ~3
: Mapping generators: ~1323/1280, ~3
 
{{Multival|legend=1| 23 23 -23 -17 -101 -118 }}


[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 1\23, ~64/63 = 29.3586
[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 1\23, ~64/63 = 29.3586
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{{Mapping|legend=1| 1 4 3 3 | 0 -25 -7 -2 }}
{{Mapping|legend=1| 1 4 3 3 | 0 -25 -7 -2 }}


: mapping generators: ~2, ~343/320
: Mapping generators: ~2, ~343/320
 
{{Multival|legend=1| 25 7 2 -47 -67 -15 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/320 = 115.9169
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/320 = 115.9169
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{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}
{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}


: mapping generators: ~972/875, ~3
: Mapping generators: ~972/875, ~3


[[Optimal tuning]] ([[POTE]]): ~972/875 = 1\7, ~3/2 = 700.5854 (or ~10/9 = 186.2997)
[[Optimal tuning]] ([[POTE]]): ~972/875 = 1\7, ~3/2 = 700.5854 (or ~10/9 = 186.2997)
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Badness: 0.022291
Badness: 0.022291
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
{{Mapping|legend=1| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}
Optimal tuning ([[CTE]]): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
=== 29-limit ===
{{ See also | Fifth-chroma temperaments }}
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 261/260, 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
{{Mapping|legend=1| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}
Optimal tuning ([[CTE]]): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


== Dodifo ==
== Dodifo ==
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[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Porwell| ]] <!-- key article -->
[[Category:Porwell| ]] <!-- key article -->
[[Category:Hendecatonic]]
[[Category:Hendecatonic]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Porwell_temperaments"