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Porwell temperaments: Difference between revisions

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== Hendecatonic ==
== Hendecatonic ==
{{see also|11th-octave temperaments}}
: ''For the 5-limit version, see [[11th-octave temperaments #Hendecapent]].''


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
[[Optimal tuning]] ([[POTE]]): ~16/15 = 109.091{{c}}, ~3/2 = 703.054{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\11, ~3/2 = 703.054


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


[[Badness]]: 0.041081
[[Badness]] (Smith): 0.041081


=== 11-limit ===
=== 11-limit ===
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Comma list: 121/120, 176/175, 10976/10935
Comma list: 121/120, 176/175, 10976/10935


{{Mapping|legend=1| 11 0 43 -4 38 | 0 1 -1 2 0 }}
Mapping: {{mapping| 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 = 109.091{{c}}, ~3/2 = 702.636{{c}}


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


Badness: 0.046088
Badness (Smith): 0.046088


==== 13-limit ====
==== 13-limit ====
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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|legend=1| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}
Mapping: {{mapping| 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 = 109.091{{c}}, ~3/2 = 702.291{{c}}


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


Badness: 0.040099
Badness (Smith): 0.040099


==== 17-limit ====
==== 17-limit ====
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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|legend=1| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}
Mapping: {{mapping| 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 = 109.091{{c}}, ~3/2 = 702.301{{c}}


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


Badness: 0.029054
Badness (Smith): 0.029054


=== Cohendecatonic ===
=== Cohendecatonic ===
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Comma list: 540/539, 896/891, 4375/4356
Comma list: 540/539, 896/891, 4375/4356


{{Mapping|legend=1| 11 0 43 -4 73 | 0 1 -1 2 -2 }}
Mapping: {{mapping| 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 = 109.091{{c}}, ~3/2 = 703.686{{c}}


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


Badness: 0.038042
Badness (Smith): 0.038042


==== 13-limit ====
==== 13-limit ====
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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|legend=1| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}
Mapping: {{mapping| 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 = 109.091{{c}}, ~3/2 = 703.888{{c}}


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


Badness: 0.036112
Badness (Smith): 0.036112


==== 17-limit ====
==== 17-limit ====
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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|legend=1| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}
Mapping: {{mapping| 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 = 109.091{{c}}, ~3/2 = 703.877{{c}}


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


Badness: 0.022590
Badness (Smith): 0.022590


=== Icosidillic ===
=== Icosidillic ===
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Comma list: 3388/3375, 6144/6125, 9801/9800
Comma list: 3388/3375, 6144/6125, 9801/9800


{{Mapping|legend=1| 22 0 86 -8 111 | 0 1 -1 2 -1 }}
Mapping: {{mapping| 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 = 54.545{{c}}, ~3/2 = 702.914{{c}}
 
Optimal tuning (POTE): ~33/32 = 1\22, ~3/2 = 702.914


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


Badness: 0.057725
Badness (Smith): 0.057725


== Twothirdtonic ==
== Twothirdtonic ==
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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
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~15/14 = 130.401{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 130.401


{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}


[[Badness]]: 0.099601
[[Badness]] (Smith): 0.099601


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 3 2 4 4 | 0 -13 3 -11 -5 }}
Mapping: {{mapping| 1 3 2 4 4 | 0 -13 3 -11 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.430
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 130.430{{c}}


{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}


Badness: 0.040768
Badness (Smith): 0.040768


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 3 2 4 4 5 | 0 -13 3 -11 -5 -12 }}
Mapping: {{mapping| 1 3 2 4 4 5 | 0 -13 3 -11 -5 -12 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 130.409
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/13 = 130.409{{c}}


{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}


Badness: 0.025941
Badness (Smith): 0.025941


== Semaja ==
== Semaja ==
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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
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~8/7 = 226.4834{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 226.4834


{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}
{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}


[[Badness]]: 0.107023
[[Badness]] (Smith): 0.107023


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}
Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4856
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 226.4856{{c}}


{{Optimal ET sequence|legend=1| 16, 37, 53 }}
{{Optimal ET sequence|legend=0| 16, 37, 53 }}


Badness: 0.059838
Badness (Smith): 0.059838


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }}
Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4794
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 226.4794{{c}}


{{Optimal ET sequence|legend=1| 16, 37, 53 }}
{{Optimal ET sequence|legend=0| 16, 37, 53 }}


Badness: 0.032564
Badness (Smith): 0.032564


== Nessafof ==
== Nessafof ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit 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
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~35/32 = 157.480{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~35/32 = 157.480


{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}
{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}


[[Badness]]: 0.045048
[[Badness]] (Smith): 0.045048


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 3 2 5 10 8 | 0 7 5 -4 6 }}
Mapping: {{mapping| 3 2 5 10 8 | 0 7 5 -4 6 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~12/11 = 157.520
Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~12/11 = 157.520{{c}}


{{Optimal ET sequence|legend=1| 15, 54be, 69e, 84e, 99 }}
{{Optimal ET sequence|legend=0| 15, 54be, 69e, 84e, 99 }}


Badness: 0.068427
Badness (Smith): 0.068427


=== Nessa ===
=== Nessa ===
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Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}
Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.539
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~35/32 = 157.539{{c}}


{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99e }}
{{Optimal ET sequence|legend=0| 15, 54b, 69, 84, 99e }}


Badness: 0.048836
Badness (Smith): 0.048836


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 3 2 5 10 10 6 | 0 7 5 -4 1 13 }}
Mapping: {{mapping| 3 2 5 10 10 6 | 0 7 5 -4 1 13 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.429
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~35/32 = 157.429{{c}}


{{Optimal ET sequence|legend=1| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}
{{Optimal ET sequence|legend=0| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}


Badness: 0.037409
Badness (Smith): 0.037409


== Aufo ==
== Aufo ==
:''For the 5-limit version, see [[High badness temperaments #Untriton]].''  
:''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''  


Also named by [[Petr Pařízek]] in 2011, ''aufo'' refers to the augmented fourth, which is a generator of this temperament<ref name="petr's long post"/>.  
Also named by [[Petr Pařízek]] in 2011, ''aufo'' refers to the augmented fourth, which is a generator of this temperament<ref name="petr's long post"/>.  
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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
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~45/32 = 588.782{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 588.782


{{Optimal ET sequence|legend=1| 53, 161, 214 }}
{{Optimal ET sequence|legend=1| 53, 161, 214 }}


[[Badness]]: 0.121428
[[Badness]] (Smith): 0.121428


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 6 -7 19 1 | 0 -9 19 -33 5 }}
Mapping: {{mapping| 1 6 -7 19 1 | 0 -9 19 -33 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.811
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.811{{c}}


{{Optimal ET sequence|legend=1| 53, 108e, 161e }}
{{Optimal ET sequence|legend=0| 53, 108e, 161e }}


Badness: 0.088631
Badness (Smith): 0.088631


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 6 -7 19 1 -12 | 0 -9 19 -33 5 32 }}
Mapping: {{mapping| 1 6 -7 19 1 -12 | 0 -9 19 -33 5 32 }}


Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.788
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.788{{c}}


{{Optimal ET sequence|legend=1| 53, 108e, 161e, 214ee }}
{{Optimal ET sequence|legend=0| 53, 108e, 161e, 214ee }}


Badness: 0.058507
Badness (Smith): 0.058507


=== Aufic ===
=== Aufic ===
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Mapping: {{mapping| 1 6 -7 19 -25 | 0 -9 19 -33 58 }}
Mapping: {{mapping| 1 6 -7 19 -25 | 0 -9 19 -33 58 }}


Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.800
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.800{{c}}


{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375 }}
{{Optimal ET sequence|legend=0| 53, 108, 161, 214, 375 }}


Badness: 0.075149
Badness (Smith): 0.075149


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 6 -7 19 -25 -12 | 0 -9 19 -33 58 32 }}
Mapping: {{mapping| 1 6 -7 19 -25 -12 | 0 -9 19 -33 58 32 }}


Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.796
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.796{{c}}


{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375, 589be }}
{{Optimal ET sequence|legend=0| 53, 108, 161, 214, 375, 589be }}


Badness: 0.039050
Badness (Smith): 0.039050


== Whoops ==
== Whoops ==
:''For the 5-limit version, see [[Very high accuracy temperaments #Whoosh]].''  
: ''For the 5-limit version, see [[Very high accuracy temperaments #Whoosh]].''  


Also named by [[Petr Pařízek]] in 2011, ''whoops'' is a relatively simple extension to the otherwise very accurate microtemperament known as ''whoosh''<ref name="petr's long post"/>.  
Also named by [[Petr Pařízek]] in 2011, ''whoops'' is a relatively simple extension to the otherwise very accurate microtemperament known as ''whoosh''<ref name="petr's long post"/>.  
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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
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~441/320 = 560.519{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~441/320 = 560.519


{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608d, 623bd, 775bcd }}
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608d, 623bd, 775bcd }}


[[Badness]]: 0.175840
[[Badness]] (Smith): 0.175840


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 17 14 -7 10 | 0 -33 -25 21 -14 }}
Mapping: {{mapping| 1 17 14 -7 10 | 0 -33 -25 21 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~242/175 = 560.519
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~242/175 = 560.519{{c}}


{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
{{Optimal ET sequence|legend=0| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}


Badness: 0.043743
Badness (Smith): 0.043743


== Polypyth ==
== Polypyth ==
:''For the 5-limit version, see [[High badness temperaments #Leapday]].''  
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''  


Polypyth (46 &amp; 121) tempers out the same 5-limit comma as the [[Hemifamity temperaments #Leapday|leapday temperament]] (29 &amp; 46), but with the porwell (6144/6125) rather than the hemifamity (5120/5103) tempered out.
Polypyth (46 & 121) tempers out the same 5-limit comma as the [[leapday]] temperament (29 & 46), but with the porwell (6144/6125) rather than the hemifamity (5120/5103) tempered out.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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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 = 1200.000{{c}}, ~3/2 = 704.174{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.174


{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}


[[Badness]]: 0.137995
[[Badness]] (Smith): 0.137995


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}
Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.177
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.177{{c}}


{{Optimal ET sequence|legend=1| 46, 121, 167, 288be, 455bcde }}
{{Optimal ET sequence|legend=0| 46, 121, 167, 288be, 455bcde }}


Badness: 0.051131
Badness (Smith): 0.051131


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 0 -31 52 59 64 | 0 1 21 -31 -35 -38 }}
Mapping: {{mapping| 1 0 -31 52 59 64 | 0 1 21 -31 -35 -38 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.168
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.168{{c}}


{{Optimal ET sequence|legend=1| 46, 121, 167, 288be }}
{{Optimal ET sequence|legend=0| 46, 121, 167, 288be }}


Badness: 0.030292
Badness (Smith): 0.030292


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }}
Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.168
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.168{{c}}


{{Optimal ET sequence|legend=1| 46, 121, 167, 288beg }}
{{Optimal ET sequence|legend=0| 46, 121, 167, 288beg }}


Badness: 0.019051
Badness (Smith): 0.019051


== Icositritonic ==
== Icositritonic ==
{{ See also | 23rd-octave temperaments }}
{{See also| 23rd-octave temperaments }}
The icositritonic temperament (46 &amp; 161) has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]].
 
The icositritonic temperament (46 & 161) has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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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
[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 52.1739{{c}}, ~64/63 = 29.3586{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 1\23, ~64/63 = 29.3586


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}


[[Badness]]: 0.196622
[[Badness]] (Smith): 0.196622


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}
Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}


Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3980
Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3980{{c}}


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness: 0.064613
Badness (Smith): 0.064613


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}
Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}


Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2830
Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.2830{{c}}


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness: 0.040484
Badness (Smith): 0.040484


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}
Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}


Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2800
Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.2800{{c}}


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness: 0.024676
Badness (Smith): 0.024676


=== 19-limit ===
=== 19-limit ===
Line 504: Line 496:
Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}
Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}


Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3760
Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3760{{c}}


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness: 0.021579
Badness (Smith): 0.021579


=== 23-limit ===
=== 23-limit ===
Line 517: Line 509:
Mapping: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}
Mapping: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}


Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3471
Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3471{{c}}


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368ci }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368ci }}


Badness: 0.017745
Badness (Smith): 0.017745


== Absurdity ==
== Absurdity ==
: ''For the 5-limit version, see [[Syntonic–chromatic_equivalence_continuum#Absurdity_(5-limit)|Syntonic–chromatic equivalence continuum#Absurdity (5-limit)]].''
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Absurdity (5-limit)]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 531: Line 523:


{{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 = 171.4286{{c}}, ~3/2 = 700.5854{{c}} (or ~10/9 = 186.2997{{c}})
 
[[Optimal tuning]] ([[POTE]]): ~972/875 = 1\7, ~3/2 = 700.5854 (or ~10/9 = 186.2997)


{{Optimal ET sequence|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=1| 77, 84, 161 }}


[[Badness]]: 0.133520
[[Badness]] (Smith): 0.133520


=== 11-limit ===
=== 11-limit ===
Line 545: Line 536:
Comma list: 441/440, 6144/6125, 72171/71680
Comma list: 441/440, 6144/6125, 72171/71680


{{Mapping|legend=1| 7 0 -17 64 124 | 0 1 3 -4 -9 }}
Mapping: {{mapping| 7 0 -17 64 124 | 0 1 3 -4 -9 }}


Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 700.6354 (or ~10/9 = 186.3497)
Optimal tuning (POTE): ~495/448 = 171.4286{{c}}, ~3/2 = 700.6354{{c}} (or ~10/9 = 186.3497{{c}})


{{Optimal ET sequence|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


Badness: 0.081564
Badness (Smith): 0.081564


=== 13-limit ===
=== 13-limit ===
Line 558: Line 549:
Comma list: 351/350, 441/440, 1188/1183, 3584/3575
Comma list: 351/350, 441/440, 1188/1183, 3584/3575


{{Mapping|legend=1| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}
Mapping: {{mapping| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}


Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6291 (or ~10/9 = 186.3434)
Optimal tuning (POTE): ~72/65 = 171.4286{{c}}, ~3/2 = 700.6291{{c}} (or ~10/9 = 186.3434{{c}})


{{Optimal ET sequence|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


Badness: 0.041600
Badness (Smith): 0.041600


=== 17-limit ===
=== 17-limit ===
Line 571: Line 562:
Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625
Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625


{{Mapping|legend=1| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}
Mapping: {{mapping| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}


Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6524 (or ~10/9 = 186.3667)
Optimal tuning (POTE): ~72/65 = 171.4286{{c}}, ~3/2 = 700.6524{{c}} (or ~10/9 = 186.3667{{c}})


{{Optimal ET sequence|legend=1| 77, 161 }}
{{Optimal ET sequence|legend=0| 77, 161 }}


Badness: 0.031783
Badness (Smith): 0.031783


=== 19-limit ===
=== 19-limit ===
Line 584: Line 575:
Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494


{{Mapping|legend=1| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}


Optimal tuning (POTE): ~21/19 = 1\7, ~3/2 = 700.6565 (or ~10/9 = 186.3708)
Optimal tuning (POTE): ~21/19 = 171.4286{{c}}, ~3/2 = 700.6565{{c}} (or ~10/9 = 186.3708{{c}})


{{Optimal ET sequence|legend=1| 77, 161 }}
{{Optimal ET sequence|legend=0| 77, 161 }}


Badness: 0.022291
Badness (Smith): 0.022291


=== 23-limit ===
=== 23-limit ===
Line 597: Line 588:
Comma list: 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
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 }}
Mapping: {{mapping| 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 tuning ([[CTE]]): ~21/19 = 171.429{{c}}, ~3/2 = 700.629{{c}} (or ~10/9 = 186.343{{c}})


{{Optimal ET sequence|legend=0| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


=== 29-limit ===
=== 29-limit ===
{{ See also | Fifth-chroma temperaments }}
{{See also| Fifth-chroma temperaments }}
Subgroup: 2.3.5.7.11.13.17.19.23
 
Subgroup: 2.3.5.7.11.13.17.19.23.29


Comma list: 261/260, 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
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 }}
Mapping: {{mapping| 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 tuning ([[CTE]]): ~21/19 = 171.429{{c}}, ~3/2 = 700.629{{c}} (or ~10/9 = 186.343{{c}})


{{Optimal ET sequence|legend=0| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


== Dodifo ==
== Dodifo ==
: ''For the 5-limit version, see [[High badness temperaments #Dodifo]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Dodifo]].''


Also named by [[Petr Pařízek]] in 2011, ''dodifo'' refers to the (tetraptolemaic) double-diminished fourth, which is a generator of this temperament<ref name="petr's long post"/>. The extension here is a less accurate 7-limit intepretation.  
Also named by [[Petr Pařízek]] in 2011, ''dodifo'' refers to the (tetraptolemaic) double-diminished fourth, which is a generator of this temperament<ref name="petr's long post"/>. The extension here is a less accurate 7-limit intepretation.  
Line 626: Line 618:
{{Mapping|legend=1| 1 12 5 4 | 0 -35 -9 -4 }}
{{Mapping|legend=1| 1 12 5 4 | 0 -35 -9 -4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 357.070
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/40 = 357.070{{c}}


{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}
{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}


[[Badness]]: 0.179692
[[Badness]] (Smith): 0.179692


=== 11-limit ===
=== 11-limit ===
Line 639: Line 631:
Mapping: {{mapping| 1 12 5 4 -1 | 0 -35 -9 -4 15 }}
Mapping: {{mapping| 1 12 5 4 -1 | 0 -35 -9 -4 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.048
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 357.048{{c}}


{{Optimal ET sequence|legend=1| 37, 84, 121, 326dee }}
{{Optimal ET sequence|legend=0| 37, 84, 121, 326dee }}


Badness: 0.081923
Badness (Smith): 0.081923


=== 13-limit ===
=== 13-limit ===
Line 652: Line 644:
Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}
Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 357.049
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/13 = 357.049{{c}}


{{Optimal ET sequence|legend=1| 37, 84, 121, 326deef }}
{{Optimal ET sequence|legend=0| 37, 84, 121, 326deef }}


Badness: 0.039533
Badness (Smith): 0.039533


== Notes ==
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Hendecatonic]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Porwell_temperaments"