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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 [[Würschmidt family #Hemiwürschmidt|hemiwürschmidt]], [[Semicomma family #Orwell|orwell]], [[Amity family #Amity|amity]], [[Starling temperaments #Valentine|valentine]], [[Porcupine family #Porcupine|porcupine]], [[Diaschismic family #Shrutar|shrutar]], [[Pelogic family #Armodue|hexadecimal]], [[Mirkwai clan #Grendel|grendel]], [[Kleismic family #Hemikleismic|hemikleismic]], and [[Meantone family #Mohajira|mohajira]].
Discussed elsewhere are:
* ''[[Armodue]]'' (+36/35) → [[Pelogic family #Armodue|Pelogic family]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Porcupine|Porcupine family]]
* [[Mohajira]] (+81/80) → [[Meantone family #Mohajira|Meantone family]]
* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]]
* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]
* [[Shrutar]] (+245/243) → [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]]
* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* [[Amity]] (+4375/4374 or 5120/5103) → [[Amity family #Septimal amity|Amity family]]
* ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]]
* ''[[Grendel]]'' (+16875/16807) → [[Mirkwai clan #Grendel|Mirkwai clan]]
* ''[[Hemischis]]'' (+19683/19600) → [[Schismatic family #Hemischis|Schismatic family]]
* ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Alphatrident]]'' (+14348907/14336000) → [[Alphatricot family #Alphatrident|Alphatricot family]]
* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]
* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]
* ''[[Twilight]]'' (+{{monzo| 19 -22 2 4 }}) → [[Undim family #Twilight|Undim family]]


= Hendecatonic =
== Hendecatonic ==
The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represents [[9/7]].
{{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]].
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 10976/10935
[[Comma list]]: 6144/6125, 10976/10935


[[Mapping]]: [<11 0 43 -4], <0 1 -1 2|]
{{Mapping|legend=1| 11 0 43 -4 | 0 1 -1 2 }}


[[Wedgie]]: <<11 -11 22 -43 4 82||
: Mapping generators: ~16/15, ~3


[[POTE tuning|POTE generator]]: ~3/2 = 703.054
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\11, ~3/2 = 703.054


[[EDO|Vals]]: {{Val list| 22, 55, 77, 99 }}
{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}


[[Badness]]: 0.041081
[[Badness]]: 0.041081


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 10976/10935
Comma list: 121/120, 176/175, 10976/10935


Mapping: [<11 0 43 -4 38], <0 1 -1 2 0|]
{{Mapping|legend=1| 11 0 43 -4 38 | 0 1 -1 2 0 }}


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


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


Badness: 0.046088
Badness: 0.046088


== Icosidillic ==
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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 }}
 
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.291
 
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176e }}
 
Badness: 0.040099
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
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 }}
 
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301
 
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176eg }}
 
Badness: 0.029054
 
=== Cohendecatonic ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 896/891, 4375/4356
 
{{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 ET sequence|legend=0| 22, 77e, 99e, 121, 220e }}
 
Badness: 0.038042
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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 }}
 
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.888
 
{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 341bdeeff }}
 
Badness: 0.036112
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
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 }}
 
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.877
 
{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}
 
Badness: 0.022590
 
=== Icosidillic ===
Subgroup: 2.3.5.7.11
 
Comma list: 3388/3375, 6144/6125, 9801/9800
Comma list: 3388/3375, 6144/6125, 9801/9800


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


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


Vals: {{Val list| 22, 154, 176, 198 }}
{{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}


Badness: 0.057725
Badness: 0.057725


= Hemischis =
== Twothirdtonic ==
{{see also| Schismatic family }}
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 686/675, 6144/6125
 
{{Mapping|legend=1| 1 3 2 4 | 0 -13 3 -11 }}
 
: Mapping generators: ~2, ~15/14
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 130.401
 
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
 
[[Badness]]: 0.099601
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 686/675
 
Mapping: {{mapping| 1 3 2 4 4 | 0 -13 3 -11 -5 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.430
 
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
 
Badness: 0.040768
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 121/120, 169/168, 176/175
 
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 ET sequence|legend=1| 9, 28b, 37, 46 }}
 
Badness: 0.025941
 
== Semaja ==
Cryptically named by [[Petr Pařízek]] in 2011, semaja adds the [[gariboh comma]] to the comma list. The name actually refers to the fact that two of its ~8/7 generator steps reach a 13/10<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3125/3087, 6144/6125
 
{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}
 
: Mapping generators: ~2, ~8/7
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 226.4834
 
{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}
 
[[Badness]]: 0.107023
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 3125/3087
 
Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4856
 
{{Optimal ET sequence|legend=1| 16, 37, 53 }}
 
Badness: 0.059838
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 169/168, 176/175, 275/273
 
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 ET sequence|legend=1| 16, 37, 53 }}
 
Badness: 0.032564
 
== 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"/>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 250047/250000


Commas: 6144/6125, 19683/19600
{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}


POTE generator: ~81/70 = 249.203
: Mapping generators: ~63/50, ~35/32


Map: [&lt;1 0 15 -17|, &lt;0 2 -16 25|]
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~35/32 = 157.480


Wedgie: &lt;&lt;2 -16 25 -30 34 103||
{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}


EDOs: {{EDOs| 24, 53, 130, 183, 313 }}
[[Badness]]: 0.045048


Badness: 0.0458
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 121/120, 176/175, 250047/250000
Commas: 540/539, 8019/8000, 5632/5625


POTE generator: ~81/70 = 249.199
Mapping: {{mapping| 3 2 5 10 8 | 0 7 5 -4 6 }}


Map: [&lt;1 0 15 -17 51|, &lt;0 2 -16 25 -60|]
Optimal tuning (POTE): ~63/50 = 1\3, ~12/11 = 157.520


EDOs: {{EDOs| 24e, 53, 130, 183, 313 }}
{{Optimal ET sequence|legend=1| 15, 54be, 69e, 84e, 99 }}


Badness: 0.0363
Badness: 0.068427


== 13-limit ==
=== Nessa ===
Commas: 351/350, 540/539, 676/675, 4096/4095
Subgroup: 2.3.5.7.11


POTE generator: ~15/13 = 249.199
Comma list: 441/440, 1344/1331, 4375/4356


Map: [&lt;1 0 15 -17 51 14|, &lt;0 2 -16 25 -60 -13|]
Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}


EDOs: {{EDOs| 24e, 53, 130, 183, 313 }}
Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.539


Badness: 0.0208
{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99e }}


== 17-limit ==
Badness: 0.048836
Commas: 351/350, 442/441, 561/560, 676/675, 4096/4095


POTE generator: ~15/13 = 249.190
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 15 -17 51 14 -49|, &lt;0 2 -16 25 -60 -13 67|]
Comma list: 144/143, 364/363, 441/440, 625/624


EDOs: {{EDOs| 24egg, 53, 130, 183, 679df }}
Mapping: {{mapping| 3 2 5 10 10 6 | 0 7 5 -4 1 13 }}


Badness: 0.0211
Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.429


= Twothirdtonic =
{{Optimal ET sequence|legend=1| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}
Commas: 686/675, 6144/6125


POTE generator: ~15/14 = 130.401
Badness: 0.037409


Map: [&lt;1 3 2 4|, &lt;0 -13 3 -11|]
== Aufo ==
:''For the 5-limit version, see [[High badness temperaments #Untriton]].''


Wedgie: &lt;&lt;13 -3 11 -35 -19 34||
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"/>.


EDOs: {{EDOs| 9, 28b, 37, 46 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0996
[[Comma list]]: 6144/6125, 177147/175616


== 11-limit ==
{{Mapping|legend=1| 1 6 -7 19 | 0 -9 19 -33 }}
Commas: 121/120, 176/175, 686/675


POTE generator: ~15/14 = 130.430
: Mapping generators: ~2, ~45/32


Map: [&lt;1 3 2 4 4|, &lt;0 -13 3 -11 -5|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 588.782


EDOs: {{EDOs| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 53, 161, 214 }}


Badness: 0.0408
[[Badness]]: 0.121428


== 13-limit ==
=== 11-limit ===
Commas: 91/90, 121/120, 169/168, 176/175
Subgroup: 2.3.5.7.11


POTE generator: ~15/14 = 130.409
Comma list: 121/120, 176/175, 177147/175616


Map: [&lt;1 3 2 4 4 5|, &lt;0 -13 3 -11 -5 -12|]
Mapping: {{mapping| 1 6 -7 19 1 | 0 -9 19 -33 5 }}


EDOs: {{EDOs| 9, 28b, 37, 46 }}
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.811


Badness: 0.0259
{{Optimal ET sequence|legend=1| 53, 108e, 161e }}


= Nessafof =
Badness: 0.088631
Commas: 6144/6125, 250047/250000


POTE generator: ~35/32 = 157.420
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;3 2 5 10|, &lt;0 7 5 -4|]
Comma list: 121/120, 176/175, 351/350, 58806/57967


Wedgie: &lt;&lt;21 15 -12 -25 -78 -70||
Mapping: {{mapping| 1 6 -7 19 1 -12 | 0 -9 19 -33 5 32 }}


EDOs: {{EDOs| 15, 54b, 69, 84, 99, 282, 381 }}
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.788


Badness: 0.0450
{{Optimal ET sequence|legend=1| 53, 108e, 161e, 214ee }}


= Septisuperfourth =
Badness: 0.058507
Commas: 6144/6125, 118098/117649


POTE generator: ~48/35 = 544.680
=== Aufic ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 4 4 7|, &lt;0 -9 7 -15|]
Comma list: 540/539, 5632/5625, 72171/71680


Wedgie: &lt;&lt;18 -14 30 -64 -3 109||
Mapping: {{mapping| 1 6 -7 19 -25 | 0 -9 19 -33 58 }}


EDOs: {{EDOs| 22, 86, 108, 130, 152, 282 }}
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.800


Badness: 0.0592
{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375 }}


== 11-limit ==
Badness: 0.075149
Commas: 540/539, 4000/3993, 5632/5625


POTE generator: ~48/35 = 544.696
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 4 4 7 6|, &lt;0 -9 7 -15 10|]
Comma list: 351/350, 540/539, 847/845, 4096/4095


EDOs: {{EDOs| 22, 86, 108, 130, 152, 282, 434de, 716de }}
Mapping: {{mapping| 1 6 -7 19 -25 -12 | 0 -9 19 -33 58 32 }}


Badness: 0.0246
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.796


=== 13-limit ===
{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375, 589be }}
Commas: 540/539, 729/728, 4000/3993, 21168/21125


POTE generator: ~48/35 = 544.675
Badness: 0.039050


Map: [&lt;2 4 4 7 6 11|, &lt;0 -9 7 -15 10 -39|]
== Whoops ==
:''For the 5-limit version, see [[Very high accuracy temperaments #Whoosh]].''


EDOs: {{EDOs| 22f, 108f, 130, 282 }}
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"/>.


Badness: 0.0229
[[Subgroup]]: 2.3.5.7


=== Septisuperquad ===
[[Comma list]]: 6144/6125, 244140625/243045684
Commas: 351/350, 364/363, 540/539, 5632/5625


POTE generator: ~48/35 = 544.641
{{Mapping|legend=1| 1 17 14 -7 | 0 -33 -25 21 }}


Map: [&lt;2 4 4 7 6 5|, &lt;0 -9 7 -15 10 26|]
: Mapping generators: ~2, ~441/320


EDOs: {{EDOs| 22, 86f, 108, 130 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~441/320 = 560.519


Badness: 0.0330
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608d, 623bd, 775bcd }}


= Whoops =
[[Badness]]: 0.175840
{{see also| Very high accuracy temperaments #Whoosh }}


Commas: 6144/6125, 244140625/243045684
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~441/320 = 560.519
Comma list: 3025/3024, 4000/3993, 6144/6125


Map: [&lt;1 17 14 -7|, &lt;0 -33 -25 21|]
Mapping: {{mapping| 1 17 14 -7 10 | 0 -33 -25 21 -14 }}


Wedgie: &lt;&lt;33 25 -21 -37 -126 -119||
Optimal tuning (POTE): ~2 = 1\1, ~242/175 = 560.519


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


Badness: 0.1758
Badness: 0.043743


== 11-limit ==
== Polypyth ==
Commas: 6144/6125, 3025/3024, 4000/3993
:''For the 5-limit version, see [[High badness temperaments #Leapday]].''


POTE generator: ~242/175 = 560.519
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.


Map: [&lt;1 17 14 -7 10|, &lt;0 -33 -25 21 -14|]
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
[[Comma list]]: 6144/6125, 179200/177147


Badness: 0.0437
{{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}


= Polypyth =
: Mapping generators: ~2, ~3
Commas: 6144/6125, 179200/177147


POTE generator: ~3/2 = 704.174
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.174


Map: [&lt;1 0 -31 52|, &lt;0 1 21 -31|]
{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}


EDOs: {{Val list| 46, 121, 167, 288b, 455bcd, 743bcd }}
[[Badness]]: 0.137995


Badness: 0.137995
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 896/891, 2200/2187, 6144/6125
Commas: 896/891, 2200/2187, 6144/6125


POTE generator: ~3/2 = 704.177
Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}


Map: [&lt;1 0 -31 52 59|, &lt;0 1 21 -31 -35|]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.177


EDOs: {{Val list| 46, 121, 167, 288be, 455bcde }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288be, 455bcde }}


Badness: 0.051131
Badness: 0.051131


== 13-limit ==
=== 13-limit ===
Commas: 325/324, 352/351, 364/363, 1716/1715
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 704.168
Comma list: 325/324, 352/351, 364/363, 1716/1715


Map: [&lt;1 0 -31 52 59 64|, &lt;0 1 21 -31 -35 -38|]
Mapping: {{mapping| 1 0 -31 52 59 64 | 0 1 21 -31 -35 -38 }}


EDOs: {{Val list| 46, 121, 167, 288be }}
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.168
 
{{Optimal ET sequence|legend=1| 46, 121, 167, 288be }}


Badness: 0.030292
Badness: 0.030292


== 17-limit ==
=== 17-limit ===
Commas: 256/255, 325/324, 352/351, 364/363, 1716/1715
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~3/2 = 704.168
Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715


Map: [&lt;1 0 -31 52 59 64 39|, &lt;0 1 21 -31 -35 -38 -22|]
Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }}


EDOs: {{Val list| 46, 121, 167, 288beg }}
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.168
 
{{Optimal ET sequence|legend=1| 46, 121, 167, 288beg }}


Badness: 0.019051
Badness: 0.019051


= Icositritonic =
== Icositritonic ==
The ''icositritonic'' temperament (46&amp;161, named by [[User:Xenllium|Xenllium]]) has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]].
{{ 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]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 9920232/9765625
 
{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}


Commas: 6144/6125, 9920232/9765625
: Mapping generators: ~1323/1280, ~3


Map: [&lt;23 37 54 64|, &lt;0 -1 -1 1|]
[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 1\23, ~64/63 = 29.3586


POTE generator: ~64/63 = 29.3586
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}


EDOs: 23, 46, 69, 115, 161, 207
[[Badness]]: 0.196622


Badness: 0.196622
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 441/440, 6144/6125, 35937/35840
Commas: 441/440, 6144/6125, 35937/35840


Map: [&lt;23 37 54 64 79|, &lt;0 -1 -1 1 1|]
Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}


POTE generator: ~64/63 = 29.3980
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3980


EDOs: 23, 46, 69, 115, 161, 207
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}


Badness: 0.064613
Badness: 0.064613


== 13-limit ==
=== 13-limit ===
Commas: 351/350, 441/440, 847/845, 3584/3575
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 441/440, 847/845, 3584/3575


Map: [&lt;23 37 54 64 79 84|, &lt;0 -1 -1 1 1 2|]
Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}


POTE generator: ~64/63 = 29.2830
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2830


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


Badness: 0.040484
Badness: 0.040484


== 17-limit ==
=== 17-limit ===
Commas: 351/350, 441/440, 561/560, 847/845, 1089/1088
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;23 37 54 64 79 84 94|, &lt;0 -1 -1 1 1 2 0|]
Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088


POTE generator: ~64/63 = 29.2800
Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}


EDOs: 46, 115, 161, 207
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2800
 
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}


Badness: 0.024676
Badness: 0.024676


== 19-limit ==
=== 19-limit ===
Commas: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;23 37 54 64 79 84 94 96|, &lt;0 -1 -1 1 1 2 0 3|]
Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845


POTE generator: ~64/63 = 29.3760
Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}


EDOs: 46, 115, 161, 207
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3760
 
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}


Badness: 0.021579
Badness: 0.021579


== 23-limit ==
=== 23-limit ===
Commas: 276/275, 351/350, 391/390, 441/440, 456/455, 476/475, 847/845
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 276/275, 351/350, 391/390, 441/440, 456/455, 476/475, 847/845


Map: [&lt;23 37 54 64 79 84 94 96 104|, &lt;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 }}


POTE generator: ~64/63 = 29.3471
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3471


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


Badness: 0.017745
Badness: 0.017745


[[Category:Theory]]
== Countermiracle ==
[[Category:Temperament]]
The ''countermiracle'' temperament (31 &amp; 145) tempers out the trimyna, 50421/50000 and the [[quince comma]], 823543/819200.
[[Category:Porwell]]
 
[[Category:Hemischis]]
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 50421/50000
 
{{Mapping|legend=1| 1 4 3 3 | 0 -25 -7 -2 }}
 
: Mapping generators: ~2, ~343/320
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/320 = 115.9169
 
{{Optimal ET sequence|legend=1| 31, 114, 145, 176, 559cc, 735cc }}
 
[[Badness]]: 0.102326
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3388/3375, 6144/6125
 
Mapping: {{mapping| 1 4 3 3 8 | 0 -25 -7 -2 -47 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9158
 
{{Optimal ET sequence|legend=1| 31, 114e, 145, 176 }}
 
Badness: 0.039162
 
==== Countermiraculous ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 1001/1000, 6144/6125
 
Mapping: {{mapping| 1 4 3 3 8 1 | 0 -25 -7 -2 -47 28 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8803
 
{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145, 321ceff }}
 
Badness: 0.039271
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224
 
Mapping: {{mapping| 1 4 3 3 8 1 1 | 0 -25 -7 -2 -47 28 32 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8756
 
{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145 }}
 
Badness: 0.029496
 
==== Counterbenediction ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 441/440, 3146/3125, 3584/3575
 
Mapping: {{mapping| 1 4 3 3 8 -2 | 0 -25 -7 -2 -47 59 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9335
 
{{Optimal ET sequence|legend=1| 31, 114ef, 145f, 176, 207, 383c, 590cc }}
 
Badness: 0.045569
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125
 
Mapping: {{mapping| 1 4 3 3 8 -2 -2 | 0 -25 -7 -2 -47 59 63 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9391
 
{{Optimal ET sequence|legend=1| 31, 114efg, 145fg, 176, 207 }}
 
Badness: 0.036289
 
==== Countermanna ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 3388/3375, 6144/6125
 
Mapping: {{mapping| 1 4 3 3 8 15  0 -25 -7 -2 -47 -117 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8898
 
{{Optimal ET sequence|legend=1| 145, 176, 321ce }}
 
Badness: 0.053409
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
 
Mapping: {{mapping| 1 4 3 3 8 15 15 | 0 -25 -7 -2 -47 -117 -113 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8832
 
{{Optimal ET sequence|legend=1| 145, 321ce }}
 
Badness: 0.040898
 
=== Counterrevelation ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 50421/50000
 
Mapping: {{mapping| 1 4 3 3 5 | 0 -25 -7 -2 -16 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9192
 
{{Optimal ET sequence|legend=1| 31, 114, 145e, 176e }}
 
Badness: 0.064070
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 196/195, 13750/13689
 
Mapping: {{mapping| 1 4 3 3 5 1 | 0 -25 -7 -2 -16 28 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~273/256 = 115.8624
 
{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
 
Badness: 0.057497
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625
 
Mapping: {{mapping| 1 4 3 3 5 1 1 | 0 -25 -7 -2 -16 28 32 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~91/85 = 115.8527
 
{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
 
Badness: 0.044043
 
== Absurdity ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Absurdity]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 177147/175000
 
{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}
 
: Mapping generators: ~972/875, ~3
 
[[Optimal tuning]] ([[POTE]]): ~972/875 = 1\7, ~3/2 = 700.5854 (or ~10/9 = 186.2997)
 
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
 
[[Badness]]: 0.133520
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 6144/6125, 72171/71680
 
{{Mapping|legend=1| 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 ET sequence|legend=1| 77, 84, 161 }}
 
Badness: 0.081564
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 }}
 
Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6291 (or ~10/9 = 186.3434)
 
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
 
Badness: 0.041600
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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 }}
 
Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6524 (or ~10/9 = 186.3667)
 
{{Optimal ET sequence|legend=1| 77, 161 }}
 
Badness: 0.031783
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
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 }}
 
Optimal tuning (POTE): ~21/19 = 1\7, ~3/2 = 700.6565 (or ~10/9 = 186.3708)
 
{{Optimal ET sequence|legend=1| 77, 161 }}
 
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 ==
: ''For the 5-limit version, see [[High badness 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 2500000/2470629
 
{{Mapping|legend=1| 1 12 5 4 | 0 -35 -9 -4 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 357.070
 
{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}
 
[[Badness]]: 0.179692
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 2560/2541, 4375/4356
 
Mapping: {{mapping| 1 12 5 4 -1 | 0 -35 -9 -4 15 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.048
 
{{Optimal ET sequence|legend=1| 37, 84, 121, 326dee }}
 
Badness: 0.081923
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 625/624, 640/637, 1375/1372
 
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 ET sequence|legend=1| 37, 84, 121, 326deef }}
 
Badness: 0.039533
 
== Notes ==
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Porwell| ]] <!-- key article -->
[[Category:Hendecatonic]]
[[Category:Hendecatonic]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Porwell_temperaments"