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Porwell temperaments: Difference between revisions

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{{Technical data page}}
{{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.  
This is a collection of [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[porwell comma]] ({{monzo|legend=1| 11 1 -3 -2 }}, [[ratio]]: [[6144/6125]]).  


Discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Armodue (temperament)|Armodue]]'' (+36/35) → [[Mavila family #Armodue|Mavila family]]
* ''[[Armodue (temperament)|Armodue]]'' (+36/35) → [[Mavila family #Armodue|Mavila family]]
* [[Mohajira]] (+81/80) → [[Meantone family #Mohajira|Meantone family]]
* ''[[Hemischis]]'' (+19683/19600) → [[Schismatic family #Hemischis|Schismatic family]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Porcupine|Porcupine family]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Porcupine|Porcupine family]]
* [[Mohajira]] (+81/80) → [[Meantone family #Mohajira|Meantone family]]
* ''[[Alphatrident]]'' (+14348907/14336000) → [[Alphatricot family #Alphatrident|Alphatricot family]]
* ''[[Shrutar]]'' (+245/243) → [[Diaschismic family #Shrutar|Diaschismic family]]
* [[Amity]] (+4375/4374 or 5120/5103) → [[Amity family #Septimal amity|Amity family]]
* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Twilight]]'' (+{{monzo| 19 -22 2 4 }}) → [[Undim family #Twilight|Undim family]]
* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]]
* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]]
* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]]
* [[Shrutar]] (+245/243) → [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap 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]]
* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* [[Amity]] (+4375/4374 or 5120/5103) → [[Amity family #Septimal amity|Amity family]]
* ''[[Twothirdtonic]]'' (+686/675) → [[Sengic temperaments #Twothirdtonic|Sengic temperaments]]
* ''[[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]]
* ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[Quinkee]]'' (+1029/1000) → [[Keegic temperaments #Quinkee|Keegic temperaments]]
* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Semaja]]'' (+3125/3087) → [[Gariboh clan #Semaja|Gariboh clan]]
* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Countermiracle]]'' (+823543/819200) → [[Quince clan #Countermiracle|Quince clan]]
* ''[[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]]
 
* ''[[Twilight]]'' (+{{monzo| 19 -22 2 4 }}) → [[Undim family #Twilight|Undim family]]
Considered below are hendecatonic, nessafof, grendel, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing [[badness]].
* ''[[Countermiracle]]'' (+823543/819200) → [[Quince clan #Countermiracle|Quince clan]]


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


The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]]. It tempers out 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 22 & 99 }} temperament, with [[99edo]] giving an almost perfect tuning.  
The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]]. It tempers out 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 22 & 99 }} temperament, with [[99edo]] giving an almost perfect tuning.  
Line 47: Line 50:
[[Badness]] (Sintel): 1.04
[[Badness]] (Sintel): 1.04


=== 11-limit ===
=== Hendecaton ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 153: Line 156:
Badness (Sintel): 1.84
Badness (Sintel): 1.84


== Twothirdtonic ==
== Nessafof ==
Twothirdtonic tempers out 686/675, the [[senga]], in addition to the porwell comma, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot. [[46edo]] may be recommended as a tuning.  
: ''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 five times, makes 5/1<ref name="petr's long post"/>.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 686/675, 6144/6125
[[Comma list]]: 6144/6125, 250047/250000


{{Mapping|legend=1| 1 -10 5 -7 | 0 13 -3 11 }}
{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}
: mapping generators: ~2, ~28/15
: mapping generators: ~63/50, ~35/32


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}
* [[WE]]: ~63/50 = 399.9023{{c}}, ~35/32 = 157.4418{{c}}
: [[error map]]: {{val| -0.693 +1.736 +3.278 -5.176 }}
: [[error map]]: {{val| -0.293 -0.057 +0.407 +0.430 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~35/32 = 157.4658{{c}}
: error map: {{val| 0.000 +2.515 +4.962 -3.505 }}
: error map: {{val| 0.000 +0.306 1.016 +1.311 }}


{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}


[[Badness]] (Sintel): 2.52
[[Badness]] (Sintel): 1.14


=== 11-limit ===
=== Nessa ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 686/675
Comma list: 441/440, 1344/1331, 4375/4356


Mapping: {{mapping| 1 -10 5 -7 -1 | 0 13 -3 11 5 }}
Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}
* WE: ~44/35 = 399.7815{{c}}, ~35/32 = 157.4527{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~35/32 = 157.5109{{c}}


{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=0| 15, 69, 84, 99e }}


Badness (Sintel): 1.35
Badness (Sintel): 1.61


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 121/120, 169/168, 176/175
Comma list: 144/143, 364/363, 441/440, 625/624


Mapping: {{mapping| 1 -10 5 -7 -1 -7 | 0 13 -3 11 5 12 }}
Mapping: {{mapping| 3 2 5 10 10 6 | 0 7 5 -4 1 13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}
* WE: ~44/35 = 399.7595{{c}}, ~35/32 = 157.3348{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~35/32 = 157.3955{{c}}
 
{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
 
Badness (Sintel): 1.07
 
== 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 = 1200.0000{{c}}, ~8/7 = 226.4834{{c}}
{{Optimal ET sequence|legend=0| 15, 69, 84, 99ef, 183ef, 282eeff }}


{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}
Badness (Sintel): 1.55


[[Badness]] (Smith): 0.107023
=== Fof ===
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 3125/3087
Comma list: 121/120, 176/175, 250047/250000


Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}
Mapping: {{mapping| 3 2 5 10 8 | 0 7 5 -4 6 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 226.4856{{c}}
Optimal tunings:  
* WE: ~63/50 = 400.0266{{c}}, ~12/11 = 157.5301{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~12/11 = 157.5240{{c}}


{{Optimal ET sequence|legend=0| 16, 37, 53 }}
{{Optimal ET sequence|legend=0| 15, 69e, 84e, 99 }}


Badness (Smith): 0.059838
Badness (Sintel): 2.26


=== 13-limit ===
== Grendel ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Counterwürschmidt]].''


Comma list: 121/120, 169/168, 176/175, 275/273
Grendel tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 31 & 152 }} temperament. [[152edo]], [[183edo]] and especially [[335edo]] serve as good tunings.


Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }}
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 226.4794{{c}}
[[Comma list]]: 6144/6125, 16875/16807


{{Optimal ET sequence|legend=0| 16, 37, 53 }}
{{Mapping|legend=1| 1 -14 3 -6 | 0 23 -1 13 }}
: mapping generators: ~2, ~8/5


Badness (Smith): 0.032564
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7348{{c}}, ~8/5 = 812.9574{{c}}
: [[error map]]: {{val| -0.265 -0.220 -0.067 +1.212 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 813.1311{{c}}
: error map: {{val| 0.000 +0.059 +0.555 +1.878 }}


== Nessafof ==
{{Optimal ET sequence|legend=1| 31, 90, 121, 152, 335d, 822dd }}
: ''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"/>.  
[[Badness]] (Sintel): 1.31


[[Subgroup]]: 2.3.5.7
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Comma list]]: 6144/6125, 250047/250000
Comma list: 540/539, 1375/1372, 5632/5625


{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}
Mapping: {{mapping| 1 -14 3 -6 -25 | 0 23 -1 13 42 }}
: mapping generators: ~63/50, ~35/32


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~35/32 = 157.480{{c}}
Optimal tunings:  
* WE: ~2 = 1199.7355{{c}}, ~8/5 = 812.9622{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1353{{c}}


{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152, 335d, 487d }}


[[Badness]] (Smith): 0.045048
Badness (Sintel): 0.656


=== 11-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 250047/250000
Comma list: 352/351, 540/539, 625/624, 1375/1372


Mapping: {{mapping| 3 2 5 10 8 | 0 7 5 -4 6 }}
Mapping: {{mapping| 1 -14 3 -6 -25 22 | 0 23 -1 13 42 -27 }}


Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~12/11 = 157.520{{c}}
Optimal tunings:  
* WE: ~2 = 1199.4412{{c}}, ~8/5 = 812.7956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1209{{c}}


{{Optimal ET sequence|legend=0| 15, 54be, 69e, 84e, 99 }}
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152f, 273def, 425deff }}


Badness (Smith): 0.068427
Badness (Sintel): 1.03


=== Nessa ===
=== 17-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 441/440, 1344/1331, 4375/4356
Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274


Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}
Mapping: {{mapping| 1 -14 3 -6 -25 22 19 | 0 23 -1 13 42 -27 -22 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~35/32 = 157.539{{c}}
Optimal tunings:  
* WE: ~2 = 1199.3029{{c}}, ~8/5 = 812.7156{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1843{{c}}


{{Optimal ET sequence|legend=0| 15, 54b, 69, 84, 99e }}
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152fg, 273defgg }}


Badness (Smith): 0.048836
Badness (Sintel): 1.09


==== 13-limit ====
=== 19-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 144/143, 364/363, 441/440, 625/624
Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714


Mapping: {{mapping| 3 2 5 10 10 6 | 0 7 5 -4 1 13 }}
Mapping: {{mapping| 1 -14 3 -6 -25 22 19 30 | 0 23 -1 13 42 -27 -22 -38 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~35/32 = 157.429{{c}}
Optimal tunings:  
* WE: ~2 = 1199.3587{{c}}, ~8/5 = 812.7462{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1796{{c}}


{{Optimal ET sequence|legend=0| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152fg, 273defgg }}


Badness (Smith): 0.037409
Badness (Sintel): 1.12


== Aufo ==
== Aufo ==
:''For the 5-limit version, see [[Miscellaneous 5-limit 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">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>. The functional generator however is the [[64/45]] diminished fifth, and like its [[untriton]] variant, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 311: Line 314:
[[Comma list]]: 6144/6125, 177147/175616
[[Comma list]]: 6144/6125, 177147/175616


{{Mapping|legend=1| 1 6 -7 19 | 0 -9 19 -33 }}
{{Mapping|legend=1| 1 -3 12 -14 | 0 9 -19 33 }}
: mapping generators: ~2, ~45/32
: mapping generators: ~2, ~64/45


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~45/32 = 588.782{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9758{{c}}, ~64/45 = 611.2055{{c}}
: [[error map]]: {{val| -0.024 -1.303 +0.491 +1.295 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/45 = 611.2177{{c}}
: error map: {{val| 0.000 -0.996 +0.551 +1.357 }}


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


[[Badness]] (Smith): 0.121428
[[Badness]] (Sintel): 3.07


=== 11-limit ===
=== 11-limit ===
Line 325: Line 332:
Comma list: 121/120, 176/175, 177147/175616
Comma list: 121/120, 176/175, 177147/175616


Mapping: {{mapping| 1 6 -7 19 1 | 0 -9 19 -33 5 }}
Mapping: {{mapping| 1 -3 12 -14 6 | 0 9 -19 33 -5 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.811{{c}}
Optimal tunings:
* WE: ~2 = 1200.4500{{c}}, ~64/45 = 611.4185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.1918{{c}}


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


Badness (Smith): 0.088631
Badness (Sintel): 2.93


==== 13-limit ====
==== 13-limit ====
Line 338: Line 347:
Comma list: 121/120, 176/175, 351/350, 58806/57967
Comma list: 121/120, 176/175, 351/350, 58806/57967


Mapping: {{mapping| 1 6 -7 19 1 -12 | 0 -9 19 -33 5 32 }}
Mapping: {{mapping| 1 -3 12 -14 6 20 | 0 9 -19 33 -5 -32 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.788{{c}}
Optimal tunings:
* WE: ~2 = 1200.3134{{c}}, ~64/45 = 611.3715{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2118{{c}}


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


Badness (Smith): 0.058507
Badness (Sintel): 2.42


=== Aufic ===
=== Aufic ===
Line 351: Line 362:
Comma list: 540/539, 5632/5625, 72171/71680
Comma list: 540/539, 5632/5625, 72171/71680


Mapping: {{mapping| 1 6 -7 19 -25 | 0 -9 19 -33 58 }}
Mapping: {{mapping| 1 -3 12 -14 33 | 0 9 -19 33 -58 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.800{{c}}
Optimal tunings:
* WE: ~2 = 1200.0668{{c}}, ~64/45 = 611.2342{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2000{{c}}


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


Badness (Smith): 0.075149
Badness (Sintel): 2.48


==== 13-limit ====
==== 13-limit ====
Line 364: Line 377:
Comma list: 351/350, 540/539, 847/845, 4096/4095
Comma list: 351/350, 540/539, 847/845, 4096/4095


Mapping: {{mapping| 1 6 -7 19 -25 -12 | 0 -9 19 -33 58 32 }}
Mapping: {{mapping| 1 -3 12 -14 33 20 | 0 9 -19 33 -58 -32 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.796{{c}}
Optimal tunings:
* WE: ~2 = 1200.0177{{c}}, ~64/45 = 611.2130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2039{{c}}


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


Badness (Smith): 0.039050
Badness (Sintel): 1.61


== Whoops ==
== Absurdity ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Whoosh]].''  
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Absurdity (5-limit)]].''
 
{{See also| Fifth-chroma temperaments }}
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"/>.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 244140625/243045684
[[Comma list]]: 6144/6125, 177147/175000


{{Mapping|legend=1| 1 17 14 -7 | 0 -33 -25 21 }}
{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}
: mapping generators: ~2, ~441/320
: mapping generators: ~972/875, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~441/320 = 560.519{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~972/875 = 171.4382{{c}}, ~3/2 = 700.6247{{c}}
: [[error map]]: {{val| +0.067 -1.263 +1.313 +0.450 }}
* [[CWE]]: ~972/875 = 171.4286{{c}}, ~3/2 = 700.5871{{c}}
: error map: {{val| 0.000 -1.368 +1.162 +0.254 }}


{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608d, 623bd, 775bcd }}
{{Optimal ET sequence|legend=1| 77, 84, 161 }}


[[Badness]] (Smith): 0.175840
[[Badness]] (Sintel): 3.38


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4000/3993, 6144/6125
Comma list: 441/440, 6144/6125, 72171/71680
 
Mapping: {{mapping| 7 0 -17 64 124 | 0 1 3 -4 -9 }}
 
Optimal tunings:
* WE: ~495/448 = 171.4346{{c}}, ~3/2 = 700.6602{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 700.6339{{c}}
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
 
Badness (Sintel): 2.70


Mapping: {{mapping| 1 17 14 -7 10 | 0 -33 -25 21 -14 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~242/175 = 560.519{{c}}
Comma list: 351/350, 441/440, 1188/1183, 3584/3575


{{Optimal ET sequence|legend=0| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
Mapping: {{mapping| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}


Badness (Smith): 0.043743
Optimal tunings:  
* WE: ~72/65 = 171.4223{{c}}, ~3/2 = 700.6036{{c}}
* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6306{{c}}


== Polypyth ==
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''


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.
Badness (Sintel): 1.72


[[Subgroup]]: 2.3.5.7
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 6144/6125, 179200/177147
Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625


{{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}
Mapping: {{mapping| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 704.174{{c}}
Optimal tunings:  
* WE: ~72/65 = 171.4263{{c}}, ~3/2 = 700.6429{{c}}
* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6525{{c}}


{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}
{{Optimal ET sequence|legend=0| 77, 161 }}


[[Badness]] (Smith): 0.137995
Badness (Sintel): 1.62


=== 11-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 896/891, 2200/2187, 6144/6125
Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494


Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.177{{c}}
Optimal tunings:  
* WE: ~21/19 = 171.4244{{c}}, ~3/2 = 700.6395{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6568{{c}}


{{Optimal ET sequence|legend=0| 46, 121, 167, 288be, 455bcde }}
{{Optimal ET sequence|legend=0| 77, 161 }}


Badness (Smith): 0.051131
Badness (Sintel): 1.36


=== 13-limit ===
=== 23-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 325/324, 352/351, 364/363, 1716/1715
Comma list: 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494


Mapping: {{mapping| 1 0 -31 52 59 64 | 0 1 21 -31 -35 -38 }}
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.168{{c}}
Optimal tunings:  
* WE: ~21/19 = 171.4321{{c}}, ~3/2 = 700.6475{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6325{{c}}


{{Optimal ET sequence|legend=0| 46, 121, 167, 288be }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


Badness (Smith): 0.030292
Badness (Sintel): 1.34


=== 17-limit ===
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.19.23.29


Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715
Comma list: 261/260, 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494


Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }}
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.168{{c}}
Optimal tunings:  
* WE: ~21/19 = 171.4348{{c}}, ~3/2 = 700.6612{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6351{{c}}


{{Optimal ET sequence|legend=0| 46, 121, 167, 288beg }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


Badness (Smith): 0.019051
Badness (Sintel): 1.25


== Icositritonic ==
== Polypyth ==
{{See also| 23rd-octave temperaments }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''


The icositritonic temperament (46 & 161) has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]].
Polypyth tempers out the same 5-limit comma as [[leapday]], with which it shares the similarly sharp [[3/2|perfect-fifth]] generator, but the porwell comma (6144/6125) rather than the hemifamity comma (5120/5103) is tempered out here. It may be described as the {{nowrap| 46 & 121 }} temperament, and [[121edo]] and [[167edo]] make for good tunings.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 9920232/9765625
[[Comma list]]: 6144/6125, 179200/177147


{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}
{{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}
: mapping generators: ~1323/1280, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 52.1739{{c}}, ~64/63 = 29.3586{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3465{{c}}, ~3/2 = 703.7905{{c}}
: [[error map]]: {{val| -0.654 +1.182 -0.177 -0.056 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1749{{c}}
: error map: {{val| 0.000 +2.220 +1.359 +1.752 }}


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd }}


[[Badness]] (Smith): 0.196622
[[Badness]] (Sintel): 3.49


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 6144/6125, 35937/35840
Comma list: 896/891, 2200/2187, 6144/6125


Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}
Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}


Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3980{{c}}
Optimal tunings:  
* WE: ~2 = 1199.3335{{c}}, ~3/2 = 703.7856{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1812{{c}}


{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 121, 167, 288be, 455bcde }}


Badness (Smith): 0.064613
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 441/440, 847/845, 3584/3575
Comma list: 325/324, 352/351, 364/363, 1716/1715


Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}
Mapping: {{mapping| 1 0 -31 52 59 64 | 0 1 21 -31 -35 -38 }}


Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.2830{{c}}
Optimal tunings:  
* WE: ~2 = 1199.3768{{c}}, ~3/2 = 703.8018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1731{{c}}


{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 75e, 121, 167, 288be }}


Badness (Smith): 0.040484
Badness (Sintel): 1.25


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088
Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715
 
Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }}
 
Optimal tunings:
* WE: ~2 = 1199.3518{{c}}, ~3/2 = 703.7880{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1747{{c}}
 
{{Optimal ET sequence|legend=0| 46, 75e, 121, 167, 288beg }}


Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}
Badness (Sintel): 0.971


Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.2800{{c}}
== Whoops ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Whoosh]].''


{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
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 (Smith): 0.024676
[[Subgroup]]: 2.3.5.7


=== 19-limit ===
[[Comma list]]: 6144/6125, 244140625/243045684
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845
{{Mapping|legend=1| 1 -16 -11 14 | 0 33 25 -21 }}
: mapping generators: ~2, ~640/441


Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5944{{c}}, ~640/441 = 639.2648{{c}}
: [[error map]]: {{val| -0.406 +0.272 -0.233 +0.936 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~640/441 = 639.4769{{c}}
: error map: {{val| 0.000 +0.783 +0.609 +2.159 }}


Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3760{{c}}
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 623bdd, 775bcdd, 927bcddd, 1079bcddd }}


{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
[[Badness]] (Sintel): 4.45


Badness (Smith): 0.021579
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 23-limit ===
Comma list: 3025/3024, 4000/3993, 6144/6125
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
Mapping: {{mapping| 1 -16 -11 14 -4 | 0 33 25 -21 14 }}


Mapping: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}
Optimal tunings:  
* WE: ~2 = 1199.5936{{c}}, ~175/121 = 639.264{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~175/121 = 639.4770{{c}}


Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3471{{c}}
{{Optimal ET sequence|legend=0| 15, 122d, 137, 152, 623bdde, 775bcdde, 927bcdddee, 1079bcdddee }}


{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368ci }}
Badness (Sintel): 1.45


Badness (Smith): 0.017745
== Dodifo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Dodifo]].''


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 177147/175000
[[Comma list]]: 6144/6125, 2500000/2470629


{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}
{{Mapping|legend=1| 1 -23 -4 0 | 0 35 9 4 }}
: mapping generators: ~972/875, ~3
: mapping generators: ~2, ~80/49


[[Optimal tuning]] ([[POTE]]): ~972/875 = 171.4286{{c}}, ~3/2 = 700.5854{{c}} (or ~10/9 = 186.2997{{c}})
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6429{{c}}, ~80/49 = 842.6790{{c}}
: [[error map]]: {{val| -0.357 +0.228 -0.774 +1.890 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9243{{c}}
: error map: {{val| 0.000 +0.396 +0.005 +2.871 }}


{{Optimal ET sequence|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}


[[Badness]] (Smith): 0.133520
[[Badness]] (Sintel): 4.55


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 6144/6125, 72171/71680
Comma list: 1375/1372, 2560/2541, 4375/4356


Mapping: {{mapping| 7 0 -17 64 124 | 0 1 3 -4 -9 }}
Mapping: {{mapping| 1 -23 -4 0 14 | 0 35 9 4 -15 }}


Optimal tuning (POTE): ~495/448 = 171.4286{{c}}, ~3/2 = 700.6354{{c}} (or ~10/9 = 186.3497{{c}})
Optimal tunings:
* WE: ~2 = 1199.3401{{c}}, ~80/49 = 842.4880{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/49 = 842.9457{{c}}


{{Optimal ET sequence|legend=0| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 37, 84, 121, 326dee }}


Badness (Smith): 0.081564
Badness (Sintel): 2.71


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 441/440, 1188/1183, 3584/3575
Comma list: 364/363, 625/624, 640/637, 1375/1372


Mapping: {{mapping| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}
Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}


Optimal tuning (POTE): ~72/65 = 171.4286{{c}}, ~3/2 = 700.6291{{c}} (or ~10/9 = 186.3434{{c}})
Optimal tunings:
* WE: ~2 = 1199.3410{{c}}, ~13/8 = 842.4885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9466{{c}}


{{Optimal ET sequence|legend=0| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 37, 84, 121, 326deef }}


Badness (Smith): 0.041600
Badness (Sintel): 1.63


=== 17-limit ===
== Icositritonic ==
Subgroup: 2.3.5.7.11.13.17
{{See also| 23rd-octave temperaments }}


Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625
Icositritonic has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]]. It may be described as {{nowrap| 46 & 161 }}. It was named by [[Xenllium]] in 2019 for its number of periods per octave.


Mapping: {{mapping| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~72/65 = 171.4286{{c}}, ~3/2 = 700.6524{{c}} (or ~10/9 = 186.3667{{c}})
[[Comma list]]: 6144/6125, 9920232/9765625


{{Optimal ET sequence|legend=0| 77, 161 }}
{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}
: mapping generators: ~1323/1280, ~3


Badness (Smith): 0.031783
[[Optimal tuning]]s:  
* [[WE]]: ~1323/1280 = 52.1732{{c}}, ~3/2 = 701.0660{{c}}
: [[error map]]: {{val| -0.017 -0.906 +1.679 -0.386 }}
* [[CWE]]: ~1323/1280 = 52.1739{{c}}, ~3/2 = 701.0722{{c}}
: error map: {{val| 0.000 -0.883 +1.715 -0.333 }}


=== 19-limit ===
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
[[Badness]] (Sintel): 4.98


Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~21/19 = 171.4286{{c}}, ~3/2 = 700.6565{{c}} (or ~10/9 = 186.3708{{c}})
Comma list: 441/440, 6144/6125, 35937/35840


{{Optimal ET sequence|legend=0| 77, 161 }}
Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}


Badness (Smith): 0.022291
Optimal tunings:  
* WE: ~33/32 = 52.1740{{c}}, ~3/2 = 701.0379{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0370{{c}}


=== 23-limit ===
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
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
Badness (Sintel): 2.14


Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal tuning ([[CTE]]): ~21/19 = 171.429{{c}}, ~3/2 = 700.629{{c}} (or ~10/9 = 186.343{{c}})
Comma list: 351/350, 441/440, 847/845, 3584/3575


{{Optimal ET sequence|legend=0| 77, 84, 161 }}
Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}


=== 29-limit ===
Optimal tunings:
{{See also| Fifth-chroma temperaments }}
* WE: ~33/32 = 52.1724{{c}}, ~3/2 = 701.1310{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.1524{{c}}


Subgroup: 2.3.5.7.11.13.17.19.23.29
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Comma list: 261/260, 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
Badness (Sintel): 1.67


Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Optimal tuning ([[CTE]]): ~21/19 = 171.429{{c}}, ~3/2 = 700.629{{c}} (or ~10/9 = 186.343{{c}})
Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088


{{Optimal ET sequence|legend=0| 77, 84, 161 }}
Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}


== Dodifo ==
Optimal tunings:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Dodifo]].''
* WE: ~33/32 = 52.1735{{c}}, ~3/2 = 701.1493{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.1549{{c}}


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


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 1.26


[[Comma list]]: 6144/6125, 2500000/2470629
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


{{Mapping|legend=1| 1 12 5 4 | 0 -35 -9 -4 }}
Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/40 = 357.070{{c}}
Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}


{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}
Optimal tunings:
* WE: ~33/32 = 52.1744{{c}}, ~3/2 = 701.0649{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0582{{c}}


[[Badness]] (Smith): 0.179692
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


=== 11-limit ===
Badness (Sintel): 1.31
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 2560/2541, 4375/4356
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Mapping: {{mapping| 1 12 5 4 -1 | 0 -35 -9 -4 15 }}
Comma list: 276/275, 351/350, 391/390, 441/440, 456/455, 476/475, 847/845


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 357.048{{c}}
Mapping: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}


{{Optimal ET sequence|legend=0| 37, 84, 121, 326dee }}
Optimal tunings:
* WE: ~33/32 = 52.1768{{c}}, ~3/2 = 701.1259{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0841{{c}}


Badness (Smith): 0.081923
{{Optimal ET sequence|legend=0| 46, 115, 161, 207 }}


=== 13-limit ===
Badness (Sintel): 1.27
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 = 1200.000{{c}}, ~16/13 = 357.049{{c}}
 
{{Optimal ET sequence|legend=0| 37, 84, 121, 326deef }}
 
Badness (Smith): 0.039533


== References ==
== References ==


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