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]]).  
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]]).  


Temperaments 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]]
* ''[[Countermiracle]]'' (+823543/819200) → [[Quince clan #Countermiracle|Quince clan]]


Considered below are hendecatonic, twothirdtonic, nessafof, aufo, whoops, polypyth, icositritonic, absurdity, and dodifo.  
Considered below are hendecatonic, nessafof, grendel, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing [[badness]].  


== 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 49: 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 155: 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 }}
{{Optimal ET sequence|legend=0| 15, 69, 84, 99ef, 183ef, 282eeff }}


Badness (Sintel): 1.07
Badness (Sintel): 1.55
 
== Semaja ==
Cryptically named by [[Petr Pařízek]] in 2011, semaja adds the [[gariboh comma]] to the comma list, and may be described as the {{nowrap| 37 & 53 }} temperament. Its [[ploidacot]] is gamma-19-cot. 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]]s:
* [[WE]]: ~2 = 1199.4860{{c}}, ~8/7 = 226.3864{{c}}
: [[error map]]: {{val| -0.514 +0.415 -2.123 +3.246 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.4697{{c}}
: error map: {{val| 0.000 +0.970 -1.026 +4.704 }}
 
{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}


[[Badness]] (Sintel): 2.71
=== 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 tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9818{{c}}, ~8/7 = 226.4821{{c}}
* WE: ~63/50 = 400.0266{{c}}, ~12/11 = 157.5301{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4851{{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 (Sintel): 1.98
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 tunings:  
[[Comma list]]: 6144/6125, 16875/16807
* WE: ~2 = 1200.1020{{c}}, ~8/7 = 226.4987{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4822{{c}}


{{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 (Sintel): 1.35
[[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 five 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]]s:  
Optimal tunings:  
* [[WE]]: ~63/50 = 399.9023{{c}}, ~35/32 = 157.4418{{c}}
* WE: ~2 = 1199.7355{{c}}, ~8/5 = 812.9622{{c}}
: [[error map]]: {{val| -0.293 -0.057 +0.407 +0.430 }}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1353{{c}}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~35/32 = 157.4658{{c}}
: error map: {{val| 0.000 +0.306 1.016 +1.311 }}


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


[[Badness]] (Sintel): 1.14
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 tunings:  
Optimal tunings:  
* WE: ~63/50 = 400.0266{{c}}, ~12/11 = 157.5301{{c}}
* WE: ~2 = 1199.4412{{c}}, ~8/5 = 812.7956{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~12/11 = 157.5240{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1209{{c}}


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


Badness (Sintel): 2.26
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 tunings:  
Optimal tunings:  
* WE: ~44/35 = 399.7815{{c}}, ~35/32 = 157.4527{{c}}
* WE: ~2 = 1199.3029{{c}}, ~8/5 = 812.7156{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~35/32 = 157.5109{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1843{{c}}


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


Badness (Sintel): 1.61
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 tunings:  
Optimal tunings:  
* WE: ~44/35 = 399.7595{{c}}, ~35/32 = 157.3348{{c}}
* WE: ~2 = 1199.3587{{c}}, ~8/5 = 812.7462{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~35/32 = 157.3955{{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 (Sintel): 1.55
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 331: 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 345: 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 358: 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 371: 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 384: 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 }}


Mapping: {{mapping| 1 17 14 -7 10 | 0 -33 -25 21 -14 }}
Badness (Sintel): 2.70


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~242/175 = 560.519{{c}}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=0| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
Comma list: 351/350, 441/440, 1188/1183, 3584/3575


Badness (Smith): 0.043743
Mapping: {{mapping| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}


== Polypyth ==
Optimal tunings:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''
* WE: ~72/65 = 171.4223{{c}}, ~3/2 = 700.6036{{c}}
* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6306{{c}}
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


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]]

Latest revision as of 08:00, 15 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of temperaments that temper out the porwell comma (monzo: [11 1 -3 -2⟩, ratio: 6144/6125).

Temperaments discussed elsewhere are:

Considered below are hendecatonic, nessafof, grendel, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing badness.

Hendecatonic

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 22 & 99 temperament, with 99edo giving an almost perfect tuning.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 10976/10935

Mapping: [⟨11 0 43 -4], ⟨0 1 -1 2]]

mapping generators: ~16/15, ~3

Optimal tunings:

  • WE: ~16/15 = 109.0526 ¢, ~3/2 = 702.8069 ¢
error map: ⟨-0.421 +0.431 +0.563 -0.265]
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.9705 ¢
error map: ⟨0.000 +1.015 +1.625 +0.751]

Optimal ET sequence: 22, 55, 77, 99

Badness (Sintel): 1.04

Hendecaton

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 10976/10935

Mapping: [⟨11 0 43 -4 38], ⟨0 1 -1 2 0]]

Optimal tunings:

  • WE: ~16/15 = 109.0977 ¢, ~3/2 = 702.6801 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.6484 ¢

Optimal ET sequence: 22, 55, 77, 99

Badness (Sintel): 1.52

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨11 0 43 -4 38 93], ⟨0 1 -1 2 0 -3]]

Optimal tunings:

  • WE: ~16/15 = 109.1092 ¢, ~3/2 = 702.4093 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.2930 ¢

Optimal ET sequence: 22, 55, 77, 99

Badness (Sintel): 1.66

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨11 0 43 -4 38 93 45], ⟨0 1 -1 2 0 -3 0]]

Optimal tunings:

  • WE: ~16/15 = 109.0933 ¢, ~3/2 = 702.3170 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.3017 ¢

Optimal ET sequence: 22, 55, 77, 99, 176eg

Badness (Sintel): 1.48

Cohendecatonic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨11 0 43 -4 73], ⟨0 1 -1 2 -2]]

Optimal tunings:

  • WE: ~16/15 = 109.0237 ¢, ~3/2 = 703.2522 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.6563 ¢

Optimal ET sequence: 22, 77e, 99e, 121, 220e

Badness (Sintel): 1.26

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨11 0 43 -4 73 128], ⟨0 1 -1 2 -2 -5]]

Optimal tunings:

  • WE: ~16/15 = 109.0189 ¢, ~3/2 = 703.4228 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.9248 ¢

Optimal ET sequence: 22, 99ef, 121, 341bdeeff

Badness (Sintel): 1.49

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨11 0 43 -4 73 128 45], ⟨0 1 -1 2 -2 -5 0]]

Optimal tunings:

  • WE: ~16/15 = 109.0159 ¢, ~3/2 = 703.3932 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.9110 ¢

Optimal ET sequence: 22, 99ef, 121, 220efg, 341bdeeffgg

Badness (Sintel): 1.15

Icosidillic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨22 0 86 -8 111], ⟨0 1 -1 2 -1]]

mapping generators: ~33/32, ~3

Optimal tunings:

  • WE: ~33/32 = 54.5305 ¢, ~3/2 = 702.7206 ¢
  • CWE: ~33/32 = 54.5455 ¢, ~3/2 = 702.8829 ¢

Optimal ET sequence: 22, 154, 176, 198

Badness (Sintel): 1.84

Nessafof

For the 5-limit version, see Miscellaneous 5-limit temperaments #Nessafof.

Cryptically named by Petr Pařízek in 2011[1], 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[2].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 250047/250000

Mapping: [⟨3 2 5 10], ⟨0 7 5 -4]]

mapping generators: ~63/50, ~35/32

Optimal tunings:

  • WE: ~63/50 = 399.9023 ¢, ~35/32 = 157.4418 ¢
error map: ⟨-0.293 -0.057 +0.407 +0.430]
  • CWE: ~63/50 = 400.0000 ¢, ~35/32 = 157.4658 ¢
error map: ⟨0.000 +0.306 1.016 +1.311]

Optimal ET sequence: 15, 54b, 69, 84, 99, 282, 381

Badness (Sintel): 1.14

Nessa

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1344/1331, 4375/4356

Mapping: [⟨3 2 5 10 10], ⟨0 7 5 -4 1]]

Optimal tunings:

  • WE: ~44/35 = 399.7815 ¢, ~35/32 = 157.4527 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~35/32 = 157.5109 ¢

Optimal ET sequence: 15, 69, 84, 99e

Badness (Sintel): 1.61

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 364/363, 441/440, 625/624

Mapping: [⟨3 2 5 10 10 6], ⟨0 7 5 -4 1 13]]

Optimal tunings:

  • WE: ~44/35 = 399.7595 ¢, ~35/32 = 157.3348 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~35/32 = 157.3955 ¢

Optimal ET sequence: 15, 69, 84, 99ef, 183ef, 282eeff

Badness (Sintel): 1.55

Fof

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 2 5 10 8], ⟨0 7 5 -4 6]]

Optimal tunings:

  • WE: ~63/50 = 400.0266 ¢, ~12/11 = 157.5301 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~12/11 = 157.5240 ¢

Optimal ET sequence: 15, 69e, 84e, 99

Badness (Sintel): 2.26

Grendel

For the 5-limit version, see Syntonic–31 equivalence continuum #Counterwürschmidt.

Grendel tempers out 16875/16807, the mirkwai comma, and may be described as the 31 & 152 temperament. 152edo, 183edo and especially 335edo serve as good tunings.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 16875/16807

Mapping: [⟨1 -14 3 -6], ⟨0 23 -1 13]]

mapping generators: ~2, ~8/5

Optimal tunings:

  • WE: ~2 = 1199.7348 ¢, ~8/5 = 812.9574 ¢
error map: ⟨-0.265 -0.220 -0.067 +1.212]
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1311 ¢
error map: ⟨0.000 +0.059 +0.555 +1.878]

Optimal ET sequence: 31, 90, 121, 152, 335d, 822dd

Badness (Sintel): 1.31

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 5632/5625

Mapping: [⟨1 -14 3 -6 -25], ⟨0 23 -1 13 42]]

Optimal tunings:

  • WE: ~2 = 1199.7355 ¢, ~8/5 = 812.9622 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1353 ¢

Optimal ET sequence: 31, 90e, 121, 152, 335d, 487d

Badness (Sintel): 0.656

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 625/624, 1375/1372

Mapping: [⟨1 -14 3 -6 -25 22], ⟨0 23 -1 13 42 -27]]

Optimal tunings:

  • WE: ~2 = 1199.4412 ¢, ~8/5 = 812.7956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1209 ¢

Optimal ET sequence: 31, 90e, 121, 152f, 273def, 425deff

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274

Mapping: [⟨1 -14 3 -6 -25 22 19], ⟨0 23 -1 13 42 -27 -22]]

Optimal tunings:

  • WE: ~2 = 1199.3029 ¢, ~8/5 = 812.7156 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1843 ¢

Optimal ET sequence: 31, 90e, 121, 152fg, 273defgg

Badness (Sintel): 1.09

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714

Mapping: [⟨1 -14 3 -6 -25 22 19 30], ⟨0 23 -1 13 42 -27 -22 -38]]

Optimal tunings:

  • WE: ~2 = 1199.3587 ¢, ~8/5 = 812.7462 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1796 ¢

Optimal ET sequence: 31, 90e, 121, 152fg, 273defgg

Badness (Sintel): 1.12

Aufo

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[2]. The functional generator however is the 64/45 diminished fifth, and like its untriton variant, nine generator steps give the interval class of 3. The ploidacot for this temperament is delta-enneacot.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175616

Mapping: [⟨1 -3 12 -14], ⟨0 9 -19 33]]

mapping generators: ~2, ~64/45

Optimal tunings:

  • WE: ~2 = 1199.9758 ¢, ~64/45 = 611.2055 ¢
error map: ⟨-0.024 -1.303 +0.491 +1.295]
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2177 ¢
error map: ⟨0.000 -0.996 +0.551 +1.357]

Optimal ET sequence: 53, 161, 214

Badness (Sintel): 3.07

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 177147/175616

Mapping: [⟨1 -3 12 -14 6], ⟨0 9 -19 33 -5]]

Optimal tunings:

  • WE: ~2 = 1200.4500 ¢, ~64/45 = 611.4185 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.1918 ¢

Optimal ET sequence: 53, 108e, 161e

Badness (Sintel): 2.93

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350, 58806/57967

Mapping: [⟨1 -3 12 -14 6 20], ⟨0 9 -19 33 -5 -32]]

Optimal tunings:

  • WE: ~2 = 1200.3134 ¢, ~64/45 = 611.3715 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2118 ¢

Optimal ET sequence: 53, 108e

Badness (Sintel): 2.42

Aufic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625, 72171/71680

Mapping: [⟨1 -3 12 -14 33], ⟨0 9 -19 33 -58]]

Optimal tunings:

  • WE: ~2 = 1200.0668 ¢, ~64/45 = 611.2342 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2000 ¢

Optimal ET sequence: 53, 108, 161, 214, 375

Badness (Sintel): 2.48

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 4096/4095

Mapping: [⟨1 -3 12 -14 33 20], ⟨0 9 -19 33 -58 -32]]

Optimal tunings:

  • WE: ~2 = 1200.0177 ¢, ~64/45 = 611.2130 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2039 ¢

Optimal ET sequence: 53, 108, 161, 214, 375

Badness (Sintel): 1.61

Absurdity

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Absurdity (5-limit).
See also: Fifth-chroma temperaments

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175000

Mapping: [⟨7 0 -17 64], ⟨0 1 3 -4]]

mapping generators: ~972/875, ~3

Optimal tunings:

  • WE: ~972/875 = 171.4382 ¢, ~3/2 = 700.6247 ¢
error map: ⟨+0.067 -1.263 +1.313 +0.450]
  • CWE: ~972/875 = 171.4286 ¢, ~3/2 = 700.5871 ¢
error map: ⟨0.000 -1.368 +1.162 +0.254]

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 3.38

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 72171/71680

Mapping: [⟨7 0 -17 64 124], ⟨0 1 3 -4 -9]]

Optimal tunings:

  • WE: ~495/448 = 171.4346 ¢, ~3/2 = 700.6602 ¢
  • CWE: ~495/448 = 171.4286 ¢, ~3/2 = 700.6339 ¢

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 2.70

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 1188/1183, 3584/3575

Mapping: [⟨7 0 -17 64 124 37], ⟨0 1 3 -4 -9 -1]]

Optimal tunings:

  • WE: ~72/65 = 171.4223 ¢, ~3/2 = 700.6036 ¢
  • CWE: ~72/65 = 171.4286 ¢, ~3/2 = 700.6306 ¢

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 1.72

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625

Mapping: [⟨7 0 -17 64 124 37 -49], ⟨0 1 3 -4 -9 -1 7]]

Optimal tunings:

  • WE: ~72/65 = 171.4263 ¢, ~3/2 = 700.6429 ¢
  • CWE: ~72/65 = 171.4286 ¢, ~3/2 = 700.6525 ¢

Optimal ET sequence: 77, 161

Badness (Sintel): 1.62

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: [⟨7 0 -17 64 124 37 -49 63], ⟨0 1 3 -4 -9 -1 7 -3]]

Optimal tunings:

  • WE: ~21/19 = 171.4244 ¢, ~3/2 = 700.6395 ¢
  • CWE: ~21/19 = 171.4286 ¢, ~3/2 = 700.6568 ¢

Optimal ET sequence: 77, 161

Badness (Sintel): 1.36

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: [⟨7 0 -17 64 124 37 -49 63 76], ⟨0 1 3 -4 -9 -1 7 -3 -4]]

Optimal tunings:

  • WE: ~21/19 = 171.4321 ¢, ~3/2 = 700.6475 ¢
  • CWE: ~21/19 = 171.4286 ¢, ~3/2 = 700.6325 ¢

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 1.34

29-limit

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

Mapping: [⟨7 0 -17 64 124 37 -49 63 76 34], ⟨0 1 3 -4 -9 -1 7 -3 -4 0]]

Optimal tunings:

  • WE: ~21/19 = 171.4348 ¢, ~3/2 = 700.6612 ¢
  • CWE: ~21/19 = 171.4286 ¢, ~3/2 = 700.6351 ¢

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 1.25

Polypyth

For the 5-limit version, see Miscellaneous 5-limit temperaments #Leapday.

Polypyth tempers out the same 5-limit comma as leapday, with which it shares the similarly sharp 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 46 & 121 temperament, and 121edo and 167edo make for good tunings.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 179200/177147

Mapping: [⟨1 0 -31 52], ⟨0 1 21 -31]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.3465 ¢, ~3/2 = 703.7905 ¢
error map: ⟨-0.654 +1.182 -0.177 -0.056]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1749 ¢
error map: ⟨0.000 +2.220 +1.359 +1.752]

Optimal ET sequence: 46, 121, 167, 288b, 455bcd

Badness (Sintel): 3.49

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 6144/6125

Mapping: [⟨1 0 -31 52 59], ⟨0 1 21 -31 -35]]

Optimal tunings:

  • WE: ~2 = 1199.3335 ¢, ~3/2 = 703.7856 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1812 ¢

Optimal ET sequence: 46, 121, 167, 288be, 455bcde

Badness (Sintel): 1.69

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 1716/1715

Mapping: [⟨1 0 -31 52 59 64], ⟨0 1 21 -31 -35 -38]]

Optimal tunings:

  • WE: ~2 = 1199.3768 ¢, ~3/2 = 703.8018 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1731 ¢

Optimal ET sequence: 46, 75e, 121, 167, 288be

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715

Mapping: [⟨1 0 -31 52 59 64 39], ⟨0 1 21 -31 -35 -38 -22]]

Optimal tunings:

  • WE: ~2 = 1199.3518 ¢, ~3/2 = 703.7880 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1747 ¢

Optimal ET sequence: 46, 75e, 121, 167, 288beg

Badness (Sintel): 0.971

Whoops

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[2].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 244140625/243045684

Mapping: [⟨1 -16 -11 14], ⟨0 33 25 -21]]

mapping generators: ~2, ~640/441

Optimal tunings:

  • WE: ~2 = 1199.5944 ¢, ~640/441 = 639.2648 ¢
error map: ⟨-0.406 +0.272 -0.233 +0.936]
  • CWE: ~2 = 1200.0000 ¢, ~640/441 = 639.4769 ¢
error map: ⟨0.000 +0.783 +0.609 +2.159]

Optimal ET sequence: 15, 122d, 137, 152, 623bdd, 775bcdd, 927bcddd, 1079bcddd

Badness (Sintel): 4.45

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 6144/6125

Mapping: [⟨1 -16 -11 14 -4], ⟨0 33 25 -21 14]]

Optimal tunings:

  • WE: ~2 = 1199.5936 ¢, ~175/121 = 639.264 ¢
  • CWE: ~2 = 1200.0000 ¢, ~175/121 = 639.4770 ¢

Optimal ET sequence: 15, 122d, 137, 152, 623bdde, 775bcdde, 927bcdddee, 1079bcdddee

Badness (Sintel): 1.45

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[2]. The extension here is a less accurate 7-limit interpretation.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 2500000/2470629

Mapping: [⟨1 -23 -4 0], ⟨0 35 9 4]]

mapping generators: ~2, ~80/49

Optimal tunings:

  • WE: ~2 = 1199.6429 ¢, ~80/49 = 842.6790 ¢
error map: ⟨-0.357 +0.228 -0.774 +1.890]
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 842.9243 ¢
error map: ⟨0.000 +0.396 +0.005 +2.871]

Optimal ET sequence: 37, 84, 121, 205

Badness (Sintel): 4.55

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2560/2541, 4375/4356

Mapping: [⟨1 -23 -4 0 14], ⟨0 35 9 4 -15]]

Optimal tunings:

  • WE: ~2 = 1199.3401 ¢, ~80/49 = 842.4880 ¢
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 842.9457 ¢

Optimal ET sequence: 37, 84, 121, 326dee

Badness (Sintel): 2.71

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 625/624, 640/637, 1375/1372

Mapping: [⟨1 12 5 4 -1 4], ⟨0 -35 -9 -4 15 -1]]

Optimal tunings:

  • WE: ~2 = 1199.3410 ¢, ~13/8 = 842.4885 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.9466 ¢

Optimal ET sequence: 37, 84, 121, 326deef

Badness (Sintel): 1.63

Icositritonic

See also: 23rd-octave temperaments

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 46 & 161. It was named by Xenllium in 2019 for its number of periods per octave.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 9920232/9765625

Mapping: [⟨23 0 17 101], ⟨0 1 1 -1]]

mapping generators: ~1323/1280, ~3

Optimal tunings:

  • WE: ~1323/1280 = 52.1732 ¢, ~3/2 = 701.0660 ¢
error map: ⟨-0.017 -0.906 +1.679 -0.386]
  • CWE: ~1323/1280 = 52.1739 ¢, ~3/2 = 701.0722 ¢
error map: ⟨0.000 -0.883 +1.715 -0.333]

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Sintel): 4.98

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 35937/35840

Mapping: [⟨23 0 17 101 116], ⟨0 1 1 -1 -1]]

Optimal tunings:

  • WE: ~33/32 = 52.1740 ¢, ~3/2 = 701.0379 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.0370 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Sintel): 2.14

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 847/845, 3584/3575

Mapping: [⟨23 0 17 101 116 158], ⟨0 1 1 -1 -1 -2]]

Optimal tunings:

  • WE: ~33/32 = 52.1724 ¢, ~3/2 = 701.1310 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.1524 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Sintel): 1.67

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088

Mapping: [⟨23 0 17 101 116 158 94], ⟨0 1 1 -1 -1 -2 0]]

Optimal tunings:

  • WE: ~33/32 = 52.1735 ¢, ~3/2 = 701.1493 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.1549 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Sintel): 1.26

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845

Mapping: [⟨23 0 17 101 116 158 94 207], ⟨0 1 1 -1 -1 -2 0 -3]]

Optimal tunings:

  • WE: ~33/32 = 52.1744 ¢, ~3/2 = 701.0649 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.0582 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Sintel): 1.31

23-limit

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: [⟨23 0 17 101 116 158 94 207 104], ⟨0 1 1 -1 -1 -2 0 -3 0]]

Optimal tunings:

  • WE: ~33/32 = 52.1768 ¢, ~3/2 = 701.1259 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.0841 ¢

Optimal ET sequence: 46, 115, 161, 207

Badness (Sintel): 1.27

References

  1. ↑ Yahoo! Tuning Group | Some more unclassified temperaments
  2. ↑ 2.0 2.1 2.2 2.3 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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