Porwell temperaments: Difference between revisions

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:  

* ''[[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]]

* ''[[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]]

* ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]]

* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]

* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]

* ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]]

* ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]]

* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]

* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]

* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]

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

* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]

 

 

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.  

[[Subgroup]]: 2.3.5.7

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

[[Optimal tuning]]s:

* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 702.9705{{c}}

: error map: {{val| 0.000 +1.015 +1.625 +0.751 }}

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

[[Badness]] (Sintel): 1.04

=== Hendecaton ===

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

Optimal tunings:  

* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.6484{{c}}

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

Badness (Sintel): 1.52

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

Optimal tunings:  

* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.2930{{c}}

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

Badness (Sintel): 1.66

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

Optimal tunings:  

* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.3017{{c}}

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

Badness (Sintel): 1.48

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

Optimal tunings:  

* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.6563{{c}}

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

Badness (Sintel): 1.26

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

Optimal tunings:  

* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.9248{{c}}

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

Badness (Sintel): 1.49

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

Optimal tunings:  

* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.9110{{c}}

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

Badness (Sintel): 1.15

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

Optimal tunings:  

* CWE: ~33/32 = 54.5455{{c}}, ~3/2 = 702.8829{{c}}

{{Optimal ET sequence|legend=0| 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<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

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

{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}

[[Optimal tuning]]s:  

* [[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 }}

[[Badness]] (Sintel): 1.14

=== Nessa ===

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

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

Optimal tunings:  

* CWE: ~44/35 = 400.0000{{c}}, ~35/32 = 157.5109{{c}}

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

Badness (Sintel): 1.61

==== 13-limit ====

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

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

Optimal tunings:  

* CWE: ~44/35 = 400.0000{{c}}, ~35/32 = 157.3955{{c}}

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

Badness (Sintel): 1.55

 

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

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

Optimal tunings:  

* CWE: ~63/50 = 400.0000{{c}}, ~12/11 = 157.5240{{c}}

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

Badness (Sintel): 2.26

: ''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 {{nowrap| 31 & 152 }} temperament. [[152edo]], [[183edo]] and especially [[335edo]] serve as good tunings.  

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 6144/6125, 16875/16807

{{Mapping|legend=1| 1 -14 3 -6 | 0 23 -1 13 }}

[[Optimal tuning]]s:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 813.1311{{c}}

: error map: {{val| 0.000 +0.059 +0.555 +1.878 }}

{{Optimal ET sequence|legend=1| 31, 90, 121, 152, 335d, 822dd }}

[[Badness]] (Sintel): 1.31

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

Mapping: {{mapping| 1 -14 3 -6 -25 | 0 23 -1 13 42 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1353{{c}}

{{Optimal ET sequence|legend=0| 31, 90e, 121, 152, 335d, 487d }}

Badness (Sintel): 0.656

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

 

 

 

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

 

 

 

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

Mapping: {{mapping| 1 -14 3 -6 -25 22 19 | 0 23 -1 13 42 -27 -22 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1843{{c}}

{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -14 3 -6 -25 22 19 30 | 0 23 -1 13 42 -27 -22 -38 }}

* 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| 31, 90e, 121, 152fg, 273defgg }}

Badness (Sintel): 1.12

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.  

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 -10 5 -7 | 0 13 -3 11 }}

[[Optimal tuning]]s:  

* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}

: error map: {{val| 0.000 +2.515 +4.962 -3.505 }}

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

[[Badness]] (Sintel): 2.52

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

Optimal tunings:  

* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}

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

Badness (Sintel): 1.35

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 176/175

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

Optimal tunings:  

* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}

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

Badness (Sintel): 1.07

{{See also| Llywelynsmic clan }}

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 (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). 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 }}

[[Optimal tuning]]s:  

* [[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

=== 11-limit ===

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

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4851{{c}}

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

Badness (Sintel): 1.98

=== 13-limit ===

Comma list: 121/120, 169/168, 176/175, 275/273

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4822{{c}}

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

Badness (Sintel): 1.35

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

 

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 -3 12 -14 | 0 9 -19 33 }}

: [[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 }}

[[Badness]] (Sintel): 3.07

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

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 }}

Badness (Sintel): 2.93

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

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 }}

Badness (Sintel): 2.42

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

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 }}

Badness (Sintel): 2.48

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

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 }}

Badness (Sintel): 1.61

== Absurdity ==

{{See also| Fifth-chroma temperaments }}

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}

[[Optimal tuning]]s:  

* [[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| 77, 84, 161 }}

[[Badness]] (Sintel): 3.38

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

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

Optimal tunings:  

* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 700.6339{{c}}

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

Badness (Sintel): 2.70

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

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

Optimal tunings:  

* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6306{{c}}

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

Badness (Sintel): 1.72

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

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

Optimal tunings:  

* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6525{{c}}

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

* 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| 77, 161 }}

Badness (Sintel): 1.36

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: {{mapping| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}

Optimal tunings:  

* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6325{{c}}

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

Badness (Sintel): 1.34

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: {{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{{c}}, ~3/2 = 700.6612{{c}}

* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6351{{c}}

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

Badness (Sintel): 1.25

: ''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 [[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

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

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

: mapping generators: ~2, ~3

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.3465{{c}}, ~3/2 = 703.7905{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1749{{c}}

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

[[Badness]] (Sintel): 3.49

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

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1812{{c}}

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

Badness (Sintel): 1.69

=== 13-limit ===

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

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1731{{c}}

{{Optimal ET sequence|legend=0| 46, 75e, 121, 167, 288be }}

Badness (Sintel): 1.25

=== 17-limit ===

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:  

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1747{{c}}

{{Optimal ET sequence|legend=0| 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''<ref name="petr's long post"/>.

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 -16 -11 14 | 0 33 25 -21 }}

[[Optimal tuning]]s:  

: [[error map]]: {{val| -0.406 +0.272 -0.233 +0.936 }}

{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 623bdd, 775bcdd, 927bcddd, 1079bcddd }}

[[Badness]] (Sintel): 4.45

Subgroup: 2.3.5.7.11

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

Mapping: {{mapping| 1 -16 -11 14 -4 | 0 33 25 -21 14 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~175/121 = 639.4770{{c}}

{{Optimal ET sequence|legend=0| 15, 122d, 137, 152, 623bdde, 775bcdde, 927bcdddee, 1079bcdddee }}

Badness (Sintel): 1.45

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

Also named by [[Petr Pařízek]] in 2011, ''dodifo'' refers to the (tetraptolemaic) double-diminished fourth, which is a generator of this temperament<ref name="petr's long post"/>. The extension here is a less accurate 7-limit interpretation.  

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 1 -23 -4 0 | 0 35 9 4 }}

: mapping generators: ~2, ~80/49

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.6429{{c}}, ~80/49 = 842.6790{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9243{{c}}

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

[[Badness]] (Sintel): 4.55

=== 11-limit ===

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

Mapping: {{mapping| 1 -23 -4 0 14 | 0 35 9 4 -15 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~80/49 = 842.9457{{c}}

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

Badness (Sintel): 2.71

=== 13-limit ===

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 tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9466{{c}}

{{Optimal ET sequence|legend=0| 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 {{nowrap| 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|legend=1| 23 0 17 101 | 0 1 1 -1 }}

[[Optimal tuning]]s:  

: error map: {{val| 0.000 -0.883 +1.715 -0.333 }}

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

[[Badness]] (Sintel): 4.98

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

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

Optimal tunings:  

* WE: ~33/32 = 52.1740{{c}}, ~3/2 = 701.0379{{c}}