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Porwell temperaments: Difference between revisions

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__FORCETOC__
{{Technical data page}}
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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 family of temperaments tempers out the ''porwell comma'', |11 1 -3 -2> = 6144/6125, and includes hemiwuerschmidt, orwell, amity, valentine, porcupine, hendecatonic, shrutar, hexadecimal, grendel, hemikleismic, mohajira, twothirdtonic and nessafof.


=Hendecatonic=
Discussed elsewhere are:
Commas: 6144/6125, 10976/10935
* ''[[Armodue]]'' (+36/35) → [[Pelogic family #Armodue|Pelogic family]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Porcupine|Porcupine family]]
* [[Mohajira]] (+81/80) → [[Meantone family #Mohajira|Meantone family]]
* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]]
* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]
* [[Shrutar]] (+245/243) → [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]]
* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* [[Amity]] (+4375/4374 or 5120/5103) → [[Amity family #Septimal amity|Amity family]]
* ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]]
* ''[[Grendel]]'' (+16875/16807) → [[Mirkwai clan #Grendel|Mirkwai clan]]
* ''[[Hemischis]]'' (+19683/19600) → [[Schismatic family #Hemischis|Schismatic family]]
* ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Alphatrident]]'' (+14348907/14336000) → [[Alphatricot family #Alphatrident|Alphatricot family]]
* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]
* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]
* ''[[Twilight]]'' (+{{monzo| 19 -22 2 4 }}) → [[Undim family #Twilight|Undim family]]


[[POTE_tuning|POTE generator]]: ~3/2 = 703.054
== Hendecatonic ==
{{see also|11th-octave temperaments}}


Map: [<11 0 43 -4], <0 1 -1 2|]
The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]].


Wedgie: <<11 -11 22 -43 4 82||
[[Subgroup]]: 2.3.5.7


EDOs: 22, 55, 77, [[99edo|99]]
[[Comma list]]: 6144/6125, 10976/10935


Badness: 0.0411
{{Mapping|legend=1| 11 0 43 -4 | 0 1 -1 2 }}


==11-limit==
: Mapping generators: ~16/15, ~3
Commas: 121/120, 176/175, 10976/10935


[[POTE_tuning|POTE generator]]: ~3/2 = 702.636
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\11, ~3/2 = 703.054


Map: [<11 0 43 -4 38], <0 1 -1 2 0|]
{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}


EDOs: 22, 55, 77, 99, 176e, 275e
[[Badness]]: 0.041081


Badness: 0.0461
=== 11-limit ===
Subgroup: 2.3.5.7.11


==Icosidillic==
Comma list: 121/120, 176/175, 10976/10935
Commas: 3388/3375, 6144/6125, 9801/9800


[[POTE_tuning|POTE generator]]: 702.914
{{Mapping|legend=1| 11 0 43 -4 38 | 0 1 -1 2 0 }}


Map: [<22 0 86 -8 111|, <0 1 -1 2 -1|]
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.636


EDOs: 22, 154, 176, 198, 968, 1166
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176e, 275e }}


Badness: 0.0577
Badness: 0.046088


=Hemischis=
==== 13-limit ====
Commas: 6144/6125, 19683/19600
Subgroup: 2.3.5.7.11.13


POTE generator: ~81/70 = 249.203
Comma list: 121/120, 176/175, 351/350, 4459/4455


Map: [<1 0 15 -17|, <0 2 -16 25|]
{{Mapping|legend=1| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}


Wedgie: <<2 -16 25 -30 34 103||
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.291


EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 313
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176e }}


Badness: 0.0458
Badness: 0.040099


==11-limit==
==== 17-limit ====
Commas: 540/539, 8019/8000, 5632/5625
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~81/70 = 249.199
Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023


Map: [<1 0 15 -17 51|, <0 2 -16 25 -60|]
{{Mapping|legend=1| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}


EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 313
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301


Badness: 0.0363
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176eg }}


==13-limit==
Badness: 0.029054
Commas: 351/350, 540/539, 676/675, 4096/4095


POTE generator: ~15/13 = 249.199
=== Cohendecatonic ===
Subgroup: 2.3.5.7.11


Map: [<1 0 15 -17 51 14|, <0 2 -16 25 -60 -13|]
Comma list: 540/539, 896/891, 4375/4356


EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 313
{{Mapping|legend=1| 11 0 43 -4 73 | 0 1 -1 2 -2 }}


Badness: 0.0208
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.686


==17-limit==
{{Optimal ET sequence|legend=0| 22, 77e, 99e, 121, 220e }}
Commas: 351/350, 442/441, 561/560, 676/675, 4096/4095


POTE generator: ~15/13 = 249.190
Badness: 0.038042


Map: [<1 0 15 -17 51 14|, <0 2 -16 25 -60 -13|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 679
Comma list: 352/351, 364/363, 540/539, 625/624


Badness: 0.0211
{{Mapping|legend=1| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}


=Twothirdtonic=
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.888
Commas: 686/675, 6144/6125


POTE generator: ~15/14 = 130.401
{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 341bdeeff }}


Map: [<1 3 2 4|, <0 -13 3 -11|]
Badness: 0.036112


Wedgie: <<13 -3 11 -35 -19 34||
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDOs: 9, 10, 19, 28, 37, 46
Comma list: 256/255, 352/351, 364/363, 375/374, 540/539


Badness: 0.0996
{{Mapping|legend=1| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}


==11-limit==
Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.877
Commas: 121/120, 176/175, 686/675


POTE generator: ~15/14 = 130.430
{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}


Map: [<1 3 2 4 4|, <0 -13 3 -11 -5|]
Badness: 0.022590


EDOs: 9, 10, 19, 28, 37, 46
=== Icosidillic ===
Subgroup: 2.3.5.7.11


Badness: 0.0408
Comma list: 3388/3375, 6144/6125, 9801/9800


==13-limit==
{{Mapping|legend=1| 22 0 86 -8 111 | 0 1 -1 2 -1 }}
Commas: 91/90, 121/120, 169/168, 176/175


POTE generator: ~15/14 = 130.409
: Mapping generators: ~33/32, ~3


Map: [<1 3 2 4 4 5|, <0 -13 3 -11 -5 -12|]
Optimal tuning (POTE): ~33/32 = 1\22, ~3/2 = 702.914


EDOs: 9, 10, 19, 28, 37, 46
{{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}


Badness: 0.0259
Badness: 0.057725


=Nessafof=
== Twothirdtonic ==
Commas: 6144/6125, 250047/250000
[[Subgroup]]: 2.3.5.7


POTE generator: ~35/32 = 157.420
[[Comma list]]: 686/675, 6144/6125


Map: [<3 2 5 10|, <0 7 5 -4|]
{{Mapping|legend=1| 1 3 2 4 | 0 -13 3 -11 }}


Wedgie: <<21 15 -12 -25 -78 -70||
: Mapping generators: ~2, ~15/14


EDOs: 15, 69, 84, 99, 282, 381
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 130.401


Badness: 0.0450
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}


=Septisuperfourth=
[[Badness]]: 0.099601
Commas: 6144/6125, 118098/117649


POTE generator: ~48/35 = 544.680
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [<2 4 4 7|, <0 -9 7 -15|]
Comma list: 121/120, 176/175, 686/675


Wedgie: <<18 -14 30 -64 -3 109||
Mapping: {{mapping| 1 3 2 4 4 | 0 -13 3 -11 -5 }}


EDOs: 22, 86, 108, 130, 152, 282
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.430


Badness: 0.0592
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}


==11-limit==
Badness: 0.040768
Commas: 540/539, 4000/3993, 5632/5625


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


Map: [<2 4 4 7 6|, <0 -9 7 -15 10|]
Comma list: 91/90, 121/120, 169/168, 176/175


EDOs: 22, 86, 108, 130, 152, 282, 434de, 716de
Mapping: {{mapping| 1 3 2 4 4 5 | 0 -13 3 -11 -5 -12 }}


Badness: 0.0246
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 130.409


==13-limit==
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
Commas: 540/539 729/728 4000/3993 21168/21125


POTE generator: ~48/35 = 544.675
Badness: 0.025941


Map: [<2 4 4 7 6 11|, <0 -9 7 -15 10 -39|]
== 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>.


EDOs: 130, 282
[[Subgroup]]: 2.3.5.7


Badness: 0.0229
[[Comma list]]: 3125/3087, 6144/6125


==Septisuperquad==
{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}
Commas: 351/350, 364/363, 540/539, 5632/5625


POTE generator: ~48/35 = 544.641
: Mapping generators: ~2, ~8/7


Map: [&lt;2 4 4 7 6 5|, &lt;0 -9 7 -15 10 26|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 226.4834


EDOs: 22, 108, 130
{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}


Badness: 0.0330
[[Badness]]: 0.107023


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


POTE generator: ~441/320 = 560.519
Comma list: 121/120, 176/175, 3125/3087


Map: [&lt;1 17 14 -7|, &lt;0 -33 -25 21|]
Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}


Wedgie: &lt;&lt;33 25 -21 -37 -126 -119||
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4856


EDOs: 15, 122d, 137, 152, 608d, 623bd, 775bcd
{{Optimal ET sequence|legend=1| 16, 37, 53 }}


Badness: 0.1758
Badness: 0.059838


==11-limit==
=== 13-limit ===
Commas: 6144/6125, 3025/3024, 4000/3993
Subgroup: 2.3.5.7.11.13


POTE generator: ~242/175 = 560.519
Comma list: 121/120, 169/168, 176/175, 275/273


Map: [&lt;1 17 14 -7 10|, &lt;0 -33 -25 21 -14|]
Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }}


EDOs: 15, 122d, 137, 152, 608de, 623bde, 775bcde
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4794


Badness: 0.0437
{{Optimal ET sequence|legend=1| 16, 37, 53 }}


=Polypyth=
Badness: 0.032564
Commas: 6144/6125, 179200/177147


POTE generator: ~3/2 = 704.174
== Nessafof ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments#Nessafof]].''


Map: [&lt;1 0 -31 52|, &lt;0 1 21 -31|]
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"/>.


EDOs: 46, 121, 167, 288b, 455bcd, 743bcd
[[Subgroup]]: 2.3.5.7


Badness: 0.13800
[[Comma list]]: 6144/6125, 250047/250000


==11-limit==
{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}
Commas: 896/891, 2200/2187, 6144/6125


POTE generator: ~3/2 = 704.177
: Mapping generators: ~63/50, ~35/32


Map: [&lt;1 0 -31 52 59|, &lt;0 1 21 -31 -35|]
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~35/32 = 157.480


EDOs: 46, 121, 167, 288be, 455bcde
{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}


Badness: 0.0511
[[Badness]]: 0.045048


==13-limit==
=== 11-limit ===
Commas: 325/324, 352/351, 364/363, 1716/1715
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 704.168
Comma list: 121/120, 176/175, 250047/250000


Map: [&lt;1 0 -31 52 59 64|, &lt;0 1 21 -31 -35 -38|]
Mapping: {{mapping| 3 2 5 10 8 | 0 7 5 -4 6 }}


EDOs: 46, 121, 167, 288be
Optimal tuning (POTE): ~63/50 = 1\3, ~12/11 = 157.520


Badness: 0.0303
{{Optimal ET sequence|legend=1| 15, 54be, 69e, 84e, 99 }}


==17-limit==
Badness: 0.068427
Commas: 256/255, 325/324, 352/351, 364/363, 1716/1715


POTE generator: ~3/2 = 704.168
=== Nessa ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -31 52 59 64 39|, &lt;0 1 21 -31 -35 -38 -22|]
Comma list: 441/440, 1344/1331, 4375/4356


EDOs: 46, 121, 167, 288beg
Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}


Badness: 0.0191
Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.539
[[Category:hemischis]]
 
[[Category:hendecatonic]]
{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99e }}
[[Category:porwell]]
 
[[Category:rank_2]]
Badness: 0.048836
[[Category:temperament]]
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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 tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.429
 
{{Optimal ET sequence|legend=1| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}
 
Badness: 0.037409
 
== Aufo ==
:''For the 5-limit version, see [[High badness 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"/>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 177147/175616
 
{{Mapping|legend=1| 1 6 -7 19 | 0 -9 19 -33 }}
 
: Mapping generators: ~2, ~45/32
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 588.782
 
{{Optimal ET sequence|legend=1| 53, 161, 214 }}
 
[[Badness]]: 0.121428
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 177147/175616
 
Mapping: {{mapping| 1 6 -7 19 1 | 0 -9 19 -33 5 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.811
 
{{Optimal ET sequence|legend=1| 53, 108e, 161e }}
 
Badness: 0.088631
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 351/350, 58806/57967
 
Mapping: {{mapping| 1 6 -7 19 1 -12 | 0 -9 19 -33 5 32 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.788
 
{{Optimal ET sequence|legend=1| 53, 108e, 161e, 214ee }}
 
Badness: 0.058507
 
=== Aufic ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 5632/5625, 72171/71680
 
Mapping: {{mapping| 1 6 -7 19 -25 | 0 -9 19 -33 58 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.800
 
{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375 }}
 
Badness: 0.075149
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 847/845, 4096/4095
 
Mapping: {{mapping| 1 6 -7 19 -25 -12 | 0 -9 19 -33 58 32 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.796
 
{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375, 589be }}
 
Badness: 0.039050
 
== 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 17 14 -7 | 0 -33 -25 21 }}
 
: Mapping generators: ~2, ~441/320
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~441/320 = 560.519
 
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608d, 623bd, 775bcd }}
 
[[Badness]]: 0.175840
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4000/3993, 6144/6125
 
Mapping: {{mapping| 1 17 14 -7 10 | 0 -33 -25 21 -14 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~242/175 = 560.519
 
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
 
Badness: 0.043743
 
== Polypyth ==
:''For the 5-limit version, see [[High badness temperaments #Leapday]].''
 
Polypyth (46 &amp; 121) tempers out the same 5-limit comma as the [[Hemifamity temperaments #Leapday|leapday temperament]] (29 &amp; 46), but with the porwell (6144/6125) rather than the hemifamity (5120/5103) tempered out.
 
[[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]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.174
 
{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}
 
[[Badness]]: 0.137995
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 896/891, 2200/2187, 6144/6125
 
Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.177
 
{{Optimal ET sequence|legend=1| 46, 121, 167, 288be, 455bcde }}
 
Badness: 0.051131
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 tuning (POTE): ~2 = 1\1, ~3/2 = 704.168
 
{{Optimal ET sequence|legend=1| 46, 121, 167, 288be }}
 
Badness: 0.030292
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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 tuning (POTE): ~2 = 1\1, ~3/2 = 704.168
 
{{Optimal ET sequence|legend=1| 46, 121, 167, 288beg }}
 
Badness: 0.019051
 
== Icositritonic ==
{{ See also | 23rd-octave temperaments }}
The icositritonic temperament (46 &amp; 161) has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 9920232/9765625
 
{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}
 
: Mapping generators: ~1323/1280, ~3
 
[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 1\23, ~64/63 = 29.3586
 
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
 
[[Badness]]: 0.196622
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 6144/6125, 35937/35840
 
Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}
 
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3980
 
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
 
Badness: 0.064613
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 441/440, 847/845, 3584/3575
 
Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}
 
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2830
 
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
 
Badness: 0.040484
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088
 
Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}
 
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.2800
 
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
 
Badness: 0.024676
 
=== 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: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}
 
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3760
 
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
 
Badness: 0.021579
 
=== 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: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}
 
Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3471
 
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368ci }}
 
Badness: 0.017745
 
== Countermiracle ==
The ''countermiracle'' temperament (31 &amp; 145) tempers out the trimyna, 50421/50000 and the [[quince comma]], 823543/819200.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 50421/50000
 
{{Mapping|legend=1| 1 4 3 3 | 0 -25 -7 -2 }}
 
: Mapping generators: ~2, ~343/320
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/320 = 115.9169
 
{{Optimal ET sequence|legend=1| 31, 114, 145, 176, 559cc, 735cc }}
 
[[Badness]]: 0.102326
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3388/3375, 6144/6125
 
Mapping: {{mapping| 1 4 3 3 8 | 0 -25 -7 -2 -47 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9158
 
{{Optimal ET sequence|legend=1| 31, 114e, 145, 176 }}
 
Badness: 0.039162
 
==== Countermiraculous ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 1001/1000, 6144/6125
 
Mapping: {{mapping| 1 4 3 3 8 1 | 0 -25 -7 -2 -47 28 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8803
 
{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145, 321ceff }}
 
Badness: 0.039271
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 196/195, 256/255, 352/351, 1001/1000, 1225/1224
 
Mapping: {{mapping| 1 4 3 3 8 1 1 | 0 -25 -7 -2 -47 28 32 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8756
 
{{Optimal ET sequence|legend=1| 31, 83e, 114e, 145 }}
 
Badness: 0.029496
 
==== Counterbenediction ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 441/440, 3146/3125, 3584/3575
 
Mapping: {{mapping| 1 4 3 3 8 -2 | 0 -25 -7 -2 -47 59 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9335
 
{{Optimal ET sequence|legend=1| 31, 114ef, 145f, 176, 207, 383c, 590cc }}
 
Badness: 0.045569
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 441/440, 561/560, 1632/1625, 3146/3125
 
Mapping: {{mapping| 1 4 3 3 8 -2 -2 | 0 -25 -7 -2 -47 59 63 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.9391
 
{{Optimal ET sequence|legend=1| 31, 114efg, 145fg, 176, 207 }}
 
Badness: 0.036289
 
==== Countermanna ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 3388/3375, 6144/6125
 
Mapping: {{mapping| 1 4 3 3 8 15  0 -25 -7 -2 -47 -117 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8898
 
{{Optimal ET sequence|legend=1| 145, 176, 321ce }}
 
Badness: 0.053409
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 1632/1625, 3388/3375
 
Mapping: {{mapping| 1 4 3 3 8 15 15 | 0 -25 -7 -2 -47 -117 -113 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.8832
 
{{Optimal ET sequence|legend=1| 145, 321ce }}
 
Badness: 0.040898
 
=== Counterrevelation ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 50421/50000
 
Mapping: {{mapping| 1 4 3 3 5 | 0 -25 -7 -2 -16 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~343/320 = 115.9192
 
{{Optimal ET sequence|legend=1| 31, 114, 145e, 176e }}
 
Badness: 0.064070
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 196/195, 13750/13689
 
Mapping: {{mapping| 1 4 3 3 5 1 | 0 -25 -7 -2 -16 28 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~273/256 = 115.8624
 
{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
 
Badness: 0.057497
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 121/120, 154/153, 176/175, 196/195, 10647/10625
 
Mapping: {{mapping| 1 4 3 3 5 1 1 | 0 -25 -7 -2 -16 28 32 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~91/85 = 115.8527
 
{{Optimal ET sequence|legend=1| 31, 83, 114, 145e }}
 
Badness: 0.044043
 
== Absurdity ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Absurdity]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 177147/175000
 
{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}
 
: Mapping generators: ~972/875, ~3
 
[[Optimal tuning]] ([[POTE]]): ~972/875 = 1\7, ~3/2 = 700.5854 (or ~10/9 = 186.2997)
 
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
 
[[Badness]]: 0.133520
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 6144/6125, 72171/71680
 
{{Mapping|legend=1| 7 0 -17 64 124 | 0 1 3 -4 -9 }}
 
Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 700.6354 (or ~10/9 = 186.3497)
 
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
 
Badness: 0.081564
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 441/440, 1188/1183, 3584/3575
 
{{Mapping|legend=1| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}
 
Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6291 (or ~10/9 = 186.3434)
 
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
 
Badness: 0.041600
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625
 
{{Mapping|legend=1| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}
 
Optimal tuning (POTE): ~72/65 = 1\7, ~3/2 = 700.6524 (or ~10/9 = 186.3667)
 
{{Optimal ET sequence|legend=1| 77, 161 }}
 
Badness: 0.031783
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
 
{{Mapping|legend=1| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}
 
Optimal tuning (POTE): ~21/19 = 1\7, ~3/2 = 700.6565 (or ~10/9 = 186.3708)
 
{{Optimal ET sequence|legend=1| 77, 161 }}
 
Badness: 0.022291
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
 
{{Mapping|legend=1| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}
 
Optimal tuning ([[CTE]]): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
 
=== 29-limit ===
{{ See also | Fifth-chroma temperaments }}
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 261/260, 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
 
{{Mapping|legend=1| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}
 
Optimal tuning ([[CTE]]): ~21/19 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
 
== Dodifo ==
: ''For the 5-limit version, see [[High badness temperaments #Dodifo]].''
 
Also named by [[Petr Pařízek]] in 2011, ''dodifo'' refers to the (tetraptolemaic) double-diminished fourth, which is a generator of this temperament<ref name="petr's long post"/>. The extension here is a less accurate 7-limit intepretation.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 2500000/2470629
 
{{Mapping|legend=1| 1 12 5 4 | 0 -35 -9 -4 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 357.070
 
{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}
 
[[Badness]]: 0.179692
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 2560/2541, 4375/4356
 
Mapping: {{mapping| 1 12 5 4 -1 | 0 -35 -9 -4 15 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.048
 
{{Optimal ET sequence|legend=1| 37, 84, 121, 326dee }}
 
Badness: 0.081923
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 625/624, 640/637, 1375/1372
 
Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 357.049
 
{{Optimal ET sequence|legend=1| 37, 84, 121, 326deef }}
 
Badness: 0.039533
 
== Notes ==
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Porwell| ]] <!-- key article -->
[[Category:Hendecatonic]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Porwell_temperaments"