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]]

* ''[[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]]

* ''[[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]]

 

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

 

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 }}

* [[WE]]: ~16/15 = 109.0526{{c}}, ~3/2 = 702.8069{{c}}

: [[error map]]: {{val| -0.421 +0.431 +0.563 -0.265 }}

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

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

[[Badness]] (Sintel): 1.04

Subgroup: 2.3.5.7.11

 

 

 

 

 

==== 13-limit ====

 

 

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

 

 

 

 

==== 17-limit ====

 

 

 

 

 

 

=== Cohendecatonic ===

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

 

 

 

 

 

 

 

 

 

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

 

 

 

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

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 }}

 

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

 

{{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}

 

Badness (Sintel): 1.84

 

 

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"/>.

 

 

 

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

* [[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 ====

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[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

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1209{{c}}

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

Badness (Sintel): 1.03

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

 

 

* WE: ~2 = 1199.3029{{c}}, ~8/5 = 812.7156{{c}}

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

 

 

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 }}

 

 

Badness (Sintel): 1.12

 

[[Subgroup]]: 2.3.5.7

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

* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}

: [[error map]]: {{val| -0.693 +1.736 +3.278 -5.176 }}

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

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

[[Badness]] (Sintel): 2.52

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

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

Optimal tunings:  

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

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

Badness (Sintel): 1.07

== Semaja ==

Cryptically named by [[Petr Pařízek]] in 2011, semaja adds the [[gariboh comma]] to the comma list, 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

: mapping generators: ~2, ~8/7

: [[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 }}

[[Badness]] (Sintel): 2.71

Subgroup: 2.3.5.7.11

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

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

Optimal 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 ===

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:  

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

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

Badness (Sintel): 1.35

== 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<ref name="petr's long post"/>. 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

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

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

[[Optimal tuning]]s:  

: [[error map]]: {{val| -0.024 -1.303 +0.491 +1.295 }}

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

[[Badness]] (Sintel): 3.07

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.1918{{c}}

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

Badness (Sintel): 2.93

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2118{{c}}

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

Badness (Sintel): 2.42

=== Aufic ===

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

 

 

 

 

 

 

 

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2039{{c}}

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

Badness (Sintel): 1.61

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

[[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

 

 

 

 

 

 

 

 

* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6568{{c}}

 

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

 

 

 

Comma list: 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494

 

 

 

 

 

 

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:  

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

{{Optimal ET sequence|legend=0| 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 [[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 }}

[[Optimal tuning]]s:

* [[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, 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

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

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

 

 

* 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 }}

Badness (Sintel): 0.971

: ''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:  

* [[CWE]]: ~2 = 1200.0000{{c}}, ~640/441 = 639.4769{{c}}

: error map: {{val| 0.000 +0.783 +0.609 +2.159 }}

{{Optimal ET sequence|legend=1| 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: {{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

== 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<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 }}

[[Optimal tuning]]s:

* [[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| 37, 84, 121, 205 }}

[[Badness]] (Sintel): 4.55

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 ==

 

 

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}

* [[CWE]]: ~1323/1280 = 52.1739{{c}}, ~3/2 = 701.0722{{c}}

: 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

=== 11-limit ===

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

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

Optimal tunings:  

* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0370{{c}}

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

Badness (Sintel): 2.14

=== 13-limit ===

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Optimal tunings:  

* WE: ~33/32 = 52.1768{{c}}, ~3/2 = 701.1259{{c}}

* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0841{{c}}

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

Badness (Sintel): 1.27

[[Category:Temperament collections]]

[[Category:Porwell temperaments| ]] <!-- main article -->