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]]).  

* ''[[Alphatrident]]'' (+14348907/14336000) → [[Alphatricot family #Alphatrident|Alphatricot 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]]

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

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

=== Hendecaton ===

 

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>.  

== 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.  

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

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

: mapping generators: ~2, ~64/45

* [[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 }}

[[Badness]] (Sintel): 3.07

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

 

 

 

 

 

 

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

* 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

=== Aufic ===

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

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

* 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

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

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

* 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 ==

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

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

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

: mapping generators: ~972/875, ~3

[[Optimal tuning]]s:

* [[WE]]: ~972/875 = 171.4382{{c}}, ~3/2 = 700.6247{{c}}

* [[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:  

* WE: ~495/448 = 171.4346{{c}}, ~3/2 = 700.6602{{c}}

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

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

Badness (Sintel): 2.70

=== 13-limit ===

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:  

* 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 }}

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

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=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 }}

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| 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: {{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

=== 29-limit ===

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

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.  

[[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

Optimal tunings:

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

Badness (Sintel): 1.69

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

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"/>.  

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

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

: mapping generators: ~2, ~640/441

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.5944{{c}}, ~640/441 = 639.2648{{c}}

* [[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

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

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

Optimal tunings:  

* WE: ~2 = 1199.5936{{c}}, ~175/121 = 639.264{{c}}

* 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:  

: error map: {{val| 0.000 +0.396 +0.005 +2.871 }}

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

[[Badness]] (Sintel): 4.55

Subgroup: 2.3.5.7.11

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

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 tunings:  

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

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

Badness (Sintel): 1.63

{{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.  

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

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

: mapping generators: ~1323/1280, ~3

[[Optimal tuning]]s:

* [[WE]]: ~1323/1280 = 52.1732{{c}}, ~3/2 = 701.0660{{c}}

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

{{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}}

* 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

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 tunings:

* WE: ~33/32 = 52.1724{{c}}, ~3/2 = 701.1310{{c}}

* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.1524{{c}}

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

Badness (Sintel): 1.67

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 tunings:

* WE: ~33/32 = 52.1735{{c}}, ~3/2 = 701.1493{{c}}

* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.1549{{c}}

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

Badness (Sintel): 1.26

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 tunings:

* WE: ~33/32 = 52.1744{{c}}, ~3/2 = 701.0649{{c}}

* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0582{{c}}

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

Badness (Sintel): 1.31

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 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