Porwell temperaments: Difference between revisions

This is a collection of [[regular temperament|temperaments]] that [[tempering out|tempers out]] the porwell comma, {{monzo| 11 1 -3 -2 }} ([[6144/6125]]).  

* [[Mohajira]] (+81/80) → [[Meantone family #Mohajira|Meantone family]]

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

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

* ''[[Hemischis]]'' (+19683/19600) → [[Schismatic family #Hemischis|Schismatic family]]

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

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

Considered below are hendecatonic, twothirdtonic, nessafof, aufo, whoops, polypyth, icositritonic, absurdity, and dodifo.  

: ''For the 5-limit version, see [[11th-octave temperaments #Hendecapent]].''  

=== 11-limit ===

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

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

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

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

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

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

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

== Twothirdtonic ==

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.  

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

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

: mapping generators: ~2, ~28/15

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

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

* [[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 ===

Comma list: 121/120, 176/175, 686/675

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

* 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

=== 13-limit ===

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

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

* 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

== Semaja ==

 

 

 

 

 

 

 

=== 11-limit ===

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

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

* WE: ~2 = 1199.9818{{c}}, ~8/7 = 226.4821{{c}}

* 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:  

* 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 }}

Badness (Sintel): 1.35

== 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:  

* [[WE]]: ~63/50 = 399.9023{{c}}, ~35/32 = 157.4418{{c}}

: [[error map]]: {{val| -0.293 -0.057 +0.407 +0.430 }}

* [[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

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

* WE: ~63/50 = 400.0266{{c}}, ~12/11 = 157.5301{{c}}

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

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

Badness (Sintel): 2.26

=== Nessa ===

Subgroup: 2.3.5.7.11

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

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

* WE: ~44/35 = 399.7815{{c}}, ~35/32 = 157.4527{{c}}

* 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 ====

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 }}

* WE: ~44/35 = 399.7595{{c}}, ~35/32 = 157.3348{{c}}

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

{{Optimal ET sequence|legend=0| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}

Badness (Sintel): 1.55

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.  

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

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

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

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

== Whoops ==

: ''For the 5-limit version, see [[Very high accuracy temperaments #Whoosh]].''  

 

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

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

: mapping generators: ~2, ~640/441

* [[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 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 }}

* 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

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.

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

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

Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }}

== Icositritonic ==

{{See also| 23rd-octave temperaments }}

The icositritonic temperament (46 & 161) has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]].

[[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 = 52.1739{{c}}, ~64/63 = 29.3586{{c}}

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

[[Badness]] (Smith): 0.196622

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 = 52.1739{{c}}, ~64/63 = 29.3980{{c}}

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

Badness (Smith): 0.064613

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 = 52.1739{{c}}, ~64/63 = 29.2830{{c}}

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

Badness (Smith): 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 = 52.1739{{c}}, ~64/63 = 29.2800{{c}}

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

Badness (Smith): 0.024676

=== 19-limit ===

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 = 52.1739{{c}}, ~64/63 = 29.3760{{c}}

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

Badness (Smith): 0.021579

=== 23-limit ===

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 = 52.1739{{c}}, ~64/63 = 29.3471{{c}}

 

 

 

: ''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]] ([[POTE]]): ~972/875 = 171.4286{{c}}, ~3/2 = 700.5854{{c}} (or ~10/9 = 186.2997{{c}})

{{Optimal ET sequence|legend=1| 77, 84, 161 }}

[[Badness]] (Smith): 0.133520

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

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

Optimal tuning (POTE): ~495/448 = 171.4286{{c}}, ~3/2 = 700.6354{{c}} (or ~10/9 = 186.3497{{c}})

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

Badness (Smith): 0.081564

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 tuning (POTE): ~72/65 = 171.4286{{c}}, ~3/2 = 700.6291{{c}} (or ~10/9 = 186.3434{{c}})

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

Badness (Smith): 0.041600

=== 17-limit ===

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 tuning (POTE): ~72/65 = 171.4286{{c}}, ~3/2 = 700.6524{{c}} (or ~10/9 = 186.3667{{c}})

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

Badness (Smith): 0.031783

=== 19-limit ===

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 tuning (POTE): ~21/19 = 171.4286{{c}}, ~3/2 = 700.6565{{c}} (or ~10/9 = 186.3708{{c}})

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

Badness (Smith): 0.022291

=== 23-limit ===

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 tuning ([[CTE]]): ~21/19 = 171.429{{c}}, ~3/2 = 700.629{{c}} (or ~10/9 = 186.343{{c}})

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

=== 29-limit ===

{{See also| Fifth-chroma temperaments }}

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 tuning ([[CTE]]): ~21/19 = 171.429{{c}}, ~3/2 = 700.629{{c}} (or ~10/9 = 186.343{{c}})

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

== 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 intepretation.

[[Subgroup]]: 2.3.5.7

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/40 = 357.070{{c}}

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

[[Badness]] (Smith): 0.179692

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 = 1200.000{{c}}, ~49/40 = 357.048{{c}}

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

 

Badness (Smith): 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 = 1200.000{{c}}, ~16/13 = 357.049{{c}}

 

Badness (Smith): 0.039533

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

[[Category:Rank 2]]