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]]), and includes hendecatonic, hemischis, twothirdtonic, nessafof, septisuperfourth, whoops, and polypyth.  

Discussed elsewhere are:  

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

* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]

* [[Shrutar]] (+245/243) → [[Diaschismic family #Shrutar|Diaschismic family]]

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

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

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

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

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

The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]].

[[Optimal tuning]] ([[POTE]]): ~16/15 = 109.091{{c}}, ~3/2 = 703.054{{c}}

[[Badness]] (Smith): 0.041081

=== 11-limit ===

Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 702.636{{c}}

{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176e, 275e }}

Badness (Smith): 0.046088

Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 702.291{{c}}

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

Badness (Smith): 0.040099

Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 702.301{{c}}

Badness (Smith): 0.029054

Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 703.686{{c}}

Badness (Smith): 0.038042

Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 703.888{{c}}

{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 341bdeeff }}

Badness (Smith): 0.036112

Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 703.877{{c}}

{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}

Badness (Smith): 0.022590

Optimal tuning (POTE): ~33/32 = 54.545{{c}}, ~3/2 = 702.914{{c}}

Badness (Smith): 0.057725

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

{{Mapping|legend=1| 1 3 2 4 | 0 -13 3 -11 }}

: mapping generators: ~2, ~15/14

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~15/14 = 130.401{{c}}

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

[[Badness]] (Smith): 0.099601

=== 11-limit ===

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

Mapping: {{mapping| 1 3 2 4 4 | 0 -13 3 -11 -5 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 130.430{{c}}

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

Badness (Smith): 0.040768

=== 13-limit ===

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

Mapping: {{mapping| 1 3 2 4 4 5 | 0 -13 3 -11 -5 -12 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/13 = 130.409{{c}}

Badness (Smith): 0.025941

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>.  

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

{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}

: mapping generators: ~2, ~8/7

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~8/7 = 226.4834{{c}}

{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}

[[Badness]] (Smith): 0.107023

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

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

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 226.4856{{c}}

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

Badness (Smith): 0.059838

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 tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 226.4794{{c}}

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

Badness (Smith): 0.032564

: ''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 5 times, makes 5/1<ref name="petr's long post"/>.  

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

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

: mapping generators: ~63/50, ~35/32

[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~35/32 = 157.480{{c}}

{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}

[[Badness]] (Smith): 0.045048

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

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

Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~12/11 = 157.520{{c}}

{{Optimal ET sequence|legend=0| 15, 54be, 69e, 84e, 99 }}

Badness (Smith): 0.068427

=== Nessa ===

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

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

Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~35/32 = 157.539{{c}}

{{Optimal ET sequence|legend=0| 15, 54b, 69, 84, 99e }}

Badness (Smith): 0.048836

==== 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 tuning (POTE): ~44/35 = 400.000{{c}}, ~35/32 = 157.429{{c}}

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

Badness (Smith): 0.037409

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"/>.  

{{Mapping|legend=1| 1 6 -7 19 | 0 -9 19 -33 }}

: mapping generators: ~2, ~45/32

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~45/32 = 588.782{{c}}

[[Badness]] (Smith): 0.121428

Mapping: {{mapping| 1 6 -7 19 1 | 0 -9 19 -33 5 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.811{{c}}

Badness (Smith): 0.088631

Mapping: {{mapping| 1 6 -7 19 1 -12 | 0 -9 19 -33 5 32 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.788{{c}}

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

Badness (Smith): 0.058507

Mapping: {{mapping| 1 6 -7 19 -25 | 0 -9 19 -33 58 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.800{{c}}

Badness (Smith): 0.075149

Mapping: {{mapping| 1 6 -7 19 -25 -12 | 0 -9 19 -33 58 32 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.796{{c}}

{{Optimal ET sequence|legend=0| 53, 108, 161, 214, 375, 589be }}

Badness (Smith): 0.039050

== Whoops ==

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

 

[[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 = 1200.000{{c}}, ~441/320 = 560.519{{c}}

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

[[Badness]] (Smith): 0.175840

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

Mapping: {{mapping| 1 17 14 -7 10 | 0 -33 -25 21 -14 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~242/175 = 560.519{{c}}

{{Optimal ET sequence|legend=0| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}

Badness (Smith): 0.043743

== Polypyth ==

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

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.

[[Subgroup]]: 2.3.5.7

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

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 704.174{{c}}

{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}

[[Badness]] (Smith): 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 = 1200.000{{c}}, ~3/2 = 704.177{{c}}

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

Badness (Smith): 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 = 1200.000{{c}}, ~3/2 = 704.168{{c}}

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

Badness (Smith): 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 = 1200.000{{c}}, ~3/2 = 704.168{{c}}

{{Optimal ET sequence|legend=0| 46, 121, 167, 288beg }}

Badness (Smith): 0.019051

== 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

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

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

Badness (Smith): 0.021579

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

Badness (Smith): 0.017745

: ''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 }}

 

 

 

 

 

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

Badness (Smith): 0.039533