Porwell temperaments: Difference between revisions

Revision as of 08:12, 5 April 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of temperaments that tempers out the porwell comma, [11 1 -3 -2⟩ (6144/6125), and includes hendecatonic, hemischis, twothirdtonic, nessafof, septisuperfourth, whoops, and polypyth.

Discussed elsewhere are:

Armodue (+36/35) → Mavila family
Porcupine (+64/63) → Porcupine family
Mohajira (+81/80) → Meantone family
Valentine (+126/125) → Starling temperaments
Orwell (+225/224) → Semicomma family
Shrutar (+245/243) → Diaschismic family
Quinkee (+1029/1000) → Cloudy clan
Hemiwürschmidt (+2401/2400 or 3136/3125) → Hemimean clan
Hemikleismic (+4000/3969) → Kleismic family
Amity (+4375/4374 or 5120/5103) → Amity family
Freivald (+6272/6075) → Passion family
Grendel (+16875/16807) → Mirkwai clan
Hemischis (+19683/19600) → Schismatic family
Bison (+78732/78125) → Sensipent family
Hemimabila (+117649/116640) → Mabila family
Septisuperfourth (+118098/117649) → Escapade family
Alphatrident (+14348907/14336000) → Alphatricot family
Hemimaquila (+[-5 10 5 -8⟩) → Maquila family
Decimaleap (+[15 -18 1 4⟩) → Quintaleap family
Twilight (+[19 -22 2 4⟩) → Undim family
Countermiracle (+823543/819200) → Quince clan

Hendecatonic

For the 5-limit version, see 11th-octave temperaments #Hendecapent.

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

Subgroup: 2.3.5.7

Comma list: 6144/6125, 10976/10935

Mapping: [⟨11 0 43 -4], ⟨0 1 -1 2]]

mapping generators: ~16/15, ~3

Optimal tuning (POTE): ~16/15 = 109.091 ¢, ~3/2 = 703.054 ¢

Optimal ET sequence: 22, 55, 77, 99

Badness (Smith): 0.041081

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 10976/10935

Mapping: [⟨11 0 43 -4 38], ⟨0 1 -1 2 0]]

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

Optimal ET sequence: 22, 55, 77, 99, 176e, 275e

Badness (Smith): 0.046088

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350, 4459/4455

Mapping: [⟨11 0 43 -4 38 93], ⟨0 1 -1 2 0 -3]]

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

Optimal ET sequence: 22, 55, 77, 99, 176e

Badness (Smith): 0.040099

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023

Mapping: [⟨11 0 43 -4 38 93 45], ⟨0 1 -1 2 0 -3 0]]

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

Optimal ET sequence: 22, 55, 77, 99, 176eg

Badness (Smith): 0.029054

Cohendecatonic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨11 0 43 -4 73], ⟨0 1 -1 2 -2]]

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

Optimal ET sequence: 22, 77e, 99e, 121, 220e

Badness (Smith): 0.038042

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539, 625/624

Mapping: [⟨11 0 43 -4 73 128], ⟨0 1 -1 2 -2 -5]]

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

Optimal ET sequence: 22, 77eff, 99ef, 121, 341bdeeff

Badness (Smith): 0.036112

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨11 0 43 -4 73 128 45], ⟨0 1 -1 2 -2 -5 0]]

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

Optimal ET sequence: 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg

Badness (Smith): 0.022590

Icosidillic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨22 0 86 -8 111], ⟨0 1 -1 2 -1]]

mapping generators: ~33/32, ~3

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

Optimal ET sequence: 22, 154, 176, 198

Badness (Smith): 0.057725

Twothirdtonic

Subgroup: 2.3.5.7

Comma list: 686/675, 6144/6125

Mapping: [⟨1 3 2 4], ⟨0 -13 3 -11]]

mapping generators: ~2, ~15/14

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 130.401 ¢

Optimal ET sequence: 9, 28b, 37, 46

Badness (Smith): 0.099601

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 2 4 4], ⟨0 -13 3 -11 -5]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 130.430 ¢

Optimal ET sequence: 9, 28b, 37, 46

Badness (Smith): 0.040768

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 2 4 4 5], ⟨0 -13 3 -11 -5 -12]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~14/13 = 130.409 ¢

Optimal ET sequence: 9, 28b, 37, 46

Badness (Smith): 0.025941

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^[1].

Subgroup: 2.3.5.7

Comma list: 3125/3087, 6144/6125

Mapping: [⟨1 -2 1 3], ⟨0 19 7 -1]]

mapping generators: ~2, ~8/7

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~8/7 = 226.4834 ¢

Optimal ET sequence: 16, 37, 53, 196d

Badness (Smith): 0.107023

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 1 3 1], ⟨0 19 7 -1 13]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~8/7 = 226.4856 ¢

Optimal ET sequence: 16, 37, 53

Badness (Smith): 0.059838

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 1 3 1 2], ⟨0 19 7 -1 13 9]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~8/7 = 226.4794 ¢

Optimal ET sequence: 16, 37, 53

Badness (Smith): 0.032564

Nessafof

For the 5-limit version, see Miscellaneous 5-limit temperaments #Nessafof.

Cryptically named by Petr Pařízek in 2011^[2], 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^[1].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 250047/250000

Mapping: [⟨3 2 5 10], ⟨0 7 5 -4]]

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

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~35/32 = 157.480 ¢

Optimal ET sequence: 15, 54b, 69, 84, 99, 282, 381

Badness (Smith): 0.045048

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 2 5 10 8], ⟨0 7 5 -4 6]]

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

Optimal ET sequence: 15, 54be, 69e, 84e, 99

Badness (Smith): 0.068427

Nessa

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 2 5 10 10], ⟨0 7 5 -4 1]]

Optimal tuning (POTE): ~44/35 = 400.000 ¢, ~35/32 = 157.539 ¢

Optimal ET sequence: 15, 54b, 69, 84, 99e

Badness (Smith): 0.048836

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 364/363, 441/440, 625/624

Mapping: [⟨3 2 5 10 10 6], ⟨0 7 5 -4 1 13]]

Optimal tuning (POTE): ~44/35 = 400.000 ¢, ~35/32 = 157.429 ¢

Optimal ET sequence: 15, 54bf, 69, 84, 99ef, 183ef, 282eeff

Badness (Smith): 0.037409

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^[1].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175616

Mapping: [⟨1 6 -7 19], ⟨0 -9 19 -33]]

mapping generators: ~2, ~45/32

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~45/32 = 588.782 ¢

Optimal ET sequence: 53, 161, 214

Badness (Smith): 0.121428

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 -7 19 1], ⟨0 -9 19 -33 5]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~45/32 = 588.811 ¢

Optimal ET sequence: 53, 108e, 161e

Badness (Smith): 0.088631

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 6 -7 19 1 -12], ⟨0 -9 19 -33 5 32]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~45/32 = 588.788 ¢

Optimal ET sequence: 53, 108e, 161e, 214ee

Badness (Smith): 0.058507

Aufic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 -7 19 -25], ⟨0 -9 19 -33 58]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~45/32 = 588.800 ¢

Optimal ET sequence: 53, 108, 161, 214, 375

Badness (Smith): 0.075149

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 6 -7 19 -25 -12], ⟨0 -9 19 -33 58 32]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~45/32 = 588.796 ¢

Optimal ET sequence: 53, 108, 161, 214, 375, 589be

Badness (Smith): 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^[1].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 244140625/243045684

Mapping: [⟨1 17 14 -7], ⟨0 -33 -25 21]]

mapping generators: ~2, ~441/320

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~441/320 = 560.519 ¢

Optimal ET sequence: 15, 122d, 137, 152, 608d, 623bd, 775bcd

Badness (Smith): 0.175840

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 17 14 -7 10], ⟨0 -33 -25 21 -14]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~242/175 = 560.519 ¢

Optimal ET sequence: 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: [⟨1 0 -31 52], ⟨0 1 21 -31]]

mapping generators: ~2, ~3

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 704.174 ¢

Optimal ET sequence: 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: [⟨1 0 -31 52 59], ⟨0 1 21 -31 -35]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 704.177 ¢

Optimal ET sequence: 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: [⟨1 0 -31 52 59 64], ⟨0 1 21 -31 -35 -38]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 704.168 ¢

Optimal ET sequence: 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: [⟨1 0 -31 52 59 64 39], ⟨0 1 21 -31 -35 -38 -22]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~3/2 = 704.168 ¢

Optimal ET sequence: 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.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 9920232/9765625

Mapping: [⟨23 0 17 101], ⟨0 1 1 -1]]

mapping generators: ~1323/1280, ~3

Optimal tuning (POTE): ~1323/1280 = 52.1739 ¢, ~64/63 = 29.3586 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Smith): 0.196622

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨23 0 17 101 116], ⟨0 1 1 -1 -1]]

Optimal tuning (POTE): ~33/32 = 52.1739 ¢, ~64/63 = 29.3980 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Smith): 0.064613

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨23 0 17 101 116 158], ⟨0 1 1 -1 -1 -2]]

Optimal tuning (POTE): ~33/32 = 52.1739 ¢, ~64/63 = 29.2830 ¢

Optimal ET sequence: 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: [⟨23 0 17 101 116 158 94], ⟨0 1 1 -1 -1 -2 0]]

Optimal tuning (POTE): ~33/32 = 52.1739 ¢, ~64/63 = 29.2800 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Smith): 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: [⟨23 0 17 101 116 158 94 207], ⟨0 1 1 -1 -1 -2 0 -3]]

Optimal tuning (POTE): ~33/32 = 52.1739 ¢, ~64/63 = 29.3760 ¢

Optimal ET sequence: 46, 115, 161, 207, 368c

Badness (Smith): 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: [⟨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 ¢, ~64/63 = 29.3471 ¢

Optimal ET sequence: 46, 115, 161, 207, 368ci

Badness (Smith): 0.017745

Absurdity

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: [⟨7 0 -17 64], ⟨0 1 3 -4]]

mapping generators: ~972/875, ~3

Optimal tuning (POTE): ~972/875 = 171.4286 ¢, ~3/2 = 700.5854 ¢ (or ~10/9 = 186.2997 ¢)

Optimal ET sequence: 77, 84, 161

Badness (Smith): 0.133520

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨7 0 -17 64 124], ⟨0 1 3 -4 -9]]

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

Optimal ET sequence: 77, 84, 161

Badness (Smith): 0.081564

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 1188/1183, 3584/3575

Mapping: [⟨7 0 -17 64 124 37], ⟨0 1 3 -4 -9 -1]]

Optimal tuning (POTE): ~72/65 = 171.4286 ¢, ~3/2 = 700.6291 ¢ (or ~10/9 = 186.3434 ¢)

Optimal ET sequence: 77, 84, 161

Badness (Smith): 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: [⟨7 0 -17 64 124 37 -49], ⟨0 1 3 -4 -9 -1 7]]

Optimal tuning (POTE): ~72/65 = 171.4286 ¢, ~3/2 = 700.6524 ¢ (or ~10/9 = 186.3667 ¢)

Optimal ET sequence: 77, 161

Badness (Smith): 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: [⟨7 0 -17 64 124 37 -49 63], ⟨0 1 3 -4 -9 -1 7 -3]]

Optimal tuning (POTE): ~21/19 = 171.4286 ¢, ~3/2 = 700.6565 ¢ (or ~10/9 = 186.3708 ¢)

Optimal ET sequence: 77, 161

Badness (Smith): 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: [⟨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 ¢, ~3/2 = 700.629 ¢ (or ~10/9 = 186.343 ¢)

Optimal ET sequence: 77, 84, 161

29-limit

See also: Fifth-chroma temperaments

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 261/260, 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494

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 ¢, ~3/2 = 700.629 ¢ (or ~10/9 = 186.343 ¢)

Optimal ET sequence: 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^[1]. The extension here is a less accurate 7-limit intepretation.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 2500000/2470629

Mapping: [⟨1 12 5 4], ⟨0 -35 -9 -4]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/40 = 357.070 ¢

Optimal ET sequence: 37, 84, 121, 205

Badness (Smith): 0.179692

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2560/2541, 4375/4356

Mapping: [⟨1 12 5 4 -1], ⟨0 -35 -9 -4 15]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/40 = 357.048 ¢

Optimal ET sequence: 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: [⟨1 12 5 4 -1 4], ⟨0 -35 -9 -4 15 -1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~16/13 = 357.049 ¢

Optimal ET sequence: 37, 84, 121, 326deef

Badness (Smith): 0.039533

References

[petr's_long_post-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 Yahoo! Tuning Group | Suggested names for the unclasified temperaments

[petr's_short_post-2] Yahoo! Tuning Group | Some more unclassified temperaments

[1]

[2]

@@ Line 26: / Line 26: @@
 == Hendecatonic ==
-{{see also|11th-octave temperaments}}
+: ''For the 5-limit version, see [[11th-octave temperaments #Hendecapent]].''
 The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]].
@@ Line 35: / Line 35: @@
 {{Mapping|legend=1| 11 0 43 -4 | 0 1 -1 2 }}
+: mapping generators: ~16/15, ~3
-: Mapping generators: ~16/15, ~3
+[[Optimal tuning]] ([[POTE]]): ~16/15 = 109.091{{c}}, ~3/2 = 703.054{{c}}
-[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\11, ~3/2 = 703.054
 {{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}
-[[Badness]]: 0.041081
+[[Badness]] (Smith): 0.041081
 === 11-limit ===
@@ Line 49: / Line 48: @@
 Comma list: 121/120, 176/175, 10976/10935
-{{Mapping|legend=1| 11 0 43 -4 38 | 0 1 -1 2 0 }}
+Mapping: {{mapping| 11 0 43 -4 38 | 0 1 -1 2 0 }}
-Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.636
+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: 0.046088
+Badness (Smith): 0.046088
 ==== 13-limit ====
@@ Line 62: / Line 61: @@
 Comma list: 121/120, 176/175, 351/350, 4459/4455
-{{Mapping|legend=1| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}
+Mapping: {{mapping| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}
-Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.291
+Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 702.291{{c}}
 {{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176e }}
-Badness: 0.040099
+Badness (Smith): 0.040099
 ==== 17-limit ====
@@ Line 75: / Line 74: @@
 Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023
-{{Mapping|legend=1| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}
+Mapping: {{mapping| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}
-Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 702.301
+Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 702.301{{c}}
 {{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176eg }}
-Badness: 0.029054
+Badness (Smith): 0.029054
 === Cohendecatonic ===
@@ Line 88: / Line 87: @@
 Comma list: 540/539, 896/891, 4375/4356
-{{Mapping|legend=1| 11 0 43 -4 73 | 0 1 -1 2 -2 }}
+Mapping: {{mapping| 11 0 43 -4 73 | 0 1 -1 2 -2 }}
-Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.686
+Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 703.686{{c}}
 {{Optimal ET sequence|legend=0| 22, 77e, 99e, 121, 220e }}
-Badness: 0.038042
+Badness (Smith): 0.038042
 ==== 13-limit ====
@@ Line 101: / Line 100: @@
 Comma list: 352/351, 364/363, 540/539, 625/624
-{{Mapping|legend=1| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}
+Mapping: {{mapping| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}
-Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.888
+Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 703.888{{c}}
 {{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 341bdeeff }}
-Badness: 0.036112
+Badness (Smith): 0.036112
 ==== 17-limit ====
@@ Line 114: / Line 113: @@
 Comma list: 256/255, 352/351, 364/363, 375/374, 540/539
-{{Mapping|legend=1| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}
+Mapping: {{mapping| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}
-Optimal tuning (POTE): ~16/15 = 1\11, ~3/2 = 703.877
+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: 0.022590
+Badness (Smith): 0.022590
 === Icosidillic ===
@@ Line 127: / Line 126: @@
 Comma list: 3388/3375, 6144/6125, 9801/9800
-{{Mapping|legend=1| 22 0 86 -8 111 | 0 1 -1 2 -1 }}
+Mapping: {{mapping| 22 0 86 -8 111 | 0 1 -1 2 -1 }}
+: mapping generators: ~33/32, ~3
-: Mapping generators: ~33/32, ~3
+Optimal tuning (POTE): ~33/32 = 54.545{{c}}, ~3/2 = 702.914{{c}}
-Optimal tuning (POTE): ~33/32 = 1\22, ~3/2 = 702.914
 {{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}
-Badness: 0.057725
+Badness (Smith): 0.057725
 == Twothirdtonic ==
@@ Line 143: / Line 141: @@
 {{Mapping|legend=1| 1 3 2 4 | 0 -13 3 -11 }}
+: mapping generators: ~2, ~15/14
-: Mapping generators: ~2, ~15/14
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~15/14 = 130.401{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 130.401
 {{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
-[[Badness]]: 0.099601
+[[Badness]] (Smith): 0.099601
 === 11-limit ===
@@ Line 159: / Line 156: @@
 Mapping: {{mapping| 1 3 2 4 4 | 0 -13 3 -11 -5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 130.430
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 130.430{{c}}
-{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
+{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
-Badness: 0.040768
+Badness (Smith): 0.040768
 === 13-limit ===
@@ Line 172: / Line 169: @@
 Mapping: {{mapping| 1 3 2 4 4 5 | 0 -13 3 -11 -5 -12 }}
-Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 130.409
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/13 = 130.409{{c}}
-{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
+{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
-Badness: 0.025941
+Badness (Smith): 0.025941
 == Semaja ==
@@ Line 186: / Line 183: @@
 {{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}
+: mapping generators: ~2, ~8/7
-: Mapping generators: ~2, ~8/7
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~8/7 = 226.4834{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 226.4834
 {{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}
-[[Badness]]: 0.107023
+[[Badness]] (Smith): 0.107023
 === 11-limit ===
@@ Line 202: / Line 198: @@
 Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}
-Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4856
+Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 226.4856{{c}}
-{{Optimal ET sequence|legend=1| 16, 37, 53 }}
+{{Optimal ET sequence|legend=0| 16, 37, 53 }}
-Badness: 0.059838
+Badness (Smith): 0.059838
 === 13-limit ===
@@ Line 215: / Line 211: @@
 Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }}
-Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.4794
+Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 226.4794{{c}}
-{{Optimal ET sequence|legend=1| 16, 37, 53 }}
+{{Optimal ET sequence|legend=0| 16, 37, 53 }}
-Badness: 0.032564
+Badness (Smith): 0.032564
 == Nessafof ==
-: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments#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 5 times, makes 5/1<ref name="petr's long post"/>.
@@ Line 231: / Line 227: @@
 {{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}
+: mapping generators: ~63/50, ~35/32
-: Mapping generators: ~63/50, ~35/32
+[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~35/32 = 157.480{{c}}
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~35/32 = 157.480
 {{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}
-[[Badness]]: 0.045048
+[[Badness]] (Smith): 0.045048
 === 11-limit ===
@@ Line 247: / Line 242: @@
 Mapping: {{mapping| 3 2 5 10 8 | 0 7 5 -4 6 }}
-Optimal tuning (POTE): ~63/50 = 1\3, ~12/11 = 157.520
+Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~12/11 = 157.520{{c}}
-{{Optimal ET sequence|legend=1| 15, 54be, 69e, 84e, 99 }}
+{{Optimal ET sequence|legend=0| 15, 54be, 69e, 84e, 99 }}
-Badness: 0.068427
+Badness (Smith): 0.068427
 === Nessa ===
@@ Line 260: / Line 255: @@
 Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~35/32 = 157.539
+Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~35/32 = 157.539{{c}}
-{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99e }}
+{{Optimal ET sequence|legend=0| 15, 54b, 69, 84, 99e }}
-Badness: 0.048836
+Badness (Smith): 0.048836
 ==== 13-limit ====
@@ Line 273: / Line 268: @@
 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 tuning (POTE): ~44/35 = 400.000{{c}}, ~35/32 = 157.429{{c}}
-{{Optimal ET sequence|legend=1| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}
+{{Optimal ET sequence|legend=0| 15, 54bf, 69, 84, 99ef, 183ef, 282eeff }}
-Badness: 0.037409
+Badness (Smith): 0.037409
 == Aufo ==
-:''For the 5-limit version, see [[High badness temperaments #Untriton]].''
+:''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"/>.
@@ Line 289: / Line 284: @@
 {{Mapping|legend=1| 1 6 -7 19 | 0 -9 19 -33 }}
+: mapping generators: ~2, ~45/32
-: Mapping generators: ~2, ~45/32
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~45/32 = 588.782{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 588.782
 {{Optimal ET sequence|legend=1| 53, 161, 214 }}
-[[Badness]]: 0.121428
+[[Badness]] (Smith): 0.121428
 === 11-limit ===
@@ Line 305: / Line 299: @@
 Mapping: {{mapping| 1 6 -7 19 1 | 0 -9 19 -33 5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.811
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.811{{c}}
-{{Optimal ET sequence|legend=1| 53, 108e, 161e }}
+{{Optimal ET sequence|legend=0| 53, 108e, 161e }}
-Badness: 0.088631
+Badness (Smith): 0.088631
 ==== 13-limit ====
@@ Line 318: / Line 312: @@
 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 tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.788{{c}}
-{{Optimal ET sequence|legend=1| 53, 108e, 161e, 214ee }}
+{{Optimal ET sequence|legend=0| 53, 108e, 161e, 214ee }}
-Badness: 0.058507
+Badness (Smith): 0.058507
 === Aufic ===
@@ Line 331: / Line 325: @@
 Mapping: {{mapping| 1 6 -7 19 -25 | 0 -9 19 -33 58 }}
-Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 588.800
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.800{{c}}
-{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375 }}
+{{Optimal ET sequence|legend=0| 53, 108, 161, 214, 375 }}
-Badness: 0.075149
+Badness (Smith): 0.075149
 ==== 13-limit ====
@@ Line 344: / Line 338: @@
 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 tuning (POTE): ~2 = 1200.000{{c}}, ~45/32 = 588.796{{c}}
-{{Optimal ET sequence|legend=1| 53, 108, 161, 214, 375, 589be }}
+{{Optimal ET sequence|legend=0| 53, 108, 161, 214, 375, 589be }}
-Badness: 0.039050
+Badness (Smith): 0.039050
 == Whoops ==
-:''For the 5-limit version, see [[Very high accuracy temperaments #Whoosh]].''
+: ''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"/>.
@@ Line 360: / Line 354: @@
 {{Mapping|legend=1| 1 17 14 -7 | 0 -33 -25 21 }}
+: mapping generators: ~2, ~441/320
-: Mapping generators: ~2, ~441/320
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~441/320 = 560.519{{c}}
-[[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
+[[Badness]] (Smith): 0.175840
 === 11-limit ===
@@ Line 376: / Line 369: @@
 Mapping: {{mapping| 1 17 14 -7 10 | 0 -33 -25 21 -14 }}
-Optimal tuning (POTE): ~2 = 1\1, ~242/175 = 560.519
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~242/175 = 560.519{{c}}
-{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
+{{Optimal ET sequence|legend=0| 15, 122d, 137, 152, 608de, 623bde, 775bcde }}
-Badness: 0.043743
+Badness (Smith): 0.043743
 == Polypyth ==
-:''For the 5-limit version, see [[High badness temperaments #Leapday]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit 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.
+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
@@ Line 392: / Line 385: @@
 {{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}
+: mapping generators: ~2, ~3
-: Mapping generators: ~2, ~3
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 704.174{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.174
 {{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd, 743bcd }}
-[[Badness]]: 0.137995
+[[Badness]] (Smith): 0.137995
 === 11-limit ===
@@ Line 408: / Line 400: @@
 Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.177
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.177{{c}}
-{{Optimal ET sequence|legend=1| 46, 121, 167, 288be, 455bcde }}
+{{Optimal ET sequence|legend=0| 46, 121, 167, 288be, 455bcde }}
-Badness: 0.051131
+Badness (Smith): 0.051131
 === 13-limit ===
@@ Line 421: / Line 413: @@
 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 tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.168{{c}}
-{{Optimal ET sequence|legend=1| 46, 121, 167, 288be }}
+{{Optimal ET sequence|legend=0| 46, 121, 167, 288be }}
-Badness: 0.030292
+Badness (Smith): 0.030292
 === 17-limit ===
@@ Line 434: / Line 426: @@
 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 tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.168{{c}}
-{{Optimal ET sequence|legend=1| 46, 121, 167, 288beg }}
+{{Optimal ET sequence|legend=0| 46, 121, 167, 288beg }}
-Badness: 0.019051
+Badness (Smith): 0.019051
 == Icositritonic ==
-{{ See also | 23rd-octave temperaments }}
+{{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]].
+The icositritonic temperament (46 & 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
@@ Line 449: / Line 442: @@
 {{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}
+: mapping generators: ~1323/1280, ~3
-: Mapping generators: ~1323/1280, ~3
+[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 52.1739{{c}}, ~64/63 = 29.3586{{c}}
-[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 1\23, ~64/63 = 29.3586
 {{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
-[[Badness]]: 0.196622
+[[Badness]] (Smith): 0.196622
 === 11-limit ===
@@ Line 465: / Line 457: @@
 Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}
-Optimal tuning (POTE): ~33/32 = 1\23, ~64/63 = 29.3980
+Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3980{{c}}
-{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
+{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
-Badness: 0.064613
+Badness (Smith): 0.064613
 === 13-limit ===
@@ Line 478: / Line 470: @@
 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 tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.2830{{c}}
-{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
+{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
-Badness: 0.040484
+Badness (Smith): 0.040484
 === 17-limit ===
@@ Line 491: / Line 483: @@
 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 tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.2800{{c}}
-{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
+{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
-Badness: 0.024676
+Badness (Smith): 0.024676
 === 19-limit ===
@@ Line 504: / Line 496: @@
 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 tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3760{{c}}
-{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
+{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
-Badness: 0.021579
+Badness (Smith): 0.021579
 === 23-limit ===
@@ Line 517: / Line 509: @@
 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 tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3471{{c}}
-{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368ci }}
+{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368ci }}
-Badness: 0.017745
+Badness (Smith): 0.017745
 == Absurdity ==
-: ''For the 5-limit version, see [[Syntonic–chromatic_equivalence_continuum#Absurdity_(5-limit)|Syntonic–chromatic equivalence continuum#Absurdity (5-limit)]].''
+: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Absurdity (5-limit)]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 531: / Line 523: @@
 {{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}
+: mapping generators: ~972/875, ~3
-: 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 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
+[[Badness]] (Smith): 0.133520
 === 11-limit ===
@@ Line 545: / Line 536: @@
 Comma list: 441/440, 6144/6125, 72171/71680
-{{Mapping|legend=1| 7 0 -17 64 124 | 0 1 3 -4 -9 }}
+Mapping: {{mapping| 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 tuning (POTE): ~495/448 = 171.4286{{c}}, ~3/2 = 700.6354{{c}} (or ~10/9 = 186.3497{{c}})
-{{Optimal ET sequence|legend=1| 77, 84, 161 }}
+{{Optimal ET sequence|legend=0| 77, 84, 161 }}
-Badness: 0.081564
+Badness (Smith): 0.081564
 === 13-limit ===
@@ Line 558: / Line 549: @@
 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 }}
+Mapping: {{mapping| 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 tuning (POTE): ~72/65 = 171.4286{{c}}, ~3/2 = 700.6291{{c}} (or ~10/9 = 186.3434{{c}})
-{{Optimal ET sequence|legend=1| 77, 84, 161 }}
+{{Optimal ET sequence|legend=0| 77, 84, 161 }}
-Badness: 0.041600
+Badness (Smith): 0.041600
 === 17-limit ===
@@ Line 571: / Line 562: @@
 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 }}
+Mapping: {{mapping| 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 tuning (POTE): ~72/65 = 171.4286{{c}}, ~3/2 = 700.6524{{c}} (or ~10/9 = 186.3667{{c}})
-{{Optimal ET sequence|legend=1| 77, 161 }}
+{{Optimal ET sequence|legend=0| 77, 161 }}
-Badness: 0.031783
+Badness (Smith): 0.031783
 === 19-limit ===
@@ Line 584: / Line 575: @@
 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 }}
+Mapping: {{mapping| 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 tuning (POTE): ~21/19 = 171.4286{{c}}, ~3/2 = 700.6565{{c}} (or ~10/9 = 186.3708{{c}})
-{{Optimal ET sequence|legend=1| 77, 161 }}
+{{Optimal ET sequence|legend=0| 77, 161 }}
-Badness: 0.022291
+Badness (Smith): 0.022291
 === 23-limit ===
@@ Line 597: / Line 588: @@
 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 }}
+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 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)
+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 }}
+{{See also| Fifth-chroma temperaments }}
-Subgroup: 2.3.5.7.11.13.17.19.23
+Subgroup: 2.3.5.7.11.13.17.19.23.29
 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 }}
+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 = 1\7, ~3/2 = 700.629 (or ~10/9 = 186.343)
+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 [[High badness temperaments #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.
@@ Line 626: / Line 618: @@
 {{Mapping|legend=1| 1 12 5 4 | 0 -35 -9 -4 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 357.070
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/40 = 357.070{{c}}
 {{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}
-[[Badness]]: 0.179692
+[[Badness]] (Smith): 0.179692
 === 11-limit ===
@@ Line 639: / Line 631: @@
 Mapping: {{mapping| 1 12 5 4 -1 | 0 -35 -9 -4 15 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 357.048
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 357.048{{c}}
-{{Optimal ET sequence|legend=1| 37, 84, 121, 326dee }}
+{{Optimal ET sequence|legend=0| 37, 84, 121, 326dee }}
-Badness: 0.081923
+Badness (Smith): 0.081923
 === 13-limit ===
@@ Line 652: / Line 644: @@
 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 tuning (POTE): ~2 = 1200.000{{c}}, ~16/13 = 357.049{{c}}
-{{Optimal ET sequence|legend=1| 37, 84, 121, 326deef }}
+{{Optimal ET sequence|legend=0| 37, 84, 121, 326deef }}
-Badness: 0.039533
+Badness (Smith): 0.039533
-== Notes ==
+== References ==
 [[Category:Temperament collections]]
 [[Category:Porwell temperaments| ]] <!-- main article -->
-[[Category:Hendecatonic]]
 [[Category:Rank 2]]

Porwell temperaments: Difference between revisions

Revision as of 08:12, 5 April 2026

Hendecatonic

11-limit

13-limit

17-limit

Cohendecatonic

13-limit

17-limit

Icosidillic

Twothirdtonic

11-limit

13-limit

Semaja

11-limit

13-limit

Nessafof

11-limit

Nessa

13-limit

Aufo

11-limit

13-limit

Aufic

13-limit

Whoops

11-limit

Polypyth

11-limit

13-limit

17-limit

Icositritonic

11-limit

13-limit

17-limit

19-limit

23-limit

Absurdity

11-limit

13-limit

17-limit

19-limit

23-limit

29-limit

Dodifo

11-limit

13-limit

References

Navigation menu

Search