Porwell temperaments

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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 temper out the porwell comma (monzo: [11 1 -3 -2⟩, ratio: 6144/6125).

Temperaments discussed elsewhere are:

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

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. It tempers out 10976/10935, the hemimage comma, and may be described as the 22 & 99 temperament, with 99edo giving an almost perfect tuning.

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 tunings:

  • WE: ~16/15 = 109.0526 ¢, ~3/2 = 702.8069 ¢
error map: ⟨-0.421 +0.431 +0.563 -0.265]
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.9705 ¢
error map: ⟨0.000 +1.015 +1.625 +0.751]

Optimal ET sequence: 22, 55, 77, 99

Badness (Sintel): 1.04

Hendecaton

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 tunings:

  • WE: ~16/15 = 109.0977 ¢, ~3/2 = 702.6801 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.6484 ¢

Optimal ET sequence: 22, 55, 77, 99

Badness (Sintel): 1.52

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 tunings:

  • WE: ~16/15 = 109.1092 ¢, ~3/2 = 702.4093 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.2930 ¢

Optimal ET sequence: 22, 55, 77, 99

Badness (Sintel): 1.66

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 tunings:

  • WE: ~16/15 = 109.0933 ¢, ~3/2 = 702.3170 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 702.3017 ¢

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

Badness (Sintel): 1.48

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 tunings:

  • WE: ~16/15 = 109.0237 ¢, ~3/2 = 703.2522 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.6563 ¢

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

Badness (Sintel): 1.26

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 tunings:

  • WE: ~16/15 = 109.0189 ¢, ~3/2 = 703.4228 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.9248 ¢

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

Badness (Sintel): 1.49

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 tunings:

  • WE: ~16/15 = 109.0159 ¢, ~3/2 = 703.3932 ¢
  • CWE: ~16/15 = 109.0909 ¢, ~3/2 = 703.9110 ¢

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

Badness (Sintel): 1.15

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 tunings:

  • WE: ~33/32 = 54.5305 ¢, ~3/2 = 702.7206 ¢
  • CWE: ~33/32 = 54.5455 ¢, ~3/2 = 702.8829 ¢

Optimal ET sequence: 22, 154, 176, 198

Badness (Sintel): 1.84

Nessafof

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

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

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 tunings:

  • WE: ~63/50 = 399.9023 ¢, ~35/32 = 157.4418 ¢
error map: ⟨-0.293 -0.057 +0.407 +0.430]
  • CWE: ~63/50 = 400.0000 ¢, ~35/32 = 157.4658 ¢
error map: ⟨0.000 +0.306 1.016 +1.311]

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

Badness (Sintel): 1.14

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 tunings:

  • WE: ~44/35 = 399.7815 ¢, ~35/32 = 157.4527 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~35/32 = 157.5109 ¢

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

Optimal tunings:

  • WE: ~44/35 = 399.7595 ¢, ~35/32 = 157.3348 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~35/32 = 157.3955 ¢

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

Badness (Sintel): 1.55

Fof

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 tunings:

  • WE: ~63/50 = 400.0266 ¢, ~12/11 = 157.5301 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~12/11 = 157.5240 ¢

Optimal ET sequence: 15, 69e, 84e, 99

Badness (Sintel): 2.26

Grendel

For the 5-limit version, see Syntonic–31 equivalence continuum #Counterwürschmidt.

Grendel tempers out 16875/16807, the mirkwai comma, and may be described as the 31 & 152 temperament. 152edo, 183edo and especially 335edo serve as good tunings.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 16875/16807

Mapping: [⟨1 -14 3 -6], ⟨0 23 -1 13]]

mapping generators: ~2, ~8/5

Optimal tunings:

  • WE: ~2 = 1199.7348 ¢, ~8/5 = 812.9574 ¢
error map: ⟨-0.265 -0.220 -0.067 +1.212]
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1311 ¢
error map: ⟨0.000 +0.059 +0.555 +1.878]

Optimal ET sequence: 31, 90, 121, 152, 335d, 822dd

Badness (Sintel): 1.31

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 5632/5625

Mapping: [⟨1 -14 3 -6 -25], ⟨0 23 -1 13 42]]

Optimal tunings:

  • WE: ~2 = 1199.7355 ¢, ~8/5 = 812.9622 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1353 ¢

Optimal ET sequence: 31, 90e, 121, 152, 335d, 487d

Badness (Sintel): 0.656

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 625/624, 1375/1372

Mapping: [⟨1 -14 3 -6 -25 22], ⟨0 23 -1 13 42 -27]]

Optimal tunings:

  • WE: ~2 = 1199.4412 ¢, ~8/5 = 812.7956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1209 ¢

Optimal ET sequence: 31, 90e, 121, 152f, 273def, 425deff

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -14 3 -6 -25 22 19], ⟨0 23 -1 13 42 -27 -22]]

Optimal tunings:

  • WE: ~2 = 1199.3029 ¢, ~8/5 = 812.7156 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1843 ¢

Optimal ET sequence: 31, 90e, 121, 152fg, 273defgg

Badness (Sintel): 1.09

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714

Mapping: [⟨1 -14 3 -6 -25 22 19 30], ⟨0 23 -1 13 42 -27 -22 -38]]

Optimal tunings:

  • WE: ~2 = 1199.3587 ¢, ~8/5 = 812.7462 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 813.1796 ¢

Optimal ET sequence: 31, 90e, 121, 152fg, 273defgg

Badness (Sintel): 1.12

Twothirdtonic

Twothirdtonic tempers out 686/675, the senga, in addition to the porwell comma, and may be described as the 37 & 46 temperament, generated by one third of a 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 reduced make the perfect fifth; it follows that the ploidacot for this temperament is 11-sheared 13-cot. 46edo may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 686/675, 6144/6125

Mapping: [⟨1 -10 5 -7], ⟨0 13 -3 11]]

mapping generators: ~2, ~28/15

Optimal tunings:

  • WE: ~2 = 1199.3074 ¢, ~28/15 = 1068.9820 ¢
error map: ⟨-0.693 +1.736 +3.278 -5.176]
  • CWE: ~2 = 1200.0000 ¢, ~28/15 = 1069.5746 ¢
error map: ⟨0.000 +2.515 +4.962 -3.505]

Optimal ET sequence: 9, 28b, 37, 46

Badness (Sintel): 2.52

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -10 5 -7 -1], ⟨0 13 -3 11 5]]

Optimal tunings:

  • WE: ~2 = 1199.7068 ¢, ~28/15 = 1069.3084 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/15 = 1069.5600 ¢

Optimal ET sequence: 9, 28b, 37, 46

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -10 5 -7 -1 -7], ⟨0 13 -3 11 5 12]]

Optimal tunings:

  • WE: ~2 = 1199.9531 ¢, ~13/7 = 1069.5492 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/7 = 1069.5893 ¢

Optimal ET sequence: 9, 28b, 37, 46

Badness (Sintel): 1.07

Semaja

See also: Llywelynsmic clan

Cryptically named by Petr Pařízek in 2011, semaja adds the gariboh comma to the comma list, and may be described as the 37 & 53 temperament. Its ploidacot is gamma-19-cot (or alpha-heptaseph due to a much simpler 2.5.7-subgroup restriction). The name actually refers to the fact that two of its ~8/7 generator steps reach a ~13/10[2].

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 tunings:

  • WE: ~2 = 1199.4860 ¢, ~8/7 = 226.3864 ¢
error map: ⟨-0.514 +0.415 -2.123 +3.246]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.4697 ¢
error map: ⟨0.000 +0.970 -1.026 +4.704]

Optimal ET sequence: 16, 37, 53, 196d

Badness (Sintel): 2.71

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 tunings:

  • WE: ~2 = 1199.9818 ¢, ~8/7 = 226.4821 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.4851 ¢

Optimal ET sequence: 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: [⟨1 -2 1 3 1 2], ⟨0 19 7 -1 13 9]]

Optimal tunings:

  • WE: ~2 = 1200.1020 ¢, ~8/7 = 226.4987 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.4822 ¢

Optimal ET sequence: 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[2]. The functional generator however is the 64/45 diminished fifth, and like its untriton variant, nine generator steps give the interval class of 3. The ploidacot for this temperament is delta-enneacot.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 177147/175616

Mapping: [⟨1 -3 12 -14], ⟨0 9 -19 33]]

mapping generators: ~2, ~64/45

Optimal tunings:

  • WE: ~2 = 1199.9758 ¢, ~64/45 = 611.2055 ¢
error map: ⟨-0.024 -1.303 +0.491 +1.295]
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2177 ¢
error map: ⟨0.000 -0.996 +0.551 +1.357]

Optimal ET sequence: 53, 161, 214

Badness (Sintel): 3.07

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -3 12 -14 6], ⟨0 9 -19 33 -5]]

Optimal tunings:

  • WE: ~2 = 1200.4500 ¢, ~64/45 = 611.4185 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.1918 ¢

Optimal ET sequence: 53, 108e, 161e

Badness (Sintel): 2.93

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -3 12 -14 6 20], ⟨0 9 -19 33 -5 -32]]

Optimal tunings:

  • WE: ~2 = 1200.3134 ¢, ~64/45 = 611.3715 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2118 ¢

Optimal ET sequence: 53, 108e

Badness (Sintel): 2.42

Aufic

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -3 12 -14 33], ⟨0 9 -19 33 -58]]

Optimal tunings:

  • WE: ~2 = 1200.0668 ¢, ~64/45 = 611.2342 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2000 ¢

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

Badness (Sintel): 2.48

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -3 12 -14 33 20], ⟨0 9 -19 33 -58 -32]]

Optimal tunings:

  • WE: ~2 = 1200.0177 ¢, ~64/45 = 611.2130 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 611.2039 ¢

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

Badness (Sintel): 1.61

Absurdity

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Absurdity (5-limit).
See also: Fifth-chroma temperaments

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 tunings:

  • WE: ~972/875 = 171.4382 ¢, ~3/2 = 700.6247 ¢
error map: ⟨+0.067 -1.263 +1.313 +0.450]
  • CWE: ~972/875 = 171.4286 ¢, ~3/2 = 700.5871 ¢
error map: ⟨0.000 -1.368 +1.162 +0.254]

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 3.38

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 tunings:

  • WE: ~495/448 = 171.4346 ¢, ~3/2 = 700.6602 ¢
  • CWE: ~495/448 = 171.4286 ¢, ~3/2 = 700.6339 ¢

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 2.70

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 tunings:

  • WE: ~72/65 = 171.4223 ¢, ~3/2 = 700.6036 ¢
  • CWE: ~72/65 = 171.4286 ¢, ~3/2 = 700.6306 ¢

Optimal ET sequence: 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: [⟨7 0 -17 64 124 37 -49], ⟨0 1 3 -4 -9 -1 7]]

Optimal tunings:

  • WE: ~72/65 = 171.4263 ¢, ~3/2 = 700.6429 ¢
  • CWE: ~72/65 = 171.4286 ¢, ~3/2 = 700.6525 ¢

Optimal ET sequence: 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: [⟨7 0 -17 64 124 37 -49 63], ⟨0 1 3 -4 -9 -1 7 -3]]

Optimal tunings:

  • WE: ~21/19 = 171.4244 ¢, ~3/2 = 700.6395 ¢
  • CWE: ~21/19 = 171.4286 ¢, ~3/2 = 700.6568 ¢

Optimal ET sequence: 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: [⟨7 0 -17 64 124 37 -49 63 76], ⟨0 1 3 -4 -9 -1 7 -3 -4]]

Optimal tunings:

  • WE: ~21/19 = 171.4321 ¢, ~3/2 = 700.6475 ¢
  • CWE: ~21/19 = 171.4286 ¢, ~3/2 = 700.6325 ¢

Optimal ET sequence: 77, 84, 161

Badness (Sintel): 1.34

29-limit

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 tunings:

  • WE: ~21/19 = 171.4348 ¢, ~3/2 = 700.6612 ¢
  • CWE: ~21/19 = 171.4286 ¢, ~3/2 = 700.6351 ¢

Optimal ET sequence: 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 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 46 & 121 temperament, and 121edo and 167edo make for good tunings.

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 tunings:

  • WE: ~2 = 1199.3465 ¢, ~3/2 = 703.7905 ¢
error map: ⟨-0.654 +1.182 -0.177 -0.056]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1749 ¢
error map: ⟨0.000 +2.220 +1.359 +1.752]

Optimal ET sequence: 46, 121, 167, 288b, 455bcd

Badness (Sintel): 3.49

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 tunings:

  • WE: ~2 = 1199.3335 ¢, ~3/2 = 703.7856 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1812 ¢

Optimal ET sequence: 46, 121, 167, 288be, 455bcde

Badness (Sintel): 1.69

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 tunings:

  • WE: ~2 = 1199.3768 ¢, ~3/2 = 703.8018 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1731 ¢

Optimal ET sequence: 46, 75e, 121, 167, 288be

Badness (Sintel): 1.25

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 tunings:

  • WE: ~2 = 1199.3518 ¢, ~3/2 = 703.7880 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1747 ¢

Optimal ET sequence: 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[2].

Subgroup: 2.3.5.7

Comma list: 6144/6125, 244140625/243045684

Mapping: [⟨1 -16 -11 14], ⟨0 33 25 -21]]

mapping generators: ~2, ~640/441

Optimal tunings:

  • WE: ~2 = 1199.5944 ¢, ~640/441 = 639.2648 ¢
error map: ⟨-0.406 +0.272 -0.233 +0.936]
  • CWE: ~2 = 1200.0000 ¢, ~640/441 = 639.4769 ¢
error map: ⟨0.000 +0.783 +0.609 +2.159]

Optimal ET sequence: 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: [⟨1 -16 -11 14 -4], ⟨0 33 25 -21 14]]

Optimal tunings:

  • WE: ~2 = 1199.5936 ¢, ~175/121 = 639.264 ¢
  • CWE: ~2 = 1200.0000 ¢, ~175/121 = 639.4770 ¢

Optimal ET sequence: 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[2]. The extension here is a less accurate 7-limit interpretation.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 2500000/2470629

Mapping: [⟨1 -23 -4 0], ⟨0 35 9 4]]

mapping generators: ~2, ~80/49

Optimal tunings:

  • WE: ~2 = 1199.6429 ¢, ~80/49 = 842.6790 ¢
error map: ⟨-0.357 +0.228 -0.774 +1.890]
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 842.9243 ¢
error map: ⟨0.000 +0.396 +0.005 +2.871]

Optimal ET sequence: 37, 84, 121, 205

Badness (Sintel): 4.55

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -23 -4 0 14], ⟨0 35 9 4 -15]]

Optimal tunings:

  • WE: ~2 = 1199.3401 ¢, ~80/49 = 842.4880 ¢
  • CWE: ~2 = 1200.0000 ¢, ~80/49 = 842.9457 ¢

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

Badness (Sintel): 2.71

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 tunings:

  • WE: ~2 = 1199.3410 ¢, ~13/8 = 842.4885 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 842.9466 ¢

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

Badness (Sintel): 1.63

Icositritonic

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 46 & 161. It was named by Xenllium in 2019 for its number of periods per octave.

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 tunings:

  • WE: ~1323/1280 = 52.1732 ¢, ~3/2 = 701.0660 ¢
error map: ⟨-0.017 -0.906 +1.679 -0.386]
  • CWE: ~1323/1280 = 52.1739 ¢, ~3/2 = 701.0722 ¢
error map: ⟨0.000 -0.883 +1.715 -0.333]

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

Badness (Sintel): 4.98

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 tunings:

  • WE: ~33/32 = 52.1740 ¢, ~3/2 = 701.0379 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.0370 ¢

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

Badness (Sintel): 2.14

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 tunings:

  • WE: ~33/32 = 52.1724 ¢, ~3/2 = 701.1310 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.1524 ¢

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

Badness (Sintel): 1.67

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 tunings:

  • WE: ~33/32 = 52.1735 ¢, ~3/2 = 701.1493 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.1549 ¢

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

Badness (Sintel): 1.26

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 tunings:

  • WE: ~33/32 = 52.1744 ¢, ~3/2 = 701.0649 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.0582 ¢

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

Badness (Sintel): 1.31

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 tunings:

  • WE: ~33/32 = 52.1768 ¢, ~3/2 = 701.1259 ¢
  • CWE: ~33/32 = 52.1739 ¢, ~3/2 = 701.0841 ¢

Optimal ET sequence: 46, 115, 161, 207

Badness (Sintel): 1.27

References

  1. ↑ Yahoo! Tuning Group | Some more unclassified temperaments
  2. ↑ 2.0 2.1 2.2 2.3 2.4 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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