Gamelismic clan

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The 2.3.7-subgroup comma for the gamelismic clan is the gamelisma, 1029/1024, with monzo [-10 1 0 3⟩. For any member of the clan, for the rank-3 gamelismic temperament itself, and for the rank-2 2.3.7 temperament slendric (a.k.a. gamelic), this means three ~8/7 intervals give a fifth, 3/2. In fact, we find that 3/2 = (8/7)3 × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. 36edo is a good tuning for slendric, though if the full 7-limit is desired, 72edo, 77edo or 118edo might be preferred.

To the gamelisma itself we need to add the comma which appears next on the modified normal comma list for the full 7-limit. The second comma on the list for mothra is 81/80, for rodan 245/243, for guiron 32805/32768, for gorgo 36/35, and for gidorah 256/245. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.

Miracle adds 33075/32768 and uses the secor, half an ~8/7, as generator. Lemba adds 525/512 to the list, and has a half-octave period. Valentine adds 6144/6125 with a generator of ~21/20 and superkleismic adds 875/864 with a generator of ~6/5. Unidec adds 4375/4374, and has a generator of ~10/9 with a half-octave period. Hemithirds adds 65625/65536 with a generator half of a classical major third. Finally, tritikleismic adds 15625/15552 and has a generator of 6/5 with a 1/3-octave period.

Full 7-limit temperaments discussed elsewhere are:

The rest are considered below.

No-five subgroup extensions of slendric include radon, a 2.3.7.11 extension that may be viewed as no-five rodan, and baladic, a 2.3.7.13.17 extension, considered below. Dicussed elsewhere is gigapyth in the 2.3.7.85 subgroup.

Slendric

Main article: Slendric

Subgroup: 2.3.7

Comma list: 1029/1024

Sval mapping: [⟨1 1 3], ⟨0 3 -1]]

sval mapping generators: ~2, ~8/7

Gencom mapping: [⟨1 1 0 3], ⟨0 3 0 -1]]

gencom: [2 8/7; 1029/1024]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 233.889
error map: ⟨0.000 -0.288 -2.715]
  • POTE: ~2 = 1200.000, ~8/7 = 233.688
error map: ⟨0.000 -0.892 -2.513]

Optimal ET sequence: 36, 77, 113, 190

Radon

Subgroup: 2.3.7.11

Comma list: 896/891, 1029/1024

Sval mapping: [⟨1 1 3 6], ⟨0 3 -1 -13]]

Gencom mapping: [⟨1 1 0 3 6], ⟨0 3 0 -1 -13]]

gencom: [2 8/7; 896/891 1029/1024]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.384
error map: ⟨0.000 +1.197 -3.210 +1.691]
  • POTE: ~2 = 1200.000, ~8/7 = 234.381
error map: ⟨0.000 +1.187 -3.206 +1.735]

Optimal ET sequence: 36, 41, 87, 128

Baladic

Baladic is a 2.3.7.13.17 subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. 36edo is an excellent baladic tuning.

Subgroup: 2.3.7.13.17

Comma list: 169/168, 273/272, 289/288

Sval mapping: [⟨2 2 6 7 7], ⟨0 3 -1 1 3]]

sval mapping generators: ~17/12, ~8/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.138
  • POTE: ~2 = 1200.000, ~8/7 = 233.616

Optimal ET sequence: 10, 26, 36, 154f, 190ffg, 226ffg

Rodan

Main article: Rodan
For the 5-limit version, see Syntonic–diatonic equivalence continuum #Rodan (5-limit).

Rodan tempers out 245/243 and can be described as the 41 & 46 temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.

Subgroup: 2.3.5.7

Comma list: 245/243, 1029/1024

Mapping: [⟨1 1 -1 3], ⟨0 3 17 -1]]

Wedgie: ⟨⟨ 3 17 -1 20 -10 -50 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.450
error map: ⟨0.000 +1.396 -0.660 -3.276]
  • POTE: ~2 = 1200.000, ~8/7 = 234.417
error map: ⟨0.000 +1.295 -1.229 -3.243]

Minimax tuning:

[[1 0 0 0⟩, [5/3 0 1/6 -1/6⟩, [25/9 0 17/18 -17/18⟩, [25/9 0 -1/18 1/18⟩]
eigenmonzo (unchanged-interval) basis: 2.7/5

Algebraic generator: larger root of 20x2 - 36x + 15, or (9 + √6)/10.

Optimal ET sequence: 41, 87, 128, 215d

Badness (Smith): 0.037112

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 441/440

Mapping: [⟨1 1 -1 3 6], ⟨0 3 17 -1 -13]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.463
  • POTE: ~2 = 1200.000, ~8/7 = 234.459

Minimax tuning:

  • 11-odd-limit: ~8/7 = [4/19 2/19 0 0 -1/19⟩
[[1 0 0 0 0⟩, [31/19 6/19 0 0 -3/19⟩, [49/19 34/19 0 0 -17/19⟩, [53/19 -2/19 0 0 1/19⟩, [62/19 -26/19 0 0 13/19⟩]
eigenmonzo (unchanged-interval) basis: 2.11/9

Algebraic generator: positive root of x2 + 16x - 31, or √95 - 8.

Optimal ET sequence: 41, 87

Badness (Smith): 0.023093

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 352/351, 364/363

Mapping: [⟨1 1 -1 3 6 8], ⟨0 3 17 -1 -13 -22]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.482
  • POTE: ~2 = 1200.000, ~8/7 = 234.482

Minimax tuning:

  • 13- and 15-odd-limit: ~8/7 = [3/14 1/14 0 0 0 -1/28⟩
eigenmonzo (unchanged-interval) basis: 2.13/9

Algebraic generator: Gatetone, positive root of 4x6 - 7x - 1. Recurrence converges slowly.

Optimal ET sequence: 41, 46, 87

Badness (Smith): 0.018448

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 154/153, 196/195, 245/243, 256/255, 273/272

Mapping: [⟨1 1 -1 3 6 8 8], ⟨0 3 17 -1 -13 -22 -20]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.532
  • POTE: ~2 = 1200.000, ~8/7 = 234.524

Minimax tuning:

  • 17-odd-limit: ~8/7 = [3/13 1/13 0 0 0 0 -1/26⟩
eigenmonzo (unchanged-interval) basis: 2.17/9

Optimal ET sequence: 41, 46, 87

Badness (Smith): 0.016743

Aerodactyl

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 245/243, 385/384, 441/440

Mapping: [⟨1 1 -1 3 6 -1], ⟨0 3 17 -1 -13 24]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.670
  • POTE: ~2 = 1200.000, ~8/7 = 234.639

Optimal ET sequence: 5, 41f, 46

Badness (Smith): 0.033986

Aerodino

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243, 1029/1024

Mapping: [⟨1 1 -1 3 -3], ⟨0 3 17 -1 33]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.719
  • POTE: ~2 = 1200.000, ~8/7 = 234.728

Optimal ET sequence: 5e, 41e, 46

Badness (Smith): 0.054294

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243, 847/845

Mapping: [⟨1 1 -1 3 -3 -1], ⟨0 3 17 -1 33 24]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.786
  • POTE: ~2 = 1200.000, ~8/7 = 234.782

Optimal ET sequence: 5e, 41ef, 46

Badness (Smith): 0.035836

Varan

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243, 1029/1024

Mapping: [⟨1 1 -1 3 -2], ⟨0 3 17 -1 28]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.197
  • POTE: ~2 = 1200.000, ~8/7 = 234.145

Optimal ET sequence: 5e, 36ce, 41

Badness (Smith): 0.044937

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 245/243, 352/351

Mapping: [⟨1 1 -1 3 -2 0], ⟨0 3 17 -1 28 19]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 234.111
  • POTE: ~2 = 1200.000, ~8/7 = 234.089

Optimal ET sequence: 5e, 36ce, 41

Badness (Smith): 0.032284

Guiron

Guiron tempers out the schisma, and finds the prime 5 at the diminished fourth as does any temperament in the schismatic family. It can be described as 36 & 41. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 10976/10935

Mapping: [⟨1 1 7 3], ⟨0 3 -24 -1]]

mapping generators: ~2, ~8/7

Wedgie: ⟨⟨ 3 -24 -1 -45 -10 65 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 233.903
error map: ⟨0.000 -0.246 +0.012 -2.729]
  • POTE: ~2 = 1200.000, ~8/7 = 233.930
error map: ⟨0.000 -0.165 -0.637 -2.756]

Minimax tuning:

[[1 0 0 0⟩, [15/8 0 -1/8 0⟩, [0 0 1 0⟩, [65/24 0 1/24 0⟩]
eigenmonzo (unchanged-interval) basis: 2.5

Optimal ET sequence: 36, 41, 77, 118, 277d

Badness (Smith): 0.047544

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 10976/10935

Mapping: [⟨1 1 7 3 -2], ⟨0 3 -24 -1 28]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 233.930
  • POTE: ~2 = 1200.000, ~8/7 = 233.931

Minimax tuning:

  • 11-odd-limit: ~8/7 = [7/24 0 -1/24⟩
[[1 0 0 0 0⟩, [15/8 0 -1/8 0 0⟩, [0 0 1 0 0⟩, [65/24 0 1/24 0 0⟩, [37/6 0 -7/6 0 0⟩]
eigenmonzo (unchanged-interval) basis: 2.5

Optimal ET sequence: 36e, 41, 77, 118, 159, 277d

Badness (Smith): 0.026648

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 729/728

Mapping: [⟨1 1 7 3 -2 0], ⟨0 3 -24 -1 28 19]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 233.902
  • POTE: ~2 = 1200.000, ~8/7 = 233.899

Optimal ET sequence: 36e, 41, 77, 118

Badness (Smith): 0.028444

Mothra

Mothra tempers out 81/80 and finds the prime 5 at a stack of four fifths as does any temperament in the meantone family. It also tempers out 1728/1715, the orwellisma. It can be described as 26 & 31. Using 31edo with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential[1], so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo in the article on strictly proper 7-tone 31edo scales.

Note that mothra is also called cynder in the 7-limit, which can be a little confusing sometimes.

Its S-expression-based comma list is {S6/S7, S7/S8(, S6/S8 = S9)}, taking advantage of the fact that 81/80 is a semiparticular.

Subgroup: 2.3.5.7

Comma list: 81/80, 1029/1024

Mapping: [⟨1 1 0 3], ⟨0 3 12 -1]]

mapping generators: ~2, ~8/7

Wedgie: ⟨⟨ 3 12 -1 12 -10 -36 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 232.400
error map: ⟨0.000 -4.756 +2.482 -1.226]
  • POTE: ~2 = 1200.000, ~8/7 = 232.193
error map: ⟨0.000 -5.375 +0.005 -1.019]

Algebraic generator: Rabrindanath, largest real root of x8 - 3x2 + 1, or 232.0774 cents.

Minimax tuning:

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3 0 -1/12 0⟩]
eigenmonzo (unchanged-interval) basis: 2.5

Optimal ET sequence: 5, 21c, 26, 31

Badness (Smith): 0.037146

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 385/384

Mapping: [⟨1 1 0 3 5], ⟨0 3 12 -1 -8]]

Wedgie: ⟨⟨ 3 12 -1 -8 12 -10 -23 -36 -60 -19 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 232.203
  • POTE: ~2 = 1200.000, ~8/7 = 232.031

Optimal ET sequence: 5, 26, 31, 88, 119be, 150be

Badness (Smith): 0.025642

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 105/104, 144/143

Mapping: [⟨1 1 0 3 5 1], ⟨0 3 12 -1 -8 14]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 231.993
  • POTE: ~2 = 1200.000, ~8/7 = 231.811

Optimal ET sequence: 5, 26, 31, 57, 88

Badness (Smith): 0.023954

Music

Cynder

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 1029/1024

Mapping: [⟨1 1 0 3 0], ⟨0 3 12 -1 18]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 231.566
  • POTE: ~2 = 1200.000, ~8/7 = 231.317

Optimal ET sequence: 5e, 21ce, 26

Badness (Smith): 0.055706

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 78/77, 81/80, 640/637

Mapping: [⟨1 1 0 3 0 1], ⟨0 3 12 -1 18 14]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 231.546
  • POTE: ~2 = 1200.000, ~8/7 = 231.293

Optimal ET sequence: 5e, 21cef, 26

Badness (Smith): 0.034124

Mosura

The S-expression-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {S6/S7, S7/S8, (S6/S8 = S9,) S8/S10}.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 540/539

Mapping: [⟨1 1 0 3 -1], ⟨0 3 12 -1 23]]

Wedgie: ⟨⟨ 3 12 -1 23 12 -10 26 -36 12 68 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 232.557
  • POTE: ~2 = 1200.000, ~8/7 = 232.419

Optimal ET sequence: 5e, 26e, 31, 129

Badness (Smith): 0.031334

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 144/143, 176/175, 196/195

Mapping: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 232.635
  • POTE: ~2 = 1200.000, ~8/7 = 232.640

Optimal ET sequence: 31, 67, 98

Badness (Smith): 0.036857

Gorgo

For the 5-limit version, see Syntonic–diatonic equivalence continuum #Laconic.

Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the 16 & 21 temperament.

Subgroup: 2.3.5.7

Comma list: 36/35, 1029/1024

Mapping: [⟨1 1 1 3], ⟨0 3 7 -1]]

Wedgie: ⟨⟨ 3 7 -1 4 -10 -22 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 228.724
error map: ⟨0.000 -15.782 +14.756 +2.450]
  • POTE: ~2 = 1200.000, ~8/7 = 228.334
error map: ⟨0.000 -16.954 +12.022 +2.840]

Optimal ET sequence: 5, 11c, 16, 21

Badness (Smith): 0.060663

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 1029/1024

Mapping: [⟨1 1 1 3 1], ⟨0 3 7 -1 13]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 227.833
  • POTE: ~2 = 1200.000, ~8/7 = 227.373

Optimal ET sequence: 5e, 16, 21, 37b

Badness (Smith): 0.049500

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 507/500

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

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 227.633
  • POTE: ~2 = 1200.000, ~8/7 = 227.230

Optimal ET sequence: 5e, 16, 21, 37b

Badness (Smith): 0.032664

Spartan

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 1029/1024

Mapping: [⟨1 1 1 3 5], ⟨0 3 7 -1 -8]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 229.420
  • POTE: ~2 = 1200.000, ~8/7 = 229.535

Optimal ET sequence: 5, 16e, 21

Badness (Smith): 0.062683

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 56/55, 507/500

Mapping: [⟨1 1 1 3 5 2], ⟨0 3 7 -1 -8 9]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 228.758
  • POTE: ~2 = 1200.000, ~8/7 = 229.059

Optimal ET sequence: 5, 16e, 21

Badness (Smith): 0.047071

Music

Gidorah

For the 5-limit version, see Syntonic–diatonic equivalence continuum #University.

Gidorah is a very low-accuracy temperament where the generator of ~8/7 is lumped together with ~6/5. 16c-, 21cc-, and 26ccc-edo are among the possible tunings.

Subgroup: 2.3.5.7

Comma list: 21/20, 144/125

Mapping: [⟨1 1 2 3], ⟨0 3 2 -1]]

Wedgie: ⟨⟨ 3 2 -1 -4 -10 -8 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 227.100
error map: ⟨0.000 -20.655 +67.886 +4.074]
  • POTE: ~2 = 1200.000, ~8/7 = 230.762
error map: ⟨0.000 -9.668 +75.211 +0.412]

Optimal ET sequence: 1b, 5

Badness (Smith): 0.062262

Oncle

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

Oncle can be described as the 31 & 36c temperamnet.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 2430/2401

Mapping: [⟨1 1 6 3], ⟨0 3 -19 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 232.383
error map: ⟨0.000 -4.807 -1.585 -1.209]
  • POTE: ~2 = 1200.000, ~8/7 = 232.498
error map: ⟨0.000 -4.461 -3.778 -1.324]

Optimal ET sequence: 31, 98c, 129c, 160bc

Badness (Smith): 0.088384

Archaeotherium

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

Archaeotherium can be described as the 21 & 26 temperamnet.

Subgroup: 2.3.5.7

Comma list: 405/392, 1029/1024

Mapping: [⟨1 1 5 3], ⟨0 3 -14 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 229.951
error map: ⟨0.000 -12.102 -5.626 +1.223]
  • POTE: ~2 = 1200.000, ~8/7 = 230.258
error map: ⟨0.000 -11.180 -9.933 +0.916]

Optimal ET sequence: 21, 26, 47, 73bc, 99bc

Badness (Smith): 0.146306

Clyndro

See also: Mavila family

Clyndro tempers out 135/128 and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the mavila family. It can be described as the 11 & 16 temperamnet.

Subgroup: 2.3.5.7

Comma list: 135/128, 360/343

Mapping: [⟨1 1 4 3], ⟨0 3 -9 -1]]

Wedgie: ⟨⟨ 3 -9 -1 -21 -10 23 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 225.752
error map: ⟨0.000 -24.699 -18.081 +5.422]
  • POTE: ~2 = 1200.000, ~8/7 = 226.469
error map: ⟨0.000 -22.548 -24.534 +4.705]

Optimal ET sequence: 5c, 11, 16

Badness (Smith): 0.159179

11-limit

Subgroup: 2.3.5.7.11

Comma list: 33/32, 45/44, 352/343

Mapping: [⟨1 1 4 3 4], ⟨0 3 -9 -1 -3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 225.384
  • POTE: ~2 = 1200.000, ~8/7 = 226.428

Optimal ET sequence: 5c, 11, 16

Badness (Smith): 0.069703

Miracle

Main article: Miracle
For the 5-limit version, see Syntonic–31 equivalence continuum #Ampersand.

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1024

Mapping: [⟨1 1 3 3], ⟨0 6 -7 -2]]

mapping generator: ~2, ~15/14

Wedgie: ⟨⟨ 6 -7 -2 -25 -20 15 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 116.677
error map: ⟨0.000 -1.892 -3.054 -2.180]
  • POTE: ~2 = 1200.000, ~15/14 = 116.675
error map: ⟨0.000 -1.904 -3.040 -2.176]

Minimax tuning:

[[1 0 0 0⟩, [25/13 6/13 -6/13 0⟩, [25/13 -7/13 7/13 0⟩, [35/13 -2/13 2/13 0⟩]
eigenmonzo (unchanged-interval) basis: 2.5/3
[[1 0 0 0⟩, [25/19 12/19 -6/19 0⟩, [50/19 -14/19 7/19 0⟩, [55/19 -4/19 2/19 0⟩]
eigenmonzo (unchanged-interval) basis: 2.9/5

Tuning ranges:

  • 7-odd-limit diamond monotone: ~15/14 = [114.286, 120.000] (2\21 to 1\10)
  • 9-odd-limit diamond monotone: ~15/14 = [116.129, 120.000] (3\31 to 1\10)
  • 7- and 9-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]

Algebraic generator: Secor59, positive root of 15x6 - 8x4 - 12

Optimal ET sequence: 10, 21, 31, 41, 72

Badness (Smith): 0.016742

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 385/384

Mapping: [⟨1 1 3 3 2], ⟨0 6 -7 -2 15]]

Wedgie: ⟨⟨ 6 -7 -2 15 -25 -20 3 15 59 49 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 116.711
  • POTE: ~2 = 1200.000, ~15/14 = 116.633

Minimax tuning:

  • 11-odd-limit: ~15/14 = [1/19 2/19 -1/19⟩
[[1 0 0 0 0⟩, [25/19 12/19 -6/19 0 0⟩, [50/19 -14/19 7/19 0 0⟩, [55/19 -4/19 2/19 0 0⟩, [53/19 30/19 -15/19 0 0⟩]
eigenmonzo (unchanged-interval) basis: 2.9/5

Tuning ranges:

  • 11-odd-limit diamond monotone: ~15/14 = [116.129, 117.073] (3\31 to 4\41)
  • 11-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]

Algebraic generator: Secor59

Optimal ET sequence: 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde

Badness (Smith): 0.010684

Miraculous

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 243/242

Mapping: [⟨1 1 3 3 2 4], ⟨0 6 -7 -2 15 -3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 116.758
  • POTE: ~2 = 1200.000, ~15/14 = 116.747

Optimal ET sequence: 10, 21e, 31, 41, 72f, 113f, 185cff

Badness (Smith): 0.018669

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 120/119, 144/143, 154/153, 170/169

Mapping: [⟨1 1 3 3 2 4 4], ⟨0 6 -7 -2 15 -3 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 116.742
  • POTE: ~2 = 1200.000, ~15/14 = 116.769

Optimal ET sequence: 10, 21e, 31, 41, 72fg

Badness (Smith): 0.017084

Benediction

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350, 385/384

Mapping: [⟨1 1 3 3 2 7], ⟨0 6 -7 -2 15 -34]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 116.541
  • POTE: ~2 = 1200.000, ~15/14 = 116.574

Optimal ET sequence: 31, 72, 103, 175f

Badness (Smith): 0.015715

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 351/350, 375/374

Mapping: [⟨1 1 3 3 2 7 7], ⟨0 6 -7 -2 15 -34 -30]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 116.529
  • POTE: ~2 = 1200.000, ~15/14 = 116.585

Optimal ET sequence: 31, 72, 103, 175f, 422bcdefffg

Badness (Smith): 0.012537

Manna

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 325/324, 385/384

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 38]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 116.814
  • POTE: ~2 = 1200.000, ~15/14 = 116.739

Optimal ET sequence: 31f, 41, 72, 185cf, 257cff

Badness (Smith): 0.017012

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 325/324, 385/384

Mapping: [⟨1 1 3 3 2 0 0], ⟨0 6 -7 -2 15 38 42]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 116.802
  • POTE: ~2 = 1200.000, ~15/14 = 116.727

Optimal ET sequence: 31fg, 41, 72, 185cf, 257cff

Badness (Smith): 0.014680

Semimiracle

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 243/242, 385/384

Mapping: [⟨2 2 6 6 4 7], ⟨0 6 -7 -2 15 2]]

mapping generators: ~55/39, ~15/14

Optimal tunings:

  • CTE: ~55/39 = 600.000, ~15/14 = 116.735
  • POTE: ~55/39 = 600.000, ~15/14 = 116.624

Optimal ET sequence: 10, 62, 72

Badness (Smith): 0.024622

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 243/242, 273/272

Mapping: [⟨2 2 6 6 4 7 7], ⟨0 6 -7 -2 15 2 6]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~15/14 = 116.771
  • POTE: ~17/12 = 600.000, ~15/14 = 116.628

Optimal ET sequence: 10, 62, 72

Badness (Smith): 0.016130

Hemisecordite

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 847/845

Mapping: [⟨1 1 3 3 2 2], ⟨0 12 -14 -4 30 35]]

mapping generators: ~2, ~27/26

Optimal tunings:

  • CTE: ~2 = 1200.000, ~27/26 = 58.337
  • POTE: ~2 = 1200.000, ~27/26 = 58.288

Optimal ET sequence: 41, 62, 103, 247c, 350bcde

Badness (Smith): 0.025589

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 385/384, 847/845

Mapping: [⟨1 1 3 3 2 2 2], ⟨0 12 -14 -4 30 35 43]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~27/26 = 58.312
  • POTE: ~2 = 1200.000, ~27/26 = 58.261

Optimal ET sequence: 41, 62, 103

Badness (Smith): 0.022535

Semihemisecordite

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 289/288, 385/384, 847/845

Mapping: [⟨2 2 6 6 4 4 7], ⟨0 12 -14 -4 30 35 12]]

mapping generators: ~17/12, ~27/26

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~27/26 = 58.350
  • POTE: ~17/12 = 600.000, ~27/26 = 58.288

Optimal ET sequence: 62, 144g, 206begg

Badness (Smith): 0.046958

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384

Mapping: [⟨2 2 6 6 4 4 7 8], ⟨0 12 -14 -4 30 35 12 5]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~27/26 = 58.356
  • POTE: ~17/12 = 600.000, ~27/26 = 58.283

Optimal ET sequence: 62, 144gh, 206begghh

Badness (Smith): 0.035057

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384

Mapping: [⟨2 2 6 6 4 4 7 8 7], ⟨0 12 -14 -4 30 35 12 5 21]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~27/26 = 58.366
  • POTE: ~17/12 = 600.000, ~27/26 = 58.283

Optimal ET sequence: 62, 144gh, 206begghhi

Badness (Smith): 0.026421

Phicordial

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 2200/2197

Mapping: [⟨1 7 -4 1 17 4], ⟨0 -18 21 6 -45 -1]]

mapping generators: ~2, ~16/13

Optimal tunings:

  • CTE: ~2 = 1200.000, ~16/13 = 361.096
  • POTE: ~2 = 1200.000, ~16/13 = 361.121

Optimal ET sequence: 103, 216c, 319bcde, 535bccdef

Badness (Smith): 0.033198

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 441/440, 2200/2197

Mapping: [⟨1 7 -4 1 17 4 8], ⟨0 -18 21 6 -45 -1 -13]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~16/13 = 361.098
  • POTE: ~2 = 1200.000, ~16/13 = 361.123

Optimal ET sequence: 103, 216c, 319bcde

Badness (Smith): 0.024705

Revelation

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1029/1024

Mapping: [⟨1 1 3 3 5], ⟨0 6 -7 -2 -16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 116.142
  • POTE: ~2 = 1200.000, ~15/14 = 116.277

Optimal ET sequence: 10e, 21, 31

Badness (Smith): 0.032946

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 105/104, 512/507

Mapping: [⟨1 1 3 3 5 4], ⟨0 6 -7 -2 -16 -3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 116.194
  • POTE: ~2 = 1200.000, ~15/14 = 116.268

Optimal ET sequence: 10e, 21, 31

Badness (Smith): 0.029452

Hemimiracle

Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 1029/1024

Mapping: [⟨1 1 3 3 4], ⟨0 12 -14 -4 -11]]

mapping generators: ~2, ~33/32

Optimal tunings:

  • CTE: ~2 = 1200.000, ~33/32 = 58.399
  • POTE: ~2 = 1200.000, ~33/32 = 58.408

Optimal ET sequence: 20, 21, 41

Badness (Smith): 0.059232

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 196/195, 245/242, 512/507

Mapping: [⟨1 1 3 3 4 4], ⟨0 12 -14 -4 -11 -6]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~33/32 = 58.436
  • POTE: ~2 = 1200.000, ~33/32 = 58.430

Optimal ET sequence: 20, 21, 41

Badness (Smith): 0.043151

Oracle

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 1029/1024

Mapping: [⟨1 7 -4 1 3], ⟨0 -12 14 4 1]]

mapping generators: ~2, ~11/8

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/8 = 541.670
  • POTE: ~2 = 1200.000, ~11/8 = 541.668

Optimal ET sequence: 11, 20, 31, 82e, 113e, 144ee

Badness (Smith): 0.042687

Hemiseven

Hemiseven can be described as the 72 & 77 temperament. 149edo is an obvious tuning.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 19683/19600

Mapping: [⟨1 4 14 2], ⟨0 -6 -29 2]]

mapping generators: ~2, ~320/243

Wedgie: ⟨⟨ 6 29 -2 32 -20 -86 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~320/243 = 483.215
error map: ⟨0.000 -1.247 +0.441 -2.395]
  • POTE: ~2 = 1200.000, ~320/243 = 483.267
error map: ⟨0.000 -1.554 -1.043 -2.293]

Optimal ET sequence: 72, 149, 221, 514bd, 735bcdd

Badness (Smith): 0.056557

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 19683/19600

Mapping: [⟨1 4 14 2 -5], ⟨0 -6 -29 2 21]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~320/243 = 483.247
  • POTE: ~2 = 1200.000, ~320/243 = 483.276

Optimal ET sequence: 72, 149, 221e, 293de

Badness (Smith): 0.028467

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384, 441/440, 676/675

Mapping: [⟨1 4 14 2 -5 19], ⟨0 -6 -29 2 21 -38]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~120/91 = 483.213
  • POTE: ~2 = 1200.000, ~120/91 = 483.255

Optimal ET sequence: 72, 149, 221ef

Badness (Smith): 0.021900

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 351/350, 385/384, 441/440, 676/675

Mapping: [⟨1 4 14 2 -5 19 21], ⟨0 -6 -29 2 21 -38 -42]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~45/34 = 483.213
  • POTE: ~2 = 1200.000, ~45/34 = 483.261

Optimal ET sequence: 72, 149, 221ef

Badness (Smith): 0.015701

Unidec

Main article: Unidec

5-limit (unidecmic)

Subgroup: 2.3.5

Comma list: 31381059609/31250000000

Mapping: [⟨2 5 8], ⟨0 -6 -11]]

mapping generators: ~177147/125000, ~10/9

Optimal tunings:

  • CTE: ~177147/125000 = 600.000, ~10/9 = 183.041
error map: ⟨0.000 -0.201 +0.235]
  • POTE: ~177147/125000 = 600.000, ~10/9 = 183.047
error map: ⟨0.000 -0.236 +0.172]

Optimal ET sequence: 26, 46, 72, 118, 2524, 2642, 2760

Badness (Smith): 0.082423

7-limit

Subgroup: 2.3.5.7

Comma list: 1029/1024, 4375/4374

Mapping: [⟨2 5 8 5], ⟨0 -6 -11 2]]

Wedgie: ⟨⟨ 12 22 -4 7 -40 -71 ]]

Optimal tunings:

  • CTE: ~1225/864 = 600.000, ~10/9 = 183.060
error map: ⟨0.000 -0.313 +0.030 -2.707]
  • POTE: ~1225/864 = 600.000, ~10/9 = 183.161
error map: ⟨0.000 -0.924 -1.090 -2.503]

Minimax tuning:

[[1 0 0 0⟩, [47/26 0 6/13 -6/13⟩, [71/26 0 11/13 -11/13⟩, [71/26 0 -2/13 2/13⟩]
eigenmonzo (unchanged-interval) basis: 2.7/5
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [57/28 11/7 0 -11/14⟩, [20/7 -2/7 0 1/7⟩]
eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 26, 46, 72, 118, 190

Badness (Smith): 0.038393

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 4375/4374

Mapping: [⟨2 5 8 5 6], ⟨0 -6 -11 2 3]]

Optimal tunings:

  • CTE: ~99/70 = 600.000, ~10/9 = 183.074
  • CWE: ~99/70 = 600.000, ~10/9 = 183.146

Minimax tuning:

[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [57/28 11/7 0 -11/14 0⟩, [20/7 -2/7 0 1/7 0⟩, [99/28 -3/7 0 3/14 0⟩]
eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 26, 46, 72, 118, 190

Badness (Smith): 0.015479

Ekadash

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 625/624, 729/728

Mapping: [⟨2 5 8 5 6 19], ⟨0 -6 -11 2 3 -38]]

Optimal tunings:

  • CTE: ~99/70 = 600.000, ~10/9 = 183.125
  • POTE: ~99/70 = 600.000, ~10/9 = 183.187

Optimal ET sequence: 46f, 72, 118, 190, 262df, 452cdef

Badness (Smith): 0.020381

Hendec

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 385/384

Mapping: [⟨2 5 8 5 6 8], ⟨0 -6 -11 2 3 -2]]

Optimal tunings:

  • CTE: ~91/64 = 600.000, ~10/9 = 183.048
  • POTE: ~91/64 = 600.000, ~10/9 = 183.198

Optimal ET sequence: 26, 46, 72, 190ff

Badness (Smith): 0.017707

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 273/272, 325/324, 364/363

Mapping: [⟨2 5 8 5 6 8 10], ⟨0 -6 -11 2 3 -2 -6]]

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~10/9 = 183.020
  • POTE: ~17/12 = 600.000, ~10/9 = 183.196

Optimal ET sequence: 26, 46, 72, 190ffg

Badness (Smith): 0.011676

Lagaca

Subgroup: 2.3.5.7

Comma list: 1029/1024, 11529602/11390625

Mapping: [⟨2 5 2 5], ⟨0 -9 13 3]]

mapping generators: ~3375/2401, ~15/14

Wedgie: ⟨⟨ 18 -26 -6 -83 -60 59 ]]

Optimal tunings:

  • CTE: ~3375/2401 = 600.000, ~15/14 = 122.031
error map: ⟨0.000 -0.232 +0.087 -2.734]
  • POTE: ~3375/2401 = 600.000, ~15/14 = 122.027
error map: ⟨0.000 -0.195 +0.033 -2.746]

Optimal ET sequence: 10, 98, 108, 118

Badness (Smith): 0.144345

Necromanteion

Subgroup: 2.3.5.7

Comma list: 1029/1024, 5103/5000

Mapping: [⟨1 7 10 1], ⟨0 -12 -17 4]]

mapping generators: ~2, ~48/35

Wedgie: ⟨⟨ 12 17 -4 -1 -40 -57 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~48/35 = 541.743
error map: ⟨0.000 -2.872 +4.053 -1.853]
  • POTE: ~2 = 1200.000, ~48/35 = 541.779
error map: ⟨0.000 -3.304 +3.442 -1.710]

Optimal ET sequence: 11c, 20c, 31, 144c, 175c

Badness (Smith): 0.117680

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242, 1029/1024

Mapping: [⟨1 7 10 1 17], ⟨0 -12 -17 4 -30]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/11 = 541.695
  • POTE: ~2 = 1200.000, ~15/11 = 541.729

Optimal ET sequence: 20ce, 31, 113c, 144c

Badness (Smith): 0.053459

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242, 343/338

Mapping: [⟨1 7 10 1 17 1], ⟨0 -12 -17 4 -30 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/11 = 541.673
  • POTE: ~2 = 1200.000, ~15/11 = 541.606

Optimal ET sequence: 20ce, 31, 82cf, 113cf

Badness (Smith): 0.047015

Restles

See also: Lesser tendoneutralic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 153664/151875

Mapping: [⟨1 -2 8 4], ⟨0 12 -19 -4]]

mapping generators: ~2. ~315/256

Wedgie: ⟨⟨ 12 -19 -4 -58 -40 44 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~315/256 = 358.548
error map: ⟨0.000 +0.620 +1.275 -3.018]
  • POTE: ~2 = 1200.000, ~315/256 = 358.548
error map: ⟨0.000 +0.627 +1.265 -3.020]

Optimal ET sequence: 77, 87, 164

Badness (Smith): 0.108011

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 153664/151875

Mapping: [⟨1 -2 8 4 -7], ⟨0 12 -19 -4 35]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~27/22 = 358.575
  • POTE: ~2 = 1200.000, ~27/22 = 358.571

Optimal ET sequence: 77, 87, 164, 251d

Badness (Smith): 0.054655

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 676/675

Mapping: [⟨1 -2 8 4 -7 4], ⟨0 12 -19 -4 35 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~16/13 = 358.576
  • POTE: ~2 = 1200.000, ~16/13 = 358.574

Optimal ET sequence: 77, 87, 164, 251d

Badness (Smith): 0.028187

Quartemka

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1250000/1240029

Mapping: [⟨1 4 6 2], ⟨0 -21 -32 7]]

mapping generators: ~2, ~27/25

Wedgie: ⟨⟨ 21 32 -7 2 -70 -106 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~27/25 = 137.971
error map: ⟨0.000 +0.658 -1.380 -3.030]
  • POTE: ~2 = 1200.000, ~27/25 = 138.006
error map: ⟨0.000 -0.075 -2.496 -2.786]

Optimal ET sequence: 26, 61, 87, 113, 200

Badness (Smith): 0.152287

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 800000/793881

Mapping: [⟨1 4 6 2 3], ⟨0 -21 -32 7 4]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~27/25 = 137.970
  • POTE: ~2 = 1200.000, ~27/25 = 137.990

Optimal ET sequence: 26, 61, 87, 200, 287d

Badness (Smith): 0.057307

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 2200/2197

Mapping: [⟨1 4 6 2 3 6], ⟨0 -21 -32 7 4 -20]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~13/12 = 137.971
  • POTE: ~2 = 1200.000, ~13/12 = 137.990

Optimal ET sequence: 26, 61, 87, 200

Badness (Smith): 0.028393

Tritriple

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1959552/1953125

Mapping: [⟨1 -11 -7 7], ⟨0 27 20 -9]]

mapping generators: ~2, ~864/625

Wedgie: ⟨⟨ 27 20 -9 -31 -90 -77 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~864/625 = 559.320
error map: ⟨0.000 -0.317 +0.085 -2.705]
  • POTE: ~2 = 1200.000, ~864/625 = 559.295
error map: ⟨0.000 -1.003 -0.423 -2.477]

Optimal ET sequence: 15, …, 88, 103, 118, 221, 339d

Badness (Smith): 0.118640

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 43923/43750

Mapping: [⟨1 -11 -7 7 -4], ⟨0 27 20 -9 16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~242/175 = 559.327
  • POTE: ~2 = 1200.000, ~242/175 = 559.293

Optimal ET sequence: 15, …, 88, 103, 118, 221e, 339de

Badness (Smith): 0.035350

Widefourth

Subgroup: 2.3.5.7

Comma list: 1029/1024, 48828125/48771072

Mapping: [⟨1 16 8 -2], ⟨0 -33 -13 11]]

Wedgie: ⟨⟨ 33 13 -11 -56 -110 -62 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3125/2304 = 524.188
error map: ⟨0.000 -0.154 -0.756 -2.759]
  • POTE: ~2 = 1200.000, ~3125/2304 = 524.210
error map: ⟨0.000 -0.892 -1.047 -2.513]

Optimal ET sequence: 16, 71, 87, 103, 190

Badness (Smith): 0.154117

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 234375/234256

Mapping: [⟨1 16 8 -2 17], ⟨0 -33 -13 11 -31]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~847/625 = 524.183
  • POTE: ~2 = 1200.000, ~847/625 = 524.210

Optimal ET sequence: 16, 71, 87, 103, 190

Badness (Smith): 0.040785

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 625/624, 847/845

Mapping: [⟨1 16 8 -2 17 12], ⟨0 -33 -13 11 -31 -19]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~65/48 = 524.183
  • POTE: ~2 = 1200.000, ~65/48 = 524.209

Optimal ET sequence: 16, 71, 87, 103, 190

Badness (Smith): 0.021636

Notes

  1. ↑ 31-EDO Music Theory: Supermajor Hexatonic Scale by Zhea Erose
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