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Gamelismic clan

Revision as of 17:47, 7 May 2023 by Cmloegcmluin (talk | contribs) ("optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence)

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 three gamelan temperament itself, and for the rank two 2.3.7 temperament slendric, 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 7/4, or if they do not, the other of the pair must be flattened even more. 36edo is a good tuning for gamelismic itself, 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's 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 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, the 2.3.7.11 extension that may be viewed as no-five rodan, and baladic, the 2.3.7.13.17 extension, considered below.

Slendric

Main article: Slendric
See also: No-fives subgroup temperaments § Slendric

Subgroup: 2.3.7

Comma list: 1029/1024

Sval mapping: [⟨1 1 3], ⟨0 3 -1]]

Mapping generators: ~2, ~8/7

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

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

POTE generator: ~8/7 = 233.688

Optimal ET sequence: 36, 77, 113, 190

Scales: slendric5, slendric6, slendric11, slendric16

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]]

Mapping generators: ~17/12, ~8/7

POTE generator: ~8/7 = 233.6155

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

Rodan

Main article: Rodan

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 ]]

POTE generator: ~8/7 = 234.417

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⟩]
Eigenmonzos (unchanged-intervals): 2, 7/5

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

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

Badness: 0.037112

Scales: Rodan26opt, Rodan31opt, Rodan41opt

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]]

POTE generator: ~8/7 = 234.459

Minimax tuning:

[[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⟩]
Eigenmonzos (unchanged-intervals): 2, 11/9

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

Optimal ET sequence: 41, 46, 87

Badness: 0.023093

Scales: Rodan26opt, Rodan31opt, Rodan41opt

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]]

POTE generator: ~8/7 = 234.482

Minimax tuning:

  • 13- and 15-odd-limit: ~8/7 = [3/14 1/14 0 0 0 -1/28⟩
Eigenmonzos (unchanged-intervals): 2, 13/9

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

Optimal ET sequence: 41, 46, 87

Badness: 0.018448

Scales: Rodan26opt, Rodan31opt, Rodan41opt

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]]

POTE generator: ~8/7 = 234.524

Minimax tuning:

  • 17-odd-limit: ~8/7 = [3/13 1/13 0 0 0 0 -1/26⟩
Eigenmonzos (unchanged-intervals): 2, 18/17

Optimal ET sequence: 41, 46, 87, 220dg, 307dgg

Badness: 0.016743

Scales: Rodan26opt, Rodan31opt, Rodan41opt

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]]

POTE generator: ~8/7 = 234.639

Optimal ET sequence: 5, 41f, 46, 133ff

Badness: 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]]

POTE generator: ~8/7 = 234.728

Optimal ET sequence: 41e, 46

Badness: 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]]

POTE generator: ~8/7 = 234.782

Optimal ET sequence: 41ef, 46

Badness: 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]]

POTE generator: ~8/7 = 234.145

Optimal ET sequence: 36ce, 41

Badness: 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]]

POTE generator: ~8/7 = 234.089

Optimal ET sequence: 36ce, 41

Badness: 0.032284

Guiron

See also: Schismatic family

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 ]]

POTE generator: ~8/7 = 233.930

Minimax tuning:

[[1 0 0 0⟩, [15/8 0 -1/8 0⟩, [0 0 1 0⟩, [65/24 0 1/24 0⟩]
Eigenmonzos (unchanged-intervals): 2, 5

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

Badness: 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

POTE generator: ~8/7 = 233.931

Minimax tuning:

[[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⟩]
Eigenmonzos (unchanged-intervals): 2, 5

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

Badness: 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

POTE generator: ~8/7 = 233.890

Optimal ET sequence: 36e, 41, 77, 118

Badness: 0.028444

Gorgo

In the 5-limit, gorgo tempers out the laconic comma, 2187/2000, which is the difference between three 10/9's and a 3/2. Although a higher-error temperament, it does pop up enough in the low-numbered EDOs to be useful, most notably in 16EDO and 21EDO. The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list.

5-limit (laconic)

Subgroup: 2.3.5

Comma list: 2187/2000

Mapping: [⟨1 1 1], ⟨0 3 7]]

Wedgie: ⟨⟨ 3 7 4 ]]

POTE generator: ~10/9 = 227.426

Optimal ET sequence: 5, 16, 21, 37b

Badness: 0.161799

7-limit

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 ]]

POTE generator: ~8/7 = 228.334

Optimal ET sequence: 5, 11c, 16, 21

Badness: 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]]

POTE generator: ~8/7 = 227.373

Optimal ET sequence: 16, 21, 37b

Badness: 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]]

POTE generator: ~8/7 = 227.230

Optimal ET sequence: 16, 21, 37b

Badness: 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]]

POTE generator: ~8/7 = 229.535

Optimal ET sequence: 5, 16e, 21, 47c, 68bcce

Badness: 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]]

POTE generator: ~8/7 = 229.059

Optimal ET sequence: 5, 16e, 21, 68bccef

Badness: 0.047071

Music

Gidorah

Main article: University temperament

5-limit (university)

Subgroup: 2.3.5

Comma list: 144/125

Mapping: [⟨1 1 2], ⟨0 3 2]]

POTE generator: ~6/5 = 235.4416

Optimal ET sequence: 5, 31cccc, 36…, 41…, 46…, 51…

Badness: 0.101806

7-limit

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 ]]

POTE generator: ~8/7 = 230.762

Optimal ET sequence: 5, 16c, 21cc, 26ccc

Badness: 0.062262

Oncle

For the 5-limit version of this temperament, see High badness temperaments #Oncle.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 2430/2401

Mapping: [⟨1 1 6 3], ⟨0 3 -19 -1]]

POTE generator: ~8/7 = 232.498

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

Badness: 0.088384

Archaeotherium

For the 5-limit version of this temperament, see High badness temperaments #Archaeotherium.

Subgroup: 2.3.5.7

Comma list: 405/392, 1029/1024

Mapping: [⟨1 1 5 3], ⟨0 3 -14 -1]]

POTE generator: ~8/7 = 230.258

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

Badness: 0.146306

Clyndro

See also: Pelogic family

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 ]]

POTE generator: ~8/7 = 226.469

Optimal ET sequence: 5c, 11, 16

Badness: 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]]

POTE generator: ~8/7 = 226.428

Optimal ET sequence: 5c, 11, 16

Badness: 0.069703

Miracle

Main article: Miracle

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 ]]

POTE generator: ~15/14 = 116.675

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⟩]
Eigenmonzos (unchanged-intervals): 2, 6/5
[[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⟩]
Eigenmonzos (unchanged-intervals): 2, 10/9

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]
  • 7-odd-limit diamond monotone and tradeoff: ~15/14 = [115.587, 116.993]
  • 9-odd-limit diamond monotone and tradeoff: ~15/14 = [116.129, 116.993]

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

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

Badness: 0.016742

Scales: Miracle 10, Blackjack

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]]

POTE generator: ~15/14 = 116.633

Minimax tuning:

[[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⟩]
Eigenmonzos (unchanged-intervals): 2, 10/9

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]
  • 11-odd-limit diamond monotone and tradeoff: ~15/14 = [116.129, 116.993]

Algebraic generator: Secor59

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

Badness: 0.010684

Scales: Miracle 10, Blackjack

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]]

POTE generator: ~14/13 = 116.747

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

Badness: 0.018669

Scales: Miracle 10, Blackjack

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]]

POTE generator: ~14/13 = 116.769

Optimal ET sequence: 10, 21e, 31, 41, 72fg, 113fgg

Badness: 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]]

POTE generator: ~15/14 = 116.574

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

Badness: 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]]

POTE generator: ~15/14 = 116.585

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

Badness: 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]]

POTE generator: ~15/14 = 116.739

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

Badness: 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]]

POTE generator: ~15/14 = 116.727

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

Badness: 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]]

POTE generator: ~15/14 = 116.624

Optimal ET sequence: 10, 62, 72

Badness: 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]]

POTE generator: ~15/14 = 116.628

Optimal ET sequence: 10, 62, 72

Badness: 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]]

POTE generator: ~27/26 = 58.288

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

Badness: 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]]

POTE generator: ~27/26 = 58.261

Optimal ET sequence: 41, 62, 103

Badness: 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]]

POTE generator: ~27/26 = 58.288

Optimal ET sequence: 62, 144g, 206begg, 350bcdeggg

Badness: 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]]

POTE generator: ~27/26 = 58.283

Optimal ET sequence: 62, 144gh, 206begghh

Badness: 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]]

POTE generator: ~27/26 = 58.283

Optimal ET sequence: 62, 144gh, 206begghhi

Badness: 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]]

POTE generator: ~16/13 = 361.121

Optimal ET sequence: 103, 113, 216c

Badness: 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]]

POTE generator: ~16/13 = 361.123

Optimal ET sequence: 103, 113, 216c

Badness: 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]]

POTE generator: ~15/14 = 116.277

Optimal ET sequence: 10e, 21, 31

Badness: 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]]

POTE generator: ~15/14 = 116.268

Optimal ET sequence: 10e, 21, 31

Badness: 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]]

POTE generator: ~33/32 = 58.408

Optimal ET sequence: 20, 21, 41, 144e, 185cee, 226cee

Badness: 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]]

POTE generator: ~33/32 = 58.430

Optimal ET sequence: 20, 21, 41, 144eff, 185ceeff

Badness: 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]]

POTE generator: ~11/8 = 541.668

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

Badness: 0.042687

Hemiseven

Subgroup: 2.3.5.7

Comma list: 1029/1024, 19683/19600

Mapping: [⟨1 4 14 2], ⟨0 -6 -29 2]]

Wedgie: ⟨⟨ 6 29 -2 32 -20 -86 ]]

POTE generator: ~320/243 = 483.267

Optimal ET sequence: 72, 77, 149, 221, 514bd, 735bcdd

Badness: 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]]

POTE generator: ~320/243 = 483.276

Optimal ET sequence: 72, 77, 149, 221e, 293de

Badness: 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]]

POTE generator: ~120/91 = 483.256

Optimal ET sequence: 72, 77, 149, 221ef

Badness: 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]]

POTE generator: ~45/34 = 483.261

Optimal ET sequence: 72, 77, 149, 221ef

Badness: 0.015701

Unidec

Main article: Unidec

Subgroup: 2.3.5

Comma list: 31381059609/31250000000

Mapping: [⟨2 5 8], ⟨0 -6 -11]]

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

POTE generator: ~10/9 = 183.047

Optimal ET sequence: 26, 46, 72, 118, 2524, 2642, 2760, 2878b, …, 5002bc

Badness: 0.082423

Scales: Unidec26

7-limit

Subgroup: 2.3.5.7

Comma list: 1029/1024, 4375/4374

Mapping: [⟨2 5 8 5], ⟨0 -6 -11 2]]

Mapping generators: ~1225/864, ~10/9

Wedgie: ⟨⟨ 12 22 -4 7 -40 -71 ]]

POTE generator: ~10/9 = 183.161

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⟩]
Eigenmonzos (unchanged-intervals): 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⟩]
Eigenmonzos (unchanged-intervals): 2, 9/7

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

Badness: 0.038393

Scales: Unidec26

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]]

Mapping generators: ~99/70, ~10/9

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⟩]
Eigenmonzos (unchanged-intervals): 2, 9/7

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

Badness: 0.015479

Scales: Unidec26

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]]

Mapping generators: ~99/70, ~10/9

POTE generator: ~10/9 = 183.187

Optimal ET sequence: 26f, 46f, 72, 118, 190, 262df, 452cdef

Badness: 0.020381

Scales: Unidec26

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]]

Mapping generators: ~91/64, ~10/9

POTE generator: ~10/9 = 183.198

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

Badness: 0.017707

Scales: Unidec26

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]]

Mapping generators: ~17/12, ~10/9

POTE generator: ~10/9 = 183.196

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

Badness: 0.011676

Scales: Unidec26

Superkleismic

Main article: Superkleismic
See also: Shibboleth family § Superkleismic

Subgroup: 2.3.5.7

Comma list: 875/864, 1029/1024

Mapping: [⟨1 4 5 2], ⟨0 -9 -10 3]]

Wedgie: ⟨⟨ 9 10 -3 -5 -30 -35 ]]

POTE generator: ~6/5 = 321.930

Optimal ET sequence: 11c, 15, 26, 41

Badness: 0.047932

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 385/384

Mapping: [⟨1 4 5 2 4], ⟨0 -9 -10 3 -2]]

POTE generator: ~6/5 = 321.847

Optimal ET sequence: 11c, 15, 26, 41, 179cde, 220cde, 261ccdee

Badness: 0.025659

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 245/243

Mapping: [⟨1 4 5 2 4 8], ⟨0 -9 -10 3 -2 -16]]

POTE generator: ~6/5 = 321.994

Optimal ET sequence: 11cf, 15, 26, 41

Badness: 0.021478

Lagaca

Subgroup: 2.3.5.7

Comma list: 1029/1024, 11529602/11390625

Mapping: [⟨2 5 2 5], ⟨0 -9 13 3]]

Wedgie: ⟨⟨ 18 -26 -6 -83 -60 59 ]]

POTE generator: ~15/14 = 122.027

Optimal ET sequence: 10, 98, 108, 118

Badness: 0.144345

Necromanteion

Subgroup: 2.3.5.7

Comma list: 1029/1024, 5103/5000

Mapping: [⟨1 7 10 1], ⟨0 -12 -17 4]]

Wedgie: ⟨⟨ 12 17 -4 -1 -40 -57 ]]

POTE generator: ~48/35 = 541.779

Optimal ET sequence: 11c, 20c, 31, 144c, 175c, 206bc, 237bc, 505bbccd

Badness: 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]]

POTE generator: ~15/11 = 541.729

Optimal ET sequence: 20ce, 31, 113c, 144c, 175c, 381bccdee

Badness: 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]]

POTE generator: ~15/11 = 541.606

Optimal ET sequence: 20ce, 31, 51ce, 82cf, 113cf, 144cf

Badness: 0.047015

Restles

Subgroup: 2.3.5.7

Comma list: 1029/1024, 153664/151875

Mapping: [⟨1 -2 8 4], ⟨0 12 -19 -4]]

Wedgie: ⟨⟨ 12 -19 -4 -58 -40 44 ]]

POTE generator: ~315/256 = 358.5485

Optimal ET sequence: 10, 77, 87, 164

Badness: 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]]

POTE generator: ~27/22 = 358.5713

Optimal ET sequence: 10, 77, 87, 164

Badness: 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]]

POTE generator: ~16/13 = 358.5739

Optimal ET sequence: 10, 77, 87, 164

Badness: 0.028187

Quartemka

For the 5-limit version of this temperament, see High badness temperaments #Quartemka.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1250000/1240029

Mapping: [⟨1 4 6 2], ⟨0 -21 -32 7]]

Wedgie: ⟨⟨ 21 32 -7 2 -70 -106 ]]

POTE generator: ~27/25 = 138.006

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

Badness: 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]]

POTE generator: ~27/25 = 137.990

Optimal ET sequence: 26, 61, 87, 200, 287d, 487cdd

Badness: 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]]

POTE generator: ~13/12 = 137.990

Optimal ET sequence: 26, 61, 87, 200, 487cdd

Badness: 0.028393

Tritriple

For the 5-limit version of this temperament, see High badness temperaments #Tritriple.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1959552/1953125

Mapping: [⟨1 -11 -7 7], ⟨0 27 20 -9]]

Wedgie: ⟨⟨ 27 20 -9 -31 -90 -77 ]]

POTE generator: ~864/625 = 559.295

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

Badness: 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]]

POTE generator: ~242/175 = 559.293

Optimal ET sequence: 15, 88, 103, 118, 339de

Badness: 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 ]]

POTE generator: ~3125/2304 = 524.210

Optimal ET sequence: 16, 55b, 71, 87, 103, 190

Badness: 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]]

POTE generator: ~847/625 = 524.210

Optimal ET sequence: 16, 55be, 71, 87, 103, 190

Badness: 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]]

POTE generator: ~65/48 = 524.209

Optimal ET sequence: 16, 55be, 71, 87, 103, 190

Badness: 0.021636

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