Gamelismic clan

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:

• Blacksmith, {28/27, 49/48} → Limmic temperaments
• Lemba, {50/49, 525/512} → Jubilismic clan
• Mothra, {81/80, 1029/1024} → Meantone family
• Valentine, {126/125, 1029/1024} → Starling temperaments
• Echidnic, {686/675, 1029/1024} → Diaschismic family
• Trismegistus, {1029/1024, 3125/3072} → Magic family
• Hemithirds, {1029/1024, 3136/3125} → Hemimean clan
• Tritikleismic, {1029/1024, 15625/15552} → Kleismic family
• Heinz, {1029/1024, 78732/78125} → Sensipent family
• Decades, {1029/1024, 118098/117649} → Compton family
• Triwell, {1029/1024, 235298/234375} → Semicomma family
• Gamity, {1029/1024, 1071875/1062882} → Amity family

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

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

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:

• 7- and 9-odd-limit: ~8/7 = [2/9 0 1/18 -1/18⟩

[[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.

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:

• 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⟩]

Eigenmonzos (unchanged intervals): 2, 11/9

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

Optimal GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 0.054294

13-limit

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 GPV sequence: Template:Val list

Badness: 0.035836

Varan

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 GPV sequence: Template:Val list

Badness: 0.044937

13-limit

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 GPV sequence: Template:Val list

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:

• 7- and 9-odd-limit: ~8/7 = [7/24 0 -1/24⟩

[[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

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:

• 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⟩]

Eigenmonzos (unchanged intervals): 2, 5

Optimal GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 0.047071

Music

• Gorgo Example by Herman Miller

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

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

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

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

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

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 GPV sequence: Template:Val list

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:

• 7-odd-limit: ~15/14 = [2/13 1/13 -1/13⟩

[[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

• 9-odd-limit: ~15/14 = [1/19 2/19 -1/19⟩

[[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

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:

• 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⟩]

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 GPV sequence: Template:Val list

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: ~15/14 = 116.747

Optimal GPV sequence: Template:Val list

Badness: 0.018669

Scales: Miracle 10, Blackjack

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 0.017012

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 0.022535

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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:

• 7-odd-limit: ~10/9 = [3/26 0 -1/13 1/13⟩

[[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

• 9-odd-limit: ~10/9 = [5/28 -1/7 0 1/14⟩

[[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

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:

• 11-odd-limit: ~10/9 = [5/28 -1/7 0 1/14⟩

[[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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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

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 GPV sequence: Template:Val list

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

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 0.021636