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
Optimal GPV 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 GPV 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
- 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.
Optimal GPV 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:
- 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: 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 GPV 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 GPV 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 GPV 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
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 GPV 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 GPV 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 GPV 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
- 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
Optimal GPV 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:
- 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: 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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
- 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
- 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 GPV 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:
- 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: 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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
- 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
Optimal GPV 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:
- 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: 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV 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 GPV sequence: 16, 55be, 71, 87, 103, 190
Badness: 0.021636