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-3 gamelismic temperament itself, and for the rank-2 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 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:
- Lemba (+50/49) → Jubilismic clan
- Echidnic (+686/675} → Diaschismic family
- Valentine (+126/125) → Starling temperaments
- Blacksmith (+28/27) → Limmic temperaments
- Trismegistus (+3125/3072) → Magic family
- Hemithirds (+3136/3125) → Hemimean clan
- Gamity (+1071875/1062882) → Amity family
- Tritikleismic (+15625/15552) → Kleismic family
- Heinz (+78732/78125) → Sensipent family
- Triwell (+235298/234375) → Semicomma family
- Decades (+118098/117649) → Compton family
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 aka Gamelic
- See also: No-fives subgroup temperaments #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 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 tuning (POTE): ~2 = 1\1, ~8/7 = 234.381
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 tuning (POTE): ~2 = 1\1, ~8/7 = 233.6155
Optimal ET sequence: 10, 26, 36, 154f, 190ffg
Rodan
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 tuning (POTE): ~2 = 1\1, ~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⟩]
- 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: 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 tuning (POTE): ~2 = 1\1, ~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, 46, 87
Badness: 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 tuning (POTE): ~2 = 1\1, ~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
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 tuning (POTE): ~2 = 1\1, ~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
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 tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.782
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]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.145
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]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 234.089
Badness: 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 tuning (POTE): ~2 = 1\1, ~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⟩]
- Eigenmonzo (unchanged-interval) basis: 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
Optimal tuning (POTE): ~2 = 1\1, ~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: 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 tuning (POTE): ~2 = 1\1, ~8/7 = 233.890
Optimal ET sequence: 36e, 41, 77, 118
Badness: 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 tuning (POTE): ~2 = 1\1, ~8/7 = 232.193
Algebraic generator: Rabrindanath, largest real root of x8 - 3x2 + 1, or 232.0774 cents.
- 7- and 9-odd-limit: ~8/7 = [0 0 1/12⟩
- [[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, 26, 31
Badness: 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 tuning (POTE): ~2 = 1\1, ~8/7 = 232.031
Optimal ET sequence: 5, 26, 31, 88, 150be, 181bee
Badness: 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 tuning (POTE): ~2 = 1\1, ~8/7 = 231.811
Optimal ET sequence: 5, 26, 31, 57, 88
Badness: 0.023954
- Music
- Prelude for solo piano in mothra16, brat 4 tuning (dead link) by Chris Vaisvil (blog post)
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 tuning (POTE): ~2 = 1\1, ~8/7 = 231.317
Optimal ET sequence: 5e, 26, 57e, 83bce
Badness: 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 tuning (POTE): ~2 = 1\1, ~8/7 = 231.293
Optimal ET sequence: 5e, 26, 57e, 83bce
Badness: 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 tuning (POTE): ~2 = 1\1, ~8/7 = 232.419
Optimal ET sequence: 31, 129, 160be, 191bce, 222bce, 253bcee
Badness: 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 tuning (POTE): ~2 = 1\1, ~8/7 = 232.640
Optimal ET sequence: 31, 36, 67, 98
Badness: 0.036857
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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 229.059
Optimal ET sequence: 5, 16e, 21, 68bccef
Badness: 0.047071
- Music
Gidorah
5-limit (university)
Subgroup: 2.3.5
Comma list: 144/125
Mapping: [⟨1 1 2], ⟨0 3 2]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 226.428
Optimal ET sequence: 5c, 11, 16
Badness: 0.069703
Miracle
For the 5-limit temperament, see Ampersand comma#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 tuning (POTE): ~2 = 1\1, ~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⟩]
- Eigenmonzo (unchanged-interval) basis: 2.5/3
- 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⟩]
- Eigenmonzo (unchanged-interval) basis: 2.9/5
- 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
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 tuning (POTE): ~2 = 1\1, ~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]
- 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
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 tuning (POTE): ~2 = 1\1, ~15/14 = 116.747
Optimal ET sequence: 10, 21e, 31, 41, 72f, 113f, 185cff
Badness: 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 tuning (POTE): ~2 = 1\1, ~15/14 = 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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~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]]
Optimal tuning (POTE): ~2 = 17\12, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~17/12 = 1\2, ~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]]
Optimal tuning (POTE): ~17/12 = 1\2, ~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]]
Optimal tuning (POTE): ~17/12 = 1\2, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~45/34 = 483.261
Optimal ET sequence: 72, 77, 149, 221ef
Badness: 0.015701
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 tuning (POTE): ~177147/125000 = 1\2, ~10/9 = 183.047
Optimal ET sequence: 26, 46, 72, 118, 2524, 2642, 2760, 2878b, …, 5002bc
Badness: 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 tuning (POTE): ~1225/864 = 1\2, ~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⟩]
- Eigenmonzo (unchanged-interval) basis: 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⟩]
- Eigenmonzo (unchanged-interval) basis: 2.9/7
Optimal ET sequence: 26, 46, 72, 118, 190
Badness: 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]]
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⟩]
- Eigenmonzo (unchanged-interval) basis: 2.9/7
Optimal ET sequence: 26, 46, 72, 118, 190
Badness: 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 tuning (POTE): ~99/70 = 1\2, ~10/9 = 183.187
Optimal ET sequence: 26f, 46f, 72, 118, 190, 262df, 452cdef
Badness: 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 tuning (POTE): ~91/64 = 1\2, ~10/9 = 183.198
Optimal ET sequence: 26, 46, 72, 190ff
Badness: 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 tuning (POTE): ~17/12 = 1\2, ~10/9 = 183.196
Optimal ET sequence: 26, 46, 72, 190ffg
Badness: 0.011676
Superkleismic
- See also: Shibboleth family #Superkleismic
The S-expression-based comma list of superkleismic is {S5/S6, S7/S8, S10, S12(, S21)}, from which (through careful observation of the equivalences therein) one can derive that a sharpened ~6/5 is the generator as well as the mapping of the full 13-limit.
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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~3375/2401 = 1\2, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~65/48 = 524.209
Optimal ET sequence: 16, 55be, 71, 87, 103, 190
Badness: 0.021636