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 (a.k.a. gamelic), this means three ~8/7 intervals give a fifth, 3/2. In fact, we find that 3/2 = (8/7)³ × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. 36edo is a good tuning for slendric, though if the full 7-limit is desired, 72edo, 77edo or 118edo might be preferred.

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

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

Full 7-limit temperaments discussed elsewhere are:

• Lemba (+50/49) → Jubilismic clan
• Echidnic (+686/675} → Diaschismic family
• Valentine (+126/125) → Starling temperaments
• Superkleismic (+875/864) → Shibboleth family
• 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

Main article: Slendric

Subgroup: 2.3.7

Comma list: 1029/1024

Subgroup-val mapping: [⟨1 1 3], ⟨0 3 -1]]

sval mapping generators: ~2, ~8/7

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

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

Optimal tunings:

• CTE: ~2 = 1200.000, ~8/7 = 233.889

error map: ⟨0.000 -0.288 -2.715]

• POTE: ~2 = 1200.000, ~8/7 = 233.688

error map: ⟨0.000 -0.892 -2.513]

Optimal ET sequence: 36, 77, 113, 190

Radon

Subgroup: 2.3.7.11

Comma list: 896/891, 1029/1024

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

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

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

Optimal tunings:

• CTE: ~2 = 1200.000, ~8/7 = 234.384

error map: ⟨0.000 +1.197 -3.210 +1.691]

• POTE: ~2 = 1200.000, ~8/7 = 234.381

error map: ⟨0.000 +1.187 -3.206 +1.735]

Optimal ET sequence: 36, 41, 87, 128

Baladic

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

Subgroup: 2.3.7.13.17

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

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

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

Optimal tunings:

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

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

Rodan

Main article: Rodan

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

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

Subgroup: 2.3.5.7

Comma list: 245/243, 1029/1024

Mapping: [⟨1 1 -1 3], ⟨0 3 17 -1]]

Wedgie: ⟨⟨ 3 17 -1 20 -10 -50 ]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~8/7 = 234.450

error map: ⟨0.000 +1.396 -0.660 -3.276]

• POTE: ~2 = 1200.000, ~8/7 = 234.417

error map: ⟨0.000 +1.295 -1.229 -3.243]

Minimax tuning:

• 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 (Smith): 0.037112

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Minimax tuning:

• 11-odd-limit: ~8/7 = [4/19 2/19 0 0 -1/19⟩

[[1 0 0 0 0⟩, [31/19 6/19 0 0 -3/19⟩, [49/19 34/19 0 0 -17/19⟩, [53/19 -2/19 0 0 1/19⟩, [62/19 -26/19 0 0 13/19⟩]

eigenmonzo (unchanged-interval) basis: 2.11/9

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

Optimal ET sequence: 41, 87

Badness (Smith): 0.023093

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Minimax tuning:

• 13- and 15-odd-limit: ~8/7 = [3/14 1/14 0 0 0 -1/28⟩

eigenmonzo (unchanged-interval) basis: 2.13/9

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

Optimal ET sequence: 41, 46, 87

Badness (Smith): 0.018448

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

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

Minimax tuning:

• 17-odd-limit: ~8/7 = [3/13 1/13 0 0 0 0 -1/26⟩

eigenmonzo (unchanged-interval) basis: 2.17/9

Optimal ET sequence: 41, 46, 87

Badness (Smith): 0.016743

Aerodactyl

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 5, 41f, 46

Badness (Smith): 0.033986

Aerodino

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 5e, 41e, 46

Badness (Smith): 0.054294

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 5e, 41ef, 46

Badness (Smith): 0.035836

Varan

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 5e, 36ce, 41

Badness (Smith): 0.044937

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 5e, 36ce, 41

Badness (Smith): 0.032284

Guiron

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 10976/10935

Mapping: [⟨1 1 7 3], ⟨0 3 -24 -1]]

mapping generators: ~2, ~8/7

Wedgie: ⟨⟨ 3 -24 -1 -45 -10 65 ]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~8/7 = 233.903

error map: ⟨0.000 -0.246 +0.012 -2.729]

• POTE: ~2 = 1200.000, ~8/7 = 233.930

error map: ⟨0.000 -0.165 -0.637 -2.756]

Minimax tuning:

• 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 (Smith): 0.047544

11-limit

Subgroup: 2.3.5.7.11

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

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

mapping generators: ~2, ~8/7

Optimal tunings:

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

Minimax tuning:

• 11-odd-limit: ~8/7 = [7/24 0 -1/24⟩

[[1 0 0 0 0⟩, [15/8 0 -1/8 0 0⟩, [0 0 1 0 0⟩, [65/24 0 1/24 0 0⟩, [37/6 0 -7/6 0 0⟩]

eigenmonzo (unchanged-interval) basis: 2.5

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

Badness (Smith): 0.026648

13-limit

Subgroup: 2.3.5.7.11.13

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

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

mapping generators: ~2, ~8/7

Optimal tunings:

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

Optimal ET sequence: 36e, 41, 77, 118

Badness (Smith): 0.028444

Mothra

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

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

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

Subgroup: 2.3.5.7

Comma list: 81/80, 1029/1024

Mapping: [⟨1 1 0 3], ⟨0 3 12 -1]]

mapping generators: ~2, ~8/7

Wedgie: ⟨⟨ 3 12 -1 12 -10 -36 ]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~8/7 = 232.400

error map: ⟨0.000 -4.756 +2.482 -1.226]

• POTE: ~2 = 1200.000, ~8/7 = 232.193

error map: ⟨0.000 -5.375 +0.005 -1.019]

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

Minimax tuning:

• 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, 21c, 26, 31

Badness (Smith): 0.037146

11-limit

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

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

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

Badness (Smith): 0.025642

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Smith): 0.023954

Music

• Prelude for solo piano (2014) by Chris Vaisvil – play | blog – in Mothra[16], brat 4 tuning

Cynder

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 5e, 21ce, 26

Badness (Smith): 0.055706

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 5e, 21cef, 26

Badness (Smith): 0.034124

Mosura

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

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

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

Optimal ET sequence: 5e, 26e, 31, 129

Badness (Smith): 0.031334

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 31, 67, 98

Badness (Smith): 0.036857

Gorgo

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 tunings:

• CTE: ~2 = 1200.000, ~8/7 = 228.700

error map: ⟨0.000 -15.856 +14.584]

• POTE: ~2 = 1200.000, ~10/9 = 227.426

error map: ⟨0.000 -19.679 +5.664]

Optimal ET sequence: 5, 11c, 16, 21, 37b

Badness (Smith): 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 tunings:

• CTE: ~2 = 1200.000, ~8/7 = 228.724

error map: ⟨0.000 -15.782 +14.756 +2.450]

• POTE: ~2 = 1200.000, ~8/7 = 228.334

error map: ⟨0.000 -16.954 +12.022 +2.840]

Optimal ET sequence: 5, 11c, 16, 21

Badness (Smith): 0.060663

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 1 3 1], ⟨0 3 7 -1 13]]

Optimal tunings:

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

Optimal ET sequence: 5e, 16, 21, 37b

Badness (Smith): 0.049500

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 5e, 16, 21, 37b

Badness (Smith): 0.032664

Spartan

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 1 3 5], ⟨0 3 7 -1 -8]]

Optimal tunings:

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

Optimal ET sequence: 5, 16e, 21

Badness (Smith): 0.062683

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 5, 16e, 21

Badness (Smith): 0.047071

Music

• Gorgo Example by Herman Miller

Gidorah

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

16c-, 21cc-, and 26ccc-edo are among the possible tunings.

Subgroup: 2.3.5.7

Comma list: 21/20, 144/125

Mapping: [⟨1 1 2 3], ⟨0 3 2 -1]]

Wedgie: ⟨⟨ 3 2 -1 -4 -10 -8 ]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~8/7 = 227.100

error map: ⟨0.000 -20.655 +67.886 +4.074]

• POTE: ~2 = 1200.000, ~8/7 = 230.762

error map: ⟨0.000 -9.668 +75.211 +0.412]

Optimal ET sequence: 1b, 5

Badness (Smith): 0.062262

Oncle

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 2430/2401

Mapping: [⟨1 1 6 3], ⟨0 3 -19 -1]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~8/7 = 232.383

error map: ⟨0.000 -4.807 -1.585 -1.209]

• POTE: ~2 = 1200.000, ~8/7 = 232.498

error map: ⟨0.000 -4.461 -3.778 -1.324]

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

Badness (Smith): 0.088384

Archaeotherium

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

Subgroup: 2.3.5.7

Comma list: 405/392, 1029/1024

Mapping: [⟨1 1 5 3], ⟨0 3 -14 -1]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~8/7 = 229.951

error map: ⟨0.000 -12.102 -5.626 +1.223]

• POTE: ~2 = 1200.000, ~8/7 = 230.258

error map: ⟨0.000 -11.180 -9.933 +0.916]

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

Badness (Smith): 0.146306

Clyndro

See also: Mavila family

Subgroup: 2.3.5.7

Comma list: 135/128, 360/343

Mapping: [⟨1 1 4 3], ⟨0 3 -9 -1]]

Wedgie: ⟨⟨ 3 -9 -1 -21 -10 23 ]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~8/7 = 225.752

error map: ⟨0.000 -24.699 -18.081 +5.422]

• POTE: ~2 = 1200.000, ~8/7 = 226.469

error map: ⟨0.000 -22.548 -24.534 +4.705]

Optimal ET sequence: 5c, 11, 16

Badness (Smith): 0.159179

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 5c, 11, 16

Badness (Smith): 0.069703

Miracle

Main article: Miracle

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

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1024

Mapping: [⟨1 1 3 3], ⟨0 6 -7 -2]]

mapping generator: ~2, ~15/14

Wedgie: ⟨⟨ 6 -7 -2 -25 -20 15 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~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⟩]

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

Tuning ranges:

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

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

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

Badness (Smith): 0.016742

11-limit

Subgroup: 2.3.5.7.11

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

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.633

Minimax tuning:

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

[[1 0 0 0 0⟩, [25/19 12/19 -6/19 0 0⟩, [50/19 -14/19 7/19 0 0⟩, [55/19 -4/19 2/19 0 0⟩, [53/19 30/19 -15/19 0 0⟩]

eigenmonzo (unchanged-interval) basis: 2.9/5

Tuning ranges:

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

Algebraic generator: Secor59

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

Badness (Smith): 0.010684

Miraculous

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.747

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

Badness (Smith): 0.018669

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.769

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

Badness (Smith): 0.017084

Benediction

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.574

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

Badness (Smith): 0.015715

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.585

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

Badness (Smith): 0.012537

Manna

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.739

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

Badness (Smith): 0.017012

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.727

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

Badness (Smith): 0.014680

Semimiracle

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~99/70 = 600.000, ~15/14 = 116.624

Optimal ET sequence: 10, 62, 72

Badness (Smith): 0.024622

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tuning (POTE): ~17\12 = 600.000, ~15/14 = 116.628

Optimal ET sequence: 10, 62, 72

Badness (Smith): 0.016130

Hemisecordite

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~27/26 = 58.288

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

Badness (Smith): 0.025589

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~27/26 = 58.261

Optimal ET sequence: 41, 62, 103

Badness (Smith): 0.022535

Semihemisecordite

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tuning (POTE): ~17/12 = 600.000, ~27/26 = 58.288

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

Badness (Smith): 0.046958

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tuning (POTE): ~17/12 = 600.000, ~27/26 = 58.283

Optimal ET sequence: 62, 144gh, 206begghh

Badness (Smith): 0.035057

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

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

Optimal tuning (POTE): ~17/12 = 600.000, ~27/26 = 58.283

Optimal ET sequence: 62, 144gh, 206begghhi

Badness (Smith): 0.026421

Phicordial

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~16/13 = 361.121

Optimal ET sequence: 103, 113, 216c

Badness (Smith): 0.033198

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~16/13 = 361.123

Optimal ET sequence: 103, 113, 216c

Badness (Smith): 0.024705

Revelation

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.277

Optimal ET sequence: 10e, 21, 31

Badness (Smith): 0.032946

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.268

Optimal ET sequence: 10e, 21, 31

Badness (Smith): 0.029452

Hemimiracle

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 58.408

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

Badness (Smith): 0.059232

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 58.430

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

Badness (Smith): 0.043151

Oracle

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 541.668

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

Badness (Smith): 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 = 1200.000, ~320/243 = 483.267

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

Badness (Smith): 0.056557

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~320/243 = 483.276

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

Badness (Smith): 0.028467

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~120/91 = 483.256

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

Badness (Smith): 0.021900

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~45/34 = 483.261

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

Badness (Smith): 0.015701

Unidec

Main article: Unidec

5-limit (unidecmic)

Subgroup: 2.3.5

Comma list: 31381059609/31250000000

Mapping: [⟨2 5 8], ⟨0 -6 -11]]

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

Optimal tuning (POTE): ~177147/125000 = 600.000, ~10/9 = 183.047

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

Badness (Smith): 0.082423

7-limit

Subgroup: 2.3.5.7

Comma list: 1029/1024, 4375/4374

Mapping: [⟨2 5 8 5], ⟨0 -6 -11 2]]

Wedgie: ⟨⟨ 12 22 -4 7 -40 -71 ]]

Optimal tuning (POTE): ~1225/864 = 600.000, ~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⟩]

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 (Smith): 0.038393

11-limit

Subgroup: 2.3.5.7.11

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

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

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 (Smith): 0.015479

Ekadash

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~99/70 = 600.000, ~10/9 = 183.187

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

Badness (Smith): 0.020381

Hendec

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~91/64 = 600.000, ~10/9 = 183.198

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

Badness (Smith): 0.017707

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tuning (POTE): ~17/12 = 600.000, ~10/9 = 183.196

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

Badness (Smith): 0.011676

Lagaca

Subgroup: 2.3.5.7

Comma list: 1029/1024, 11529602/11390625

Mapping: [⟨2 5 2 5], ⟨0 -9 13 3]]

Wedgie: ⟨⟨ 18 -26 -6 -83 -60 59 ]]

Optimal tuning (POTE): ~3375/2401 = 600.000, ~15/14 = 122.027

Optimal ET sequence: 10, 98, 108, 118

Badness (Smith): 0.144345

Necromanteion

Subgroup: 2.3.5.7

Comma list: 1029/1024, 5103/5000

Mapping: [⟨1 7 10 1], ⟨0 -12 -17 4]]

Wedgie: ⟨⟨ 12 17 -4 -1 -40 -57 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~48/35 = 541.779

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

Badness (Smith): 0.117680

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/11 = 541.729

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

Badness (Smith): 0.053459

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~15/11 = 541.606

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

Badness (Smith): 0.047015

Restles

See also: Lesser tendoneutralic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 153664/151875

Mapping: [⟨1 -2 8 4], ⟨0 12 -19 -4]]

Wedgie: ⟨⟨ 12 -19 -4 -58 -40 44 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~315/256 = 358.548

Optimal ET sequence: 10, 77, 87, 164

Badness (Smith): 0.108011

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~27/22 = 358.571

Optimal ET sequence: 10, 77, 87, 164

Badness (Smith): 0.054655

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~16/13 = 358.574

Optimal ET sequence: 10, 77, 87, 164

Badness (Smith): 0.028187

Quartemka

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1250000/1240029

Mapping: [⟨1 4 6 2], ⟨0 -21 -32 7]]

Wedgie: ⟨⟨ 21 32 -7 2 -70 -106 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~27/25 = 138.006

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

Badness (Smith): 0.152287

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~27/25 = 137.990

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

Badness (Smith): 0.057307

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 137.990

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

Badness (Smith): 0.028393

Tritriple

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

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1959552/1953125

Mapping: [⟨1 -11 -7 7], ⟨0 27 20 -9]]

Wedgie: ⟨⟨ 27 20 -9 -31 -90 -77 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~864/625 = 559.295

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

Badness (Smith): 0.118640

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~242/175 = 559.293

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

Badness (Smith): 0.035350

Widefourth

Subgroup: 2.3.5.7

Comma list: 1029/1024, 48828125/48771072

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

Wedgie: ⟨⟨ 33 13 -11 -56 -110 -62 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~3125/2304 = 524.210

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

Badness (Smith): 0.154117

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~847/625 = 524.210

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

Badness (Smith): 0.040785

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1200.000, ~65/48 = 524.209

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

Badness (Smith): 0.021636