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.

Slendric

Main article: Slendric

Period: 1\1

Optimal (POTE) generator: ~8/7 = 233.688

EDO generator: 7\36, 8\41, 15\77

Scales (Scala files): slendric5, slendric6, slendric11, slendric16

Technical data

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]

Full seven limit extensions

To the gamelisma itself we need to add the comma which appears next on the modified normal comma list, which is often a 5-limit comma. 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 15625/15536 and has a generator of 6/5 with a 1/3 octave period.

Discussed elsewhere are blacksmith, mothra, guiron and heinz. The rest are considered below.

Subgroup extensions

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.

Miracle

Main article: Miracle

Period: 1\1

Optimal (POTE) generator: ~15/14 = 116.675

EDO generators: 3\31, 4\41, 7\72

Scales (Scala files): Miracle 10, Blackjack

Technical data

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1024

Mapping: [⟨1 1 3 3], ⟨0 6 -7 -2]]

Wedgie: ⟨⟨ 6 -7 -2 -25 -20 15 ]]

Minimax tuning:

• 7-odd-limit:

[[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: 2, 6/5

• 9-odd-limit:

[[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: 2, 10/9

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

Badness: 0.0167

11-limit

Period: 1\1

Optimal (POTE) generator: ~15/14 = 116.633

EDO generators: 3\31, 4\41, 7\72

Scales (Scala files): Miracle 10, Blackjack

Technical data

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]]

Minimax tuning:

• 11-odd-limit:

[[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: 2, 10/9

Algebraic generator: Secor59

Badness: 0.0107

Miraculous

Period: 1\1

Optimal (POTE) generator: ~15/14 = 116.747

EDO generators: 3\31, 4\41

Scales (Scala files): Miracle 10, Blackjack

Technical data

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]]

Badness: 0.0187

Benediction

Period: 1\1

Optimal (POTE) generator: ~15/14 = 116.574

EDO generators: 3\31, 7\72, 10\103

Scales (Scala files):

Technical data

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]]

Badness: 0.0157

17-limit

Period: 1\1

Optimal (POTE) generator: ~15/14 = 116.585

EDO generators: 3\31, 7\72, 10\103

Scales (Scala files):

Technical data

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]]

Manna

Period: 1\1

Optimal (POTE) generator: ~15/14 = 116.739

EDO generators: 4\41, 7\72, 11\113

Scales (Scala files):

Technical data

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]]

Badness: 0.0170

Semimiracle

Period: 1\2

Optimal (POTE) generator: ~15/14 = 116.624

EDO generators: 6\62, 7\72, 8\82

Scales (Scala files):

Technical data

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]]

Badness: 0.0246

17-limit

Period: 1\2

Optimal (POTE) generator: ~15/14 = 116.628

EDO generators: 6\62, 7\72, 8\82

Scales (Scala files):

Technical data

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]]

Badness: 0.0161

Hemisecordite

Period: 1\1

Optimal (POTE) generator: ~27/26 = 58.288

EDO generators: 2\41, 3\62, 5\103

Scales (Scala files):

Technical data

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]]

Badness: 0.0256

17-limit

Period: 1\1

Optimal (POTE) generator: ~27/26 = 58.261

EDO generators: 2\41, 3\62, 5\103

Scales (Scala files):

Technical data

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]]

Badness: 0.0225

Phicordial

Period: 1\1

Optimal (POTE) generator: ~16/13 = 361.121

EDO generators: 31\103, 34\113

Scales (Scala files):

Technical data

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]]

Badness: 0.0332

17-limit

Period: 1\1

Optimal (POTE) generator: ~16/13 = 361.123

EDO generators: 31\103, 34\113

Scales (Scala files):

Technical data

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]]

Badness: 0.0247

Revelation

Period: 1\1

Optimal (POTE) generator: ~15/14 = 116.277

EDO generators: 2\21, 3\31

Scales (Scala files):

Technical data

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]]

Badness: 0.0329

13-limit

Period: 1\1

Optimal (POTE) generator: ~15/14 = 116.268

EDO generators: 2\21, 3\31

Scales (Scala files):

Technical data

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]]

Badness: 0.0295

Hemimiracle

Period: 1\1

Optimal (POTE) generator: ~33/32 = 58.408

EDO generators: 1\20, 1\21, 2\41

Scales (Scala files):

Technical data

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]]

Badness: 0.0592

13-limit

Period: 1\1

Optimal (POTE) generator: ~33/32 = 58.430

EDO generators: 1\20, 1\21, 2\41

Scales (Scala files):

Technical data

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]]

Badness: 0.0432

Oracle

Period: 1\1

Optimal (POTE) generator: ~11/8 = 541.668

EDO generators: 5\11, 9\20, 14\31

Scales (Scala files):

Technical data

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]]

Badness: 0.0427

Rodan

Main article: Rodan

Period: 1\1

Optimal (POTE) generator: ~8/7 = 234.417

EDO generators: 8\41, 9\46, 17\87

Scales (Scala files): rodan26opt, rodan31opt, rodan41opt

Technical data

Subgroup: 2.3.5.7

Comma list: 245/243, 1029/1024

Mapping: [⟨1 1 -1 3], ⟨0 3 17 -1]]

Minimax tuning:

• 7- and 9-odd-limit:

[[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: 2, 7/5

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

Badness: 0.0371

11-limit

Period: 1\1

Optimal (POTE) generator: ~8/7 = 234.459

EDO generators: 8\41, 9\46, 17\87

Scales (Scala files): rodan26opt, rodan31opt, rodan41opt

Technical data

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]]

Minimax tuning:

• 11-odd-limit:

[[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: 2, 11/9

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

Badness: 0.0231

13-limit

Period: 1\1

Optimal (POTE) generator: ~8/7 = 234.482

EDO generators: 8\41, 9\46, 17\87

Scales (Scala files): rodan26opt, rodan31opt, rodan41opt

Technical data

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]]

Minimax tuning:

• 13- and 15-odd-limit:

[[1 0 0 0 0 0⟩, [23/14 3/14 0 0 0 -3/28⟩, [37/14 17/14 0 0 0 -17/28⟩, [39/14 -1/14 0 0 0 1/28⟩, [45/14 -13/14 0 0 0 13/28⟩, [23/7 -11/7 0 0 0 11/14⟩]

Eigenmonzos: 2, 13/9

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

Badness: 0.0184

17-limit

Period: 1\1

Optimal (POTE) generator: ~8/7 = 234.524

EDO generators: 8\41, 9\46, 17\87

Scales (Scala files): rodan26opt, rodan31opt, rodan41opt

Technical data

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]]

Minimax tuning:

• 17-odd-limit eigenmonzos: 2, 18/17

Badness: 0.0167

Aerodactyl

Period: 1\1

Optimal (POTE) generator: ~8/7 = 234.639

EDO generators: 8\41, 9\46, 10\51

Scales (Scala files):

Technical data

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]]

Badness: 0.0340

Aerodino

Period: 1\1

Optimal (POTE) generator: ~8/7 = 234.728

EDO generators: 8\41, 9\46, 10\51

Scales (Scala files):

Technical data

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]]

Badness: 0.0543

13-limit

Period: 1\1

Optimal (POTE) generator: ~8/7 = 234.782

EDO generators: 8\41, 9\46, 10\51

Scales (Scala files):

Technical data

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

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

Badness: 0.0358

Varan

Period: 1\1

Optimal (POTE) generator: ~8/7 = 234.145

EDO generators: 7\36, 8\41, 9\46

Scales (Scala files):

Technical data

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

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

Badness: 0.0449

13-limit

Period: 1\1

Optimal (POTE) generator: ~8/7 = 234.089

EDO generators: 7\36, 8\41, 9\46

Scales (Scala files):

Technical data

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

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

Badness: 0.0323

Valentine

Main article: Valentine

See also: Starling temperaments § Valentine

5-limit

Period: 1\1

Optimal (POTE) generator: ~25/24 = 78.039

EDO generators: 1\15, 1\16, 2\31, 3\46, 5\77

Scales (Scala files):

Technical data

Subgroup: 2.3.5

Comma list: 1990656/1953125

Mapping: [⟨1 1 2], ⟨0 9 5]]

Badness: 0.1228

7-limit

Period: 1\1

Optimal (POTE) generator: ~21/20 = 77.864

EDO generators: 1\15, 1\16, 2\31, 3\46, 5\77

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7

Comma list: 126/125, 1029/1024

Mapping: [⟨1 1 2 3], ⟨0 9 5 -3]]

Mapping generators: ~2, ~21/20

Minimax tuning:

• 7-odd-limit:

[[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [17/6 5/12 0 -5/12⟩, [5/2 -1/4 0 1/4⟩]

Eigenmonzos: 2, 7/6

• 9-odd-limit:

[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [47/21 10/21 0 -5/21⟩, [20/7 -2/7 0 1/7⟩]

Eigenmonzos: 2, 9/7

Algebraic generator: smaller root of x² - 89x + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.

Badness: 0.0311

11-limit

Period: 1\1

Optimal (POTE) generator: ~21/20 = 77.881

EDO generators: 1\15, 1\16, 2\31, 3\46, 5\77

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125, 176/175

Mapping: [⟨1 1 2 3 3], ⟨0 9 5 -3 7]]

Mapping generators: ~2, ~21/20

Minimax tuning:

• 11-odd-limit:

[[1 0 0 0 0⟩, [1 0 0 -9/10 9/10⟩, [2 0 0 -1/2 1/2⟩, [3 0 0 3/10 -3/10⟩, [3 0 0 -7/10 7/10⟩]

Eigenmonzos: 2, 11/7

Algebraic generator: positive root of 4x³ + 15x² - 21, or else Gontrand2, the smallest positive root of 4x⁷ - 8x⁶ + 5.

Badness: 0.0167

Unidec

Main article: Unidec

5-limit

Period: 1\2

Optimal (POTE) generator: ~10/9 = 183.047

EDO generator: 4\26, 7\46, 11\72, 18\118

Scales (Scala files): unidec26

Technical data

Subgroup: 2.3.5

Comma list: 31381059609/31250000000

Mapping: [⟨2 5 8], ⟨0 -6 -11]]

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

Badness: 0.0824

7-limit

Period: 1\2

Optimal (POTE) generator: ~10/9 = 183.161

EDO generator: 4\26, 7\46, 11\72, 18\118

Scales (Scala files): unidec26

Technical data

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]]

Minimax tuning:

• 7-odd-limit:

[[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: 2, 7/5

• 9-odd-limit:

[[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: 2, 9/7

Badness: 0.0384

11-limit

Period: 1\2

Optimal (POTE) generator: ~10/9 = 183.165

EDO generator: 4\26, 7\46, 11\72, 18\118

Scales (Scala files): unidec26

Technical data

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:

[[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: 2, 9/7

Badness: 0.0155

Ekadash

Period: 1\2

Optimal (POTE) generator: ~10/9 = 183.187

EDO generator: 11\72, 18\118, 29\190

Scales (Scala files): unidec26

Technical data

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

Badness: 0.0204

Hendec

Period: 1\2

Optimal (POTE) generator: ~10/9 = 183.198

EDO generator: 4\26, 7\46, 11\72, 18\118

Scales (Scala files): unidec26

Technical data

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 1716/1715

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

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

Badness: 0.0177

17-limit

Period: 1\2

Optimal (POTE) generator: ~10/9 = 183.196

EDO generator: 4\26, 7\46, 11\72, 18\118

Scales (Scala files): unidec26

Technical data

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

Hemithirds

Main article: Hemithirds

See also: Luna family § Hemithirds

Period: 1\1

Optimal (POTE) generator: ~28/25 = 193.244

EDO generator: 5\31, 9\56, 14\87, 23\118

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3136/3125

Mapping: [⟨1 4 2 2], ⟨0 -15 2 5]]

Wedgie: ⟨⟨ 15 -2 -5 -38 -50 -6 ]]

Minimax tuning:

• 7-odd-limit:

[[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [11/5 -1/10 0 1/10⟩, [5/2 -1/4 0 1/4⟩]

Eigenmonzos: 2, 7/6

• 9-odd-limit:

[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [82/35 -4/35 0 2/35⟩, [20/7 -2/7 0 1/7⟩]

Eigenmonzos: 2, 7/6

Badness: 0.0443

11-limit

Period: 1\1

Optimal (POTE) generator: ~28/25 = 193.227

EDO generator: 5\31, 9\56, 14\87, 23\118

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 3136/3125

Mapping: [⟨1 4 2 2 7], ⟨0 -15 2 5 -22]]

Minimax tuning:

• 11-odd-limit:

[[1 0 0 0 0⟩, [11/9 0 0 -5/9 5/9⟩, [64/27 0 0 2/27 -2/27⟩, [79/27 0 0 5/27 -5/27⟩, [79/27 0 0 -22/27 22/27⟩]

Eigenmonzos: 2, 11/7

Badness: 0.0190

13-limit

Period: 1\1

Optimal (POTE) generator: ~28/25 = 193.166

EDO generator: 5\31, 9\56, 14\87, 23\118

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 625/624

Mapping: [⟨1 4 2 2 7 0], ⟨0 -15 2 5 -22 23]]

Badness: 0.0217

Hemiseven

Period: 1\1

Optimal (POTE) generator: ~320/243 = 483.267

EDO generator: 29\72, 31\77, 60\149

Scales (Scala files):

Technical data

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]]

Badness: 0.0566

11-limit

Period: 1\1

Optimal (POTE) generator: ~320/243 = 483.276

EDO generator: 29\72, 31\77, 60\149

Scales (Scala files):

Technical data

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]]

Badness: 0.0285

13-limit

Period: 1\1

Optimal (POTE) generator: ~120/91 = 483.256

EDO generator: 29\72, 31\77, 60\149

Scales (Scala files):

Technical data

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]]

17-limit

Period: 1\1

Optimal (POTE) generator: ~45/34 = 483.261

EDO generator: 29\72, 31\77, 60\149

Scales (Scala files):

Technical data

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]]

Tritikleismic

See also: Kleismic family § Tritikleismic

Period: 1\3

Optimal (POTE) generator: ~6/5 = 316.872

EDO generator: 15\57, 19\72, 23\87

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7

Comma list: 1029/1024, 15625/15552

Mapping: [⟨3 0 3 10], ⟨0 6 5 -2]]

Mapping generators: ~63/50, ~6/5

Minimax tuning:

• 7-odd-limit:

[[1 0 0 0⟩, [2 0 6/7 -6/7⟩, [8/3 0 5/7 -5/7⟩, [8/3 0 -2/7 2/7⟩]

Eigenmonzos: 2, 7/5

• 9-odd-limit:

[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [46/21 5/7 0 -5/14⟩, [20/7 -2/7 0 1/7⟩]

Eigenmonzos: 2, 9/7

Badness: 0.0563

11-limit

Period: 1\3

Optimal (POTE) generator: ~6/5 = 316.881

EDO generator: 15\57, 19\72, 23\87

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 4000/3993

Mapping: [⟨3 0 3 10 8], ⟨0 6 5 -2 3]]

Mapping generators: ~44/35, ~6/5

Minimax tuning:

• 11-odd-limit:

[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [46/21 5/7 0 -5/14 0⟩, [20/7 -2/7 0 1/7 0⟩, [71/21 3/7 0 -3/14 0⟩]

Eigenmonzos: 2, 9/7

Badness: 0.0193

13-limit

Period: 1\3

Optimal (POTE) generator: ~6/5 = 316.9585

EDO generator: 19\72, 23\87, 42\159

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 625/624

Mapping: [⟨3 0 3 10 8 0], ⟨0 6 5 -2 3 14]]

Mapping generators: ~44/35, ~6/5

17-limit

Period: 1\3

Optimal (POTE) generator: ~6/5 = 316.9082

EDO generator: 19\72, 23\87, 42\159

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 375/374, 385/384

Mapping: [⟨3 0 3 10 8 0 -2], ⟨0 6 5 -2 3 14 18]]

Mapping generators: ~34/27, ~6/5

Superkleismic

See also: Shibboleth family § Superkleismic

Period: 1\1

Optimal (POTE) generator: ~6/5 = 321.930

EDO generator: 4\15, 7\26, 11\41

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7

Comma list: 875/864, 1029/1024

Mapping: [⟨1 4 5 2], ⟨0 -9 -10 3]]

Badness: 0.0479

11-limit

Period: 1\1

Optimal (POTE) generator: ~6/5 = 321.847

EDO generator: 4\15, 7\26, 11\41

Scales (Scala files):

Technical data

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]]

Badness: 0.0257

13-limit

Period: 1\1

Optimal (POTE) generator: ~6/5 = 321.994

EDO generator: 4\15, 7\26, 11\41

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 245/243, 1188/1183

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

Badness: 0.0215

Gorgo

See also: Laconic family § Gorgo

Period: 1\1

Optimal (POTE) generator: ~8/7 = 228.334

EDO generator: 3\16, 4\21

Scales (Scala files):

Technical data

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]]

Badness: 0.0607

11-limit

Period: 1\1

Optimal (POTE) generator: ~8/7 = 229.535

EDO generator: 3\16, 4\21

Scales (Scala files):

Technical data

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]]

Badness: 0.0627

13-limit

Period: 1\1

Optimal (POTE) generator: ~8/7 = 229.059

EDO generator: 3\16, 4\21

Scales (Scala files):

Technical data

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]]

Badness: 0.0471

Music

• Gorgo Example by Herman Miller

Lemba

Main article: Lemba

See also: Jubilismic clan § Lemba

Period: 1\2

Optimal (POTE) generator: ~8/7 = 232.089

EDO generator: 2\10, 3\16, 5\26

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7

Comma list: 50/49, 525/512

Mapping: [⟨2 2 5 6], ⟨0 3 -1 -1]]

Mapping generators: ~7/5, ~8/7

Wedgie: ⟨⟨ 6 -2 -2 -17 -20 1 ]]

Badness: 0.0622

Gidorah

See also: University temperament

Period: 1\1

Optimal (POTE) generator: ~8/7 = 230.762

EDO generator: 1\5, 2\11, 3\16

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7

Comma list: 21/20, 144/125

Mapping: [⟨1 1 2 3], ⟨0 3 2 -1]]

Badness: 0.0623

Clyndro

Period: 1\1

Optimal (POTE) generator: ~8/7 = 226.469

EDO generator: 1\5, 2\11, 3\16

Scales (Scala files):

Technical data

Subgroup: 2.3.5.7

Comma list: 135/128, 360/343

Mapping: [⟨1 1 4 3], ⟨0 3 -9 -1]]

Badness: 0.1592

11-limit

Period: 1\1

Optimal (POTE) generator: ~8/7 = 226.428

EDO generator: 1\5, 2\11, 3\16

Scales (Scala files):

Technical data

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]]

Badness: 0.0697

Necromanteion

Comma list: 1029/1024, 5103/5000

POTE generator: ~48/35 = 541.779

Mapping: [⟨1 7 10 1], ⟨0 -12 -17 4]]

Badness: 0.1177

11-limit

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

POTE generator: ~15/11 = 541.729

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

Badness: 0.0535

13-limit

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

POTE generator: ~15/11 = 541.606

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

Badness: 0.0470

Widefourth

Comma list: 1029/1024, 48828125/48771072

POTE generator: ~3125/2304 = 524.210

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

Wedgie: ⟨⟨33 13 -11 -56 -110 -62]]

Badness: 0.1541

11-limit

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

POTE generator: ~3125/2304 = 524.210

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

Badness: 0.0408

13-limit

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

POTE generator: ~65/48 = 524.209

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

Badness: 0.0216

Tritriple

5-limit

Comma list: [31 20 -27⟩

POTE generator: ~864/625 = 559.332

Mapping: [⟨1 -11 -7], ⟨0 27 20]]

Badness: 0.2836

7-limit

Comma list: 1029/1024, 1959552/1953125

POTE generator: ~864/625 = 559.295

Mapping: [⟨1 -11 -7 7], ⟨0 27 20 -9]]

Badness: 0.1186

11-limit

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

POTE generator: ~242/175 = 559.293

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

Badness: 0.0353

Restles

Comma list: 1029/1024, 153664/151875

POTE generator: ~315/256 = 358.5485

Mapping: [⟨1 -2 8 4], ⟨0 12 -19 -4]]

Badness: 0.1080

11-limit

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

POTE generator: ~27/22 = 358.5713

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

Badness: 0.0547

13-limit

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

POTE generator: ~16/13 = 358.5739

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

Badness: 0.0282

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.

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

Period: 1\2

POTE generator: ~8/7 = 233.6155

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