Hemimean clan

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The hemimean clan tempers out the no-threes hemimean comma 3136/3125. The head of this clan is the 2.5.7 subgroup temperament didacus. Full 7-limit extensions of didacus, in general, split the syntonic comma into two, each for 126/125~225/224, as 3136/3125 = (126/125)/(225/224). These include hemithirds, semisept, emka, decipentic, arch, sengagen, subpental, mowglic, quitagar, and undetrita, considered below, as well as these considered elsewhere:

A notable subgroup extension of didacus is roulette.

Didacus

Subgroup: 2.5.7

Comma list: 3136/3125

Sval mapping: [1 0 -3], 0 2 5]]

Sval mapping generators: ~2, ~56/25

Gencom mapping: [1 0 0 -3], 0 0 2 5]]

Gencom: [2 56/25; 3136/3125]

POTE generator: ~28/25 = 193.772

Vals6, 19, 25, 31, 99, 130, 161, 353, 514c

Hemithirds

Main article: Hemithirds
See also: Luna family #Hemithirds

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]]

POTE generator: ~28/25 = 193.244

Minimax tuning:

[[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 (unchanged intervals): 2, 7/6
[[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 (unchanged intervals): 2, 7/6

Vals25, 31, 87, 118

Badness: 0.044284

11-limit

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]]

POTE generator: ~28/25 = 193.227

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 (unchanged intervals): 2, 11/7

Vals: 25e, 31, 87, 118

Badness: 0.019003

13-limit

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]]

POTE generator: ~28/25 = 193.166

Vals: 31, 56, 87, 118, 205d

Badness: 0.021738

Semisept

Subgroup: 2.3.5

Comma list: 782757789696/762939453125

Mapping: [1 12 6], 0 -17 -6]]

POTE generator: ~192/125 = 735.146

Vals18, 31, 80, 111

Badness: 0.630576

7-limit

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3136/3125

Mapping: [1 12 6 12], 0 -17 -6 -15]]

Wedgie⟨⟨17 6 15 -30 -24 18]]

POTE generator: ~75/49 = 735.155

Vals18, 31, 80, 111

Badness: 0.050472

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 1331/1323

Mapping: [1 12 6 12 20], 0 -17 -6 -15 -27]]

POTE generator: ~55/36 = 735.125

Vals: 18e, 31, 80, 111, 364cd

Badness: 0.022476

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539, 1375/1372

Mapping: [1 12 6 12 20 -11], 0 -17 -6 -15 -27 24]]

POTE generator: ~55/36 = 735.126

Vals: 31, 80, 111

Badness: 0.025204

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 640/637, 715/714

Mapping: [1 12 6 12 20 -11 -10], 0 -17 -6 -15 -27 24 23]]

POTE generator: ~26/17 = 735.125

Vals: 31, 80, 111

Badness: 0.019919

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323

Mapping: [1 12 6 12 20 -11 -10 -8], 0 -17 -6 -15 -27 24 23 20]]

POTE generator: ~26/17 = 735.116

Vals: 31, 80, 111

Badness: 0.016301

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459

Mapping: [1 12 6 12 20 -11 -10 -8 18], 0 -17 -6 -15 -27 24 23 20 -22]]

POTE generator: ~26/17 = 735.106

Vals: 31, 80, 111, 191cdh, 302cdgh

Badness: 0.014957

Semishly

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195, 275/273

Mapping: [1 12 6 12 20 8], 0 -17 -6 -15 -27 -7]]

POTE generator: ~13/10 = 464.980

Vals: 31, 49f, 80f

Badness: 0.028408

Emka

Emka tempers out [-50 -8 27 in the 5-limit. This temperament can be described as 37&50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension emkay (87&224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.


Subgroup: 2.3.5

Comma list: [-50 -8 27

Mapping: [1 14 6], 0 -27 -8]]

POTE generator: ~9765625/7077888 = 551.784

Vals37, 50, 87, 137, 224

Badness: 0.544752

7-limit

Subgroup: 2.3.5.7

Comma list: 3136/3125, 84035/82944

Mapping: [1 14 6 12], 0 -27 -8 -20]]

Wedgie⟨⟨27 8 20 -50 -44 24]]

POTE generator: ~48/35 = 551.782

Vals37, 50, 87, 137d, 224d

Badness: 0.144338

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2401/2376, 3136/3125

Mapping: [1 14 6 12 3], 0 -27 -8 -20 1]]

POTE generator: ~11/8 = 551.765

Vals: 37, 50, 87, 224d, 311d

Badness: 0.054744

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 625/624

Mapping: [1 14 6 12 3 6], 0 -27 -8 -20 1 -5]]

POTE generator: ~11/8 = 551.758

Vals: 37, 50, 87, 224d, 311d, 398d

Badness: 0.029741

Decipentic

The generator for the decipentic temperament (43&56) is tenth root of fifth harmonic (5/1, pentave), 51/10, tuned between 75/64 and 20/17 (close to 27/23). Aside from the hemimean comma, this temperament tempers out the bronzisma, 2097152/2083725 (satriru-agugu). 99EDO is a good tuning for decipentic, with generator 23\99, and MOS of 9, 13, 17, 30, 43 or 56 notes are available.


Subgroup: 2.3.5.7

Comma list: 3136/3125, 2097152/2083725

Mapping: [1 6 0 -3], 0 -19 10 25]]

Wedgie⟨⟨19 -10 -25 -60 -93 -30]]

POTE generator: ~75/64 = 278.800

Vals13, 43, 56, 99

Badness: 0.087325

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1344/1331, 3136/3125

Mapping: [1 6 0 -3 3], 0 -19 10 25 2]]

POTE generator: ~75/64 = 278.799

Vals: 13, 43, 56, 99e

Badness: 0.061413

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 441/440, 832/825, 975/968

Mapping: [1 6 0 -3 3 3], 0 -19 10 25 2 3]]

POTE generator: ~13/11 = 278.802

Vals: 13, 43, 56, 99e

Badness: 0.047611

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 256/255, 273/272, 375/374

Mapping: [1 6 0 -3 3 3 2], 0 -19 10 25 2 3 9]]

POTE generator: ~13/11 = 278.798

Vals: 13, 43, 56, 99e

Badness: 0.031191

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285

Mapping: [1 6 0 -3 3 3 2 1], 0 -19 10 25 2 3 9 14]]

POTE generator: ~13/11 = 278.790

Vals: 13, 43, 56, 99e

Badness: 0.023899

Quasijerome

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 15488/15435, 16384/16335

Mapping: [1 6 0 -3 3], 0 -38 20 50 47]]

POTE generator: ~896/825 = 139.403

Vals: 43, 112, 155, 198, 439cd, 637cd

Badness: 0.092996

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435

Mapping: [1 6 0 -3 3 8], 0 -38 20 50 47 -37]]

POTE generator: ~13/12 = 139.403

Vals: 43, 155, 198, 439cdf, 637cdf

Badness: 0.044328

Arch

See also: Escapade family

Subgroup: 2.3.5.7

Comma list: 3136/3125, 5250987/5242880

Mapping: [1 2 2 2], 0 -18 14 35]]

Wedgie⟨⟨18 -14 -35 -64 -106 -42]]

POTE generator: ~64/63 = 27.668

Vals43, 87, 130, 217, 347, 824c, 1171c, 1518cd

Badness: 0.094345

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4000/3993

Mapping: [1 2 2 2 3], 0 -18 14 35 20]]

POTE generator: ~64/63 = 27.663

Vals: 43, 87, 130, 217, 347e, 911cde

Badness: 0.036541

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 3136/3125

Mapping: [1 2 2 2 3 4], 0 -18 14 35 20 -13]]

POTE generator: ~64/63 = 27.660

Vals: 43, 87, 130, 217, 347e, 564e

Badness: 0.019504

Sengagen

Subgroup: 2.3.5.7

Comma list: 3136/3125, 420175/419904

Mapping: [1 1 2 2], 0 29 16 40]]

Wedgie⟨⟨29 16 40 -42 -18 48]]

POTE generator: ~686/675 = 24.217

Vals49, 50, 99, 248, 347, 446

Badness: 0.057978

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1344/1331, 3136/3125

Mapping: [1 1 2 2 3], 0 29 16 40 23]]

POTE generator: ~99/98 = 24.235

Vals: 49, 50, 99e

Badness: 0.053828

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 975/968, 1344/1331

Mapping: [1 1 2 2 3 4], 0 29 16 40 23 -15]]

POTE generator: ~99/98 = 24.181

Vals: 49, 50, 99e, 149e

Badness: 0.053531

Sengage

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 364/363, 625/624

Mapping: [1 1 2 2 3 3], 0 29 16 40 23 35]]

POTE generator: ~99/98 = 24.234

Vals: 49f, 50, 99ef

Badness: 0.037416

Subpental

See also: Sensipent family

Subgroup: 2.3.5.7

Comma list: 3136/3125, 19683/19600

Mapping: [1 6 8 17], 0 -14 -18 -45]]

Wedgie⟨⟨14 18 45 -4 32 54]]

POTE generator: ~56/45 = 378.467

Vals19, 111, 130, 929c, 1059c, 1189bc, 1319bc

Badness: 0.054303

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 8019/8000

Mapping: [1 6 8 17 -6], 0 -14 -18 -45 30]]

POTE generator: ~56/45 = 378.440

Vals: 19, 111, 130, 241, 371ce, 501cde, 872cde

Badness: 0.045352

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 3136/3125

Mapping: [1 6 8 17 -6 16], 0 -14 -18 -45 30 -39]]

POTE generator: ~56/45 = 378.437

Vals: 19, 111, 130, 241, 371ce

Badness: 0.023940

Mowglic

The mowglic temperament (19&161) is an extension of the mowgli temperament which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.


Subgroup: 2.3.5.7

Comma list: 3136/3125, 177147/175000

Mapping: [1 0 0 -3], 0 15 22 55]]

Wedgie⟨⟨15 22 55 0 45 66]]

POTE generator: ~27/25 = 126.706

Vals19, 123d, 142, 161

Badness: 0.129915

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 72171/71680

Mapping: [1 0 0 -3 8], 0 15 22 55 -43]]

POTE generator: ~27/25 = 126.711

Vals: 19, 123de, 142, 161

Badness: 0.094032

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1701/1690, 3136/3125

Mapping: [1 0 0 -3 8 -2], 0 15 22 55 -43 54]]

POTE generator: ~14/13 = 126.705

Vals: 19, 123def, 142f, 161

Badness: 0.051571

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125

Mapping: [1 0 0 -3 8 -2 10], 0 15 22 55 -43 54 -56]]

POTE generator: ~14/13 = 126.703

Vals: 19, 123defg, 142f, 161

Badness: 0.041918

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690

Mapping: [1 0 0 -3 8 -2 10 9], 0 15 22 55 -43 54 -56 -45]]

POTE generator: ~14/13 = 126.705

Vals: 19, 123defg, 142f, 161

Badness: 0.032168

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: [1 0 0 -3 8 -2 10 9 6], 0 15 22 55 -43 54 -56 -45 -14]]

POTE generator: ~14/13 = 126.703

Vals: 19, 123defg, 142f, 161

Badness: 0.026117

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: [1 0 0 -3 8 -2 10 9 6 0], 0 15 22 55 -43 54 -56 -45 -14 46]]

POTE generator: ~14/13 = 126.704

Vals: 19, 123defg, 142f, 161

Badness: 0.021398

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: [1 0 0 -3 8 -2 10 9 6 0 2], 0 15 22 55 -43 54 -56 -45 -14 46 28]]

POTE generator: ~14/13 = 126.703

Vals: 19, 123defgk, 142fk, 161

Badness: 0.019331

Quintagar

See also: 28ed5 #Regular temperaments

The quintagar temperament (12&217) tempers out the hemimean comma (3136/3125) and the garischisma (33554432/33480783) in the 7-limit. In the 2.3.5.7.17.19 subgroup, 256/255 (the difference between 16/15 and 17/16), 289/288 (between 17/16 and 18/17), 324/323 (between 18/17 and 19/18), 361/360 (between 19/18 and 20/19), and 400/399 (between 20/19 and 21/20) are equated together, and 476/475 (between 28/25 and 19/17) is tempered out. Immediate 2.3.5.7.11.17.19 extensions include quintoneum (12&217, tempering out 441/440), quintasandra (217&229, equating 385/384 with 400/399), and quintasandroid (12&229, equating 400/399 with 441/440). Full 19-limit extensions include quintoneum (12f&217), quintoneoid (12&217), quintasandra (217&229), quintasandroid (12f&229), and quintasand (12&229). The name quintagar is so named because the generator is 1/5 of the garibaldi fourth (~4/3, about 497.8 cents) aside from tempering out the garischisma.


Subgroup: 2.3.5.7

Comma list: 3136/3125, 33554432/33480783

Mapping: [1 2 0 -3], 0 -5 28 70]]

Wedgie⟨⟨5 -28 -70 -56 -125 -84]]

POTE generator: ~200/189 = 99.555

Vals12, 217, 229, 446, 675c

Badness: 0.142897

Quintoneum

The name quintoneum is a play on the words "quintans" (Latin for "one fifth") and "cotoneum".

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 7168000/7144929

Mapping: [1 2 0 -3 -5], 0 -5 28 70 102]]

POTE generator: ~35/33 = 99.539

Vals: 12, 205d, 217

Badness: 0.087157

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3136/3125, 13720/13689

Mapping: [1 2 0 -3 -5 -7], 0 -5 28 70 102 129]]

POTE generator: ~35/33 = 99.541

Vals: 12f, 205df, 217

Badness: 0.052361

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3136/3125, 3757/3750

Mapping: [1 2 0 -3 -5 -7 5], 0 -5 28 70 102 129 -11]]

POTE generator: ~18/17 = 99.540

Vals: 12f, 205df, 217

Badness: 0.035653

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 441/440, 476/475, 595/594, 1216/1215, 3757/3750

Mapping: [1 2 0 -3 -5 -7 5 4], 0 -5 28 70 102 129 -11 3]]

POTE generator: ~18/17 = 99.541

Vals: 12f, 205df, 217

Badness: 0.025782

Quintoneoid

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 59150/59049

Mapping: [1 2 0 -3 -5 11], 0 -5 28 70 102 -88]]

POTE generator: ~35/33 = 99.537

Vals: 12, 205d, 217

Badness: 0.072826

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 595/594, 1001/1000, 2601/2600, 3136/3125

Mapping: [1 2 0 -3 -5 11 5], 0 -5 28 70 102 -88 -11]]

POTE generator: ~18/17 = 99.537

Vals: 12, 205d, 217

Badness: 0.042339

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 441/440, 476/475, 595/594, 1001/1000, 1216/1215, 2601/2600

Mapping: [1 2 0 -3 -5 11 5 4], 0 -5 28 70 102 -88 -11 3]]

POTE generator: ~18/17 = 99.537

Vals: 12, 205d, 217

Badness: 0.028983

Quintasandra

The name quintasandra is a play on the words "quintans" and "cassandra". This temperament tempers out 19712/19683 and 41503/41472 in the 2.3.7.11 subgroup as the cassandra temperament, but with the hemimean comma rather than the schisma tempered out.

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 19712/19683, 41503/41472

Mapping: [1 2 0 -3 13], 0 -5 28 70 -115]]

POTE generator: ~200/189 = 99.551

Vals: 12e, 217, 446

Badness: 0.109908

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3136/3125, 4096/4095, 19712/19683

Mapping: [1 2 0 -3 13 11], 0 -5 28 70 -115 -88]]

POTE generator: ~55/52 = 99.548

Vals: 12e, 217, 446, 663c

Badness: 0.067730

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 1156/1155, 1377/1375, 3136/3125, 4096/4095

Mapping: [1 2 0 -3 13 11 5], 0 -5 28 70 -115 -88 -11]]

POTE generator: ~18/17 = 99.548

Vals: 12e, 217, 446, 663c

Badness: 0.038153

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 936/935, 1156/1155, 1216/1215, 1377/1375, 1729/1728

Mapping: [1 2 0 -3 13 11 5 4], 0 -5 28 70 -115 -88 -11 3]]

POTE generator: ~18/17 = 99.547

Vals: 12e, 217, 446, 663ch

Badness: 0.026654

Quintasandroid

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 8019/8000, 15488/15435

Mapping: [1 2 0 -3 -6], 0 -5 28 70 114]]

POTE generator: ~200/189 = 99.570

Vals: 12, 217e, 229, 470cd, 699cd

Badness: 0.093971

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 2080/2079, 3136/3125, 10648/10647

Mapping: [1 2 0 -3 -6 -8], 0 -5 28 70 114 141]]

POTE generator: ~55/52 = 99.578

Vals: 12f, 217ef, 229, 241, 470cd, 711ccd

Badness: 0.065701

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 561/560, 3136/3125, 7744/7735

Mapping: [1 2 0 -3 -6 -8 5], 0 -5 28 70 114 141 -11]]

POTE generator: ~18/17 = 99.574

Vals: 12f, 217ef, 229, 241, 470cd

Badness: 0.046624

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 442/441, 476/475, 561/560, 627/625, 6144/6137

Mapping: [1 2 0 -3 -6 -8 5 4], 0 -5 28 70 114 141 -11 3]]

POTE generator: ~18/17 = 99.575

Vals: 12f, 217ef, 229, 241, 470cd

Badness: 0.033145

Quintasand

Subgroup: 2.3.5.7.11.13

Comma list: 1573/1568, 3136/3125, 4096/4095, 4459/4455

Mapping: [1 2 0 -3 -6 11], 0 -5 28 70 114 -88]]

POTE generator: ~200/189 = 99.556

Vals: 12, 217e, 229, 446e, 675ceef

Badness: 0.100195

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 715/714, 1701/1700, 3136/3125, 4096/4095

Mapping: [1 2 0 -3 -6 11 5], 0 -5 28 70 114 -88 -11]]

POTE generator: ~18/17 = 99.556

Vals: 12, 217e, 229, 446e, 675ceef

Badness: 0.057851

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 476/475, 561/560, 627/625, 1216/1215, 1729/1728

Mapping: [1 2 0 -3 -6 11 5 4], 0 -5 28 70 114 -88 -11 3]]

POTE generator: ~18/17 = 99.557

Vals: 12, 217e, 229, 446e, 675ceefh

Badness: 0.040410

Undetrita

The undetrita temperament (111&118) tempers out the hemimean comma (3136/3125) and skeetsma (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to 11EDT tuning, and the name undetrita is a play on the words "undecimus" (Latin for "eleventh") and "tritave" (third harmonic). It is also related to the twentcufo temperament, which is no-sevens version of 111&118.


Subgroup: 2.3.5.7

Comma list: 3136/3125, 14348907/14336000

Mapping: [1 0 -2 -8], 0 11 30 75]]

Wedgie⟨⟨11 30 75 22 88 90]]

POTE generator: ~448/405 = 172.917

Vals111, 118, 229, 347, 576c

Badness: 0.114188

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 8019/8000

Mapping: [1 0 -2 -8 0], 0 11 30 75 24]]

POTE generator: ~400/363 = 172.912

Vals: 111, 118, 229, 347

Badness: 0.043883

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 729/728, 1001/1000, 3025/3024

Mapping: [1 0 -2 -8 0 5], 0 11 30 75 24 -9]]

POTE generator: ~72/65 = 172.930

Vals: 111, 229f

Badness: 0.038771

Undetritoid

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125

Mapping: [1 0 -2 -8 0 -11], 0 11 30 75 24 102]]

POTE generator: ~400/363 = 172.933

Vals: 111, 229

Badness: 0.042744

Isra

Isra results from taking every other generator of septimal meantone. It is named after the Isrāʾ (iss-RAH) night journey in the Qur'an, because it's similar to luna.


Subgroup: 2.9.5.7

Comma list: 81/80, 126/125

Mapping: [1 3 2 2], 0 1 2 5]]

Mapping generators: ~2/1, ~9/8

POTE generator: ~9/8 = 192.9898

Vals: 6, 19, 25, 31, 56b, 87b

Tutone

See also: Chromatic pairs #Tutone

Deutone

See also: Chromatic pairs #Deutone

Leantone

See also: Chromatic pairs #Leantone