Hemimean clan

The hemimean clan tempers out the no-threes hemimean comma 3136/3125.

Didacus

Subgroup: 2.5.7

Comma list: 3136/3125

Sval mapping: [⟨1 0 -3], ⟨0 2 5]]

Mapping generators: ~2, ~28/25

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

Gencom: [2 28/25; 3136/3125]

POTE generator: ~28/25 = 193.772

Extensions

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, misty, semisept, sengagen, subpental, arch, and emka, considered below, as well as these considered elsewhere:

• hexe, {50/49, 128/125} → Augmented family #Hexe
• passion, {64/63, 3125/3087} → Archytas clan #Passion
• meantone, {81/80, 126/125} → Meantone family #Septimal meantone
• clyde, {245/243, 3136/3125} → Kleismic family #Clyde
• sycamore, {686/675, 875/864} → Sycamore family #Septimal sycamore
• bidia, {2048/2025, 3136/3125} → Diaschismic family #Bidia
• hemiwürschmidt, {2401/2400, 3136/3125} → Würschmidt family #Hemiwürschmidt
• parakleismic, {3136/3125, 4375/4374} → Ragismic microtemperaments #Parakleismic
• bischismic, {3136/3125, 32805/32768} → Schismatic family #Bischismic
• decistearn, {3136/3125, 118098/117649} → Stearnsmic clan #Decistearn

A notable subgroup extension of didacus is roulette.

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:

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

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: Template:Val list

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: Template:Val list

Badness: 0.021738

Misty

See also: Misty family

Subgroup: 2.3.5.7

Comma list: 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56], ⟨0 1 -4 -10]]

Wedgie: ⟨⟨ 3 -12 -30 -26 -56 -36 ]]

POTE generator: ~3/2 = 703.021

Badness: 0.036802

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4375/4356

Mapping: [⟨3 0 26 56 77], ⟨0 1 -4 -10 -14]]

POTE generator: ~3/2 = 703.210

Vals: Template:Val list

Badness: 0.035932

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 625/624

Mapping: [⟨3 0 26 56 77 92], ⟨0 1 -4 -10 -14 -17]]

POTE generator: ~3/2 = 703.303

Vals: Template:Val list

Badness: 0.027820

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 352/351, 364/363, 375/374, 595/594

Mapping: [⟨3 0 26 56 77 92 -2], ⟨0 1 -4 -10 -14 -17 3]]

POTE generator: ~3/2 = 703.233

Vals: Template:Val list

Badness: 0.023908

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594

Mapping: [⟨3 0 26 56 77 92 -2 8], ⟨0 1 -4 -10 -14 -17 3 1]]

POTE generator: ~3/2 = 703.253

Vals: Template:Val list

Badness: 0.019878

Mystic

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 441/440, 896/891, 1001/1000

Mapping: [⟨3 0 26 56 77 -46], ⟨0 1 -4 -10 -14 12]]

POTE generator: ~3/2 = 703.311

Vals: Template:Val list

Badness: 0.036872

Hemimist

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 5120/5103

Mapping: [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]]

POTE generator: ~11/8 = 551.527

Vals: Template:Val list

Badness: 0.047439

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845, 3025/3024

Mapping: [⟨3 0 26 56 8 23], ⟨0 2 -8 -20 1 -5]]

POTE generator: ~11/8 = 551.525

Vals: Template:Val list

Badness: 0.022923

Semisept

Subgroup: 2.3.5

Comma list: 782757789696/762939453125

Mapping: [⟨1 12 6], ⟨0 -17 -6]]

POTE generator: ~192/125 = 735.146

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

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: [{{val| 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: Template:Val list

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: Template:Val list

Badness: 0.028408

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

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: Template:Val list

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

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: Template:Val list

Badness: 0.023940

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

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: Template:Val list

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: Template:Val list

Badness: 0.019504

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

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

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: Template:Val list

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: Template:Val list

Badness: 0.029741

Quintoneum

See also: 28ed5 § 28ed5 as a generator

The quintoneum 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 (the difference between 17/16 and 18/17), 324/323 (the difference between 18/17 and 19/18), 361/360 (the difference between 19/18 and 20/19), and 400/399 (the difference between 20/19 and 21/20) are equated together, and 476/475 (the difference 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). The name quintoneum is so named because the generator is 1/5 of the cotoneum fourth (~4/3, about 497.7 cents).

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

Badness: 0.142897

11-limit

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

Badness: 0.028983

Quintasandra

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

Badness: 0.040410