Archytas clan

The archytas clan tempers out the Archytas comma, 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to 81/80 in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of 22edo.

Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.

Discussed under their respective 5-limit families are mother, dominant, augene, porcupine, pajara, catalan, modus, and immunized. The rest are considered below.

Archy

Subgroup: 2.3.7

Comma list: 64/63

Sval mapping: [⟨1 0 6], ⟨0 1 -2]]

Sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 0 4], ⟨0 1 0 -2]]

Gencom: [2 3/2; 64/63]

POTE generator: ~3/2 = 709.321

Scales: archy5, archy7, archy12

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]

Sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 0 4 7], ⟨0 1 0 -2 -6]]

Gencom: [2 3/2; 64/63 99/98]

POTE generator: ~3/2 = 707.192

Scales: supra7, supra12

Supraphon

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 99/98

Sval mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]

Sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 0 4 7 9], ⟨0 1 0 -2 -6 -9]]

Gencom: [2 3/2; 64/63 78/77 99/98]

POTE generator: ~3/2 = 706.137

Scales: supra7, supra12

Suhajira

Subgroup: 2.3.7.11

Comma list: 64/63, 243/242

Sval mapping: [⟨1 1 4 2], ⟨0 2 -4 5]]

Sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 0 4 2], ⟨0 1 0 -4 5]]

Gencom: [2 3/2; 64/63 99/98]

POTE generator: ~11/9 = 353.958

Scales: suhajira7, suhajira10, suhajira17

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 144/143

Sval mapping: [⟨1 1 4 2 4], ⟨0 2 -4 5 -1]]

Sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 0 4 2 4], ⟨0 1 0 -4 5 -1]]

Gencom: [2 3/2; 64/63 78/77 99/98]

POTE generator: ~11/9 = 353.775

Scales: suhajira7, suhajira10, suhajira17

Superpyth

Main article: Superpyth

Subgroup: 2.3.5.7

Comma list: 64/63, 245/243

Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]

Wedgie: ⟨⟨ 1 9 -2 12 -6 -30 ]]

POTE generator: ~3/2 = 710.291

Badness: 0.0323

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 245/243

Mapping: [⟨1 0 -12 6 -22], ⟨0 1 9 -2 16]]

POTE generator: ~3/2 = 710.175

Badness: 0.0250

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 100/99

Mapping: [⟨1 0 -12 6 -22 -17], ⟨0 1 9 -2 16 13]]

POTE generator: ~3/2 = 710.479

Badness: 0.0247

Suprapyth

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 99/98

Mapping: [⟨1 0 -12 6 13], ⟨0 1 9 -2 -6]]

POTE generator: ~3/2 = 709.495

Badness: 0.0328

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 65/63, 99/98

Mapping: [⟨1 0 -12 6 13 18], ⟨0 1 9 -2 -6 -9]]

POTE generator: ~3/2 = 708.703

Badness: 0.0363

Quasisuper

Subgroup: 2.3.5.7

Comma list: 64/63, 2430/2401

Mapping: [⟨1 0 23 6], ⟨0 1 -13 -2]]

Wedgie: ⟨⟨ 1 -13 -2 -23 -2 -6 32 ]]

POTE generator: ~3/2 = 708.328

Badness: 0.0638

Quasisupra

Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament supra, with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).

Subgroup: 2.3.5.7.11

Comma list: 64/63, 99/98, 121/120

Mapping: [⟨1 2 -3 2 1], ⟨0 -1 13 2 6]]

POTE generator: ~3/2 = 708.205

Badness: 0.0322

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 121/120

Mapping: [⟨1 0 23 6 13 18], ⟨0 1 -13 -2 -6 -9]]

POTE generator: ~3/2 = 708.004

Badness: 0.0302

Quasisoup

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2430/2401

Mapping: [⟨1 0 23 6 -22], ⟨0 1 -13 -2 16]]

POTE generator: ~3/2 = 709.021

Badness: 0.0835

Schism

See also: Schismatic family § Schism

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]

Wedgie: ⟨⟨ 1 -8 -2 -15 -6 18 ]]

POTE generator: ~3/2 = 701.556

Badness: 0.0566

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 64/63, 99/98

Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]

POTE generator: ~3/2 = 702.136

Badness: 0.0375

Blacksmith

Lattice of blacksmith

5-limit (blackwood)

Subgroup: 2.3.5

Comma list: 256/243

Mapping: [⟨5 8 0], ⟨0 0 1]]

Mapping generators: ~9/8, ~5

POTE generator: ~5/4 = 399.594

Badness: 0.0638

7-limit

Subgroup: 2.3.5.7

Comma list: 28/27, 49/48

Mapping: [⟨5 8 0 14], ⟨0 0 1 0]]

Mapping generators: ~7/6, ~5

Wedgie: ⟨⟨ 0 5 0 8 0 -14 ]]

POTE generator: ~5/4 = 392.767

Badness: 0.0256

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 49/48, 55/54

Mapping: [⟨5 8 0 14 29], ⟨0 0 1 0 -1]]

POTE generator: ~5/4 = 394.948

Badness: 0.0246

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 49/48, 55/54

Mapping: [⟨5 8 0 14 29 7], ⟨0 0 1 0 -1 1]]

POTE generator: ~5/4 = 391.0367

Badness: 0.0205

Farrier

Subgroup: 2.3.5.7.11

Comma list: 28/27, 49/48, 77/75

Mapping: [⟨5 8 0 14 -6], ⟨0 0 1 0 2]]

POTE generator: ~5/4 = 398.070

Badness: 0.0292

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 49/48, 66/65

Mapping: [⟨5 8 0 14 -6 7], ⟨0 0 1 0 2 1]]

POTE generator: ~5/4 = 396.812

Badness: 0.0223

Ferrum

Subgroup: 2.3.5.7.11

Comma list: 28/27, 35/33, 49/48

Mapping: [⟨5 8 0 14 6], ⟨0 0 1 0 1]]

POTE generator: ~5/4 = 374.763

Badness: 0.0309

Beatles

5-limit

Subgroup: 2.3.5

Comma list: 524288/492075

Mapping: [⟨1 1 5], ⟨0 2 -9]]

POTE generator: ~512/405 = 355.930

Badness: 0.3585

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 686/675

Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4]]

Wedgie: ⟨⟨ 2 -9 -4 -19 -12 16 ]]

POTE generator: ~49/40 = 355.904

Badness: 0.0459

Music

• Beatles Improv by Herman Miller

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 686/675

Mapping: [⟨1 1 5 4 10], ⟨0 2 -9 -4 -22]]

POTE generator: ~49/40 = 356.140

Badness: 0.0456

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 91/90, 100/99, 169/168

Mapping: [⟨1 1 5 4 10 4], ⟨0 2 -9 -4 -22 -1]]

POTE generator: ~16/13 = 356.229

Badness: 0.0302

Ringo

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 540/539

Mapping: [⟨1 1 5 4 2], ⟨0 2 -9 -4 5]]

POTE generator: ~11/9 = 355.419

Badness: 0.0329

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 91/90

Mapping: [⟨1 1 5 4 2 4], ⟨0 2 -9 -4 5 -1]]

POTE generator: ~11/9 = 355.456

Badness: 0.0226

Passion

5-limit

Subgroup: 2.3.5

Comma list: 262144/253125

Mapping: [⟨1 2 2], ⟨0 -5 4]]

Mapping generators: 2, 16/15

POTE generator: ~16/15 = 98.670

Badness: 0.1686

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/3087

Mapping: [⟨1 2 2 2], ⟨0 -5 4 10]]

Mapping generators: 2, 16/15

Wedgie: ⟨⟨ 5 -4 -10 -18 -30 -12 ]]

POTE generator: ~16/15 = 98.153

Badness: 0.0623

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 1375/1372

Mapping: [⟨1 2 2 2 2], ⟨0 -5 4 10 18]]

POTE generator: ~16/15 = 98.019

Badness: 0.0408

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 196/195, 275/273

Mapping: [⟨1 2 2 2 2 2], ⟨0 -5 4 10 18 21]]

POTE generator: ~16/15 = 97.910

Badness: 0.0309

Fervor

5-limit

Subgroup: 2.3.5

Comma list: 67108864/61509375

Mapping: [⟨1 4 -2], ⟨0 -5 9]]

POTE generator: ~64/45 = 577.705

Badness: 0.8526

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 9604/9375

Mapping: [⟨1 4 -2 -2], ⟨0 -5 9 10]]

Wedgie: ⟨⟨ 5 -9 -10 -26 -30 2 ]]

POTE generator: ~7/5 = 577.777

Badness: 0.1085

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 1350/1331

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

POTE generator: ~7/5 = 577.850

Badness: 0.0521

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 507/500

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

POTE generator: ~7/5 = 578.060

Badness: 0.0397

Progress

5-limit

Subgroup: 2.3.5

Comma list: 32768/30375

Mapping: [⟨1 0 5], ⟨0 3 -5]]

POTE generator: ~64/45 = 561.264

Badness: 0.2461

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 392/375

Mapping: [⟨1 0 5 6], ⟨0 3 -5 -6]]

Wedgie: ⟨⟨ 3 -5 -6 -15 -18 0 ]]

POTE generator: ~7/5 = 562.122

Badness: 0.0664

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 77/75

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

POTE generator: ~7/5 = 562.085

Badness: 0.0310

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 66/65, 77/75

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

POTE generator: ~7/5 = 562.365

Badness: 0.0262

Progressive

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 64/63, 77/75

Mapping: [⟨1 0 5 6 4 9], ⟨0 3 -5 -6 -1 -10]]

POTE generator: ~7/5 = 563.239

Badness: 0.0327

Sixix

See also: Dual-fifth temperaments#Dual-3 sixix

5-limit

Subgroup: 2.3.5

Comma list: 3125/2916

Mapping: [⟨1 3 4], ⟨0 -5 -6]]

POTE generator: ~6/5 = 338.365

Badness: 0.1531

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/2916

Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]

POTE generator: ~6/5 = 337.4419