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 subgroup temperaments is the 2.3.7 archy. Under their respective 5-limit families are mother, dominant, augene, porcupine, pajara, modus, and immunized. The rest are considered below.

Blacksmith

Lattice of blacksmith

5-limit (blackwood)

Period: 1\5

Optimal (POTE) generator: ~5/4 = 399.594

EDO generators: 3\10, 4\15

Scales (Scala files):

Technical data

Comma list: 256/243

Mapping: [⟨5 8 0], ⟨0 0 1]]

Mapping generators: ~9/8, ~5

Badness: 0.0638

7-limit

Period: 1\5

Optimal (POTE) generator: ~5/4 = 392.767

EDO generators: 3\10, 4\15

Scales (Scala files):

Technical data

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]]

Badness: 0.0256

11-limit

Period: 1\5

Optimal (POTE) generator: ~5/4 = 394.948

EDO generators: 3\10, 4\15

Scales (Scala files):

Technical data

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

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

Badness: 0.0246

13-limit

Period: 1\5

Optimal (POTE) generator: ~5/4 = 391.0367

EDO generators: 3\10, 4\15

Scales (Scala files):

Technical data

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

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

Badness: 0.0205

Farrier

Period: 1\5

Optimal (POTE) generator: ~5/4 = 398.070

EDO generators: 3\10, 4\15

Scales (Scala files):

Technical data

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

Mapping: [⟨5 8 0 14 -6], ⟨0 0 1 0 2]]

Badness: 0.0292

13-limit

Period: 1\5

Optimal (POTE) generator: ~5/4 = 396.812

EDO generators: 3\10, 4\15

Scales (Scala files):

Technical data

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

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

Badness: 0.0223

Ferrum

Period: 1\5

Optimal (POTE) generator: ~5/4 = 374.763

EDO generators: 3\10

Scales (Scala files):

Technical data

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

Mapping: [⟨5 8 0 14 6], ⟨0 0 1 0 1]]

Badness: 0.0309

Superpyth

Main article: Superpyth

Comma list: 64/63, 245/243

POTE generator: ~3/2 = 710.291

Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]

Wedgie: ⟨⟨1 9 -2 12 -6 -30]]

Badness: 0.0323

11-limit

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

POTE generator: ~3/2 = 710.175

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

Badness: 0.0250

13-limit

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

POTE generator: ~3/2 = 710.479

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

Badness: 0.0247

Suprapyth

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

POTE generator: ~3/2 = 709.495

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

Badness: 0.0328

13-limit

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

POTE generator: ~3/2 = 708.703

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

Badness: 0.0363

Beatles

5-limit

Comma list: 524288/492075

POTE generator: ~512/405 = 355.930

Mapping: [⟨1 1 5], ⟨0 2 -9]]

Badness: 0.3585

7-limit

Comma list: 64/63, 686/675

POTE generator: ~49/40 = 355.904

Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4]]

Wedgie: ⟨⟨2 -9 -4 -19 -12 16]]

Badness: 0.0459

Music: Beatles Improv by Herman Miller

11-limit

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

POTE generator: ~49/40 = 356.140

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

Badness: 0.0456

13-limit

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

POTE generator: ~16/13 = 356.229

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

Badness: 0.0302

Ringo

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

POTE generator: ~11/9 = 355.419

Mapping: [⟨1 1 5 4 2], ⟨0 2 -9 -4 5]]

Badness: 0.0329

13-limit

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

POTE generator: ~11/9 = 355.456

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

Badness: 0.0226

Schism

See also: Schismatic family § Schism

Comma list: 64/63, 360/343

POTE generator: ~3/2 = 701.556

Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]

Wedgie: ⟨⟨1 -8 -2 -15 -6 18]]

Badness: 0.0566

11-limit

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

POTE generator ~3/2 = 702.136

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

Badness: 0.0375

Passion

5-limit

Comma list: 262144/253125

POTE generator: ~16/15 = 98.670

Mapping: [⟨1 2 2], ⟨0 -5 4]]

Badness: 0.1686

7-limit

Comma list: 64/63, 3125/3087

POTE generator: ~16/15 = 98.153

Mapping: [⟨1 2 2 2], ⟨0 -5 4 10]]

Mapping generators: 2, 16/15

Wedgie: ⟨⟨5 -4 -10 -18 -30 -12]]

Badness: 0.0623

11-limit

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

POTE generator: ~16/15 = 98.019

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

Badness: 0.0408

13-limit

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

POTE generator: ~16/15 = 97.910

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

Badness: 0.0309

Fervor

5-limit

Comma list: 67108864/61509375

POTE generator: ~64/45 = 577.705

Mapping: [⟨1 4 -2], ⟨0 -5 9]]

Badness: 0.8526

7-limit

Comma list: 64/63, 9604/9375

POTE generator: ~7/5 = 577.777

Mapping: [⟨1 4 -2 -2], ⟨0 -5 9 10]]

Wedgie: ⟨⟨5 -9 -10 -26 -30 2]]

Badness: 0.1085

11-limit

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

POTE generator: ~7/5 = 577.850

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

Badness: 0.0521

13-limit

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

POTE generator: ~7/5 = 578.060

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

Badness: 0.0397

Quasisuper

Comma list: 64/63, 2430/2401

POTE generator: 708.328

Mapping: [⟨1 0 23 6], ⟨0 1 -13 -2]]

Wedgie: ⟨⟨1 -13 -2 -23 -2 -6 32]]

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).

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

POTE generator: ~3/2 = 708.205

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

Badness: 0.0322

13-limit

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

POTE generator: ~3/2 = 708.004

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

Badness: 0.0302

Quasisoup

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

POTE generator: ~3/2 = 709.021

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

Badness: 0.0835

Progress

5-limit

Comma list: 32768/30375

POTE generator: ~64/45 = 561.264

Mapping: [⟨1 0 5], ⟨0 3 -5]]

Badness: 0.2461

7-limit

Comma list: 64/63, 392/375

POTE generator: ~7/5 = 562.122

Mapping: [⟨1 0 5 6], ⟨0 3 -5 -6]]

Wedgie: ⟨⟨3 -5 -6 -15 -18 0]]

Badness: 0.0664

11-limit

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

POTE generator: ~7/5 = 562.085

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

Badness: 0.0310

13-limit

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

POTE generator: ~7/5 = 562.365

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

Badness: 0.0262

Progressive

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

POTE generator: ~7/5 = 563.239

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

Badness: 0.0327

Sixix

5-limit

Comma list: 3125/2916

POTE generator: ~6/5 = 338.365

Mapping: [⟨1 3 4], ⟨0 -5 -6]]

Badness: 0.1531

7-limit

Comma list: 3125/2916, 64/63

POTE generator: ~6/5 = 337.4419

Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]