Archytas clan

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The archytas clan (or archy family) tempers out the Archytas' comma, 64/63. This means a stack of two 3/2 fifths octave-reduced equals a whole tone of 8/7~9/8 tempered together; two of these tones or equivalently four stacked fifths octave-reduced equal a 9/7 major third. Note the similarity in function to 81/80 in meantone, where four stacked fifths octave-reduced equal a 5/4 major third. This leads to tunings with 3's and 7's quite sharp, such as those of 22edo.

This article focuses on rank-2 temperaments. See Archytas family for the rank-3 temperament resulting from tempering out 64/63 alone in the full 7-limit.

Archy

Main article: Superpyth

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]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 709.595
error map: ⟨0.000 +7.640 +11.984]
  • POTE: ~2 = 1200.000, ~3/2 = 709.321
error map: ⟨0.000 +7.366 +12.532]

Optimal ET sequence: 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd

Badness (Smith): 0.00464

Scales: archy5, archy7, archy12

Overview to extensions

Adding 245/243 to the list of commas gives superpyth; 2430/2401 gives quasisuper; 36/35 gives dominant; 360/343 gives schism; 6860/6561 gives ultrapyth; 33614/32805 gives quasiultra; 16/15 gives mother. These all use the same generators as archy.

686/675 gives beatles, splitting the fifth in two. 8748/8575 gives immunized, splitting the twelfth in two. 50/49 gives pajara with a semioctave period. 392/375 gives progress, splitting the twelfth in three. 250/243 gives porcupine, splitting the fourth in three. 126/125 gives augene with a 1/3-octave period. 4375/4374 gives modus, splitting the fifth in four. 3125/3024 gives brightstone. 9604/9375 gives fervor. 3125/2916 gives sixix. 3125/3087 gives passion. Those split the generator in five in various ways. 28/27 gives blacksmith with a 1/5-octave period. Finally, 15625/15552 gives catalan, splitting the twelfth in six.

Temperaments discussed elsewhere are:

Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

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

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 708.456
  • POTE: ~2 = 1200.000, ~3/2 = 707.192

Optimal ET sequence: 5, 12, 17, 39d, 56d

Badness (Smith): 0.00933

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]]

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 707.344
  • POTE: ~2 = 1200.000, ~3/2 = 706.137

Optimal ET sequence: 12f, 17

Badness (Smith): 0.0111

Scales: supra7, supra12

Superpyth

Main article: Superpyth
For the 5-limit version, see Superpyth–22 equivalence continuum #Superpyth (5-limit).

Superpyth adds 245/243 and 1728/1715 to the comma list and can be described as 22 & 27. ~5/4 is found at +9 generator steps, as an augmented second (C–D#).

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 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 709.591
error map: ⟨0.000 +7.636 +0.002 +11.993]
  • POTE: ~2 = 1200.000, ~3/2 = 710.291
error map: ⟨0.000 +8.336 +6.305 +10.592]

Optimal ET sequence: 5, 17, 22, 27, 49, 174bbcddd

Badness (Smith): 0.032318

11-limit

The canonical extension to the 13-limit finds the ~11/8 at +16 generator steps, as a double augmented second (C–Dx) and finds the ~13/8 at +13 generator steps, as a double augmented fourth (C–Fx).

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 709.514
  • POTE: ~2 = 1200.000, ~3/2 = 710.175

Optimal ET sequence: 22, 27e, 49

Badness (Smith): 0.024976

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 709.836
  • POTE: ~2 = 1200.000, ~3/2 = 710.479

Optimal ET sequence: 22, 27e, 49, 76bcde

Badness (Smith): 0.024673

Thomas

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 169/168, 245/243

Mapping: [⟨1 1 -3 4 -6 4], ⟨0 2 18 -4 32 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~16/13 = 354.759
  • POTE: ~2 = 1200.000, ~16/13 = 355.036

Optimal ET sequence: 27e, 44, 71d, 98bde

Badness (Smith): 0.049183

Suprapyth

Suprapyth finds the ~11/8 at the diminished fifth (C–Gb), and finds the ~13/8 at the diminished seventh (C–Bbb).

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 709.356
  • POTE: ~2 = 1200.000, ~3/2 = 709.495

Optimal ET sequence: 5, 17, 22

Badness (Smith): 0.032768

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 708.702
  • POTE: ~2 = 1200.000, ~3/2 = 708.703

Optimal ET sequence: 5f, 17, 22

Badness (Smith): 0.036336

Quasisuper

Quasisuper can be described as 17c & 22, with the ~5/4 mapped to -13 generator steps, as a double diminished fifth (C–Gbb).

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 -6 32 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 708.769
error map: ⟨0.000 +6.814 -0.310 +13.636]
  • POTE: ~2 = 1200.000, ~3/2 = 708.238
error map: ⟨0.000 +6.283 +6.586 +14.697]

Optimal ET sequence: 17c, 22, 61d

Badness (Smith): 0.063794

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 0 23 6 13], ⟨0 1 -13 -2 -6]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 708.713
  • POTE: ~2 = 1200.000, ~3/2 = 708.205

Optimal ET sequence: 17c, 22, 39d, 61d

Badness (Smith): 0.032203

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 708.411
  • POTE: ~2 = 1200.000, ~3/2 = 708.004

Optimal ET sequence: 17c, 22, 39d, 61df, 100bcdf

Badness (Smith): 0.030219

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 709.043
  • POTE: ~2 = 1200.000, ~3/2 = 709.021

Optimal ET sequence: 22

Badness (Smith): 0.083490

Ultrapyth

Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 oceanfront temperament, mapping the ~5/4 to +14 fifths as a double augmented unison (C–Cx).

Subgroup: 2.3.5.7

Comma list: 64/63, 6860/6561

Mapping: [⟨1 0 -20 6], ⟨0 1 14 -2]]

Wedgie: ⟨⟨ 1 14 -2 20 -6 -44 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 713.218
error map: ⟨0.000 +11.263 -1.264 +4.738]
  • POTE: ~2 = 1200.000, ~3/2 = 713.651
error map: ⟨0.000 +11.696 +4.800 +3.872]

Optimal ET sequence: 5, 27c, 32, 37

Badness (Smith): 0.108466

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2401/2376

Mapping: [⟨1 0 -20 6 21], ⟨0 1 14 -2 -11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 713.282
  • POTE: ~2 = 1200.000, ~3/2 = 713.395

Optimal ET sequence: 5, 32, 37

Badness (Smith): 0.068238

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 1573/1568

Mapping: [⟨1 0 -20 6 21 -25], ⟨0 1 14 -2 -11 18]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 713.309
  • POTE: ~2 = 1200.000, ~3/2 = 713.500

Optimal ET sequence: 5, 32, 37

Badness (Smith): 0.049172

Ultramarine

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 3773/3645

Mapping: [⟨1 0 -20 6 -38], ⟨0 1 14 -2 26]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 713.395
  • POTE: ~2 = 1200.000, ~3/2 = 713.791

Optimal ET sequence: 5e, 32e, 37, 79bce

Badness (Smith): 0.078068

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 91/90, 100/99, 847/845

Mapping: [⟨1 0 -20 6 -38 -25], ⟨0 1 14 -2 26 18]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 713.708
  • POTE: ~2 = 1200.000, ~3/2 = 713.811

Optimal ET sequence: 5e, 32e, 37, 79bcef

Badness (Smith): 0.045653

Quasiultra

Quasiultra is to ultrapyth what quasisuper is to superpyth. It is the 27 & 32 temperament, mapping the ~5/4 to -18 fifths as a double diminished sixth (C–Abbb).

Subgroup: 2.3.5.7

Comma list: 64/63, 33614/32805

Mapping: [⟨1 0 31 6], ⟨0 1 -18 -2]]

Wedgie: ⟨⟨ 1 -18 -2 -31 -6 46 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 711.838
error map: ⟨0.000 +9.883 +0.608 +7.499]
  • CWE: ~2 = 1200.000, ~3/2 = 711.543
error map: ⟨0.000 +9.588 +5.914 +8.088]

Optimal ET sequence: 27, 86bd, 113bcd, 140bbcd

Badness (Smith): 0.132

Schism

See also: Schismatic family § Schism

Schism tempers out the schisma, mapping the ~5/4 to -8 fifths as a diminished fourth (C–Fb) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 702.270
error map: ⟨0.000 +0.315 -4.471 +26.635]
  • POTE: ~2 = 1200.000, ~3/2 = 701.556
error map: ⟨0.000 -0.399 +1.237 +28.062]

Wedgie: ⟨⟨ 1 -8 -2 -15 -6 18 ]]

Optimal ET sequence: 5c, 7c, 12

Badness (Smith): 0.056648

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 703.383
  • POTE: ~2 = 1200.000, ~3/2 = 702.136

Optimal ET sequence: 5c, 7ce, 12, 29de

Badness (Smith): 0.037482

Beatles

For the 5-limit version, see Miscellaneous 5-limit temperaments #Beatles.

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 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~49/40 = 356.634
error map: ⟨0.000 +11.312 +3.984 +4.640]
  • POTE: ~2 = 1200.000, ~49/40 = 355.904
error map: ⟨0.000 +9.853 +10.549 +7.558]

Optimal ET sequence: 10, 17c, 27, 64b, 91bcd, 118bccd

Badness (Smith): 0.045872

Music

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~49/40 = 356.719
  • POTE: ~2 = 1200.000, ~49/40 = 356.140

Optimal ET sequence: 10e, 17cee, 27e, 64be, 91bcdee

Badness (Smith): 0.045639

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~16/13 = 356.722
  • POTE: ~2 = 1200.000, ~16/13 = 356.229

Optimal ET sequence: 10e, 27e, 37, 64be

Badness (Smith): 0.030161

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/9 = 355.992
  • POTE: ~2 = 1200.000, ~11/9 = 355.419

Optimal ET sequence: 10, 17c, 27e

Badness (Smith): 0.032863

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/9 = 356.004
  • POTE: ~2 = 1200.000, ~11/9 = 355.456

Optimal ET sequence: 10, 17c, 27e

Badness (Smith): 0.022619

Beetle

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 686/675

Mapping: [⟨1 1 5 4 -1], ⟨0 2 -9 -4 15]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~49/40 = 356.694
  • POTE: ~2 = 1200.000, ~49/40 = 356.710

Optimal ET sequence: 10, 27, 37

Badness (Smith): 0.058084

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 169/168

Mapping: [⟨1 1 5 4 -1 4], ⟨0 2 -9 -4 15 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~16/13 = 356.700
  • POTE: ~2 = 1200.000, ~16/13 = 356.701

Optimal ET sequence: 10, 27, 37

Badness (Smith): 0.033971

Progress

For the 5-limit version, see Miscellaneous 5-limit temperaments #Progress.

Subgroup: 2.3.5.7

Comma list: 64/63, 392/375

Mapping: [⟨1 0 5 6], ⟨0 3 -5 -6]]

mapping generators: ~2, ~10/7

Wedgie: ⟨⟨ 3 -5 -6 -15 -18 0 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/7 = 638.782
error map: ⟨0.000 +14.391 +19.777 -1.518]
  • POTE: ~2 = 1200.000, ~10/7 = 637.878
error map: ⟨0.000 +11.679 +24.297 +3.907]

Optimal ET sequence: 2, 13, 15, 32c

Badness (Smith): 0.066400

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/7 = 638.846
  • POTE: ~2 = 1200.000, ~10/7 = 637.915

Optimal ET sequence: 2, 13, 15, 32c, 47bc

Badness (Smith): 0.031036

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/7 = 637.871
  • POTE: ~2 = 1200.000, ~10/7 = 637.635

Optimal ET sequence: 15, 17c, 32cf

Badness (Smith): 0.026214

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/7 = 637.797
  • POTE: ~2 = 1200.000, ~10/7 = 636.761

Optimal ET sequence: 2f, 15f, 17c

Badness (Smith): 0.032721

Fervor

For the 5-limit version, see Miscellaneous 5-limit temperaments #Fervor.

Subgroup: 2.3.5.7

Comma list: 64/63, 9604/9375

Mapping: [⟨1 4 -2 -2], ⟨0 -5 9 10]]

mapping generators: ~2, ~7/5

Wedgie: ⟨⟨ 5 -9 -10 -26 -30 2 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/5 = 577.353
error map: ⟨0.000 +11.278 +9.867 +4.709]
  • POTE: ~2 = 1200.000, ~7/5 = 577.776
error map: ⟨0.000 +9.163 +13.673 +8.937]

Optimal ET sequence: 2, 25, 27

Badness (Smith): 0.108455

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/5 = 577.296
  • POTE: ~2 = 1200.000, ~7/5 = 577.850

Optimal ET sequence: 2, 25e, 27e

Badness (Smith): 0.052054

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/5 = 577.374
  • POTE: ~2 = 1200.000, ~7/5 = 578.060

Optimal ET sequence: 2f, 27e

Badness (Smith): 0.039705

Sixix

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Sixix (5-limit).
See also: Dual-fifth temperaments § Dual-3 Sixix

Sixix is related to the kleismic family in a way similar to the one between meantone and mavila. In both cases the generator is nominally a 6/5 and the complexity to generate major and minor chords is the same, but in sixix it is tuned extremely sharply, to the point where the 3rd and 5th harmonics are reached by going down instead of up, inverting the logic of chord construction.

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/2916

Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]

Wedgie: ⟨⟨ 5 6 -10 -2 -30 -40 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 337.519
error map: ⟨0.000 +10.449 -11.429 +6.366]
  • POTE: ~2 = 1200.000, ~6/5 = 337.442
error map: ⟨0.000 +10.835 -10.965 +5.594]

Optimal ET sequence: 7, 18d, 25, 32

Badness (Smith): 0.158903

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 125/121

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

Optimal tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 337.749
  • POTE: ~2 = 1200.000, ~6/5 = 337.564

Optimal ET sequence: 7, 25e, 32

Badness (Smith): 0.070799

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 55/54, 64/63, 125/121

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

Optimal tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 337.793
  • POTE: ~2 = 1200.000, ~6/5 = 337.483

Optimal ET sequence: 7, 25e, 32f

Badness (Smith): 0.046206

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 55/54, 64/63, 85/84, 125/121

Mapping: [⟨1 3 4 0 6 4 1], ⟨0 -5 -6 10 -9 -1 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 337.629
  • POTE: ~2 = 1200.000, ~6/5 = 337.513

Optimal ET sequence: 7, 25e, 32f

Badness (Smith): 0.039224

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