Archytas clan

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The archytas clan (or archy family) 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 3's and 7's quite sharp, such as those of 22edo.

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 tuning (POTE): ~3/2 = 709.321

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

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; 16/15 gives mother. These all use the same generators as archy.

50/49 gives pajara with a semioctave period. 126/125 gives augene with a 1/3-octave period. 28/27 gives blacksmith with a 1/5-octave period. 686/675 gives beatles, splitting the fifth in two. 250/243 gives porcupine, splitting the fourth in three. 4375/4374 gives modus, splitting the fifth in four. 3125/3087 gives passion, splitting the fourth in five.

Discussed under their respective 5-limit families are:

The rest are considered below.

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 tuning (POTE): ~3/2 = 707.192

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

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 tuning (POTE): ~3/2 = 706.137

Optimal ET sequence: 12f, 17

Scales: supra7, supra12


Superpyth

Main article: Superpyth

In the 5-limit, superpyth tempers out 20480/19683. This temperament has a fifth generator of ~3/2 = ~710¢ and ~5/4 is found at +9 generator steps, as an augmented second (C-D#). It also has a weak extension, bipyth (10cd & 22), tempering out the same 5-limit comma as the superpyth, but with a half-octave period and the jubilisma (50/49) rather than the Archytas comma tempered out.

Subgroup: 2.3.5

Comma list: 20480/19683

Mapping: [⟨1 0 -12], ⟨0 1 9]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 710.078

Optimal ET sequence: 5, 17, 22, 49, 120b, 169bbc

Badness: 0.135141

7-limit

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 tuning (POTE): ~2 = 1\1, ~3/2 = 710.291

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

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 710.175

Optimal ET sequence: 22, 27e, 49

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 710.479

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

Badness: 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 tuning (POTE): ~2 = 1\1, ~16/13 = 355.036

Optimal ET sequence: 17e, 27e, 44, 71d

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 709.495

Optimal ET sequence: 5, 12c, 17, 22

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 708.703

Optimal ET sequence: 17, 22, 83cdf

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 708.328

Optimal ET sequence: 17c, 22, 61d

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 708.205

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

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 708.004

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

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 709.021

Optimal ET sequence: 5ce, 17ce, 22

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 713.651

Optimal ET sequence: 5, 32, 37

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 713.395

Optimal ET sequence: 5, 32, 37

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 713.500

Optimal ET sequence: 5, 32, 37

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 713.791

Optimal ET sequence: 5e, 32e, 37, 79bce, 116bbce

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 713.811

Optimal ET sequence: 5e, 32e, 37, 79bcef, 116bbcef

Badness: 0.045653

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.

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.556

Optimal ET sequence: 12, 41d, 53d

Badness: 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 tuning (POTE): ~2 = 1\1, ~3/2 = 702.136

Optimal ET sequence: 12, 29de, 41de

Badness: 0.037482

Beatles

For the 5-limit version of this temperament, see High badness 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 tuning (POTE): ~2 = 1\1, ~49/40 = 355.904

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

Badness: 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 tuning (POTE): ~2 = 1\1, ~49/40 = 356.140

Optimal ET sequence: 27e, 37, 64be, 91bcde

Badness: 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 tuning (POTE): ~2 = 1\1, ~16/13 = 356.229

Optimal ET sequence: 27e, 37, 64be

Badness: 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 tuning (POTE): ~2 = 1\1, ~11/9 = 355.419

Optimal ET sequence: 10, 17c, 27e

Badness: 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 tuning (POTE): ~2 = 1\1, ~11/9 = 355.456

Optimal ET sequence: 10, 17c, 27e

Badness: 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 tuning (POTE): ~2 = 1\1, ~49/40 = 356.710

Optimal ET sequence: 10, 27, 37

Badness: 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 tuning (POTE): ~2 = 1\1, ~16/13 = 356.701

Optimal ET sequence: 10, 27, 37

Badness: 0.033971

Fervor

For the 5-limit version of this temperament, see High badness temperaments #Fervor.

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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 577.776

Optimal ET sequence: 2, 25, 27

Badness: 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 tuning (POTE): ~2 = 1\1, ~7/5 = 577.850

Optimal ET sequence: 2, 25e, 27e

Badness: 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 tuning (POTE): ~2 = 1\1, ~7/5 = 578.060

Optimal ET sequence: 2f, 25ef, 27e

Badness: 0.039705

Progress

For the 5-limit version of this temperament, see High badness temperaments #Progress.

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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.122

Optimal ET sequence: 2, 13, 15, 32c, 79bcc, 111bcc

Badness: 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 tuning (POTE): ~2 = 1\1, ~7/5 = 562.085

Optimal ET sequence: 2, 13, 15, 32c, 47bc, 79bcce

Badness: 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 tuning (POTE): ~2 = 1\1, ~7/5 = 562.365

Optimal ET sequence: 15, 17c, 32cf

Badness: 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 tuning (POTE): ~2 = 1\1, ~7/5 = 563.239

Optimal ET sequence: 15f, 17c, 32c, 49c

Badness: 0.032721

Sixix

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

Subgroup: 2.3.5

Comma list: 3125/2916

Mapping: [⟨1 3 4], ⟨0 -5 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 338.365

Optimal ET sequence: 7, 25, 32

Badness: 0.153088

7-limit

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 tuning (POTE): ~2 = 1\1, ~6/5 = 337.442

Optimal ET sequence: 7, 25, 32

Badness: 0.158903

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