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]]

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

Wedgie: ⟨⟨1 -8 -2 -15 -6 18]]

Optimal ET sequence: 12, 29d, 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

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

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

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 tuning (POTE): ~2 = 1\1, ~6/5 = 337.564

Optimal ET sequence: 7, 25e, 32

Badness: 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 tuning (POTE): ~2 = 1\1, ~6/5 = 337.483

Optimal ET sequence: 7, 25e, 32f

Badness: 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 tuning (POTE): ~2 = 1\1, ~6/5 = 337.513

Optimal ET sequence: 7, 25e, 32f

Badness: 0.039224

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