# 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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