Archytas clan: Difference between revisions

Revision as of 14:06, 21 December 2024

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
POTE: ~2 = 1200.000, ~3/2 = 709.321

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. 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, splitting the twelfth in five. 3125/3087 gives passion, splitting the fourth in five. 28/27 gives blacksmith with a 1/5-octave period. Finally, 15625/15552 gives catalan, splitting the twelfth in six.

Discussed under their respective 5-limit families are:

Mother (+16/15) → Father family
Dominant (+36/35) → Meantone family
Medusa (+15/14) → Very low accuracy temperaments
Suhajira (+243/242) → Neutral clan
Immunized (+8748/8575) → Immunity family
Pajara (+50/49) → Diaschismic family
Augene (+126/125) → Augmented family
Porcupine (+250/243) → Porcupine family
Modus (+4375/4374) → Tetracot family
Brightstone (+3125/3024) → Magic family
Passion (+3125/3087) → Passion family
Blacksmith (+28/27) → Limmic temperaments
Catalan (+15625/15552) → Kleismic family

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 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

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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 709.393
POTE: ~2 = 1200.000, ~3/2 = 710.078

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

Badness (Smith): 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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 709.591
POTE: ~2 = 1200.000, ~3/2 = 710.291

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
POTE: ~2 = 1200.000, ~3/2 = 708.328

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
POTE: ~2 = 1200.000, ~3/2 = 713.651

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
CWE: ~2 = 1200.000, ~3/2 = 711.5429

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
POTE: ~2 = 1200.000, ~3/2 = 701.556

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 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 tunings:

CTE: ~2 = 1200.000, ~49/40 = 356.634
POTE: ~2 = 1200.000, ~49/40 = 355.904

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

Badness (Smith): 0.045872

Music

Beatles Improv by Herman Miller

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 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]]

mapping generators: ~2, ~10/7

Wedgie: ⟨⟨ 3 -5 -6 -15 -18 0 ]]

Optimal tunings:

CTE: ~2 = 1200.000, ~10/7 = 638.782
POTE: ~2 = 1200.000, ~10/7 = 637.878

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 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]]

mapping generators: ~2, ~7/5

Wedgie: ⟨⟨ 5 -9 -10 -26 -30 2 ]]

Optimal tunings:

CTE: ~2 = 1200.000, ~7/5 = 577.353
POTE: ~2 = 1200.000, ~7/5 = 577.776

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

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]]

mapping generators: ~2, ~6/5

Optimal tunings:

CTE: ~2 = 1200.000, ~6/5 = 338.005
POTE: ~2 = 1200.000, ~6/5 = 338.365

Optimal ET sequence: 7, 25, 32, 39c

Badness (Smith): 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 tunings:

CTE: ~2 = 1200.000, ~6/5 = 337.519
POTE: ~2 = 1200.000, ~6/5 = 337.442

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

@@ Line 19: / Line 19: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~3/2 = 709.595
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.595
-* [[POTE]]: ~3/2 = 709.321
+* [[POTE]]: ~2 = 1200.000, ~3/2 = 709.321
 {{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
-[[Badness]]: 0.00464
+[[Badness]] (Smith): 0.00464
 Scales: [[archy5]], [[archy7]], [[archy12]]
@@ Line 35: / Line 35: @@
 Discussed under their respective 5-limit families are:
 * ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
+* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
 * ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
-* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
 * ''[[Suhajira]]'' (+243/242) → [[Neutral clan #Suhajira|Neutral clan]]
 * ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
@@ Line 62: / Line 62: @@
 Optimal tunings:
-* CTE: ~3/2 = 708.456
+* CTE: ~2 = 1200.000, ~3/2 = 708.456
-* POTE: ~3/2 = 707.192
+* POTE: ~2 = 1200.000, ~3/2 = 707.192
-Optimal ET sequence: {{Optimal ET sequence| 5, 12, 17, 39d, 56d }}
+Optimal ET sequence: {{optimal ET sequence| 5, 12, 17, 39d, 56d }}
-Badness: 0.00933
+Badness (Smith): 0.00933
 Scales: [[supra7]], [[supra12]]
@@ Line 83: / Line 83: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 707.344
+* CTE: ~2 = 1200.000, ~3/2 = 707.344
-* POTE: ~2 = 1\1, ~3/2 = 706.137
+* POTE: ~2 = 1200.000, ~3/2 = 706.137
-Optimal ET sequence: {{Optimal ET sequence| 12f, 17 }}
+Optimal ET sequence: {{optimal ET sequence| 12f, 17 }}
-Badness: 0.0111
+Badness (Smith): 0.0111
 Scales: [[supra7]], [[supra12]]
@@ Line 95: / Line 95: @@
 {{Main| 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, [[Jubilismic clan #Bipyth|bipyth]] (10cd &amp; 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.
+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, [[Jubilismic clan #Bipyth|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
@@ Line 104: / Line 104: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~3/2 = 709.393
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.393
-* [[POTE]]: ~2 = 1\1, ~3/2 = 710.078
+* [[POTE]]: ~2 = 1200.000, ~3/2 = 710.078
 {{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
-[[Badness]]: 0.135141
+[[Badness]] (Smith): 0.135141
 === 7-limit ===
@@ Line 121: / Line 121: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~3/2 = 709.591
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.591
-* [[POTE]]: ~2 = 1\1, ~3/2 = 710.291
+* [[POTE]]: ~2 = 1200.000, ~3/2 = 710.291
 {{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
-[[Badness]]: 0.032318
+[[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).
+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
@@ Line 138: / Line 138: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 709.514
+* CTE: ~2 = 1200.000, ~3/2 = 709.514
-* POTE: ~2 = 1\1, ~3/2 = 710.175
+* POTE: ~2 = 1200.000, ~3/2 = 710.175
-Optimal ET sequence: {{Optimal ET sequence| 22, 27e, 49 }}
+Optimal ET sequence: {{optimal ET sequence| 22, 27e, 49 }}
-Badness: 0.024976
+Badness (Smith): 0.024976
 ==== 13-limit ====
@@ Line 153: / Line 153: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 709.836
+* CTE: ~2 = 1200.000, ~3/2 = 709.836
-* POTE: ~2 = 1\1, ~3/2 = 710.479
+* POTE: ~2 = 1200.000, ~3/2 = 710.479
-Optimal ET sequence: {{Optimal ET sequence| 22, 27e, 49, 76bcde }}
+Optimal ET sequence: {{optimal ET sequence| 22, 27e, 49, 76bcde }}
-Badness: 0.024673
+Badness (Smith): 0.024673
 ==== Thomas ====
@@ Line 168: / Line 168: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~16/13 = 354.759
+* CTE: ~2 = 1200.000, ~16/13 = 354.759
-* POTE: ~2 = 1\1, ~16/13 = 355.036
+* POTE: ~2 = 1200.000, ~16/13 = 355.036
-Optimal ET sequence: {{Optimal ET sequence| 27e, 44, 71d, 98bde }}
+Optimal ET sequence: {{optimal ET sequence| 27e, 44, 71d, 98bde }}
-Badness: 0.049183
+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).
+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
@@ Line 185: / Line 185: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 709.356
+* CTE: ~2 = 1200.000, ~3/2 = 709.356
-* POTE: ~2 = 1\1, ~3/2 = 709.495
+* POTE: ~2 = 1200.000, ~3/2 = 709.495
-Optimal ET sequence: {{Optimal ET sequence| 5, 17, 22 }}
+Optimal ET sequence: {{optimal ET sequence| 5, 17, 22 }}
-Badness: 0.032768
+Badness (Smith): 0.032768
 ==== 13-limit ====
@@ Line 200: / Line 200: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 708.702
+* CTE: ~2 = 1200.000, ~3/2 = 708.702
-* POTE: ~2 = 1\1, ~3/2 = 708.703
+* POTE: ~2 = 1200.000, ~3/2 = 708.703
-Optimal ET sequence: {{Optimal ET sequence| 5f, 17, 22 }}
+Optimal ET sequence: {{optimal ET sequence| 5f, 17, 22 }}
-Badness: 0.036336
+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).
+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
@@ Line 219: / Line 219: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~3/2 = 708.769
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 708.769
-* [[POTE]]: ~2 = 1\1, ~3/2 = 708.328
+* [[POTE]]: ~2 = 1200.000, ~3/2 = 708.328
 {{Optimal ET sequence|legend=1| 17c, 22, 61d }}
-[[Badness]]: 0.063794
+[[Badness]] (Smith): 0.063794
 === Quasisupra ===
@@ Line 236: / Line 236: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 708.713
+* CTE: ~2 = 1200.000, ~3/2 = 708.713
-* POTE: ~2 = 1\1, ~3/2 = 708.205
+* POTE: ~2 = 1200.000, ~3/2 = 708.205
-Optimal ET sequence: {{Optimal ET sequence| 17c, 22, 39d, 61d }}
+Optimal ET sequence: {{optimal ET sequence| 17c, 22, 39d, 61d }}
-Badness: 0.032203
+Badness (Smith): 0.032203
 ==== 13-limit ====
@@ Line 251: / Line 251: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 708.411
+* CTE: ~2 = 1200.000, ~3/2 = 708.411
-* POTE: ~2 = 1\1, ~3/2 = 708.004
+* POTE: ~2 = 1200.000, ~3/2 = 708.004
-Optimal ET sequence: {{Optimal ET sequence| 17c, 22, 39d, 61df, 100bcdf }}
+Optimal ET sequence: {{optimal ET sequence| 17c, 22, 39d, 61df, 100bcdf }}
-Badness: 0.030219
+Badness (Smith): 0.030219
 === Quasisoup ===
@@ Line 266: / Line 266: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 709.043
+* CTE: ~2 = 1200.000, ~3/2 = 709.043
-* POTE: ~2 = 1\1, ~3/2 = 709.021
+* POTE: ~2 = 1200.000, ~3/2 = 709.021
-Optimal ET sequence: {{Optimal ET sequence| 22 }}
+Optimal ET sequence: {{optimal ET sequence| 22 }}
-Badness: 0.083490
+Badness (Smith): 0.083490
 == Ultrapyth ==
-Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[The Biosphere #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double augmented unison (C-Cx).
+Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[The Biosphere #Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double augmented unison (C–Cx).
 [[Subgroup]]: 2.3.5.7
@@ Line 285: / Line 285: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~3/2 = 713.218
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 713.218
-* [[POTE]]: ~2 = 1\1, ~3/2 = 713.651
+* [[POTE]]: ~2 = 1200.000, ~3/2 = 713.651
 {{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
-[[Badness]]: 0.108466
+[[Badness]] (Smith): 0.108466
 === 11-limit ===
@@ Line 300: / Line 300: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 713.282
+* CTE: ~2 = 1200.000, ~3/2 = 713.282
-* POTE: ~2 = 1\1, ~3/2 = 713.395
+* POTE: ~2 = 1200.000, ~3/2 = 713.395
-Optimal ET sequence: {{Optimal ET sequence| 5, 32, 37 }}
+Optimal ET sequence: {{optimal ET sequence| 5, 32, 37 }}
-Badness: 0.068238
+Badness (Smith): 0.068238
 ==== 13-limit ====
@@ Line 315: / Line 315: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 713.309
+* CTE: ~2 = 1200.000, ~3/2 = 713.309
-* POTE: ~2 = 1\1, ~3/2 = 713.500
+* POTE: ~2 = 1200.000, ~3/2 = 713.500
-Optimal ET sequence: {{Optimal ET sequence| 5, 32, 37 }}
+Optimal ET sequence: {{optimal ET sequence| 5, 32, 37 }}
-Badness: 0.049172
+Badness (Smith): 0.049172
 === Ultramarine ===
@@ Line 330: / Line 330: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 713.395
+* CTE: ~2 = 1200.000, ~3/2 = 713.395
-* POTE: ~2 = 1\1, ~3/2 = 713.791
+* POTE: ~2 = 1200.000, ~3/2 = 713.791
-Optimal ET sequence: {{Optimal ET sequence| 5e, 32e, 37, 79bce }}
+Optimal ET sequence: {{optimal ET sequence| 5e, 32e, 37, 79bce }}
-Badness: 0.078068
+Badness (Smith): 0.078068
 ==== 13-limit ====
@@ Line 345: / Line 345: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 713.708
+* CTE: ~2 = 1200.000, ~3/2 = 713.708
-* POTE: ~2 = 1\1, ~3/2 = 713.811
+* POTE: ~2 = 1200.000, ~3/2 = 713.811
-Optimal ET sequence: {{Optimal ET sequence| 5e, 32e, 37, 79bcef }}
+Optimal ET sequence: {{optimal ET sequence| 5e, 32e, 37, 79bcef }}
-Badness: 0.045653
+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).
+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
@@ Line 364: / Line 364: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~3/2 = 711.838
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 711.838
-* [[CWE]]: ~2 = 1\1, ~3/2 = 711.5429
+* [[CWE]]: ~2 = 1200.000, ~3/2 = 711.5429
 {{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
-[[Badness]]: 0.132
+[[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.
+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
@@ Line 383: / Line 383: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~3/2 = 702.270
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 702.270
-* [[POTE]]: ~2 = 1\1, ~3/2 = 701.556
+* [[POTE]]: ~2 = 1200.000, ~3/2 = 701.556
 {{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
@@ Line 390: / Line 390: @@
 {{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
-[[Badness]]: 0.056648
+[[Badness]] (Smith): 0.056648
 === 11-limit ===
@@ Line 400: / Line 400: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~3/2 = 703.383
+* CTE: ~2 = 1200.000, ~3/2 = 703.383
-* POTE: ~2 = 1\1, ~3/2 = 702.136
+* POTE: ~2 = 1200.000, ~3/2 = 702.136
-Optimal ET sequence: {{Optimal ET sequence| 5c, 7ce, 12, 29de }}
+Optimal ET sequence: {{optimal ET sequence| 5c, 7ce, 12, 29de }}
-Badness: 0.037482
+Badness (Smith): 0.037482
 == Beatles ==
@@ Line 419: / Line 419: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~49/40 = 356.634
+* [[CTE]]: ~2 = 1200.000, ~49/40 = 356.634
-* [[POTE]]: ~2 = 1\1, ~49/40 = 355.904
+* [[POTE]]: ~2 = 1200.000, ~49/40 = 355.904
 {{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
-[[Badness]]: 0.045872
+[[Badness]] (Smith): 0.045872
 ; Music
@@ Line 437: / Line 437: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~49/40 = 356.719
+* CTE: ~2 = 1200.000, ~49/40 = 356.719
-* POTE: ~2 = 1\1, ~49/40 = 356.140
+* POTE: ~2 = 1200.000, ~49/40 = 356.140
-Optimal ET sequence: {{Optimal ET sequence| 10e, 17cee, 27e, 64be, 91bcdee }}
+Optimal ET sequence: {{optimal ET sequence| 10e, 17cee, 27e, 64be, 91bcdee }}
-Badness: 0.045639
+Badness (Smith): 0.045639
 ==== 13-limit ====
@@ Line 452: / Line 452: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~16/13 = 356.722
+* CTE: ~2 = 1200.000, ~16/13 = 356.722
-* POTE: ~2 = 1\1, ~16/13 = 356.229
+* POTE: ~2 = 1200.000, ~16/13 = 356.229
-Optimal ET sequence: {{Optimal ET sequence| 10e, 27e, 37, 64be }}
+Optimal ET sequence: {{optimal ET sequence| 10e, 27e, 37, 64be }}
-Badness: 0.030161
+Badness (Smith): 0.030161
 === Ringo ===
@@ Line 467: / Line 467: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~11/9 = 355.992
+* CTE: ~2 = 1200.000, ~11/9 = 355.992
-* POTE: ~2 = 1\1, ~11/9 = 355.419
+* POTE: ~2 = 1200.000, ~11/9 = 355.419
-Optimal ET sequence: {{Optimal ET sequence| 10, 17c, 27e }}
+Optimal ET sequence: {{optimal ET sequence| 10, 17c, 27e }}
-Badness: 0.032863
+Badness (Smith): 0.032863
 ==== 13-limit ====
@@ Line 482: / Line 482: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~11/9 = 356.004
+* CTE: ~2 = 1200.000, ~11/9 = 356.004
-* POTE: ~2 = 1\1, ~11/9 = 355.456
+* POTE: ~2 = 1200.000, ~11/9 = 355.456
-Optimal ET sequence: {{Optimal ET sequence| 10, 17c, 27e }}
+Optimal ET sequence: {{optimal ET sequence| 10, 17c, 27e }}
-Badness: 0.022619
+Badness (Smith): 0.022619
 === Beetle ===
@@ Line 497: / Line 497: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~49/40 = 356.694
+* CTE: ~2 = 1200.000, ~49/40 = 356.694
-* POTE: ~2 = 1\1, ~49/40 = 356.710
+* POTE: ~2 = 1200.000, ~49/40 = 356.710
-Optimal ET sequence: {{Optimal ET sequence| 10, 27, 37 }}
+Optimal ET sequence: {{optimal ET sequence| 10, 27, 37 }}
-Badness: 0.058084
+Badness (Smith): 0.058084
 ==== 13-limit ====
@@ Line 512: / Line 512: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~16/13 = 356.700
+* CTE: ~2 = 1200.000, ~16/13 = 356.700
-* POTE: ~2 = 1\1, ~16/13 = 356.701
+* POTE: ~2 = 1200.000, ~16/13 = 356.701
-Optimal ET sequence: {{Optimal ET sequence| 10, 27, 37 }}
+Optimal ET sequence: {{optimal ET sequence| 10, 27, 37 }}
-Badness: 0.033971
+Badness (Smith): 0.033971
 == Progress ==
@@ Line 533: / Line 533: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~10/7 = 638.782
+* [[CTE]]: ~2 = 1200.000, ~10/7 = 638.782
-* [[POTE]]: ~2 = 1\1, ~10/7 = 637.878
+* [[POTE]]: ~2 = 1200.000, ~10/7 = 637.878
 {{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
-[[Badness]]: 0.066400
+[[Badness]] (Smith): 0.066400
 === 11-limit ===
@@ Line 548: / Line 548: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/7 = 638.846
+* CTE: ~2 = 1200.000, ~10/7 = 638.846
-* POTE: ~2 = 1\1, ~10/7 = 637.915
+* POTE: ~2 = 1200.000, ~10/7 = 637.915
-Optimal ET sequence: {{Optimal ET sequence| 2, 13, 15, 32c, 47bc }}
+Optimal ET sequence: {{optimal ET sequence| 2, 13, 15, 32c, 47bc }}
-Badness: 0.031036
+Badness (Smith): 0.031036
 ==== 13-limit ====
@@ Line 563: / Line 563: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/7 = 637.871
+* CTE: ~2 = 1200.000, ~10/7 = 637.871
-* POTE: ~2 = 1\1, ~10/7 = 637.635
+* POTE: ~2 = 1200.000, ~10/7 = 637.635
-Optimal ET sequence: {{Optimal ET sequence| 15, 17c, 32cf }}
+Optimal ET sequence: {{optimal ET sequence| 15, 17c, 32cf }}
-Badness: 0.026214
+Badness (Smith): 0.026214
 ==== Progressive ====
@@ Line 578: / Line 578: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/7 = 637.797
+* CTE: ~2 = 1200.000, ~10/7 = 637.797
-* POTE: ~2 = 1\1, ~10/7 = 636.761
+* POTE: ~2 = 1200.000, ~10/7 = 636.761
-Optimal ET sequence: {{Optimal ET sequence| 2f, 15f, 17c }}
+Optimal ET sequence: {{optimal ET sequence| 2f, 15f, 17c }}
-Badness: 0.032721
+Badness (Smith): 0.032721
 == Fervor ==
@@ Line 599: / Line 599: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~7/5 = 577.353
+* [[CTE]]: ~2 = 1200.000, ~7/5 = 577.353
-* [[POTE]]: ~2 = 1\1, ~7/5 = 577.776
+* [[POTE]]: ~2 = 1200.000, ~7/5 = 577.776
 {{Optimal ET sequence|legend=1| 2, 25, 27 }}
-[[Badness]]: 0.108455
+[[Badness]] (Smith): 0.108455
 === 11-limit ===
@@ Line 614: / Line 614: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~7/5 = 577.296
+* CTE: ~2 = 1200.000, ~7/5 = 577.296
-* POTE: ~2 = 1\1, ~7/5 = 577.850
+* POTE: ~2 = 1200.000, ~7/5 = 577.850
-Optimal ET sequence: {{Optimal ET sequence| 2, 25e, 27e }}
+Optimal ET sequence: {{optimal ET sequence| 2, 25e, 27e }}
-Badness: 0.052054
+Badness (Smith): 0.052054
 === 13-limit ===
@@ Line 629: / Line 629: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~7/5 = 577.374
+* CTE: ~2 = 1200.000, ~7/5 = 577.374
-* POTE: ~2 = 1\1, ~7/5 = 578.060
+* POTE: ~2 = 1200.000, ~7/5 = 578.060
-Optimal ET sequence: {{Optimal ET sequence| 2f, 27e }}
+Optimal ET sequence: {{optimal ET sequence| 2f, 27e }}
-Badness: 0.039705
+Badness (Smith): 0.039705
 == 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.
+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
@@ Line 650: / Line 650: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~6/5 = 338.005
+* [[CTE]]: ~2 = 1200.000, ~6/5 = 338.005
-* [[POTE]]: ~2 = 1\1, ~6/5 = 338.365
+* [[POTE]]: ~2 = 1200.000, ~6/5 = 338.365
 {{Optimal ET sequence|legend=1| 7, 25, 32, 39c }}
-[[Badness]]: 0.153088
+[[Badness]] (Smith): 0.153088
 === 7-limit ===
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 64/63, 3125/2916
@@ Line 667: / Line 667: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~6/5 = 337.519
+* [[CTE]]: ~2 = 1200.000, ~6/5 = 337.519
-* [[POTE]]: ~2 = 1\1, ~6/5 = 337.442
+* [[POTE]]: ~2 = 1200.000, ~6/5 = 337.442
 {{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
-[[Badness]]: 0.158903
+[[Badness]] (Smith): 0.158903
 === 11-limit ===
@@ Line 682: / Line 682: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~6/5 = 337.749
+* CTE: ~2 = 1200.000, ~6/5 = 337.749
-* POTE: ~2 = 1\1, ~6/5 = 337.564
+* POTE: ~2 = 1200.000, ~6/5 = 337.564
-Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32 }}
+Optimal ET sequence: {{optimal ET sequence| 7, 25e, 32 }}
-Badness: 0.070799
+Badness (Smith): 0.070799
 === 13-limit ===
@@ Line 697: / Line 697: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~6/5 = 337.793
+* CTE: ~2 = 1200.000, ~6/5 = 337.793
-* POTE: ~2 = 1\1, ~6/5 = 337.483
+* POTE: ~2 = 1200.000, ~6/5 = 337.483
-Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
+Optimal ET sequence: {{optimal ET sequence| 7, 25e, 32f }}
-Badness: 0.046206
+Badness (Smith): 0.046206
 === 17-limit ===
@@ Line 712: / Line 712: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~6/5 = 337.629
+* CTE: ~2 = 1200.000, ~6/5 = 337.629
-* POTE: ~2 = 1\1, ~6/5 = 337.513
+* POTE: ~2 = 1200.000, ~6/5 = 337.513
-Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
+Optimal ET sequence: {{optimal ET sequence| 7, 25e, 32f }}
-Badness: 0.039224
+Badness (Smith): 0.039224
 [[Category:Archytas clan| ]] <!-- main article -->
 [[Category:Temperament clans]]
 [[Category:Rank 2]]

Archytas clan: Difference between revisions

Revision as of 14:06, 21 December 2024

Archy

Overview to extensions

Supra

Supraphon

Superpyth

7-limit

11-limit

13-limit

Thomas

Suprapyth

13-limit

Quasisuper

Quasisupra

13-limit

Quasisoup

Ultrapyth

11-limit

13-limit

Ultramarine

13-limit

Quasiultra

Schism

11-limit

Beatles

11-limit

13-limit

Ringo

13-limit

Beetle

13-limit

Progress

11-limit

13-limit

Progressive

Fervor

11-limit

13-limit

Sixix

7-limit

11-limit

13-limit

17-limit

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