Archytas clan: Difference between revisions

Latest revision as of 12:52, 22 July 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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, 27edo, or 49edo.

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 0 0 6], ⟨0 1 0 -2]]

gencom: [2 3; 64/63]

Optimal tunings:

WE: ~2 = 1196.9552 ¢, ~3/2 = 707.5215 ¢

error map: ⟨-3.045 +2.522 +3.952]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.3901 ¢

error map: ⟨0.000 +7.435 +12.394]

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

Badness (Sintel): 0.159

Scales: archy5, archy7, archy12

Overview to extensions

7-limit extensions

The second comma in the comma list defines which 7-limit family member we are looking at:

Schism adds 360/343, for a tuning around 12edo;
Dominant adds 36/35, for a tuning between 12edo and 17c-edo;
Quasisuper adds 2430/2401, for a tuning between 17c-edo and 22edo;
Superpyth adds 245/243, for a tuning between 22edo and 27edo;
Quasiultra adds 33614/32805, for a tuning between 27edo and 32edo;
Ultrapyth adds 6860/6561, for a tuning sharp of 32edo;
Mother adds 16/15, for an exotemperament well tuned around 5edo.

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:

Mother (+16/15) → Father family
Dominant (+36/35) → Meantone family
Medusa (+15/14) → Very low accuracy temperaments
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
Blackwood (+28/27) → Limmic temperaments
Catalan (+15625/15552) → Kleismic family

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

Subgroup extensions

Omitting prime 5, archy can be extended to the 2.3.7.11 subgroup by identifying 11/8 as a diminished fourth (C–Gb). This is called supra, given right below. Discussed elsewhere is suhajira of the neutral clan.

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 0 0 6 13], ⟨0 1 0 -2 -6]]

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

Optimal tunings:

WE: ~2 = 1197.2650 ¢, ~3/2 = 705.5803 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 707.4981 ¢

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

Badness (Sintel): 0.352

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 0 0 6 13 18], ⟨0 1 0 -2 -6 -9]]

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

Optimal tunings:

WE: ~2 = 1197.1909 ¢, ~3/2 = 704.4836 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 706.4289 ¢

Optimal ET sequence: 12f, 17

Badness (Sintel): 0.498

Scales: supra7, supra12

Superpyth

Main article: Superpyth

For the 5-limit version, see Syntonic–diatonic equivalence continuum #Superpyth (5-limit).

Superpyth, virtually the canonical extension, 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#). 49edo remains an obvious tuning choice.

Subgroup: 2.3.5.7

Comma list: 64/63, 245/243

Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]

Optimal tunings:

WE: ~2 = 1197.0549 ¢, ~3/2 = 708.5478 ¢

error map: ⟨-2.945 +3.648 -0.548 +2.298]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.1193 ¢

error map: ⟨0.000 +8.164 +4.760 +10.935]

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

Badness (Sintel): 0.818

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:

WE: ~2 = 1197.0673 ¢, ~3/2 = 708.4391 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.0129 ¢

Optimal ET sequence: 22, 27e, 49

Badness (Sintel): 0.826

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:

WE: ~2 = 1197.3011 ¢, ~3/2 = 708.8813 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.3219 ¢

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

Badness (Sintel): 1.02

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:

WE: ~2 = 1197.4942 ¢, ~16/13 = 354.2950 ¢
CWE: ~2 = 1200.0000 ¢, ~16/13 = 354.9824 ¢

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

Badness (Sintel): 2.03

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:

WE: ~2 = 1198.6960 ¢, ~3/2 = 708.7235 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.4699 ¢

Optimal ET sequence: 5, 17, 22

Badness (Sintel): 1.08

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:

WE: ~2 = 1199.9871 ¢, ~3/2 = 708.6952 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.7028 ¢

Optimal ET sequence: 5f, 17, 22

Badness (Sintel): 1.50

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]]

Optimal tunings:

WE: ~2 = 1196.9830 ¢, ~3/2 = 706.4578 ¢

error map: ⟨-3.017 +1.486 -0.435 +6.190]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.3716 ¢

error map: ⟨0.000 +6.417 +4.855 +14.431]

Optimal ET sequence: 17c, 22, 61d

Badness (Sintel): 1.61

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:

WE: ~2 = 1197.5675 ¢, ~3/2 = 706.7690 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.3200 ¢

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

Badness (Sintel): 1.06

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:

WE: ~2 = 1198.2543 ¢, ~3/2 = 706.9736 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.0936 ¢

Optimal ET sequence: 17c, 22, 39d

Badness (Sintel): 1.25

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:

WE: ~2 = 1198.8446 ¢, ~3/2 = 708.3388 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.0252 ¢

Optimal ET sequence: 22

Badness (Sintel): 2.76

Ultrapyth

Main article: 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]]

Optimal tunings:

WE: ~2 = 1197.2673 ¢, ~3/2 = 712.0258 ¢

error map: ⟨-2.733 +7.338 -1.557 -3.808]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.5430 ¢

error map: ⟨0.000 +11.588 +3.288 +4.088]

Optimal ET sequence: 5, 27c, 32, 37

Badness (Sintel): 2.74

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:

WE: ~2 = 1198.0290 ¢, ~3/2 = 712.2235 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.3754 ¢

Optimal ET sequence: 5, 32, 37

Badness (Sintel): 2.26

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:

WE: ~2 = 1198.1911 ¢, ~3/2 = 712.4243 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.4684 ¢

Optimal ET sequence: 5, 32, 37

Badness (Sintel): 2.03

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:

WE: ~2 = 1197.2230 ¢, ~3/2 = 712.1393 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.6928 ¢

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

Badness (Sintel): 2.58

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:

WE: ~2 = 1197.2739 ¢, ~3/2 = 712.1893 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.7079 ¢

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

Badness (Sintel): 1.89

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]]

Optimal tunings:

WE: ~2 = 1196.9257 ¢, ~3/2 = 709.6211 ¢

error map: ⟨0.000 +9.883 +0.608 +7.499]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.5429 ¢

error map: ⟨0.000 +9.588 +5.914 +8.088]

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

Badness (Sintel): 3.34

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 (53dd 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:

WE: ~2 = 1197.3598 ¢, ~3/2 = 700.0126 ¢

error map: ⟨-2.640 -4.583 -4.896 +20.588]

CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7376 ¢

error map: ⟨0.000 -0.217 -0.214 +27.699]

Optimal ET sequence: 5c, 7c, 12

Badness (Sintel): 1.43

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:

WE: ~2 = 1196.1607 ¢, ~3/2 = 699.8897 ¢
CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.4385 ¢

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

Badness (Sintel): 1.24

Beatles

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

Beatles tempers out 686/675, which may also be characterized by saying it tempers out 2401/2400. It may be described as the 10 & 17c temperament. It splits the fifth into two neutral-third generators of 49/40~60/49; its ploidacot is dicot. 5/4 may be found at -9 generator steps, as a semidiminished fourth (C–Fd). 27edo is an obvious tuning, though 17c-edo and 37edo are among the possibilities.

Beatles extends easily to the no-11 13-limit, as the generator can be interpreted as ~16/13, tempering out 91/90, 169/168, and 196/195.

Subgroup: 2.3.5.7

Comma list: 64/63, 686/675

Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4]]

Optimal tunings:

WE: ~2 = 1196.6244 ¢, ~49/40 = 354.9029 ¢

error map: ⟨-3.376 +4.475 +2.682 -1.940]

CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.0819 ¢

error map: ⟨0.000 +10.209 +8.949 +6.847]

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

Badness (Sintel): 1.16

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:

WE: ~2 = 1196.7001 ¢, ~49/40 = 355.1606 ¢
CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.2795 ¢

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

Badness (Sintel): 1.51

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:

WE: ~2 = 1197.2504 ¢, ~16/13 = 355.4132 ¢
CWE: ~2 = 1200.0000 ¢, ~16/13 = 356.3273 ¢

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

Badness (Sintel): 1.25

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:

WE: ~2 = 1195.4102 ¢, ~11/9 = 354.0597 ¢
CWE: ~2 = 1200.0000 ¢, ~11/9 = 355.5207 ¢

Optimal ET sequence: 10, 17c, 27e

Badness (Sintel): 1.09

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:

WE: ~2 = 1195.9943 ¢, ~11/9 = 354.2695 ¢
CWE: ~2 = 1200.0000 ¢, ~11/9 = 355.5398 ¢

Optimal ET sequence: 10, 17c, 27e

Badness (Sintel): 0.935

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:

WE: ~2 = 1197.9660 ¢, ~49/40 = 356.1056 ¢
CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.7075 ¢

Optimal ET sequence: 10, 27, 37

Badness (Sintel): 1.92

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:

WE: ~2 = 1198.1741 ¢, ~16/13 = 356.1582 ¢
CWE: ~2 = 1200.0000 ¢, ~16/13 = 356.7008 ¢

Optimal ET sequence: 10, 27, 37

Badness (Sintel): 1.40

Progress

Not to be confused with Progression.

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

Progress tempers out 392/375 and may be described as 15 & 17c. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 32c-edo gives an obvious tuning.

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

Optimal tunings:

WE: ~2 = 1195.1377 ¢, ~10/7 = 635.2932 ¢

error map: ⟨-4.862 +3.925 +12.908 -9.759]

CWE: ~2 = 1200.0000 ¢, ~10/7 = 638.0791 ¢

error map: ⟨0.000 +12.282 +23.291 +2.700]

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

Badness (Sintel): 1.68

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:

WE: ~2 = 1195.4920 ¢, ~10/7 = 635.5183 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 638.0884 ¢

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

Badness (Sintel): 1.03

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:

WE: ~2 = 1195.0786 ¢, ~10/7 = 635.0197 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 637.6691 ¢

Optimal ET sequence: 15, 17c, 32cf

Badness (Sintel): 1.08

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:

WE: ~2 = 1196.0245 ¢, ~10/7 = 634.6516 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 636.9528 ¢

Optimal ET sequence: 2f, 15f, 17c

Badness (Sintel): 1.35

Fervor

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

Fervor tempers out 9704/9375 and may be described as 25 & 27. It splits the 6th harmonic into five generators of ~10/7; its ploidacot is beta-pentacot. 27edo is about as accurate as it can be tuned.

Subgroup: 2.3.5.7

Comma list: 64/63, 9604/9375

Mapping: [⟨1 -1 7 8], ⟨0 5 -9 -10]]

mapping generators: ~2, ~10/7

Optimal tunings:

WE: ~2 = 1196.2742 ¢, ~10/7 = 620.2918 ¢

error map: ⟨-3.726 +3.230 +4.980 -1.550]

CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.3179 ¢

error map: ⟨0.000 +9.634 +12.826 +7.996]

Optimal ET sequence: 2, 25, 27

Badness (Sintel): 2.74

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 1350/1331

Mapping: [⟨1 -1 7 8 4], ⟨0 5 -9 -10 -1]]

Optimal tunings:

WE: ~2 = 1195.4148 ¢, ~10/7 = 619.7729 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.2525 ¢

Optimal ET sequence: 2, 25e, 27e

Badness (Sintel): 1.72

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 507/500

Mapping: [⟨1 -1 7 8 4 12], ⟨0 5 -9 -10 -1 -16]]

Optimal tunings:

WE: ~2 = 1195.6284 ¢, ~10/7 = 619.6738 ¢
CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.0631 ¢

Optimal ET sequence: 2f, 27e

Badness (Sintel): 1.64

Sixix

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Sixix (5-limit).

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

Sixix tempers out 3125/2916 and may be described as 25 & 32. It 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. Its ploidacot is gamma-pentacot.

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/2916

Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]

Optimal tunings:

WE: ~2 = 1198.9028 ¢, ~6/5 = 337.1334 ¢

error map: ⟨-1.097 +9.086 -13.503 +2.508]

CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.4588 ¢

error map: ⟨0.000 +10.751 -11.066 +5.762]

Optimal ET sequence: 7, 18d, 25, 32

Badness (Sintel): 4.02

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:

WE: ~2 = 1198.5480 ¢, ~6/5 = 337.1557 ¢
CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.6000 ¢

Optimal ET sequence: 7, 25e, 32

Badness (Sintel): 2.34

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:

WE: ~2 = 1197.7111 ¢, ~6/5 = 336.8391 ¢
CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.5336 ¢

Optimal ET sequence: 7, 25e, 32f

Badness (Sintel): 1.91

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:

WE: ~2 = 1197.7807 ¢, ~6/5 = 336.8884 ¢
CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.5279 ¢

Optimal ET sequence: 7, 25e, 32f

Badness (Sintel): 2.00

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-The '''archytas clan''' (or '''archy family''') [[tempering out|tempers out]] the [[64/63|Archytas' comma]], 64/63. This means a stack of two [[3/2]] fifths [[octave reduction|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]].
+The '''archytas clan''' (or '''archy family''') [[tempering out|tempers out]] the [[64/63|Archytas' comma]], 64/63. This means a stack of two [[3/2]] fifths [[octave reduction|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]], [[27edo]], or [[49edo]].
 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.
@@ Line 15: / Line 15: @@
 : sval mapping generators: ~2, ~3
-{{Mapping|legend=3| 1 1 0 4 | 0 1 0 -2 }}
+{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}
-: [[gencom]]: [2 3/2; 64/63]
+: [[gencom]]: [2 3; 64/63]
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.595
+* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}
-: [[error map]]: {{val| 0.000 +7.640 +11.984 }}
+: [[error map]]: {{val| -3.045 +2.522 +3.952 }}
-* [[POTE]]: ~2 = 1200.000, ~3/2 = 709.321
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}
-: error map: {{val| 0.000 +7.366 +12.532 }}
+: error map: {{val| 0.000 +7.435 +12.394 }}
 {{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
-[[Badness]] (Smith): 0.00464
+[[Badness]] (Sintel): 0.159
 Scales: [[archy5]], [[archy7]], [[archy12]]
@@ Line 34: / Line 34: @@
 ==== 7-limit extensions ====
 The second comma in the comma list defines which [[7-limit]] family member we are looking at:
-* [[#Schism|Schism]] adds 360/343, for an [[exotemperament]] well tuned around [[12edo]];
+* [[#Schism|Schism]] adds 360/343, for a tuning around [[12edo]];
-* Dominant adds [[36/35]], for a tuning flat of [[17edo|17c-edo]];
+* Dominant adds [[36/35]], for a tuning between [[12edo]] and [[17edo|17c-edo]];
 * [[#Quasisuper|Quasisuper]] adds [[2430/2401]], for a tuning between 17c-edo and [[22edo]];
 * [[#Superpyth|Superpyth]] adds [[245/243]], for a tuning between 22edo and [[27edo]];
@@ Line 50: / Line 50: @@
 * [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
 * ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
-* ''[[Suhajira]]'' (+243/242) → [[Neutral clan #Suhajira|Neutral clan]]
 * ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
 * [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
@@ Line 58: / Line 57: @@
 * ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
 * ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
-* ''[[Blacksmith]]'' (+28/27) → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
+* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
 * ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
@@ Line 64: / Line 63: @@
 ==== Subgroup extensions ====
-Omitting prime 5, archy can be extended to the 2.3.7.11 subgroup by identifying 11/8 as a diminished fourth (C–Gb). This is called supra, given right below.
+Omitting prime 5, archy can be extended to the 2.3.7.11 subgroup by identifying 11/8 as a diminished fourth (C–Gb). This is called supra, given right below. Discussed elsewhere is [[suhajira]] of the [[neutral clan #Suhajira|neutral clan]].
 === Supra ===
@@ Line 73: / Line 72: @@
 Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
-Gencom mapping: {{mapping| 1 1 0 4 7 | 0 1 0 -2 -6 }}
+Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}
-: gencom: [2 3/2; 64/63 99/98]
+: gencom: [2 3; 64/63 99/98]
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 708.456
+* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 707.192
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}
 {{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}
-Badness (Smith): 0.00933
+Badness (Sintel): 0.352
 Scales: [[supra7]], [[supra12]]
@@ Line 94: / Line 93: @@
 Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
-Gencom mapping: {{mapping| 1 1 0 4 7 9 | 0 1 0 -2 -6 -9 }}
+Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
-: gencom: [2 3/2; 64/63 78/77 99/98]
+: gencom: [2 3; 64/63 78/77 99/98]
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 707.344
+* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 706.137
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}
 {{Optimal ET sequence|legend=0| 12f, 17 }}
-Badness (Smith): 0.0111
+Badness (Sintel): 0.498
 Scales: [[supra7]], [[supra12]]
@@ Line 112: / Line 111: @@
 : ''For the 5-limit version, see [[Syntonic–diatonic 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#).
+Superpyth, virtually the canonical extension, adds [[245/243]] and [[1728/1715]] to the comma list and can be described as {{nowrap| 22 & 27 }}. ~5/4 is found at +9 generator steps, as an augmented second (C–D#). 49edo remains an obvious tuning choice.
 [[Subgroup]]: 2.3.5.7
@@ Line 119: / Line 118: @@
 {{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}
-{{Multival|legend=1| 1 9 -2 12 -6 -30 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.591
+* [[WE]]: ~2 = 1197.0549{{c}}, ~3/2 = 708.5478{{c}}
-: [[error map]]: {{val| 0.000 +7.636 +0.002 +11.993 }}
+: [[error map]]: {{val| -2.945 +3.648 -0.548 +2.298 }}
-* [[POTE]]: ~2 = 1200.000, ~3/2 = 710.291
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1193{{c}}
-: error map: {{val| 0.000 +8.336 +6.305 +10.592 }}
+: error map: {{val| 0.000 +8.164 +4.760 +10.935 }}
 {{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
-[[Badness]] (Smith): 0.032318
+[[Badness]] (Sintel): 0.818
 === 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 142: / Line 139: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 709.514
+* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 710.175
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}
 {{Optimal ET sequence|legend=0| 22, 27e, 49 }}
-Badness (Smith): 0.024976
+Badness (Sintel): 0.826
 ==== 13-limit ====
@@ Line 157: / Line 154: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 709.836
+* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 710.479
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}
 {{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
-Badness (Smith): 0.024673
+Badness (Sintel): 1.02
 ==== Thomas ====
@@ Line 172: / Line 169: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~16/13 = 354.759
+* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
-* POTE: ~2 = 1200.000, ~16/13 = 355.036
+* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}
 {{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}
-Badness (Smith): 0.049183
+Badness (Sintel): 2.03
 === Suprapyth ===
@@ Line 189: / Line 186: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 709.356
+* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 709.495
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}
 {{Optimal ET sequence|legend=0| 5, 17, 22 }}
-Badness (Smith): 0.032768
+Badness (Sintel): 1.08
 ==== 13-limit ====
@@ Line 204: / Line 201: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 708.702
+* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 708.703
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}
 {{Optimal ET sequence|legend=0| 5f, 17, 22 }}
-Badness (Smith): 0.036336
+Badness (Sintel): 1.50
 == 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 {{nowrap| 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 216: @@
 {{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}
-{{Multival|legend=1| 1 -13 -2 -23 -6 32 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3/2 = 708.769
+* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}
-: [[error map]]: {{val| 0.000 +6.814 -0.310 +13.636 }}
+: [[error map]]: {{val| -3.017 +1.486 -0.435 +6.190 }}
-* [[POTE]]: ~2 = 1200.000, ~3/2 = 708.238
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 708.3716{{c}}
-: error map: {{val| 0.000 +6.283 +6.586 +14.697 }}
+: error map: {{val| 0.000 +6.417 +4.855 +14.431 }}
 {{Optimal ET sequence|legend=1| 17c, 22, 61d }}
-[[Badness]] (Smith): 0.063794
+[[Badness]] (Sintel): 1.61
 === Quasisupra ===
@@ Line 242: / Line 237: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 708.713
+* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 708.205
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}
 {{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}
-Badness (Smith): 0.032203
+Badness (Sintel): 1.06
 ==== 13-limit ====
@@ Line 257: / Line 252: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 708.411
+* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 708.004
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}
-{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61df, 100bcdf }}
+{{Optimal ET sequence|legend=0| 17c, 22, 39d }}
-Badness (Smith): 0.030219
+Badness (Sintel): 1.25
 === Quasisoup ===
@@ Line 272: / Line 267: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 709.043
+* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 709.021
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}
 {{Optimal ET sequence|legend=0| 22 }}
-Badness (Smith): 0.083490
+Badness (Sintel): 2.76
 == 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).
+{{Main| 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).
 [[Subgroup]]: 2.3.5.7
@@ Line 287: / Line 284: @@
 {{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
-{{Multival|legend=1| 1 14 -2 20 -6 -44 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3/2 = 713.218
+* [[WE]]: ~2 = 1197.2673{{c}}, ~3/2 = 712.0258{{c}}
-: [[error map]]: {{val| 0.000 +11.263 -1.264 +4.738 }}
+: [[error map]]: {{val| -2.733 +7.338 -1.557 -3.808 }}
-* [[POTE]]: ~2 = 1200.000, ~3/2 = 713.651
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5430{{c}}
-: error map: {{val| 0.000 +11.696 +4.800 +3.872 }}
+: error map: {{val| 0.000 +11.588 +3.288 +4.088 }}
 {{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
-[[Badness]] (Smith): 0.108466
+[[Badness]] (Sintel): 2.74
 === 11-limit ===
@@ Line 308: / Line 303: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 713.282
+* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 713.395
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}
 {{Optimal ET sequence|legend=0| 5, 32, 37 }}
-Badness (Smith): 0.068238
+Badness (Sintel): 2.26
 ==== 13-limit ====
@@ Line 323: / Line 318: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 713.309
+* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 713.500
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}
 {{Optimal ET sequence|legend=0| 5, 32, 37 }}
-Badness (Smith): 0.049172
+Badness (Sintel): 2.03
 === Ultramarine ===
@@ Line 338: / Line 333: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 713.395
+* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 713.791
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}
 {{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}
-Badness (Smith): 0.078068
+Badness (Sintel): 2.58
 ==== 13-limit ====
@@ Line 353: / Line 348: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 713.708
+* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 713.811
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}
 {{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}
-Badness (Smith): 0.045653
+Badness (Sintel): 1.89
 == 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 {{nowrap| 27 & 32 }} temperament, mapping the ~5/4 to -18 fifths as a double diminished sixth (C–Abbb).
 [[Subgroup]]: 2.3.5.7
@@ Line 368: / Line 363: @@
 {{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
-{{Multival|legend=1| 1 -18 -2 -31 -6 46 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3/2 = 711.838
+* [[WE]]: ~2 = 1196.9257{{c}}, ~3/2 = 709.6211{{c}}
 : [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}
-* [[CWE]]: ~2 = 1200.000, ~3/2 = 711.543
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5429{{c}}
 : error map: {{val| 0.000 +9.588 +5.914 +8.088 }}
 {{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
-[[Badness]] (Smith): 0.132
+[[Badness]] (Sintel): 3.34
 == 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 (53dd val) can be used.
 [[Subgroup]]: 2.3.5.7
@@ Line 393: / Line 386: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3/2 = 702.270
+* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}
-: [[error map]]: {{val| 0.000 +0.315 -4.471 +26.635 }}
+: [[error map]]: {{val| -2.640 -4.583 -4.896 +20.588 }}
-* [[POTE]]: ~2 = 1200.000, ~3/2 = 701.556
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7376{{c}}
-: error map: {{val| 0.000 -0.399 +1.237 +28.062 }}
+: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}
-{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
 {{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
-[[Badness]] (Smith): 0.056648
+[[Badness]] (Sintel): 1.43
 === 11-limit ===
@@ Line 412: / Line 403: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 703.383
+* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 702.136
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}
 {{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}
-Badness (Smith): 0.037482
+Badness (Sintel): 1.24
 == Beatles ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''
+Beatles tempers out 686/675, which may also be characterized by saying it tempers out [[2401/2400]]. It may be described as the {{nowrap| 10 & 17c }} temperament. It splits the fifth into two neutral-third generators of 49/40~60/49; its [[ploidacot]] is dicot. 5/4 may be found at -9 generator steps, as a semidiminished fourth (C–Fd). 27edo is an obvious tuning, though 17c-edo and 37edo are among the possibilities.
+Beatles extends easily to the no-11 13-limit, as the generator can be interpreted as ~16/13, tempering out 91/90, 169/168, and 196/195.
 [[Subgroup]]: 2.3.5.7
@@ Line 427: / Line 422: @@
 {{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
-{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~49/40 = 356.634
+* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}
-: [[error map]]: {{val| 0.000 +11.312 +3.984 +4.640 }}
+: [[error map]]: {{val| -3.376 +4.475 +2.682 -1.940 }}
-* [[POTE]]: ~2 = 1200.000, ~49/40 = 355.904
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}
-: error map: {{val| 0.000 +9.853 +10.549 +7.558 }}
+: error map: {{val| 0.000 +10.209 +8.949 +6.847 }}
 {{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
-[[Badness]] (Smith): 0.045872
+[[Badness]] (Sintel): 1.16
 ; Music
-* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 ''Beatles Improv''] by [[Herman Miller]]
+* [https://web.archive.org/web/20201127013829/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 ''Beatles Improv''] by [[Herman Miller]]
 === 11-limit ===
@@ Line 451: / Line 444: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~49/40 = 356.719
+* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
-* POTE: ~2 = 1200.000, ~49/40 = 356.140
+* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}
 {{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}
-Badness (Smith): 0.045639
+Badness (Sintel): 1.51
 ==== 13-limit ====
@@ Line 466: / Line 459: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~16/13 = 356.722
+* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
-* POTE: ~2 = 1200.000, ~16/13 = 356.229
+* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}
 {{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}
-Badness (Smith): 0.030161
+Badness (Sintel): 1.25
 === Ringo ===
@@ Line 481: / Line 474: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/9 = 355.992
+* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
-* POTE: ~2 = 1200.000, ~11/9 = 355.419
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}
 {{Optimal ET sequence|legend=0| 10, 17c, 27e }}
-Badness (Smith): 0.032863
+Badness (Sintel): 1.09
 ==== 13-limit ====
@@ Line 496: / Line 489: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/9 = 356.004
+* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
-* POTE: ~2 = 1200.000, ~11/9 = 355.456
+* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}
 {{Optimal ET sequence|legend=0| 10, 17c, 27e }}
-Badness (Smith): 0.022619
+Badness (Sintel): 0.935
 === Beetle ===
@@ Line 508: / Line 501: @@
 Comma list: 55/54, 64/63, 686/675
-Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15}}
+Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~49/40 = 356.694
+* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
-* POTE: ~2 = 1200.000, ~49/40 = 356.710
+* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}
 {{Optimal ET sequence|legend=0| 10, 27, 37 }}
-Badness (Smith): 0.058084
+Badness (Sintel): 1.92
 ==== 13-limit ====
@@ Line 526: / Line 519: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~16/13 = 356.700
+* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
-* POTE: ~2 = 1200.000, ~16/13 = 356.701
+* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}
 {{Optimal ET sequence|legend=0| 10, 27, 37 }}
-Badness (Smith): 0.033971
+Badness (Sintel): 1.40
 == Progress ==
+{{Distinguish| Progression }}
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
+Progress tempers out 392/375 and may be described as {{nowrap| 15 & 17c }}. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 32c-edo gives an obvious tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 543: / Line 539: @@
 : mapping generators: ~2, ~10/7
-{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~10/7 = 638.782
+* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}
-: [[error map]]: {{val| 0.000 +14.391 +19.777 -1.518 }}
+: [[error map]]: {{val| -4.862 +3.925 +12.908 -9.759 }}
-* [[POTE]]: ~2 = 1200.000, ~10/7 = 637.878
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 638.0791{{c}}
-: error map: {{val| 0.000 +11.679 +24.297 +3.907 }}
+: error map: {{val| 0.000 +12.282 +23.291 +2.700 }}
 {{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
-[[Badness]] (Smith): 0.066400
+[[Badness]] (Sintel): 1.68
 === 11-limit ===
@@ Line 564: / Line 558: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~10/7 = 638.846
+* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
-* POTE: ~2 = 1200.000, ~10/7 = 637.915
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}
 {{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}
-Badness (Smith): 0.031036
+Badness (Sintel): 1.03
 ==== 13-limit ====
@@ Line 579: / Line 573: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~10/7 = 637.871
+* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
-* POTE: ~2 = 1200.000, ~10/7 = 637.635
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}
 {{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
-Badness (Smith): 0.026214
+Badness (Sintel): 1.08
 ==== Progressive ====
@@ Line 594: / Line 588: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~10/7 = 637.797
+* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
-* POTE: ~2 = 1200.000, ~10/7 = 636.761
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}
 {{Optimal ET sequence|legend=0| 2f, 15f, 17c }}
-Badness (Smith): 0.032721
+Badness (Sintel): 1.35
 == Fervor ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
+Fervor tempers out 9704/9375 and may be described as {{nowrap| 25 & 27 }}. It splits the 6th harmonic into five generators of ~10/7; its ploidacot is beta-pentacot. 27edo is about as accurate as it can be tuned.
 [[Subgroup]]: 2.3.5.7
@@ Line 608: / Line 604: @@
 [[Comma list]]: 64/63, 9604/9375
-{{Mapping|legend=1| 1 4 -2 -2 | 0 -5 9 10 }}
+{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
-: mapping generators: ~2, ~7/5
+: mapping generators: ~2, ~10/7
-{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~7/5 = 577.353
+* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}
-: [[error map]]: {{val| 0.000 +11.278 +9.867 +4.709 }}
+: [[error map]]: {{val| -3.726 +3.230 +4.980 -1.550 }}
-* [[POTE]]: ~2 = 1200.000, ~7/5 = 577.776
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 622.3179{{c}}
-: error map: {{val| 0.000 +9.163 +13.673 +8.937 }}
+: error map: {{val| 0.000 +9.634 +12.826 +7.996 }}
 {{Optimal ET sequence|legend=1| 2, 25, 27 }}
-[[Badness]] (Smith): 0.108455
+[[Badness]] (Sintel): 2.74
 === 11-limit ===
@@ Line 629: / Line 623: @@
 Comma list: 56/55, 64/63, 1350/1331
-Mapping: {{mapping| 1 4 -2 -2 3 | 0 -5 9 10 1 }}
+Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~7/5 = 577.296
+* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
-* POTE: ~2 = 1200.000, ~7/5 = 577.850
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}
 {{Optimal ET sequence|legend=0| 2, 25e, 27e }}
-Badness (Smith): 0.052054
+Badness (Sintel): 1.72
 === 13-limit ===
@@ Line 644: / Line 638: @@
 Comma list: 56/55, 64/63, 78/77, 507/500
-Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}
+Mapping: {{mapping| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~7/5 = 577.374
+* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
-* POTE: ~2 = 1200.000, ~7/5 = 578.060
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}
 {{Optimal ET sequence|legend=0| 2f, 27e }}
-Badness (Smith): 0.039705
+Badness (Sintel): 1.64
 == Sixix ==
@@ Line 658: / Line 652: @@
 {{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 tempers out 3125/2916 and may be described as {{nowrap| 25 & 32 }}. It 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. Its ploidacot is gamma-pentacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 665: / Line 659: @@
 {{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}
-{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~6/5 = 337.519
+* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}
-: [[error map]]: {{val| 0.000 +10.449 -11.429 +6.366 }}
+: [[error map]]: {{val| -1.097 +9.086 -13.503 +2.508 }}
-* [[POTE]]: ~2 = 1200.000, ~6/5 = 337.442
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 337.4588{{c}}
-: error map: {{val| 0.000 +10.835 -10.965 +5.594 }}
+: error map: {{val| 0.000 +10.751 -11.066 +5.762 }}
 {{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
-[[Badness]] (Smith): 0.158903
+[[Badness]] (Sintel): 4.02
 === 11-limit ===
@@ Line 686: / Line 678: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~6/5 = 337.749
+* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
-* POTE: ~2 = 1200.000, ~6/5 = 337.564
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}
 {{Optimal ET sequence|legend=0| 7, 25e, 32 }}
-Badness (Smith): 0.070799
+Badness (Sintel): 2.34
 === 13-limit ===
@@ Line 701: / Line 693: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~6/5 = 337.793
+* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
-* POTE: ~2 = 1200.000, ~6/5 = 337.483
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}
 {{Optimal ET sequence|legend=0| 7, 25e, 32f }}
-Badness (Smith): 0.046206
+Badness (Sintel): 1.91
 === 17-limit ===
@@ Line 716: / Line 708: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~6/5 = 337.629
+* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
-* POTE: ~2 = 1200.000, ~6/5 = 337.513
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}
 {{Optimal ET sequence|legend=0| 7, 25e, 32f }}
-Badness (Smith): 0.039224
+Badness (Sintel): 2.00
 [[Category:Archytas clan| ]] <!-- main article -->
 [[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Rank 2]]

Archytas clan: Difference between revisions

Latest revision as of 12:52, 22 July 2025

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Ringo

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