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These all use the same generators as archy.  
These all use the same generators as archy.  


[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split 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. [[15625/15552]] gives catalan, splitting the twelfth in six. Finally, [[28/27]] gives blackwood and [[17496/16807]] gives [[gothic]], reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.  
[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a semi-octave period. [[392/375]] gives progress. [[6125/5832]] gives kangaroo. Those split 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. [[15625/15552]] gives catalan, splitting the twelfth in six. Finally, [[28/27]] gives blackwood and [[17496/16807]] gives [[gothic]], reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.  


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
Line 503: Line 503:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
: ''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.  
Progress tempers out [[392/375]] and may be described as {{nowrap| 15 & 17c }}. It splits the [[3/1|perfect twelfth]] into three generators of [[~]][[10/7]]; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]], and the 13-limit extension further equates it to [[13/9]]. [[32edo|32c-edo]] gives an obvious tuning.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 510: Line 510:


{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
: mapping generators: ~2, ~10/7
: mapping generators: ~2, ~10/7


Line 567: Line 566:


Badness (Sintel): 1.35
Badness (Sintel): 1.35
== Kangaroo ==
: ''For 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
Kangaroo splits the [[3/1|perfect twelfth]] into three [[35/24]][[~]][[40/27]] generators; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]]. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat.
Originally the canonical extension of 5-limit kangaroo was [[trienstonic clan #Parakangaroo|parakangaroo]], but it was decanonicalized in 2024 in favor of this extension because parakangaroo is an [[exotemperament]] that only supports the [[patent val]] of [[15edo]], while this extension is a bit more practical since it is at least low-medium accuracy and supports the [[patent val]]s of 15edo, [[17edo]] and [[32edo]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 64/63, 6125/5832
{{Mapping|legend=1| 1 0 -3 6 | 0 3 10 -6 }}
: mapping generators: ~2, ~35/24
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.1767{{c}}, ~35/24 = 637.2914{{c}}
: [[error map]]: {{val| -1.823 +9.919 -7.930 -3.514 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 638.1977{{c}}
: error map: {{val| 0.000 +12.638 -4.336 +1.988 }}
{{Optimal ET sequence|legend=1| 15, 32, 47b }}
[[Badness]] (Sintel): 2.91
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 245/242
{{Mapping|legend=0| 1 0 -3 6 4 | 0 3 10 -6 -1 }}
Optimal tunings:
* WE: ~2 = 1197.8893{{c}}, ~16/11 = 637.1922{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 638.2605{{c}}
{{Optimal ET sequence|legend=0| 15, 32, 47b }}
Badness (Sintel): 1.65


== Fervor ==
== Fervor ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #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.  
Fervor tempers out [[9604/9375]] and may be described as {{nowrap| 25 & 27 }}. It splits the [[6/1|6th harmonic]] into five generators of [[~]][[10/7]]; its ploidacot is beta-pentacot. The 11-limit extension simply equates the generator to [[16/11]]. [[27edo]] is about as accurate as it can be tuned.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 578: Line 616:


{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
: mapping generators: ~2, ~10/7
: mapping generators: ~2, ~10/7


Latest revision as of 17:56, 15 October 2026

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

Subgroup-val mapping: [⟨1 0 6], ⟨0 1 -2]]

Gencom mapping: [⟨1 0 0 6], ⟨0 1 0 -2]]

mapping generators: ~2, ~3

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:

These all use the same generators as archy.

25/24 gives dichotic. 686/675 gives beatles. Those split the fifth in two. 8748/8575 gives immunized, splitting the twelfth in two. 50/49 gives pajara with a semi-octave period. 392/375 gives progress. 6125/5832 gives kangaroo. Those split 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. 15625/15552 gives catalan, splitting the twelfth in six. Finally, 28/27 gives blackwood and 17496/16807 gives gothic, reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.

Temperaments discussed elsewhere are:

The rest are considered below.

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–G♭). This is called supra, given in #Subgroup extensions. Discussed elsewhere is suhajira of the rastmic clan.

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♯). In the 11-limit it finds the ~11/8 at +16 generator steps, as a double-augmented second (C–D𝄪). 49edo remains an obvious tuning choice in either case.

Extending superpyth to the 13-limit is more diffcult. Tridecimal superpyth finds the ~13/8 at +13 generator steps, as a double-augmented fourth (C–F𝄪), for which 27edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning. The other extension, called uberpyth, is more flexible with its tunings, but unfortunately tends to tune the 13 very sharp.

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

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

Uberpyth

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 144/143, 245/243

Mapping: [⟨1 0 -12 6 -22 26], ⟨0 1 9 -2 16 -14]]

Optimal tunings:

  • WE: ~2 = 1196.6666 ¢, ~3/2 = 708.3602 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.2878 ¢

Optimal ET sequence: 22f, 27e, 49f, 125bcddeeeff, 174bbcdddeeeeffff

Badness (Sintel): 1.11

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–G♭), and finds the ~13/8 at the diminished seventh (C–B𝄫).

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

Main article: Quasisuper

Quasisuper can be described as 17c & 22, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–G𝄫). The 11-limit version, quasisupra, can be viewed as an extension of the excellent 2.3.7.11-subgroup 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

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

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

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–C𝄪).

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–A𝄫♭).

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–F♭) 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

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. The 11-limit extension simply equates the generator to 16/11, and the 13-limit extension further equates it to 13/9. 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

Kangaroo

For 5-limit version, see Miscellaneous 5-limit temperaments #Kangaroo.

Kangaroo splits the perfect twelfth into three 35/24~40/27 generators; its ploidacot is alpha-tricot. The 11-limit extension simply equates the generator to 16/11. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~5/4 will be too flat.

Originally the canonical extension of 5-limit kangaroo was parakangaroo, but it was decanonicalized in 2024 in favor of this extension because parakangaroo is an exotemperament that only supports the patent val of 15edo, while this extension is a bit more practical since it is at least low-medium accuracy and supports the patent vals of 15edo, 17edo and 32edo.

Subgroup: 2.3.5.7

Comma list: 64/63, 6125/5832

Mapping: [⟨1 0 -3 6], ⟨0 3 10 -6]]

mapping generators: ~2, ~35/24

Optimal tunings:

  • WE: ~2 = 1198.1767 ¢, ~35/24 = 637.2914 ¢
error map: ⟨-1.823 +9.919 -7.930 -3.514]
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 638.1977 ¢
error map: ⟨0.000 +12.638 -4.336 +1.988]

Optimal ET sequence: 15, 32, 47b

Badness (Sintel): 2.91

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 245/242

Mapping: [⟨1 0 -3 6 4], ⟨0 3 10 -6 -1]]

Optimal tunings:

  • WE: ~2 = 1197.8893 ¢, ~16/11 = 637.1922 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 638.2605 ¢

Optimal ET sequence: 15, 32, 47b

Badness (Sintel): 1.65

Fervor

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

Fervor tempers out 9604/9375 and may be described as 25 & 27. It splits the 6th harmonic into five generators of ~10/7; its ploidacot is beta-pentacot. The 11-limit extension simply equates the generator to 16/11. 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

Subgroup extensions

Supra (2.3.7.11)

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Subgroup-val mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]

Gencom mapping: [⟨1 0 0 6 13], ⟨0 1 0 -2 -6]]

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

This extension maps 13/11 to the minor third (C–E♭), 12/11~14/13 to the augmented unison (C–C♯), and 13/12 to the diminished third (C–E𝄫).

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 99/98

Subgroup-val 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]]

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

Oceanfront (2.3.7.13/5)

This is the restriction of ultrapyth to the 2.3.7.13/5 subgroup, mapping 13/10 to the major third.

Subgroup: 2.3.7.13/5

Comma list: 64/63 ([6 -2 -1 0⟩), 91/90 ([-1 -2 1 1⟩)

Subgroup-val mapping: [⟨1 0 6 -5], ⟨0 1 -2 4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1197.2656 ¢, ~3/2 = 710.1993 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.7413 ¢

Optimal ET sequence: 5, 17, 22, 27, 32, 59

RMS error: 2.063 cents

Scales: Oceanfront scales

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