Archytas clan: Difference between revisions

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.

{{Main| Superpyth }}

[[Subgroup]]: 2.3.7

[[Comma list]]: 64/63

{{Mapping|legend=3| 1 0 6 | 0 1 -2 }}

{{Mapping|legend=5| 1 0 0 6 | 0 1 0 -2 }}

: mapping generators: ~2, ~3

* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}

: [[error map]]: {{val| -3.045 +2.522 +3.952 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}

: error map: {{val| 0.000 +7.435 +12.394 }}

{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}

[[Badness]] (Sintel): 0.159

Scales: [[archy5]], [[archy7]], [[archy12]]

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

* [[#Superpyth|Superpyth]] adds [[245/243]], for a tuning between 22edo and [[27edo]];

* Mother adds [[16/15]], for an exotemperament well tuned around [[5edo]].  

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:  

* ''[[Phlegyas]]'' (+10/9) → [[Very low accuracy temperaments #Phlegyas|Very low accuracy temperaments]]

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_2|#Subgroup extensions]]. Discussed elsewhere is [[suhajira]] of the [[rastmic clan #Suhajira|rastmic clan]].

: ''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 {{nowrap| [[22edo|22]] [[&]] [[27edo|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.  

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 64/63, 245/243

[[Optimal tuning]]s:

: [[error map]]: {{val| -2.945 +3.648 -0.548 +2.298 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1193{{c}}

: error map: {{val| 0.000 +8.164 +4.760 +10.935 }}

{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}

[[Badness]] (Sintel): 0.818

Subgroup: 2.3.5.7.11

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

{{Mapping|legend=0| 1 0 -12 6 -22 | 0 1 9 -2 16 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}

{{Optimal ET sequence|legend=0| 22, 27e, 49 }}

Badness (Sintel): 0.826

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 100/99

{{Mapping|legend=0| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}

Optimal tunings:  

* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}

{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}

Badness (Sintel): 1.02

Subgroup: 2.3.5.7.11.13

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

{{Mapping|legend=0| 1 0 -12 6 -22 26 | 0 1 9 -2 16 -14 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.2878{{c}}

{{Optimal ET sequence|legend=0| 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|legend=0| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}

* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}

{{Optimal ET sequence|legend=0| 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|legend=0| 1 0 -12 6 13 | 0 1 9 -2 -6 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}

{{Optimal ET sequence|legend=0| 5, 17, 22 }}

Badness (Sintel): 1.08

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 65/63, 99/98

{{Mapping|legend=0| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}

{{Optimal ET sequence|legend=0| 5f, 17, 22 }}

Badness (Sintel): 1.50

{{Main|Quasisuper}}

Quasisuper can be described as {{nowrap| 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|legend=1| 1 0 23 6 | 0 1 -13 -2 }}

* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}

: [[error map]]: {{val| -3.017 +1.486 -0.435 +6.190 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 708.3716{{c}}

: error map: {{val| 0.000 +6.417 +4.855 +14.431 }}

{{Optimal ET sequence|legend=1| 17c, 22, 61d }}

[[Badness]] (Sintel): 1.61

=== Quasisupra ===

Subgroup: 2.3.5.7.11

Comma list: 64/63, 99/98, 121/120

{{Mapping|legend=0| 1 0 23 6 13 | 0 1 -13 -2 -6 }}

* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}

{{Optimal ET sequence|legend=0| 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|legend=0| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}

* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}

{{Optimal ET sequence|legend=0| 17c, 22, 39d }}

Badness (Sintel): 1.25

=== Quasisoup ===

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2430/2401

{{Mapping|legend=0| 1 0 23 6 -22 | 0 1 -13 -2 16 }}

* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}

Badness (Sintel): 2.76

: ''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|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|legend=1| 1 0 -20 6 | 0 1 14 -2 }}

[[Optimal tuning]]s:  

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5430{{c}}

: error map: {{val| 0.000 +11.588 +3.288 +4.088 }}

{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}

[[Badness]] (Sintel): 2.74

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2401/2376

{{Mapping|legend=0| 1 0 -20 6 21 | 0 1 14 -2 -11 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}

{{Optimal ET sequence|legend=0| 5, 32, 37 }}

Badness (Sintel): 2.26

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 1573/1568

{{Mapping|legend=0| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}

{{Optimal ET sequence|legend=0| 5, 32, 37 }}

 

Badness (Sintel): 2.03

 

=== Ultramarine ===

Subgroup: 2.3.5.7.11

 

Comma list: 64/63, 100/99, 3773/3645

 

{{Mapping|legend=0| 1 0 -20 6 -38 | 0 1 14 -2 26 }}

 

Optimal tunings:  

* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}

 

 

 

==== 13-limit ====

 

Comma list: 64/63, 91/90, 100/99, 847/845

 

{{Mapping|legend=0| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}

 

Optimal tunings:  

* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}

 

 

Badness (Sintel): 1.89

 

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–A𝄫♭).

 

[[Subgroup]]: 2.3.5.7

 

[[Comma list]]: 64/63, 33614/32805

 

{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}

 

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1196.9257{{c}}, ~3/2 = 709.6211{{c}}

: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5429{{c}}

: error map: {{val| 0.000 +9.588 +5.914 +8.088 }}

 

 

[[Badness]] (Sintel): 3.34

 

 

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|legend=1| 1 0 15 6 | 0 1 -8 -2 }}

 

* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}

: [[error map]]: {{val| -2.640 -4.583 -4.896 +20.588 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7376{{c}}

: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}

 

 

[[Badness]] (Sintel): 1.43

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 45/44, 64/63, 99/98

 

{{Mapping|legend=0| 1 0 15 6 13 | 0 1 -8 -2 -6 }}

 

Optimal tunings:  

* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}

 

 

Badness (Sintel): 1.24

 

: ''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

 

[[Comma list]]: 64/63, 686/675

 

{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}

: [[error map]]: {{val| -3.376 +4.475 +2.682 -1.940 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}

: error map: {{val| 0.000 +10.209 +8.949 +6.847 }}

 

 

[[Badness]] (Sintel): 1.16

 

; Music

* [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 ===

Subgroup: 2.3.5.7.11

 

Comma list: 64/63, 100/99, 686/675

 

{{Mapping|legend=0| 1 1 5 4 10 | 0 2 -9 -4 -22 }}

 

Optimal tunings:  

* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}

 

 

Badness (Sintel): 1.51

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

Comma list: 64/63, 91/90, 100/99, 169/168

 

{{Mapping|legend=0| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}

 

Optimal tunings:  

* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}

 

 

Badness (Sintel): 1.25

 

Subgroup: 2.3.5.7.11

 

Comma list: 56/55, 64/63, 540/539

 

{{Mapping|legend=0| 1 1 5 4 2 | 0 2 -9 -4 5 }}

 

Optimal tunings:  

* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}

 

 

Badness (Sintel): 1.09

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

Comma list: 56/55, 64/63, 78/77, 91/90

 

{{Mapping|legend=0| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}

 

Optimal tunings:  

* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}

 

{{Optimal ET sequence|legend=0| 10, 17c, 27e }}

 

Badness (Sintel): 0.935

 

Subgroup: 2.3.5.7.11

 

Comma list: 55/54, 64/63, 686/675

 

{{Mapping|legend=0| 1 1 5 4 -1 | 0 2 -9 -4 15 }}

 

Optimal tunings:  

* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}

 

 

Badness (Sintel): 1.92

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

Comma list: 55/54, 64/63, 91/90, 169/168

 

{{Mapping|legend=0| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}

 

Optimal tunings:  

* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}

 

 

Badness (Sintel): 1.40

 

: ''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 [[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

 

[[Comma list]]: 64/63, 392/375

 

{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}

: [[error map]]: {{val| -4.862 +3.925 +12.908 -9.759 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 638.0791{{c}}

: error map: {{val| 0.000 +12.282 +23.291 +2.700 }}

 

 

[[Badness]] (Sintel): 1.68

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 56/55, 64/63, 77/75

 

{{Mapping|legend=0| 1 0 5 6 4 | 0 3 -5 -6 -1 }}

 

Optimal tunings:  

* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}

 

 

Badness (Sintel): 1.03

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

Comma list: 56/55, 64/63, 66/65, 77/75

 

{{Mapping|legend=0| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}

 

Optimal tunings:  

* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}

 

 

Badness (Sintel): 1.08

 

Subgroup: 2.3.5.7.11.13

 

Comma list: 26/25, 56/55, 64/63, 77/75

 

{{Mapping|legend=0| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}

 

Optimal tunings:  

* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}

 

{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}

 

 

: ''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 decanonized 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]].

 

 

[[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}}

 

 

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 {{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

 

[[Comma list]]: 64/63, 9604/9375

 

{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}

: mapping generators: ~2, ~10/7

 

* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}

: [[error map]]: {{val| -3.726 +3.230 +4.980 -1.550 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 622.3179{{c}}

: error map: {{val| 0.000 +9.634 +12.826 +7.996 }}

 

{{Optimal ET sequence|legend=1| 2, 25, 27 }}

 

[[Badness]] (Sintel): 2.74

 

=== 11-limit ===

 

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

 

 

* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}

 

{{Optimal ET sequence|legend=0| 2, 25e, 27e }}

 

Badness (Sintel): 1.72

 

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

 

 

{{Mapping|legend=0| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}

 

Optimal tunings:  

* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}

 

{{Optimal ET sequence|legend=0| 2f, 27e }}

 

Badness (Sintel): 1.64

 

: ''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 {{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. An important val not in the [[optimal ET sequence]] is 39c, that is, [[39edo]] using the second-best approximation of [[5/4]] (the 369-cent one inherited from [[13edo]]). It is also supported by [[57edo|57b-edo]], which uses the sharper of [[57edo]]'s two diatonic fifths.

 

[[Subgroup]]: 2.3.5.7

 

[[Comma list]]: 64/63, 3125/2916

 

{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}

: [[error map]]: {{val| -1.097 +9.086 -13.503 +2.508 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 337.4588{{c}}

: error map: {{val| 0.000 +10.751 -11.066 +5.762 }}

 

 

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 55/54, 64/63, 125/121

 

{{Mapping|legend=0| 1 3 4 0 6 | 0 -5 -6 10 -9 }}

 

* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}

 

 

Badness (Sintel): 2.34

 

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

 

Comma list: 40/39, 55/54, 64/63, 125/121

 

{{Mapping|legend=0| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}

 

* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}

 

{{Optimal ET sequence|legend=0| 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|legend=0| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}

 

Optimal tunings:  

* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}

 

{{Optimal ET sequence|legend=0| 7, 25e, 32f }}

 

Badness (Sintel): 2.00

 

 

 

{{Mapping|legend=2| 1 0 6 13 | 0 1 -2 -6 }}

 

{{Mapping|legend=4| 1 0 0 6 13 | 0 1 0 -2 -6 }}

 

* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}

 

 

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

 

{{Mapping|legend=2| 1 0 6 13 18 | 0 1 -2 -6 -9 }}

 

{{Mapping|legend=4| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}