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The '''archytas clan''' tempers out the [[64/63|Archytas comma]], 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to [[81/80]] in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of [[22edo]]. Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.  
{{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]], [[27edo]], or [[49edo]].  


Discussed under subgroup temperaments is the 2.3.7 [[Subgroup temperaments #Archy|archy]]. Under their respective 5-limit families are [[Diaschismic family #Pajara|pajara]], [[Meantone family #Dominant|dominant]], [[Augmented family #Augene|augene]], [[porcupine family|porcupine]], and [[Tetracot family #Modus|modus]]. The rest are considered below.
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.  


= Mother =
== Archy ==
{{see also| Father family }}
{{Main| Superpyth }}


Commas: 16/15, 21/20
[[Subgroup]]: 2.3.7


[[POTE generator]]: 721.569
[[Comma list]]: 64/63


Map: [<1 0 4 6|, <0 1 -1 -2|]
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}


Wedgie: <<1 -1 -2 -4 -6 -2||
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}


{{EDOs|legend=1| 5, 148, 153 }}
: mapping generators: ~2, ~3


Badness: 0.0242
[[Optimal tuning]]s:  
* [[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 }}


= Blacksmith =
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
== 5-limit (blackwood) ==
Comma: 256/243


[[POTE generator]]: 399.594
[[Badness]] (Sintel): 0.159


Map: [<5 8 0|, <0 0 1|]
Scales: [[archy5]], [[archy7]], [[archy12]]


{{EDOs|legend=1| 5, 10, 15}}
=== Overview to extensions ===
==== 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 a tuning around [[12edo]];
* 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]];
* [[#Quasiultra|Quasiultra]] adds 33614/32805, for a tuning between 27edo and [[32edo]];
* [[#Ultrapyth|Ultrapyth]] adds 6860/6561, for a tuning sharp of 32edo;
* Mother adds [[16/15]], for an exotemperament well tuned around [[5edo]].


Badness: 0.0638
These all use the same generators as archy.  


== 7-limit ==
[[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.
Commas: 28/27, 49/48


[[POTE_tuning|POTE generator]]: ~5/4 = 392.767
Temperaments discussed elsewhere are:
* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Augene]] (+126/125) → [[Augmented family #Augene|Augmented family]]
* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* ''[[Modus]]'' (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]


Map: [<5 8 0 14|, <0 0 1 0|]
Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.


Wedgie: <<0 5 0 8 0 -14||
==== 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 #Suhajira|neutral clan]].


{{EDOs|legend=1| 5, 10, 15, 40b, 55b}}
=== Supra ===
Subgroup: 2.3.7.11


Badness: 0.0256
Comma list: 64/63, 99/98


== 11-limit ==
Subgroup-val mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
Commas: 28/27, 49/48, 55/54


POTE generator: ~5/4 = 394.948
Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}


Map: [<5 8 0 14 29|, <0 0 1 0 -1|]
Optimal tunings:  
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}


{{EDOs|legend=1| 5, 10, 15, 40be, 55be, 70bde, 85bcde}}
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}


Badness: 0.0246
Badness (Sintel): 0.352


=== 13-limit ===
Scales: [[supra7]], [[supra12]]
Commas: 28/27, 40/39, 49/48, 55/54
 
==== Supraphon ====
Subgroup: 2.3.7.11.13
 
Comma list: 64/63, 78/77, 99/98
 
Subgroup-val mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
 
Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
 
Optimal tunings:
* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 17 }}
 
Badness (Sintel): 0.498
 
Scales: [[supra7]], [[supra12]]
 
== Superpyth ==
{{Main| 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 {{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
 
[[Comma list]]: 64/63, 245/243
 
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.0549{{c}}, ~3/2 = 708.5478{{c}}
: [[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
 
=== 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: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}
 
Optimal tunings:
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}
 
Optimal tunings:
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}
 
Optimal tunings:
* 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–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: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}
 
Optimal tunings:
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}
 
Optimal tunings:
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}
 
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}
 
Badness (Sintel): 1.50
 
== Quasisuper ==
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
 
[[Comma list]]: 64/63, 2430/2401
 
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}
 
[[Optimal tuning]]s:
* [[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 ===
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


POTE generator: ~5/4 = 391.0367
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}


Map: [<5 8 0 14 29 7|, <0 0 1 0 -1 1|]
Optimal tunings:  
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


{{EDOs|legend=1| 5, 10, 15, 25e, 40bef}}
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}


Badness: 0.0205
Badness (Sintel): 1.06


== Farrier ==
==== 13-limit ====
Commas: 28/27, 49/48, 77/75
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 398.070
Comma list: 64/63, 78/77, 91/90, 121/120


Map: [<5 8 0 14 -6|, <0 0 1 0 2|]
Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}


{{EDOs|legend=1| 5e, 15 }}
Optimal tunings:
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


Badness: 0.0292
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


=== 13-limit ===
Badness (Sintel): 1.25
Commas: 28/27, 40/39, 49/48, 66/65


POTE generator: ~5/4 = 396.812
=== Quasisoup ===
Subgroup: 2.3.5.7.11


Map: [<5 8 0 14 -6 7|, <0 0 1 0 2 1|]
Comma list: 55/54, 64/63, 2430/2401


{{EDOs|legend=1| 5e, 10e, 15 }}
Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}


Badness: 0.0223
Optimal tunings:  
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


== Ferrum ==
{{Optimal ET sequence|legend=0| 22 }}
Commas: 28/27, 35/33, 49/48


POTE generator: ~5/4 = 374.763
Badness (Sintel): 2.76


Map: [<5 8 0 14 6|, <0 0 1 0 1|]
== Ultrapyth ==
{{Main| Ultrapyth }}


{{EDOs|legend=1| 10 }}
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).


Badness: 0.0309
[[Subgroup]]: 2.3.5.7


[[File:blacksmith10.jpg|alt=blacksmith10.jpg|blacksmith10.jpg]]
[[Comma list]]: 64/63, 6860/6561


= Superpyth =
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
{{main| Superpyth }}


Commas: 64/63, 245/243
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.2673{{c}}, ~3/2 = 712.0258{{c}}
: [[error map]]: {{val| -2.733 +7.338 -1.557 -3.808 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5430{{c}}
: error map: {{val| 0.000 +11.588 +3.288 +4.088 }}


[[POTE generator]]: 710.291
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}


Map: [<1 0 -12 6|, <0 1 9 -2|]
[[Badness]] (Sintel): 2.74


Wedgie: <<1 9 -2 12 -6 -30||
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{EDOs|legend=1| 5, 17, 22, 27, 49 }}
Comma list: 55/54, 64/63, 2401/2376


Badness: 0.0323
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}


== 11-limit ==
Optimal tunings:
Commas: 64/63, 100/99, 245/243
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


[[POTE_tuning|POTE generator]]: 710.175
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]
Badness (Sintel): 2.26


{{EDOs|legend=1| 22, 49 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0250
Comma list: 55/54, 64/63, 91/90, 1573/1568


=== 13-limit ===
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
Commas: 64/63, 78/77, 91/90, 100/99


POTE generator: ~3/2 = 710.479
Optimal tunings:  
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|]
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


{{EDOs|legend=1| 22, 27e, 49, 76bcde }}
Badness (Sintel): 2.03


Badness: 0.0247
=== Ultramarine ===
Subgroup: 2.3.5.7.11


== Suprapyth ==
Comma list: 64/63, 100/99, 3773/3645
Commas: 55/54, 64/63, 99/98


POTE generator: ~3/2 = 709.495
Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}


Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]
Optimal tunings:  
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


{{EDOs|legend=1| 5, 7, 12, 17, 22 }}
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}


Badness: 0.0328
Badness (Sintel): 2.58


=== 13-limit ===
==== 13-limit ====
Commas: 55/54, 64/63, 65/63, 364/363
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 708.703
Comma list: 64/63, 91/90, 100/99, 847/845


Map: [<1 0 -12 6 13 18|, <0 1 9 -2 -6 -9|]
Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}


{{EDOs|legend=1| 17, 22, 83cdf }}
Optimal tunings:
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}


Badness: 0.0363
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}


= Beatles =
Badness (Sintel): 1.89
== 5-limit ==
Comma: 524288/492075


POTE generator: ~512/405 = 355.930
== Quasiultra ==
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).  


Map: [<1 1 5|,<0 2 -9|]
[[Subgroup]]: 2.3.5.7


{{EDOs|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}
[[Comma list]]: 64/63, 33614/32805


Badness: 0.3585
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}


== 7-limit ==
[[Optimal tuning]]s:
Commas: 64/63, 686/675
* [[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 }}


[[POTE generator]]: ~49/40 = 355.904
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}


Map: [<1 1 5 4|,<0 2 -9 -4|]
[[Badness]] (Sintel): 3.34


Wedgie: <<2 -9 -4 -19 -12 16||
== Schism ==
{{See also| Schismatic family #Schism }}


{{EDOs|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
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.


Badness: 0.0459
[[Subgroup]]: 2.3.5.7


Music: [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 Beatles Improv] by Herman Miller
[[Comma list]]: 64/63, 360/343


== 11-limit ==
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}
Commas: 64/63, 100/99, 686/675


POTE generator: ~49/40 = 356.140
[[Optimal tuning]]s:  
* [[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 }}


Map: [<1 1 5 4 10|,<0 2 -9 -4 -22|]
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}


{{EDOs|legend=1| 27e, 37, 64be, 91bcde }}
[[Badness]] (Sintel): 1.43


Badness: 0.0456
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 45/44, 64/63, 99/98
Commas: 64/63, 91/90, 100/99, 169/168


POTE generator: ~16/13 = 356.229
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


Map: [<1 1 5 4 10 4|,<0 2 -9 -4 -22 -1|]
Optimal tunings:  
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


{{EDOs|legend=1| 27e, 37, 64be }}
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}


Badness: 0.0302
Badness (Sintel): 1.24


== Ringo ==
== Beatles ==
Commas: 56/55, 64/63, 540/539
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''


POTE generator: ~11/9 = 355.419
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.  


Map: [<1 1 5 4 2|,<0 2 -9 -4 5|]
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.


{{EDOs|legend=1| 10, 17c, 27e }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0329
[[Comma list]]: 64/63, 686/675


=== 13-limit ===
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
Commas: 56/55, 64/63, 78/77, 91/90


POTE generator: ~11/9 = 355.456
[[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 }}


Map: [<1 1 5 4 2 4|,<0 2 -9 -4 5 -1|]
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}


{{EDOs|legend=1| 10, 17c, 27e }}
[[Badness]] (Sintel): 1.16


Badness: 0.0226
; 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]]


= Schism =
=== 11-limit ===
{{see also|Schismatic family #Schism}}
Subgroup: 2.3.5.7.11


Commas: 64/63, 360/343
Comma list: 64/63, 100/99, 686/675


[[POTE generator]]: ~3/2 = 701.556
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}


Map: [<1 0 15 6|, <0 1 -8 -2|]
Optimal tunings:  
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


Wedgie: <<1 -8 -2 -15 -6 18||
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}


{{EDOs|legend=1| 12, 41d, 53d }}
Badness (Sintel): 1.51


Badness: 0.0566
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== 11-limit ==
Comma list: 64/63, 91/90, 100/99, 169/168
Commas: 45/44, 64/63, 99/98


POTE generator ~3/2 = 702.136
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}


Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]
Optimal tunings:  
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


{{EDOs|legend=1| 12, 29de, 41de }}
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}


Badness: 0.0375
Badness (Sintel): 1.25


= Passion =
=== Ringo ===
== 5-limit ==
Subgroup: 2.3.5.7.11
Comma: 262144/253125


POTE generator: ~16/15 = 98.670
Comma list: 56/55, 64/63, 540/539


Map: [<1 2 2|, <0 -5 4|]
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}


{{EDOs|legend=1| 11, 12, 49, 61, 73 }}
Optimal tunings:
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


Badness: 0.1686
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


=== Passive ===
Badness (Sintel): 1.09
Commas: 225/224, 256/245


POTE generator: ~16/15 = 98.809
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [<1 2 2 3|, <0 -5 4 -2|]
Comma list: 56/55, 64/63, 78/77, 91/90


{{EDOs|legend=1| 1, 11, 12, 49d }}
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}


Badness: 0.0751
Optimal tunings:  
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
Commas: 64/63, 3125/3087


[[POTE generator]]: ~16/15 = 98.153
Badness (Sintel): 0.935


Map: [<1 2 2 2|, <0 -5 4 10|]
=== Beetle ===
Subgroup: 2.3.5.7.11


Wedgie: <<5 -4 -10 -18 -30 -12||
Comma list: 55/54, 64/63, 686/675


Generators: 2, 16/15
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}


{{EDOs|legend=1| 12, 37, 49, 110bcd }}
Optimal tunings:
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


Badness: 0.0623
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


== 11-limit ==
Badness (Sintel): 1.92
Commas: 64/63, 100/99, 1375/1372


POTE generator: ~16/15 = 98.019
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [<1 2 2 2 2|, <0 -5 4 10 18|]
Comma list: 55/54, 64/63, 91/90, 169/168


{{EDOs|legend=1| 12, 37, 49 }}
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}


Badness: 0.0408
Optimal tunings:  
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
Commas: 64/63, 100/99, 196/195, 275/273


POTE generator: ~16/15 = 97.910
Badness (Sintel): 1.40


Map: [<1 2 2 2 2 2|, <0 -5 4 10 18 21|]
== Progress ==
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''


{{EDOs|legend=1| 12f, 37, 49f }}
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.


Badness: 0.0309
[[Subgroup]]: 2.3.5.7


= Fervor =
[[Comma list]]: 64/63, 392/375
== 5-limit ==
Comma: 67108864/61509375


POTE generator: ~64/45 = 577.705
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}


Map: [<1 4 -2|, <0 -5 9|]
: mapping generators: ~2, ~10/7


{{EDOs|legend=1| 25, 27 }}
[[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: 0.8526
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


== 7-limit ==
[[Badness]] (Sintel): 1.68
Commas: 64/63, 9604/9375


POTE generator: ~7/5 = 577.777
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [<1 4 -2 -2|, <0 -5 9 10|]
Comma list: 56/55, 64/63, 77/75


Wedgie: <<5 -9 -10 -26 -30 2||
Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}


{{EDOs|legend=1| 25, 27 }}
Optimal tunings:
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}


Badness: 0.1085
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}


== 11-limit ==
Badness (Sintel): 1.03
Commas: 56/55, 64/63, 1350/1331


POTE generator: ~7/5 = 577.850
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [<1 4 -2 -2 3|, <0 -5 9 10 1|]
Comma list: 56/55, 64/63, 66/65, 77/75


{{EDOs|legend=1| 25e, 27e }}
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}


Badness: 0.0521
Optimal tunings:  
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
Commas: 56/55, 64/63, 78/77, 507/500


POTE generator: ~7/5 = 578.060
Badness (Sintel): 1.08


Map: [<1 4 -2 -2 3 -4|, <0 -5 9 10 1 16|]
==== Progressive ====
Subgroup: 2.3.5.7.11.13


{{EDOs|legend=1| 27e }}
Comma list: 26/25, 56/55, 64/63, 77/75


Badness: 0.0397
Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}


= Quasisuper =
Optimal tunings:
Commas: 64/63, 2430/2401
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}


[[POTE_tuning|POTE generator]]: 708.328
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}


Map: [<1 0 23 6|, <0 1 -13 -2|]
Badness (Sintel): 1.35


Wedgie: <<1 -13 -2 -23 -2 -6 32||
== Fervor ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''


{{EDOs|legend=1| 22, 61 }}
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.


Badness: 0.0638
[[Subgroup]]: 2.3.5.7


== Quasisupra ==
[[Comma list]]: 64/63, 9604/9375
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).


Commas: 64/63, 99/98, 121/120
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}


POTE generator: ~3/2 = 708.205
: mapping generators: ~2, ~10/7


Map: [<1 2 -3 2 1|, <0 -1 13 2 6|]
[[Optimal tuning]]s:  
* [[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 }}


{{EDOs|legend=1| 17c, 22, 27c, 39d, 61d }}
{{Optimal ET sequence|legend=1| 2, 25, 27 }}


Badness: 0.0322
[[Badness]] (Sintel): 2.74


=== 13-limit ===
=== 11-limit ===
Commas: 64/63, 78/77, 91/90, 121/120
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 708.004
Comma list: 56/55, 64/63, 1350/1331


Map: [<1 0 23 6 13 18|, <0 1 -13 -2 -6 -9|]
Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}


{{EDOs|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
Optimal tunings:
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}


Badness: 0.0302
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}


== Quasisoup ==
Badness (Sintel): 1.72
Commas: 55/54, 64/63, 2430/2401


POTE generator: ~3/2 = 709.021
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [<1 0 23 6 -22|, <0 1 -13 -2 16|]
Comma list: 56/55, 64/63, 78/77, 507/500


{{EDOs|legend=1| 22 }}
Mapping: {{mapping| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}


Badness: 0.0835
Optimal tunings:  
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}


= Progress =
{{Optimal ET sequence|legend=0| 2f, 27e }}
== 5-limit ==
Comma: 32768/30375


POTE generator: ~64/45 = 561.264
Badness (Sintel): 1.64


Map: [<1 0 5|, <0 3 -5|]
== Sixix ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sixix (5-limit)]].''
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}


{{EDOs|legend=1| 4, 13, 15, 32c, 47bc, 62bc }}
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.


Badness: 0.2461
[[Subgroup]]: 2.3.5.7


== 7-limit ==
[[Comma list]]: 64/63, 3125/2916
Commas: 64/63, 392/375


POTE generator: ~7/5 = 562.122
{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}


Map: [<1 0 5 6|, <0 3 -5 -6|]
[[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 }}


Wedgie: <<3 -5 -6 -15 -18 0||
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}


{{EDOs|legend=1| 13, 15, 32c, 79bcc, 111bcc }}
[[Badness]] (Sintel): 4.02


Badness: 0.0664
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 55/54, 64/63, 125/121
Commas: 56/55, 64/63, 77/75


POTE generator: ~7/5 = 562.085
Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}


Map: [<1 0 5 6 4|, <0 3 -5 -6 -1|]
Optimal tunings:  
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}


{{EDOs|legend=1| 13, 15, 32c, 47bc, 79bcce }}
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}


Badness: 0.0310
Badness (Sintel): 2.34


=== 13-limit ===
=== 13-limit ===
Commas: 56/55, 64/63, 66/65, 77/75
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 55/54, 64/63, 125/121
 
Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}


POTE generator: ~7/5 = 562.365
Optimal tunings:  
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}


Map: [<1 0 5 6 4 0|, <0 3 -5 -6 -1 7|]
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


{{EDOs|legend=1| 15, 17c, 32cf }}
Badness (Sintel): 1.91


Badness: 0.0262
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=== Progressive ===
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121
Commas: 26/25, 56/55, 64/63, 77/75


POTE generator: ~7/5 = 563.239
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}


Map: [<1 0 5 6 4 9|, <0 3 -5 -6 -1 -10|]
Optimal tunings:  
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}


{{EDOs|legend=1| 15f, 17c, 32c, 49c }}
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


Badness: 0.0327
Badness (Sintel): 2.00


[[Category:Theory]]
[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Temperament clan]]
[[Category:Temperament clans]]
[[Category:Archytas]]
[[Category:Pages with mostly numerical content]]
[[Category:Rank 2]]
[[Category:Rank 2]]
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