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{{Technical data page}}
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|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|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, and 2430/2401 gives quasisuper. Pajara is discussed under [[Diaschismic_family|diaschismic family]], dominant under [[Meantone_family|meantone family]], augene under [[Augmented_family|augmented family]] and porcupine under [[Porcupine_family|porcupine family]]. The rest are considered below.
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]].  


=Mother=
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.
Commas: 16/15, 21/20


[[POTE_tuning|POTE generator]]: 721.569
== Archy ==
{{Main| Superpyth }}


Map: [<1 0 4 6|, <0 1 -1 -2|]
[[Subgroup]]: 2.3.7


Wedgie: <<1 -1 -2 -4 -6 -2||
[[Comma list]]: 64/63


EDOs: 5, 148, 153
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}


Badness: 0.0242
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}
: mapping generators: ~2, ~3


=Blacksmith=
[[Optimal tuning]]s:
Commas: 28/27, 49/48
* [[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 }}


[[POTE_tuning|POTE generator]]: ~5/4 = 392.767
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}


Map: [<5 8 0 14|, <0 0 1 0|]
[[Badness]] (Sintel): 0.159


Wedgie: <<0 5 0 8 0 -14||
Scales: [[archy5]], [[archy7]], [[archy12]]


EDOs: 5, 10, 15, 40b, 55b
=== 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]];
* [[Meantone family #Dominant|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: 0.0256
These all use the same generators as archy.  


==11-limit==
[[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.
Commas: 28/27, 49/48, 55/54


POTE generator: ~5/4 = 394.948
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]]
* ''[[Dichotic]]'' (+25/24) → [[Dicot family #Dichotic|Dicot family]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Augene]] (+126/125) → [[Augmented family #Septimal augmented (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]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
* ''[[Gothic]]'' (+17496/16807) → [[17th-octave temperaments #Gothic|17th-octave temperaments]]


Map: [<5 8 0 14 29|, <0 0 1 0 -1|]
Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.


EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde
==== 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 right below. Discussed elsewhere is [[suhajira]] of the [[rastmic clan #Suhajira|rastmic clan]].


Badness: 0.0246
=== Supra ===
Subgroup: 2.3.7.11


==13-limit==
Comma list: 64/63, 99/98
Commas: 28/27, 40/39, 49/48, 55/54


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


Map: [<5 8 0 14 29 7|, <0 0 1 0 -1 1|]
Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}


EDOs: 5, 10, 15, 25e, 40bef
Optimal tunings:  
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}


Badness: 0.0205
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}


==Farrier==
Badness (Sintel): 0.352
Commas: 28/27, 49/48, 77/75


POTE generator: ~5/4 = 398.070
Scales: [[supra7]], [[supra12]]


Map: [<5 8 0 14 -6|, <0 0 1 0 2|]
==== 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𝄫).


EDOs: 5e, 15
Subgroup: 2.3.7.11.13


Badness: 0.0292
Comma list: 64/63, 78/77, 99/98


===13-limit===
Subgroup-val mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
Commas: 28/27, 40/39, 49/48, 66/65


POTE generator: ~5/4 = 396.812
Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}


Map: [<5 8 0 14 -6 7|, <0 0 1 0 2 1|]
Optimal tunings:  
* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}


EDOs: 5e, 10e, 15
{{Optimal ET sequence|legend=0| 12f, 17 }}


Badness: 0.0223
Badness (Sintel): 0.498


==Ferrum==
Scales: [[supra7]], [[supra12]]
Commas: 28/27, 35/33, 49/48


POTE generator: ~5/4 = 374.763
== Superpyth ==
{{Main| Superpyth }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


Map: [<5 8 0 14 6|, <0 0 1 0 1|]
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.


EDOs: 10
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.


Badness: 0.0309
[[Subgroup]]: 2.3.5.7


[[File:blacksmith10.jpg|alt=blacksmith10.jpg|blacksmith10.jpg]]
[[Comma list]]: 64/63, 245/243


==Blackwood==
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}
''Blackwood (5-limit)''


Comma: 256/243
[[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 }}


[[POTE_tuning|POTE generator]]: 399.594
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}


Map: [<5 8 0|, <0 0 1|]
[[Badness]] (Sintel): 0.818


EDOs: 5, 10, 15
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0638
Comma list: 64/63, 100/99, 245/243


=Superpyth=
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}
[[Superpyth|Main article]]


Commas: 64/63, 245/243
Optimal tunings:  
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


[[POTE_tuning|POTE generator]]: 710.291
{{Optimal ET sequence|legend=0| 22, 27e, 49 }}


Map: [<1 0 -12 6|, <0 1 9 -2|]
Badness (Sintel): 0.826


Wedgie: <<1 9 -2 12 -6 -30||
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 5, 17, 22, 27, 49
Comma list: 64/63, 78/77, 91/90, 100/99


Badness: 0.0323
Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}


==11-limit==
Optimal tunings:
Commas: 64/63, 100/99, 245/243
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}


[[POTE_tuning|POTE generator]]: 710.175
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}


Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]
Badness (Sintel): 1.02


EDOs: 22, 49
==== Uberpyth ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0250
Comma list: 64/63, 100/99, 144/143, 245/243


==13-limit==
Mapping: {{mapping| 1 0 -12 6 -22 26 | 0 1 9 -2 16 -14 }}
Commas: 64/63, 78/77, 91/90, 100/99


POTE generator: ~3/2 = 710.479
Optimal tunings:  
* WE: ~2 = 1196.6666{{c}}, ~3/2 = 708.3602{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.2878{{c}}


Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|]
{{Optimal ET sequence|legend=0| 22f, 27e, 49f, 125bcddeeeff, 174bbcdddeeeeffff }}


EDOs: 22, 27e, 49, 76bcde
Badness (Sintel): 1.11


Badness: 0.0247
==== Thomas ====
Subgroup: 2.3.5.7.11.13


==Suprapyth==
Comma list: 64/63, 100/99, 169/168, 245/243
Commas: 55/54, 64/63, 99/98


POTE generator: ~3/2 = 709.495
Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}


Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]
Optimal tunings:  
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


EDOs: 5, 7, 12, 17, 22
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}


Badness: 0.0328
Badness (Sintel): 2.03


===13-limit===
=== Suprapyth ===
Commas: 55/54, 64/63, 65/63, 364/363
Suprapyth finds the ~11/8 at the diminished fifth (C–G♭), and finds the ~13/8 at the diminished seventh (C–B𝄫).


POTE generator: ~3/2 = 708.703
Subgroup: 2.3.5.7.11


Map: [<1 0 -12 6 13 18|, <0 1 9 -2 -6 -9|]
Comma list: 55/54, 64/63, 99/98


EDOs: 17, 22, 83cdf
Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}


Badness: 0.0363
Optimal tunings:  
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


=Beatles=
{{Optimal ET sequence|legend=0| 5, 17, 22 }}
Comma: 524288/492075


POTE generator: ~512/405 = 355.930
Badness (Sintel): 1.08


Map: [<1 1 5|,<0 2 -9|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 10, 17c, 27, 64b, 91bc, 118bc
Comma list: 55/54, 64/63, 65/63, 99/98


Badness: 0.3585
Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}


==7-limit==
Optimal tunings:
Commas: 64/63, 686/675
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}


[[POTE_tuning|POTE generator]]: ~49/40 = 355.904
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}


Map: [<1 1 5 4|,<0 2 -9 -4|]
Badness (Sintel): 1.50


Wedgie: <<2 -9 -4 -19 -12 16||
== Quasisuper ==
{{Main|Quasisuper}}


EDOs: 10, 17c, 27, 64b, 91bcd, 118bcd
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).


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, 2430/2401


==11-limit==
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}
Commas: 64/63, 100/99, 686/675


POTE generator: ~49/40 = 356.140
[[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 }}


Map: [<1 1 5 4 10|,<0 2 -9 -4 -22|]
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}


EDOs: 27e, 37, 64be, 91bcde
[[Badness]] (Sintel): 1.61


Badness: 0.0456
=== Quasisupra ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 64/63, 99/98, 121/120
Commas: 64/63, 91/90, 100/99, 169/168


POTE generator: ~16/13 = 356.229
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}


Map: [<1 1 5 4 10 4|,<0 2 -9 -4 -22 -1|]
Optimal tunings:  
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


EDOs: 27e, 37, 64be
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}


Badness: 0.0302
Badness (Sintel): 1.06


==Ringo==
==== 13-limit ====
Commas: 56/55, 64/63, 540/539
Subgroup: 2.3.5.7.11.13


POTE generator: ~11/9 = 355.419
Comma list: 64/63, 78/77, 91/90, 121/120


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


EDOs: 10, 17c, 27e
Optimal tunings:  
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


Badness: 0.0329
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


===13-limit===
Badness (Sintel): 1.25
Commas: 56/55, 64/63, 78/77, 91/90


POTE generator: ~11/9 = 355.456
=== Quasisoup ===
Subgroup: 2.3.5.7.11


Map: [<1 1 5 4 2 4|,<0 2 -9 -4 5 -1|]
Comma list: 55/54, 64/63, 2430/2401


EDOs: 10, 17c, 27e
Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}


Badness: 0.0226
Optimal tunings:  
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


=Schism=
{{Optimal ET sequence|legend=0| 22 }}
Commas: 64/63, 360/343


[[POTE_tuning|POTE generator]]: ~3/2 = 701.556
Badness (Sintel): 2.76


Map: [<1 0 15 6|, <0 1 -8 -2|]
== Ultrapyth ==
{{Main| Ultrapyth }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Ultrapyth (5-limit)]].''


Wedgie: <<1 -8 -2 -15 -6 18||
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–C𝄪).


EDOs: 12, 41d, 53d
[[Subgroup]]: 2.3.5.7


Badness: 0.0566
[[Comma list]]: 64/63, 6860/6561


==11-limit==
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
Commas: 45/44, 64/63, 99/98


POTE generator ~3/2 = 702.136
[[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 }}


Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}


EDOs: 12, 29de, 41de
[[Badness]] (Sintel): 2.74


Badness: 0.0375
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


==5-limit==
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}
Comma: 262144/253125


POTE generator: ~16/15 = 98.670
Optimal tunings:  
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


Map: [<1 2 2|, <0 -5 4|]
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


EDOs: 11, 12, 49, 61, 73
Badness (Sintel): 2.26


Badness: 0.1686
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


===Passive===
Comma list: 55/54, 64/63, 91/90, 1573/1568
Commas: 225/224, 256/245


POTE generator: ~16/15 = 98.809
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}


Map: [<1 2 2 3|, <0 -5 4 -2|]
Optimal tunings:  
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


EDOs: 1, 11, 12, 49d
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Badness: 0.0751
Badness (Sintel): 2.03


==7-limit==
=== Ultramarine ===
Commas: 64/63, 3125/3087
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~16/15 = 98.153
Comma list: 64/63, 100/99, 3773/3645


Map: [<1 2 2 2|, <0 -5 4 10|]
Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}


Wedgie: <<5 -4 -10 -18 -30 -12||
Optimal tunings:  
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


Generators: 2, 16/15
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}


EDOs: 12, 37, 49, 110bcd
Badness (Sintel): 2.58


Badness: 0.0623
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 64/63, 91/90, 100/99, 847/845
Commas: 64/63, 100/99, 1375/1372


POTE generator: ~16/15 = 98.019
Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}


Map: [<1 2 2 2 2|, <0 -5 4 10 18|]
Optimal tunings:  
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}


EDOs: 12, 37, 49
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}


Badness: 0.0408
Badness (Sintel): 1.89


==13-limit==
== Quasiultra ==
Commas: 64/63 100/99 196/195 275/273
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𝄫♭).


POTE generator: ~16/15 = 97.910
[[Subgroup]]: 2.3.5.7


Map: [<1 2 2 2 2 2|, <0 -5 4 10 18 21|]
[[Comma list]]: 64/63, 33614/32805


EDOs: 12f, 37, 49f
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}


Badness: 0.0309
[[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 }}


=Fervor=
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}
Comma: 67108864/61509375


POTE generator: ~64/45 = 577.705
[[Badness]] (Sintel): 3.34


Map: [<1 4 -2|, <0 -5 9|]
== Schism ==
{{See also| Schismatic family #Schism }}


EDOs: 25, 27
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.


Badness: 0.8526
[[Subgroup]]: 2.3.5.7


==7-limit==
[[Comma list]]: 64/63, 360/343
Commas: 64/63, 9604/9375


POTE generator: ~7/5 = 577.777
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}


Map: [<1 4 -2 -2|, <0 -5 9 10|]
[[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 }}


Wedgie: <<5 -9 -10 -26 -30 2||
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}


EDOs: 25, 27
[[Badness]] (Sintel): 1.43


Badness: 0.1085
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 45/44, 64/63, 99/98
Commas: 56/55, 64/63, 1350/1331


POTE generator: ~7/5 = 577.850
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


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


EDOs: 25e, 27e
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}


Badness: 0.0521
Badness (Sintel): 1.24


==13-limit==
== Beatles ==
Commas: 56/55, 64/63, 78/77, 507/500
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''


POTE generator: ~7/5 = 578.060
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 4 -2 -2 3 -4|, <0 -5 9 10 1 16|]
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: 27e
[[Subgroup]]: 2.3.5.7


Badness: 0.0397
[[Comma list]]: 64/63, 686/675


=Quasisuper=
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}
Commas: 64/63, 2430/2401


[[POTE_tuning|POTE generator]]: 708.328
[[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 0 23 6|, <0 1 -13 -2|]
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}


Wedgie: <<1 -13 -2 -23 -2 -6 32||
[[Badness]] (Sintel): 1.16


EDOs: 22, 61
; 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]]


Badness: 0.0638
=== 11-limit ===
Subgroup: 2.3.5.7.11


==Quasisupra==
Comma list: 64/63, 100/99, 686/675
Commas: 64/63, 99/98, 121/120


Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament [[Supra|supra]], with the quasisuper mapping of 5 thrown in (rather than the superpyth mapping of 5, which results in suprapyth).
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}


POTE generator: ~3/2 = 708.205
Optimal tunings:
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


Map: [<1 2 -3 2 1|, <0 -1 13 2 6|]
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}


EDOs: 17c, 22, 27c, 39d, 61d
Badness (Sintel): 1.51


Badness: 0.0322
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


===13-limit===
Comma list: 64/63, 91/90, 100/99, 169/168
Commas: 64/63, 78/77, 91/90, 121/120


POTE generator: ~3/2 = 708.004
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}


Map: [<1 0 23 6 13 18|, <0 1 -13 -2 -6 -9|]
Optimal tunings:  
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


EDOs: 17c, 22, 39d, 61df, 100bcdf
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}


Badness: 0.0302
Badness (Sintel): 1.25


==Quasisoup==
=== Ringo ===
Commas: 55/54, 64/63, 2430/2401
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 709.021
Comma list: 56/55, 64/63, 540/539


Map: [<1 0 23 6 -22|, <0 1 -13 -2 16|]
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}


EDOs: 22
Optimal tunings:  
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


Badness: 0.0835
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


=Progress=
Badness (Sintel): 1.09
Comma: 32768/30375


POTE generator: ~64/45 = 561.264
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


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


EDOs: 4, 13, 15, 32c, 47bc, 62bc
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}


Badness: 0.2461
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, 392/375


POTE generator: ~7/5 = 562.122
Badness (Sintel): 0.935


Map: [<1 0 5 6|, <0 3 -5 -6|]
=== Beetle ===
Subgroup: 2.3.5.7.11


Wedgie: <<3 -5 -6 -15 -18 0||
Comma list: 55/54, 64/63, 686/675


EDOs: 13, 15, 32c, 79bcc, 111bcc
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}


Badness: 0.0664
Optimal tunings:  
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
Commas: 56/55, 64/63, 77/75


POTE generator: ~7/5 = 562.085
Badness (Sintel): 1.92


Map: [<1 0 5 6 4|, <0 3 -5 -6 -1|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 13, 15, 32c, 47bc, 79bcce
Comma list: 55/54, 64/63, 91/90, 169/168


Badness: 0.0310
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}


==13-limit==
Optimal tunings:
Commas: 56/55, 64/63, 66/65, 77/75
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


POTE generator: ~7/5 = 562.365
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Map: [<1 0 5 6 4 0|, <0 3 -5 -6 -1 7|]
Badness (Sintel): 1.40


EDOs: 15, 17c, 32cf
== Progress ==
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''


Badness: 0.0262
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.  


==Progressive==
[[Subgroup]]: 2.3.5.7
Commas: 26/25, 56/55, 64/63, 77/75


POTE generator: ~7/5 = 563.239
[[Comma list]]: 64/63, 392/375


Map: [<1 0 5 6 4 9|, <0 3 -5 -6 -1 -10|]
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}


EDOs: 15f, 17c, 32c, 49c
: mapping generators: ~2, ~10/7


Badness: 0.0327
[[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 }}


[[Category:Theory]]
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
[[Category:Archytas]]
 
[[Category:Clan]]
[[Badness]] (Sintel): 1.68
[[Category:Temperament]]
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 64/63, 77/75
 
Mapping: {{mapping| 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}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 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}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 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 }}
 
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
 
[[Comma list]]: 64/63, 9604/9375
 
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
 
: mapping generators: ~2, ~10/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 2, 25, 27 }}
 
[[Badness]] (Sintel): 2.74
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 64/63, 1350/1331
 
Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}
 
Optimal tunings:
* 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
 
Comma list: 56/55, 64/63, 78/77, 507/500
 
Mapping: {{mapping| 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
 
== 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 {{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
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
 
Optimal tunings:
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
 
Optimal tunings:
* 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: {{mapping| 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
 
[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Archytas_clan"