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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
: sval mapping generators: ~2, ~3


=Blacksmith=
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}
Commas: 28/27, 49/48


[[POTE_tuning|POTE generator]]: ~5/4 = 392.767
: [[gencom]]: [2 3; 64/63]


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


Wedgie: <<0 5 0 8 0 -14||
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}


EDOs: 5, 10, 15, 40b, 55b
[[Badness]] (Sintel): 0.159


Badness: 0.0256
Scales: [[archy5]], [[archy7]], [[archy12]]


==11-limit==
=== Overview to extensions ===
Commas: 28/27, 49/48, 55/54
==== 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]].


POTE generator: ~5/4 = 394.948
These all use the same generators as archy.  


Map: [<5 8 0 14 29|, <0 0 1 0 -1|]
[[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.


EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde
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]]


Badness: 0.0246
Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.  


==13-limit==
==== Subgroup extensions ====
Commas: 28/27, 40/39, 49/48, 55/54
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]].


POTE generator: ~5/4 = 391.0367
=== Supra ===
Subgroup: 2.3.7.11


Map: [<5 8 0 14 29 7|, <0 0 1 0 -1 1|]
Comma list: 64/63, 99/98


EDOs: 5, 10, 15, 25e, 40bef
Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}


Badness: 0.0205
Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}


==Farrier==
: gencom: [2 3; 64/63 99/98]
Commas: 28/27, 49/48, 77/75


POTE generator: ~5/4 = 398.070
Optimal tunings:  
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}


Map: [<5 8 0 14 -6|, <0 0 1 0 2|]
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}


EDOs: 5e, 15
Badness (Sintel): 0.352


Badness: 0.0292
Scales: [[supra7]], [[supra12]]


===13-limit===
==== Supraphon ====
Commas: 28/27, 40/39, 49/48, 66/65
Subgroup: 2.3.7.11.13


POTE generator: ~5/4 = 396.812
Comma list: 64/63, 78/77, 99/98


Map: [<5 8 0 14 -6 7|, <0 0 1 0 2 1|]
Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}


EDOs: 5e, 10e, 15
Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}


Badness: 0.0223
: gencom: [2 3; 64/63 78/77 99/98]


==Ferrum==
Optimal tunings:
Commas: 28/27, 35/33, 49/48
* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}


POTE generator: ~5/4 = 374.763
{{Optimal ET sequence|legend=0| 12f, 17 }}


Map: [<5 8 0 14 6|, <0 0 1 0 1|]
Badness (Sintel): 0.498


EDOs: 10
Scales: [[supra7]], [[supra12]]


Badness: 0.0309
== Superpyth ==
{{Main| Superpyth }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


[[File:blacksmith10.jpg|alt=blacksmith10.jpg|blacksmith10.jpg]]
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.  


==Blackwood==
[[Subgroup]]: 2.3.5.7
''Blackwood (5-limit)''


Comma: 256/243
[[Comma list]]: 64/63, 245/243


[[POTE_tuning|POTE generator]]: 399.594
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}


Map: [<5 8 0|, <0 0 1|]
[[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 }}


EDOs: 5, 10, 15
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}


Badness: 0.0638
[[Badness]] (Sintel): 0.818


=Superpyth=
=== 11-limit ===
[[Superpyth|Main article]]
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).


Commas: 64/63, 245/243
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: 710.291
Comma list: 64/63, 100/99, 245/243


Map: [<1 0 -12 6|, <0 1 9 -2|]
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}


Wedgie: <<1 9 -2 12 -6 -30||
Optimal tunings:  
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


EDOs: 5, 17, 22, 27, 49
{{Optimal ET sequence|legend=0| 22, 27e, 49 }}


Badness: 0.0323
Badness (Sintel): 0.826


==11-limit==
==== 13-limit ====
Commas: 64/63, 100/99, 245/243
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: 710.175
Comma list: 64/63, 78/77, 91/90, 100/99


Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]
Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}


EDOs: 22, 49
Optimal tunings:  
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}


Badness: 0.0250
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}


==13-limit==
Badness (Sintel): 1.02
Commas: 64/63, 78/77, 91/90, 100/99


POTE generator: ~3/2 = 710.479
==== Thomas ====
Subgroup: 2.3.5.7.11.13


Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|]
Comma list: 64/63, 100/99, 169/168, 245/243


EDOs: 22, 27e, 49, 76bcde
Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}


Badness: 0.0247
Optimal tunings:  
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


==Suprapyth==
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}
Commas: 55/54, 64/63, 99/98


POTE generator: ~3/2 = 709.495
Badness (Sintel): 2.03


Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]
=== Suprapyth ===
Suprapyth finds the ~11/8 at the diminished fifth (C–Gb), and finds the ~13/8 at the diminished seventh (C–Bbb).


EDOs: 5, 7, 12, 17, 22
Subgroup: 2.3.5.7.11


Badness: 0.0328
Comma list: 55/54, 64/63, 99/98


===13-limit===
Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}
Commas: 55/54, 64/63, 65/63, 364/363


POTE generator: ~3/2 = 708.703
Optimal tunings:  
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


Map: [<1 0 -12 6 13 18|, <0 1 9 -2 -6 -9|]
{{Optimal ET sequence|legend=0| 5, 17, 22 }}


EDOs: 17, 22, 83cdf
Badness (Sintel): 1.08


Badness: 0.0363
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=Beatles=
Comma list: 55/54, 64/63, 65/63, 99/98
Comma: 524288/492075


POTE generator: ~512/405 = 355.930
Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}


Map: [<1 1 5|,<0 2 -9|]
Optimal tunings:  
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}


EDOs: 10, 17c, 27, 64b, 91bc, 118bc
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}


Badness: 0.3585
Badness (Sintel): 1.50


==7-limit==
== Quasisuper ==
Commas: 64/63, 686/675
Quasisuper can be described as {{nowrap| 17c & 22 }}, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–Gbb).


[[POTE_tuning|POTE generator]]: ~49/40 = 355.904
[[Subgroup]]: 2.3.5.7


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


Wedgie: <<2 -9 -4 -19 -12 16||
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}


EDOs: 10, 17c, 27, 64b, 91bcd, 118bcd
[[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 }}


Badness: 0.0459
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}


Music: [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 Beatles Improv] by Herman Miller
[[Badness]] (Sintel): 1.61


==11-limit==
=== Quasisupra ===
Commas: 64/63, 100/99, 686/675
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).


POTE generator: ~49/40 = 356.140
Subgroup: 2.3.5.7.11


Map: [<1 1 5 4 10|,<0 2 -9 -4 -22|]
Comma list: 64/63, 99/98, 121/120


EDOs: 27e, 37, 64be, 91bcde
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}


Badness: 0.0456
Optimal tunings:  
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}
Commas: 64/63, 91/90, 100/99, 169/168


POTE generator: ~16/13 = 356.229
Badness (Sintel): 1.06


Map: [<1 1 5 4 10 4|,<0 2 -9 -4 -22 -1|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 27e, 37, 64be
Comma list: 64/63, 78/77, 91/90, 121/120


Badness: 0.0302
Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}


==Ringo==
Optimal tunings:
Commas: 56/55, 64/63, 540/539
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


POTE generator: ~11/9 = 355.419
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


Map: [<1 1 5 4 2|,<0 2 -9 -4 5|]
Badness (Sintel): 1.25


EDOs: 10, 17c, 27e
=== Quasisoup ===
Subgroup: 2.3.5.7.11


Badness: 0.0329
Comma list: 55/54, 64/63, 2430/2401


===13-limit===
Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
Commas: 56/55, 64/63, 78/77, 91/90


POTE generator: ~11/9 = 355.456
Optimal tunings:  
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


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


EDOs: 10, 17c, 27e
Badness (Sintel): 2.76


Badness: 0.0226
== Ultrapyth ==
{{Main| Ultrapyth }}


=Schism=
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).
Commas: 64/63, 360/343


[[POTE_tuning|POTE generator]]: ~3/2 = 701.556
[[Subgroup]]: 2.3.5.7


Map: [<1 0 15 6|, <0 1 -8 -2|]
[[Comma list]]: 64/63, 6860/6561


Wedgie: <<1 -8 -2 -15 -6 18||
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}


EDOs: 12, 41d, 53d
[[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 }}


Badness: 0.0566
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}


==11-limit==
[[Badness]] (Sintel): 2.74
Commas: 45/44, 64/63, 99/98


POTE generator ~3/2 = 702.136
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]
Comma list: 55/54, 64/63, 2401/2376


EDOs: 12, 29de, 41de
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}


Badness: 0.0375
Optimal tunings:  
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


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


==5-limit==
Badness (Sintel): 2.26
Comma: 262144/253125


POTE generator: ~16/15 = 98.670
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [<1 2 2|, <0 -5 4|]
Comma list: 55/54, 64/63, 91/90, 1573/1568


EDOs: 11, 12, 49, 61, 73
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}


Badness: 0.1686
Optimal tunings:  
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


===Passive===
{{Optimal ET sequence|legend=0| 5, 32, 37 }}
Commas: 225/224, 256/245


POTE generator: ~16/15 = 98.809
Badness (Sintel): 2.03


Map: [<1 2 2 3|, <0 -5 4 -2|]
=== Ultramarine ===
Subgroup: 2.3.5.7.11


EDOs: 1, 11, 12, 49d
Comma list: 64/63, 100/99, 3773/3645


Badness: 0.0751
Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}


==7-limit==
Optimal tunings:
Commas: 64/63, 3125/3087
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


[[POTE_tuning|POTE generator]]: ~16/15 = 98.153
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}


Map: [<1 2 2 2|, <0 -5 4 10|]
Badness (Sintel): 2.58


Wedgie: <<5 -4 -10 -18 -30 -12||
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Generators: 2, 16/15
Comma list: 64/63, 91/90, 100/99, 847/845


EDOs: 12, 37, 49, 110bcd
Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}


Badness: 0.0623
Optimal tunings:  
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}
Commas: 64/63, 100/99, 1375/1372


POTE generator: ~16/15 = 98.019
Badness (Sintel): 1.89


Map: [<1 2 2 2 2|, <0 -5 4 10 18|]
== 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).


EDOs: 12, 37, 49
[[Subgroup]]: 2.3.5.7


Badness: 0.0408
[[Comma list]]: 64/63, 33614/32805


==13-limit==
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
Commas: 64/63 100/99 196/195 275/273


POTE generator: ~16/15 = 97.910
[[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 }}


Map: [<1 2 2 2 2 2|, <0 -5 4 10 18 21|]
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}


EDOs: 12f, 37, 49f
[[Badness]] (Sintel): 3.34


Badness: 0.0309
== Schism ==
{{See also| Schismatic family #Schism }}


=Fervor=
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.
Comma: 67108864/61509375


POTE generator: ~64/45 = 577.705
[[Subgroup]]: 2.3.5.7


Map: [<1 4 -2|, <0 -5 9|]
[[Comma list]]: 64/63, 360/343


EDOs: 25, 27
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}


Badness: 0.8526
[[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 }}


==7-limit==
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
Commas: 64/63, 9604/9375


POTE generator: ~7/5 = 577.777
[[Badness]] (Sintel): 1.43


Map: [<1 4 -2 -2|, <0 -5 9 10|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: <<5 -9 -10 -26 -30 2||
Comma list: 45/44, 64/63, 99/98


EDOs: 25, 27
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


Badness: 0.1085
Optimal tunings:  
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}
Commas: 56/55, 64/63, 1350/1331


POTE generator: ~7/5 = 577.850
Badness (Sintel): 1.24


Map: [<1 4 -2 -2 3|, <0 -5 9 10 1|]
== Beatles ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''


EDOs: 25e, 27e
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.


Badness: 0.0521
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.  


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 56/55, 64/63, 78/77, 507/500


POTE generator: ~7/5 = 578.060
[[Comma list]]: 64/63, 686/675


Map: [<1 4 -2 -2 3 -4|, <0 -5 9 10 1 16|]
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}


EDOs: 27e
[[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: 0.0397
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}


=Quasisuper=
[[Badness]] (Sintel): 1.16
Commas: 64/63, 2430/2401


[[POTE_tuning|POTE generator]]: 708.328
; 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]]


Map: [<1 0 23 6|, <0 1 -13 -2|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: <<1 -13 -2 -23 -2 -6 32||
Comma list: 64/63, 100/99, 686/675


EDOs: 22, 61
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}


Badness: 0.0638
Optimal tunings:  
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


==Quasisupra==
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}
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).
Badness (Sintel): 1.51


POTE generator: ~3/2 = 708.205
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [<1 2 -3 2 1|, <0 -1 13 2 6|]
Comma list: 64/63, 91/90, 100/99, 169/168


EDOs: 17c, 22, 27c, 39d, 61d
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}


Badness: 0.0322
Optimal tunings:  
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}
Commas: 64/63, 78/77, 91/90, 121/120


POTE generator: ~3/2 = 708.004
Badness (Sintel): 1.25


Map: [<1 0 23 6 13 18|, <0 1 -13 -2 -6 -9|]
=== Ringo ===
Subgroup: 2.3.5.7.11


EDOs: 17c, 22, 39d, 61df, 100bcdf
Comma list: 56/55, 64/63, 540/539


Badness: 0.0302
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}


==Quasisoup==
Optimal tunings:
Commas: 55/54, 64/63, 2430/2401
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


POTE generator: ~3/2 = 709.021
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


Map: [<1 0 23 6 -22|, <0 1 -13 -2 16|]
Badness (Sintel): 1.09


EDOs: 22
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0835
Comma list: 56/55, 64/63, 78/77, 91/90


=Progress=
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
Comma: 32768/30375


POTE generator: ~64/45 = 561.264
Optimal tunings:  
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}


Map: [<1 0 5|, <0 3 -5|]
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


EDOs: 4, 13, 15, 32c, 47bc, 62bc
Badness (Sintel): 0.935


Badness: 0.2461
=== Beetle ===
Subgroup: 2.3.5.7.11


==7-limit==
Comma list: 55/54, 64/63, 686/675
Commas: 64/63, 392/375


POTE generator: ~7/5 = 562.122
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}


Map: [<1 0 5 6|, <0 3 -5 -6|]
Optimal tunings:  
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


Wedgie: <<3 -5 -6 -15 -18 0||
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


EDOs: 13, 15, 32c, 79bcc, 111bcc
Badness (Sintel): 1.92


Badness: 0.0664
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 55/54, 64/63, 91/90, 169/168
Commas: 56/55, 64/63, 77/75


POTE generator: ~7/5 = 562.085
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}


Map: [<1 0 5 6 4|, <0 3 -5 -6 -1|]
Optimal tunings:  
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


EDOs: 13, 15, 32c, 47bc, 79bcce
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Badness: 0.0310
Badness (Sintel): 1.40


==13-limit==
== Progress ==
Commas: 56/55, 64/63, 66/65, 77/75
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''


POTE generator: ~7/5 = 562.365
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.  


Map: [<1 0 5 6 4 0|, <0 3 -5 -6 -1 7|]
[[Subgroup]]: 2.3.5.7


EDOs: 15, 17c, 32cf
[[Comma list]]: 64/63, 392/375


Badness: 0.0262
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}


==Progressive==
: mapping generators: ~2, ~10/7
Commas: 26/25, 56/55, 64/63, 77/75


POTE generator: ~7/5 = 563.239
[[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 }}


Map: [<1 0 5 6 4 9|, <0 3 -5 -6 -1 -10|]
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


EDOs: 15f, 17c, 32c, 49c
[[Badness]] (Sintel): 1.68


Badness: 0.0327
=== 11-limit ===
[[Category:archytas]]
Subgroup: 2.3.5.7.11
[[Category:clan]]
 
[[Category:overview]]
Comma list: 56/55, 64/63, 77/75
[[Category:temperaments]]
 
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:Pages with mostly numerical content]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Archytas_clan"