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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]].  
{{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]].  


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.  
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.  


Discussed under subgroup temperaments is the 2.3.7 [[Subgroup temperaments #Archy|archy]]. Under their respective 5-limit families are [[Father family #Mother|mother]], [[Meantone family #Dominant|dominant]], [[Augmented family #Augene|augene]], [[Porcupine family|porcupine]], [[Diaschismic family #Pajara|pajara]], [[Tetracot family #Modus|modus]], and [[Immunity family #Immunized|immunized]]. The rest are considered below.
== Archy ==
{{Main| Superpyth }}


= Blacksmith =
[[Subgroup]]: 2.3.7
== 5-limit (blackwood) ==
Comma: 256/243


[[POTE generator]]: 399.594
[[Comma list]]: 64/63


Map: [<5 8 0|, <0 0 1|]
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}


{{EDOs|legend=1| 5, 10, 15}}
: sval mapping generators: ~2, ~3


Badness: 0.0638
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}


== 7-limit ==
: [[gencom]]: [2 3; 64/63]
Commas: 28/27, 49/48


[[POTE_tuning|POTE generator]]: ~5/4 = 392.767
[[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 }}


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


Wedgie: <<0 5 0 8 0 -14||
[[Badness]] (Sintel): 0.159


{{EDOs|legend=1| 5, 10, 15, 40b, 55b}}
Scales: [[archy5]], [[archy7]], [[archy12]]


Badness: 0.0256
=== 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]].  


== 11-limit ==
These all use the same generators as archy.
Commas: 28/27, 49/48, 55/54


POTE generator: ~5/4 = 394.948
[[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.  


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


{{EDOs|legend=1| 5, 10, 15, 40be, 55be, 70bde, 85bcde}}
Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.


Badness: 0.0246
==== 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]].


=== 13-limit ===
=== Supra ===
Commas: 28/27, 40/39, 49/48, 55/54
Subgroup: 2.3.7.11
 
Comma list: 64/63, 99/98
 
Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}
 
Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}
 
: gencom: [2 3; 64/63 99/98]
 
Optimal tunings:
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}
 
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}
 
Badness (Sintel): 0.352
 
Scales: [[supra7]], [[supra12]]
 
==== Supraphon ====
Subgroup: 2.3.7.11.13
 
Comma list: 64/63, 78/77, 99/98
 
Sval 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 }}
 
: gencom: [2 3; 64/63 78/77 99/98]
 
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


POTE generator: ~5/4 = 391.0367
Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}


Map: [<5 8 0 14 29 7|, <0 0 1 0 -1 1|]
Optimal tunings:  
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


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


Badness: 0.0205
Badness (Sintel): 1.08


== 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: 55/54, 64/63, 65/63, 99/98


Map: [<5 8 0 14 -6|, <0 0 1 0 2|]
Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}


{{EDOs|legend=1| 5e, 15 }}
Optimal tunings:
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}


Badness: 0.0292
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}


=== 13-limit ===
Badness (Sintel): 1.50
Commas: 28/27, 40/39, 49/48, 66/65


POTE generator: ~5/4 = 396.812
== 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).  


Map: [<5 8 0 14 -6 7|, <0 0 1 0 2 1|]
[[Subgroup]]: 2.3.5.7


{{EDOs|legend=1| 5e, 10e, 15 }}
[[Comma list]]: 64/63, 2430/2401


Badness: 0.0223
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}


== Ferrum ==
[[Optimal tuning]]s:
Commas: 28/27, 35/33, 49/48
* [[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 }}


POTE generator: ~5/4 = 374.763
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}


Map: [<5 8 0 14 6|, <0 0 1 0 1|]
[[Badness]] (Sintel): 1.61


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


Badness: 0.0309
Subgroup: 2.3.5.7.11


[[File:blacksmith10.jpg|alt=blacksmith10.jpg|blacksmith10.jpg]]
Comma list: 64/63, 99/98, 121/120


= Superpyth =
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}
{{main| Superpyth }}


Commas: 64/63, 245/243
Optimal tunings:  
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


[[POTE generator]]: 710.291
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}


Map: [<1 0 -12 6|, <0 1 9 -2|]
Badness (Sintel): 1.06


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


{{EDOs|legend=1| 5, 17, 22, 27, 49 }}
Comma list: 64/63, 78/77, 91/90, 121/120


Badness: 0.0323
Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}


== 11-limit ==
Optimal tunings:
Commas: 64/63, 100/99, 245/243
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


[[POTE_tuning|POTE generator]]: 710.175
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]
Badness (Sintel): 1.25


{{EDOs|legend=1| 22, 49 }}
=== Quasisoup ===
Subgroup: 2.3.5.7.11


Badness: 0.0250
Comma list: 55/54, 64/63, 2430/2401


=== 13-limit ===
Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
Commas: 64/63, 78/77, 91/90, 100/99


POTE generator: ~3/2 = 710.479
Optimal tunings:  
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


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


{{EDOs|legend=1| 22, 27e, 49, 76bcde }}
Badness (Sintel): 2.76


Badness: 0.0247
== Ultrapyth ==
{{Main| Ultrapyth }}


== Suprapyth ==
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: 55/54, 64/63, 99/98


POTE generator: ~3/2 = 709.495
[[Subgroup]]: 2.3.5.7


Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]
[[Comma list]]: 64/63, 6860/6561


{{EDOs|legend=1| 5, 7, 12, 17, 22 }}
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}


Badness: 0.0328
[[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 }}


=== 13-limit ===
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
Commas: 55/54, 64/63, 65/63, 364/363


POTE generator: ~3/2 = 708.703
[[Badness]] (Sintel): 2.74


Map: [<1 0 -12 6 13 18|, <0 1 9 -2 -6 -9|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{EDOs|legend=1| 17, 22, 83cdf }}
Comma list: 55/54, 64/63, 2401/2376


Badness: 0.0363
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}


= Beatles =
Optimal tunings:
== 5-limit ==
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
Comma: 524288/492075
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


POTE generator: ~512/405 = 355.930
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Map: [<1 1 5|,<0 2 -9|]
Badness (Sintel): 2.26


{{EDOs|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.3585
Comma list: 55/54, 64/63, 91/90, 1573/1568


== 7-limit ==
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
Commas: 64/63, 686/675


[[POTE generator]]: ~49/40 = 355.904
Optimal tunings:  
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


Map: [<1 1 5 4|,<0 2 -9 -4|]
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Wedgie: <<2 -9 -4 -19 -12 16||
Badness (Sintel): 2.03


{{EDOs|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
=== Ultramarine ===
Subgroup: 2.3.5.7.11


Badness: 0.0459
Comma list: 64/63, 100/99, 3773/3645


Music: [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 Beatles Improv] by Herman Miller
Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}


== 11-limit ==
Optimal tunings:
Commas: 64/63, 100/99, 686/675
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


POTE generator: ~49/40 = 356.140
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}


Map: [<1 1 5 4 10|,<0 2 -9 -4 -22|]
Badness (Sintel): 2.58


{{EDOs|legend=1| 27e, 37, 64be, 91bcde }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0456
Comma list: 64/63, 91/90, 100/99, 847/845


=== 13-limit ===
Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
Commas: 64/63, 91/90, 100/99, 169/168


POTE generator: ~16/13 = 356.229
Optimal tunings:  
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}


Map: [<1 1 5 4 10 4|,<0 2 -9 -4 -22 -1|]
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}


{{EDOs|legend=1| 27e, 37, 64be }}
Badness (Sintel): 1.89


Badness: 0.0302
== 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).  


== Ringo ==
[[Subgroup]]: 2.3.5.7
Commas: 56/55, 64/63, 540/539


POTE generator: ~11/9 = 355.419
[[Comma list]]: 64/63, 33614/32805


Map: [<1 1 5 4 2|,<0 2 -9 -4 5|]
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}


{{EDOs|legend=1| 10, 17c, 27e }}
[[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: 0.0329
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}


=== 13-limit ===
[[Badness]] (Sintel): 3.34
Commas: 56/55, 64/63, 78/77, 91/90


POTE generator: ~11/9 = 355.456
== Schism ==
{{See also| Schismatic family #Schism }}


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


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


Badness: 0.0226
[[Comma list]]: 64/63, 360/343


= Schism =
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}
{{see also|Schismatic family #Schism}}


Commas: 64/63, 360/343
[[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 }}


[[POTE generator]]: ~3/2 = 701.556
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}


Map: [<1 0 15 6|, <0 1 -8 -2|]
[[Badness]] (Sintel): 1.43


Wedgie: <<1 -8 -2 -15 -6 18||
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{EDOs|legend=1| 12, 41d, 53d }}
Comma list: 45/44, 64/63, 99/98


Badness: 0.0566
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


== 11-limit ==
Optimal tunings:
Commas: 45/44, 64/63, 99/98
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


POTE generator ~3/2 = 702.136
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}


Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]
Badness (Sintel): 1.24


{{EDOs|legend=1| 12, 29de, 41de }}
== Beatles ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''


Badness: 0.0375
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.  


= Passion =
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.
== 5-limit ==
Comma: 262144/253125


POTE generator: ~16/15 = 98.670
[[Subgroup]]: 2.3.5.7


Map: [<1 2 2|, <0 -5 4|]
[[Comma list]]: 64/63, 686/675


{{EDOs|legend=1| 11, 12, 49, 61, 73 }}
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}


Badness: 0.1686
[[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 }}


=== Passive ===
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
Commas: 225/224, 256/245


POTE generator: ~16/15 = 98.809
[[Badness]] (Sintel): 1.16


Map: [<1 2 2 3|, <0 -5 4 -2|]
; 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]]


{{EDOs|legend=1| 1, 11, 12, 49d }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0751
Comma list: 64/63, 100/99, 686/675


== 7-limit ==
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
Commas: 64/63, 3125/3087


[[POTE generator]]: ~16/15 = 98.153
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 2 2|, <0 -5 4 10|]
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}


Wedgie: <<5 -4 -10 -18 -30 -12||
Badness (Sintel): 1.51


Generators: 2, 16/15
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{EDOs|legend=1| 12, 37, 49, 110bcd }}
Comma list: 64/63, 91/90, 100/99, 169/168


Badness: 0.0623
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}


== 11-limit ==
Optimal tunings:
Commas: 64/63, 100/99, 1375/1372
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


POTE generator: ~16/15 = 98.019
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}


Map: [<1 2 2 2 2|, <0 -5 4 10 18|]
Badness (Sintel): 1.25


{{EDOs|legend=1| 12, 37, 49 }}
=== Ringo ===
Subgroup: 2.3.5.7.11


Badness: 0.0408
Comma list: 56/55, 64/63, 540/539


== 13-limit ==
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}
Commas: 64/63, 100/99, 196/195, 275/273


POTE generator: ~16/15 = 97.910
Optimal tunings:  
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


Map: [<1 2 2 2 2 2|, <0 -5 4 10 18 21|]
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


{{EDOs|legend=1| 12f, 37, 49f }}
Badness (Sintel): 1.09


Badness: 0.0309
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


= Fervor =
Comma list: 56/55, 64/63, 78/77, 91/90
== 5-limit ==
Comma: 67108864/61509375


POTE generator: ~64/45 = 577.705
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}


Map: [<1 4 -2|, <0 -5 9|]
Optimal tunings:  
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}


{{EDOs|legend=1| 25, 27 }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


Badness: 0.8526
Badness (Sintel): 0.935


== 7-limit ==
=== Beetle ===
Commas: 64/63, 9604/9375
Subgroup: 2.3.5.7.11


POTE generator: ~7/5 = 577.777
Comma list: 55/54, 64/63, 686/675


Map: [<1 4 -2 -2|, <0 -5 9 10|]
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}


Wedgie: <<5 -9 -10 -26 -30 2||
Optimal tunings:  
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


{{EDOs|legend=1| 25, 27 }}
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Badness: 0.1085
Badness (Sintel): 1.92


== 11-limit ==
==== 13-limit ====
Commas: 56/55, 64/63, 1350/1331
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 577.850
Comma list: 55/54, 64/63, 91/90, 169/168


Map: [<1 4 -2 -2 3|, <0 -5 9 10 1|]
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}


{{EDOs|legend=1| 25e, 27e }}
Optimal tunings:
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


Badness: 0.0521
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


== 13-limit ==
Badness (Sintel): 1.40
Commas: 56/55, 64/63, 78/77, 507/500


POTE generator: ~7/5 = 578.060
== Progress ==
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''


Map: [<1 4 -2 -2 3 -4|, <0 -5 9 10 1 16|]
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.


{{EDOs|legend=1| 27e }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0397
[[Comma list]]: 64/63, 392/375


= Quasisuper =
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
Commas: 64/63, 2430/2401


[[POTE_tuning|POTE generator]]: 708.328
: mapping generators: ~2, ~10/7


Map: [<1 0 23 6|, <0 1 -13 -2|]
[[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 }}


Wedgie: <<1 -13 -2 -23 -2 -6 32||
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


{{EDOs|legend=1| 22, 61 }}
[[Badness]] (Sintel): 1.68


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


== Quasisupra ==
Comma list: 56/55, 64/63, 77/75
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: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}


POTE generator: ~3/2 = 708.205
Optimal tunings:
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}


Map: [<1 2 -3 2 1|, <0 -1 13 2 6|]
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}


{{EDOs|legend=1| 17c, 22, 27c, 39d, 61d }}
Badness (Sintel): 1.03


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


=== 13-limit ===
Comma list: 56/55, 64/63, 66/65, 77/75
Commas: 64/63, 78/77, 91/90, 121/120


POTE generator: ~3/2 = 708.004
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}


Map: [<1 0 23 6 13 18|, <0 1 -13 -2 -6 -9|]
Optimal tunings:  
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}


{{EDOs|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}


Badness: 0.0302
Badness (Sintel): 1.08


== Quasisoup ==
==== Progressive ====
Commas: 55/54, 64/63, 2430/2401
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 709.021
Comma list: 26/25, 56/55, 64/63, 77/75


Map: [<1 0 23 6 -22|, <0 1 -13 -2 16|]
Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}


{{EDOs|legend=1| 22 }}
Optimal tunings:
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}


Badness: 0.0835
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}


= Progress =
Badness (Sintel): 1.35
== 5-limit ==
Comma: 32768/30375


POTE generator: ~64/45 = 561.264
== Fervor ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''


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


{{EDOs|legend=1| 4, 13, 15, 32c, 47bc, 62bc }}
[[Subgroup]]: 2.3.5.7


Badness: 0.2461
[[Comma list]]: 64/63, 9604/9375


== 7-limit ==
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
Commas: 64/63, 392/375


POTE generator: ~7/5 = 562.122
: mapping generators: ~2, ~10/7


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


Wedgie: <<3 -5 -6 -15 -18 0||
{{Optimal ET sequence|legend=1| 2, 25, 27 }}


{{EDOs|legend=1| 13, 15, 32c, 79bcc, 111bcc }}
[[Badness]] (Sintel): 2.74


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


== 11-limit ==
Comma list: 56/55, 64/63, 1350/1331
Commas: 56/55, 64/63, 77/75


POTE generator: ~7/5 = 562.085
Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}


Map: [<1 0 5 6 4|, <0 3 -5 -6 -1|]
Optimal tunings:  
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}


{{EDOs|legend=1| 13, 15, 32c, 47bc, 79bcce }}
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}


Badness: 0.0310
Badness (Sintel): 1.72


=== 13-limit ===
=== 13-limit ===
Commas: 56/55, 64/63, 66/65, 77/75
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 }}


POTE generator: ~7/5 = 562.365
[[Badness]] (Sintel): 4.02


Map: [<1 0 5 6 4 0|, <0 3 -5 -6 -1 7|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{EDOs|legend=1| 15, 17c, 32cf }}
Comma list: 55/54, 64/63, 125/121


Badness: 0.0262
Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}


=== Progressive ===
Optimal tunings:
Commas: 26/25, 56/55, 64/63, 77/75
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}


POTE generator: ~7/5 = 563.239
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}


Map: [<1 0 5 6 4 9|, <0 3 -5 -6 -1 -10|]
Badness (Sintel): 2.34


{{EDOs|legend=1| 15f, 17c, 32c, 49c }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0327
Comma list: 40/39, 55/54, 64/63, 125/121


= Sixix =
Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
== 5-limit ==
Comma: 3125/2916


POTE generator: ~6/5 = 338.365
Optimal tunings:  
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}


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


EDOs: {{EDOs| 7, 25, 32 }}
Badness (Sintel): 1.91


Badness: 0.1531
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 7-limit ==
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121
Commas: 3125/2916, 64/63


POTE generator: ~6/5 = 337.4419
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}


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


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


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]]

Latest revision as of 12:52, 22 July 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The archytas clan (or archy family) tempers out the Archytas' comma, 64/63. This means a stack of two 3/2 fifths octave-reduced equals a whole tone of 8/7~9/8 tempered together; two of these tones or equivalently four stacked fifths octave-reduced equal a 9/7 major third. Note the similarity in function to 81/80 in meantone, where four stacked fifths octave-reduced equal a 5/4 major third. This leads to tunings with 3's and 7's quite sharp, such as those of 22edo, 27edo, or 49edo.

This article focuses on rank-2 temperaments. See Archytas family for the rank-3 temperament resulting from tempering out 64/63 alone in the full 7-limit.

Archy

Main article: Superpyth

Subgroup: 2.3.7

Comma list: 64/63

Sval mapping: [⟨1 0 6], ⟨0 1 -2]]

sval mapping generators: ~2, ~3

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

gencom: [2 3; 64/63]

Optimal tunings:

  • WE: ~2 = 1196.9552 ¢, ~3/2 = 707.5215 ¢
error map: ⟨-3.045 +2.522 +3.952]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.3901 ¢
error map: ⟨0.000 +7.435 +12.394]

Optimal ET sequence: 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd

Badness (Sintel): 0.159

Scales: archy5, archy7, archy12

Overview to extensions

7-limit extensions

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

These all use the same generators as archy.

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.

Temperaments discussed elsewhere are:

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

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.

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

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

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

gencom: [2 3; 64/63 99/98]

Optimal tunings:

  • WE: ~2 = 1197.2650 ¢, ~3/2 = 705.5803 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 707.4981 ¢

Optimal ET sequence: 5, 12, 17, 39d, 56d

Badness (Sintel): 0.352

Scales: supra7, supra12

Supraphon

Subgroup: 2.3.7.11.13

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

Sval mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]

Gencom mapping: [⟨1 0 0 6 13 18], ⟨0 1 0 -2 -6 -9]]

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

Optimal tunings:

  • WE: ~2 = 1197.1909 ¢, ~3/2 = 704.4836 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 706.4289 ¢

Optimal ET sequence: 12f, 17

Badness (Sintel): 0.498

Scales: supra7, supra12

Superpyth

Main article: Superpyth
For the 5-limit version, see Syntonic–diatonic equivalence continuum #Superpyth (5-limit).

Superpyth, virtually the canonical extension, adds 245/243 and 1728/1715 to the comma list and can be described as 22 & 27. ~5/4 is found at +9 generator steps, as an augmented second (C–D#). 49edo remains an obvious tuning choice.

Subgroup: 2.3.5.7

Comma list: 64/63, 245/243

Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]

Optimal tunings:

  • WE: ~2 = 1197.0549 ¢, ~3/2 = 708.5478 ¢
error map: ⟨-2.945 +3.648 -0.548 +2.298]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.1193 ¢
error map: ⟨0.000 +8.164 +4.760 +10.935]

Optimal ET sequence: 5, 17, 22, 27, 49, 174bbcddd

Badness (Sintel): 0.818

11-limit

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: [⟨1 0 -12 6 -22], ⟨0 1 9 -2 16]]

Optimal tunings:

  • WE: ~2 = 1197.0673 ¢, ~3/2 = 708.4391 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.0129 ¢

Optimal ET sequence: 22, 27e, 49

Badness (Sintel): 0.826

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -12 6 -22 -17], ⟨0 1 9 -2 16 13]]

Optimal tunings:

  • WE: ~2 = 1197.3011 ¢, ~3/2 = 708.8813 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.3219 ¢

Optimal ET sequence: 22, 27e, 49, 76bcde

Badness (Sintel): 1.02

Thomas

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 169/168, 245/243

Mapping: [⟨1 1 -3 4 -6 4], ⟨0 2 18 -4 32 -1]]

Optimal tunings:

  • WE: ~2 = 1197.4942 ¢, ~16/13 = 354.2950 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/13 = 354.9824 ¢

Optimal ET sequence: 27e, 44, 71d, 98bde

Badness (Sintel): 2.03

Suprapyth

Suprapyth finds the ~11/8 at the diminished fifth (C–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: [⟨1 0 -12 6 13], ⟨0 1 9 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1198.6960 ¢, ~3/2 = 708.7235 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.4699 ¢

Optimal ET sequence: 5, 17, 22

Badness (Sintel): 1.08

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -12 6 13 18], ⟨0 1 9 -2 -6 -9]]

Optimal tunings:

  • WE: ~2 = 1199.9871 ¢, ~3/2 = 708.6952 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.7028 ¢

Optimal ET sequence: 5f, 17, 22

Badness (Sintel): 1.50

Quasisuper

Quasisuper can be described as 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: [⟨1 0 23 6], ⟨0 1 -13 -2]]

Optimal tunings:

  • WE: ~2 = 1196.9830 ¢, ~3/2 = 706.4578 ¢
error map: ⟨-3.017 +1.486 -0.435 +6.190]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.3716 ¢
error map: ⟨0.000 +6.417 +4.855 +14.431]

Optimal ET sequence: 17c, 22, 61d

Badness (Sintel): 1.61

Quasisupra

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

Mapping: [⟨1 0 23 6 13], ⟨0 1 -13 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1197.5675 ¢, ~3/2 = 706.7690 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.3200 ¢

Optimal ET sequence: 17c, 22, 39d, 61d

Badness (Sintel): 1.06

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 121/120

Mapping: [⟨1 0 23 6 13 18], ⟨0 1 -13 -2 -6 -9]]

Optimal tunings:

  • WE: ~2 = 1198.2543 ¢, ~3/2 = 706.9736 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.0936 ¢

Optimal ET sequence: 17c, 22, 39d

Badness (Sintel): 1.25

Quasisoup

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 23 6 -22], ⟨0 1 -13 -2 16]]

Optimal tunings:

  • WE: ~2 = 1198.8446 ¢, ~3/2 = 708.3388 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 708.0252 ¢

Optimal ET sequence: 22

Badness (Sintel): 2.76

Ultrapyth

Main article: Ultrapyth

Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 oceanfront temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–Cx).

Subgroup: 2.3.5.7

Comma list: 64/63, 6860/6561

Mapping: [⟨1 0 -20 6], ⟨0 1 14 -2]]

Optimal tunings:

  • WE: ~2 = 1197.2673 ¢, ~3/2 = 712.0258 ¢
error map: ⟨-2.733 +7.338 -1.557 -3.808]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.5430 ¢
error map: ⟨0.000 +11.588 +3.288 +4.088]

Optimal ET sequence: 5, 27c, 32, 37

Badness (Sintel): 2.74

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -20 6 21], ⟨0 1 14 -2 -11]]

Optimal tunings:

  • WE: ~2 = 1198.0290 ¢, ~3/2 = 712.2235 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.3754 ¢

Optimal ET sequence: 5, 32, 37

Badness (Sintel): 2.26

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -20 6 21 -25], ⟨0 1 14 -2 -11 18]]

Optimal tunings:

  • WE: ~2 = 1198.1911 ¢, ~3/2 = 712.4243 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.4684 ¢

Optimal ET sequence: 5, 32, 37

Badness (Sintel): 2.03

Ultramarine

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -20 6 -38], ⟨0 1 14 -2 26]]

Optimal tunings:

  • WE: ~2 = 1197.2230 ¢, ~3/2 = 712.1393 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.6928 ¢

Optimal ET sequence: 5e, 32e, 37, 79bce

Badness (Sintel): 2.58

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -20 6 -38 -25], ⟨0 1 14 -2 26 18]]

Optimal tunings:

  • WE: ~2 = 1197.2739 ¢, ~3/2 = 712.1893 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.7079 ¢

Optimal ET sequence: 5e, 32e, 37, 79bcef

Badness (Sintel): 1.89

Quasiultra

Quasiultra is to ultrapyth what quasisuper is to superpyth. It is the 27 & 32 temperament, mapping the ~5/4 to -18 fifths as a double diminished sixth (C–Abbb).

Subgroup: 2.3.5.7

Comma list: 64/63, 33614/32805

Mapping: [⟨1 0 31 6], ⟨0 1 -18 -2]]

Optimal tunings:

  • WE: ~2 = 1196.9257 ¢, ~3/2 = 709.6211 ¢
error map: ⟨0.000 +9.883 +0.608 +7.499]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.5429 ¢
error map: ⟨0.000 +9.588 +5.914 +8.088]

Optimal ET sequence: 27, 86bd, 113bcd, 140bbcd

Badness (Sintel): 3.34

Schism

See also: Schismatic family § Schism

Schism tempers out the schisma, mapping the ~5/4 to -8 fifths as a diminished fourth (C–Fb) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53dd val) can be used.

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]

Optimal tunings:

  • WE: ~2 = 1197.3598 ¢, ~3/2 = 700.0126 ¢
error map: ⟨-2.640 -4.583 -4.896 +20.588]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7376 ¢
error map: ⟨0.000 -0.217 -0.214 +27.699]

Optimal ET sequence: 5c, 7c, 12

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1196.1607 ¢, ~3/2 = 699.8897 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.4385 ¢

Optimal ET sequence: 5c, 7ce, 12, 29de

Badness (Sintel): 1.24

Beatles

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

Beatles tempers out 686/675, which may also be characterized by saying it tempers out 2401/2400. It may be described as the 10 & 17c temperament. It splits the fifth into two neutral-third generators of 49/40~60/49; its ploidacot is dicot. 5/4 may be found at -9 generator steps, as a semidiminished fourth (C–Fd). 27edo is an obvious tuning, though 17c-edo and 37edo are among the possibilities.

Beatles extends easily to the no-11 13-limit, as the generator can be interpreted as ~16/13, tempering out 91/90, 169/168, and 196/195.

Subgroup: 2.3.5.7

Comma list: 64/63, 686/675

Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4]]

Optimal tunings:

  • WE: ~2 = 1196.6244 ¢, ~49/40 = 354.9029 ¢
error map: ⟨-3.376 +4.475 +2.682 -1.940]
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.0819 ¢
error map: ⟨0.000 +10.209 +8.949 +6.847]

Optimal ET sequence: 10, 17c, 27, 64b, 91bcd, 118bccd

Badness (Sintel): 1.16

Music

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 5 4 10], ⟨0 2 -9 -4 -22]]

Optimal tunings:

  • WE: ~2 = 1196.7001 ¢, ~49/40 = 355.1606 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.2795 ¢

Optimal ET sequence: 10e, 17cee, 27e, 64be, 91bcdee

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 5 4 10 4], ⟨0 2 -9 -4 -22 -1]]

Optimal tunings:

  • WE: ~2 = 1197.2504 ¢, ~16/13 = 355.4132 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/13 = 356.3273 ¢

Optimal ET sequence: 10e, 27e, 37, 64be

Badness (Sintel): 1.25

Ringo

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 5 4 2], ⟨0 2 -9 -4 5]]

Optimal tunings:

  • WE: ~2 = 1195.4102 ¢, ~11/9 = 354.0597 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 355.5207 ¢

Optimal ET sequence: 10, 17c, 27e

Badness (Sintel): 1.09

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 5 4 2 4], ⟨0 2 -9 -4 5 -1]]

Optimal tunings:

  • WE: ~2 = 1195.9943 ¢, ~11/9 = 354.2695 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 355.5398 ¢

Optimal ET sequence: 10, 17c, 27e

Badness (Sintel): 0.935

Beetle

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 5 4 -1], ⟨0 2 -9 -4 15]]

Optimal tunings:

  • WE: ~2 = 1197.9660 ¢, ~49/40 = 356.1056 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/40 = 356.7075 ¢

Optimal ET sequence: 10, 27, 37

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 5 4 -1 4], ⟨0 2 -9 -4 15 -1]]

Optimal tunings:

  • WE: ~2 = 1198.1741 ¢, ~16/13 = 356.1582 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/13 = 356.7008 ¢

Optimal ET sequence: 10, 27, 37

Badness (Sintel): 1.40

Progress

Not to be confused with Progression.
For the 5-limit version, see Miscellaneous 5-limit temperaments #Progress.

Progress tempers out 392/375 and may be described as 15 & 17c. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. 32c-edo gives an obvious tuning.

Subgroup: 2.3.5.7

Comma list: 64/63, 392/375

Mapping: [⟨1 0 5 6], ⟨0 3 -5 -6]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1195.1377 ¢, ~10/7 = 635.2932 ¢
error map: ⟨-4.862 +3.925 +12.908 -9.759]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 638.0791 ¢
error map: ⟨0.000 +12.282 +23.291 +2.700]

Optimal ET sequence: 2, 13, 15, 32c

Badness (Sintel): 1.68

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 5 6 4], ⟨0 3 -5 -6 -1]]

Optimal tunings:

  • WE: ~2 = 1195.4920 ¢, ~10/7 = 635.5183 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 638.0884 ¢

Optimal ET sequence: 2, 13, 15, 32c, 47bc

Badness (Sintel): 1.03

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 5 6 4 0], ⟨0 3 -5 -6 -1 7]]

Optimal tunings:

  • WE: ~2 = 1195.0786 ¢, ~10/7 = 635.0197 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 637.6691 ¢

Optimal ET sequence: 15, 17c, 32cf

Badness (Sintel): 1.08

Progressive

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 5 6 4 9], ⟨0 3 -5 -6 -1 -10]]

Optimal tunings:

  • WE: ~2 = 1196.0245 ¢, ~10/7 = 634.6516 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 636.9528 ¢

Optimal ET sequence: 2f, 15f, 17c

Badness (Sintel): 1.35

Fervor

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

Fervor tempers out 9704/9375 and may be described as 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: [⟨1 -1 7 8], ⟨0 5 -9 -10]]

mapping generators: ~2, ~10/7

Optimal tunings:

  • WE: ~2 = 1196.2742 ¢, ~10/7 = 620.2918 ¢
error map: ⟨-3.726 +3.230 +4.980 -1.550]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.3179 ¢
error map: ⟨0.000 +9.634 +12.826 +7.996]

Optimal ET sequence: 2, 25, 27

Badness (Sintel): 2.74

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 7 8 4], ⟨0 5 -9 -10 -1]]

Optimal tunings:

  • WE: ~2 = 1195.4148 ¢, ~10/7 = 619.7729 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.2525 ¢

Optimal ET sequence: 2, 25e, 27e

Badness (Sintel): 1.72

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 507/500

Mapping: [⟨1 -1 7 8 4 12], ⟨0 5 -9 -10 -1 -16]]

Optimal tunings:

  • WE: ~2 = 1195.6284 ¢, ~10/7 = 619.6738 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 622.0631 ¢

Optimal ET sequence: 2f, 27e

Badness (Sintel): 1.64

Sixix

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Sixix (5-limit).
See also: Dual-fifth temperaments § Dual-3 Sixix

Sixix tempers out 3125/2916 and may be described as 25 & 32. It is related to the kleismic family in a way similar to the one between meantone and mavila. In both cases the generator is nominally a 6/5 and the complexity to generate major and minor chords is the same, but in sixix it is tuned extremely sharply, to the point where the 3rd and 5th harmonics are reached by going down instead of up, inverting the logic of chord construction. Its ploidacot is gamma-pentacot.

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/2916

Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]

Optimal tunings:

  • WE: ~2 = 1198.9028 ¢, ~6/5 = 337.1334 ¢
error map: ⟨-1.097 +9.086 -13.503 +2.508]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.4588 ¢
error map: ⟨0.000 +10.751 -11.066 +5.762]

Optimal ET sequence: 7, 18d, 25, 32

Badness (Sintel): 4.02

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 4 0 6], ⟨0 -5 -6 10 -9]]

Optimal tunings:

  • WE: ~2 = 1198.5480 ¢, ~6/5 = 337.1557 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.6000 ¢

Optimal ET sequence: 7, 25e, 32

Badness (Sintel): 2.34

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 4 0 6 4], ⟨0 -5 -6 10 -9 -1]]

Optimal tunings:

  • WE: ~2 = 1197.7111 ¢, ~6/5 = 336.8391 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.5336 ¢

Optimal ET sequence: 7, 25e, 32f

Badness (Sintel): 1.91

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 55/54, 64/63, 85/84, 125/121

Mapping: [⟨1 3 4 0 6 4 1], ⟨0 -5 -6 10 -9 -1 11]]

Optimal tunings:

  • WE: ~2 = 1197.7807 ¢, ~6/5 = 336.8884 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 337.5279 ¢

Optimal ET sequence: 7, 25e, 32f

Badness (Sintel): 2.00

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