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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) ==


[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]
[[Comma list]]: 64/63


Comma list: 256/243
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}


[[POTE generator]]: ~5/4 = 399.594
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}


Mapping: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
: mapping generators: ~2, ~3


{{Val list|legend=1| 5, 10, 15 }}
[[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 }}


Badness: 0.0638
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}


== 7-limit ==
[[Badness]] (Sintel): 0.159
Comma list: 28/27, 49/48


[[POTE generator]]: ~5/4 = 392.767
Scales: [[archy5]], [[archy7]], [[archy12]]


Mapping: [{{val| 5 8 0 14 }}, {{val| 0 0 1 0 }}]
=== 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]].


Wedgie: {{wedgie| 0 5 0 8 0 -14 }}
These all use the same generators as archy.


{{Val list|legend=1| 5, 10, 15, 40b, 55b }}
[[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.


Badness: 0.0256
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]]


== 11-limit ==
Considered below are superpyth, quasisuper, ultrapyth, quasiultra, schism, beatles, progress, fervor, and sixix.
Comma list: 28/27, 49/48, 55/54


POTE generator: ~5/4 = 394.948
==== 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]].


Mapping: [{{val| 5 8 0 14 29 }}, {{val| 0 0 1 0 -1 }}]
=== Supra ===
Subgroup: 2.3.7.11


{{Val list|legend=1| 5, 10, 15, 40be, 55be, 70bde, 85bcde}}
Comma list: 64/63, 99/98


Badness: 0.0246
Subgroup-val mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}


=== 13-limit ===
Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}
Comma list: 28/27, 40/39, 49/48, 55/54
 
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


POTE generator: ~5/4 = 391.0367
Comma list: 64/63, 78/77, 99/98


Mapping: [{{val| 5 8 0 14 29 7 }}, {{val| 0 0 1 0 -1 1 }}]
Subgroup-val mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}


{{Val list|legend=1| 5, 10, 15, 25e, 40bef}}
Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}


Badness: 0.0205
Optimal tunings:  
* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}


== Farrier ==
{{Optimal ET sequence|legend=0| 12f, 17 }}
Comma list: 28/27, 49/48, 77/75


POTE generator: ~5/4 = 398.070
Badness (Sintel): 0.498


Mapping: [{{val| 5 8 0 14 -6 }}, {{val| 0 0 1 0 2 }}]
Scales: [[supra7]], [[supra12]]


{{Val list|legend=1| 5e, 15 }}
== Superpyth ==
{{Main| Superpyth }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


Badness: 0.0292
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.  


=== 13-limit ===
[[Subgroup]]: 2.3.5.7
Comma list: 28/27, 40/39, 49/48, 66/65


POTE generator: ~5/4 = 396.812
[[Comma list]]: 64/63, 245/243


Mapping: [{{val| 5 8 0 14 -6 7 }}, {{val| 0 0 1 0 2 1 }}]
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}


{{Val list|legend=1| 5e, 10e, 15 }}
[[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 }}


Badness: 0.0223
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}


== Ferrum ==
[[Badness]] (Sintel): 0.818
Comma list: 28/27, 35/33, 49/48


POTE generator: ~5/4 = 374.763
=== 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).  


Mapping: [{{val| 5 8 0 14 6 }}, {{val| 0 0 1 0 1 }}]
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 10 }}
Comma list: 64/63, 100/99, 245/243


Badness: 0.0309
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}


= Superpyth =
Optimal tunings:
{{main| Superpyth }}
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


Comma list: 64/63, 245/243
{{Optimal ET sequence|legend=0| 22, 27e, 49 }}


[[POTE generator]]: ~3/2 = 710.291
Badness (Sintel): 0.826


Mapping: [{{val| 1 0 -12 6 }}, {{val| 0 1 9 -2 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Wedgie: {{wedgie| 1 9 -2 12 -6 -30 }}
Comma list: 64/63, 78/77, 91/90, 100/99


{{Val list|legend=1| 5, 17, 22, 27, 49 }}
Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}


Badness: 0.0323
Optimal tunings:  
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}
Comma list: 64/63, 100/99, 245/243


[[POTE generator]]: ~3/2 = 710.175
Badness (Sintel): 1.02


Mapping: [{{val| 1 0 -12 6 -22 }}, {{val| 0 1 9 -2 16 }}]
==== Thomas ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 22, 27e, 49 }}
Comma list: 64/63, 100/99, 169/168, 245/243


Badness: 0.0250
Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}


=== 13-limit ===
Optimal tunings:
Comma list: 64/63, 78/77, 91/90, 100/99
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


POTE generator: ~3/2 = 710.479
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}


Mapping: [{{val| 1 0 -12 6 -22 -17 }}, {{val| 0 1 9 -2 16 13 }}]
Badness (Sintel): 2.03


{{Val list|legend=1| 22, 27e, 49, 76bcde }}
=== Suprapyth ===
Suprapyth finds the ~11/8 at the diminished fifth (C–Gb), and finds the ~13/8 at the diminished seventh (C–Bbb).


Badness: 0.0247
Subgroup: 2.3.5.7.11


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


POTE generator: ~3/2 = 709.495
Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}
 
Optimal tunings:
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


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


{{Val list|legend=1| 17, 22 }}
Badness (Sintel): 1.08


Badness: 0.0328
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 55/54, 64/63, 65/63, 99/98
Comma list: 55/54, 64/63, 65/63, 99/98


POTE generator: ~3/2 = 708.703
Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}
 
Optimal tunings:
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}
 
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}
 
Badness (Sintel): 1.50
 
== Quasisuper ==
Quasisuper can be described as {{nowrap| 17c & 22 }}, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–Gbb).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 64/63, 2430/2401
 
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}
: [[error map]]: {{val| -3.017 +1.486 -0.435 +6.190 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 708.3716{{c}}
: error map: {{val| 0.000 +6.417 +4.855 +14.431 }}
 
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}
 
[[Badness]] (Sintel): 1.61
 
=== Quasisupra ===
Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament [[supra]], with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).
 
Subgroup: 2.3.5.7.11
 
Comma list: 64/63, 99/98, 121/120
 
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}
 
Optimal tunings:
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


Mapping: [{{val| 1 0 -12 6 13 18 }}, {{val| 0 1 9 -2 -6 -9 }}]
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}


{{Val list|legend=1| 17, 22, 83cdf }}
Badness (Sintel): 1.06


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


= Beatles =
Comma list: 64/63, 78/77, 91/90, 121/120
== 5-limit ==
Comma list: 524288/492075


POTE generator: ~512/405 = 355.930
Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}


Mapping: [{{val| 1 1 5 }}, {{val| 0 2 -9 }}]
Optimal tunings:  
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


{{Val list|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


Badness: 0.3585
Badness (Sintel): 1.25


== 7-limit ==
=== Quasisoup ===
Comma list: 64/63, 686/675
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~49/40 = 355.904
Comma list: 55/54, 64/63, 2430/2401


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


Wedgie: {{wedgie| 2 -9 -4 -19 -12 16 }}
Optimal tunings:  
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


{{Val list|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
{{Optimal ET sequence|legend=0| 22 }}


Badness: 0.0459
Badness (Sintel): 2.76


Music: [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 Beatles Improv] by Herman Miller
== Ultrapyth ==
{{Main| Ultrapyth }}


== 11-limit ==
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).
Comma list: 64/63, 100/99, 686/675


POTE generator: ~49/40 = 356.140
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 1 5 4 10 }}, {{val| 0 2 -9 -4 -22 }}]
[[Comma list]]: 64/63, 6860/6561


{{Val list|legend=1| 27e, 37, 64be, 91bcde }}
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}


Badness: 0.0456
[[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 }}
Comma list: 64/63, 91/90, 100/99, 169/168


POTE generator: ~16/13 = 356.229
[[Badness]] (Sintel): 2.74


Mapping: [{{val| 1 1 5 4 10 4 }}, {{val| 0 2 -9 -4 -22 -1 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 27e, 37, 64be }}
Comma list: 55/54, 64/63, 2401/2376


Badness: 0.0302
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}


== Ringo ==
Optimal tunings:
Comma list: 56/55, 64/63, 540/539
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


POTE generator: ~11/9 = 355.419
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Mapping: [{{val| 1 1 5 4 2 }}, {{val| 0 2 -9 -4 5 }}]
Badness (Sintel): 2.26


{{Val list|legend=1| 10, 17c, 27e }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0329
Comma list: 55/54, 64/63, 91/90, 1573/1568


=== 13-limit ===
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
Comma list: 56/55, 64/63, 78/77, 91/90


POTE generator: ~11/9 = 355.456
Optimal tunings:  
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


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


{{Val list|legend=1| 10, 17c, 27e }}
Badness (Sintel): 2.03


Badness: 0.0226
=== Ultramarine ===
Subgroup: 2.3.5.7.11


= Schism =
Comma list: 64/63, 100/99, 3773/3645
{{see also|Schismatic family #Schism}}


Comma list: 64/63, 360/343
Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}


[[POTE generator]]: ~3/2 = 701.556
Optimal tunings:  
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


Mapping: [{{val| 1 0 15 6 }}, {{val| 0 1 -8 -2 }}]
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}


Wedgie: {{wedgie| 1 -8 -2 -15 -6 18 }}
Badness (Sintel): 2.58


{{Val list|legend=1| 12, 41d, 53d }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0566
Comma list: 64/63, 91/90, 100/99, 847/845


== 11-limit ==
Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
Comma list: 45/44, 64/63, 99/98


POTE generator ~3/2 = 702.136
Optimal tunings:
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}


Mapping: [{{val| 1 0 15 6 13 }}, {{val| 0 1 -8 -2 -6 }}]
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}


{{Val list|legend=1| 12, 29de, 41de }}
Badness (Sintel): 1.89


Badness: 0.0375
== 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).  


= Passion =
[[Subgroup]]: 2.3.5.7
== 5-limit ==
Comma list: 262144/253125


POTE generator: ~16/15 = 98.670
[[Comma list]]: 64/63, 33614/32805


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


{{Val list|legend=1| 11, 12, 49, 61, 73 }}
[[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: 0.1686
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}


== 7-limit ==
[[Badness]] (Sintel): 3.34
Comma list: 64/63, 3125/3087


[[POTE generator]]: ~16/15 = 98.153
== Schism ==
{{See also| Schismatic family #Schism }}


Mapping: [{{val| 1 2 2 2 }}, {{val| 0 -5 4 10 }}]
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.


Mapping generators: 2, 16/15
[[Subgroup]]: 2.3.5.7


Wedgie: {{wedgie| 5 -4 -10 -18 -30 -12 }}
[[Comma list]]: 64/63, 360/343


{{Val list|legend=1| 12, 37, 49, 110bcd }}
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}


Badness: 0.0623
[[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 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
Comma list: 64/63, 100/99, 1375/1372


POTE generator: ~16/15 = 98.019
[[Badness]] (Sintel): 1.43


Mapping: [{{val| 1 2 2 2 2 }}, {{val| 0 -5 4 10 18 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 12, 37, 49 }}
Comma list: 45/44, 64/63, 99/98


Badness: 0.0408
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


== 13-limit ==
Optimal tunings:
Comma list: 64/63, 100/99, 196/195, 275/273
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


POTE generator: ~16/15 = 97.910
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}


Mapping: [{{val| 1 2 2 2 2 2 }}, {{val| 0 -5 4 10 18 21 }}]
Badness (Sintel): 1.24


{{Val list|legend=1| 12f, 37, 49f }}
== Beatles ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''


Badness: 0.0309
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.  


= Fervor =
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 list: 67108864/61509375


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


Mapping: [{{val| 1 4 -2 }}, {{val| 0 -5 9 }}]
[[Comma list]]: 64/63, 686/675


{{Val list|legend=1| 25, 27 }}
{{Mapping|legend=1| 1 1 5 4 | 0 2 -9 -4 }}


Badness: 0.8526
[[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 }}


== 7-limit ==
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
Comma list: 64/63, 9604/9375


POTE generator: ~7/5 = 577.777
[[Badness]] (Sintel): 1.16


Mapping: [{{val| 1 4 -2 -2 }}, {{val| 0 -5 9 10 }}]
; 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]]


Wedgie: {{wedgie| 5 -9 -10 -26 -30 2 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 25, 27 }}
Comma list: 64/63, 100/99, 686/675


Badness: 0.1085
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}


== 11-limit ==
Optimal tunings:
Comma list: 56/55, 64/63, 1350/1331
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


POTE generator: ~7/5 = 577.850
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}


Mapping: [{{val| 1 4 -2 -2 3 }}, {{val| 0 -5 9 10 1 }}]
Badness (Sintel): 1.51


{{Val list|legend=1| 25e, 27e }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0521
Comma list: 64/63, 91/90, 100/99, 169/168


== 13-limit ==
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
Comma list: 56/55, 64/63, 78/77, 507/500


POTE generator: ~7/5 = 578.060
Optimal tunings:  
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


Mapping: [{{val| 1 4 -2 -2 3 -4 }}, {{val| 0 -5 9 10 1 16 }}]
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}


{{Val list|legend=1| 27e }}
Badness (Sintel): 1.25


Badness: 0.0397
=== Ringo ===
Subgroup: 2.3.5.7.11


= Quasisuper =
Comma list: 56/55, 64/63, 540/539
Comma list: 64/63, 2430/2401


[[POTE generator]]: 708.328
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}


Mapping: [{{val| 1 0 23 6 }}, {{val| 0 1 -13 -2 }}]
Optimal tunings:  
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


Wedgie: {{wedgie| 1 -13 -2 -23 -2 -6 32 }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


{{Val list|legend=1| 22, 61 }}
Badness (Sintel): 1.09


Badness: 0.0638
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== Quasisupra ==
Comma list: 56/55, 64/63, 78/77, 91/90
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).


Comma list: 64/63, 99/98, 121/120
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}


POTE generator: ~3/2 = 708.205
Optimal tunings:
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}


Mapping: [{{val| 1 2 -3 2 1 }}, {{val| 0 -1 13 2 6 }}]
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


{{Val list|legend=1| 17c, 22, 27c, 39d, 61d }}
Badness (Sintel): 0.935


Badness: 0.0322
=== Beetle ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 55/54, 64/63, 686/675
Comma list: 64/63, 78/77, 91/90, 121/120


POTE generator: ~3/2 = 708.004
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}


Mapping: [{{val| 1 0 23 6 13 18 }}, {{val| 0 1 -13 -2 -6 -9 }}]
Optimal tunings:  
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


{{Val list|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


Badness: 0.0302
Badness (Sintel): 1.92


== Quasisoup ==
==== 13-limit ====
Comma list: 55/54, 64/63, 2430/2401
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 709.021
Comma list: 55/54, 64/63, 91/90, 169/168


Mapping: [{{val| 1 0 23 6 -22 }}, {{val| 0 1 -13 -2 16 }}]
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}


{{Val list|legend=1| 22 }}
Optimal tunings:
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


Badness: 0.0835
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


= Progress =
Badness (Sintel): 1.40
== 5-limit ==
Comma list: 32768/30375


POTE generator: ~64/45 = 561.264
== Progress ==
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''


Mapping: [{{val| 1 0 5 }}, {{val| 0 3 -5 }}]
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.


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


Badness: 0.2461
[[Comma list]]: 64/63, 392/375


== 7-limit ==
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
Comma list: 64/63, 392/375


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


Mapping: [{{val| 1 0 5 6 }}, {{val| 0 3 -5 -6 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}
: [[error map]]: {{val| -4.862 +3.925 +12.908 -9.759 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 638.0791{{c}}
: error map: {{val| 0.000 +12.282 +23.291 +2.700 }}


Wedgie: {{wedgie| 3 -5 -6 -15 -18 0 }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


{{Val list|legend=1| 13, 15, 32c, 79bcc, 111bcc }}
[[Badness]] (Sintel): 1.68


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


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


POTE generator: ~7/5 = 562.085
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}}


Mapping: [{{val| 1 0 5 6 4 }}, {{val| 0 3 -5 -6 -1 }}]
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}


{{Val list|legend=1| 13, 15, 32c, 47bc, 79bcce }}
Badness (Sintel): 1.03


Badness: 0.0310
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 56/55, 64/63, 66/65, 77/75
Comma list: 56/55, 64/63, 66/65, 77/75


POTE generator: ~7/5 = 562.365
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}


Mapping: [{{val| 1 0 5 6 4 0 }}, {{val| 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}}


{{Val list|legend=1| 15, 17c, 32cf }}
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}


Badness: 0.0262
Badness (Sintel): 1.08
 
==== Progressive ====
Subgroup: 2.3.5.7.11.13


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


POTE generator: ~7/5 = 563.239
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}}


Mapping: [{{val| 1 0 5 6 4 9 }}, {{val| 0 3 -5 -6 -1 -10 }}]
{{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


{{Val list|legend=1| 15f, 17c, 32c, 49c }}
Comma list: 40/39, 55/54, 64/63, 125/121


Badness: 0.0327
Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}


= Sixix =
Optimal tunings:
== 5-limit ==
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
Comma list: 3125/2916
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}


POTE generator: ~6/5 = 338.365
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


Mapping: [{{val| 1 3 4 }}, {{val| 0 -5 -6 }}]
Badness (Sintel): 1.91


{{Val list|legend=1| 7, 25, 32 }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.1531
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121


== 7-limit ==
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
Comma list: 3125/2916, 64/63


POTE generator: ~6/5 = 337.4419
Optimal tunings:  
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}


Mapping: [{{val| 1 3 4 0 }}, {{val| 0 -5 -6 10 }}]
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


{{Val list|legend=1| 7, 25, 32 }}
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 11:31, 23 August 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

Subgroup-val mapping: [⟨1 0 6], ⟨0 1 -2]]

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

mapping generators: ~2, ~3

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

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

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

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

Subgroup-val 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]]

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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