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The '''archytas clan''' (or '''archy family''') 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|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 their respective 5-limit families are [[Father family #Mother|mother]], [[Meantone family #Dominant|dominant]], [[Augmented family #Augene|augene]], [[Porcupine family #Porcupine|porcupine]], [[Diaschismic family #Pajara|pajara]], [[Kleismic family #Catalan|catalan]], [[Tetracot family #Modus|modus]], and [[Immunity family #Immunized|immunized]]. The rest are considered below.
== Archy ==
{{Main| Superpyth }}


== Archy ==
[[Subgroup]]: 2.3.7
Subgroup: 2.3.7


[[Comma list]]: 64/63
[[Comma list]]: 64/63


[[Sval]] [[mapping]]: [{{val| 1 0 6 }}, {{val| 0 1 -2 }}]
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}


Sval mapping generators: ~2, ~3
: sval mapping generators: ~2, ~3


Gencom mapping: [{{val| 1 1 0 4 }}, {{val| 0 1 0 -2 }}]
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}


[[Gencom]]: [2 3/2; 64/63]
: [[gencom]]: [2 3; 64/63]


[[POTE generator]]: ~3/2 = 709.321
[[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 }}


{{Val list|legend=1| 2, 3, 5, 12, 17, 22 }}
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
 
[[Badness]] (Sintel): 0.159


Scales: [[archy5]], [[archy7]], [[archy12]]
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:
* [[#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]].
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:
* ''[[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]]
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 #Suhajira|neutral clan]].


=== Supra ===
=== Supra ===
Line 29: Line 70:
Comma list: 64/63, 99/98
Comma list: 64/63, 99/98


Sval mapping: [{{val| 1 0 6 13 }}, {{val| 0 1 -2 -6 }}]
Sval mapping: {{mapping| 1 0 6 13 | 0 1 -2 -6 }}


Sval mapping generators: ~2, ~3
Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}


Gencom mapping: [{{val| 1 1 0 4 7 }}, {{val| 0 1 0 -2 -6 }}]
: gencom: [2 3; 64/63 99/98]


Gencom: [2 3/2; 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}}


POTE generator: ~3/2 = 707.192
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}


Vals: {{Val list| 5, 12, 17, 39d, 56d }}
Badness (Sintel): 0.352


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]
Line 48: Line 91:
Comma list: 64/63, 78/77, 99/98
Comma list: 64/63, 78/77, 99/98


Sval mapping: [{{val| 1 0 6 13 18 }}, {{val| 0 1 -2 -6 -9 }}]
Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}


Sval mapping generators: ~2, ~3
Gencom mapping: {{mapping| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}


Gencom mapping: [{{val| 1 1 0 4 7 9 }}, {{val| 0 1 0 -2 -6 -9 }}]
: gencom: [2 3; 64/63 78/77 99/98]


Gencom: [2 3/2; 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}}


POTE generator: ~3/2 = 706.137
{{Optimal ET sequence|legend=0| 12f, 17 }}


Vals: {{Val list| 12f, 17 }}
Badness (Sintel): 0.498


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]


=== Suhajira ===
== Superpyth ==
Subgroup: 2.3.7.11
{{Main| Superpyth }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''


Comma list: 64/63, 243/242
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.


Sval mapping: [{{val| 1 1 4 2 }}, {{val| 0 2 -4 5 }}]
[[Subgroup]]: 2.3.5.7


Sval mapping generators: ~2, ~11/9
[[Comma list]]: 64/63, 245/243


Gencom mapping: [{{val| 1 1 0 4 2 }}, {{val| 0 1 0 -4 5 }}]
{{Mapping|legend=1| 1 0 -12 6 | 0 1 9 -2 }}


Gencom: [2 3/2; 64/63 99/98]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.0549{{c}}, ~3/2 = 708.5478{{c}}
: [[error map]]: {{val| -2.945 +3.648 -0.548 +2.298 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1193{{c}}
: error map: {{val| 0.000 +8.164 +4.760 +10.935 }}


POTE generator: ~11/9 = 353.958
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}


Vals: {{Val list| 7, 10, 17, 44e, 61de }}
[[Badness]] (Sintel): 0.818


Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
=== 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).


==== 2.3.7.11.13 ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.7.11.13


Comma list: 64/63, 78/77, 144/143
Comma list: 64/63, 100/99, 245/243


Sval mapping: [{{val| 1 1 4 2 4 }}, {{val| 0 2 -4 5 -1 }}]
Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}


Sval mapping generators: ~2, ~11/9
Optimal tunings:
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


Gencom mapping: [{{val| 1 1 0 4 2 4 }}, {{val| 0 1 0 -4 5 -1 }}]
{{Optimal ET sequence|legend=0| 22, 27e, 49 }}


Gencom: [2 3/2; 64/63 78/77 99/98]
Badness (Sintel): 0.826


POTE generator: ~11/9 = 353.775
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 7, 10, 17, 44e, 61de }}
Comma list: 64/63, 78/77, 91/90, 100/99
 
Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
 
== Superpyth ==
{{main| Superpyth }}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 64/63, 245/243
 
[[Mapping]]: [{{val| 1 0 -12 6 }}, {{val| 0 1 9 -2 }}]
 
{{Multival|legend=1| 1 9 -2 12 -6 -30 }}


[[POTE generator]]: ~3/2 = 710.291
Mapping: {{mapping| 1 0 -12 6 -22 -17 | 0 1 9 -2 16 13 }}


{{Val list|legend=1| 5, 17, 22, 27, 49 }}
Optimal tunings:
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}


[[Badness]]: 0.032318
{{Optimal ET sequence|legend=0| 22, 27e, 49, 76bcde }}


=== 11-limit ===
Badness (Sintel): 1.02
Subgroup: 2.3.5.7.11


Comma list: 64/63, 100/99, 245/243
==== Thomas ====
 
Mapping: [{{val| 1 0 -12 6 -22 }}, {{val| 0 1 9 -2 16 }}]
 
POTE generator: ~3/2 = 710.175
 
Vals: {{Val list| 22, 27e, 49 }}
 
Badness: 0.024976
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 64/63, 78/77, 91/90, 100/99
Comma list: 64/63, 100/99, 169/168, 245/243


Mapping: [{{val| 1 0 -12 6 -22 -17 }}, {{val| 0 1 9 -2 16 13 }}]
Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}


POTE generator: ~3/2 = 710.479
Optimal tunings:
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


Vals: {{Val list| 22, 27e, 49, 76bcde }}
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}


Badness: 0.024673
Badness (Sintel): 2.03


=== Suprapyth ===
=== 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
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 0 -12 6 13 }}, {{val| 0 1 9 -2 -6 }}]
Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}


POTE generator: ~3/2 = 709.495
Optimal tunings:  
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


Vals: {{Val list| 17, 22 }}
{{Optimal ET sequence|legend=0| 5, 17, 22 }}


Badness: 0.032768
Badness (Sintel): 1.08


==== 13-limit ====
==== 13-limit ====
Line 161: Line 198:
Comma list: 55/54, 64/63, 65/63, 99/98
Comma list: 55/54, 64/63, 65/63, 99/98


Mapping: [{{val| 1 0 -12 6 13 18 }}, {{val| 0 1 9 -2 -6 -9 }}]
Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}


POTE generator: ~3/2 = 708.703
Optimal tunings:  
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}


Vals: {{Val list| 17, 22, 83cdf }}
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}


Badness: 0.036336
Badness (Sintel): 1.50


== Quasisuper ==
== Quasisuper ==
Subgroup: 2.3.5.7
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
[[Comma list]]: 64/63, 2430/2401


[[Mapping]]: [{{val| 1 0 23 6 }}, {{val| 0 1 -13 -2 }}]
{{Mapping|legend=1| 1 0 23 6 | 0 1 -13 -2 }}


{{Multival|legend=1| 1 -13 -2 -23 -6 32 }}
[[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 }}


[[POTE generator]]: ~3/2 = 708.328
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}


{{Val list|legend=1| 17c, 22, 61d }}
[[Badness]] (Sintel): 1.61
 
[[Badness]]: 0.063794


=== Quasisupra ===
=== Quasisupra ===
Line 191: Line 234:
Comma list: 64/63, 99/98, 121/120
Comma list: 64/63, 99/98, 121/120


Mapping: [{{val| 1 0 23 6 13 }}, {{val| 0 1 -13 -2 -6 }}]
Mapping: {{mapping| 1 0 23 6 13 | 0 1 -13 -2 -6 }}


POTE generator: ~3/2 = 708.205
Optimal tunings:  
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


Vals: {{Val list| 17c, 22, 39d, 61d }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}


Badness: 0.032203
Badness (Sintel): 1.06


==== 13-limit ====
==== 13-limit ====
Line 204: Line 249:
Comma list: 64/63, 78/77, 91/90, 121/120
Comma list: 64/63, 78/77, 91/90, 121/120


Mapping: [{{val| 1 0 23 6 13 18 }}, {{val| 0 1 -13 -2 -6 -9 }}]
Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}


POTE generator: ~3/2 = 708.004
Optimal tunings:  
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


Vals: {{Val list| 17c, 22, 39d, 61df, 100bcdf }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}


Badness: 0.030219
Badness (Sintel): 1.25


=== Quasisoup ===
=== Quasisoup ===
Line 217: Line 264:
Comma list: 55/54, 64/63, 2430/2401
Comma list: 55/54, 64/63, 2430/2401


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


POTE generator: ~3/2 = 709.021
Optimal tunings:  
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


Vals: {{Val list| 5ce, 17ce, 22 }}
{{Optimal ET sequence|legend=0| 22 }}


Badness: 0.083490
Badness (Sintel): 2.76


== Ultrapyth ==
== Ultrapyth ==
Subgroup: 2.3.5.7
{{Main| Ultrapyth }}
 
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).
 
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 0 -20 6 }}, {{val| 0 1 14 -2 }}]
{{Mapping|legend=1| 1 0 -20 6 | 0 1 14 -2 }}
 
{{Multival|legend=1| 1 14 -2 20 -6 -44 }}


[[POTE generator]]: ~3/2 = 713.651
[[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 }}


{{Val list|legend=1| 5, 32, 37 }}
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}


[[Badness]]: 0.108466
[[Badness]] (Sintel): 2.74


=== 11-limit ===
=== 11-limit ===
Line 245: Line 300:
Comma list: 55/54, 64/63, 2401/2376
Comma list: 55/54, 64/63, 2401/2376


Mapping: [{{val| 1 0 -20 6 21 }}, {{val| 0 1 14 -2 -11 }}]
Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}


POTE generator: ~3/2 = 713.395
Optimal tunings:  
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


Vals: {{Val list| 5, 32, 37 }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Badness: 0.068238
Badness (Sintel): 2.26


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 -20 6 21 -25 }}, {{val| 0 1 14 -2 -11 18 }}]
Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}


POTE generator: ~3/2 = 713.500
Optimal tunings:  
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


Vals: {{Val list| 5, 32, 37 }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}


Badness: 0.049172
Badness (Sintel): 2.03


== Schism ==
=== Ultramarine ===
{{see also|Schismatic family #Schism}}
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7
Comma list: 64/63, 100/99, 3773/3645


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


[[Mapping]]: [{{val| 1 0 15 6 }}, {{val| 0 1 -8 -2 }}]
Optimal tunings:  
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}


[[POTE generator]]: ~3/2 = 701.556
Badness (Sintel): 2.58


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


[[Badness]]: 0.056648
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 }}
Subgroup: 2.3.5.7.11


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


POTE generator: ~3/2 = 702.136
Badness (Sintel): 1.89


Vals: {{Val list| 12, 29de, 41de }}
== 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).


Badness: 0.037482
[[Subgroup]]: 2.3.5.7


== Blacksmith ==
[[Comma list]]: 64/63, 33614/32805
[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]


=== 5-limit (blackwood) ===
{{Mapping|legend=1| 1 0 31 6 | 0 1 -18 -2 }}
Subgroup: 2.3.5


[[Comma]]: 256/243
[[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 }}


[[Mapping]]: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
{{Optimal ET sequence|legend=1| 27, 86bd, 113bcd, 140bbcd }}


Mapping generators: ~9/8, ~5
[[Badness]] (Sintel): 3.34


[[POTE generator]]: ~5/4 = 399.594
== Schism ==
{{See also| Schismatic family #Schism }}


{{Val list|legend=1| 5, 10, 15 }}
Schism tempers out the [[schisma]], mapping the ~5/4 to -8 fifths as a diminished fourth (C–Fb) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53dd val) can be used.


[[Badness]]: 0.063760
[[Subgroup]]: 2.3.5.7


=== 7-limit ===
[[Comma list]]: 64/63, 360/343
Subgroup: 2.3.5.7


[[Comma list]]: 28/27, 49/48
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}


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


Mapping generators: ~7/6, ~5
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}


{{Multival|legend=1| 0 5 0 8 0 -14 }}
[[Badness]] (Sintel): 1.43
 
[[POTE generator]]: ~5/4 = 392.767
 
{{Val list|legend=1| 5, 10, 15, 40b, 55b }}
 
[[Badness]]: 0.025640


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 28/27, 49/48, 55/54
Comma list: 45/44, 64/63, 99/98
 
Mapping: [{{val| 5 8 0 14 29 }}, {{val| 0 0 1 0 -1 }}]
 
POTE generator: ~5/4 = 394.948
 
Vals: {{Val list| 5, 10, 15, 40be, 55be, 70bde, 85bcde}}
 
Badness: 0.024641
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 28/27, 40/39, 49/48, 55/54
 
Mapping: [{{val| 5 8 0 14 29 7 }}, {{val| 0 0 1 0 -1 1 }}]
 
POTE generator: ~5/4 = 391.037
 
Vals: {{Val list| 5, 10, 15, 25e, 40bef}}
 
Badness: 0.020498
 
=== Farrier ===
Subgroup: 2.3.5.7.11
 
Comma list: 28/27, 49/48, 77/75
 
Mapping: [{{val| 5 8 0 14 -6 }}, {{val| 0 0 1 0 2 }}]
 
POTE generator: ~5/4 = 398.070
 
Vals: {{Val list| 5e, 10e, 15 }}
 
Badness: 0.029200
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 28/27, 40/39, 49/48, 66/65
 
Mapping: [{{val| 5 8 0 14 -6 7 }}, {{val| 0 0 1 0 2 1 }}]
 
POTE generator: ~5/4 = 396.812
 
Vals: {{Val list| 5e, 10e, 15 }}
 
Badness: 0.022325
 
=== Ferrum ===
Subgroup: 2.3.5.7.11
 
Comma list: 28/27, 35/33, 49/48


Mapping: [{{val| 5 8 0 14 6 }}, {{val| 0 0 1 0 1 }}]
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


POTE generator: ~5/4 = 374.763
Optimal tunings:  
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


Vals: {{Val list| 5e, 10 }}
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}


Badness: 0.030883
Badness (Sintel): 1.24


== Beatles ==
== Beatles ==
Subgroup: 2.3.5
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Beatles]].''
 
[[Comma]]: 524288/492075


[[Mapping]]: [{{val| 1 1 5 }}, {{val| 0 2 -9 }}]
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.


[[POTE generator]]: ~512/405 = 355.930
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.  


{{Val list|legend=1| 10, 17c, 27, 64b, 91bc, 118bc }}
[[Subgroup]]: 2.3.5.7
 
[[Badness]]: 0.358542
 
=== 7-limit ===
Subgroup: 2.3.5.7


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


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


{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}
[[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 }}


[[POTE generator]]: ~49/40 = 355.904
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}


{{Val list|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
[[Badness]] (Sintel): 1.16
 
[[Badness]]: 0.045872


; Music
; Music
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/beatles-improv.mp3 Beatles Improv] by Herman Miller
* [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]]


=== 11-limit ===
=== 11-limit ===
Line 432: Line 441:
Comma list: 64/63, 100/99, 686/675
Comma list: 64/63, 100/99, 686/675


Mapping: [{{val| 1 1 5 4 10 }}, {{val| 0 2 -9 -4 -22 }}]
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}


POTE generator: ~49/40 = 356.140
Optimal tunings:  
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


Vals: {{Val list| 27e, 37, 64be, 91bcde }}
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}


Badness: 0.045639
Badness (Sintel): 1.51


==== 13-limit ====
==== 13-limit ====
Line 445: Line 456:
Comma list: 64/63, 91/90, 100/99, 169/168
Comma list: 64/63, 91/90, 100/99, 169/168


Mapping: [{{val| 1 1 5 4 10 4 }}, {{val| 0 2 -9 -4 -22 -1 }}]
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}


POTE generator: ~16/13 = 356.229
Optimal tunings:  
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


Vals: {{Val list| 27e, 37, 64be }}
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}


Badness: 0.030161
Badness (Sintel): 1.25


=== Ringo ===
=== Ringo ===
Line 458: Line 471:
Comma list: 56/55, 64/63, 540/539
Comma list: 56/55, 64/63, 540/539


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


POTE generator: ~11/9 = 355.419
Optimal tunings:  
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


Vals: {{Val list| 10, 17c, 27e }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}


Badness: 0.032863
Badness (Sintel): 1.09


==== 13-limit ====
==== 13-limit ====
Line 471: Line 486:
Comma list: 56/55, 64/63, 78/77, 91/90
Comma list: 56/55, 64/63, 78/77, 91/90


Mapping: [{{val| 1 1 5 4 2 4 }}, {{val| 0 2 -9 -4 5 -1 }}]
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}
 
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
 
Badness (Sintel): 0.935
 
=== Beetle ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 64/63, 686/675


POTE generator: ~11/9 = 355.456
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}


Vals: {{Val list| 10, 17c, 27e }}
Optimal tunings:  
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


Badness: 0.022619
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


== Passion ==
Badness (Sintel): 1.92
Passion tempers out 262144/253125 (saquingu) in the 5-limit. This temperament can be described as 12&49 temperament, which tempers out 64/63 and the gariboh (3125/3087). Alternative extension [[Marvel temperaments #Passive|passive]] (1&11) tempers out the same 5-limit comma as the passion, but with the marvel (225/224) rather than 64/63 tempered out.


Subgroup: 2.3.5
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Comma]]: 262144/253125
Comma list: 55/54, 64/63, 91/90, 169/168


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


Mapping generators: 2, 16/15
Optimal tunings:  
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


[[POTE generator]]: ~16/15 = 98.670
{{Optimal ET sequence|legend=0| 10, 27, 37 }}


{{Val list|legend=1| 12, 49, 61, 73 }}
Badness (Sintel): 1.40


[[Badness]]: 0.168550
== Progress ==
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''


=== 7-limit ===
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.  
Subgroup: 2.3.5.7


[[Comma list]]: 64/63, 3125/3087
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 2 2 }}, {{val| 0 -5 4 10 }}]
[[Comma list]]: 64/63, 392/375


Mapping generators: 2, 16/15
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}


{{Multival|legend=1| 5 -4 -10 -18 -30 -12 }}
: mapping generators: ~2, ~10/7


[[POTE generator]]: ~16/15 = 98.153
[[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 }}


{{Val list|legend=1| 12, 37, 49, 110bcd }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


[[Badness]]: 0.062327
[[Badness]] (Sintel): 1.68


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 64/63, 100/99, 1375/1372
Comma list: 56/55, 64/63, 77/75


Mapping: [{{val| 1 2 2 2 2 }}, {{val| 0 -5 4 10 18 }}]
Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}


POTE generator: ~16/15 = 98.019
Optimal tunings:  
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}


Vals: {{Val list| 12, 37, 49 }}
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}


Badness: 0.040809
Badness (Sintel): 1.03


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 64/63, 100/99, 196/195, 275/273
Comma list: 56/55, 64/63, 66/65, 77/75


Mapping: [{{val| 1 2 2 2 2 2 }}, {{val| 0 -5 4 10 18 21 }}]
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}


POTE generator: ~16/15 = 97.910
Optimal tunings:  
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}


Vals: {{Val list| 12f, 25f, 37, 49f }}
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}


Badness: 0.030886
Badness (Sintel): 1.08


== Fervor ==
==== Progressive ====
Subgroup: 2.3.5
Subgroup: 2.3.5.7.11.13
 
Comma list: 26/25, 56/55, 64/63, 77/75


[[Comma]]: 67108864/61509375
Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}


[[Mapping]]: [{{val| 1 4 -2 }}, {{val| 0 -5 9 }}]
Optimal tunings:  
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}


[[POTE generator]]: ~64/45 = 577.705
{{Optimal ET sequence|legend=0| 2f, 15f, 17c }}


{{Val list|legend=1| 2, 25, 27 }}
Badness (Sintel): 1.35


[[Badness]]: 0.852612
== Fervor ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''


=== 7-limit ===
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
 
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 4 -2 -2 }}, {{val| 0 -5 9 10 }}]
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}


{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}
: mapping generators: ~2, ~10/7


[[POTE generator]]: ~7/5 = 577.776
[[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 }}


{{Val list|legend=1| 2, 25, 27 }}
{{Optimal ET sequence|legend=1| 2, 25, 27 }}


[[Badness]]: 0.108455
[[Badness]] (Sintel): 2.74


=== 11-limit ===
=== 11-limit ===
Line 572: Line 623:
Comma list: 56/55, 64/63, 1350/1331
Comma list: 56/55, 64/63, 1350/1331


Mapping: [{{val| 1 4 -2 -2 3 }}, {{val| 0 -5 9 10 1 }}]
Mapping: {{mapping| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}


POTE generator: ~7/5 = 577.850
Optimal tunings:  
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}


Vals: {{Val list| 2, 25e, 27e }}
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}


Badness: 0.052054
Badness (Sintel): 1.72


=== 13-limit ===
=== 13-limit ===
Line 585: Line 638:
Comma list: 56/55, 64/63, 78/77, 507/500
Comma list: 56/55, 64/63, 78/77, 507/500


Mapping: [{{val| 1 4 -2 -2 3 -4 }}, {{val| 0 -5 9 10 1 16 }}]
Mapping: {{mapping| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}


POTE generator: ~7/5 = 578.060
Optimal tunings:  
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}


Vals: {{Val list| 2f, 25ef, 27e }}
{{Optimal ET sequence|legend=0| 2f, 27e }}


Badness: 0.039705
Badness (Sintel): 1.64


== Progress ==
== Sixix ==
Subgroup: 2.3.5
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sixix (5-limit)]].''
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}


[[Comma]]: 32768/30375
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.


[[Mapping]]: [{{val| 1 0 5 }}, {{val| 0 3 -5 }}]
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~64/45 = 561.264
[[Comma list]]: 64/63, 3125/2916
 
{{Val list|legend=1| 2, 13, 15, 32c, 47bc, 62bc }}


Badness: 0.246073
{{Mapping|legend=1| 1 3 4 0 | 0 -5 -6 10 }}


=== 7-limit ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.7
* [[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 }}


[[Comma list]]: 64/63, 392/375
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}


[[Mapping]]: [{{val| 1 0 5 6 }}, {{val| 0 3 -5 -6 }}]
[[Badness]] (Sintel): 4.02
 
{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
 
[[POTE generator]]: ~7/5 = 562.122
 
{{Val list|legend=1| 2, 13, 15, 32c, 79bcc, 111bcc }}
 
[[Badness]]: 0.066400


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 64/63, 77/75
Comma list: 55/54, 64/63, 125/121


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


POTE generator: ~7/5 = 562.085
Optimal tunings:  
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}


Vals: {{Val list| 2, 13, 15, 32c, 47bc, 79bcce }}
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}


Badness: 0.031036
Badness (Sintel): 2.34


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 64/63, 66/65, 77/75
Comma list: 40/39, 55/54, 64/63, 125/121


Mapping: [{{val| 1 0 5 6 4 0 }}, {{val| 0 3 -5 -6 -1 7 }}]
Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}


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


Vals: {{Val list| 15, 17c, 32cf }}
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


Badness: 0.026214
Badness (Sintel): 1.91


=== Progressive ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 26/25, 56/55, 64/63, 77/75
 
Mapping: [{{val| 1 0 5 6 4 9 }}, {{val| 0 3 -5 -6 -1 -10 }}]
 
POTE generator: ~7/5 = 563.239
 
Vals: {{Val list| 15f, 17c, 32c, 49c }}
 
Badness: 0.032721
 
== Sixix ==
:''See also: [[Dual-fifth temperaments#Dual-3 Sixix]]''
 
Subgroup: 2.3.5
 
[[Comma]]: 3125/2916
 
[[Mapping]]: [{{val| 1 3 4 }}, {{val| 0 -5 -6 }}]
 
[[POTE generator]]: ~6/5 = 338.365
 
{{Val list|legend=1| 7, 25, 32 }}
 
[[Badness]]: 0.153088
 
=== 7-limit ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 64/63, 3125/2916


[[Mapping]]: [{{val| 1 3 4 0 }}, {{val| 0 -5 -6 10 }}]
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121


{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}


[[POTE generator]]: ~6/5 = 337.442
Optimal tunings:  
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}


{{Val list|legend=1| 7, 25, 32 }}
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}


[[Badness]]: 0.158903
Badness (Sintel): 2.00


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