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


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


: sval mapping generators: ~2, ~3
{{Mapping|legend=5| 1 0 0 6 | 0 1 0 -2 }}
 
: mapping generators: ~2, ~3
{{Mapping|legend=3| 1 0 0 6 | 0 1 0 -2 }}
 
: [[gencom]]: [2 3; 64/63]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.595
* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}
: [[error map]]: {{val| 0.000 +7.640 +11.984 }}
: [[error map]]: {{val| -3.045 +2.522 +3.952 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 709.321
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}
: error map: {{val| 0.000 +7.366 +12.532 }}
: error map: {{val| 0.000 +7.435 +12.394 }}


{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
{{Optimal ET sequence|legend=1| 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd }}
Line 34: Line 31:
==== 7-limit extensions ====
==== 7-limit extensions ====
The second comma in the comma list defines which [[7-limit]] family member we are looking at:  
The second comma in the comma list defines which [[7-limit]] family member we are looking at:  
* [[#Schism|Schism]] adds 360/343, for an [[exotemperament]] well tuned around [[12edo]];  
* [[#Schism|Schism]] adds [[360/343]], for a tuning around [[12edo]];  
* Dominant adds [[36/35]], for a tuning flat of [[17edo|17c-edo]];  
* [[Meantone family #Dominant|Dominant]] adds [[36/35]], for a tuning between [[12edo]] and [[17edo|17c-edo]];  
* [[#Quasisuper|Quasisuper]] adds [[2430/2401]], for a tuning between 17c-edo and [[22edo]];  
* [[#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]];  
* [[#Superpyth|Superpyth]] adds [[245/243]], for a tuning between 22edo and [[27edo]];  
* [[#Quasiultra|Quasiultra]] adds 33614/32805, for a tuning between 27edo and [[32edo]];  
* [[#Quasiultra|Quasiultra]] adds [[33614/32805]], for a tuning between 27edo and [[32edo]];  
* [[#Ultrapyth|Ultrapyth]] adds 6860/6561, for a tuning sharp of 32edo;  
* [[#Ultrapyth|Ultrapyth]] adds [[6860/6561]], for a tuning sharp of 32edo;  
* Mother adds [[16/15]], for an exotemperament well tuned around [[5edo]].  
* Mother adds [[16/15]], for an exotemperament well tuned around [[5edo]].  


These all use the same generators as archy.  
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.  
[[25/24]] gives dichotic. [[686/675]] gives beatles. Those split the fifth in two. [[8748/8575]] gives immunized, splitting the twelfth in two. [[50/49]] gives pajara with a semi-octave period. [[392/375]] gives progress. [[6125/5832]] gives kangaroo. Those split the twelfth in three. [[250/243]] gives porcupine, splitting the fourth in three. [[126/125]] gives augene with a 1/3-octave period. [[4375/4374]] gives modus, splitting the fifth in four. [[3125/3024]] gives brightstone. [[9604/9375]] gives fervor. [[3125/2916]] gives sixix. [[3125/3087]] gives passion. Those split the generator in five in various ways. [[15625/15552]] gives catalan, splitting the twelfth in six. Finally, [[28/27]] gives blackwood and [[17496/16807]] gives [[gothic]], reducing the intervals of 3 and 7 to the same chain of 1/5- and 1/17-octave periods, respectively.  


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Phlegyas]]'' (+10/9) → [[Very low accuracy temperaments #Phlegyas|Very low accuracy temperaments]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
* ''[[Mother]]'' (+16/15) → [[Father family #Mother|Father family]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* ''[[Dichotic]]'' (+25/24) → [[Dicot family #Dichotic|Dicot family]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* ''[[Immunized]]'' (+8748/8575) → [[Immunity family #Immunized|Immunity family]]
* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Pajara]] (+50/49) → [[Diaschismic family #Pajara|Diaschismic family]]
* [[Augene]] (+126/125) → [[Augmented family #Augene|Augmented family]]
* [[Augene]] (+126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* [[Porcupine]] (+250/243) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* ''[[Modus]]'' (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
* [[Modus]] (+4375/4374) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
* ''[[Brightstone]]'' (+3125/3024) → [[Magic family #Brightstone|Magic family]]
* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
* ''[[Passion]]'' (+3125/3087) → [[Passion family #Septimal passion|Passion family]]
* ''[[Blacksmith]]'' (+28/27) → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
* ''[[Gothic]]'' (+17496/16807) → [[17th-octave temperaments #Gothic|17th-octave temperaments]]


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


==== Subgroup extensions ====
==== 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]].
Omitting prime 5, archy can be extended to the [[2.3.7.11 subgroup]] by identifying 11/8 as a diminished fourth (C–G♭). This is called supra, given in [[#Subgroup extensions_2|#Subgroup extensions]]. Discussed elsewhere is [[suhajira]] of the [[rastmic clan #Suhajira|rastmic clan]].
 
=== Supra ===
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:
* CTE: ~2 = 1200.000, ~3/2 = 708.456
* POTE: ~2 = 1200.000, ~3/2 = 707.192
 
{{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:
* CTE: ~2 = 1200.000, ~3/2 = 707.344
* POTE: ~2 = 1200.000, ~3/2 = 706.137
 
{{Optimal ET sequence|legend=0| 12f, 17 }}
 
Badness (Sintel): 0.498
 
Scales: [[supra7]], [[supra12]]


== Superpyth ==
== Superpyth ==
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: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Superpyth (5-limit)]].''
: ''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.  
Superpyth, virtually the canonical extension, adds [[245/243]] and [[1728/1715]] to the comma list and can be described as {{nowrap| [[22edo|22]] [[&]] [[27edo|27]] }}. ~[[5/4]] is found at +9 generator steps, as an augmented second (C–D♯). In the [[11-limit]] it finds the ~[[11/8]] at +16 generator steps, as a double-augmented second (C–D𝄪). [[49edo]] remains an obvious tuning choice in either case.
 
Extending superpyth to the [[13-limit]] is more diffcult. Tridecimal superpyth finds the ~[[13/8]] at +13 generator steps, as a double-augmented fourth (C–F𝄪), for which 27edo can be recommended as a tuning since it is the only [[13-odd-limit]] [[diamond monotone]] tuning. The other extension, called uberpyth, is more flexible with its tunings, but unfortunately tends to tune the 13 very sharp.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 120: Line 80:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 709.591
* [[WE]]: ~2 = 1197.0549{{c}}, ~3/2 = 708.5478{{c}}
: [[error map]]: {{val| 0.000 +7.636 +0.002 +11.993 }}
: [[error map]]: {{val| -2.945 +3.648 -0.548 +2.298 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 710.291
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1193{{c}}
: error map: {{val| 0.000 +8.336 +6.305 +10.592 }}
: error map: {{val| 0.000 +8.164 +4.760 +10.935 }}


{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 49, 174bbcddd }}
Line 130: Line 90:


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


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


Mapping: {{mapping| 1 0 -12 6 -22 | 0 1 9 -2 16 }}
{{Mapping|legend=0| 1 0 -12 6 -22 | 0 1 9 -2 16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 709.514
* WE: ~2 = 1197.0673{{c}}, ~3/2 = 708.4391{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 710.175
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.0129{{c}}


{{Optimal ET sequence|legend=0| 22, 27e, 49 }}
{{Optimal ET sequence|legend=0| 22, 27e, 49 }}
Line 151: Line 109:
Comma list: 64/63, 78/77, 91/90, 100/99
Comma list: 64/63, 78/77, 91/90, 100/99


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 709.836
* WE: ~2 = 1197.3011{{c}}, ~3/2 = 708.8813{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 710.479
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.3219{{c}}


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


Badness (Sintel): 1.02
Badness (Sintel): 1.02
==== Uberpyth ====
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 100/99, 144/143, 245/243
{{Mapping|legend=0| 1 0 -12 6 -22 26 | 0 1 9 -2 16 -14 }}
Optimal tunings:
* WE: ~2 = 1196.6666{{c}}, ~3/2 = 708.3602{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 710.2878{{c}}
{{Optimal ET sequence|legend=0| 22f, 27e, 49f, 125bcddeeeff, 174bbcdddeeeeffff }}
Badness (Sintel): 1.11


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


Mapping: {{mapping| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}
{{Mapping|legend=0| 1 1 -3 4 -6 4 | 0 2 18 -4 32 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~16/13 = 354.759
* WE: ~2 = 1197.4942{{c}}, ~16/13 = 354.2950{{c}}
* POTE: ~2 = 1200.000, ~16/13 = 355.036
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 354.9824{{c}}


{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}
{{Optimal ET sequence|legend=0| 27e, 44, 71d, 98bde }}
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=== Suprapyth ===
=== Suprapyth ===
Suprapyth finds the ~11/8 at the diminished fifth (C–Gb), and finds the ~13/8 at the diminished seventh (C–Bbb).  
Suprapyth finds the ~11/8 at the diminished fifth (C–G♭), and finds the ~13/8 at the diminished seventh (C–B𝄫).  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Comma list: 55/54, 64/63, 99/98
Comma list: 55/54, 64/63, 99/98


Mapping: {{mapping| 1 0 -12 6 13 | 0 1 9 -2 -6 }}
{{Mapping|legend=0| 1 0 -12 6 13 | 0 1 9 -2 -6 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 709.356
* WE: ~2 = 1198.6960{{c}}, ~3/2 = 708.7235{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 709.495
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 709.4699{{c}}


{{Optimal ET sequence|legend=0| 5, 17, 22 }}
{{Optimal ET sequence|legend=0| 5, 17, 22 }}
Line 198: Line 171:
Comma list: 55/54, 64/63, 65/63, 99/98
Comma list: 55/54, 64/63, 65/63, 99/98


Mapping: {{mapping| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}
{{Mapping|legend=0| 1 0 -12 6 13 18 | 0 1 9 -2 -6 -9 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 708.702
* WE: ~2 = 1199.9871{{c}}, ~3/2 = 708.6952{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 708.703
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.7028{{c}}


{{Optimal ET sequence|legend=0| 5f, 17, 22 }}
{{Optimal ET sequence|legend=0| 5f, 17, 22 }}
Line 209: Line 182:


== Quasisuper ==
== 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).  
{{Main|Quasisuper}}
 
Quasisuper can be described as {{nowrap| 17c & 22 }}, with the ~5/4 mapped to -13 generator steps, as a double-diminished fifth (C–G𝄫). The 11-limit version, quasisupra, can be viewed as an extension of the excellent 2.3.7.11-subgroup temperament [[supra]], with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 218: Line 193:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 708.769
* [[WE]]: ~2 = 1196.9830{{c}}, ~3/2 = 706.4578{{c}}
: [[error map]]: {{val| 0.000 +6.814 -0.310 +13.636 }}
: [[error map]]: {{val| -3.017 +1.486 -0.435 +6.190 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 708.238
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 708.3716{{c}}
: error map: {{val| 0.000 +6.283 +6.586 +14.697 }}
: error map: {{val| 0.000 +6.417 +4.855 +14.431 }}


{{Optimal ET sequence|legend=1| 17c, 22, 61d }}
{{Optimal ET sequence|legend=1| 17c, 22, 61d }}
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=== Quasisupra ===
=== 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
Subgroup: 2.3.5.7.11


Comma list: 64/63, 99/98, 121/120
Comma list: 64/63, 99/98, 121/120


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 708.713
* WE: ~2 = 1197.5675{{c}}, ~3/2 = 706.7690{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 708.205
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.3200{{c}}


{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d, 61d }}
Line 249: Line 222:
Comma list: 64/63, 78/77, 91/90, 121/120
Comma list: 64/63, 78/77, 91/90, 121/120


Mapping: {{mapping| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}
{{Mapping|legend=0| 1 0 23 6 13 18 | 0 1 -13 -2 -6 -9 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 708.411
* WE: ~2 = 1198.2543{{c}}, ~3/2 = 706.9736{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 708.004
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0936{{c}}


{{Optimal ET sequence|legend=0| 17c, 22, 39d }}
{{Optimal ET sequence|legend=0| 17c, 22, 39d }}
Line 264: Line 237:
Comma list: 55/54, 64/63, 2430/2401
Comma list: 55/54, 64/63, 2430/2401


Mapping: {{mapping| 1 0 23 6 -22 | 0 1 -13 -2 16 }}
{{Mapping|legend=0| 1 0 23 6 -22 | 0 1 -13 -2 16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 709.043
* WE: ~2 = 1198.8446{{c}}, ~3/2 = 708.3388{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 709.021
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 708.0252{{c}}


{{Optimal ET sequence|legend=0| 22 }}
{{Optimal ET sequence|legend=0| 22 }}
Line 275: Line 248:


== Ultrapyth ==
== Ultrapyth ==
{{main|Ultrapyth}}
{{Main| Ultrapyth }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Ultrapyth (5-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).
Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 [[#Oceanfront|oceanfront]] temperament, mapping the ~5/4 to +14 fifths as a double-augmented unison (C–C𝄪).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 286: Line 260:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 713.218
* [[WE]]: ~2 = 1197.2673{{c}}, ~3/2 = 712.0258{{c}}
: [[error map]]: {{val| 0.000 +11.263 -1.264 +4.738 }}
: [[error map]]: {{val| -2.733 +7.338 -1.557 -3.808 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 713.651
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5430{{c}}
: error map: {{val| 0.000 +11.696 +4.800 +3.872 }}
: error map: {{val| 0.000 +11.588 +3.288 +4.088 }}


{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37 }}
Line 300: Line 274:
Comma list: 55/54, 64/63, 2401/2376
Comma list: 55/54, 64/63, 2401/2376


Mapping: {{mapping| 1 0 -20 6 21 | 0 1 14 -2 -11 }}
{{Mapping|legend=0| 1 0 -20 6 21 | 0 1 14 -2 -11 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 713.282
* WE: ~2 = 1198.0290{{c}}, ~3/2 = 712.2235{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 713.395
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.3754{{c}}


{{Optimal ET sequence|legend=0| 5, 32, 37 }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}
Line 315: Line 289:
Comma list: 55/54, 64/63, 91/90, 1573/1568
Comma list: 55/54, 64/63, 91/90, 1573/1568


Mapping: {{mapping| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}
{{Mapping|legend=0| 1 0 -20 6 21 -25 | 0 1 14 -2 -11 18 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 713.309
* WE: ~2 = 1198.1911{{c}}, ~3/2 = 712.4243{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 713.500
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.4684{{c}}


{{Optimal ET sequence|legend=0| 5, 32, 37 }}
{{Optimal ET sequence|legend=0| 5, 32, 37 }}
Line 330: Line 304:
Comma list: 64/63, 100/99, 3773/3645
Comma list: 64/63, 100/99, 3773/3645


Mapping: {{mapping| 1 0 -20 6 -38 | 0 1 14 -2 26 }}
{{Mapping|legend=0| 1 0 -20 6 -38 | 0 1 14 -2 26 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 713.395
* WE: ~2 = 1197.2230{{c}}, ~3/2 = 712.1393{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 713.791
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.6928{{c}}


{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bce }}
Line 345: Line 319:
Comma list: 64/63, 91/90, 100/99, 847/845
Comma list: 64/63, 91/90, 100/99, 847/845


Mapping: {{mapping| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}
{{Mapping|legend=0| 1 0 -20 6 -38 -25 | 0 1 14 -2 26 18 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 713.708
* WE: ~2 = 1197.2739{{c}}, ~3/2 = 712.1893{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 713.811
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.7079{{c}}


{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}
{{Optimal ET sequence|legend=0| 5e, 32e, 37, 79bcef }}
Line 356: Line 330:


== Quasiultra ==
== 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).  
Quasiultra is to ultrapyth what quasisuper is to superpyth. It is the {{nowrap| 27 & 32 }} temperament, mapping the ~5/4 to -18 fifths as a double diminished sixth (C–A𝄫♭).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 365: Line 339:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 711.838
* [[WE]]: ~2 = 1196.9257{{c}}, ~3/2 = 709.6211{{c}}
: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}
: [[error map]]: {{val| 0.000 +9.883 +0.608 +7.499 }}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 711.543
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5429{{c}}
: error map: {{val| 0.000 +9.588 +5.914 +8.088 }}
: error map: {{val| 0.000 +9.588 +5.914 +8.088 }}


Line 377: Line 351:
{{See also| Schismatic family #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.  
Schism tempers out the [[schisma]], mapping the ~5/4 to -8 fifths as a diminished fourth (C–F♭) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53dd val) can be used.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 386: Line 360:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 702.270
* [[WE]]: ~2 = 1197.3598{{c}}, ~3/2 = 700.0126{{c}}
: [[error map]]: {{val| 0.000 +0.315 -4.471 +26.635 }}
: [[error map]]: {{val| -2.640 -4.583 -4.896 +20.588 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 701.556
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7376{{c}}
: error map: {{val| 0.000 -0.399 +1.237 +28.062 }}
: error map: {{val| 0.000 -0.217 -0.214 +27.699 }}


{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
Line 400: Line 374:
Comma list: 45/44, 64/63, 99/98
Comma list: 45/44, 64/63, 99/98


Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}
{{Mapping|legend=0| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 703.383
* WE: ~2 = 1196.1607{{c}}, ~3/2 = 699.8897{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 702.136
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4385{{c}}


{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}
{{Optimal ET sequence|legend=0| 5c, 7ce, 12, 29de }}
Line 412: Line 386:
== Beatles ==
== Beatles ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #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 {{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.
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
[[Subgroup]]: 2.3.5.7
Line 420: Line 398:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~49/40 = 356.634
* [[WE]]: ~2 = 1196.6244{{c}}, ~49/40 = 354.9029{{c}}
: [[error map]]: {{val| 0.000 +11.312 +3.984 +4.640 }}
: [[error map]]: {{val| -3.376 +4.475 +2.682 -1.940 }}
* [[POTE]]: ~2 = 1200.000, ~49/40 = 355.904
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 356.0819{{c}}
: error map: {{val| 0.000 +9.853 +10.549 +7.558 }}
: error map: {{val| 0.000 +10.209 +8.949 +6.847 }}


{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}
Line 437: Line 415:
Comma list: 64/63, 100/99, 686/675
Comma list: 64/63, 100/99, 686/675


Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
{{Mapping|legend=0| 1 1 5 4 10 | 0 2 -9 -4 -22 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~49/40 = 356.719
* WE: ~2 = 1196.7001{{c}}, ~49/40 = 355.1606{{c}}
* POTE: ~2 = 1200.000, ~49/40 = 356.140
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.2795{{c}}


{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}
{{Optimal ET sequence|legend=0| 10e, 17cee, 27e, 64be, 91bcdee }}
Line 452: Line 430:
Comma list: 64/63, 91/90, 100/99, 169/168
Comma list: 64/63, 91/90, 100/99, 169/168


Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
{{Mapping|legend=0| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~16/13 = 356.722
* WE: ~2 = 1197.2504{{c}}, ~16/13 = 355.4132{{c}}
* POTE: ~2 = 1200.000, ~16/13 = 356.229
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.3273{{c}}


{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}
{{Optimal ET sequence|legend=0| 10e, 27e, 37, 64be }}
Line 467: Line 445:
Comma list: 56/55, 64/63, 540/539
Comma list: 56/55, 64/63, 540/539


Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}
{{Mapping|legend=0| 1 1 5 4 2 | 0 2 -9 -4 5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 355.992
* WE: ~2 = 1195.4102{{c}}, ~11/9 = 354.0597{{c}}
* POTE: ~2 = 1200.000, ~11/9 = 355.419
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5207{{c}}


{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
Line 482: Line 460:
Comma list: 56/55, 64/63, 78/77, 91/90
Comma list: 56/55, 64/63, 78/77, 91/90


Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
{{Mapping|legend=0| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/9 = 356.004
* WE: ~2 = 1195.9943{{c}}, ~11/9 = 354.2695{{c}}
* POTE: ~2 = 1200.000, ~11/9 = 355.456
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 355.5398{{c}}


{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
{{Optimal ET sequence|legend=0| 10, 17c, 27e }}
Line 497: Line 475:
Comma list: 55/54, 64/63, 686/675
Comma list: 55/54, 64/63, 686/675


Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15 }}
{{Mapping|legend=0| 1 1 5 4 -1 | 0 2 -9 -4 15 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~49/40 = 356.694
* WE: ~2 = 1197.9660{{c}}, ~49/40 = 356.1056{{c}}
* POTE: ~2 = 1200.000, ~49/40 = 356.710
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 356.7075{{c}}


{{Optimal ET sequence|legend=0| 10, 27, 37 }}
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
Line 512: Line 490:
Comma list: 55/54, 64/63, 91/90, 169/168
Comma list: 55/54, 64/63, 91/90, 169/168


Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
{{Mapping|legend=0| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~16/13 = 356.700
* WE: ~2 = 1198.1741{{c}}, ~16/13 = 356.1582{{c}}
* POTE: ~2 = 1200.000, ~16/13 = 356.701
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 356.7008{{c}}


{{Optimal ET sequence|legend=0| 10, 27, 37 }}
{{Optimal ET sequence|legend=0| 10, 27, 37 }}
Line 523: Line 501:


== Progress ==
== Progress ==
{{Distinguish| Progression }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Progress]].''
Progress tempers out [[392/375]] and may be described as {{nowrap| 15 & 17c }}. It splits the [[3/1|perfect twelfth]] into three generators of [[~]][[10/7]]; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]], and the 13-limit extension further equates it to [[13/9]]. [[32edo|32c-edo]] gives an obvious tuning.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 530: Line 511:


{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
: mapping generators: ~2, ~10/7
: mapping generators: ~2, ~10/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~10/7 = 638.782
* [[WE]]: ~2 = 1195.1377{{c}}, ~10/7 = 635.2932{{c}}
: [[error map]]: {{val| 0.000 +14.391 +19.777 -1.518 }}
: [[error map]]: {{val| -4.862 +3.925 +12.908 -9.759 }}
* [[POTE]]: ~2 = 1200.000, ~10/7 = 637.878
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 638.0791{{c}}
: error map: {{val| 0.000 +11.679 +24.297 +3.907 }}
: error map: {{val| 0.000 +12.282 +23.291 +2.700 }}


{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}
Line 548: Line 528:
Comma list: 56/55, 64/63, 77/75
Comma list: 56/55, 64/63, 77/75


Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
{{Mapping|legend=0| 1 0 5 6 4 | 0 3 -5 -6 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 638.846
* WE: ~2 = 1195.4920{{c}}, ~10/7 = 635.5183{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 637.915
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 638.0884{{c}}


{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}
{{Optimal ET sequence|legend=0| 2, 13, 15, 32c, 47bc }}
Line 563: Line 543:
Comma list: 56/55, 64/63, 66/65, 77/75
Comma list: 56/55, 64/63, 66/65, 77/75


Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
{{Mapping|legend=0| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 637.871
* WE: ~2 = 1195.0786{{c}}, ~10/7 = 635.0197{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 637.635
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 637.6691{{c}}


{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
{{Optimal ET sequence|legend=0| 15, 17c, 32cf }}
Line 578: Line 558:
Comma list: 26/25, 56/55, 64/63, 77/75
Comma list: 26/25, 56/55, 64/63, 77/75


Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
{{Mapping|legend=0| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 637.797
* WE: ~2 = 1196.0245{{c}}, ~10/7 = 634.6516{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 636.761
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 636.9528{{c}}


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


Badness (Sintel): 1.35
Badness (Sintel): 1.35
== Kangaroo ==
: ''For 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
Kangaroo splits the [[3/1|perfect twelfth]] into three [[35/24]][[~]][[40/27]] generators; its [[ploidacot]] is alpha-tricot. The 11-limit extension simply equates the generator to [[16/11]]. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat.
Originally the canonical extension of 5-limit kangaroo was [[trienstonic clan #Parakangaroo|parakangaroo]], but it was decanonized in 2024 in favor of this extension because parakangaroo is an [[exotemperament]] that only supports the [[patent val]] of [[15edo]], while this extension is a bit more practical since it is at least low-medium accuracy and supports the [[patent val]]s of 15edo, [[17edo]] and [[32edo]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 64/63, 6125/5832
{{Mapping|legend=1| 1 0 -3 6 | 0 3 10 -6 }}
: mapping generators: ~2, ~35/24
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.1767{{c}}, ~35/24 = 637.2914{{c}}
: [[error map]]: {{val| -1.823 +9.919 -7.930 -3.514 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 638.1977{{c}}
: error map: {{val| 0.000 +12.638 -4.336 +1.988 }}
{{Optimal ET sequence|legend=1| 15, 32, 47b }}
[[Badness]] (Sintel): 2.91
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 245/242
{{Mapping|legend=0| 1 0 -3 6 4 | 0 3 10 -6 -1 }}
Optimal tunings:
* WE: ~2 = 1197.8893{{c}}, ~16/11 = 637.1922{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 638.2605{{c}}
{{Optimal ET sequence|legend=0| 15, 32, 47b }}
Badness (Sintel): 1.65


== Fervor ==
== Fervor ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Fervor]].''
Fervor tempers out [[9604/9375]] and may be described as {{nowrap| 25 & 27 }}. It splits the [[6/1|6th harmonic]] into five generators of [[~]][[10/7]]; its ploidacot is beta-pentacot. The 11-limit extension simply equates the generator to [[16/11]]. [[27edo]] is about as accurate as it can be tuned.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 595: Line 616:
[[Comma list]]: 64/63, 9604/9375
[[Comma list]]: 64/63, 9604/9375


{{Mapping|legend=1| 1 4 -2 -2 | 0 -5 9 10 }}
{{Mapping|legend=1| 1 -1 7 8 | 0 5 -9 -10 }}
 
: mapping generators: ~2, ~10/7
: mapping generators: ~2, ~7/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~7/5 = 577.353
* [[WE]]: ~2 = 1196.2742{{c}}, ~10/7 = 620.2918{{c}}
: [[error map]]: {{val| 0.000 +11.278 +9.867 +4.709 }}
: [[error map]]: {{val| -3.726 +3.230 +4.980 -1.550 }}
* [[POTE]]: ~2 = 1200.000, ~7/5 = 577.776
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 622.3179{{c}}
: error map: {{val| 0.000 +9.163 +13.673 +8.937 }}
: error map: {{val| 0.000 +9.634 +12.826 +7.996 }}


{{Optimal ET sequence|legend=1| 2, 25, 27 }}
{{Optimal ET sequence|legend=1| 2, 25, 27 }}
Line 614: Line 634:
Comma list: 56/55, 64/63, 1350/1331
Comma list: 56/55, 64/63, 1350/1331


Mapping: {{mapping| 1 4 -2 -2 3 | 0 -5 9 10 1 }}
{{Mapping|legend=0| 1 -1 7 8 4 | 0 5 -9 -10 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/5 = 577.296
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* POTE: ~2 = 1200.000, ~7/5 = 577.850
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.2525{{c}}


{{Optimal ET sequence|legend=0| 2, 25e, 27e }}
{{Optimal ET sequence|legend=0| 2, 25e, 27e }}
Line 629: Line 649:
Comma list: 56/55, 64/63, 78/77, 507/500
Comma list: 56/55, 64/63, 78/77, 507/500


Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}
{{Mapping|legend=0| 1 -1 7 8 4 12 | 0 5 -9 -10 -1 -16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/5 = 577.374
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
* POTE: ~2 = 1200.000, ~7/5 = 578.060
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 622.0631{{c}}


{{Optimal ET sequence|legend=0| 2f, 27e }}
{{Optimal ET sequence|legend=0| 2f, 27e }}
Line 643: Line 663:
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}


Sixix 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.
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. An important val not in the [[optimal ET sequence]] is 39c, that is, [[39edo]] using the second-best approximation of [[5/4]] (the 369-cent one inherited from [[13edo]]).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 652: Line 672:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 337.519
* [[WE]]: ~2 = 1198.9028{{c}}, ~6/5 = 337.1334{{c}}
: [[error map]]: {{val| 0.000 +10.449 -11.429 +6.366 }}
: [[error map]]: {{val| -1.097 +9.086 -13.503 +2.508 }}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 337.442
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 337.4588{{c}}
: error map: {{val| 0.000 +10.835 -10.965 +5.594 }}
: error map: {{val| 0.000 +10.751 -11.066 +5.762 }}


{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}
Line 666: Line 686:
Comma list: 55/54, 64/63, 125/121
Comma list: 55/54, 64/63, 125/121


Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
{{Mapping|legend=0| 1 3 4 0 6 | 0 -5 -6 10 -9 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 337.749
* WE: ~2 = 1198.5480{{c}}, ~6/5 = 337.1557{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 337.564
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.6000{{c}}


{{Optimal ET sequence|legend=0| 7, 25e, 32 }}
{{Optimal ET sequence|legend=0| 7, 25e, 32 }}
Line 681: Line 701:
Comma list: 40/39, 55/54, 64/63, 125/121
Comma list: 40/39, 55/54, 64/63, 125/121


Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
{{Mapping|legend=0| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 337.793
* WE: ~2 = 1197.7111{{c}}, ~6/5 = 336.8391{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 337.483
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5336{{c}}


{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
{{Optimal ET sequence|legend=0| 7, 25e, 32f }}
Line 696: Line 716:
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 337.629
* WE: ~2 = 1197.7807{{c}}, ~6/5 = 336.8884{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 337.513
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 337.5279{{c}}


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


Badness (Sintel): 2.00
Badness (Sintel): 2.00
== Subgroup extensions ==
=== Supra (2.3.7.11) ===
Subgroup: 2.3.7.11
Comma list: 64/63, 99/98
{{Mapping|legend=2| 1 0 6 13 | 0 1 -2 -6 }}
{{Mapping|legend=4| 1 0 0 6 13 | 0 1 0 -2 -6 }}
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 ====
This extension maps [[13/11]] to the minor third (C–E♭), [[12/11]][[~]][[14/13]] to the augmented unison (C–C♯), and [[13/12]] to the diminished third (C–E𝄫).
Subgroup: 2.3.7.11.13
Comma list: 64/63, 78/77, 99/98
{{Mapping|legend=2| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
{{Mapping|legend=4| 1 0 0 6 13 18 | 0 1 0 -2 -6 -9 }}
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]]
=== Oceanfront (2.3.7.13/5) ===
This is the [[restriction]] of [[#Ultrapyth|ultrapyth]] to the 2.3.7.13/5 subgroup, mapping [[13/10]] to the major third.
[[Subgroup]]: 2.3.7.13/5
[[Comma list]]: 64/63 ({{monzo| 6 -2 -1 0 }}), 91/90 ({{monzo| -1 -2 1 1 }})
{{Mapping|legend=3| 1 0 6 -5 | 0 1 -2 4 }}
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1197.2656{{c}}, ~3/2 = 710.1993{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.7413{{c}}
{{Optimal ET sequence|legend=1| 5, 17, 22, 27, 32, 59 }}
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
Scales: [[Oceanfront scales]]


[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Archytas clan| ]] <!-- main article -->
[[Category:Archytas| ]] <!-- key article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]
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