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Archytas clan: Difference between revisions

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Finish introducing all the extensions. Prefer ploidacot form for fervor's mapping
→Archy: + dichotic
 
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{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}
{{Mapping|legend=2| 1 0 6 | 0 1 -2 }}
: sval mapping generators: ~2, ~3


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


: [[gencom]]: [2 3; 64/63]
: mapping generators: ~2, ~3


[[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 }}
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==== 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 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 blackwood with a 1/5-octave period. Finally, [[15625/15552]] gives catalan, splitting the twelfth in six.  


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
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* [[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]]
* ''[[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]]
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]
* ''[[Catalan]]'' (+15625/15552) → [[Kleismic family #Catalan|Kleismic family]]


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


==== 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 right below. Discussed elsewhere is [[suhajira]] of the [[rastmic clan #Suhajira|rastmic clan]].


=== Supra ===
=== Supra ===
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Comma list: 64/63, 99/98
Comma list: 64/63, 99/98


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


Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}
Gencom mapping: {{mapping| 1 0 0 6 13 | 0 1 0 -2 -6 }}
: gencom: [2 3; 64/63 99/98]


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 708.456
* WE: ~2 = 1197.2650{{c}}, ~3/2 = 705.5803{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 707.192
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.4981{{c}}


{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d, 56d }}
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==== Supraphon ====
==== 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
Subgroup: 2.3.7.11.13


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


Sval mapping: {{mapping| 1 0 6 13 18 | 0 1 -2 -6 -9 }}
Subgroup-val 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 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:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 707.344
* WE: ~2 = 1197.1909{{c}}, ~3/2 = 704.4836{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 706.137
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.4289{{c}}


{{Optimal ET sequence|legend=0| 12f, 17 }}
{{Optimal ET sequence|legend=0| 12f, 17 }}
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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
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[[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 }}
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=== 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


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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 }}
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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: {{mapping| 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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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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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 }}
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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 }}
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== 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
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[[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


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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 }}
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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 }}
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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 }}
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== 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 [[the Biosphere #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
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[[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 }}
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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 }}
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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 }}
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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 }}
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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 }}
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== 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
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[[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 }}


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{{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
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[[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 403: Line 416:


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 424: Line 437:


[[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 444: Line 457:


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 459: Line 472:


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 474: Line 487:


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 489: Line 502:


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 504: Line 517:


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 519: Line 532:


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 541: Line 554:


[[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 558: Line 571:


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 573: Line 586:


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 588: Line 601:


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 }}
Line 609: Line 622:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~10/7 = 622.647
* [[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, ~10/7 = 622.224
* [[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 626: Line 639:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 622.704
* WE: ~2 = 1195.4148{{c}}, ~10/7 = 619.7729{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 622.150
* 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 641: Line 654:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~10/7 = 622.626
* WE: ~2 = 1195.6284{{c}}, ~10/7 = 619.6738{{c}}
* POTE: ~2 = 1200.000, ~10/7 = 621.940
* 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 661: Line 674:


[[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 678: Line 691:


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 693: Line 706:


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 708: Line 721:


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 }}
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