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No-twos subgroup temperaments: Difference between revisions

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=== Mintra ===
=== Mintra ===
This temperament splits 27/7 (the BPS generator up a tritave) into three by means of [[11/7]] or, equivalently, [[7/1]] in three by means of [[21/11]], and is the intersection of BPS, [[#Deneb|deneb]], and [[#Mintaka|mintaka]] temperaments as well as the most natural temperament satisfied in the 3.5.7.11 subgroup in [[39edt]].
This temperament splits 27/7 (the BPS generator up a tritave) into three by means of [[11/7]] or, equivalently, [[7/1]] in three by means of [[21/11]], and is the intersection of BPS, [[#Deneb|deneb]], and [[#Mintaka|mintaka]] temperaments as well as the most natural temperament satisfied in the 3.5.7.11 subgroup in [[39edt]].
The 13-limit extension uses the canonical extension for prime 13 described at [[#Tridecimal mintaka]].


Subgroup: 3.5.7.11
Subgroup: 3.5.7.11
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==== Tridecimal mintra ====
==== Tridecimal mintra ====
This temperament uses the canonical extension for prime 13 described at [[#Tridecimal mintaka]].
Subgroup: 3.5.7.11.13
Subgroup: 3.5.7.11.13


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[[Comma list]]: 13841287201/13839609375
[[Comma list]]: 13841287201/13839609375


{{Mapping|legend=2| 1 7 5 | 0 -12 -7 }}
{{Mapping|legend=2| 1 -5 -2 | 0 12 7 }}
: mapping generators: ~3, ~16807/10125
: mapping generators: ~3, ~30375/16807


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1901.958{{c}}, ~16807/10125 = 877.283{{c}}
* [[WE]]: ~3 = 1901.9584{{c}}, ~30375/16807 = 1024.6759{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~16807/10125 = 877.281{{c}}
: [[error map]]: {{val| +0.003 +0.005 -0.012 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~30375/16807 = 1024.6743{{c}}
: error map: {{val| 0.000 +0.002 -0.016 }}


[[Optimal ET sequence]]: [[13edt|b13]], [[141edt|b141]], [[154edt|b154]], …, [[258edt|b258]], [[271edt|b271]], [[800edt|b800]], [[1071edt|b1071]], [[1342edt|b1342]], [[1613edt|b1613]], [[4568edt|b4568]], [[6181edt|b6181]]
[[Optimal ET sequence]]: [[13edt|b13]], [[141edt|b141]], [[154edt|b154]], …, [[258edt|b258]], [[271edt|b271]], [[800edt|b800]], [[1071edt|b1071]], [[1342edt|b1342]], [[1613edt|b1613]], [[4568edt|b4568]], [[6181edt|b6181]]


[[Badness]] (Sintel): 0.017
[[Badness]] (Sintel): 0.0166


== Nekkar ==
== Nekkar ==
This temperament is the no-2's [[restriction]] of [[squares]], and as such is named after a star that belonged to the obsolete constellation of Quadrans Muralis, whose name has to do with squares. However, seeing the sheer complexity and size of the commas, nekkar is much more naturally thought of as 3.5.7.11 than 3.5.7, whereupon it becomes a strong extension of [[mintaka]].
This temperament is the no-2's [[restriction]] of [[squares]], and as such is named after a star that belonged to the obsolete constellation of Quadrans Muralis, whose name has to do with squares. However, seeing the sheer complexity and size of the commas, nekkar is much more naturally thought of as a temperament of 3.5.7.11 than 3.5.7, whereupon it is a restriction of undecimal squares, and a strong extension of [[#Mintaka|mintaka]]. However, the 13-limit extension restricts to [[#Minalzidar|minalzidar]] rather than tridecimal mintaka.


[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7


[[Comma list]]: 35303692060125/33232930569601
[[Comma list]]: {{monzo| -24 -3 16 }}


{{Mapping|legend=2|1 8 3|0 -16 -3}}
{{Mapping|legend=2| 1 -8 0 | 0 16 3 }}
: mapping generators: ~3, ~16807/10935
: mapping generators: ~3, ~32805/16807


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1900.155{{c}}, ~16807/10935 = 775.963{{c}}
* [[WE]]: ~3 = 1900.1550{{c}}, ~32805/16807 = 1124.1916{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~16807/10935 = 776.767{{c}}
: [[error map]]: {{val| -1.800 -0.488 +3.749 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~32805/16807 = 1125.1876{{c}}
: error map: {{val| 0.000 +1.047 +6.737 }}


[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]]
[[Optimal ET sequence]]: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[311edt|b311dd]]


[[Badness]] (Sintel): 17.120
[[Badness]] (Sintel): 17.1


=== 3.5.7.11 subgroup ===
=== 3.5.7.11 subgroup ===
{{See also| Mintaka }}
This continues the canonical 11-limit extension of squares.
Subgroup: 3.5.7.11
Subgroup: 3.5.7.11


Comma list: 1331/1323, 120285/117649
Comma list: 1331/1323, 120285/117649


Subgroup-val mapping: {{mapping| 1 8 3 3 | 0 -16 -3 -2 }}
Subgroup-val mapping: {{mapping| 1 -8 0 1 | 0 16 3 2 }}
: mapping generators: ~3, ~11/7


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 776.781{{c}}
Optimal tunings:
* WE: ~3 = 1900.6084{{c}}, ~21/11 = 1124.4193{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.1738{{c}}


Supporting ETs: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71]], [[120edt|b120]], [[191edt|b191d]], [[262edt|b262d]]


Badness (Sintel): 1.375
Badness (Sintel): 1.37


=== 3.5.7.11.13 subgroup ===
=== 3.5.7.11.13 subgroup ===
This uses the [[no-twos subgroup temperaments #Minalzidar|minalzidar]] mapping of 13.
Subgroup: 3.5.7.11.13
Subgroup: 3.5.7.11.13


Comma list: 169/165, 351/343, 11011/10935
Comma list: 169/165, 351/343, 11011/10935


Subgroup-val mapping: {{mapping| 1 8 3 3 6 | 0 -16 -3 -2 -9 }}
Subgroup-val mapping: {{mapping| 1 -8 0 1 -3 | 0 16 3 2 9 }}
: mapping generators: ~3, ~11/7


Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~11/7 = 776.678{{c}}
Optimal tunings:
* WE: ~3 = 1902.3248{{c}}, ~21/11 = 1125.4837{{c}}
* CWE: ~3 = 1901.9550{{c}}, ~21/11 = 1125.2772{{c}}


Supporting ETs: 22, 5c, 27, 49, 71f, 17cf
Optimal ET sequence: [[22edt|b22]], [[49edt|b49]], [[71edt|b71f]]


Badness (Sintel): 1.723
Badness (Sintel): 1.72


== Procyon ==
== Procyon ==
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{{Mapping|legend=2| 1 2 2 | 0 -7 -3 }}
{{Mapping|legend=2| 1 2 2 | 0 -7 -3 }}
: mapping generators: ~3, ~17/9
: mapping generators: ~3, ~49/45


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1902.198{{c}}, ~49/45 = 145.412{{c}}
* [[WE]]: ~3 = 1902.1979{{c}}, ~49/45 = 145.4124{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~49/45 = 145.368{{c}}
: [[error map]]: {{val| +0.243 +0.195 -0.667 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~49/45 = 145.3680{{c}}
: error map: {{val| 0.000 +0.020 -1.020 }}


[[Support]]ing [[ET]]s: b13, b157, b144, b170, b131, b183, b118, b14, b105, b12c, b196, b92, b27, b79
[[Optimal ET sequence]]: [[13edt|b13]], [[92edt|b92]], [[105edt|b105]], [[118edt|b118]], [[131edt|b131]], [[144edt|b144]], [[157edt|b157]], [[327edt|b327]], [[484edt|b484]], [[641edt|b641]], [[1125edt|b1125d]]


[[Badness]] (Sintel): 0.200
[[Badness]] (Sintel): 0.200
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{{Main| Mintaka }}
{{Main| Mintaka }}


Extensions to prime 5 are covered at [[#Mintra]] and [[#Nekkar]].
Mintaka tempers out [[1331/1323]] in the 3.7.11 subgroup. It is the common [[restriction]] of [[#Mintra|mintra]] and [[#Nekkar|nekkar]].


[[Subgroup]]: 3.7.11
[[Subgroup]]: 3.7.11
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