No-twos subgroup temperaments: Difference between revisions

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~~For extensions~~ of this temperament that include the ~~prime 2, see~~ [[trienstonic clan #Opossum|opossum]], [[jubilismic clan #Crepuscular|crepuscular]], [[kleismic family #Catalan|catalan]], [[tetracot family #Bunya|bunya]], [[~~sensamagic~~ clan #~~Bohpier~~|~~bohpier~~]], and [[~~gamelismic~~ clan #~~Superkleismic~~|~~superkleismic~~]].

[[Strong extension]]s of this temperament that include the octave are [[trienstonic clan #Opossum|opossum]] and [[jubilismic clan #Crepuscular|crepuscular]]. [[Weak extension]]s are [[kleismic family #Catalan|catalan]], [[tetracot family #Bunya|bunya]], [[gamelismic clan #Superkleismic|superkleismic]], and [[sensamagic clan #Bohpier|bohpier]]. Non-octave extensions are documented below.

[[Subgroup]]: 3.5.7

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[[Optimal tuning]]s:

* [[WE]]: ~3 = 1903.~~863~~{{c}}, ~5/3 = 878.~~923~~{{c}}

* [[WE]]: ~3 = 1903.8634{{c}}, ~5/3 = 878.9230{{c}}

* [[CWE]]: ~3 = 1901.~~955~~{{c}}, ~5/3 = 878.~~291~~{{c}}

: [[error map]]: {{val| +1.908 -3.527 +0.849 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 878.2907{{c}}

: error map: {{val| 0.000 -6.068 -1.037 }}

[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]

[[Badness]] (Sintel): 0.535

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Comma list: 15625/15309, 177147/171875

Subgroup-val mapping: 1 1 -1 5 | 0 1 6 -6 }}

Subgroup-val mapping: {{mapping| 1 1 -1 5 | 0 1 6 -6 }}

~~: gencom: [3/1 5/3; 15625/15309 177147/171875]~~

Optimal ~~tuning (POTE)~~: ~5/3 = 884.~~268~~{{c}}

Optimal tunings:

* WE: ~3 = 1890.5039{{c}}, ~5/3 = 879.7029{{c}}

* CWE: ~3 = 1901.9550{{c}}, ~5/3 = 884.8724{{c}}

Optimal ET sequence: [[13edt|b13e]], [[15edt|b15]], [[28edt|b28e]], [[43edt|b43dee]]

Badness (Sintel): 2.~~507~~

Badness (Sintel): 2.51

== BPS ==

For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions ~~will be~~ documented below.

For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions are documented below.

[[Subgroup]]: 3.5.7

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{{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11/2). }}

This is a strong extension to ~~BPS in~~ the subgroup 3.5.7.11/2.13/4 that equates the "semitone" of [[27/25]]~[[49/45]] to [[13/12]], and then three of these intervals to [[14/11]].

This is a strong extension of BPS to the subgroup 3.5.7.11/2.13/4 that equates the "semitone" of [[27/25]]~[[49/45]] to [[13/12]], and then three of these intervals to [[14/11]].

Subgroup: 3.5.7.11/2.13/4

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Subgroup-val mapping: {{mapping| 1 1 2 -1 2 | 0 2 -1 11 -4 }}

: mapping generators: ~3, ~7/3

: mapping generators: ~3, ~9/7

Optimal ~~tuning (~~CWE): ~3 = 1901.~~955~~{{c}}, ~9/7 = 441.~~025~~{{c}}

Optimal tunings:

* Subgroup WE: ~3 = 1903.5584{{c}}, ~9/7 = 441.3534{{c}}

* Subgroup CWE: ~3 = 1901.9550{{c}}, ~9/7 = 441.0249{{c}}

<nowiki/>* é is used as the wart for 11/2.

: <nowiki/>* é is used as the wart for 11/2.

Badness (Sintel): 0.187

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 {{Main| Arcturus }}
-For extensions of this temperament that include the prime 2, see [[trienstonic clan #Opossum|opossum]], [[jubilismic clan #Crepuscular|crepuscular]], [[kleismic family #Catalan|catalan]], [[tetracot family #Bunya|bunya]], [[sensamagic clan #Bohpier|bohpier]], and [[gamelismic clan #Superkleismic|superkleismic]].
+[[Strong extension]]s of this temperament that include the octave are [[trienstonic clan #Opossum|opossum]] and [[jubilismic clan #Crepuscular|crepuscular]]. [[Weak extension]]s are [[kleismic family #Catalan|catalan]], [[tetracot family #Bunya|bunya]], [[gamelismic clan #Superkleismic|superkleismic]], and [[sensamagic clan #Bohpier|bohpier]]. Non-octave extensions are documented below.
 [[Subgroup]]: 3.5.7
@@ Line 33: / Line 33: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~3 = 1903.863{{c}}, ~5/3 = 878.923{{c}}
+* [[WE]]: ~3 = 1903.8634{{c}}, ~5/3 = 878.9230{{c}}
-* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 878.291{{c}}
+: [[error map]]: {{val| +1.908 -3.527 +0.849 }}
+* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 878.2907{{c}}
+: error map: {{val| 0.000 -6.068 -1.037 }}
-[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]
+[[Optimal ET sequence]]: [[2edt|b2]], [[9edt|b9d]], [[11edt|b11]], [[13edt|b13]], [[340edt|b340cc]]
 [[Badness]] (Sintel): 0.535
@@ Line 47: / Line 49: @@
 Comma list: 15625/15309, 177147/171875
-Subgroup-val mapping: 1 1 -1 5 | 0 1 6 -6 }}
+Subgroup-val mapping: {{mapping| 1 1 -1 5 | 0 1 6 -6 }}
-: gencom: [3/1 5/3; 15625/15309 177147/171875]
-Optimal tuning (POTE): ~5/3 = 884.268{{c}}
+Optimal tunings:
+* WE: ~3 = 1890.5039{{c}}, ~5/3 = 879.7029{{c}}
+* CWE: ~3 = 1901.9550{{c}}, ~5/3 = 884.8724{{c}}
 Optimal ET sequence: [[13edt|b13e]], [[15edt|b15]], [[28edt|b28e]], [[43edt|b43dee]]
-Badness (Sintel): 2.507
+Badness (Sintel): 2.51
 == BPS ==
 {{Main| BPS }}
-For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions will be documented below.
+For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions are documented below.
 [[Subgroup]]: 3.5.7
@@ Line 81: / Line 84: @@
 {{Todo|inline=1|complete section|comment= Catalog the intermediate extension (3.5.7.11/2). }}
-This is a strong extension to BPS in the subgroup 3.5.7.11/2.13/4 that equates the "semitone" of [[27/25]]~[[49/45]] to [[13/12]], and then three of these intervals to [[14/11]].
+This is a strong extension of BPS to the subgroup 3.5.7.11/2.13/4 that equates the "semitone" of [[27/25]]~[[49/45]] to [[13/12]], and then three of these intervals to [[14/11]].
 Subgroup: 3.5.7.11/2.13/4
@@ Line 88: / Line 91: @@
 Subgroup-val mapping: {{mapping| 1 1 2 -1 2 | 0 2 -1 11 -4 }}
-: mapping generators: ~3, ~7/3
+: mapping generators: ~3, ~9/7
-Optimal tuning (CWE): ~3 = 1901.955{{c}}, ~9/7 = 441.025{{c}}
+Optimal tunings:
+* Subgroup WE: ~3 = 1903.5584{{c}}, ~9/7 = 441.3534{{c}}
+* Subgroup CWE: ~3 = 1901.9550{{c}}, ~9/7 = 441.0249{{c}}
-Supporting ETs: [[13edt|b13]], [[69edt|b69]], [[56edt|b56]], [[82edt|b82]], [[43edt|b43]], [[125edt|b125]], [[30edt|b30]], [[151edt|b151]], [[95edt|b95]], [[17edt|b17é]], [[194edt|b194d]], [[99edt|b99]], [[181edt|b181d]], [[108edt|b108é]] *
+Optimal ET sequence: [[13edt|b13]], [[43edt|b43]], [[56edt|b56]], [[69edt|b69]], [[289edt|b289d]], [[358edt|b358ddé]], [[427edt|b427cddé]] *
-<nowiki/>* é is used as the wart for 11/2.
+: <nowiki/>* é is used as the wart for 11/2.
 Badness (Sintel): 0.187