Subgroup temperaments: Difference between revisions

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Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]

[[Comma list]]: [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 0 2 }}), [[847/845]] ({{monzo| 0 0 1 2 -2 }})

{{Mapping|legend=5| 1 2 -3/4 ~~-11~~/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}

: ~~[[gencom]]~~: [2 15~~/13; 352/351 676/675 847/845]~~

: mapping generators: ~2, ~26/15

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[~~POTE~~]]): ~2 = ~~1\1~~, ~15/13 = ~~248~~.~~491~~

[[Optimal tuning]]s:

* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.7072{{c}}, ~26/15 = 951.2727{{c}}

* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}

{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198~~, 565d, 763bd, 961bd~~ }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents

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=== Historical ===

{{Distinguish|History (temperament)}}~~, which is the rank-3 version of this temperament in the full 13-limit.~~

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma ~~S10/S11 =~~ [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and ~~S13/S15 =~~ [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out ~~S21 =~~ [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma [[4000/3993]] ([[S-expression|S10/S11]]) to set [[11/10]] equal to one-third of 4/3, and [[676/675]] ([[S-expression|S13/S15]]) to equate [[15/13]] to one-half of 4/3, and tempers out [[441/440]] ({{S|21}}) to split 11/10 into two instances of [[22/21]][[~]][[21/20]]. [[Schismatic family #Sextilifourths|Sextilifourths]] adds the [[schismatic family #Schismic|schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

[[Subgroup]]: 2.3.7/5.11/5.13/5

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: mapping generators: ~2, ~21/20

[[Optimal tuning]] ([[Tp tuning|subgroup ~~POTE~~]]): ~21/20 = 83.~~016~~

[[Optimal tuning]]s:

* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}

* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}

[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents

=== Direct breedsmic ===

Related ~~temperament~~: [[~~hemithirds~~]], [[newt]]

This temperament was first proposed by {{u|Royalmilktea}}. The name was established by [[Lériendil]] in 2024. Related temperaments: [[breedsmic temperaments #Hemififths|hemififths]] and [[septischismic clan #Newt|newt]].

[[Subgroup]]: 2.3.49/5

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: mapping generators: ~2, ~49/40

[[Optimal tuning]] ([[Tp tuning|subgroup ~~POTE~~]]): ~49/40 = 350.~~966~~

[[Optimal tuning]]:

* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~21/20 = 350.9724{{c}}

* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 350.9743{{c}}

{{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }}

~~[[Tp tuning #T2 tuning|RMS error]]: ?~~

<nowiki/>* Wart for 49/5

==== Bridgetown ====

Bridgetown, the 5 &~~amp;~~ 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].

[[Subgroup]]: 2.3.11/5.13/5

@@ Line 212: / Line 212: @@
 {{See also| Chromatic pairs #Haumea }}
-Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].
+Related temperaments include [[#Bridgetown|bridgetown]], [[aberschismic family #Namaka|namaka]], [[schismatic family #Hemigari|hemigari]], [[#Barbados|barbados]], and [[the Archipelago #Parizekmic|parizekmic]].
 [[Subgroup]]: 2.3.7/5.11/5.13/5
-[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]
+[[Comma list]]: [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 0 2 }}), [[847/845]] ({{monzo| 0 0 1 2 -2 }})
 {{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}
-{{Mapping|legend=5| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
+{{Mapping|legend=5| 1 0 -3/4 37/4 -27/4 -7/4 | 0 2 0 -12 9 3 }}
-: [[gencom]]: [2 15/13; 352/351 676/675 847/845]
+: mapping generators: ~2, ~26/15
-[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491
+[[Optimal tuning]]s:
+* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.7072{{c}}, ~26/15 = 951.2727{{c}}
+* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}
-{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}
+{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}
 [[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
@@ Line 231: / Line 233: @@
 === Historical ===
 {{Distinguish|Historical temperaments}}
-{{Distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.
+{{Distinguish|History (temperament)}}
-Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.
+Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma [[4000/3993]] ([[S-expression|S10/S11]]) to set [[11/10]] equal to one-third of 4/3, and [[676/675]] ([[S-expression|S13/S15]]) to equate [[15/13]] to one-half of 4/3, and tempers out [[441/440]] ({{S|21}}) to split 11/10 into two instances of [[22/21]][[~]][[21/20]]. [[Schismatic family #Sextilifourths|Sextilifourths]] adds the [[schismatic family #Schismic|schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.
 [[Subgroup]]: 2.3.7/5.11/5.13/5
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 {{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
+: mapping generators: ~2, ~21/20
-[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016
+[[Optimal tuning]]s:
+* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}
+* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}
-{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}
+{{Optimal ET sequence|legend=1| 14, 29, 101, 130, 159 }}
 [[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
 === Direct breedsmic ===
-Related temperament: [[hemithirds]], [[newt]]
+This temperament was first proposed by {{u|Royalmilktea}}. The name was established by [[Lériendil]] in 2024. Related temperaments: [[breedsmic temperaments #Hemififths|hemififths]] and [[septischismic clan #Newt|newt]].
 [[Subgroup]]: 2.3.49/5
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 {{Mapping|legend=3| 1 1 3 | 0 2 1 }}
+: mapping generators: ~2, ~49/40
-[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
+[[Optimal tuning]]:
+* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~21/20 = 350.9724{{c}}
+* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 350.9743{{c}}
-{{Optimal ET sequence|legend=1|7, 10, 17}}
+{{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }}
-[[Tp tuning #T2 tuning|RMS error]]: ?
+<nowiki/>* Wart for 49/5
 ==== Bridgetown ====
 {{See also| Chromatic pairs #Bridgetown }}
-Bridgetown, the 5 &amp; 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].
+Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].
 [[Subgroup]]: 2.3.11/5.13/5