Subgroup temperaments: Difference between revisions

Line 1,030:

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.

[[Subgroup]]: 2.5.7/3.11/3

Line 1,036:

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})

: mapping generators: ~2, ~12/7

: mapping generators: ~2, ~7/6

Line 1,043:

[[Optimal tuning]]s:

* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~12/7 = ~~929~~.~~545~~

* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455

* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~12/7 = ~~929~~.~~907~~

* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}

Line 1,050:

[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents

==== Tridecimal guanyintet ====

Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])3 = [[13/10]] and ([[15/14]])3 = [[16/13]]. [[40edo]] remains an excellent tuning.

[[Subgroup]]: 2.5.7/3.11/3.13

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})

: mapping generators: ~2, ~12/7

[[Optimal tuning]]s:

* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152

* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} using subgroup TE

: <nowiki/>* wart for 7/3

Badness (Sintel): 0.329

==== Laz ====

@@ Line 1,030: / Line 1,030: @@
 {{See also | Chromatic pairs #Guanyintet }}
-Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.
+Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.
 [[Subgroup]]: 2.5.7/3.11/3
@@ Line 1,036: / Line 1,036: @@
 [[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})
-{{Mapping|legend=2| 1 0 2 -2 | 0 3 -1 5 }}
+{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
-: mapping generators: ~2, ~12/7
+: mapping generators: ~2, ~7/6
 {{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
@@ Line 1,043: / Line 1,043: @@
 [[Optimal tuning]]s:
-* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~12/7 = 929.545
+* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
-* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~12/7 = 929.907
+* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093
 {{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
@@ Line 1,050: / Line 1,050: @@
 [[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
+==== Tridecimal guanyintet ====
+Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])<sup>3</sup> = [[13/10]] and ([[15/14]])<sup>3</sup> = [[16/13]]. [[40edo]] remains an excellent tuning.
+[[Subgroup]]: 2.5.7/3.11/3.13
+[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})
+{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
+: mapping generators: ~2, ~12/7
+[[Optimal tuning]]s:
+* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
+* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218
+{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
+: <nowiki/>* wart for 7/3
+Badness (Sintel): 0.329
 ==== Laz ====