Luna family: Difference between revisions

(3 intermediate revisions by the same user not shown)

Line 28:

=== Overview to extensions ===

[[Hemimean clan #Hemithirds|Hemithirds]], a strong extension of [[didacus]] that tempers out [[1029/1024]] alongside [[3136/3125]], might be considered the main [[7-limit]] extension to luna of practical interest, as [[7/4]] is found at only 5 generators. However, luna is an undoubted microtemperament of the [[5-limit]] in its own right, supported by systems like [[118edo]] and [[441edo]], and therefore merits consideration of other extensions to the 7-limit, which are documented on this page. Though of very high complexity, they continue the level of accuracy of 5-limit luna.

[[Hemimean clan #Hemithirds|Hemithirds]] (25 & 31), a strong extension of [[didacus]] that tempers out [[1029/1024]] alongside [[3136/3125]], might be considered the main [[7-limit]] extension to luna of practical interest, as [[7/4]] is found at only 5 generators. However, luna is an undoubted microtemperament of the [[5-limit]] in its own right, supported by systems like [[118edo]] and [[441edo]], and therefore merits consideration of other extensions to the 7-limit, which are documented on this page. Though of very high complexity, they continue the level of accuracy of 5-limit luna.

Weak extensions of luna include hemiluna (87 & 323), semiluna (118 & 292), and subneutral (31 & 441).

Temperaments discussed elsewhere include:

* ''[[Subneutral]]'' (+2401/2400) → [[Breedsmic temperaments #Subneutral|Breedsmic temperaments]]

* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]

The rest are considered below.

== Lunatic ==

Line 117:

Line 120:

Badness (Sintel): 2.24

== Subneutral ==

Subneutral tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 31 & 441 }} temperament. Despite the name, the generator is only about 3 cents flat of the exact neutral third.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 274877906944/274658203125

: mapping generators: ~2, ~46875/28672

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9998{{c}}, ~46875/28672 = 851.6994{{c}} (~57344/46875 = 348.3005{{c}})

: [[error map]]: {{val| -0.000 +0.013 +0.090 -0.132 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~46875/28672 = 851.6995{{c}} (~57344/46875 = 348.3005{{c}})

: error map: {{val| 0.000 +0.014 +0.090 -0.132 }}

{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}

[[Badness]] (Sintel): 1.16

[[Category:Luna family| ]]

[[Category:Temperament families]]

[[Category:~~Luna family| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

@@ Line 28: / Line 28: @@
 === Overview to extensions ===
-[[Hemimean clan #Hemithirds|Hemithirds]], a strong extension of [[didacus]] that tempers out [[1029/1024]] alongside [[3136/3125]], might be considered the main [[7-limit]] extension to luna of practical interest, as [[7/4]] is found at only 5 generators. However, luna is an undoubted microtemperament of the [[5-limit]] in its own right, supported by systems like [[118edo]] and [[441edo]], and therefore merits consideration of other extensions to the 7-limit, which are documented on this page. Though of very high complexity, they continue the level of accuracy of 5-limit luna.
+[[Hemimean clan #Hemithirds|Hemithirds]] (25 & 31), a strong extension of [[didacus]] that tempers out [[1029/1024]] alongside [[3136/3125]], might be considered the main [[7-limit]] extension to luna of practical interest, as [[7/4]] is found at only 5 generators. However, luna is an undoubted microtemperament of the [[5-limit]] in its own right, supported by systems like [[118edo]] and [[441edo]], and therefore merits consideration of other extensions to the 7-limit, which are documented on this page. Though of very high complexity, they continue the level of accuracy of 5-limit luna.
+Weak extensions of luna include hemiluna (87 & 323), semiluna (118 & 292), and subneutral (31 & 441).
 Temperaments discussed elsewhere include:
-* ''[[Subneutral]]'' (+2401/2400) → [[Breedsmic temperaments #Subneutral|Breedsmic temperaments]]
 * [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]
+The rest are considered below.
 == Lunatic ==
@@ Line 117: / Line 120: @@
 Badness (Sintel): 2.24
+== Subneutral ==
+Subneutral tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 31 & 441 }} temperament. Despite the name, the generator is only about 3 cents flat of the exact neutral third.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 2401/2400, 274877906944/274658203125
+{{Mapping|legend=1| 1 -41 8 -5 | 0 60 -8 11 }}
+: mapping generators: ~2, ~46875/28672
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9998{{c}}, ~46875/28672 = 851.6994{{c}} (~57344/46875 = 348.3005{{c}})
+: [[error map]]: {{val| -0.000 +0.013 +0.090 -0.132 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~46875/28672 = 851.6995{{c}} (~57344/46875 = 348.3005{{c}})
+: error map: {{val| 0.000 +0.014 +0.090 -0.132 }}
+{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}
+[[Badness]] (Sintel): 1.16
+[[Category:Luna family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Luna family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]