Extension and restriction: Difference between revisions

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An '''extension''' of a [[regular temperament]] on a [[JI subgroup]] ''G'' to a larger subgroup ''G' '' is a new temperament that contains the same intervals as the original temperament, and whose [[rank]] remains the same, with the same JI interpretations in the subgroup ''G'', but gives them new JI interpretations not in the original subgroup (but are in the larger subgroup). The opposite of extension is '''restriction'''.

An '''extension''' of a [[regular temperament]] on a [[JI subgroup]] ''G'' to a larger subgroup ''G{{'}}'' is a new temperament that contains the same intervals as the original temperament, and whose [[rank]] remains the same, with the same JI interpretations in the subgroup ''G'', but gives them new JI interpretations not in the original subgroup (but are in the larger subgroup). The opposite of extension is '''restriction'''.

For example, [[septimal meantone]] and [[flattone]] are both extensions of [[5-limit]] (2.3.5) [[meantone]] to the [[7-limit]] (2.3.5.7), because ~~C-E~~ (4 fifths) represents [[5/4]] in both. They are different extensions, because in septimal meantone, 7/4 is ~~C-A#~~ (10 fifths), while in flattone, 7/4 is ~~C-Bbb~~ (-9 fifths).

For example, [[septimal meantone]] and [[flattone]] are both extensions of [[5-limit]] (2.3.5) [[meantone]] to the [[7-limit]] (2.3.5.7), because C–E (4 fifths) represents [[5/4]] in both. They are different extensions, because in septimal meantone, 7/4 is C–A♯ (+10 fifths), while in flattone, 7/4 is C–Bbb (−9 fifths).

A '''strong extension''' is an extension in which the generators are not split compared to the original temperament. The structure of a strong extension is not changed, and no new intervals are introduced. A '''weak extension''' is one in which the generators are split. Thus an extension is strong if and only if its [[pergen]] is the same as the original temperament's pergen.

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== See also ==

* [[Subgroup ~~Temperament Families~~, ~~Relationships~~, and ~~Genes~~]] – formal definitions

* [[Subgroup temperament families, relationships, and genes]] – formal definitions

[[Category:Terms]]

[[Category:Regular temperament theory]]

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-An '''extension''' of a [[regular temperament]] on a [[JI subgroup]] ''G'' to a larger subgroup ''G' '' is a new temperament that contains the same intervals as the original temperament, and whose [[rank]] remains the same, with the same JI interpretations in the subgroup ''G'', but gives them new JI interpretations not in the original subgroup (but are in the larger subgroup). The opposite of extension is '''restriction'''.
+An '''extension''' of a [[regular temperament]] on a [[JI subgroup]] ''G'' to a larger subgroup ''G{{'}}'' is a new temperament that contains the same intervals as the original temperament, and whose [[rank]] remains the same, with the same JI interpretations in the subgroup ''G'', but gives them new JI interpretations not in the original subgroup (but are in the larger subgroup). The opposite of extension is '''restriction'''.
-For example, [[septimal meantone]] and [[flattone]] are both extensions of [[5-limit]] (2.3.5) [[meantone]] to the [[7-limit]] (2.3.5.7), because C-E (4 fifths) represents [[5/4]] in both. They are different extensions, because in septimal meantone, 7/4 is C-A# (10 fifths), while in flattone, 7/4 is C-Bbb (-9 fifths).
+For example, [[septimal meantone]] and [[flattone]] are both extensions of [[5-limit]] (2.3.5) [[meantone]] to the [[7-limit]] (2.3.5.7), because C–E (4 fifths) represents [[5/4]] in both. They are different extensions, because in septimal meantone, 7/4 is C–A♯ (+10 fifths), while in flattone, 7/4 is C–Bbb (−9 fifths).
 A '''strong extension''' is an extension in which the generators are not split compared to the original temperament. The structure of a strong extension is not changed, and no new intervals are introduced. A '''weak extension''' is one in which the generators are split. Thus an extension is strong if and only if its [[pergen]] is the same as the original temperament's pergen.
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 == See also ==
-* [[Subgroup Temperament Families, Relationships, and Genes]] – formal definitions
+* [[Subgroup temperament families, relationships, and genes]] – formal definitions
 [[Category:Terms]]
 [[Category:Regular temperament theory]]