Extension and restriction: Difference between revisions

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

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 that the [[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).

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.

For example, both septimal meantone and flattone are strong extensions of 5-limit meantone since they all share the same period and generator. [[Godzilla]] is a weak extension of meantone but a strong extension of [[semaphore]].

== See also ==

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[[Category:Terms]]

[[Category:Regular temperament theory]]

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-An '''extension''' of a [[temperament]] on a [[JI subgroup]] ''G'' to a larger subgroup ''G' '' is a new temperament that contains the same intervals as the original temperament (in particular the [[rank]] remains the same), with the same JI interpretations in the subgroup ''G'', but gives the same intervals new JI interpretations not in the original subgroup (but are in the larger subgroup).
+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 that the [[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).
+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.
+For example, both septimal meantone and flattone are strong extensions of 5-limit meantone since they all share the same period and generator. [[Godzilla]] is a weak extension of meantone but a strong extension of [[semaphore]].
 == See also ==
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 [[Category:Terms]]
+[[Category:Regular temperament theory]]