Extension and restriction: Difference between revisions

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An '''extension''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup 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 original subgroup, 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]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the same intervals as the original temperament, and whose [[rank]] remains the same, with the same representations of JI intervals of the original subgroup, but adding new JI interpretations not in the original subgroup (but which 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).

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A '''weak extension''' is one in which the generators are split, implying that their structure is novel but uses the original temperament as "scaffolding". A weak extension of a notable temperament often is also a strong extension of another notable temperament in a different subgroup, and therefore shares more affinity with that; however, this is not always the case, as either its strong restriction is ridiculous (by the aforementioned criterion of it making little sense to speak of such a restriction), or (in rare cases, such as with [[cohemimabila]]) it has no strong restriction in any subgroup with prime basis elements.

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]].

For example, both septimal meantone and flattone are strong extensions of 5-limit meantone since they all share the same period ([[2/1]]) and generator ([[4/3]]). [[Godzilla]] is a weak extension of meantone, since it splits [[4/3]] in two and uses half 4/3 as the generator, but a strong extension of [[semaphore]] since in the 2.3.7 subgroup it is identical to semaphore, while adding a mapping of 5 from meantone.

In any case, a strong extension can be identified by having a [[mapping]] identical to that of the original temperament on the (formal) primes the original temperament ~~covers~~, while weak extensions have a mapping that either subdivides the [[equave]] into more [[period]]s or the elements of whose second row that cover the original set of primes are a common multiple of those of the original temperament. Additionally, a strong extension's [[pergen]] is the same as the original temperament's pergen.

In any case, a strong extension can be identified by having a [[mapping]] identical to that of the original temperament on the (formal) primes the original temperament includes, while weak extensions have a mapping that either subdivides the [[equave]] into more [[period]]s or the elements of whose second row that cover the original set of primes are a common multiple of those of the original temperament. Additionally, a strong extension's [[pergen]] is the same as the original temperament's pergen.

== See also ==

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-An '''extension''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup 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 original subgroup, 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]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the same intervals as the original temperament, and whose [[rank]] remains the same, with the same representations of JI intervals of the original subgroup, but adding new JI interpretations not in the original subgroup (but which 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).
@@ Line 7: / Line 7: @@
 A '''weak extension''' is one in which the generators are split, implying that their structure is novel but uses the original temperament as "scaffolding". A weak extension of a notable temperament often is also a strong extension of another notable temperament in a different subgroup, and therefore shares more affinity with that; however, this is not always the case, as either its strong restriction is ridiculous (by the aforementioned criterion of it making little sense to speak of such a restriction), or (in rare cases, such as with [[cohemimabila]]) it has no strong restriction in any subgroup with prime basis elements.
-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]].
+For example, both septimal meantone and flattone are strong extensions of 5-limit meantone since they all share the same period ([[2/1]]) and generator ([[4/3]]). [[Godzilla]] is a weak extension of meantone, since it splits [[4/3]] in two and uses half 4/3 as the generator, but a strong extension of [[semaphore]] since in the 2.3.7 subgroup it is identical to semaphore, while adding a mapping of 5 from meantone.
-In any case, a strong extension can be identified by having a [[mapping]] identical to that of the original temperament on the (formal) primes the original temperament covers, while weak extensions have a mapping that either subdivides the [[equave]] into more [[period]]s or the elements of whose second row that cover the original set of primes are a common multiple of those of the original temperament. Additionally, a strong extension's [[pergen]] is the same as the original temperament's pergen.
+In any case, a strong extension can be identified by having a [[mapping]] identical to that of the original temperament on the (formal) primes the original temperament includes, while weak extensions have a mapping that either subdivides the [[equave]] into more [[period]]s or the elements of whose second row that cover the original set of primes are a common multiple of those of the original temperament. Additionally, a strong extension's [[pergen]] is the same as the original temperament's pergen.
 == See also ==