Extension and restriction: Difference between revisions

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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.

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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.