Extension and restriction: Difference between revisions

An '''extension''' of a [[regular temperament]] from a [[JI subgroup]] to a larger one is a new temperament with the same [[rank]], containing the same intervals as the original temperament, but adding new JI interpretations from the larger subgroup. The opposite of extension is '''restriction'''.  

For example, [[septimal meantone]] is an extension of [[5-limit]] (2.3.5) [[meantone]] to the [[7-limit]] (2.3.5.7). In both cases, C–E (4 perfect fifths) represents [[5/4]], but septimal meantone adds the interpretation of 7/4 as C–A♯ (+10 perfect fifths). Another extension of 5-limit meantone is [[flattone]], which adds the interpretation of 7/4 as C–B𝄫 (−9 perfect fifths). Septimal meantone and flattone are different extensions, because the new interpretation of 7/4 is different, and thus their tunings tend to differ.  

We distinguish between '''strong extensions''' and '''weak extensions'''; their inverse operations are '''strong restriction''' and '''weak restriction''' respectively. A strong extension is one in which the [[generator]]s are not split compared to the original temperament, so that the structure of a strong extension is not changed, and no new intervals are introduced; strong extensions can thus be thought of as ''extending'' the harmony of their parent temperament to incorporate new elements, most commonly at more complex positions than the original ones. A weak extension is one in which the generators are split, implying that its structure is novel. It can be thought of as using the original temperament as "scaffolding" for new intervals and new structure.  

If a strong extension is more complex than the parent temperament, it competes with other strong extensions to the same set of new harmonies, and there is generally a particular subsection of the original tuning range in which any one specific extension is best. If a strong extension is clearly better than any other extension to the primes given (in terms of both accuracy and complexity, for which [[badness]] is a heuristic) and tunes well in the parent temperament's best tunings, it can be considered the ''canonical'' extension and retain the same name as the original temperament; some extensions are so "canonical" that it makes little sense to speak of any other way to extend to their expanded subgroup, and often little sense to speak of the original temperament in the restricted subgroup (an example of that being [[11-limit]] [[miracle]]).

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.

 

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.