Extension and restriction
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 generators 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.
Properties
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.
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 includes, while weak extensions have a mapping that either subdivides the equave into more periods 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.