Extension and 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).

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 primes, most commonly at more complex positions than the original ones. 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 superkleismic).

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 (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.

See also

• Subgroup temperament families, relationships, and genes – formal definitions