Extension and restriction
Definition
An extension of a regular temperament, from a JI subgroup to a larger one, is a new temperament with the same rank and original intervals but adding new JI interpretations for the new primes. The opposite of extension is restriction.
For example, septimal meantone and flattone: they are both extensions of 5-limit (2.3.5) meantone to the 7-limit (2.3.5.7)
- C–E (4 fifths) represents 5/4 in both, the core property of meantone.
- In septimal meantone, 7/4 is C–A♯ (+10 fifths)
- In flattone, 7/4 is C–Bbb (−9 fifths).
They are different strong extensions, because the new interpretation of 7/4 is different, and thus the tunings differ. We distinguish strong extensions and weak extensions, whose inverses are strong restriction and weak restriction respectively.
- A strong extension is one in which the generators of the original temperament are not split; the structure is the same and no new intervals arise. Strong extensions can 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.
- A weak extension on the other hand splits generators of the original temperament; the structure is now different and new intervals appear in the gamut since more generators being required. Weak extensions can be thought as using the original temperament as a "scaffolding" for new intervals and new structure. An example is the rastmic extension of slendric; originally slendric has a ~8/7 generator, where three make ~3/2. The weak extension with 243/242 tempered out splits the ~8/7 into two ~77/72 generators.
Properties
A strong extension may not necessarily be better than a weak one, and they can be combined. The weak extension shown has a strong extension with 385/384 tempered out, reanalyzing 77/72 as 16/15~15/14 - miracle.
A weak extension of a notable temperament often is also a strong extension of another notable temperament in a different subgroup with which it may share more "affinity". However, this is not always the case, as either its strong restriction is ridiculous (taking the former restriction of ennealimmal as an example), or (rarely such as with cohemimabila) it has no strong restriction in any subgroup with prime basis elements.
If a strong extension is more complex than the parent temperament, it competes with other strong extensions in which tuning differences give better results depending on the extension. If a strong extension is clearly better than any other extension to the primes given (accuracy and complexity wise with badness as a heuristic) and doesn't alter the tuning, it can be deemed 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 Ennealimmal, which while technically a 5-limit temperament, it is practically always referred in its 7-limit form.)
In any case, a strong extension has the same mapping as the original temperament with the original primes, while weak extensions have a mapping that either subdivides either the equave or the/a generator. Additionally, a strong extension's pergen is the same as the original temperament's pergen.