Extension and restriction: Difference between revisions

Revision as of 23:15, 17 June 2026

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.

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-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'''.
+== 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]] 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).
+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)
-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 primes, most commonly at more complex positions than the original ones.
+* 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).
-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]]).
+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 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.
+* A strong extension is one in which the [[generator]]s 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.
-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.
+* A&nbsp;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.
-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.
+== 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 [[Ennealimma#Ennealimmal|Ennealimmal]], which while technically a 5-limit temperament, it is practically always referred in its [[Ennealimmal|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-offset property|generator]]. Additionally, a strong extension's [[pergen]] is the same as the original temperament's pergen.
 == See also ==

Extension and restriction: Difference between revisions

Revision as of 23:15, 17 June 2026

Definition

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