Extension and restriction - Revision history

FloraC: Re-incorporate part of the changes, making it more concise and clear

2026-06-18T04:39:10Z

Re-incorporate part of the changes, making it more concise and clear

← Older revision		Revision as of 04:39, 18 June 2026
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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'''.		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]] ~~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]] 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 ~~primes~~, most commonly at more complex positions than the original ones.		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.

			== 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]]).		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 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 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.		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 [[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.		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.

FloraC: Undo

2026-06-18T04:23:48Z

Undo

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	~~== Definition ==~~		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'''.
	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)		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).

	* C–E (4 fifths) represents [[~~5/4~~]] ~~in both~~, the ~~core property~~ of ~~meantone.~~		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.
	* 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~~.		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 ~~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~~.		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 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~~.		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.

	~~== Properties ==~~		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.
	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 ~~splits a~~ [[~~Generator-offset property\|generator~~]] ~~or the~~ [[~~equave~~]]. Additionally, a strong extension's [[pergen]] is the same as the original temperament's pergen~~, while a weak extension might either split the fifth, or use a completely different generator, such as [[Ploidacot/Omega-tricot\|(P8, P4/3)]]~~.

	== See also ==		== See also ==

Eufalesio: Hotfixes!

2026-06-17T23:24:17Z

Hotfixes!

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	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]].)		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.		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 splits a [[Generator-offset property\|generator]] or the [[equave]]. Additionally, a strong extension's [[pergen]] is the same as the original temperament's pergen, while a weak extension might either split the fifth, or use a completely different generator, such as [[Ploidacot/Omega-tricot\|(P8, P4/3)]].

	== See also ==		== See also ==

Eufalesio: Reworded and formatted to make it look neater

2026-06-17T23:15:36Z

Reworded and formatted to make it look neater

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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 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 ==		== See also ==

Squib: i may be biased, but i think the miracle page is more clearly an example of this than the superkleismic page

2026-03-02T20:04:50Z

i may be biased, but i think the miracle page is more clearly an example of this than the superkleismic page

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	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.		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.

	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~~]]).		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 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 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.
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	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.		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 [[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.		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.

	== See also ==		== See also ==

FloraC: + see also

2025-12-14T08:08:25Z

+ see also

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	== See also ==		== See also ==
	* [[Subgroup temperament families, relationships, and genes]] – formal definitions		* [[Subgroup temperament families, relationships, and genes]] – formal definitions
			* [[Expansion and retraction]]
	* [[Temperament naming]]		* [[Temperament naming]]

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

FloraC: Cleanup

2025-10-10T08:26:02Z

Cleanup

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	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]] 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]]).		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.

	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.		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.		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.
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	== See also ==		== See also ==
	* [[Subgroup temperament families, relationships, and genes]] – formal definitions		* [[Subgroup temperament families, relationships, and genes]] – formal definitions
			* [[Temperament naming]]

	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Lériendil at 20:25, 15 May 2025

2025-05-15T20:25:46Z

← Older revision		Revision as of 20:25, 15 May 2025
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	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.		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 ~~covers~~, 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.		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.

	== See also ==		== See also ==

Lériendil at 06:04, 15 May 2025

2025-05-15T06:04:51Z

← Older revision		Revision as of 06:04, 15 May 2025
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	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 '''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 and generator. [[Godzilla]] is a weak extension of meantone but a strong extension of [[semaphore]].		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 covers, 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.		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 covers, 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.

Lériendil at 05:59, 15 May 2025

2025-05-15T05:59:05Z

← Older revision		Revision as of 05:59, 15 May 2025
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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 JI ~~interpretations in~~ the original subgroup, but ~~gives them~~ new JI interpretations not in the original subgroup (but are in the larger subgroup). The opposite of extension is '''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).		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).