Expansion and retraction - Revision history

FloraC: + see also; - stub

2025-12-14T08:08:08Z

+ see also; - stub

← Older revision		Revision as of 08:08, 14 December 2025
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	== See also ==		== See also ==
	* [[Subgroup temperament families, relationships, and genes]] – formal definitions		* [[Subgroup temperament families, relationships, and genes]] – formal definitions
			* [[Extension and restriction]]

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


	~~{{stub}}~~

Squib at 04:56, 14 December 2025

2025-12-14T04:56:05Z

← Older revision		Revision as of 04:56, 14 December 2025
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	An '''expansion''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup. Because new generators are added, expanding a temperament always increases its rank. For example, [[didacus]] is the rank-2 temperament that tempers [[3136/3125]] in the 2.5.7 subgroup. It can be expanded to the full 7-limit by adding a generator representing 3, resulting in the rank-3 temperament [[hemimean]]. The opposite of expansion is '''retraction'''.		An '''expansion''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup. Because new generators are added, expanding a temperament always increases its [[rank]]. For example, [[didacus]] is the rank-2 temperament that tempers [[3136/3125]] in the 2.5.7 subgroup. It can be expanded to the full 7-limit by adding a generator representing 3, resulting in the rank-3 temperament [[hemimean]]. The opposite of expansion is '''retraction'''.

	For temperaments with many reasonable extensions to reach certain primes, it may be useful to consider the expansion of that temperament that includes the desired intervals and tempers any new commans shared between the extensions. Then, the extensions of the original temperament can be thought of as ways to temper the expanded temperament back to the original rank (especially when some new commas are shared between the extensions, which is common). For example, the rank-2 2.3.7-subgroup temperament [[slendric]] has several extensions to the full 7-limit, as described on that page, but all of those extensions can be further extended to the 11-limit by tempering 385/384[[~]]441/440. Therefore it is logical to first expand slendric to the 11-limit by adding one more generator and tempering 385/384~441/440, resulting in the rank-3 temperament [[portent]]. The 11-limit slendric extensions then represent methods of tempering portent back into a rank-2 temperament.		For temperaments with many reasonable extensions to reach certain primes, it may be useful to consider the expansion of that temperament that includes the desired intervals and tempers any new commans shared between the extensions. Then, the extensions of the original temperament can be thought of as ways to temper the expanded temperament back to the original rank (especially when some new commas are shared between the extensions, which is common). For example, the rank-2 2.3.7-subgroup temperament [[slendric]] has several extensions to the full 7-limit, as described on that page, but all of those extensions can be further extended to the 11-limit by tempering 385/384[[~]]441/440. Therefore it is logical to first expand slendric to the 11-limit by adding one more generator and tempering 385/384~441/440, resulting in the rank-3 temperament [[portent]]. The 11-limit slendric extensions then represent methods of tempering portent back into a rank-2 temperament.

Squib: cleaned up wording

2025-12-14T04:52:53Z

cleaned up wording

← Older revision		Revision as of 04:52, 14 December 2025
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	An '''expansion''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup. For example, [[didacus]] is the rank-2 temperament that tempers [[3136/3125]] in the 2.3.7 subgroup. It can be expanded to the full 7-limit by adding a generator representing 5, resulting in the rank-3 temperament [[hemimean]]. The opposite is a '''retraction'''.		An '''expansion''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup. Because new generators are added, expanding a temperament always increases its rank. For example, [[didacus]] is the rank-2 temperament that tempers [[3136/3125]] in the 2.5.7 subgroup. It can be expanded to the full 7-limit by adding a generator representing 3, resulting in the rank-3 temperament [[hemimean]]. The opposite of expansion is '''retraction'''.

	For temperaments with many ~~possible~~ extensions to reach certain primes, it may be useful to consider the expansion of that temperament ~~to include~~ the desired intervals, ~~and think~~ of the ~~possible extensions~~ as ~~methods of tempering it~~ back to the original rank (especially when some ~~of the~~ new commas are shared between ~~all~~ extensions, as is common). For example, the rank-2 2.3.7-subgroup temperament [[slendric]] has several extensions to the full 7-limit, as described on that page, but all of those extensions can be further extended to the 11-limit by tempering 385/384[[~]]441/440. Therefore it is logical to first expand slendric to the 11-limit by adding one more generator and tempering 385/384~441/440, resulting in the rank-3 temperament [[portent]]. The 11-limit slendric extensions then represent methods of tempering portent back into a rank-2 temperament.		For temperaments with many reasonable extensions to reach certain primes, it may be useful to consider the expansion of that temperament that includes the desired intervals and tempers any new commans shared between the extensions. Then, the extensions of the original temperament can be thought of as ways to temper the expanded temperament back to the original rank (especially when some new commas are shared between the extensions, which is common). For example, the rank-2 2.3.7-subgroup temperament [[slendric]] has several extensions to the full 7-limit, as described on that page, but all of those extensions can be further extended to the 11-limit by tempering 385/384[[~]]441/440. Therefore it is logical to first expand slendric to the 11-limit by adding one more generator and tempering 385/384~441/440, resulting in the rank-3 temperament [[portent]]. The 11-limit slendric extensions then represent methods of tempering portent back into a rank-2 temperament.

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

Fredg999: Fredg999 moved page Expansion to Expansion and retraction: Based on Extension and restriction

2025-12-14T01:06:16Z

Fredg999 moved page Expansion to Expansion and retraction: Based on Extension and restriction

← Older revision	Revision as of 01:06, 14 December 2025
(No difference)

Fredg999: Mention retraction, +see also, category, move stub template

2025-12-14T01:05:47Z

Mention retraction, +see also, category, move stub template

← Older revision		Revision as of 01:05, 14 December 2025
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	~~{{stub}}~~		An '''expansion''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup. For example, [[didacus]] is the rank-2 temperament that tempers [[3136/3125]] in the 2.3.7 subgroup. It can be expanded to the full 7-limit by adding a generator representing 5, resulting in the rank-3 temperament [[hemimean]]. The opposite is a '''retraction'''.

			For temperaments with many possible extensions to reach certain primes, it may be useful to consider the expansion of that temperament to include the desired intervals, and think of the possible extensions as methods of tempering it back to the original rank (especially when some of the new commas are shared between all extensions, as is common). For example, the rank-2 2.3.7-subgroup temperament [[slendric]] has several extensions to the full 7-limit, as described on that page, but all of those extensions can be further extended to the 11-limit by tempering 385/384[[~]]441/440. Therefore it is logical to first expand slendric to the 11-limit by adding one more generator and tempering 385/384~441/440, resulting in the rank-3 temperament [[portent]]. The 11-limit slendric extensions then represent methods of tempering portent back into a rank-2 temperament.

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

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

	An '''expansion''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup. For example, [[didacus]] is the rank-2 temperament that tempers [[3136/3125]] in the 2.3.7 subgroup. It can be expanded to the full 7-limit by adding a generator representing 5, resulting in the rank-3 temperament [[hemimean]].

	For temperaments with many possible extensions to reach certain primes, it may be useful to consider the expansion of that temperament to include the desired intervals, and think of the possible extensions as methods of tempering it back to the original rank (especially when some of the new commas are shared between all extensions, as is common). For example, the rank-2 2.3.7-subgroup temperament [[slendric]] has several extensions to the full 7-limit, as described on that page, but all of those extensions can be further extended to the 11-limit by tempering 385/384[[~]]441/440. Therefore it is logical to first expand slendric to the 11-limit by adding one more generator and tempering 385/384~441/440, resulting in the rank-3 temperament [[portent]]. The 11-limit slendric extensions then represent methods of tempering portent back into a rank-2 temperament.		{{stub}}

Squib at 01:01, 14 December 2025

2025-12-14T01:01:00Z

← Older revision		Revision as of 01:01, 14 December 2025
Line 1:		Line 1:
	{{stub}}		{{stub}}

	An '''expansion''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup.		An '''expansion''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup. For example, [[didacus]] is the rank-2 temperament that tempers [[3136/3125]] in the 2.3.7 subgroup. It can be expanded to the full 7-limit by adding a generator representing 5, resulting in the rank-3 temperament [[hemimean]].

			For temperaments with many possible extensions to reach certain primes, it may be useful to consider the expansion of that temperament to include the desired intervals, and think of the possible extensions as methods of tempering it back to the original rank (especially when some of the new commas are shared between all extensions, as is common). For example, the rank-2 2.3.7-subgroup temperament [[slendric]] has several extensions to the full 7-limit, as described on that page, but all of those extensions can be further extended to the 11-limit by tempering 385/384[[~]]441/440. Therefore it is logical to first expand slendric to the 11-limit by adding one more generator and tempering 385/384~441/440, resulting in the rank-3 temperament [[portent]]. The 11-limit slendric extensions then represent methods of tempering portent back into a rank-2 temperament.

Squib: Created page with "{{stub}} An '''expansion''' of a regular temperament from a JI subgroup to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup."

2025-12-14T00:26:07Z

Created page with "{{stub}} An '''expansion''' of a regular temperament from a JI subgroup to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup."

New page

{{stub}}

An '''expansion''' of a [[regular temperament]] from a [[JI subgroup]] to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup.