Eigenmonzo basis - Revision history

Cmloegcmluin: /* See also */

2025-02-22T16:13:14Z

FloraC: A little rework

2025-02-22T07:10:14Z

A little rework

← Older revision		Revision as of 07:10, 22 February 2025
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	~~Given~~ a [[regular temperament]] tuning T, an [[eigenmonzo~~]] (unchanged-interval) is a rational interval~~ q ~~such that~~ T(q) = q~~; that is, T tunes q justly~~. The eigenmonzos of T define a [[just intonation subgroup]], the eigenmonzo subgroup.		An [[eigenmonzo\|eigenmonzo or unchanged-interval]] is a rational interval tuned justly by a [[regular temperament]] tuning. In other words, if a tuning is ''T'', then an eigenmonzo ''q'' satisfies {{nowrap\| ''T''(''q'') {{=}} ''q'' }}. The eigenmonzos of ''T'' define a [[just intonation subgroup]], the eigenmonzo subgroup, whose basis is an '''eigenmonzo basis''' or '''unchanged-interval basis'''.

	One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2<sup>''n''</sup>} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target tuning #Minimax tuning\|minimax tunings]] of regular temperaments, where for a rank-''r'' regular temperament, the eigenmonzo subgroup is a rank-''r'' JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[projection matrix]] of the minimax tuning and hence define the tuning.		One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2<sup>''n''</sup>} of powers of 2 is the eigenmonzo subgroup.

			The idea is most useful in connection to the [[Target tuning #Minimax tuning\|minimax tunings]] of regular temperaments, where for a rank-''r'' regular temperament, the eigenmonzo subgroup is a rank-''r'' JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[projection matrix]] of the minimax tuning and hence define the tuning.

	== See also ==		== See also ==
			* [[Projection #The unchanged-interval basis]]
	* [[unchanged-interval basis]]

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

Cmloegcmluin: Undo revision 181519 by Lériendil (talk) - I think the link was only unintuitive because the anchor within the destination page had been broken by careless edits to that page

2025-02-20T00:34:57Z

Undo revision 181519 by Lériendil (talk) - I think the link was only unintuitive because the anchor within the destination page had been broken by careless edits to that page

← Older revision		Revision as of 00:34, 20 February 2025
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	== See also ==		== See also ==

	* [[~~Projection~~]]		* [[unchanged-interval basis]]

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

Lériendil: fixed unintuitive link

2025-02-19T22:35:30Z

fixed unintuitive link

← Older revision		Revision as of 22:35, 19 February 2025
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	== See also ==		== See also ==

	* [[~~unchanged-interval basis~~]]		* [[Projection]]

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

Cmloegcmluin: add link to closely-related concept

2023-05-05T22:29:56Z

add link to closely-related concept

← Older revision		Revision as of 22:29, 5 May 2023
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	Given a [[regular temperament]] tuning T, an [[eigenmonzo]] is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[just intonation subgroup]], the eigenmonzo subgroup.		Given a [[regular temperament]] tuning T, an [[eigenmonzo]] (unchanged-interval) is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[just intonation subgroup]], the eigenmonzo subgroup.

	One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2<sup>''n''</sup>} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target tuning #Minimax tuning\|minimax tunings]] of regular temperaments, where for a rank-''r'' regular temperament, the eigenmonzo subgroup is a rank-''r'' JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[projection matrix]] of the minimax tuning and hence define the tuning.		One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2<sup>''n''</sup>} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target tuning #Minimax tuning\|minimax tunings]] of regular temperaments, where for a rank-''r'' regular temperament, the eigenmonzo subgroup is a rank-''r'' JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[projection matrix]] of the minimax tuning and hence define the tuning.

			== See also ==

			* [[unchanged-interval basis]]

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

FloraC: Fix a formatting mistake

2022-08-03T15:29:50Z

Fix a formatting mistake

← Older revision		Revision as of 15:29, 3 August 2022
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	Given a [[regular temperament]] tuning T, an [[eigenmonzo]] is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[just intonation subgroup]], the eigenmonzo subgroup.		Given a [[regular temperament]] tuning T, an [[eigenmonzo]] is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[just intonation subgroup]], the eigenmonzo subgroup.

	One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2<sup>''n</sup>} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target tuning #Minimax tuning\|minimax tunings]] of regular temperaments, where for a rank-''r'' regular temperament, the eigenmonzo subgroup is a rank-''r'' JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[projection matrix]] of the minimax tuning and hence define the tuning.		One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2<sup>''n''</sup>} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target tuning #Minimax tuning\|minimax tunings]] of regular temperaments, where for a rank-''r'' regular temperament, the eigenmonzo subgroup is a rank-''r'' JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[projection matrix]] of the minimax tuning and hence define the tuning.

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

FloraC: Cleanup; links; categories

2022-08-03T15:29:29Z

Cleanup; links; categories

← Older revision		Revision as of 15:29, 3 August 2022
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	Given a [[~~Abstract_regular_temperament\|~~regular temperament]] tuning T, an [[eigenmonzo]] is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[~~Just_intonation_subgroups\|~~just intonation subgroup]], the eigenmonzo subgroup.		Given a [[regular temperament]] tuning T, an [[eigenmonzo]] is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[just intonation subgroup]], the eigenmonzo subgroup.

	One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2^n} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[~~Target_tunings~~\|minimax tunings]] of regular temperaments, where for a rank r regular temperament, the eigenmonzo subgroup is a rank r JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[~~Target_tunings\|~~projection matrix]] of the minimax tuning and hence define the tuning.		One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2<sup>''n</sup>} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target tuning #Minimax tuning\|minimax tunings]] of regular temperaments, where for a rank-''r'' regular temperament, the eigenmonzo subgroup is a rank-''r'' JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[projection matrix]] of the minimax tuning and hence define the tuning.

	[[Category:~~eigenmonzo~~]]		[[Category:Regular temperament theory]]
			[[Category:Terms]]
			[[Category:Math]]
			[[Category:Monzo]]

FloraC: FloraC moved page Eigenmonzo subgroup to Eigenmonzo basis over redirect: Align with "comma basis"

2022-08-03T15:27:03Z

FloraC moved page Eigenmonzo subgroup to Eigenmonzo basis over redirect: Align with "comma basis"

← Older revision	Revision as of 15:27, 3 August 2022
(No difference)

Cmloegcmluin: it's called projection matrix on the destination page

2021-11-05T20:05:19Z

it's called projection matrix on the destination page

← Older revision		Revision as of 20:05, 5 November 2021
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	Given a [[Abstract_regular_temperament\|regular temperament]] tuning T, an [[eigenmonzo]] is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[Just_intonation_subgroups\|just intonation subgroup]], the eigenmonzo subgroup.		Given a [[Abstract_regular_temperament\|regular temperament]] tuning T, an [[eigenmonzo]] is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[Just_intonation_subgroups\|just intonation subgroup]], the eigenmonzo subgroup.

	One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2^n} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target_tunings\|minimax tunings]] of regular temperaments, where for a rank r regular temperament, the eigenmonzo subgroup is a rank r JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[Target_tunings\|projection ~~map~~]] of the minimax tuning and hence define the tuning.		One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2^n} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target_tunings\|minimax tunings]] of regular temperaments, where for a rank r regular temperament, the eigenmonzo subgroup is a rank r JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[Target_tunings\|projection matrix]] of the minimax tuning and hence define the tuning.

	[[Category:eigenmonzo]]		[[Category:eigenmonzo]]

Cmloegcmluin: fix typo

2021-07-19T20:36:33Z

fix typo

← Older revision		Revision as of 20:36, 19 July 2021
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	Given a [[Abstract_regular_temperament\|regular temperament]] tuning T, an [[eigenmonzo]] is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[Just_intonation_subgroups\|just intonation ~~subgoup~~]], the eigenmonzo subgroup.		Given a [[Abstract_regular_temperament\|regular temperament]] tuning T, an [[eigenmonzo]] is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[Just_intonation_subgroups\|just intonation subgroup]], the eigenmonzo subgroup.

	One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2^n} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target_tunings\|minimax tunings]] of regular temperaments, where for a rank r regular temperament, the eigenmonzo subgroup is a rank r JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[Target_tunings\|projection map]] of the minimax tuning and hence define the tuning.		One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2^n} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target_tunings\|minimax tunings]] of regular temperaments, where for a rank r regular temperament, the eigenmonzo subgroup is a rank r JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[Target_tunings\|projection map]] of the minimax tuning and hence define the tuning.

	[[Category:eigenmonzo]]		[[Category:eigenmonzo]]

← Older revision		Revision as of 16:13, 22 February 2025
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	== See also ==		== See also ==
	* [[Projection #The unchanged-interval basis]]		* [[Projection #The unchanged-interval basis]], for a discussion of this concept in the context of other related temperament tuning objects

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