Equivalence continuum: Difference between revisions
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=== Inversion === | === Inversion === | ||
A continuum can be inverted by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}, with temperaments in it characterized by the relation (''q''<sub>2</sub>/''q''<sub>1</sub>)<sup>''m''</sup> ~ ''q''<sub>2</sub>. Here the stacked interval is ''q''<sub>2</sub>/''q''<sub>1</sub>, and the targeted interval remains ''q''<sub>2</sub>. For instance, the inversion of syntonic–chromatic equivalence continuum is the mavila–chromatic equivalence continuum, where temperaments satisfy (135/128)<sup>''m''</sup> ~ 2187/2048. | A continuum can be inverted by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}, with temperaments in it characterized by the relation (''q''<sub>2</sub>/''q''<sub>1</sub>)<sup>''m''</sup> ~ ''q''<sub>2</sub>. Here the stacked interval is ''q''<sub>2</sub>/''q''<sub>1</sub>, and the targeted interval remains ''q''<sub>2</sub>. For instance, the inversion of the syntonic–chromatic equivalence continuum is the mavila–chromatic equivalence continuum, where temperaments satisfy (135/128)<sup>''m''</sup> ~ 2187/2048. | ||
This ''m''-continuum, like the ''n''-continuum, also meets the requirements for a possible default choice, and raises the question which one should be the ''n''-continuum and which one should be the ''m''-continuum. In principle, we take the ''n''-continuum as the main continuum and the ''m''-continuum supplementary. If one of the candidate stacked intervals is simpler ''and'' smaller, we set it to ''q''<sub>1</sub> of the ''n''-continuum so that more useful temperaments are included in it. However, the simpler interval is sometimes the larger one, in which case the choice could be made on a heuristic basis. | This ''m''-continuum, like the ''n''-continuum, also meets the requirements for a possible default choice, and raises the question which one should be the ''n''-continuum and which one should be the ''m''-continuum. In principle, we take the ''n''-continuum as the main continuum and the ''m''-continuum supplementary. If one of the candidate stacked intervals is simpler ''and'' smaller, we set it to ''q''<sub>1</sub> of the ''n''-continuum so that more useful temperaments are included in it. However, the simpler interval is sometimes the larger one, in which case the choice could be made on a heuristic basis. | ||
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* [[Miracle]] tempers out {{nowrap|1029/1024 {{=}} '''u'''<sub>''z''</sub>}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|225/224 {{=}} '''u'''<sub>''x''</sub> − '''u'''<sub>''y''</sub>}} {{nowrap|{{=}} (1, −1, 0)}}. This corresponds to {{nowrap|'''v''' {{=}} (1, 1, 0)}}. | * [[Miracle]] tempers out {{nowrap|1029/1024 {{=}} '''u'''<sub>''z''</sub>}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|225/224 {{=}} '''u'''<sub>''x''</sub> − '''u'''<sub>''y''</sub>}} {{nowrap|{{=}} (1, −1, 0)}}. This corresponds to {{nowrap|'''v''' {{=}} (1, 1, 0)}}. | ||
== | == List of equivalence continua == | ||
{{See also| Category: Equivalence continua }} | |||
All equivalence continua currently on the wiki are rank-{{nowrap|(''n'' + 1)}} continua of rank-{{nowrap|(''n'' + 1)}} temperaments within a rank-{{nowrap|(''n'' + 2)}} subgroup that are supported by a rank-''n'' system. | All equivalence continua currently on the wiki are rank-{{nowrap|(''n'' + 1)}} continua of rank-{{nowrap|(''n'' + 1)}} temperaments within a rank-{{nowrap|(''n'' + 2)}} subgroup that are supported by a rank-''n'' system. | ||
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[[Category:Math]] | [[Category:Math]] | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||