Equivalence continuum: Difference between revisions

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This guarantees that in the corresponding temperament, ''n'' equals the order of the last formal prime in the comma, and equals the number of steps to obtain the interval class of the second formal prime in the generator chain.

=== 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''2/''q''1)''m'' ~ ''q''2. Here the stacked interval is ''q''2/''q''1, and the targeted interval remains ''q''2. For instance, the inversion of syntonic–chromatic equivalence continuum is the mavila–chromatic equivalence continuum, where temperaments satisfy (135/128)''m'' ~ 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''1 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.

== Geometric interpretation ==

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 This guarantees that in the corresponding temperament, ''n'' equals the order of the last formal prime in the comma, and equals the number of steps to obtain the interval class of the second formal prime in the generator chain.
+=== 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.
+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.
 == Geometric interpretation ==