Equivalence continuum: Difference between revisions
Undo revision 185662 by Domin (talk). This is not novelty. It's a very insightful way to classify temperaments Tag: Undo |
m Move the messagebox to the specific section that deserves it |
||
| Line 1: | Line 1: | ||
An '''equivalence continuum''' comprises all the [[regular temperament|temperaments]] where a number of a certain interval is equated with another interval. Specifically, if the first interval, which we may call the stacked interval, is ''q''<sub>1</sub>, and the second interval, which we may call the targeted interval, is ''q''<sub>2</sub>, both in [[ratio]]s, an equivalence continuum is formed by all the temperaments that satisfy {{nowrap| {{subsup|''q''|1|''n''}} ~ ''q''<sub>2</sub> }}, where ''n'' is an arbitrary rational number. An equivalence continuum creates a space of temperaments on a specified [[JI subgroup]] that are [[support]]ed by a specified temperament of a lower rank (such as an [[equal temperament]]) on the same subgroup. | An '''equivalence continuum''' comprises all the [[regular temperament|temperaments]] where a number of a certain interval is equated with another interval. Specifically, if the first interval, which we may call the stacked interval, is ''q''<sub>1</sub>, and the second interval, which we may call the targeted interval, is ''q''<sub>2</sub>, both in [[ratio]]s, an equivalence continuum is formed by all the temperaments that satisfy {{nowrap| {{subsup|''q''|1|''n''}} ~ ''q''<sub>2</sub> }}, where ''n'' is an arbitrary rational number. An equivalence continuum creates a space of temperaments on a specified [[JI subgroup]] that are [[support]]ed by a specified temperament of a lower rank (such as an [[equal temperament]]) on the same subgroup. | ||
| Line 20: | Line 20: | ||
== Geometric interpretation == | == Geometric interpretation == | ||
{{Inacc}} | |||
Mathematically, the rank-''k'' '''equivalence continuum''' C(''k'', ''T'') associated with a rank-''r'' temperament ''T'' on a rank-''n'' subgroup ''S'' is the space of [[Mathematical theory of saturation|saturated]] ({{nowrap|''n − k''}})-dimensional sublattices of the [[kernel]] (set of all intervals tempered out) of ''T'', the rank-({{nowrap|''n − r''}}) lattice of commas tempered out by ''T''. This is a set of rational points on the Grassmannian {{nowrap|'''G''' {{=}} '''Gr'''(''n − k'', ''n − r'')}} of ({{nowrap|''n − k''}})-dimensional vector subspaces of '''R'''<sup>{{nowrap|''n'' − ''r''}}</sup>, identifying '''R'''<sup>{{nowrap|''n'' − ''r''}}</sup> with the '''R'''-vector space {{nowrap|ker(''T'') ⊗ '''R'''}}. | Mathematically, the rank-''k'' '''equivalence continuum''' C(''k'', ''T'') associated with a rank-''r'' temperament ''T'' on a rank-''n'' subgroup ''S'' is the space of [[Mathematical theory of saturation|saturated]] ({{nowrap|''n − k''}})-dimensional sublattices of the [[kernel]] (set of all intervals tempered out) of ''T'', the rank-({{nowrap|''n − r''}}) lattice of commas tempered out by ''T''. This is a set of rational points on the Grassmannian {{nowrap|'''G''' {{=}} '''Gr'''(''n − k'', ''n − r'')}} of ({{nowrap|''n − k''}})-dimensional vector subspaces of '''R'''<sup>{{nowrap|''n'' − ''r''}}</sup>, identifying '''R'''<sup>{{nowrap|''n'' − ''r''}}</sup> with the '''R'''-vector space {{nowrap|ker(''T'') ⊗ '''R'''}}. | ||