Equivalence continuum: Difference between revisions

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== Geometric interpretation ==

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'''{{nowrap|''n'' ~~−~~ ''r''}}, identifying '''R'''{{nowrap|''n'' ~~−~~ ''r''}} with the '''R'''-vector space {{nowrap|ker(''T'') &otimes; '''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'''{{nowrap|''n'' − ''r''}}, identifying '''R'''{{nowrap|''n'' − ''r''}} with the '''R'''-vector space {{nowrap|ker(''T'') &otimes; '''R'''}}.

=== 1-dimensional continua ===

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* [[Septimal meantone]] tempers out {{nowrap|81/80 {{=}} '''u'''''x''}} {{nowrap|{{=}} (1, 0, 0)}} and {{nowrap|126/125 {{=}} '''u'''''y''}} {{nowrap|{{=}} (0, 1, 0)}}, thus corresponds to the plane {{nowrap|''z'' {{=}} 0}}. This corresponds to {{nowrap|'''v''' {{=}} (0, 0, 1)}}.

* [[Valentine]] tempers out {{nowrap|1029/1024 {{=}} '''u'''''z''}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|126/125 {{=}} '''u'''''y''}} {{nowrap|{{=}} (0, 1, 0)}}. This corresponds to {{nowrap|'''v''' {{=}} (1, 0, 0)}}.

* [[Mohajira]] tempers out {{nowrap|81/80 {{=}} '''u'''''x''}} {{nowrap|{{=}} (1, 0, 0)}} and {{nowrap|6144/6125 {{=}} '''u'''''y'' ~~−~~ '''u'''''z''}} {{nowrap|{{=}} (0, 1, ~~−1~~)}}. This corresponds to {{nowrap|'''v''' {{=}} (0, 1, 1)}}.

* [[Mohajira]] tempers out {{nowrap|81/80 {{=}} '''u'''''x''}} {{nowrap|{{=}} (1, 0, 0)}} and {{nowrap|6144/6125 {{=}} '''u'''''y'' − '''u'''''z''}} {{nowrap|{{=}} (0, 1, −1)}}. This corresponds to {{nowrap|'''v''' {{=}} (0, 1, 1)}}.

* [[Hemithirds]] tempers out {{nowrap|1029/1024 {{=}} '''u'''''z''}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|3136/3125 {{=}} 2'''u'''''x'' + '''u'''''y''}} {{nowrap|{{=}} (2, 1, 0)}}. This corresponds to {{nowrap|'''v''' {{=}} (1, ~~−2~~, 0)}}.

* [[Hemithirds]] tempers out {{nowrap|1029/1024 {{=}} '''u'''''z''}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|3136/3125 {{=}} 2'''u'''''x'' + '''u'''''y''}} {{nowrap|{{=}} (2, 1, 0)}}. This corresponds to {{nowrap|'''v''' {{=}} (1, −2, 0)}}.

* [[Miracle]] tempers out {{nowrap|1029/1024 {{=}} '''u'''''z''}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|225/224 {{=}} '''u'''''x'' ~~−~~ '''u'''''y''}} {{nowrap|{{=}} (1, ~~−1~~, 0)}}. This corresponds to {{nowrap|'''v''' {{=}} (1, 1, 0)}}.

* [[Miracle]] tempers out {{nowrap|1029/1024 {{=}} '''u'''''z''}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|225/224 {{=}} '''u'''''x'' − '''u'''''y''}} {{nowrap|{{=}} (1, −1, 0)}}. This corresponds to {{nowrap|'''v''' {{=}} (1, 1, 0)}}.

== Examples ==

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 == Geometric interpretation ==
-Mathematically, the rank-''k'' '''equivalence continuum''' C(''k'',&nbsp;''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 &minus; k''}})-dimensional sublattices of the [[kernel]] (set of all intervals tempered out) of ''T'', the rank-({{nowrap|''n &minus; r''}}) lattice of commas tempered out by ''T''. This is a set of rational points on the Grassmannian {{nowrap|'''G''' {{=}} '''Gr'''(''n &minus; k'', ''n &minus; r'')}} of ({{nowrap|''n &minus; k''}})-dimensional vector subspaces of '''R'''<sup>{{nowrap|''n'' &minus; ''r''}}</sup>, identifying '''R'''<sup>{{nowrap|''n'' &minus; ''r''}}</sup> with the '''R'''-vector space {{nowrap|ker(''T'') &otimes; '''R'''}}.
+Mathematically, the rank-''k'' '''equivalence continuum''' C(''k'',&nbsp;''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'') &otimes; '''R'''}}.
 === 1-dimensional continua ===
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 * [[Septimal meantone]] tempers out {{nowrap|81/80 {{=}} '''u'''<sub>''x''</sub>}} {{nowrap|{{=}} (1, 0, 0)}} and {{nowrap|126/125 {{=}} '''u'''<sub>''y''</sub>}} {{nowrap|{{=}} (0, 1, 0)}}, thus corresponds to the plane {{nowrap|''z'' {{=}} 0}}. This corresponds to {{nowrap|'''v''' {{=}} (0, 0, 1)}}.
 * [[Valentine]] tempers out {{nowrap|1029/1024 {{=}} '''u'''<sub>''z''</sub>}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|126/125 {{=}} '''u'''<sub>''y''</sub>}} {{nowrap|{{=}} (0, 1, 0)}}. This corresponds to {{nowrap|'''v''' {{=}} (1, 0, 0)}}.
-* [[Mohajira]] tempers out {{nowrap|81/80 {{=}} '''u'''<sub>''x''</sub>}} {{nowrap|{{=}} (1, 0, 0)}} and {{nowrap|6144/6125 {{=}} '''u'''<sub>''y''</sub> &minus; '''u'''<sub>''z''</sub>}} {{nowrap|{{=}} (0, 1, &minus;1)}}. This corresponds to {{nowrap|'''v''' {{=}} (0, 1, 1)}}.
+* [[Mohajira]] tempers out {{nowrap|81/80 {{=}} '''u'''<sub>''x''</sub>}} {{nowrap|{{=}} (1, 0, 0)}} and {{nowrap|6144/6125 {{=}} '''u'''<sub>''y''</sub> − '''u'''<sub>''z''</sub>}} {{nowrap|{{=}} (0, 1, −1)}}. This corresponds to {{nowrap|'''v''' {{=}} (0, 1, 1)}}.
-* [[Hemithirds]] tempers out {{nowrap|1029/1024 {{=}} '''u'''<sub>''z''</sub>}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|3136/3125 {{=}} 2'''u'''<sub>''x''</sub> + '''u'''<sub>''y''</sub>}} {{nowrap|{{=}} (2, 1, 0)}}. This corresponds to {{nowrap|'''v''' {{=}} (1, &minus;2, 0)}}.
+* [[Hemithirds]] tempers out {{nowrap|1029/1024 {{=}} '''u'''<sub>''z''</sub>}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|3136/3125 {{=}} 2'''u'''<sub>''x''</sub> + '''u'''<sub>''y''</sub>}} {{nowrap|{{=}} (2, 1, 0)}}. This corresponds to {{nowrap|'''v''' {{=}} (1, −2, 0)}}.
-* [[Miracle]] tempers out {{nowrap|1029/1024 {{=}} '''u'''<sub>''z''</sub>}} {{nowrap|{{=}} (0, 0, 1)}} and {{nowrap|225/224 {{=}} '''u'''<sub>''x''</sub> &minus; '''u'''<sub>''y''</sub>}} {{nowrap|{{=}} (1, &minus;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)}}.
 == Examples ==