Equivalence continuum: Difference between revisions

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== Mathematical theory ==

Mathematically, the rank-''k'' equivalence continuum of a rank-''r'' temperament ''T'' on a rank-''n'' subgroup ''S'' can be described as the set of rational points on the Grassmannian G = Gr(''n-k'', ker(T)), or the space of ''n-k''-dimensional subspaces of the [[kernel]] of ''T'', the space of commas tempered out by ''T''. This has a particularly simple description when ''T'' is an edo, ''n'' is 3 and ''k'' is 2, as then G = Gr(1, 2) = RP^1 (real projective space of dimension 1), which can be viewed as a circle. Then the continuum corresponds to the set of lines with rational slope passing through the origin on the Cartesian plane where the lattice of ker(''T'') lives. A rational point, i.e. a temperament on the continuum, ~~its hen parametrized by~~ ''p''/''q'', where ''u''^''p''/''v''^''q'' is tempered out by the temperament ~~and ''u'' and ''v'' are two commas in ''S'' tempered out by the edo~~.

Mathematically, the rank-''k'' equivalence continuum of a rank-''r'' temperament ''T'' on a rank-''n'' subgroup ''S'' can be described as the set of rational points on the Grassmannian G = Gr(''n-k'', ker(T)), or the space of ''n-k''-dimensional subspaces of the [[kernel]] of ''T'', the space of commas tempered out by ''T''. This has a particularly simple description when ''T'' is an edo, ''n'' is 3 and ''k'' is 2, as then G = Gr(1, 2) = RP^1 (real projective space of dimension 1), which can be viewed as a circle. Then the continuum corresponds to the set of lines with rational slope passing through the origin on the Cartesian plane where the lattice of ker(''T'') lives. The lattice of ker(''T'') is generated by a [[basis]] of some pair of two commas ''u'' and ''v'' in ''S'' tempered out by the edo. A rational point, i.e. a temperament on the continuum, then corresponds to a rational ratio ''p''/''q'', where ''u''^''p''/''v''^''q'' is tempered out by the temperament.

[[Category:Math]][[Category:Theory]]

[[Category:Equivalence continua|*]]

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 == Mathematical theory ==
-Mathematically, the rank-''k'' equivalence continuum of a rank-''r'' temperament ''T'' on a rank-''n'' subgroup ''S'' can be described as the set of rational points on the Grassmannian G = Gr(''n-k'', ker(T)), or the space of ''n-k''-dimensional subspaces of the [[kernel]] of ''T'', the space of commas tempered out by ''T''. This has a particularly simple description when ''T'' is an edo, ''n'' is 3 and ''k'' is 2, as then G = Gr(1, 2) = RP^1 (real projective space of dimension 1), which can be viewed as a circle. Then the continuum corresponds to the set of lines with rational slope passing through the origin on the Cartesian plane where the lattice of ker(''T'') lives. A rational point, i.e. a temperament on the continuum, its hen parametrized by ''p''/''q'', where ''u''^''p''/''v''^''q'' is tempered out by the temperament and ''u'' and ''v'' are two commas in ''S'' tempered out by the edo.
+Mathematically, the rank-''k'' equivalence continuum of a rank-''r'' temperament ''T'' on a rank-''n'' subgroup ''S'' can be described as the set of rational points on the Grassmannian G = Gr(''n-k'', ker(T)), or the space of ''n-k''-dimensional subspaces of the [[kernel]] of ''T'', the space of commas tempered out by ''T''. This has a particularly simple description when ''T'' is an edo, ''n'' is 3 and ''k'' is 2, as then G = Gr(1, 2) = RP^1 (real projective space of dimension 1), which can be viewed as a circle. Then the continuum corresponds to the set of lines with rational slope passing through the origin on the Cartesian plane where the lattice of ker(''T'') lives. The lattice of ker(''T'') is generated by a [[basis]] of some pair of two commas ''u'' and ''v'' in ''S'' tempered out by the edo. A rational point, i.e. a temperament on the continuum, then corresponds to a rational ratio ''p''/''q'', where ''u''^''p''/''v''^''q'' is tempered out by the temperament.
 [[Category:Math]][[Category:Theory]]
 [[Category:Equivalence continua|*]]