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 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..

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.

[[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 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..
+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.
 [[Category:Math]][[Category:Theory]]
 [[Category:Equivalence continua|*]]