Equivalence continuum: Difference between revisions
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== Mathematical theory == | == 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:Math]][[Category:Theory]] | ||
[[Category:Equivalence continua|*]] | [[Category:Equivalence continua|*]] | ||
Revision as of 04:38, 16 March 2021
An equivalence continuum is the space of all rank-k temperaments on a specified subgroup that is tempered out by a specified temperament of a lower rank on the same subgroup (such as an edo viewed on a temperament on said subgroup).
Examples:
- The syntonic-chromatic equivalence continuum is the 5-limit rank-2 equivalence continuum of 7edo.
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.