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 ''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<sup>1</sup> (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 '''R'''<sup>2</sup> 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; view the plane as having two perpendicular axes corresponding to ''u'' and ''v'' directions. A rational point, i.e. a temperament on the continuum, then corresponds to a rational ratio ''p''/''q'', where ''u''<sup>''p''</sup>/''v''<sup>''q''</sup> is tempered out by the temperament. | 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<sup>1</sup> (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 '''R'''<sup>2</sup> 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; view the plane as having two perpendicular axes corresponding to ''u'' and ''v'' directions. A rational point, i.e. a temperament on the continuum, then corresponds to a rational ratio ''p''/''q'', where ''u''<sup>''p''</sup>/''v''<sup>''q''</sup> is tempered out by the temperament. | |||
[[Category:Math]][[Category:Theory]] | [[Category:Math]][[Category:Theory]] | ||
[[Category:Equivalence continua|*]] | [[Category:Equivalence continua|*]] | ||
Revision as of 04:43, 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) = RP1 (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 R2 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; view the plane as having two perpendicular axes corresponding to u and v directions. A rational point, i.e. a temperament on the continuum, then corresponds to a rational ratio p/q, where up/vq is tempered out by the temperament.