Equivalence continuum: Difference between revisions
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== Geometric interpretation == | == 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 [[saturation|saturated]] (''n − k'')-dimensional sublattices of the [[kernel]] (set of all intervals tempered out) of ''T'', the rank-(''n − r'') lattice of commas tempered out by ''T''. This is a set of rational points on the Grassmannian '''G''' = '''Gr'''(''n − k'', ''n − r'') of (''n − k'')-dimensional vector subspaces of '''R'''<sup>''n−r''</sup>, identifying '''R'''<sup>''n−r''</sup> with the '''R'''-vector space ker(''T'') ⊗ '''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 [[saturation|saturated]] (or [[defactored]])(''n − k'')-dimensional sublattices of the [[kernel]] (set of all intervals tempered out) of ''T'', the rank-(''n − r'') lattice of commas tempered out by ''T''. This is a set of rational points on the Grassmannian '''G''' = '''Gr'''(''n − k'', ''n − r'') of (''n − k'')-dimensional vector subspaces of '''R'''<sup>''n−r''</sup>, identifying '''R'''<sup>''n−r''</sup> with the '''R'''-vector space ker(''T'') ⊗ '''R'''. | ||
=== 1-dimensional continua === | === 1-dimensional continua === | ||
This has a particularly simple description when ''r'' = 1 (i.e. when ''T'' is an edo), ''n'' = 3 (for example, when ''S'' is the [[5-limit]], 2.3.7 or 2.5.7) and ''k'' = 2 (so that we're considering the equivalence continua of rank-2 temperaments associated with an edo), as then '''G''' = '''Gr'''(1, 2) = '''R'''P<sup>1</sup> (the real projective line), which can be viewed as a circle. Then the continuum corresponds to the set of lines with rational or infinite 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 choice of two commas '''u''' and '''v''' in ''S'' tempered out by the edo; view the plane as having two perpendicular ''x'' and ''y'' axes corresponding to '''u''' and '''v''' directions. A rational point, i.e. a temperament on the continuum, then corresponds to a line | This has a particularly simple description when ''r'' = 1 (i.e. when ''T'' is an edo), ''n'' = 3 (for example, when ''S'' is the [[5-limit]], 2.3.7 or 2.5.7) and ''k'' = 2 (so that we're considering the equivalence continua of rank-2 temperaments associated with an edo), as then '''G''' = '''Gr'''(1, 2) = '''R'''P<sup>1</sup> (the real projective line), which can be viewed as a circle. Then the continuum corresponds to the set of lines with rational or infinite 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 choice of two commas '''u''' and '''v''' in ''S'' tempered out by the edo; view the plane as having two perpendicular ''x'' and ''y'' axes corresponding to '''u''' and '''v''' directions. A rational point, i.e. a temperament on the continuum, then corresponds to a line | ||