Equivalence continuum: Difference between revisions
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* The [[syntonic-chromatic equivalence continuum]] is the 5-limit rank-2 equivalence continuum of [[7edo]]. | * The [[syntonic-chromatic equivalence continuum]] is the 5-limit rank-2 equivalence continuum of [[7edo]]. | ||
== | == Geometric interpretation == | ||
Mathematically, the rank-''k'' '''equivalence continuum''' C(''k'', ''T'', ''S'') 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]] 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''), identifying '''R'''<sup>''n−r''</sup> with ker(''T'') ⊗ '''R'''. | Mathematically, the rank-''k'' '''equivalence continuum''' C(''k'', ''T'', ''S'') 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]] 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''), identifying '''R'''<sup>''n−r''</sup> with ker(''T'') ⊗ '''R'''. | ||