Equivalence continuum: Difference between revisions

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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]] (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'''''n−r'', identifying '''R'''''n−r'' with the '''R'''-vector space ker(''T'') ⊗ '''R'''.

=== 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'''P1 (the real projective line), 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'''2 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 axes corresponding to '''u''' and '''v''' directions. A rational point, i.e. a temperament on the continuum, then corresponds to the line ~~through the origin~~ of rational or infinite slope ''t'' = ''p''/''q'', where the temperament is defined by the identification ''p'''''u''' ~ ''q'''''v''' (written additively). When ''t'' = 0, this corresponds to the temperament tempering out '''v'''. When ''t'' = (unsigned) infinity, this corresponds to the temperament tempering out '''u'''.

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'''P1 (the real projective line), 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'''2 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 the line

''y = tx'', of rational or infinite slope ''t'' = ''p''/''q'', where the temperament is defined by the identification ''p'''''u''' ~ ''q'''''v''' (written additively). When ''t'' = 0, this corresponds to the temperament tempering out '''v'''. When ''t'' = (unsigned) infinity, this corresponds to the temperament tempering out '''u'''.

=== 2-dimensional continua ===

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 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 &minus; k'')-dimensional sublattices of the [[kernel]] (set of all intervals tempered out) of ''T'', the rank-(''n &minus; r'') lattice of commas tempered out by ''T''. This is a set of rational points on the Grassmannian '''G''' = '''Gr'''(''n &minus; k'', ''n &minus; r'') of (''n &minus; k'')-dimensional vector subspaces of '''R'''<sup>''n&minus;r''</sup>, identifying '''R'''<sup>''n&minus;r''</sup> with the '''R'''-vector space ker(''T'') ⊗ '''R'''.
 === 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 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 axes corresponding to '''u''' and '''v''' directions. A rational point, i.e. a temperament on the continuum, then corresponds to the line through the origin of rational or infinite slope ''t'' = ''p''/''q'', where the temperament is defined by the identification ''p'''''u''' ~ ''q'''''v''' (written additively). When ''t'' = 0, this corresponds to the temperament tempering out '''v'''. When ''t'' = (unsigned) infinity, this corresponds to the temperament tempering out '''u'''.
+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 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 the line
+''y = tx'', of rational or infinite slope ''t'' = ''p''/''q'', where the temperament is defined by the identification ''p'''''u''' ~ ''q'''''v''' (written additively). When ''t'' = 0, this corresponds to the temperament tempering out '''v'''. When ''t'' = (unsigned) infinity, this corresponds to the temperament tempering out '''u'''.
 === 2-dimensional continua ===