Equivalence continuum: Difference between revisions

Line 7:

Mathematically, the rank-''k'' equivalence continuum associated with 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'', ''n-r'') = Gr(''n-k'', ker(''T'')). This is the space of ''n-k''-dimensional sublattices of the [[kernel]] of ''T'', the rank-(''n-r'') lattice of commas tempered out by ''T''.

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), as then '''G''' = Gr(1, 2) = '''R'''P1 (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'''2 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''''p''/''v''''q'' is tempered out by the temperament.

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 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''''p''/''v''''q'' is tempered out by the temperament.

A higher-dimensional example: ''r'' = 1, ''n'' = 4 (e.g. when ''S'' is the [[7-limit]]), and ''k'' = 2, our Grassmannian becomes Gr(2, 3), which can be identified with '''R'''P2, the real projective plane. [More to come...]

[[Category:Math]][[Category:Theory]]

[[Category:Equivalence continua|*]]

@@ Line 7: / Line 7: @@
 Mathematically, the rank-''k'' equivalence continuum associated with 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'', ''n-r'') = Gr(''n-k'', ker(''T'')). This is the space of ''n-k''-dimensional sublattices of the [[kernel]] of ''T'', the rank-(''n-r'') lattice of commas tempered out by ''T''.
-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), as then '''G''' = Gr(1, 2) = '''R'''P<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.
+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 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.
+A higher-dimensional example: ''r'' = 1, ''n'' = 4 (e.g. when ''S'' is the [[7-limit]]), and ''k'' = 2, our Grassmannian becomes Gr(2, 3), which can be identified with '''R'''P<sup>2</sup>, the real projective plane. [More to come...]
 [[Category:Math]][[Category:Theory]]
 [[Category:Equivalence continua|*]]