Equivalence continuum: Difference between revisions

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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 a rational ratio ''p''/''q'', where ''u''''p''/''v''''q'' is tempered out by the temperament.

A higher-dimensional example: ~~When~~ ''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, the space of lines through the origin in 3-dimensional space) by taking the unique line '''R'''''v'' perpendicular to the plane of commas tempered out for each temperament. Say that ~~this~~ vector ''v'' has components (''v''1, ''v''2, ''v''3), so that the plane has equation ''v''1''x'' + ''v''2''y'' + ''v''3''z'' = 0. [~~More~~ to ~~come~~...]

A higher-dimensional example: Say that ''r'' = 1, ''n'' = 4 (e.g. when ''S'' is the [[7-limit]]), and ''k'' = 2, for example the set of rank-2 [[7-limit]] temperaments supported by [[31edo]]. Then our Grassmannian becomes Gr(2, 3), which can be identified with '''R'''P2 (the real projective plane, the space of lines through the origin in 3-dimensional space) by taking the unique line '''R''''''v''' perpendicular to the plane of commas tempered out for each temperament. Say that the vector '''v''' defining the unique line has components (''v''1, ''v''2, ''v''3), so that the plane has equation ''v''1''x'' + ''v''2''y'' + ''v''3''z'' = 0. [We may further assume that ''v''1, ''v''2, ''v''3 are integers, since the condition of being perpendicular to two integer vectors is defined by a system of lienar equations with integer coefficients, thus has a unique rational solution up to scaling.] If we scale '''v''' by ''v''1, then '''v'''/''v''1 = (1, ''v''2/''v''1, v3/''v''1) = (1, ''s'', ''t'') describes two rational parameters ''s'' and ''t'' which defines any temperament on 31edo's 7-limit rank-2 continuum uniquely.

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

[[Category:Equivalence continua|*]]

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 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 a rational ratio ''p''/''q'', where ''u''<sup>''p''</sup>/''v''<sup>''q''</sup> is tempered out by the temperament.
-A higher-dimensional example: When ''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, the space of lines through the origin in 3-dimensional space) by taking the unique line '''R'''''v'' perpendicular to the plane of commas tempered out for each temperament. Say that this vector ''v'' has components (''v''<sub>1</sub>, ''v''<sub>2</sub>, ''v''<sub>3</sub>), so that the plane has equation  ''v''<sub>1</sub>''x'' + ''v''<sub>2</sub>''y'' + ''v''<sub>3</sub>''z'' = 0. [More to come...]
+A higher-dimensional example: Say that ''r'' = 1, ''n'' = 4 (e.g. when ''S'' is the [[7-limit]]), and ''k'' = 2, for example the set of rank-2 [[7-limit]] temperaments supported by [[31edo]]. Then our Grassmannian becomes Gr(2, 3), which can be identified with '''R'''P<sup>2</sup> (the real projective plane, the space of lines through the origin in 3-dimensional space) by taking the unique line '''R''''''v''' perpendicular to the plane of commas tempered out for each temperament. Say that the vector '''v''' defining the unique line has components (''v''<sub>1</sub>, ''v''<sub>2</sub>, ''v''<sub>3</sub>), so that the plane has equation  ''v''<sub>1</sub>''x'' + ''v''<sub>2</sub>''y'' + ''v''<sub>3</sub>''z'' = 0. [We may further assume that ''v''<sub>1</sub>, ''v''<sub>2</sub>, ''v''<sub>3</sub> are integers, since the condition of being perpendicular to two integer vectors is defined by a system of lienar equations with integer coefficients, thus has a unique rational solution up to scaling.] If we scale '''v''' by ''v''<sub>1</sub>, then '''v'''/''v''<sub>1</sub> = (1, ''v''<sub>2</sub>/''v''<sub>1</sub>, v<sub>3</sub>/''v''<sub>1</sub>) = (1, ''s'', ''t'') describes two rational parameters ''s'' and ''t'' which defines any temperament on 31edo's 7-limit rank-2 continuum uniquely.
 [[Category:Math]][[Category:Theory]]
 [[Category:Equivalence continua|*]]