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: 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 '''Rv''' 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 associated with the rank-2 temperament 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 linear equations with integer coefficients, thus has a unique rational solution up to scaling.] One coordinate ''v''i is always guaranteed to be nonzero. If ''v''1 ≠ 0 and we scale '''v''' by 1/''v''1, then the resulting vector '''v'''/''v''1 = (1, ''v''2/''v''1, v3/''v''1) = (1, ''s'', ''t'') points in the same direction as '''v''' and describes two rational (or infinite) parameters ''s'' and ''t'' which defines any temperament with ''v''1 ≠ 0 on 31edo's 7-limit rank-2 continuum uniquely. Note that this discussion assumes specific coordinates (''x'', ''y'', ''z'') using some comma basis '''u'''x, '''u'''y, '''u'''z for 7-limit 31edo. Similarly for every temperament with ''v''2 ≠ 0 and every temperament with ''v''3 ≠ 0. Note that this continuum is actually part of a mathematical manifold with a more complicated topology and needs to be described using more than one local chart (coordinate system) constructed like this; a single circle won't define every point on this 2-dimensional continuum, just like a single circle won't define every point on a 2-dimensional sphere.

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 '''Rv''' 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 associated with the rank-2 temperament 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 linear equations with integer coefficients, thus has a unique rational solution up to scaling.] One coordinate ''v''i is always guaranteed to be nonzero. If ''v''1 ≠ 0 and we scale '''v''' by 1/''v''1, then the resulting vector '''v'''/''v''1 = (1, ''v''2/''v''1, v3/''v''1) = (1, ''s'', ''t'') points in the same direction as '''v''' and describes two rational (or infinite) parameters ''s'' and ''t'' which defines any temperament with ''v''1 ≠ 0 on 31edo's 7-limit rank-2 continuum uniquely. Note that this discussion assumes specific coordinates (''x'', ''y'', ''z'') using some comma basis '''u'''x, '''u'''y, '''u'''z for 7-limit 31edo. Similarly for every temperament with ''v''2 ≠ 0 and every temperament with ''v''3 ≠ 0. Note that this continuum is actually part of a mathematical manifold with a more complicated topology and needs to be described using more than one local chart (coordinate system) constructed like this; unlike for the ''k'' - ''r'' = 1 case, a single circle won't define every point on this 2-dimensional continuum, just like a single circle won't define every point on a 2-dimensional sphere.

=== Example (7-limit rank-2 temperaments in 31edo) ===

[[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: 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 '''Rv''' 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 associated with the rank-2 temperament 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 linear equations with integer coefficients, thus has a unique rational solution up to scaling.] One coordinate ''v''<sub>i</sub> is always guaranteed to be nonzero. If ''v''<sub>1</sub> ≠ 0 and we scale '''v''' by 1/''v''<sub>1</sub>, then the resulting vector '''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'') points in the same direction as '''v''' and describes two rational (or infinite) parameters ''s'' and ''t'' which defines any temperament with ''v''<sub>1</sub> ≠ 0 on 31edo's 7-limit rank-2 continuum uniquely. Note that this discussion assumes specific coordinates (''x'', ''y'', ''z'') using some comma basis '''u'''<sub>x</sub>, '''u'''<sub>y</sub>, '''u'''<sub>z</sub> for 7-limit 31edo. Similarly for every temperament with ''v''<sub>2</sub> ≠ 0 and every temperament with ''v''<sub>3</sub> ≠ 0. Note that this continuum is actually part of a mathematical manifold with a more complicated topology and needs to be described using more than one local chart (coordinate system) constructed like this; a single circle won't define every point on this 2-dimensional continuum, just like a single circle won't define every point on a 2-dimensional sphere.
+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 '''Rv''' 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 associated with the rank-2 temperament 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 linear equations with integer coefficients, thus has a unique rational solution up to scaling.] One coordinate ''v''<sub>i</sub> is always guaranteed to be nonzero. If ''v''<sub>1</sub> ≠ 0 and we scale '''v''' by 1/''v''<sub>1</sub>, then the resulting vector '''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'') points in the same direction as '''v''' and describes two rational (or infinite) parameters ''s'' and ''t'' which defines any temperament with ''v''<sub>1</sub> ≠ 0 on 31edo's 7-limit rank-2 continuum uniquely. Note that this discussion assumes specific coordinates (''x'', ''y'', ''z'') using some comma basis '''u'''<sub>x</sub>, '''u'''<sub>y</sub>, '''u'''<sub>z</sub> for 7-limit 31edo. Similarly for every temperament with ''v''<sub>2</sub> ≠ 0 and every temperament with ''v''<sub>3</sub> ≠ 0. Note that this continuum is actually part of a mathematical manifold with a more complicated topology and needs to be described using more than one local chart (coordinate system) constructed like this; unlike for the ''k'' - ''r'' = 1 case, a single circle won't define every point on this 2-dimensional continuum, just like a single circle won't define every point on a 2-dimensional sphere.
 === Example (7-limit rank-2 temperaments in 31edo) ===
 [[Category:Math]][[Category:Theory]]
 [[Category:Equivalence continua|*]]