Equivalence continuum: Difference between revisions
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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 with gcd 1, 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. | 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 with gcd 1, 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) === | === Example (7-limit rank-2 temperaments in 31edo) === | ||
Let ux, uy, uz = 81/80, 126/125, 1029/1024 be the | Let ux, uy, uz = 81/80, 126/125, 1029/1024 be the basis for the kernel of 7-limit 31edo. Then: | ||
* [[septimal meantone]] tempers out ux = (1, 0, 0) and uy = (0,1,0), thus corresponds to the plane z = 0. This corresponds to '''v''' = (0, 0, 1). | * [[septimal meantone]] tempers out ux = (1, 0, 0) and uy = (0,1,0), thus corresponds to the plane z = 0. This corresponds to '''v''' = (0, 0, 1). | ||
[[Category:Math]][[Category:Theory]] | [[Category:Math]][[Category:Theory]] | ||
[[Category:Equivalence continua|*]] | [[Category:Equivalence continua|*]] | ||