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 '''G''' becomes '''Gr'''(2, 3). Define a coordinate system (''x'', ''y'', ''z'') for ker(T) using some fixed comma basis '''u'''<sub>''x''</sub>, '''u'''<sub>''y''</sub>, '''u'''<sub>''z''</sub> for ker(T). Then our Grassmannian 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 (according to the dot product given by the given coordinates) to the plane of commas tempered out for each temperament. Say that the vector '''v''' (which depends on ''T'') 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, for any temperament. Assuming ''v''<sub>1</sub> ≠ 0, we can 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. Two-dimensional coordinates can similarly be assigned for the set of all temperaments such that ''v''<sub>2</sub> ≠ 0 and the set of all temperaments such that ''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 '''G''' becomes '''Gr'''(2, 3). Define a coordinate system (''x'', ''y'', ''z'') for ker(T) using some fixed comma basis '''u'''<sub>''x''</sub>, '''u'''<sub>''y''</sub>, '''u'''<sub>''z''</sub> for ker(T). Then our Grassmannian 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 (according to the dot product given by the given coordinates) to the plane of commas tempered out for each temperament. Say that the vector '''v''' (which depends on ''T'') 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, for any temperament. Assuming ''v''<sub>1</sub> ≠ 0, we can 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. Two-dimensional coordinates can similarly be assigned for the set of all temperaments such that ''v''<sub>2</sub> ≠ 0 and the set of all temperaments such that ''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.--> | ||
NB: Not all | NB: Not all equivalence continua are projective spaces, because not all Grassmannians are; for example, '''Gr'''(2, 4) is not a projective space. | ||
=== Example (7-limit rank-2 temperaments in 31edo) === | === Example (7-limit rank-2 temperaments in 31edo) === | ||
Let's look at where some well-known 7-limit rank-2 temperaments supported by [[31edo]] live in the 2-dimensional equivalence continuum C(2, 31edo, 7-limit). Let '''u'''<sub>''x''</sub>, '''u'''<sub>''y''</sub>, '''u'''<sub>''z''</sub> = 81/80, 126/125, 1029/1024 be the basis for the kernel of 7-limit [[31edo]]. Then: | Let's look at where some well-known 7-limit rank-2 temperaments supported by [[31edo]] live in the 2-dimensional equivalence continuum C(2, 31edo, 7-limit). Let '''u'''<sub>''x''</sub>, '''u'''<sub>''y''</sub>, '''u'''<sub>''z''</sub> = 81/80, 126/125, 1029/1024 be the basis for the kernel of 7-limit [[31edo]]. Then: | ||