Plücker coordinates: Difference between revisions
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{{Wikipedia|Plücker embedding}} | {{Wikipedia|Plücker embedding}} | ||
'''Plücker coordinates''' (also known as the '''[[wedgie]]'''), are a way to assign coordinates to temperaments, by viewing them as elements of some projective space. | In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the '''[[wedgie]]'''), are a way to assign coordinates to temperaments, by viewing them as elements of some projective space. | ||
== Definition == | == Definition == | ||
We have a Grassmannian variety <math>\mathrm{Gr} (k, n)</math> consisting of the k-dimensional subspaces of <math>\mathbb{R}^n</math>. | We have a Grassmannian variety <math>\mathrm{Gr} (k, n)</math> consisting of the k-dimensional subspaces of <math>\mathbb{R}^n</math>. | ||
The rational points on this variety can be identified with rank-k temperaments on a JI space with n primes. | The rational points on this variety can be identified with rank-k temperaments on a JI space with n primes. | ||
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== Plücker relations == | == Plücker relations == | ||
The coordinates must satisfy some algebraic relations called Plücker relations. Generally, the projective space is much 'larger' than the Grassmannian, and the image in the projective space is some quadric surface. | The coordinates must satisfy some algebraic relations called Plücker relations. Generally, the projective space is much 'larger' than the Grassmannian, and the image in the projective space is some quadric surface. | ||
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== Height == | == Height == | ||
A height function is a way to measure the 'arithmetic complexity' of a rational point. For example, the rational numbers <math>\frac{3}{2}</math> and <math>\frac{3001}{2001}</math> are close to eachother, but intuitively the second is much more complicated. | A height function is a way to measure the 'arithmetic complexity' of a rational point. For example, the rational numbers <math>\frac{3}{2}</math> and <math>\frac{3001}{2001}</math> are close to eachother, but intuitively the second is much more complicated. | ||
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== Projective distance == | == Projective distance == | ||
Given a temperament, we want to have some notion of distance, so that we can measure how well the temperament approximates JI. Since we are talking about linear subspaces (which all intersect at the origin), the only thing that is sensible to measure is the angle between them. | Given a temperament, we want to have some notion of distance, so that we can measure how well the temperament approximates JI. Since we are talking about linear subspaces (which all intersect at the origin), the only thing that is sensible to measure is the angle between them. | ||