Plücker coordinates: Difference between revisions

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[[File:Plucker_embedding.png|thumb|600px|right|Schematic illustration of the Plücker embedding. Linear subspaces of <math>\mathbb{R}^n</math> (here lines) get mapped to points on a quadric surface in projective space.]]

In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the [[wedgie]]) are a way to assign coordinates to abstract 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 abstract temperaments, by viewing them as elements of some projective space.

The usual way to write down an abstract temperament is via its mapping matrix, but Plücker coordinates give us a unique description that is useful for some calculations.

The definition here is given in terms of temperament matrices, but by duality, we can also embed interval spaces in the same way.

More specifically, the interval subspace spanned by the commas of some temperament can also be used to give unique coordinates to that temperament.

These two representations are related via the [[Hodge dual]].

== Definition ==

A temperament ~~on a JI subgroup with some number of primes~~ can be viewed as a point in ~~a space representing all possible temperaments sharing the temperament's rank and tempering a JI subgroup with the same number of primes. This~~ is called a Grassmannian variety, written as <math>\mathrm{Gr} (k, n)</math>~~, where k is the rank of the temperament and n is the rank of the JI subgroup~~.

A temperament can be viewed as a point in what is called a Grassmannian variety, written as <math>\mathrm{Gr} (k, n)</math>.

This variety contains all possible k-dimensional subspaces of <math>\mathbb{R}^n</math>.

~~Because temperaments can be treated as linear mappings, this~~ variety ~~thus~~ contains all possible k-dimensional subspaces of <math>\mathbb{R}^n</math>.

In musical terms, k represents the rank of the temperament (how many independent generators it has), and n is the number of primes we're considering in our [[just intonation subgroup]].

Let <math>M</math> be an element of <math>\mathrm{Gr} (k, n)</math>, spanned by basis vectors <math>m_1, \ldots, m_k</math>.

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The Plücker map takes a temperament and embeds it into a projective space by taking the wedge product of the basis vectors:

:<math>

\begin{~~align~~}

\begin{aligned}

\iota: \mathrm{Gr} (k, n)

& \to \mathbf{P}\left(\Lambda^{k} \mathbb{R}^n \right) \\

& \to \mathbf{P}\left(\Lambda^{k} \, \mathbb{R}^n \right) \\

\operatorname {span} (m_1, \ldots, m_k)

& \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, .

\end{~~align~~}

\end{aligned}

</math>

Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.

Here, <math>\Lambda^{k} \, \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.

While the original space of temperaments has dimension <math>k(n-k)</math>, the space of Plücker coordinates is typically larger, with dimension <math>\binom{n}{k} - 1</math>.

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An element <math>M</math> spanned by two lines <math>x, y</math>, can be represented as the matrix

:<math>

\begin{~~equation~~}

\begin{bmatrix}

~~\begin{bmatrix}~~

x_{1} & x_{2} & x_{3} & x_{4} \\

y_{1} & y_{2} & y_{3} & y_{4}

\end{bmatrix} \, .

~~\end{equation}~~

</math>

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:<math>

~~\begin{equation}~~

\iota (M) = \left[ x \wedge y \right] = \left[ p_{12} : p_{13} : p_{14} : p_{23} : p_{24} : p_{34} \right] \, .

~~\end{equation}~~

</math>

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The height can be easily computed using the {{w|Gram matrix}}:

:<math>

\begin{~~align~~}

\begin{aligned}

\mathrm{G}_{ij} &= \left\langle m_i, m_j \right\rangle \\

\sqrt{\det(\mathrm{G})} &= \left\| m_1 \wedge \ldots \wedge m_n \right\| = \left\| X \right\| \, .

\end{~~align~~}

\end{aligned}

</math>

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In Euclidean space, one usually takes advantage of the dot product to measure angles.

Given vectors <math>a, b \in \mathbb{R^n}</math>, we famously have

Given vectors <math>a, b \in \mathbb{R}^n</math>, we famously have

:<math>

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In projective space, there is an analogous formula, using the wedge product instead.

Given some real point <math>j \in \mathbb{R^n}</math> with homogeneous coordinates <math>y</math>, and a linear subspace <math>P \in \mathrm{Gr} (k, n)</math> with Plücker coordinates <math>X</math>, we define the projective distance as

Given some real point <math>j \in \mathbb{R}^n</math> with homogeneous coordinates <math>y</math>, and a linear subspace <math>P \in \mathrm{Gr} (k, n)</math> with Plücker coordinates <math>X</math>, we define the projective distance as

:<math>

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Since for any decent temperament this angle will be extremely small, we can take <math>\sin (\theta) \approx \theta</math>.

== See also ==

* [[Wedgie supplement]] - Supplementary page going over additional information on wedgies

* [[Exterior algebra]] - exterior product, which produces wedgies

* [[Interior product]] - interior product, dual of the exterior product

* [[Hodge dual]] - acts on wedgies

[[Category:Exterior algebra]]

@@ Line 1: / Line 1: @@
-{{Expert}}
+{{Expert|Wedgie}}
 [[File:Plucker_embedding.png|thumb|600px|right|Schematic illustration of the Plücker embedding. Linear subspaces of <math>\mathbb{R}^n</math> (here lines) get mapped to points on a quadric surface in projective space.]]
 {{Wikipedia|Plücker embedding}}
-In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the [[wedgie]]) are a way to assign coordinates to abstract 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 abstract temperaments, by viewing them as elements of some projective space.
 The usual way to write down an abstract temperament is via its mapping matrix, but Plücker coordinates give us a unique description that is useful for some calculations.
+The definition here is given in terms of temperament matrices, but by duality, we can also embed interval spaces in the same way.
+More specifically, the interval subspace spanned by the commas of some temperament can also be used to give unique coordinates to that temperament.
+These two representations are related via the [[Hodge dual]].
 == Definition ==
-A temperament on a JI subgroup with some number of primes can be viewed as a point in a space representing all possible temperaments sharing the temperament's rank and tempering a JI subgroup with the same number of primes. This is called a Grassmannian variety, written as <math>\mathrm{Gr} (k, n)</math>, where k is the rank of the temperament and n is the rank of the JI subgroup.
+A temperament can be viewed as a point in what is called a Grassmannian variety, written as <math>\mathrm{Gr} (k, n)</math>.
+This variety contains all possible k-dimensional subspaces of <math>\mathbb{R}^n</math>.
-Because temperaments can be treated as linear mappings, this variety thus contains all possible k-dimensional subspaces of <math>\mathbb{R}^n</math>.
+In musical terms, k represents the rank of the temperament (how many independent generators it has), and n is the number of primes we're considering in our [[just intonation subgroup]].
 Let <math>M</math> be an element of <math>\mathrm{Gr} (k, n)</math>, spanned by basis vectors <math>m_1, \ldots, m_k</math>.
@@ Line 17: / Line 20: @@
 The Plücker map takes a temperament and embeds it into a projective space by taking the wedge product of the basis vectors:
 :<math>
-\begin{align}
+\begin{aligned}
 \iota: \mathrm{Gr} (k, n)
-	& \to \mathbf{P}\left(\Lambda^{k} \mathbb{R}^n \right) \\
+	& \to \mathbf{P}\left(\Lambda^{k} \, \mathbb{R}^n \right) \\
 \operatorname {span} (m_1, \ldots, m_k)
 	& \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, .
-\end{align}
+\end{aligned}
 </math>
-Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.
+Here, <math>\Lambda^{k} \, \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.
 While the original space of temperaments has dimension <math>k(n-k)</math>, the space of Plücker coordinates is typically larger, with dimension <math>\binom{n}{k} - 1</math>.
@@ Line 35: / Line 38: @@
 An element <math>M</math> spanned by two lines <math>x, y</math>, can be represented as the matrix
 :<math>
-\begin{equation}
+\begin{bmatrix}
-	\begin{bmatrix}
+	x_{1} & x_{2} & x_{3} & x_{4} \\
-		x_{1} & x_{2} & x_{3} & x_{4} \\
+	y_{1} & y_{2} & y_{3} & y_{4}
-		y_{1} & y_{2} & y_{3} & y_{4}
+\end{bmatrix} \, .
-	\end{bmatrix} \, .
-\end{equation}
 </math>
@@ Line 57: / Line 58: @@
 :<math>
-\begin{equation}
 	\iota (M) = \left[ x \wedge y \right] = \left[ p_{12} : p_{13} : p_{14} : p_{23} : p_{24} : p_{34} \right] \, .
-\end{equation}
 </math>
@@ Line 104: / Line 103: @@
 The height can be easily computed using the {{w|Gram matrix}}:
 :<math>
-	\begin{align}
+	\begin{aligned}
 	\mathrm{G}_{ij} &= \left\langle m_i, m_j \right\rangle \\
 	\sqrt{\det(\mathrm{G})} &= \left\| m_1 \wedge \ldots \wedge m_n \right\| = \left\| X \right\| \, .
-	\end{align}
+	\end{aligned}
 </math>
@@ Line 116: / Line 115: @@
 In Euclidean space, one usually takes advantage of the dot product to measure angles.
-Given vectors <math>a, b \in \mathbb{R^n}</math>, we famously have
+Given vectors <math>a, b \in \mathbb{R}^n</math>, we famously have
 :<math>
@@ Line 123: / Line 122: @@
 In projective space, there is an analogous formula, using the wedge product instead.
-Given some real point <math>j \in \mathbb{R^n}</math> with homogeneous coordinates <math>y</math>, and a linear subspace <math>P \in \mathrm{Gr} (k, n)</math> with Plücker coordinates <math>X</math>, we define the projective distance as
+Given some real point <math>j \in \mathbb{R}^n</math> with homogeneous coordinates <math>y</math>, and a linear subspace <math>P \in \mathrm{Gr} (k, n)</math> with Plücker coordinates <math>X</math>, we define the projective distance as
 :<math>
@@ Line 133: / Line 132: @@
 Since for any decent temperament this angle will be extremely small, we can take <math>\sin (\theta) \approx \theta</math>.
+== See also ==
+* [[Wedgie supplement]] - Supplementary page going over additional information on wedgies
+* [[Exterior algebra]] - exterior product, which produces wedgies
+* [[Interior product]] - interior product, dual of the exterior product
+* [[Hodge dual]] - acts on wedgies
 [[Category:Exterior algebra]]

Plücker coordinates: Difference between revisions

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