Plücker coordinates - Revision history

Sintel: /* Projective distance */ Typo R^n

2026-05-06T17:27:55Z

Projective distance: Typo R^n

← Older revision		Revision as of 17:27, 6 May 2026
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	In Euclidean space, one usually takes advantage of the dot product to measure angles.		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>		:<math>
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	In projective space, there is an analogous formula, using the wedge product instead.		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>		:<math>

VectorGraphics at 19:33, 28 June 2025

2025-06-28T19:33:37Z

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-{{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}}
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 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 ==

VectorGraphics at 09:57, 28 June 2025

2025-06-28T09:57:04Z

← Older revision		Revision as of 09:57, 28 June 2025
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	And then the final entry, for 3.5, is <code>8</code>. Again, 5 (i.e. 5/3) is found at one generator, but 3/1 is split into 8 parts by blackwood temperament. So, 1 * 8 = 8.		And then the final entry, for 3.5, is <code>8</code>. Again, 5 (i.e. 5/3) is found at one generator, but 3/1 is split into 8 parts by blackwood temperament. So, 1 * 8 = 8.

			For wedgies of temperaments of larger prime subgroups, the number of entries is increased, so a rank-2 temperament of 7-limit JI would have 6 entries, for 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7. Note that the new septimal entries are inserted between the entries for the 5-limit! If it helps, think of arranging all the entries in a grid, where rows represent the first prime, and columns represent the second, and reading them off one by one.

			For wedgies of higher-rank temperaments, the number of primes per entry is increased, so that for a rank-3 temperament of the 7-limit, all possible combinations of 3 primes (2.3.5, 2.3.7, 2.5.7, and 3.5.7) would be covered.

	== Definition ==		== Definition ==

VectorGraphics at 07:22, 28 June 2025

2025-06-28T07:22:43Z

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 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 ==

Sintel: /* See also */ if anything the original page is more mathematically advanced lol

2025-06-13T10:12:01Z

See also: if anything the original page is more mathematically advanced lol

← Older revision		Revision as of 10:12, 13 June 2025
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	== See also ==		== See also ==

	* [[Wedgie supplement]] - Supplementary page going over additional information on wedgies~~, including a more mathematically advanced description~~		* [[Wedgie supplement]] - Supplementary page going over additional information on wedgies
	* [[Exterior algebra]] - exterior product, which produces wedgies		* [[Exterior algebra]] - exterior product, which produces wedgies
	* [[Interior product]] - interior product, dual of the exterior product		* [[Interior product]] - interior product, dual of the exterior product

VectorGraphics at 21:03, 12 June 2025

2025-06-12T21:03:54Z

← Older revision		Revision as of 21:03, 12 June 2025
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	{{Wikipedia\|Plücker embedding}}		{{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 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.
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	Since for any decent temperament this angle will be extremely small, we can take <math>\sin (\theta) \approx \theta</math>.		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, including a more mathematically advanced description
			* [[Exterior algebra]] - exterior product, which produces wedgies
			* [[Interior product]] - interior product, dual of the exterior product
			* [[Hodge dual]] - acts on wedgies

	[[Category:Exterior algebra]]		[[Category:Exterior algebra]]

Sintel: spacing nit

2025-06-12T00:43:26Z

spacing nit

← Older revision		Revision as of 00:43, 12 June 2025
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	\begin{align}		\begin{align}
	\iota: \mathrm{Gr} (k, n)		\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)		\operatorname {span} (m_1, \ldots, m_k)
	& \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, .		& \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, .
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	</math>		</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>.		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>.

Sintel: note about duality

2025-06-11T18:52:41Z

note about duality

← Older revision		Revision as of 18:52, 11 June 2025
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	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 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 ==		== Definition ==

Sintel: Undo revision 199179 by VectorGraphics (talk) Almost everything was wrong

2025-06-11T18:41:47Z

Undo revision 199179 by VectorGraphics (talk) Almost everything was wrong

← Older revision		Revision as of 18:41, 11 June 2025
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	== Definition ==		== 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>.		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>.

VectorGraphics: beginning to hack away at this page

2025-06-10T21:43:40Z

beginning to hack away at this page

← Older revision		Revision as of 21:43, 10 June 2025
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	== Definition ==		== Definition ==
	A temperament can be viewed as a point in ~~what~~ is called a Grassmannian variety, written as <math>\mathrm{Gr} (k, n)</math>.		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.
	~~This~~ variety 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]]~~.		Because temperaments can be treated as linear mappings, this variety thus contains all possible k-dimensional subspaces of <math>\mathbb{R}^n</math>.

	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>.		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>.