Exterior algebra - Revision history

VectorGraphics at 07:48, 21 August 2025

2025-08-21T07:48:24Z

← Older revision		Revision as of 07:48, 21 August 2025
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	=== Wedge product ===		=== Wedge product ===
			[[File:Wedgediagram .png\|thumb\|504x504px\|Wedge product diagram]]
	The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure called a ''multivector'' with entries corresponding to combinations of entries in the starting vectors. For two vectors of the same length, [a<sub>1</sub> a<sub>2</sub> a<sub>3</sub> ... a<sub>n</sub>] and [b<sub>1</sub> b<sub>2</sub> b<sub>3</sub> ... b<sub>n</sub>], we go through every pair of indices ''i, j'' up to ''n'' where ''j'' > ''i,'' and the entry c<sub>i,j</sub> of the wedge product is a<sub>i</sub>b<sub>j</sub> - b<sub>i</sub>a<sub>j</sub>. c<sub>j,i</sub> is equal to -(c<sub>i,j</sub>), and c<sub>i,i</sub> where the two indices are the same is 0.		The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure called a ''multivector'' with entries corresponding to combinations of entries in the starting vectors. For two vectors of the same length, [a<sub>1</sub> a<sub>2</sub> a<sub>3</sub> ... a<sub>n</sub>] and [b<sub>1</sub> b<sub>2</sub> b<sub>3</sub> ... b<sub>n</sub>], we go through every pair of indices ''i, j'' up to ''n'' where ''j'' > ''i,'' and the entry c<sub>i,j</sub> of the wedge product is a<sub>i</sub>b<sub>j</sub> - b<sub>i</sub>a<sub>j</sub>. c<sub>j,i</sub> is equal to -(c<sub>i,j</sub>), and c<sub>i,i</sub> where the two indices are the same is 0.

	The wedge product is equivalent to [[map-merging]] or [[Edo join\|edo joins]] in regular temperament theory (and can thus be written with the symbol &), and is used to combine [[vals]] into structures which represent higher-rank temperaments (called multivals or [[wedgies]]). For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(511-87) (516-127) (816-1211)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.		The wedge product is equivalent to [[map-merging]] or [[Edo join\|edo joins]] in regular temperament theory (and can thus be written with the symbol &), and is used to combine [[vals]] into structures which represent higher-rank temperaments (called multivals or [[wedgies]]). For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(511-87) (516-127) (816-1211)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.

	The wedge product can be generalized to combine ''n'' vals together, where instead of every pair of indices, we have every combination of ''n'' indices. This results in wedgies for rank-3 temperaments and beyond.		The wedge product can be generalized to combine ''n'' vals together, where instead of every pair of indices, we have every combination of ''n'' indices. This results in wedgies for rank-3 temperaments and beyond.

VectorGraphics at 07:18, 21 August 2025

2025-08-21T07:18:29Z

← Older revision		Revision as of 07:18, 21 August 2025
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	The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure called a ''multivector'' with entries corresponding to combinations of entries in the starting vectors. For two vectors of the same length, [a<sub>1</sub> a<sub>2</sub> a<sub>3</sub> ... a<sub>n</sub>] and [b<sub>1</sub> b<sub>2</sub> b<sub>3</sub> ... b<sub>n</sub>], we go through every pair of indices ''i, j'' up to ''n'' where ''j'' > ''i,'' and the entry c<sub>i,j</sub> of the wedge product is a<sub>i</sub>b<sub>j</sub> - b<sub>i</sub>a<sub>j</sub>. c<sub>j,i</sub> is equal to -(c<sub>i,j</sub>), and c<sub>i,i</sub> where the two indices are the same is 0.		The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure called a ''multivector'' with entries corresponding to combinations of entries in the starting vectors. For two vectors of the same length, [a<sub>1</sub> a<sub>2</sub> a<sub>3</sub> ... a<sub>n</sub>] and [b<sub>1</sub> b<sub>2</sub> b<sub>3</sub> ... b<sub>n</sub>], we go through every pair of indices ''i, j'' up to ''n'' where ''j'' > ''i,'' and the entry c<sub>i,j</sub> of the wedge product is a<sub>i</sub>b<sub>j</sub> - b<sub>i</sub>a<sub>j</sub>. c<sub>j,i</sub> is equal to -(c<sub>i,j</sub>), and c<sub>i,i</sub> where the two indices are the same is 0.

	The wedge product is ~~used~~ in regular temperament theory to combine [[vals]] into [[~~Wedgie\|multivals~~]]~~, hence why multivals are called "wedgies"~~. For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(511-87) (516-127) (816-1211)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.		The wedge product is equivalent to [[map-merging]] or [[Edo join\|edo joins]] in regular temperament theory (and can thus be written with the symbol &), and is used to combine [[vals]] into structures which represent higher-rank temperaments (called multivals or [[wedgies]]). For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(511-87) (516-127) (816-1211)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.

	The wedge product can be generalized to combine ''n'' vals together, where instead of every pair of indices, we have every combination of ''n'' indices. This results in wedgies for rank-3 temperaments and beyond.		The wedge product can be generalized to combine ''n'' vals together, where instead of every pair of indices, we have every combination of ''n'' indices. This results in wedgies for rank-3 temperaments and beyond.

VectorGraphics at 20:45, 29 June 2025

2025-06-29T20:45:37Z

← Older revision		Revision as of 20:45, 29 June 2025
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	== See also ==		== See also ==
	* [[~~Plücker coordinates~~]]		* [[Wedgie]]
	* [[Hodge dual]]		* [[Hodge dual]]
	* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]		* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]

VectorGraphics at 20:44, 29 June 2025

2025-06-29T20:44:48Z

← Older revision		Revision as of 20:44, 29 June 2025
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	In many cases, the same things can be accomplished using matrix algebra or exterior algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations).		In many cases, the same things can be accomplished using matrix algebra or exterior algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations).
	Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.		Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.

			=== Wedge product ===
			The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure called a ''multivector'' with entries corresponding to combinations of entries in the starting vectors. For two vectors of the same length, [a<sub>1</sub> a<sub>2</sub> a<sub>3</sub> ... a<sub>n</sub>] and [b<sub>1</sub> b<sub>2</sub> b<sub>3</sub> ... b<sub>n</sub>], we go through every pair of indices ''i, j'' up to ''n'' where ''j'' > ''i,'' and the entry c<sub>i,j</sub> of the wedge product is a<sub>i</sub>b<sub>j</sub> - b<sub>i</sub>a<sub>j</sub>. c<sub>j,i</sub> is equal to -(c<sub>i,j</sub>), and c<sub>i,i</sub> where the two indices are the same is 0.

			The wedge product is used in regular temperament theory to combine [[vals]] into [[Wedgie\|multivals]], hence why multivals are called "wedgies". For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(511-87) (516-127) (816-1211)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.

			The wedge product can be generalized to combine ''n'' vals together, where instead of every pair of indices, we have every combination of ''n'' indices. This results in wedgies for rank-3 temperaments and beyond.

			For a higher-dimensional multivector with elements c<sub>i, j, k...</sub>, the entries at all permutations of the same indices have the same absolute value, and swapping two indices flips the sign.

	== See also ==		== See also ==

Sintel: Undo revision 203823 by VectorGraphics (talk)

2025-06-29T20:11:54Z

Undo revision 203823 by VectorGraphics (talk)

← Older revision		Revision as of 20:11, 29 June 2025
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	~~#REDIRECT~~ [[~~Mathematical formalism#Wedgies~~ and ~~multivals~~]]		{{Wikipedia}}
			'''Exterior algebra''' is a type of {{w\|Algebra over a field\|algebra}} which has a product, called '''exterior product''' or '''wedge product''' and denoted with <math>\wedge</math>, such that <math>v \wedge v = 0</math> for every vector <math>v</math> in the vector space <math>V</math>.

			In [[regular temperament theory]], exterior algebra is typically applied to the vector space of [[val]]s (or maps). The exterior product of two or more vals is called a multival, and its canonical form is called a [[wedgie]] (or [[Plücker coordinates]]), which can be used to uniquely identify a regular temperament.

			In many cases, the same things can be accomplished using matrix algebra or exterior algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations).
			Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.

			== See also ==
			* [[Plücker coordinates]]
			* [[Hodge dual]]
			* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]
			* [[Interior product]]
			* [[Recoverability]]
			* [[User:Mike Battaglia/Exterior Norm Conjecture Table]]

	[[Category:Exterior algebra\| ]] <!-- main article -->		[[Category:Exterior algebra\| ]] <!-- main article -->

VectorGraphics: Redirected page to Mathematical formalism#Wedgies and multivals

2025-06-28T23:40:30Z

Redirected page to Mathematical formalism#Wedgies and multivals

← Older revision		Revision as of 23:40, 28 June 2025
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	~~{{Wikipedia}}~~		#REDIRECT [[Mathematical formalism#Wedgies and multivals]]
	'''Exterior algebra''' is a type of {{w\|Algebra over a field\|algebra}} which has a product, called '''exterior product''' or '''wedge product''' and denoted with <math>\wedge</math>, such that <math>v \wedge v = 0</math> for every vector <math>v</math> in the vector space <math>V</math>.

	~~In [[regular temperament theory]], exterior algebra is typically applied to the vector space of~~ [[~~val]]s (or maps). The exterior product of two or more vals is called a multival,~~ and ~~its canonical form is called a [[wedgie]] (or [[Plücker coordinates]]), which can be used to uniquely identify a regular temperament.~~

	In many cases, the same things can be accomplished using matrix algebra or exterior algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations).
	~~Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.~~

	~~== See also ==~~
	* [[Plücker coordinates]]
	* [[Hodge dual]]
	* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]
	* [[Interior product]]
	* [[Recoverability]]
	* [[User:Mike Battaglia/Exterior Norm Conjecture Table]]

	[[Category:Exterior algebra\| ]] <!-- main article -->		[[Category:Exterior algebra\| ]] <!-- main article -->

Sintel: Not a beginner page

2025-06-13T10:16:58Z

Not a beginner page

← Older revision		Revision as of 10:16, 13 June 2025
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	~~{{Beginner\|Dave Keenan & Douglas Blumeyer's guide to EA for RTT}}~~
	{{Wikipedia}}		{{Wikipedia}}
	'''Exterior algebra''' is a type of {{w\|Algebra over a field\|algebra}} which has a product, called '''exterior product''' or '''wedge product''' and denoted with <math>\wedge</math>, such that <math>v \wedge v = 0</math> for every vector <math>v</math> in the vector space <math>V</math>.		'''Exterior algebra''' is a type of {{w\|Algebra over a field\|algebra}} which has a product, called '''exterior product''' or '''wedge product''' and denoted with <math>\wedge</math>, such that <math>v \wedge v = 0</math> for every vector <math>v</math> in the vector space <math>V</math>.

Sintel: /* See also */

2025-06-11T20:59:49Z

Sintel: typo

2024-12-27T16:08:15Z

typo

← Older revision		Revision as of 16:08, 27 December 2024
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	In [[regular temperament theory]], exterior algebra is typically applied to the vector space of [[val]]s (or maps). The exterior product of two or more vals is called a multival, and its canonical form is called a [[wedgie]] (or [[Plücker coordinates]]), which can be used to uniquely identify a regular temperament.		In [[regular temperament theory]], exterior algebra is typically applied to the vector space of [[val]]s (or maps). The exterior product of two or more vals is called a multival, and its canonical form is called a [[wedgie]] (or [[Plücker coordinates]]), which can be used to uniquely identify a regular temperament.

	In many cases, the same things can be accomplished using matrix algebra or exterior algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations.		In many cases, the same things can be accomplished using matrix algebra or exterior algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations).
	Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.		Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.

Sintel: clarify reasons for (not) using EA

2024-12-27T16:02:01Z

clarify reasons for (not) using EA

← Older revision		Revision as of 16:02, 27 December 2024
Line 5:		Line 5:
	In [[regular temperament theory]], exterior algebra is typically applied to the vector space of [[val]]s (or maps). The exterior product of two or more vals is called a multival, and its canonical form is called a [[wedgie]] (or [[Plücker coordinates]]), which can be used to uniquely identify a regular temperament.		In [[regular temperament theory]], exterior algebra is typically applied to the vector space of [[val]]s (or maps). The exterior product of two or more vals is called a multival, and its canonical form is called a [[wedgie]] (or [[Plücker coordinates]]), which can be used to uniquely identify a regular temperament.

	~~Nowadays~~, ~~most theorists prefer avoiding~~ the exterior algebra approach, ~~since~~ it ~~tends to be overcomplicated with little to no extra benefit~~.~~{{clarify}}~~		In many cases, the same things can be accomplished using matrix algebra or exterior algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations.
			Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.

	== See also ==		== See also ==

← Older revision		Revision as of 20:59, 11 June 2025
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	== See also ==		== See also ==
	* [[~~Basic abstract temperament translation code~~]]		* [[Plücker coordinates]]
	* [[Dave Keenan & Douglas Blumeyer's RTT ~~library in Wolfram Language~~]]		* [[Hodge dual]]
			* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]
	* [[Interior product]]		* [[Interior product]]
	* [[Recoverability]]		* [[Recoverability]]
	* [[The dual]]
	* [[User:Mike Battaglia/Exterior Norm Conjecture Table]]		* [[User:Mike Battaglia/Exterior Norm Conjecture Table]]

	[[Category:Exterior algebra\| ]] <!-- main article -->		[[Category:Exterior algebra\| ]] <!-- main article -->