Hodge dual - Revision history

VectorGraphics at 19:21, 22 August 2025

2025-08-22T19:21:12Z

← Older revision		Revision as of 19:21, 22 August 2025
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	{{wikipedia\|Hodge star operator}}		{{wikipedia\|Hodge star operator}}

	In [[exterior algebra]] applied to [[regular temperament theory]], the '''Hodge dual''', or '''Hodge star''' is an operation that converts the [[Plücker coordinates]] (or [[wedgie]]) of a temperament into the corresponding coordinates of the [[comma basis]], and vice versa.		In [[exterior algebra]] applied to [[regular temperament theory]], the '''Hodge dual''', or '''Hodge star''' is an operation that converts the [[Plücker coordinates]] (or [[wedgie]]) of a temperament into the corresponding coordinates of the [[comma basis]] (a kind of "comma wedgie"), and vice versa.

	== Definition ==		== Definition ==

VectorGraphics at 19:16, 22 August 2025

2025-08-22T19:16:33Z

← Older revision		Revision as of 19:16, 22 August 2025
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	{{wikipedia\|Hodge star operator}}		{{wikipedia\|Hodge star operator}}

	In [[exterior algebra]] applied to [[regular temperament theory]], the '''Hodge dual''', or '''Hodge star''' is an operation that converts the [[Plücker coordinates]] (or [[wedgie]]) of a temperament into the corresponding coordinates of the [[comma basis]] ~~(a "comma basis wedgie")~~, and vice versa.		In [[exterior algebra]] applied to [[regular temperament theory]], the '''Hodge dual''', or '''Hodge star''' is an operation that converts the [[Plücker coordinates]] (or [[wedgie]]) of a temperament into the corresponding coordinates of the [[comma basis]], and vice versa.

	== Definition ==		== Definition ==
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	Let's work through the example step-by-step with matrix <math>M = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 4 \end{bmatrix}</math>, the mapping matrix of 5-limit [[meantone]].		Let's work through the example step-by-step with matrix <math>M = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 4 \end{bmatrix}</math>, the mapping matrix of 5-limit [[meantone]].

	We will write the standard basis vectors as <math>\{ e_1, \, e_2, \, e_3 \}</math>, which correspond ~~the~~ the primes 2, 3 and 5 respectively.		We will write the standard basis vectors as <math>\{ e_1, \, e_2, \, e_3 \}</math>, which correspond to the primes 2, 3 and 5 respectively.
	We already know that the kernel of this mapping should be [[81/80]], so we can write the kernel as:		We already know that the kernel of this mapping should be [[81/80]], so we can write the kernel as:
	:<math>		:<math>

VectorGraphics at 19:14, 22 August 2025

2025-08-22T19:14:57Z

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	</math>		</math>

	So we find <math>\star(v_1 \wedge v_2) = 4e_1 - 4e_2 + e_3</math>, which matches with what we expect from above, up to sign.		So we find <math>\star(v_1 \wedge v_2) = 4e_1 - 4e_2 + e_3</math> (which is the monzo for the descending syntonic comma), which matches with what we expect from above, up to sign.
	The Hodge dual \( \star(v_1 \wedge v_2) \) directly gives the generator of <math> \ker M </math>.		The Hodge dual \( \star(v_1 \wedge v_2) \) directly gives the generator of <math> \ker M </math>.

VectorGraphics at 19:13, 22 August 2025

2025-08-22T19:13:17Z

← Older revision		Revision as of 19:13, 22 August 2025
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	{{wikipedia\|Hodge star operator}}		{{wikipedia\|Hodge star operator}}

	In [[exterior algebra]] applied to [[regular temperament theory]], the '''Hodge dual''', or '''Hodge star''' is an operation that converts the [[Plücker coordinates]] of a temperament into the corresponding coordinates of the [[comma basis]], and vice versa.		In [[exterior algebra]] applied to [[regular temperament theory]], the '''Hodge dual''', or '''Hodge star''' is an operation that converts the [[Plücker coordinates]] (or [[wedgie]]) of a temperament into the corresponding coordinates of the [[comma basis]] (a "comma basis wedgie"), and vice versa.

	== Definition ==		== Definition ==

Sintel: delete wedgie entirely

2025-06-21T15:04:56Z

delete wedgie entirely

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	</math>		</math>

	The Hodge dual is <math>\star K = [6, -7, -2, -25, -20, 15]</math>, which are the Plücker coordinates ~~(a.k.a. [[wedgie]])~~ for [[miracle]].		The Hodge dual is <math>\star K = [6, -7, -2, -25, -20, 15]</math>, which are the Plücker coordinates for [[miracle]].

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

Sintel: /* Computation */ add python code

2025-06-13T11:20:07Z

Computation: add python code

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 # Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order. '''C''' will have the same length as '''V''' and '''M'''.
 # For each combination <math>C_i</math>, compute <math>S_i = \sum C_i - \frac{k (k+1)}{2}</math>.
-# Multiply the ''i''-th element of '''V''' by <math>(-1)^{S_i}</math>
+# Multiply the ''i''-th element of '''V''' by <math>(-1)^{S_i}</math>.
 # Reverse the elements of '''V'''.
 To find an unknown '''V''' from a known '''M''', first reverse '''M''' and then adjust the signs.
 == Applications ==

Sintel: /* Computation */ clarify, also replace ceil(k/2) with k-th triangular number (gives the same result)

2025-06-13T11:13:10Z

Computation: clarify, also replace ceil(k/2) with k-th triangular number (gives the same result)

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	With a basis of dimension ''n'', suppose we have a ''k''-form '''V''' and wish to find its dual '''M'''. The elements of '''V''' are associated with ''k''-combinations, and of '''M''' with {{nowrap\|(''n'' − ''k'')}}-combinations, of the basis elements. Because of the symmetry of binomial coefficients, '''V''' and '''M''' will have the same length. To find '''M''' we adjust the signs of '''V''' with the following procedure:		With a basis of dimension ''n'', suppose we have a ''k''-form '''V''' and wish to find its dual '''M'''. The elements of '''V''' are associated with ''k''-combinations, and of '''M''' with {{nowrap\|(''n'' − ''k'')}}-combinations, of the basis elements. Because of the symmetry of binomial coefficients, '''V''' and '''M''' will have the same length. To find '''M''' we adjust the signs of '''V''' with the following procedure:

	# Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order		# Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order. '''C''' will have the same length as '''V''' and '''M'''.
	# '''C''' will have the same length as '''V''' and '''M'''		# For each combination <math>C_i</math>, compute <math>S_i = \sum C_i - \frac{k (k+1)}{2}</math>.
	# ~~Sum the numbers in~~ each combination ~~'''C'''~~<~~sub~~>~~''i''~~</~~sub~~> ~~with ceil~~(''k~~''/~~2~~) to find '''S'''<sub>''i''~~</~~sub~~>		# Multiply the ''i''-th element of '''V''' by <math>(-1)^{S_i}</math>
	# Multiply the ''i''th element of ''V'' by −1<~~sup>'''S'''<sub>''i''</sub~~></~~sup~~>		# Reverse the elements of '''V'''.

	~~and then reverse~~ the elements of '''V'''.

	To find an unknown '''V''' from a known '''M''', first reverse '''M''' and then adjust the signs.		To find an unknown '''V''' from a known '''M''', first reverse '''M''' and then adjust the signs.

Sintel: /* Definition */ linking

2025-06-11T21:03:13Z

Definition: linking

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	== Definition ==		== Definition ==
	Given a rank ''k'' temperament on a JI subgroup of dimension ''n'', the mapping M can be written as a <math>k \times n</math> matrix.		Given a rank ''k'' [[abstract temperament]] on a [[JI subgroup]] of dimension ''n'', the [[mapping]] M can be written as a <math>k \times n</math> matrix.
	Writing the rows of this matrix as <math>v_1, \ldots, v_k</math>, the Plücker coordinates are <math>[v_1 \wedge \cdots \wedge v_k]</math>.		Writing the rows of this matrix as <math>v_1, \ldots, v_k</math>, the Plücker coordinates are <math>[v_1 \wedge \cdots \wedge v_k]</math>.
	This matrix can also be though of as representing a ''k''-plane, spanned by the rows.		This matrix can also be though of as representing a ''k''-plane, spanned by the rows.

	The kernel <math>\ker M</math>, representing the comma space, is an <math>(n - k)</math>-dimensional subspace of <math> \mathbb{R}^n </math>.		The kernel <math>\ker M</math>, representing the [[comma basis\|comma space]], is an <math>(n - k)</math>-dimensional subspace of <math> \mathbb{R}^n </math>.
	Similarly, if we represent <math>\ker M</math> by a matrix K, then its Plücker coordinates are <math>[w_1 \wedge \cdots \wedge w_{n - k}]</math>, where \( w_i \) are the columns of K.		Similarly, if we represent <math>\ker M</math> by a matrix K, then its Plücker coordinates are <math>[w_1 \wedge \cdots \wedge w_{n - k}]</math>, where \( w_i \) are the columns of K.

Sintel: /* Computation */ typo

2025-06-11T20:52:54Z

Computation: typo

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	== Computation ==		== Computation ==
	The Hodge dual can be computed quickly by realizing that if we write the basis in ~~lexographical~~ order, we only have to reverse the coefficients and change some signs.		The Hodge dual can be computed quickly by realizing that if we write the basis in lexicographic order, we only have to reverse the coefficients and change some signs.

	With a basis of dimension ''n'', suppose we have a ''k''-form '''V''' and wish to find its dual '''M'''. The elements of '''V''' are associated with ''k''-combinations, and of '''M''' with {{nowrap\|(''n'' − ''k'')}}-combinations, of the basis elements. Because of the symmetry of binomial coefficients, '''V''' and '''M''' will have the same length. To find '''M''' we adjust the signs of '''V''' with the following procedure:		With a basis of dimension ''n'', suppose we have a ''k''-form '''V''' and wish to find its dual '''M'''. The elements of '''V''' are associated with ''k''-combinations, and of '''M''' with {{nowrap\|(''n'' − ''k'')}}-combinations, of the basis elements. Because of the symmetry of binomial coefficients, '''V''' and '''M''' will have the same length. To find '''M''' we adjust the signs of '''V''' with the following procedure:

Sintel: /* Computing the Hodge dual */

2025-06-11T20:52:28Z

Computing the Hodge dual

← Older revision		Revision as of 20:52, 11 June 2025
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	The Hodge dual \( \star(v_1 \wedge v_2) \) directly gives the generator of <math> \ker M </math>.		The Hodge dual \( \star(v_1 \wedge v_2) \) directly gives the generator of <math> \ker M </math>.

	== ~~Computing the~~ Hodge dual ==		== Computation ==
	~~Again with~~ a basis of dimension ''n'', suppose we have a ''k''-form '''V''' and wish to find its dual '''M'''. The elements of '''V''' are associated with ''k''-combinations, and of '''M''' with {{nowrap\|(''n'' − ''k'')}}-combinations, of the basis elements. Because of the symmetry of binomial coefficients, '''V''' and '''M''' will have the same length. To find '''M''' we adjust the signs of '''V''' with the following procedure		The Hodge dual can be computed quickly by realizing that if we write the basis in lexographical order, we only have to reverse the coefficients and change some signs.

			With a basis of dimension ''n'', suppose we have a ''k''-form '''V''' and wish to find its dual '''M'''. The elements of '''V''' are associated with ''k''-combinations, and of '''M''' with {{nowrap\|(''n'' − ''k'')}}-combinations, of the basis elements. Because of the symmetry of binomial coefficients, '''V''' and '''M''' will have the same length. To find '''M''' we adjust the signs of '''V''' with the following procedure:

	# Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order		# Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order