Hodge dual: Difference between revisions

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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]] (a kind of "comma wedgie"), and vice versa.

== 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]].

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:

:<math>

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</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>.

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 {{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]] (a kind of "comma wedgie"), and vice versa.
 == Definition ==
@@ Line 40: / Line 40: @@
 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:
 :<math>
@@ Line 84: / Line 84: @@
 </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>.