Hodge dual: Difference between revisions
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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 kind of "comma wedgie"), 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 | 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> | ||
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In <math> \mathbb{R}^3 </math>, the Hodge star <math> \star: \Lambda^2 \, \mathbb{R}^3 \to \Lambda^1 \, \mathbb{R}^3 </math> acts on basis elements as: | In <math> \mathbb{R}^3 </math>, the Hodge star <math> \star: \Lambda^2 \, \mathbb{R}^3 \to \Lambda^1 \, \mathbb{R}^3 </math> acts on basis elements as: | ||
:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
\star(e_1 \wedge e_2) &= e_3 \\ | \star(e_1 \wedge e_2) &= e_3 \\ | ||
\star(e_1 \wedge e_3) &= -e_2 \\ | \star(e_1 \wedge e_3) &= -e_2 \\ | ||
\quad \star(e_2 \wedge e_3) &= e_1 | \quad \star(e_2 \wedge e_3) &= e_1 | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
Applying this to <math> v_1 \wedge v_2 </math>: | Applying this to <math> v_1 \wedge v_2 </math>: | ||
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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>. | ||
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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 | 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 == | ||