Hodge dual: Difference between revisions

Line 1:

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 ==

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 1: / Line 1: @@
 {{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 ==
@@ 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>