Hodge dual: Difference between revisions

Line 99:

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

== ~~Using the dual~~ ==

== Applications ==

The dual ~~allows one~~ to ~~find the wedgie, which is a normalized multival, by wedging together monzos~~ and ~~then taking the dual~~. For ~~instance from {{nowrap| '''M''' {{~~=~~}} {{monzo| 0 3~~ -2 ~~0 }} ∧ {{monzo|~~ -2 1 -1 ~~1 }} }}~~, ~~which is {{multimonzo| 6 -4~~ 0 -1 3 ~~-2 }}, considered above~~, we ~~may~~ find ~~the dual '''M'''° as {{nowrap| {{multimonzo|6~~ -~~4 0 -1 3~~ -2~~}}° {{~~=~~}} {{multival|~~ -2 -~~3 -1 0 4 6 }} }}. Normalizing this to a wedgie gives {{multival|~~ 2 ~~3 1 0~~ -4 -~~6 }}~~, the ~~wedgie for bug temperament~~. ~~Then if '''W''' is the wedgie for ennealimmal considered above, {{nowrap| '''W''' ∧ '''M'''° {{=}} {{wmp| '''W'''|M}} {{=}} 1 }}~~. ~~We can also take~~ a ~~multival, and use the dual to get a corresponding multimonzo, and then use the same method described on the~~ [[~~abstract regular temperament~~]] ~~page~~ for ~~extracting a normal val list from a multival to get a normal comma list from the multimonzo~~.

The Hodge dual can be used to convert between commas and temperaments generally.

For example, if we take [[225/224]], with coordinates <math>w_1 = [-5, 2, 2, -1]</math> and [[1029/1024]], with coordinates <math>w_2 = [-10, 1, 0, 3]</math>, we can find:

:<math>

K = w_1 \wedge w_2 = [15, 20, -25, -2, 7, 6] .

</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]].

== See also ==

@@ Line 99: / Line 99: @@
 To find an unknown '''V''' from a known '''M''', first reverse '''M''' and then adjust the signs.
-== Using the dual ==
+== Applications ==
-The dual allows one to find the wedgie, which is a normalized multival, by wedging together monzos and then taking the dual. For instance from {{nowrap| '''M''' {{=}} {{monzo| 0 3 -2 0 }} ∧ {{monzo| -2 1 -1 1 }} }}, which is {{multimonzo| 6 -4 0 -1 3 -2 }}, considered above, we may find the dual '''M'''° as {{nowrap| {{multimonzo|6 -4 0 -1 3 -2}}° {{=}} {{multival| -2 -3 -1 0 4 6 }} }}. Normalizing this to a wedgie gives {{multival| 2 3 1 0 -4 -6 }}, the wedgie for bug temperament. Then if '''W''' is the wedgie for ennealimmal considered above, {{nowrap| '''W''' ∧ '''M'''° {{=}} {{wmp| '''W'''|M}} {{=}} 1 }}. We can also take a multival, and use the dual to get a corresponding multimonzo, and then use the same method described on the [[abstract regular temperament]] page for extracting a normal val list from a multival to get a normal comma list from the multimonzo.
+The Hodge dual can be used to convert between commas and temperaments generally.
+For example, if we take [[225/224]], with coordinates <math>w_1 = [-5, 2, 2, -1]</math> and [[1029/1024]], with coordinates <math>w_2 = [-10, 1, 0, 3]</math>, we can find:
+:<math>
+	K = w_1 \wedge w_2 = [15, 20, -25, -2, 7, 6] .
+</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]].
 == See also ==