Hodge dual: Difference between revisions

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== Using the dual ==

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 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 {{multimonzo|6 -4 0 -1 3 -2}}° = {{wedgie|-2 -3 -1 0 4 6}}. Normalizing this to a wedgie gives {{wedgie|2 3 1 0 -4 -6}}, the wedgie for bug temperament. Then if W is the wedgie for ennealimmal considered above, ~~W∧M°~~ = ⟨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|~~abstract regular temperament]] page for extracting a normal val list from a multival to get a normal comma list from the multimonzo.

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}}° {{=}} {{wedgie|-2 -3 -1 0 4 6}}}}. Normalizing this to a wedgie gives {{wedgie|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.

[[Category:EA]]

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 == Using the dual ==
-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 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 {{multimonzo|6 -4 0 -1 3 -2}}° = {{wedgie|-2 -3 -1 0 4 6}}. Normalizing this to a wedgie gives {{wedgie|2 3 1 0 -4 -6}}, the wedgie for bug temperament. Then if W is the wedgie for ennealimmal considered above, W∧M° = ⟨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|abstract regular temperament]] page for extracting a normal val list from a multival to get a normal comma list from the multimonzo.
+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}}° {{=}} {{wedgie|-2 -3 -1 0 4 6}}}}. Normalizing this to a wedgie gives {{wedgie|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.
 [[Category:EA]]