Hodge dual: Difference between revisions

Line 27:

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]]

[[Category:Exterior algebra]]

Revision as of 20:01, 19 March 2025 view source Sintel (talk \| contribs) autopatrolled 1,547 edits Inaccessible ← Older edit		Revision as of 20:02, 19 March 2025 view source Sintel (talk \| contribs) autopatrolled 1,547 edits Categories Newer edit →
Line 27:		Line 27:
	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.		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]]		[[Category:Exterior algebra]]