Talk:Mike's lecture on vector spaces and dual spaces: Difference between revisions

← Older edit Newer edit →

VisualWikitext

Revision as of 23:11, 2 December 2020 view source SAKryukov (talk \| contribs) 648 edits m Incommensurable values part, italic fix ← Older edit		Revision as of 06:55, 3 December 2020 view source FloraC (talk \| contribs) autopatrolled, Bureaucrats, Interface administrators, Sysops 18,328 edits →No vector spaces: re Newer edit →
Line 12:		Line 12:

	At the same time, if we can consider the product of generation as a ''free Abelian group'', we can see some generalization of the vector space concept, but not the vector space ''per se''. Free Abelian group is the Abelian groups with a ''free module'' (module with a basis); and the concept of a module can be considered as a generalization of the notion of vector space. This is trivial enough only for ''linear temperaments'', in particular, EDOs, where we can consider the two-generator basis: 1 microtone (note that this is an irrational number) and 1 octave. An attempt to use the basis of rational-number generators leads to infinite sets of never repeating frequency positions within the octave, because 2 and any other rational non-unison interval are incommensurable values. — [[User:SAKryukov\|SA]], ''Wednesday 2020 December 2, 23:08 UTC''		At the same time, if we can consider the product of generation as a ''free Abelian group'', we can see some generalization of the vector space concept, but not the vector space ''per se''. Free Abelian group is the Abelian groups with a ''free module'' (module with a basis); and the concept of a module can be considered as a generalization of the notion of vector space. This is trivial enough only for ''linear temperaments'', in particular, EDOs, where we can consider the two-generator basis: 1 microtone (note that this is an irrational number) and 1 octave. An attempt to use the basis of rational-number generators leads to infinite sets of never repeating frequency positions within the octave, because 2 and any other rational non-unison interval are incommensurable values. — [[User:SAKryukov\|SA]], ''Wednesday 2020 December 2, 23:08 UTC''

			: You must view it in the logarithmic measure, in which there is definitely musical sense for both addition and multiplication. [[User:FloraC\|FloraC]] ([[User talk:FloraC\|talk]]) 06:55, 3 December 2020 (UTC)