Talk:Mike's lecture on vector spaces and dual spaces: Difference between revisions
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Also, it is not clear where a vector space comes from. A vector space is something defined over a field. But any set of intervals is not a field, it only forms an Abelian group, with * operation being the arithmetic multiplication of rational number. This is the only significant operation, as additive operation between those numbers has no meaning for music; intervals is added when a frequency is multiplied by it, and two intervals can be added or subtracted. The additive operation for frequencies is the * operation for this group. With all the similarity with vector spaces, the lack of essential vector space properties is dramatic. You can scale an interval (by an integer number), but you cannot define a basis, because there is no a way to decompose a vector against the basis. No, this is not a vector space at all. — [[User:SAKryukov|SA]], ''Wednesday 2020 December 2, 17:13 UTC'' | Also, it is not clear where a vector space comes from. A vector space is something defined over a field. But any set of intervals is not a field, it only forms an Abelian group, with * operation being the arithmetic multiplication of rational number. This is the only significant operation, as additive operation between those numbers has no meaning for music; intervals is added when a frequency is multiplied by it, and two intervals can be added or subtracted. The additive operation for frequencies is the * operation for this group. With all the similarity with vector spaces, the lack of essential vector space properties is dramatic. You can scale an interval (by an integer number), but you cannot define a basis, because there is no a way to decompose a vector against the basis. No, this is not a vector space at all. — [[User:SAKryukov|SA]], ''Wednesday 2020 December 2, 17:13 UTC'' | ||
At the same time, if we can consider the product of generation as a ''free Abelian group | 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'' | ||