View source for Talk:Mike's lecture on vector spaces and dual spaces

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== Misleading "prime" intervals ==

Intervals are not decomposed in other intervals using the Monzo formalism. Instead, the rational interval is simply shown in its factorized forms. The interval itself is not "decomposed" at all. We just consider here only the rational interval, and the numerator and denominator of the rational number are factorized. It also would be good to emphasize, that the operation of transforming a rational number into a Monzo form does nothing to the interval, only show a bit different presentation of it. &mdash;&nbsp;[[User:SAKryukov|SA]],&nbsp;''Tuesday&nbsp;2020&nbsp;December&nbsp;1,&nbsp;20:47&nbsp;UTC''

Also note that even if we considered "prime" intervals, they would be pretty much insignificant in our context. This is because only the primes 1 and 2 give the intervals inside a single octave. I found a very usual misunderstanding among musicians who discuss harmonics. Harmonics are only the frequencies *1, *2, *3, *4 and so on, and tonal systems deals with frequency ratio < 2. The "harmonic" nature of intervals is much more delicate: they are perceived as "harmony" because not only instruments, but our aural system (and any thinkable receiver) has harmonics; 3/2 is perceived as "harmonic" not because fundamental frequencies resonate, but because one oscillator resonates on 2nd harmonic with another one on its 3rd harmonic. And yet, no harmonics beyond 2 are used in most tonal systems, everything works without them. &mdash;&nbsp;[[User:SAKryukov|SA]],&nbsp;''Tuesday&nbsp;2020&nbsp;December&nbsp;1,&nbsp;20:58&nbsp;UTC''

== No vector spaces ==

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. &mdash;&nbsp;[[User:SAKryukov|SA]],&nbsp;''Wednesday&nbsp;2020&nbsp;December&nbsp;2,&nbsp;17:13&nbsp;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. &mdash;&nbsp;[[User:SAKryukov|SA]],&nbsp;''Wednesday&nbsp;2020&nbsp;December&nbsp;2,&nbsp;23:08&nbsp;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)

: Added a footnote explaining the technicality. [[User:IlL|IlL]] ([[User talk:IlL|talk]]) 23:48, 6 December 2020 (UTC)

== Clarify wedgies ==

In this article, wedgies and mapping matrices are presented as alternative paths to exploring RTT. However, in recent discussions on Facebook, Discord, and the Sagittal forum I have come to understand (from folks like Herman, Dave, Mike, Graham) that wedgies have only specialized uses and are mostly for fans of certain advanced types of math. In other words, if I'm a person who has been desperately struggling for 15 years to understand what people are talking about re: temperaments, wedgies are a bit of a dead end (or at least a quagmire of distraction) and I should cease agonizing over them. I think that since the lecture series doesn't continue far enough to give specific examples of using them to navigate and describe tuning space that might have value to non-mathematicians, it could help to make that clear briefly here. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 17:17, 16 April 2021 (UTC)