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. — SA, Tuesday 2020 December 1, 20:47 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 no 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. — SA, Tuesday 2020 December 1, 20:58 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 basics, because there is no a way to decompose a vector against the basis. No, this is not a vector space at all. — SA, Wednesday 2020 December 2, 17:13 UTC