SAKryukov

::::::: Please don't get irritated by this, but now I cannot find 1) what are those two articles, 2) what is "my last comment" :-) Here is what I want to advise: I think you greatly overestimate the ability of other people to navigate (I say only "navigate", not "understand", generally I understand you quite well). You overestimate just the vision, observation of what you said before, ability to associate it, and so on. For communication on such things, there is a great tool: links! The same goes about discussions with the numbers: most people don't imagine them well. (I personally have one of the kinds of the mathematical mind. People like me think abstractly but notoriously bad with numbers. Paradoxically, mathematics and dealing with concrete numbers are almost opposite traits :-) — [[User:SAKryukov|SA]] 18:15, 27 November 2020 (UTC)

:::::Lastly, I was thinking about those commas and have some consideration. Roughly, we need to understand that in a rational-number interval i=A/B makes deep physical sense only if A and B are small enough. This is related to the nature of aural perception of any organisms or even devices. If the numbers became 3-digit numbers or more, the accuracy of the rational number doesn't play its role. Let's see: there are two physical traits: 1) the sense of harmonics in the interval with small A and B; it is based that two oscillators in the ear come in resonance, but not necessarily on their fundamental frequencies, but some low-order harmonics, if harmonics are high, the effect is unnoticeable, 2) the perception of logarithmic distances between frequencies as equal. Now, #1 and #2 are in fundamental contradiction: 1) if all intervals are rational numbers, their system is never ever equidistant, so the equivalence of tonalities is impossible, 2) if intervals are equidistant, the ratio values are never rational, so we won't feel the perfect sense of harmony. So, the question is: with this trade-off, where is the reasonable choice? The common-practice system has chosen the compromise and gave more preference to #2 than before. 12-EDO gives amazingly good compromise, but we pretty easily can perceive the deviation from harmony. At the same time, our trait #1 is more accurate than #2. What is my conclusion? It can be a bit complicated in practice, but this is nothing but some intermediate idea. First, everything depends on the composition. Where we value the sense of equidistant notes? I don't think it is absolutely important in most cases. That said, when your calculations lead you to big natural numbers, A/B, you can easily give up having a rational number for a certain degree and use a mixture of rational numbers and real numbers. I do understand how weird it can be, this is just a vague idea. Another vague idea is that you might extend some tonal system but then classify the resulting tones and their functional roles into two classes: "degrees" and "non-degrees". A "non-degree" tone can have a limited role, it cannot be, for example, used as a root of a chord, and so on. I do understand that this is very weird and not elegant, this is just for discussion. The main point is: the contradictions and trade-offs in the tonal systems are unavoidable in principle, by the very nature of thins, sorry for possible truism. — [[User:SAKryukov|SA]] 17:05, 27 November 2020 (UTC)