Hello Jay Beard! I'm only 30s into the video so far, I will finish it, but to me, "Set Theory" is the branch of mathematical logic which describes collections of objects. An example of a set could be infinite strings of binary digits, and one can prove that set is in some sense the same size with the real numbers (decimals) using logical steps. (If you're curious what 'same size' means in the context of set theory, we say two sets are the same size when we can define a natural function which maps from one set to the other without ever having two distinct inputs map to the same output and every output has to be hit by at least some input). Cheers, I will now continue into your alternative definition of set theory! Maybe there will be some parallels into the mathematical set theory which is why it is named as such. By the way, I am deep into tuning theory and appreciate that we need to invent new analytical structures to understand structures embedded in music, so I'm all for analysis and new analysis techniques. That's probably why YT recommended this video on my front page. Cheers and I'll continue the video now.
On ordering modes: You can pick ANY objective function which projects the data from each mode into a single number, and use that single number as the order of modes. As you said you can measure their compactness and then order them based on that. But why is that the only natural choice for function to decide how to order modes? What about, say, mapping the mode into a word with characters W/H only, and ordering based on alphabetical/dictionary ordering? Obviously each mode will be unique in that sense too and would provide a distinct ordering relative to your compactness measure. And there's a lot of other ways of ordering them as well. I ran into this problem when trying to find "The" natural ordering of all possible quarter note/rest rhythms for a single bar of 4/4. You can always define a metric, and that metric produces the natural ordering of the rhythms. But there is no natural metric by which is the ultimate lens for determining how those rhythms should be ordered. There can always be another natural way of looking at it which produces a distinct ordering.
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Hello Jay Beard! I'm only 30s into the video so far, I will finish it, but to me, "Set Theory" is the branch of mathematical logic which describes collections of objects. An example of a set could be infinite strings of binary digits, and one can prove that set is in some sense the same size with the real numbers (decimals) using logical steps. (If you're curious what 'same size' means in the context of set theory, we say two sets are the same size when we can define a natural function which maps from one set to the other without ever having two distinct inputs map to the same output and every output has to be hit by at least some input). Cheers, I will now continue into your alternative definition of set theory! Maybe there will be some parallels into the mathematical set theory which is why it is named as such. By the way, I am deep into tuning theory and appreciate that we need to invent new analytical structures to understand structures embedded in music, so I'm all for analysis and new analysis techniques. That's probably why YT recommended this video on my front page. Cheers and I'll continue the video now.
This is a well established "alternative" understanding of set theory - that is as it is used in music theory.
On ordering modes: You can pick ANY objective function which projects the data from each mode into a single number, and use that single number as the order of modes. As you said you can measure their compactness and then order them based on that. But why is that the only natural choice for function to decide how to order modes? What about, say, mapping the mode into a word with characters W/H only, and ordering based on alphabetical/dictionary ordering? Obviously each mode will be unique in that sense too and would provide a distinct ordering relative to your compactness measure. And there's a lot of other ways of ordering them as well. I ran into this problem when trying to find "The" natural ordering of all possible quarter note/rest rhythms for a single bar of 4/4. You can always define a metric, and that metric produces the natural ordering of the rhythms. But there is no natural metric by which is the ultimate lens for determining how those rhythms should be ordered. There can always be another natural way of looking at it which produces a distinct ordering.