> The paradox is that you can't create a theory of music whose notes are both (a) evenly spaced and (b) contain the integer ratios.
I don't know much about this, but isn't (b) impossible even if you satisfy (a)? There is no sequence of numbers that contains any arbitrary integer ratio because there are infinitely many possible ratios but only finitely many ratios you can make out of a sequence of numbers.
(Obviously some ratios like 2:1 and 3:1 are more important than, say, 52697:16427. 12-TET chooses to permit 2^n:1 at the cost of all other ratios, which seems like a good tradeoff to me.)
(a) makes it much more restrictive though: you can't even have {f, 2f, 3f} simultaneously. (If 2f = a^m f and 3f = a^n f, then 2^n = a^{mn} = 3^m, which has no nonzero solutions. Equal temperament contains *no* integer ratios at all, other than whole-number multiples).
I don't know much about this, but isn't (b) impossible even if you satisfy (a)? There is no sequence of numbers that contains any arbitrary integer ratio because there are infinitely many possible ratios but only finitely many ratios you can make out of a sequence of numbers.
(Obviously some ratios like 2:1 and 3:1 are more important than, say, 52697:16427. 12-TET chooses to permit 2^n:1 at the cost of all other ratios, which seems like a good tradeoff to me.)