On Dec 3, 2014, at 6:42 AM, James Propp <jamespropp@gmail.com> wrote:
If 3 could be written as 2^(p/q) for some positive integers p and q, then 3/2 raised to a suitable power n would be a power of 2, so we could construct an n-note scale in which fifths were mathematically perfect.
But unique factorization (combined with the fact that 2 and 3 are prime) tells us that 3 can't be written as 2^(p/q).
I don't think unique factorization is the problem here. (As Jim may agree.) Even if Q is embedded in *any* subfield K of C, it will still be the case that 2^L = 3^M in O(K) (the ring of algebraic integers of K) is *false* for all integers L,M >= 1 . So we can never have (3/2)^12 = 2^19, or anything similar, in any subfield of C. --Dan
This reminds me of a woozy old musing of mine, namely, that just as there are extensions of Q in which rational primes split, Q might have "under-things" of some kind in which distinct rational primes merge. As far as I've ever been able to tell, this is utter nonsense. Or rather, it's the wrong kind of nonsense (the kind that doesn't lead to anything interesting) as opposed to the right kind of nonsense (which does).
Rethinking my initial high regard for the article, I now think that the author has misidentified the true source of her angst. What she really longs for is Pythagoras' dream-world, where irrational numbers don't exist and in particular log_2 (3/2) is rational.
Jim Propp
On Tuesday, December 2, 2014, Dan Asimov <dasimov@earthlink.net> wrote:
I don't find this good popular math at all.
The article never explains why a non-UFD would enable us to "tune pianos". . . .