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). 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".
The problem is that simple fractions like 3/2 are not exact powers of 2^( 1/12 <x-apple-data-detectors://5>).
How would non-unique factorization fix that?
--Dan
On Dec 2, 2014, at 5:31 AM, Henry Baker <hbaker1@pipeline.com <javascript:;>> wrote:
FYI -- Many/most of you may know this stuff, but I found it an interesting angle on the problem of musical tuning.
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-th...
The Saddest Thing I Know about the Integers
By Evelyn Lamb | November 30, 2014
The integers are a unique factorization domain, so we can’t tune
pianos. That is the saddest thing I know about the integers. . . . _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun