And this musing -- Given that math precludes perfect tuning, are there engineering work-arounds other than equal temperament and n-note scales? Perhaps vibrato, besides being an embellishment, also obscures the note so that it sounds better in conjunction with other notes? Taking that further, perhaps there is a sounding-good function that gives the best frequency perturbations for all of the notes sounding at a given moment. (Psychology experiment opportunity here -- do musicians already make subtle adjustments like this?. Might it depend on previous note and next note? Weighted by the musical "importance" of the current notes?) If so, electronic instruments could re-tune themselves based on what the current chord is, both its own and the notes other instruments or people are sounding. Real-time vocal pitch correction already exists, suggesting this is possible. Would the result sound more pleasant, or always slightly off? -- Mike ----- Original Message ----- From: "James Propp" <jamespropp@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, December 03, 2014 9:42 AM Subject: Re: [math-fun] The Saddest Thing about the Integers
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
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