Am Tue, 2 Dec 2014 23:00:05 -0800 schrieb Dan Asimov <dasimov@earthlink.net>: This topic was the main reason I asked for subscription to this mailing list many (10-12?) years ago! There may be something about this in the archives. Considering the continued fraction for log2/log(3/2) suggests 5, 12, 41, 53, ... equal subdivisions of the octave, see http://doctroidalresearch.wordpress.com/pages/recreational-mathematics/conti... But considering 3/2, 4/3, 5/4,... simultaneously is certainly more involved. (fifths or fourths come perfect/augmented/diminished, while 2nds, 3rds, 6ths and 7ths come major/minor) Dirk.
Suddenly I realize that I have no idea why powers of 2^(1/12) provide good approximations to p/q for many low p and q.
k 2^(k/12) ------------- 1 1.0594+ 2 1.1224+ 3 1.1892+ ~ 6/5 (minor third) 4 1.2599+ ~ 5/4 (major third) 5 1.3348+ ~ 4/3 (major fourth) 6 1.4142+ 7 1.4983+ ~ 3/2 (major fifth) 8 1.5874+ 9 1.6817+ 10 1.7817+ 11 1.8877+ 12 2
Okay, so maybe the approximations aren't *that* good.
But they're almost that good. All four intervals are within less than 1% of just intonation.
How come? Is there some basic number-theoretic reason?
I suppose the Pythagorean comma = (3/2)^12 / 2^7 ~ 1.01+, the discrepancy between 12 perfect fifths and 7 octaves, is somehow relevant. But I don't see how this works.
--Dan
On Dec 2, 2014, at 9:04 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
It's an interesting exercise to look for equal divisions of the octave into n intervals that provide better approximations to (say) 3rd, 5th, 7th harmonics than n = 12 .
WFL
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