> Most Western instruments produce “harmonic” sounds that, when analyzed as Fourier described, have relatively strong lower overtones f, 2f, 3f, 4f. The overtones of several of these sounds will match when their fundamental frequencies are related by simple whole-number ratios.<p>I've always had problems with existing consonance theories along these lines, probably out of ignorance. I believe that consonance probably has something to do with partial relationships. But it always has seemed the existing theory regarding consonance and partials is half baked at best.<p>Harmonic sounds have partials at f, 2f, 3f, 4f, and so on. So let's take C and G, a perfect fifth. Will say f = C. So C's sawtooth has partials of<p>f 2f 3f 4f 5f 6f ...<p>G, at a perfect fifth, is 3/2 the frequency of C. So it has partials at<p>3/2f 3f 9/2f 6f 15/2f 9f 21/2f 12f ...<p>The only overlaps are 3f, 6f, 9f, 12f. That's pretty thin gruel given how fast these partials drop of in amplitude (for a sawtooth, a partial at xf drops off as 1/x)<p>Now consider E, a major 3 and also considered highly consonant. This is 5/4. So we have<p>5/4 5/2 15/4 5 25/4 15/2 35/4 10 ...<p>That's an overlap of just 5, 10, ... with a super fast dropoff.<p>Now consider Eb, a minor 3 and also consonant. This is, wait, Eb can't be approximated in rational values at all, um....<p>And if you just have sine waves, rather than sawtooth or whatnot, they consist of a single partial so there's never any overlap -- you're back to the pythagorean square 1.