Come ON guys, help me out here: Be more clear about your terminology!!!<p>Gee, I'm plenty good with the math and applications of Fourier theory. Sure, some physics course solved the differential equation of vibrating strings. Early on my career did well with Fourier theory, the FFT (fast Fourier transform), and Navy ocean data. And with violin, I got a start in the music school of Indiana University and eventually made it through several pages of the Bach E-major preludio, the Bach Chaconne, and a transcription, as I recall, up an octave and a fifth, of that famous Bach solo cello piece. Been known to take the positive whole number powers of 2^(1/12). Dissonance? Sure, I really like the S. Barber <i>Adagio</i> for strings!!! R. Wagner's chords. The Franz Schmidt Intermezzo to his Notre Dame. The Sibelius violin concerto.<p>But, stilllll, a lot of your terminology I can't follow.<p>For an opinion: Mostly music is art, that is communication, interpretation of human experience, emotion. Given the roles of, say, vibrato, Fourier theory and 2^(1/12) do not directly explain all of the art.<p>For a <i>perfect fifth</i>, sure, as is standard, when take the violin out of its case, first tune the four strings G, D, A, E in perfect fifths and do this by listening for beats in the overtones. Then with the strings so tuned, often when the music calls for one of those 4 notes, just go ahead and play it on the <i>open</i> (look, Ma, no use of fingers on the left hand) string.<p>In particular, for<p>> perfectly in mathematical ratios<p>in real music from before Bach to after Barber, I doubt that "perfectly" happens very often or is even attempted. E.g., with violin, can't play much music if play only on the open strings, and as soon as put a finger of the left hand on a string, with vibrato or even without, there won't be much "perfectly mathematical".