Allan, what an excellent can o' worms your apprentice opens up! The related questions are 1) the nature of harmonic motion and the harmonic series and 2) why strings attached at both ends move in a harmonic series when displaced. A couple of sites I looked at are http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html http://www.pass.maths.org.uk/issue12/features/harmonic/index-gifd.html http://www.spectrummuse.com/harmonics.htm Whether these get us much further forward on WHY a string moves in a harmonic series, I am not quite sure.... But, that it DOES move in that way can of course be easily demonstrated on a piano. I've just held down A2 on my piano, and sounded E4 briefly and sharply, and heard the E4 note sound beautifully from the A2 strings, which can only happen because they are moving in thirds of their length. (Also produced nice A3, A4 and E5 notes from A2). Best, David.
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