The fun stuff

One note is a stack of waves

Press the string anywhere and let go. It does not move as one wave: because both ends are pinned, the only waves that fit are whole numbers of half-waves, and the shape you pulled it into is a sum of those. The bars are the recipe, and you can hear it change: pluck somewhere else, touch the string at a half or a third, or switch a harmonic off and find out what it was adding. That recipe is why a guitar and a flute playing the same note sound different.

1. $$\frac{{\partial }^{2}y}{\partial t^2}=c^2\frac{{\partial }^{2}y}{\partial x^2}$$
   The wave equation: how any stretched string moves.
2. $$y(x,t)=\sum _{n=1}^{\infty }{a}_n\sin \left(\frac{n\pi x}{L}\right)\cos (2\pi f_{n}t)$$
   Pinned at both ends, every motion is a sum of standing waves.
3. $$f_n=n{f}_1=\frac{n}{2L}\sqrt{\frac{T}{\mu }}$$
   Each rings at a whole multiple of the lowest. Two to one is an octave, three to two a fifth: harmony is small whole numbers.