Pythagoras based his inquiries into harmony on musical notes, but as any string player knows, no note that is actually performed is mathematically pure—every note played on an instrument is influenced by other notes, the subtle vibrations of the other strings, and even nearby instruments (when my daughter plays a D on her cello, the D string on my violin a few feet away hums, even though no one is touching it). These are musical overtones, and they create the complexity we look for in a good musical instrument.
