I am not musician and I am, IMO, barely literate in mathematics. So please dumb down the answer if you have one.

I seem to recall hearing about harmonics in music. What are they? Are they related to any mathematical concepts of harmonics, e.g. harmonic functions where 2nd gradient is equal to zero (or something like that)?

Anybody got the straight dope?

Harmonic functions are so-called because of their association with the equation of harmonic motion. Harmonics in music are overtones, whole number multiples of an original tone. So, they’re related, but not.

I’ll gladly leave the discussion of harmony to someone who is versed (heh heh) in the subject. From a physical point of view, harmonics are a result of how musical instruments produce sounds. Consider a stringed instrument like a guitar or a piano. Each string is a fixed length and under a set tension (barring changes in temperature, humidity or metal fatigue) and so has a fundamental mode of vibration where the length of the string defines exactly one-half wavelength - i.e. the ends are fixed and the greatest displacement (the anti-node) is in the middle. The next vibration mode is where one full wavelength “fits” on the string, so now the middle is still (a node) and there are two anti-nodes one-quarter of the way in from each end. This mode is the first harmonic with a wavelength one-half of and a frequency twice that of the fundamental mode. The next mode has two nodes and three anti-nodes, with wavelength one-third of and frequency three times the fundamental. And so it goes…