> that's N times the fundamental as the Nth Harmonic
It's not actually "N times", isn't it?
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If the fundamental is 100hz, then the 1st harmonic is the fundamental (100hz), the 2nd harmonic is 200hz, the 3rd harmonic is 300hz, and so on.
Sometimes the harmonics aren't exact. On a piano, if the fundamental is 100hz then the 2nd harmonic might be, say, 200.1hz or something. Some inharmonic instruments like gongs aren't anywhere close to the "ideal" harmonic series.
This may be overly pedantic (even more so than your correct comments about numbering harmonics and overtones), but in this case the overtones are not harmonics, which, as you say, are by definition multiples of the fundamental frequency (“harmonic series” is a mathematical term). That’s why gongs are “inharmonic”: they have an overtone series that is not a series of harmonics.
Membranes have a harmonic series in two axes - kind of. They have complex (not that kind of complex, although it also is, in a way) modes on a constrained surface which are calculated with Bessel functions.
In three dimensions you get atomic orbitals.
Piano is even tuned with stretched tuning to match the harmonics better.