Q.What are harmonics and overtones?

Harmonics are a theoretical list of whole-number multiples of the fundamental; overtones are whichever of those actually occur, renumbered in the order they appear.

(1) Definition basis: A harmonic of order $n$ is, by definition, a tone whose frequency is exactly $n$ times the fundamental frequency $f_0$ (i.e. $f_0, 2f_0, 3f_0,\dots$), regardless of whether that particular multiple is actually produced by the vibrating source. An overtone is a tone of frequency higher than the fundamental that is actually produced by the source; overtones are numbered serially in the order they are physically present — the 1st overtone is the next frequency actually heard after the fundamental, the 2nd overtone the one after that, and so on — not necessarily $2f_0$, $3f_0$, etc.

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Harmonics are integer multiples of the fundamental; overtones are simply the modes above the fundamental, which may or may not coincide with exact harmonics.

When a system (string, air column) vibrates, it can do so in several normal modes. The lowest frequency mode is the fundamental ($f_1$, or first harmonic). Frequencies that are exact integral multiples of the fundamental — $f_1, 2f_1, 3f_1, \dots$ — are called harmonics (first, second, third harmonic, and so on).

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Harmonics = all integer multiples of f₁; overtones = whichever of those are actually excited, renumbered from the first one up.

1. Harmonics are all the frequencies that are integral multiples of the fundamental frequency $f_1$: $f_1, 2f_1, 3f_1, \ldots$ — a mathematically complete set, whether or not each one is actually present in a given vibration. …