What is the relationship between a fundamental note and its overtones ?

All sounds (other than pure sine waves) are actually conglomerate sounds made by the vibration of a resonator (let's say a sting on a cello). The pitch produced by one complete up and down wave of the string is the pitch of the fundamental, and let's use the low C of that cello as an example. So, in this case the fundamental is the low C, two ledger lines beneath the bass staff.

Now, at the same time the string will also vibrate in multiples: half the length of the string, in thirds, in quarters, in fifths and so on, and all these vibrations are happening at the same time. Because the vibrations are shorter (half the string length, a third, etc), they produce higher pitches. These higher pitches are the overtones. Every note will have it's own set of overtones based on the fractions of length of it's fundamental. We perceive this conglomerate sound as a single note of a particular timbre.

The overtones happen in a pattern called the Harmonic Overtone Series. The first overtone is an octave above the fundamental. The second is a 5th higher, the third a 4th above this, then up in 3rds; each time the distance in pitch gets mathematically smaller. For that same low cello note the first few notes of the series would be (going up in pitch): C (beneath the staff), C (in the staff), G, middle C, E, G, Bb, C, D, E, F#, G(and on, but the tuning gets weird). This Harmonic Overtone Series is the basis for harmonics on instruments, how horns work, the nature of timbre and one of the ways the brain analyses sounds, in general.

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