Harmonics
The integer multiples of a fundamental frequency whose relative strengths give every sound its characteristic color.
Harmonics
When something vibrates with a definite pitch — a string, an air column, a bell — it rarely vibrates at just one frequency. It rings at a fundamental frequency f_0 and at a whole ladder of harmonics (also called overtones) at integer multiples 2f_0, 3f_0, 4f_0, \dots all at once. The fundamental sets the pitch you name; the harmonics, and how strong each one is, set the timbre — the quality that lets you tell instruments apart.
This ladder is exactly the set of frequencies a Fourier Series is built from. A periodic sound is its harmonics, summed.
Why a violin and a flute sound different
Play the note A at 440 Hz on a violin and on a flute and you hear the same pitch — both have their fundamental at 440 Hz — yet no one confuses them. The difference is the recipe of harmonic amplitudes. A flute is nearly a pure tone: a strong fundamental and weak overtones. A bowed violin is rich and buzzy: substantial energy in many higher harmonics. Same pitch, different spectral fingerprint.
Read left to right, each bar is the strength of the fundamental, then the 2nd harmonic, the 3rd, and so on. The two instruments draw the same first bar but utterly different silhouettes — and your ear reads that silhouette as "flute" or "violin."
Where the ladder comes from
Harmonics are not arbitrary; they are forced by geometry. A string clamped at both ends can only support a Standing Wave whose length fits a whole number of half-wavelengths, and those allowed modes have frequencies f_0, 2f_0, 3f_0,\dots — the harmonic series exactly. Driving such a system near one of these frequencies produces Resonance, which is why instruments speak so readily at their natural pitches.