overtones and harmonics

Why does a violin sound like a violin and not like a flute, even when both play the exact same note? Because no real instrument produces a single pure frequency — it produces a fundamental tone plus a whole stack of higher tones ringing along with it. The fundamental sets the pitch you name; the higher tones, called overtones (or harmonics, or partials), give the sound its character and colour.

On an ideal string the picture is beautifully simple. The normal modes vibrate at frequencies f_n = n c / (2 L) for n = 1, 2, 3, ..., so they are exact integer multiples of the lowest: f_1 (the fundamental), then 2 f_1, 3 f_1, 4 f_1, and on up. The fundamental f_1 is the first harmonic; the overtones are the higher members (the second harmonic at 2 f_1, the third at 3 f_1, and so on). Beware a common naming clash: 'the first overtone' is the SECOND harmonic — overtones are counted above the fundamental, harmonics including it. When you pluck a string, you excite many of these modes at once, and the particular recipe of how loud each one is — the Fourier coefficients — is the sound's timbre.

This integer-multiple ladder is exactly why strings and pipes sound musical and can play in harmony: the overtones of one note line up neatly with the fundamentals of related notes, which is the physical root of consonance and the whole tempered scale. But the neat ladder is special to ideal strings and air columns. A drumhead, a bell, or a stiff piano string has overtones that are NOT simple integer multiples — its higher modes are 'inharmonic', which is precisely why a drum has a less definite pitch and a bell has a complex, shimmering ring.