Harmonic series
The ladder of integer-multiple frequencies that a bounded system supports above its fundamental.
Definition
When a string or an open pipe resonates, it doesn't pick one frequency — it picks a set. The lowest is the fundamental, f₁. Above it sit integer multiples: 2·f₁, 3·f₁, 4·f₁ and so on. That sequence is the harmonic series. Which harmonics a given system emphasises is what makes a violin sound different from a flute, even on the same note — that mix is called timbre.
Not every resonator has a full harmonic series. A pipe closed at one end and open at the other supports only odd harmonics: f₁, 3·f₁, 5·f₁, … This is why a clarinet (closed–open) and a flute (open–open) of similar length sound an octave apart and have distinctly different tonal palettes.
History
Pythagoras, around 500 BCE, discovered that consonant musical intervals correspond to simple integer ratios of string length — the first empirical hint of the series. The full physical derivation had to wait until the vibrating-string analyses of Taylor (1713), Bernoulli (1750s) and d'Alembert (1746).