Fourier's theorem decomposes any periodic sound into sinusoidal partials, and their amplitudes fix its timbre

Fourier’s theorem states that any perfectly periodic signal can be expressed as a weighted sum of sinusoids, each with a frequency, amplitude, and phase. For a periodic sound with fundamental frequency f0, these components fall at integer multiples f0, 2f0, 3f0… — the harmonic series (the fundamental is the first harmonic). Overtones and harmonics are numbered differently: the second harmonic is the first overtone. Not every harmonic need be present; the set of partial frequencies and their amplitudes is the sound’s spectrum, and this spectrum is the primary determinant of timbre — two instruments at the same pitch and loudness differ mainly in their spectra. Inharmonic sounds have partials off the integer grid. The ear itself acts as a frequency analyzer, splitting incoming sound into components on the basilar membrane, a biological Fourier transform. Amplitude envelope also shapes timbre, but steady-state timbre is chiefly spectral. For non-periodic sounds (transients, noise) Fourier analysis still measures energy per frequency over a time window — the Short-Time Fourier Transform and spectrogram.

Examples

A sawtooth contains all integer harmonics; a square wave contains only odd harmonics; the clarinet’s chalumeau register favours odd harmonics, giving a hollow tone. A violin and a flute both on A4 (440 Hz) share the fundamental but differ in spectrum — the flute is near-sinusoidal, the violin rich in harmonics.

Assessment

Given a 200 Hz periodic tone, list its first four harmonic frequencies and state the difference between overtone and harmonic numbering. Given two spectra, predict which sounds brighter and why.