Harmonics

Two instruments can play the same note and still sound different, because their harmonics have different strengths. The strength of each harmonic is the main thing that sets an instrument's timbre, along with the attack, formants and noise. A clarinet's cylindrical bore weakens the even harmonics, while a saxophone's conical bore lets them sound.

The basic waveforms show it plainly. A sine has only the fundamental. A square wave adds the odd harmonics, each at 1 divided by its number, so its 3rd harmonic has one-third the amplitude of the fundamental. A sawtooth has every harmonic. That's why a square sounds hollow and buzzy and a sawtooth sounds bright and full.

Your ear can even fill in a missing fundamental. Two tones at 1000 Hz and 1200 Hz can give the pitch of a 200 Hz tone, even though nothing plays at that frequency.

An overtone is any resonant frequency above the fundamental, so the 1st overtone is the 2nd harmonic. That off by one count is why the ANSI/ASA S1.1-2013 standard deprecates the word overtone in favor of harmonic.

Overtones don't have to be harmonics. A timpani's first overtone isn't a whole number multiple of its fundamental, so it counts as an overtone but not a harmonic. The general word for any component of a sound is a partial. Harmonics are the partials that sit on the whole number pattern.

Mains hum is a harmonic story. Its fundamental is twice the line frequency, 100 Hz on 50 Hz power and 120 Hz on 60 Hz power. A stack of harmonics sits above that. If a hum sits at 100 or 120 Hz, you're hearing the 2nd harmonic of the power line.

Speakers and amplifiers add harmonics of their own. A device that isn't perfectly linear puts extra content at whole number multiples of whatever you feed it, and total harmonic distortion compares the power of those extra harmonics with the original tone. Feed it a clean sine, and anything extra you hear came from the device.