Overtone Harmonic Series Calculator

Harmonic series and overtones: formula and example

On an ideal vibrating string, or an open pipe, the partials land on integer multiples of the fundamental: f_n = n · f0. Take f0 = 220 Hz (A3). Then n=1 → 220, n=2 → 440 (A4), n=3 → 660 (E5), n=4 → 880 (A5), n=5 → 1100 (≈ C♯6), n=6 → 1320 (E6). Everything above the fundamental, the overtones with n ≥ 2, makes up the spectral content your ear reads as timbre.

Context and applications

Helmholtz worked out that timbre is the combination of harmonics. A clarinet stops the air at one end, so it reinforces odd harmonics only (1, 3, 5, 7…), which is where its hollow tone comes from. A trumpet (closed-open, with a bell) plays all harmonics. And a guitarist who rests a finger lightly over the 12th, 7th or 5th fret isolates the 2nd, 3rd or 4th harmonic, the natural harmonics. Brass valves and bugle calls work the same way, navigating the harmonic series of one pipe length.

FAQ

Are overtones the same as harmonics? Not quite. Harmonics are the integer multiples of f0. Overtones are any partials sitting above the fundamental. On ideal strings or pipes the two line up, but real instruments like the piano or bells show inharmonicity.

Why does the 7th harmonic sound out of tune? It falls about 31 cents flatter than the equal-tempered minor seventh. That clashes with tempered scales, though it does show up in barbershop singing and just intonation.

How many harmonics are audible? Hearing tops out around 20 kHz, so with f0 = 220 Hz you can fit roughly 90 harmonics before they drop out. In practice the amplitude usually fades away past the first 10–15.