What is the harmonic series?

The harmonic series is the set of frequencies that are whole-number multiples of a fundamental. If the fundamental is 100 Hz the harmonics are 200, 300, 400 Hz and so on. Almost every pitched acoustic sound is a blend of these overtones, and their relative loudness defines its timbre.

How do I calculate the nth harmonic?

The nth harmonic frequency is simply the fundamental times n. The 5th harmonic of 220 Hz is 220 times 5, which is 1100 Hz. The fundamental itself is the 1st harmonic.

Why do some harmonics not line up with piano notes?

Equal temperament divides the octave into 12 equal steps, but the harmonic series follows pure whole-number ratios. The 7th and 11th harmonics in particular fall well between piano keys, the 7th about 31 cents flat and the 11th about 49 cents sharp of the nearest equal-tempered note.

A cent is one hundredth of an equal-tempered semitone, so 1200 cents make an octave. Cents measure how far a frequency sits from a reference pitch. Differences under about 5 cents are hard to hear; differences over 20 cents are clearly out of tune.

How does the harmonic series relate to musical intervals?

The early harmonics spell out familiar intervals: harmonic 2 is an octave above the fundamental, harmonic 3 a perfect twelfth (octave plus fifth), harmonic 5 a major third two octaves up, and so on. Just intonation builds scales directly from these pure ratios.

Harmonic Series Calculator

Every pitched sound you hear is a stack of overtones sitting above a fundamental at whole-number multiples of its frequency. This calculator lists the first 16 harmonics of any note, gives their exact frequencies, and shows how far each one strays from the equal-tempered piano.

How it works

The harmonic series is defined by a single rule:

harmonic n = fundamental x n

So a 110 Hz fundamental (A2) produces 110, 220, 330, 440, 550 Hz and so on. The first harmonic is the fundamental itself.

Nearest note and cents deviation

To find the closest equal-tempered note the tool converts each harmonic frequency to a position on the chromatic scale relative to A4 = 440 Hz:

semitones from A4 = 12 x log2(freq / 440)

It rounds to the nearest semitone for the note name, then measures the leftover as cents:

cents = 100 x (exact semitones - rounded semitones)

A positive value means the harmonic is sharp of the piano note; negative means flat.

What the deviations tell you

Harmonics 1, 2, 4, 8, and 16 are pure octaves and land exactly on the note (0 cents). The 3rd and 6th (fifths) sit about +2 cents. The 5th (major third) is about -14 cents — noticeably flatter than the tempered third, which is why pure thirds sound sweeter. The 7th is about -31 cents and the 11th about +49 cents, both far enough from any key to sound exotic.

Worked example

For a fundamental of 220 Hz (A3), the 3rd harmonic is 660 Hz, nearest note E5, about +2 cents — a pure perfect fifth two octaves up. The 5th harmonic is 1100 Hz, nearest note C#6, about -14 cents — the natural major third. Stacking these pure ratios is the basis of just intonation.

All maths runs locally in your browser.