Harmonic Series Calculator
The harmonic series is the simplest thing in music and the reason tuning is complicated. Every partial is a whole-number multiple of the fundamental — two, three, four times and so on — and almost none of them lands on a note a piano can play. The seventh is about 31 cents flat, which is why a natural seventh rings beautifully on a brass instrument and sounds wrong on a keyboard.
What this generator does
Multiplies the fundamental by each whole number in turn, finds the nearest equal-tempered note to each result, and measures the gap in cents — a hundredth of a semitone.
How to use this tool
- Set a fundamental frequency and its note name.
- Read down the partials — each is simply the fundamental times its number.
- Look at the cents column: the octaves are exact and nothing else is.
- Extend past the seventh partial to meet the worst offender.
Understanding the controls
- Fundamental
- The lowest frequency of the series, between 20 and 2000 Hz. 110 Hz is a low A on a guitar; 440 is the A most orchestras tune to.
- Its note name
- Used to name the partials. It does not affect the frequencies, which come from the fundamental alone.
- How many partials
- Between 4 and 16. The seventh is the famous one, so it is worth going at least that far.
Common use cases
- Understanding why equal temperament is a compromise
- Working out the partials available on a natural horn or bugle
- Teaching overtones with real frequencies rather than a diagram
- Checking which harmonics will clash with a tempered accompaniment
- Sound design and additive synthesis reference
How this generator works
Cents are a logarithmic measure — twelve hundred to an octave — so the distance from a partial to the nearest tempered note is worked out by taking a base-two logarithm of the frequency ratio. The check confirms every partial is an exact whole multiple, that the octaves come out at zero cents, and that no partial is reported more than half a semitone from its nearest note, which is impossible by definition.
Randomness and fairness
Nothing here is random at all. The harmonic series above a given fundamental is fixed by physics, so the same inputs always produce the same series.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Sharps only in the note names, so partials are named without regard to how they would be spelled in a key.
- The nearest tempered note is taken by rounding, which is ambiguous exactly at the halfway point.
- Real instruments have partials that are not perfectly harmonic — piano strings in particular stretch sharp with height.
- Amplitude is not modelled, and it is amplitude rather than frequency that gives an instrument its character.
Privacy and your data
The series is computed in your browser. The frequency you enter is never transmitted or stored.
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