How this calculator works
A standing wave can divide a string or air column into any whole number of equal parts. The frequency of each mode is that whole number times the fundamental, so a 440 Hz fundamental has partials at 880, 1320, 1760 and onward. The second partial is an octave, the third is twelve semitones plus seven, close to a perfect fifth above the octave, the fourth is two octaves, and the fifth is a major third above that. The page compares each partial with the nearest equal-tempered note and reports the gap in cents. The third partial sits about two cents sharp and the fifth about fourteen cents flat, which is why chords sound different on real instruments than on a pure equal-tempered synthesizer.
Formula
partial n frequency = n x fundamental frequency. The nearest note comes from the MIDI number 69 + 12 x log2(f/440), rounded. Cents from that note = 100 x (exact MIDI value - nearest MIDI value).
Worked example
For A4 at 440 Hz the partials are 440, 880, 1320, 1760 and so on. The third partial at 1320 Hz is 12 x log2(3) = 19.02 semitones above A4. The nearest note is E6 at 1318.51 Hz, so the partial is 1.96 cents sharp. The fifth partial at 2200 Hz is 28.02 semitones above A4; the nearest note is C#7, and the gap is about 14 cents flat.
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Frequently asked questions
Why do the upper partials sound out of tune?
The harmonic series uses exact whole-number ratios, while equal temperament divides the octave into twelve equal steps. The two systems agree on octaves and nearly on fifths, but the major third and minor seventh differ by 14 and 31 cents. Instruments with strong upper partials therefore sound richer and slightly edgier than pure sine tones.
What is a partial and what is a harmonic?
The terms overlap. A harmonic is any partial whose frequency is a whole-number multiple of the fundamental. Some instruments, especially bells and struck strings, also produce inharmonic partials that are not whole multiples; those are partials but not harmonics.
Do real strings follow the whole-number pattern exactly?
Not quite. Stiffness makes the upper partials slightly sharp, which is called inharmonicity, and piano tuners stretch the tuning to match. Wind instruments shift their partials with the shape of the bore and the player's embouchure.
Why does the third partial sound like a fifth?
A 3:2 frequency ratio is a perfect fifth, and the third partial is three times the fundamental, which is a fifth above the second partial, an octave up. The fifth is one of the closest matches between the harmonic series and equal temperament, about two cents apart.