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 — essential reading for synthesis programmers, acoustic designers, and anyone trying to understand why instruments sound the way they do.
How it works
The harmonic series is defined by a single rule:
harmonic n = fundamental x nSo a 110 Hz fundamental (A2) produces 110, 220, 330, 440, 550 Hz and so on. The first harmonic is the fundamental itself. This isn’t an approximation — it follows directly from the physics of a vibrating string or air column, where resonant modes are exact integer multiples of the fundamental mode.
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 — a tiny difference. The 5th harmonic (a major third two octaves up) is about -14 cents — noticeably flatter than the tempered third, which is why choir singers instinctively lower their thirds to achieve beatless chords.
The 7th harmonic is about -31 cents flat of the nearest piano key; this “blue” note sits so far between keys that it inspired entire genres. The 11th harmonic falls about +49 cents sharp, placing it almost exactly between two notes — equally close to two piano keys — and is central to the tritone substitution idea in jazz theory.
Worked example
For a fundamental of 220 Hz (A3), here are the first six harmonics and what they mean musically:
| Harmonic | Frequency | Nearest note | Cents offset | Interval name |
|---|---|---|---|---|
| 1 | 220 Hz | A3 | 0 | fundamental |
| 2 | 440 Hz | A4 | 0 | octave |
| 3 | 660 Hz | E5 | +2 | perfect fifth (+ octave) |
| 4 | 880 Hz | A5 | 0 | double octave |
| 5 | 1100 Hz | C#6 | −14 | major third (+ 2 octaves) |
| 6 | 1320 Hz | E6 | +2 | perfect fifth (+ 2 octaves) |
Stacking harmonics 4, 5, and 6 gives the frequencies of a pure major chord (C#–E–A) with the fifth slightly raised and the third slightly lowered compared to an equal-tempered piano. This is the acoustic origin of the major triad and explains why barbershop quartets can “lock” chords in a way that pianos cannot.
Practical uses
- Additive synthesis: knowing which harmonics dominate a brass or string timbre tells you which partials to emphasise to match the sound.
- Acoustic tuning: knowing that the 7th partial sits -31 cents flat helps tuners of bells, organs and choir singers identify “mistuned” partials by ear.
- String and wind instrument design: air column and string resonances follow the harmonic series; deviations from it (inharmonicity) define the character of a piano.
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