Music · Acoustics
Harmonic series calculator.
List the harmonics of any fundamental frequency with the nearest equal-tempered note and cents offset for each, to see why the 7th, 11th and 13th harmonics sound out of tune.
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How do you calculate the harmonic series?
The nth harmonic of a fundamental frequency f has frequency n × f. The fundamental is the first harmonic, so the first overtone is the second harmonic. This tool lists each harmonic and finds the nearest twelve-tone equal-tempered note using the MIDI relation 69 + 12 × log2(frequency / A4), then reports how many cents the harmonic sits above or below that note. The distance above the fundamental is 1200 × log2(n) cents.
Worked example. With a fundamental of 110 Hz (A2): harmonic 2 is 220 Hz (A3), harmonic 3 is 330 Hz, an E4 about 2 cents sharp, harmonic 4 is 440 Hz (A4) and harmonic 5 is 550 Hz, a C♯5 about 14 cents flat. The 7th harmonic (770 Hz) lands about 31 cents flat of G5, and the 11th and 13th are the other harmonics that differ most from equal temperament.
Frequently asked questions
What is the harmonic series?
The set of frequencies that are whole-number multiples of a fundamental: f, 2f, 3f, 4f and so on. Musical tones from strings and pipes contain many of them.
What is the difference between a harmonic and an overtone?
The fundamental is the first harmonic but not an overtone. The first overtone is the second harmonic, so overtone numbers are one lower than harmonic numbers.
Which harmonics are out of tune with equal temperament?
The 7th, 11th and 13th differ most, by roughly -31, -49 and +41 cents.
What are the first harmonics of A2?
With A2 at 110 Hz: 110, 220, 330, 440 and 550 Hz, which are roughly A2, A3, E4, A4 and C#5.
How many cents is an octave?
1,200. The second harmonic is always exactly 1,200 cents above the fundamental.
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