Harmonic Series Calculator
Enter a fundamental note (or type a frequency directly) and choose how many partials to generate. The calculator produces the complete harmonic series: the frequency of each overtone, the nearest equal-temperament note name, and the deviation in cents from that note. Results update as you type.
What is the harmonic series?
The harmonic series is the sequence of frequencies that are whole-number multiples of a fundamental frequency (f0). If the fundamental vibrates at 110 Hz, the second harmonic is 220 Hz, the third is 330 Hz, and so on. Every acoustic instrument and voice produces a blend of these partials simultaneously, and it is the relative loudness of each partial - not the fundamental alone - that gives an instrument its characteristic timbre. A flute suppresses the odd harmonics above the first, giving it a pure tone; a violin bow excites them vigorously, creating a rich, complex sound.
Partials, overtones and harmonics - what is the difference?
These three words are often used interchangeably but have precise meanings. A "partial" is any frequency component in a complex tone, numbered from 1. A "harmonic" is a partial whose frequency is an exact integer multiple of the fundamental; most real-world pitched sounds are harmonic. An "overtone" is any partial above the fundamental, so the 2nd partial is the 1st overtone, the 3rd partial is the 2nd overtone, and so on. Idiophones such as bells and bars produce inharmonic partials that are not exact integer multiples, which is why a struck bell sounds complex and pitched differently from a blown flute.
Equal temperament versus just intonation
Equal temperament divides the octave into 12 equal semitones of 100 cents each. The harmonic series does not follow this grid. The 5th harmonic is a pure major third that sits 14 cents flat of the equal-temperament major third; the 7th harmonic is a natural minor seventh that sits 31 cents flat of the equal-temperament minor seventh - this is why it sounds "blue". The 3rd harmonic is a perfect fifth only 2 cents sharp of the equal-temperament fifth, which is why the fifth sounds so consonant: it is almost, but not exactly, in tune with 12-tone equal temperament. Keyboard instruments tuned to equal temperament compromise on every interval except the octave; a string quartet in tune with each other is actually playing close to just intonation.
How to read the partials table
Each row in the partials table shows the partial number (n), the frequency in Hz (n x f0), the nearest equal-temperament note name (including octave register, e.g. A5), and the cents deviation from that note. A deviation of 0 cents means the harmonic falls exactly on an equal-temperament pitch; a deviation of -14 cents means it is 14 cents flat; +2 cents means slightly sharp. Composers writing for brass instruments use the natural harmonic series of the instrument and exploit these deviations expressively. Audio engineers use this table to set equalizer notch frequencies precisely when suppressing a feedback harmonic.
First 16 harmonics relative to a fundamental
| Partial (n) | Ratio to f0 | Musical interval | Cents from ET |
|---|---|---|---|
| 1 | 1:1 | Fundamental (unison) | 0 |
| 2 | 2:1 | Octave | 0 |
| 3 | 3:1 | Octave + perfect 5th | +2 |
| 4 | 4:1 | 2nd octave | 0 |
| 5 | 5:1 | 2nd octave + major 3rd | -14 |
| 6 | 6:1 | 2nd octave + perfect 5th | +2 |
| 7 | 7:1 | 2nd octave + minor 7th (blue note) | -31 |
| 8 | 8:1 | 3rd octave | 0 |
| 9 | 9:1 | 3rd octave + major 2nd | +4 |
| 10 | 10:1 | 3rd octave + major 3rd | -14 |
| 11 | 11:1 | 3rd octave + augmented 4th | -49 |
| 12 | 12:1 | 3rd octave + perfect 5th | +2 |
| 13 | 13:1 | 3rd octave + major 6th | +41 |
| 14 | 14:1 | 3rd octave + minor 7th | -31 |
| 15 | 15:1 | 3rd octave + major 7th | -12 |
| 16 | 16:1 | 4th octave | 0 |
Interval names and cent deviations from equal temperament for a harmonic series built on any fundamental. Deviations repeat identically regardless of the pitch of f0.
Frequently asked questions
What is the formula for the harmonic series?
The frequency of the nth harmonic is fn = n x f0, where f0 is the fundamental frequency and n is any positive integer (1, 2, 3, ...). The first harmonic (n=1) is the fundamental itself; the second harmonic (n=2) is one octave above; the third harmonic (n=3) is an octave and a fifth above, and so on.
What is A4 and why is 440 Hz the standard?
A4 refers to the note A in the fourth octave (one-line octave), and 440 Hz is the internationally agreed concert pitch standard (ISO 16). Before standardisation, instruments across Europe used pitches ranging from around 415 Hz to 466 Hz. Modern orchestras all tune to A440 so that instruments from different manufacturers play in tune together. Some early-music ensembles use 415 Hz (baroque pitch), approximately one semitone lower.
What are cents and how are they measured?
A cent is 1/100 of a semitone and 1/1200 of an octave. Because the ear perceives pitch logarithmically, cents give a perceptually uniform unit of measurement. The formula is: cents = 1200 x log2(f2 / f1). A deviation of less than about 5 cents is generally inaudible to most listeners; above 10-15 cents the interval begins to sound out of tune.
Why does the 7th harmonic sound like a "blue note"?
The 7th harmonic (7 x f0) falls 31 cents below the equal-temperament minor seventh. This gives it a strongly expressive, slightly flat quality that does not fit neatly into Western tonal harmony. Blues and jazz musicians - especially horn players - exploit this natural harmonic by bending notes toward it, producing the characteristic "blue" feeling. Barbershop quartets deliberately tune their dominant seventh chords to the natural 7th harmonic for a sweeter resonance.
How does the harmonic series relate to timbre?
Timbre (or tone colour) is determined by which harmonics are present and how loud they are relative to the fundamental. Two instruments playing the same note (same f0) have the same pitch but different timbres because they excite different harmonic profiles. A clarinet, for physical reasons, suppresses even-numbered harmonics and emphasises odd ones, giving its characteristic hollow sound. A trumpet builds up harmonics evenly to very high partials, giving its bright, penetrating quality. Audio synthesisers recreate any timbre by adding sine waves at the harmonic frequencies in chosen proportions.