When a string or an air column vibrates it does not produce one frequency only. Alongside the fundamental f₀ it vibrates in halves, thirds, quarters and so on, adding frequencies 2f₀, 3f₀, 4f₀ … – the harmonic series. The first partial is the fundamental, the higher ones are called overtones, and their relative strengths give every instrument its timbre.
The intervals between partials shrink as you go up: an octave (2:1), a perfect fifth (3:2), a perfect fourth (4:3), a major third (5:4) and so on. These simple ratios are the basis of just intonation. Equal temperament, used on pianos, divides the octave into twelve identical semitones instead, so most partials do not match a piano key exactly: the 5th partial is 13.7 cents flat of the equal-tempered major third, the 7th partial is 31.2 cents flat of the minor seventh and the 11th falls almost exactly between two keys.
The calculator shows those differences for up to 64 partials. Brass players, guitarists playing natural harmonics, organ and synthesiser designers and singers practising overtone singing all use these numbers.
How to use the Harmonic Series Calculator
- Note
- Octave
- Number of partials
- Reference pitch A4
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How it works
The fundamental frequency is found from the note and octave with your reference pitch (A4 = 440 Hz by default). Each partial is k × f₀; it is converted to a fractional MIDI number, rounded to the nearest note, and the remainder is expressed in cents (hundredths of a semitone).
Worked example
For C4 (261.626 Hz with A4 = 440 Hz) the partials are 261.63, 523.25, 784.88, 1,046.50, 1,308.13, 1,569.75, 1,831.38 and 2,093.00 Hz. The 3rd partial (784.88 Hz) is G5 + 2.0 cents, the 5th (1,308.13 Hz) is E6 − 13.7 cents and the 7th (1,831.38 Hz) is B♭6 − 31.2 cents – the only one of the first eight that misses by more than 30 cents.
The first 16 partials
| Partial | Interval above the fundamental | Ratio | Cents from equal temperament |
|---|---|---|---|
| 1 | Unison | 1:1 | 0 |
| 2 | Octave | 2:1 | 0 |
| 3 | Octave + perfect fifth | 3:1 | +2.0 |
| 4 | Two octaves | 4:1 | 0 |
| 5 | 2 octaves + major third | 5:1 | −13.7 |
| 6 | 2 octaves + fifth | 6:1 | +2.0 |
| 7 | 2 octaves + minor seventh | 7:1 | −31.2 |
| 8 | Three octaves | 8:1 | 0 |
| 9 | 3 octaves + major second | 9:1 | +3.9 |
| 10 | 3 octaves + major third | 10:1 | −13.7 |
| 11 | 3 octaves + tritone | 11:1 | −48.7 |
| 12 | 3 octaves + fifth | 12:1 | +2.0 |
| 13 | 3 octaves + minor sixth | 13:1 | +40.5 |
| 14 | 3 octaves + minor seventh | 14:1 | −31.2 |
| 15 | 3 octaves + major seventh | 15:1 | −11.7 |
| 16 | Four octaves | 16:1 | 0 |
Formula
Partial k: fₖ = k × f₀ (k = 1, 2, 3 …)
f₀ = A4 × 2^((MIDI − 69) ÷ 12)
Nearest note = round(69 + 12 × log₂(fₖ ÷ A4)) · cents = 100 × (exact − nearest)
Worked example
Inputs — Note: C; Octave: 4 (one-line); Number of partials: 8; Reference pitch A4: 440 Hz.
Results — Fundamental frequency = 261.626 Hz; Fundamental = C4 = 261.626 Hz; Highest partial = 2,093 Hz; Out-of-tune partials = Partials more than 30 cents away from equal temperament: 7; Harmonic series = Partial Freq (Hz) Nearest Cents Above fundamental
─────────────────────────────────────────────────────────────────
1 261.63 C4 +0 Perfect unison
2 523.25 C5 +0 Perfect octave
3 784.88 G5 +2 Perfect twelfth
4 1,046.5 C6 +0 Double octave
5 1,308.13 E6 −13.7 2 octaves + major third
6 1,569.75 G6 +2 2 octaves + perfect fifth
7 1,831.38 B♭6 −31.2 2 octaves + minor seventh
8 2,093 C7 +0 3 octaves.
Key terms
- Fundamental
- The lowest frequency of a sound, partial 1.
- Partial
- Any frequency component; in the harmonic series partial k = k × f₀.
- Overtone
- Any partial above the fundamental.
- Cent
- One hundredth of an equal-tempered semitone; 1,200 cents per octave.
- Just intonation
- Tuning based on simple whole-number frequency ratios.
Frequently asked questions
What is the harmonic series of C2?
65.41, 130.81, 196.22, 261.63, 327.03, 392.44, 457.85, 523.25 Hz … – the notes C2, C3, G3, C4, E4, G4, B♭4 (flat), C5.
What is the difference between a harmonic and an overtone?
The first harmonic is the fundamental; the first overtone is the second harmonic. Partial counts like harmonics.
Why is the 7th harmonic out of tune?
Its ratio 7:4 lies 31.2 cents below the equal-tempered minor seventh, so on a piano it sounds flat.
How far is the 5th harmonic from a piano’s major third?
13.7 cents flat: the just major third 5:4 is 386.3 cents, the equal-tempered one 400 cents.
Which partials sound most dissonant against equal temperament?
The 7th, 11th, 13th and 14th partials – they miss the nearest piano note by 30–50 cents.
Where are natural harmonics on a guitar?
At 1/2 (12th fret), 1/3 (7th fret), 1/4 (5th fret), 1/5 (about the 4th fret) of the string – giving the 2nd, 3rd, 4th and 5th partials.
Do real instruments follow the series exactly?
Wind instruments closely; stiff strings such as piano strings have slightly sharp overtones (inharmonicity).