Harmonic Series Calculator
List the overtones of a fundamental frequency and the interval each harmonic adds.
The 2nd harmonic is 220 Hz — one octave above the fundamental.
| Harmonic (n) | Frequency | Interval above fundamental |
|---|---|---|
| 1 | 110 Hz | Fundamental (unison) |
| 2 | 220 Hz | Octave |
| 3 | 330 Hz | Octave + perfect fifth |
| 4 | 440 Hz | 2 octaves |
| 5 | 550 Hz | 2 octaves + major third |
| 6 | 660 Hz | 2 octaves + perfect fifth |
| 7 | 770 Hz | 2 octaves + minor seventh (flat) |
| 8 | 880 Hz | 3 octaves |
| 9 | 990 Hz | 3 octaves + major second |
| 10 | 1100 Hz | 3 octaves + major third |
| 11 | 1210 Hz | 3 octaves + tritone (very flat) |
| 12 | 1320 Hz | 3 octaves + perfect fifth |
The harmonic series is the set of overtones a vibrating string or air column produces: whole-number multiples of the fundamental. Each harmonic is fₙ = n × f₀. So a 110 Hz fundamental gives harmonics at 220, 330, 440, 550 and 660 Hz — the 2nd is one octave up, the 3rd an octave plus a fifth.
What the harmonic series is
When a guitar string or a column of air vibrates, it does not move at just one frequency. It vibrates at its full length and, at the same time, in halves, thirds, quarters and so on. Each of those divisions sounds a pitch that is a whole-number multiple of the lowest one, the fundamental. Together they form the harmonic series: f₀, 2f₀, 3f₀, 4f₀, and upward. The fundamental sets the note you hear as the pitch; the higher harmonics blend into it and shape its tone colour.
fₙ is the nth harmonic, f₀ the fundamental frequency, and n = 1, 2, 3, … a positive whole number
Worked example
Take a fundamental of 110 Hz (the open A string on a bass), and list its first few harmonics.
- 1 Start with the fundamental. The 1st harmonic is the fundamental itself: f₁ = 1 × 110 = 110 Hz.
- 2 Double it for the 2nd harmonic. f₂ = 2 × 110 = 220 Hz, exactly one octave above the fundamental.
- 3 Multiply by 3 for the 3rd. f₃ = 3 × 110 = 330 Hz — an octave plus a perfect fifth above f₀.
- 4 Keep multiplying by each whole number. 4 × 110 = 440 Hz (two octaves), 5 × 110 = 550 Hz, 6 × 110 = 660 Hz, and so on for every n.
Interval each harmonic adds
The musical interval above the fundamental that each harmonic approximates, for n = 1 to 8. Higher octaves repeat the same ladder of intervals.
| Harmonic (n) | Ratio to f₀ | Interval above fundamental |
|---|---|---|
| 1 | 1 : 1 | Fundamental (unison) |
| 2 | 2 : 1 | Octave |
| 3 | 3 : 1 | Octave + perfect fifth |
| 4 | 4 : 1 | 2 octaves |
| 5 | 5 : 1 | 2 octaves + major third |
| 6 | 6 : 1 | 2 octaves + perfect fifth |
| 7 | 7 : 1 | 2 octaves + minor seventh (flat) |
| 8 | 8 : 1 | 3 octaves |
Why harmonics matter
Timbre comes from the overtones. Two instruments playing the same note share the same fundamental, so they sound at the same pitch. What tells a flute from a violin is the relative strength of the harmonics above that fundamental — a flute leans on the low ones for a pure tone, while a violin’s richer mix of upper harmonics gives it a brighter, edgier colour. The pattern of overtone amplitudes is, in effect, an instrument’s fingerprint.
The 7th harmonic sits between the cracks. Most harmonics land close to notes of the equal-tempered scale, but the 7th does not. Its pure 7 : 1 ratio falls about 31 cents below the equal-tempered minor seventh, so it sounds noticeably flat against a piano. Barbershop and brass players sometimes lean into this “harmonic seventh” on purpose for a sweeter dominant chord; on a keyboard it would sound out of tune.