The Harmonic Series
The harmonic series contains frequencies at whole-number multiples of a fundamental: f, 2f, 3f, 4f and onward. Many pitched instruments approximate this pattern, and the balance of their partials contributes strongly to timbre.
- harmonic n = n times f
- fundamental is harmonic 1
- real spectra vary
Where you run into this
- Acoustic instruments
- Synthesis
- Tuning
- Mixing
Count harmonics and overtones
For a 100 Hz fundamental:
| Harmonic | Frequency | Approximate interval above fundamental |
|---|---|---|
| 1 | 100 Hz | unison, fundamental |
| 2 | 200 Hz | octave |
| 3 | 300 Hz | octave plus perfect fifth |
| 4 | 400 Hz | two octaves |
| 5 | 500 Hz | two octaves plus natural major third |
| 6 | 600 Hz | two octaves plus perfect fifth |
| 7 | 700 Hz | natural minor seventh, lower than equal temperament |
The first overtone is harmonic 2. Harmonic and overtone numbering therefore differ by one.
Why strings produce the pattern
A string fixed at both ends supports standing-wave modes that fit its length. The whole string gives the fundamental. Halves support harmonic 2, thirds harmonic 3 and so on.
Air columns also support modes, but boundary conditions matter. Open and closed pipes emphasise different sets of resonances. Real instruments add stiffness, radiation, body resonances and playing technique.
Harmonics are not always progressively quieter
A simple model may reduce upper partials smoothly. Real instruments do not follow one universal amplitude order. A resonance can strengthen harmonic 3 or 5 above a lower neighbour, and the balance changes during attack and decay.
Timbre depends on that changing spectral envelope, not merely on how many harmonics exist.
Natural intervals and equal temperament
Harmonics 4, 5 and 6 form frequency ratios 4:5:6, resembling a major triad. The natural major third 5:4 is about 14 cents narrower than an equal-tempered major third.
Harmonic 7 is about 31 cents below the equal-tempered minor seventh. Harmonic 11 lies between familiar equal-tempered notes. The series is therefore not a hidden equal-tempered keyboard.
Harmonic and inharmonic partials
A partial is any sinusoidal component of a complex sound. A harmonic partial sits at an integer multiple of the fundamental. An inharmonic partial does not.
Bells, cymbals and membranes often have strongly inharmonic modes. Piano-string stiffness also pushes partials slightly sharp, leading tuners to stretch octaves across the instrument.
What the series explains, and what it does not
Shared or closely aligned partials can contribute to fusion and low roughness for simple intervals. The octave 2:1 and fifth 3:2 create many coincidences in ideal harmonic spectra.
Musical consonance also depends on register, spectrum, tuning, culture, familiarity and context. The harmonic series is part of the explanation, not a complete theory of why an interval sounds stable.
Build and compare
- Fundamental onlyhear the sine-like reference.
- Add harmonic 2notice octave reinforcement.
- Add harmonic 3hear the fifth-related colour.
- Add harmonics 4 to 8track brightness and edge.
- Remove the fundamentaltest whether residue pitch still suggests the missing root.