Natural frequency
The natural frequency is the frequency at which a body capable of vibrating vibrates by itself when struck. For a string it depends on length, tension and mass, for a tuning fork on the shape of its prongs. It decides which note the body gives off.
- Meaning
- lowest frequency at which a body vibrates freely
- Also
- fundamental frequency, resonant frequency, fundamental
- String
- double the length → half the frequency
- Example
- open A string of the guitar: 110 Hz
What lets a body give “its” note
Every body that is struck and then vibrates on freely does so at a certain speed. A plucked string, a struck tuning fork, a bell, an air column in a flute: all have frequencies that their construction fixes, and the lowest of them is the natural frequency. It does not depend on how strongly you strike. Plucked harder, the string sounds louder, not higher.
One string, four lengths (distances in semitones)
The open A string (A, 110 Hz) and three fingerings on the same string: at the 5th fret D (146.83 Hz), at the 7th fret E (164.81 Hz), at the 12th fret A again, an octave higher at 220 Hz. The note heights show the sound, not the written guitar notation, which is an octave higher.
A string as a worked example
For an ideally vibrating string the frequency falls with length, rises with tension and falls with the mass per length; more precisely: rising with the square root of the tension, falling with the square root of the mass per length. The laws go back to Marin Mersenne, who described them in the 17th century. The rows of the table follow from them.
| Change to the string | Effect on the natural frequency |
|---|---|
| Halve the length (12th fret) | frequency doubles, an octave higher |
| Shorten the length to two thirds (7th fret, roughly) | frequency ×1.5, a perfect fifth higher |
| Quadruple the tension | frequency doubles |
| Quadruple the mass per length (thicker string) | frequency halves |
The guitar is a good laboratory for this. The bridge stays fixed, only the vibrating length changes when you fret a note. Each fret shortens the string so that the frequency rises by the factor of the twelfth root of two, about 1.0595. After twelve frets the length is half and the note exactly an octave higher. At the 7th fret there are seven semitones, the factor 1.498, so almost exactly a perfect fifth (3:2).
Natural frequency, overtone, resonance
A string vibrates not only at its natural frequency but at the same time in parts: in halves, thirds and so on. These are the overtones, more under Harmonic series. The natural frequency is the lowest partial and as a rule decides the perceived pitch. If another body is made to vibrate along by sound from outside, one speaks of resonance. It is strongest when the excitation matches the natural frequency.
Common mistakes
- Confusing natural frequency with loudness: a stronger attack makes the note louder, not higher.
- Taking two thirds of the length for a third of the frequency: the shorter the string, the higher the note. Two thirds of the length gives one and a half times the frequency.
- Equating natural frequency and partial: the natural frequency is the lowest partial; the overtones come in addition.
Continue on Maikstrings
Related terms
VibrationResonanceTuning forkHarmonic seriesPartialStandard pitchPitchStringOctave
Editor: Maik Krohm · reviewed on 3 October 2026
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