HesapHanem

Harmonic Frequency Calculator

Find the harmonics of a vibrating string.

How it is calculated

A string fixed at both ends vibrates at the fundamental f1 = v ÷ (2L) and at whole-number multiples of it: fn = n × v ÷ (2L). Here v is the wave speed on the string in m/s (not the speed of sound in air), L is the vibrating length in metres and n is the harmonic number (1 is the fundamental, 2 the second harmonic, and so on). The harmonic number is rounded to a whole number and must be at least 1; speed and length must be above zero. The tool shows both the chosen harmonic and the fundamental, in hertz.

Worked example

A string with wave speed 200 m/s and length 0.65 m. Fundamental: 200 ÷ (2 × 0.65) = 200 ÷ 1.3 = 153.85 Hz. Third harmonic: 3 × 153.846 = 461.54 Hz. The third harmonic is exactly three times the fundamental, an octave and a fifth higher.

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Common mistakes

  • Entering the speed of sound (about 343 m/s) instead of the wave speed along the string, which depends on tension and mass per length.
  • Giving the full string length when the string is stopped by a finger or fret. Use the vibrating length only.
  • Entering the length in centimetres. The field expects metres.
  • Applying the formula to a pipe closed at one end, whose harmonics are odd multiples only and use 4L.

About this calculator

For a string fixed at both ends, the allowed standing-wave frequencies follow fₙ = n·v / (2L), where v is the wave speed on the string, L is the string's length, and n is the harmonic number (1, 2, 3...). The n=1 term is the fundamental frequency (the note you perceive as the pitch); n=2, 3, and beyond are overtones, and together this full harmonic series determines a string's characteristic timbre.

Instrument builders (luthiers) use this relationship when choosing string length, tension, and mass to hit target pitches on guitars, violins, and pianos, musicians use it to understand why fretting or stopping a string shortens it and raises its pitch, and physics students use it as the standard worked example of standing waves and resonance on a bounded medium.

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Comments (1)

  • Editorial team

    On a guitar, the harmonics you play by touching a string lightly are these same numbers. Touching over the 12th fret splits the string in half and sounds the second harmonic, an octave up. For the guide's string, harmonic 2 at 0.65 m gives 307.69 Hz, exactly the fundamental you get by entering half the length, 0.325 m. A touch near the 7th fret isolates the third. Which string or instrument are you analysing?

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