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Vibrating String Harmonic Frequency Calculator

Calculate the frequency of a specific harmonic on a vibrating string, such as a guitar or violin string.

Result

Frequency of Harmonic 1
307.69 Hz
Fundamental Frequency (n=1)
307.69 Hz

About the String Harmonic Calculator

This calculator finds the frequency of any harmonic mode on a vibrating string, whether it's a guitar string, a violin string, or a laboratory demonstration wire, from three quantities: the string's length, the speed at which waves travel along it, and the harmonic number you want to examine. It's built for musicians tuning instruments, physics students working through standing-wave problems, and instrument builders who need to predict pitch from physical string properties.

How It Works

You enter the string length in meters, the wave speed on the string in meters per second (which depends on the string's tension and mass per unit length), and a harmonic number n. The calculator multiplies n by the wave speed and divides by twice the length to produce that harmonic's frequency, and it also reports the fundamental frequency (n = 1) alongside it so you can see how the harmonic relates to the base pitch. Whatever harmonic number you type is rounded to the nearest whole number and floored at 1, since fractional or non-positive harmonics have no physical meaning on a string fixed at both ends.

f_n = n x v / (2 x L), where n is the harmonic number, v is the wave speed on the string, and L is the string length. The fundamental frequency is the n = 1 case, f_1 = v / (2L).

Formula & Methodology

To work this out by hand, find the wave speed on the string first, either measured directly or computed from tension and linear mass density using v = sqrt(T / mu). Divide that wave speed by twice the string length to get the fundamental frequency, then multiply the fundamental by the harmonic number you want. Because the relationship is linear in n, doubling the harmonic number exactly doubles the frequency.

Examples

Fundamental pitch of a guitar-length string

A string 0.65 m long carrying waves at 400 m/s (a reasonable speed for a mid-tension guitar string) has a fundamental frequency of 400 / (2 x 0.65) = 307.7 Hz.

Third harmonic on the same string

Keeping the same 0.65 m length and 400 m/s wave speed but setting the harmonic number to 3 gives 3 x 400 / (2 x 0.65) = 923.1 Hz, exactly three times the fundamental frequency.

Advantages

  • Separates the harmonic number from wave speed and length so you can quickly see how changing tension, string gauge, or fret position shifts a specific pitch.
  • Reports the fundamental alongside the requested harmonic, making it easy to check how far above the base pitch a given overtone sits.
  • Automatically corrects non-integer or invalid harmonic inputs to the nearest valid mode instead of returning a meaningless result.

Common Mistakes

  • Confusing wave speed on the string with the speed of sound in air; they are different values, and using the wrong one produces a frequency unrelated to the actual pitch.
  • Forgetting that the harmonic number must be a positive integer, then being confused when a decimal input silently gets rounded.
  • Using the string's full vibrating length when a finger is fretting or stopping the string partway along it, which shortens the effective length used in the formula.

Edge Cases to Watch For

  • The harmonic number is rounded to the nearest integer and forced to a minimum of 1, so an input of 2.4 becomes the second harmonic and an input of 0 or a negative number still returns the fundamental.
  • String length must be greater than zero; entering zero or a negative length returns an error instead of a frequency.
  • The formula assumes an idealized string fixed rigidly at both ends with negligible stiffness. Real strings, especially thick or metal-wound ones, are slightly stiff, which pushes actual overtones sharp of the calculated values, an effect called inharmonicity that this tool doesn't account for.

Common Use Cases

  • Musicians and luthiers predicting how string length, gauge, or tension changes will affect pitch before restringing or adjusting an instrument.
  • Physics students verifying standing-wave and harmonic series problems from a textbook or lab exercise.
  • Audio engineers and acoustics hobbyists working out overtone frequencies for synthesis, tuning, or room-mode style calculations.
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

Why are only certain frequencies possible on a fixed string?

A string fixed at both ends can only support standing waves where an integer number of half-wavelengths fit exactly between the two fixed ends - these allowed frequencies (fn = nv/2L) are the harmonics, with n=1 being the fundamental (lowest) frequency and higher n values being overtones.

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Conclusion

Because the underlying physics is a direct algebraic relationship between length, wave speed, and harmonic number, this calculator gives an exact answer for the idealized case in seconds. For precise real-world tuning, treat the result as a close first estimate and expect small deviations from string stiffness and other second-order effects.