String Harmonics Calculator
The fundamental frequency and harmonics of a stretched string from its length, tension and linear density — Mersenne's laws — with the wave speed, the wavelengths and what tightening or shortening the string does.
A string fixed at both ends rings at frequencies where a whole number of half-wavelengths fit its length.
How the string harmonics calculator works
A string fixed at both ends rings at frequencies where a whole number of half-wavelengths fit its length. The fundamental is the wave speed over twice the length, and the speed is the square root of tension over mass per unit length. So a string sounds higher when shorter, tighter or lighter — the three things a guitarist controls.
The harmonics are exact multiples of the fundamental, which is why a string sounds like a note and not a noize.
Formula: v = √(T / μ); fₙ = n v / (2L); λₙ = 2L / n
Worked examples
| Inputs | Fundamental frequency (Hz) | Note |
|---|---|---|
| A guitar low E string | 82.329 | about 82 Hz |
| Fretted at the 12th | 164.659 | an octave up |
| A piano A4 | 441.942 | near 440 |
FAQFrequently asked questions
What are Mersenne's laws?
Frequency is inversely proportional to length, proportional to the square root of tension, and inversely proportional to the square root of mass per length. All three follow from f = v / 2L.
Why does the 12th fret give an octave?
It halves the vibrating length, doubling the frequency. Every fret shortens the string by a factor of the twelfth root of two.
Why are thick strings wound?
To add mass without stiffness. A heavier string at the same tension sounds lower, and winding keeps it flexible enough to play.
What are harmonics?
The modes where two, three, four half-wavelengths fit the string — exact multiples of the fundamental. Touching the string lightly at a node isolates one.
How much tension can a string take?
Steel guitar strings run 50 to 100 N each; a piano's run 700 N and more, which is why a piano frame is cast iron. The octave-up row shows how fast tension climbs.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Institute of Standards and Technology — the US measurement authority
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.