Guitar String Vibration Frequency Calculator

Vibrating string frequency: formula and example

For an ideal vibrating string, Mersenne's law puts the fundamental at f = (1 / 2L) · √(T / μ). Here L is the speaking length (m), T the tension (N), and μ the linear mass density (kg/m). On a 0.65 m scale in standard guitar tuning you get E2 = 82.4 Hz, A2 = 110 Hz, D3 = 146.83 Hz, G3 = 196 Hz, B3 = 246.94 Hz, and E4 = 329.63 Hz. If you double the tension the pitch climbs by √2, about 7 semitones; cut the length in half and the frequency doubles, which is a full octave.

Context and applications

Luthiers lean on Mersenne to choose gauges that land on the target pitch without the string feeling too tight. Scale length plays into this. A Fender 25.5" (648 mm) string tuned to E4 needs roughly 7% more tension than a Gibson 24.75" (629 mm) string of the same gauge, which is why a Fender feels stiffer under the fingers. Nylon on a classical guitar (μ ≈ 0.0004 kg/m on the high E) runs at about half the tension of steel on an acoustic (μ ≈ 0.0007 kg/m), and that gap is what you feel as the softer touch.

FAQ

Why does my low E go sharp when I bend? A bend stretches the string, which raises T and so raises f. Since f ∝ √T, even a small change in tension is enough to hear.

Does stiffness affect the result? Mersenne assumes the string is perfectly flexible. Real ones aren't, and their bending stiffness pushes the upper partials slightly sharp. That's inharmonicity, and you notice it most on thick wound bass strings.

What μ should I use? Check the maker's specs. D'Addario and Ernie Ball list the unit weight of each string in lb/in or kg/m. A plain steel 0.010" comes out around 0.00040 kg/m, while a wound 0.046" bass string is closer to 0.0067 kg/m.