String Vibrating Frequency Calculator
Compute the fundamental pitch of a vibrating string from its length, tension and mass using Mersenne’s law — and see the nearest musical note.
How it works
f = (1 ÷ 2L) × √(T ÷ μ), where L is length in metres, T is tension in newtons and μ is linear mass density in kg/m (Mersenne’s law)
Mersenne’s law says a stretched string’s pitch rises with the square root of tension, falls with the square root of its mass per unit length, and is inversely proportional to its vibrating length. That is why fretting a string (shortening L) raises the pitch, why tightening a tuning peg (raising T) sharpens it, and why the thick, heavy strings on a bass sound so low. Only the square root of tension matters, so quadrupling tension merely doubles the frequency — one octave.
Worked example
A 0.65 m string under 70 N of tension with a linear mass of 6 g/m (0.006 kg/m) vibrates at (1 ÷ 1.30) × √(70 ÷ 0.006) ≈ 0.769 × 108.0 ≈ 83.1 Hz — just above an E2 on a bass guitar.
Frequently asked questions
What is linear mass density?
It is the mass of the string per unit length (here grams per metre). Thicker or denser strings have higher linear mass, which lowers their pitch — the reason wound bass strings sound so much deeper than a thin treble string of the same length and tension.
Why does tightening a string raise the pitch?
Frequency scales with the square root of tension, so more tension means a higher note. Because it is a square-root relationship, you have to quadruple the tension to raise the pitch a full octave — which is why over-tightening to reach a high note quickly risks snapping the string.
Does this give the note I actually hear?
It gives the fundamental frequency, which sets the perceived pitch. A real string also sounds harmonics (integer multiples of the fundamental) that give it timbre, but the fundamental is the note you name.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.