Work out string vibration frequency instantly with clear inputs, formula shown and shareable results.
Waves travel along a string at sqrt(T/mu), so the fundamental is (1/2L)sqrt(T/mu). A 648 mm string at 70 N with 0.4 g/m gives 324 Hz, close to E4. Because tension appears under a square root, raising pitch by an octave needs four times the tension - which is why thicker strings are used instead.
Wave speed on a string
v = sqrt(T / mu)
Fundamental (Mersenne's law)
f1 = (1 / 2L) sqrt(T / mu)
To raise mu without raising stiffness. A heavier string vibrates more slowly at playable tension, which is what wound strings achieve compactly.
Frequency scales with the square root, so a 2% tension change shifts pitch by about 1% - roughly 17 cents.