A string you can actually hear.
Change its length and tension, and the note moves with them — press play to hear the exact pitch this equation predicts.
·
never plays on its own — press Play to hear it
Length L
L⁻¹ ∝ f
0.30 mshorter · higher pitch1.00 m
Tension T
T½ ∝ f
20 Ntighter · higher pitch150 N
Gauge μ
μ⁻½ ∝ f
0.0003 kg/mthinner · higher pitch0.01 kg/m
Try
Three relationships govern every stretched string — Mersenne’s laws, first published in 1636: pitch rises as the string gets shorter, as it's pulled tighter, and as it gets thinner (lighter per unit length). Every tuning peg, every string gauge, and every playable scale length on every stringed instrument — guitar, violin, piano, harp — is a knob on this one equation. The vibrating string model behind it assumes both ends are rigidly fixed and shows only the fundamental (lowest) mode pictured above; a real plucked or bowed string also rings with overtones layered on top, which is what actually gives an instrument its timbre — not modeled here. The illustrative T/μ pairs on the presets are tuned to land on the real note, not arbitrary round numbers.