What makes a string its pitch

Length, tightness, thickness — Mersenne's three knobs. Halve the length and the pitch doubles; tighten it and it rises; fatten it and it drops. That's why the low string is the thick one.

three knobs set one pitch — turn them and pluck

82.4 Hz≈ E2

length648 mm

tightness (tension)×1.00

thickness (mass)×1.00

f = (1 / 2L) · √(T / μ)

Length — halve it, double up

Pitch and length are inversely proportional. Cut the length in half, the pitch doubles — that's the octave, and it's exactly what the 12th fret does.

Tension — a square root

Tightening raises the pitch, but by the square root of the tension. To play twice as high you must pull four times as hard — which is why strings snap.

Thickness — why low E is fat

A heavier string vibrates slower (÷√μ). Rather than build a floppy 2-metre bass string, makers just wrap it thick. That's the fat low string on every guitar.

For the classroom

Learning goals

A string's pitch follows Mersenne's law: f = (1/2L)·√(T/μ) — set by length, tension and mass per unit length.
Length is an inverse proportion (half the length → double the pitch); tension and mass enter as square roots.
Instrument makers trade the three off against each other — the thick low string is mass standing in for impractical length.

Try this

1.Set a note, then halve the length — confirm the pitch doubles. Next, to raise the pitch an octave with tension alone, students discover they need ×4 (a square root in action).
2.Challenge: reach the lowest note you can. Which knob helps most for a short, playable string? (Thickness — that's the wound bass string.)