The physics of a tuned string
A string vibrates at a pitch set by three things: its length, its tension and its mass per unit length. Tighten it and the pitch rises; make it longer or heavier and the pitch falls. The French friar and mathematician Marin Mersenne put these relationships into equations in 1636, and they still guide string makers and instrument builders.
Mersenne’s law
f = (1 ÷ 2L) × √(T ÷ μ), rearranged as T = μ × (2 × L × f)²
- f is the frequency of the note in hertz.
- L is the vibrating length, the scale length from nut to saddle on a guitar.
- μ is the unit weight, the mass per unit length of the string.
- T is the tension.
For a plain string, μ = density × π × d² ÷ 4. Steel has a density of about 7,850 kg/m³, nylon about 1,140 and fluorocarbon about 1,780.
Worked example
A plain steel .010-inch high E string on a 25.5-inch scale, tuned to E4 at 329.63 Hz:
- Unit weight: about 0.398 g/m (0.0000223 lb/in).
- Wave speed on the string: 2 × 0.6477 m × 329.63 Hz = 427 m/s.
- Tension: 16.3 lb (72.5 N), in line with the figures string makers publish.
- Tuning it up one semitone, to F4, would raise the tension to 18.3 lb, because tension grows with the square of the frequency.
Gauge and tension
Tension grows with the square of the diameter. For the same high E on a 25.5-inch scale:
| Gauge | Tension |
|---|---|
| .009 in | 13.2 lb |
| .010 in | 16.3 lb |
| .011 in | 19.7 lb |
| .012 in | 23.5 lb |
What this means for players
- Scale length: a 25.5-inch scale has about 6% more tension than a 24.75-inch scale at the same pitch and gauge, which is why the same strings feel tighter on one guitar than on another.
- Down-tuning: dropping a whole step reduces tension by about 21%. Players who tune down often use heavier strings to restore the feel.
- Balanced sets: some string makers design sets so that every string has a similar tension, which evens out the feel when bending.
- Neck setup: a big change in total tension changes the neck relief, so a truss rod adjustment may be needed after switching gauges.
Wound strings
Wound strings have a core wire wrapped with another wire, so they weigh more than a solid wire of the same outer diameter. Using the outer diameter with the steel density underestimates their tension, sometimes by a third or more. For wound strings, use the unit weight from the string maker’s tension chart with T = μ × (2Lf)².