How this calculator works
A stretched string behaves like a simple harmonic oscillator. Its frequency is set by the tension, the mass per unit length and the vibrating length: a heavier string needs more tension to reach the same pitch, and a shorter scale needs less. The tension in newtons is four times the unit weight in kilograms per metre, times the square of the length in metres, times the square of the frequency in hertz. For a plain steel string the unit weight comes from the diameter and the density of steel: mass per metre equals density times the cross-sectional area. Wound strings are lighter than a solid wire of the same diameter because the winding traps air gaps, so use the maker's published unit weight for those.
Formula
tension (N) = 4 x unit weight (kg/m) x scale length (m)^2 x frequency (Hz)^2. unit weight = tension / (4 x length^2 x frequency^2). frequency = square root of (tension / (4 x unit weight x length^2)). For a plain steel string, unit weight = density x pi x (diameter / 2)^2 with steel at about 7,850 kg/m^3.
Worked example
A plain 0.010 inch steel string has a diameter of 0.254 mm and a unit weight of about 0.000398 kg/m. On a 25.5 inch scale, which is 0.6477 m, tuned to E4 at 329.63 Hz, the tension is 4 x 0.000398 x 0.6477^2 x 329.63^2 = 72.57 N, about 16.31 pounds-force. A set of six strings is usually designed so the tensions are within a few pounds of each other.
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Frequently asked questions
What tension should a guitar string have?
Common electric guitar sets run from about 13 to 20 pounds-force per string, with the plain strings near 16 pounds and the wound strings a little higher. Classical guitars use nylon strings at much lower tension, around 7 to 10 pounds. The important thing is that a set is balanced, so no string feels much stiffer than its neighbours.
How do I find the unit weight of a wound string?
Use the maker's published weight per metre or their tension chart. A solid-wire estimate from the outer diameter overstates the mass of a wound string, because the winding leaves gaps and uses a lighter core. If you only know the gauge, compute the plain-string figure as an upper bound and treat the result as approximate.
Why does a longer scale give more tension?
Pitch depends on tension divided by mass per length, and the vibrating length enters the equation twice. To reach the same frequency on a longer string you need more tension, which is why baritone and bass guitars use heavier gauges or lower tunings. The relationship is quadratic in length.
Can I use this for other stringed instruments?
Yes. The formula applies to any vibrating string: violin, mandolin, bass, harp or piano. Enter the scale length, the pitch and the maker's unit weight, and convert units as needed. Piano strings use very high tensions, often several hundred newtons per string.