Instrument Scale Length Calculator
The pitch of a vibrating string is fixed by three things: its speaking length, the tension stretching it, and its mass per unit length. Mersenne's law ties them together, and rearranging it lets you find the string length that will sound a chosen frequency at a given tension and gauge. This calculator takes a target frequency in hertz, tension in newtons, and linear mass density in kilograms per metre, then returns the vibrating length in metres, centimetres, and inches. Luthiers and string designers use it to choose scale lengths and gauges that put a comfortable tension on every string at the intended pitch.
Scale length formula
Wave speed = sqrt(tension / linear mass density)
Length = wave speed / (2 * frequency)
Centimetres = length * 100
Inches = length / 0.0254
This is Mersenne's law rearranged for length. Tension is in newtons and linear mass density in kilograms per metre, giving wave speed in metres per second and length in metres.
String physics context
- Open low E on a guitar is about 82.41 Hz; A4 reference is 440 Hz.
- Quadrupling tension only doubles pitch, since tension is under a square root.
- Heavier strings (higher mass per metre) sound lower for the same length and tension.
- Convert a manufacturer's unit weight in pounds per inch to kilograms per metre before entering it.
- Length has the strongest effect: halving the vibrating length raises the pitch a full octave.
Scale length: frequently asked questions
What is Mersenne's law?
Mersenne's law states that the fundamental frequency of a vibrating string equals one over twice the length, multiplied by the square root of tension divided by linear mass density. It links a string's pitch to its length, how tightly it is stretched, and how heavy it is per unit length.
How do I find the scale length for a given pitch?
Rearrange Mersenne's law: the vibrating length equals one over twice the frequency, multiplied by the square root of tension divided by linear mass density. Enter your target frequency in hertz, the string tension in newtons, and the mass per metre in kilograms, and the calculator returns the length in metres.
What is linear mass density?
Linear mass density is the mass of the string per unit length, in kilograms per metre. A thicker or denser string has a higher value and vibrates at a lower pitch for the same length and tension. String manufacturers publish this figure, often as the unit weight in pounds per inch, which you can convert to kilograms per metre.
Does tension or mass matter more?
Both appear under a square root, so quadrupling the tension only doubles the pitch, and quadrupling the mass per length only halves it. Length has a stronger, inverse-proportional effect: halving the length doubles the pitch, which is why frets move closer together going up the neck.
Is this for the speaking length or the full string?
It is the speaking, or vibrating, length: the distance between the two points that anchor the vibration, such as the nut and bridge on a guitar. The portion of string wound around the tuning post does not vibrate and is not part of the scale length.
Official sources
- UNSW School of Physics: Strings, standing waves and harmonics.
- UNSW School of Physics: Music acoustics.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.