How this calculator works
Each fret shortens the vibrating string by a fixed ratio. Twelve frets halve the length, which raises the pitch by an octave, so each fret is the twelfth root of two shorter than the last: about 0.9439 of the remaining length. Measuring from the nut, fret n sits at scale length times (1 minus 2 to the power of minus n over 12). The twelfth fret is therefore exactly halfway along the string, and the twenty-fourth is three quarters of the way. Real instruments add a small compensation at the bridge because pressing a string stretches it slightly, which is why the saddle is set back from the theoretical position.
Formula
distance from the nut to fret n = scale length x (1 - 2^(-n/12)). The distance between frets n-1 and n is the difference of the two values. Scale length from a measured fret = distance / (1 - 2^(-n/12)).
Worked example
A 25.5 inch scale, which is 647.7 mm: the first fret is 647.7 x (1 - 2^(-1/12)) = 647.7 x 0.05613 = 36.35 mm from the nut. The twelfth fret is 647.7 x (1 - 2^-1) = 323.85 mm, exactly half the scale. The twenty-fourth fret is 647.7 x 0.75 = 485.78 mm, leaving a quarter of the string beyond it.
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Frequently asked questions
Why is the twelfth fret exactly in the middle?
Twelve semitones make an octave, and an octave is a frequency ratio of two to one. A string shortens by half to double its frequency, so the twelfth fret must sit at half the scale length. This is a useful check when measuring an instrument or laying out a fretboard.
What is scale length and where is it measured?
Scale length is the vibrating length of the open string, from the nut to the bridge saddle. It is not the same as the overall instrument length. Measure half of the twelfth fret distance from the nut and double it for a quick check, or measure from the nut to the saddle on the centre line.
Does this account for intonation?
No. The page gives the theoretical equal-tempered positions. Real instruments compensate for string stiffness and action height by moving the saddle slightly back, and each string may need a different offset. The fret slots themselves follow the theoretical series closely, with small variations from the manufacturer.
Why does a shorter scale feel easier to play?
Frets are closer together and the string tension is lower for the same tuning and gauge, so bends and chords need less force. A longer scale has wider fret spacing and more tension, which gives a brighter tone and better tuning stability. The ratios here scale with the chosen length.