How this instrument works
On a fretted instrument tuned to 12-tone equal temperament, each fret shortens the vibrating string length by a fixed ratio: the twelfth root of two, about 1.05946. Placing a fret at distance d from the nut on a string of scale length L means the remaining string length is L divided by 2 raised to the power of (fret number / 12) — so the distance from the nut is the scale length minus that remaining length.
This is the same geometric rule behind every fretted instrument's layout, from guitars and basses to banjos, mandolins, and ukuleles, because it's the rule that makes each fret produce a note exactly one semitone higher than the one before it. Because the formula is purely geometric, doubling the fret's power-of-two term always halves the remaining string, which is why the 12th fret — one full octave up — always sits at exactly half the scale length, on any instrument, at any scale length.
Instrument builders and DIY luthiers use this formula (or a fret-position table generated from it) to mark fret slot positions before cutting a fingerboard, and it's equally useful for checking an existing instrument's intonation or verifying a fret placement looks correct.
- Enter Scale length (mm) — the vibrating string length from nut to bridge saddle.
- Enter Fret number — which fret you want the distance for, counting from 1 at the nut end.
- Read Distance from the nut (mm) directly beneath the inputs.
- To find the gap between two adjacent frets rather than distance from the nut, calculate each fret's distance separately and subtract.
- Scale length must be greater than zero and fret number must be at least 1.
Worked example — 12th fret on a 648 mm scale
Enter Scale length = 648 mm (a common classical-guitar scale) and Fret number = 12. That's 648 - (648 / 2^(12/12)) = 648 - 324 = 324.0 mm. Distance from the nut reads 324.0.
324 mm is exactly half of 648 mm — no coincidence. The 12th fret is one full octave above the open string, and 2^(12/12) = 2, so the remaining string length is always exactly half the scale length at that fret, regardless of what the scale length itself is.
Questions
Why does the 12th fret always sit exactly halfway?
Because 12 semitones is one octave, and an octave up means the vibrating string length is halved. Plugging fret number 12 into the formula gives 2^(12/12) = 2^1 = 2, so the remaining string past the 12th fret is always scale length divided by 2 — half the scale length — no matter what that scale length is.
Does this formula work for instruments other than guitar?
Yes. Any fretted instrument tuned to standard 12-tone equal temperament — bass guitar, banjo, mandolin, ukulele — follows the same twelfth-root-of-two spacing rule, since the physics of a vibrating string and the definition of an equal-tempered semitone don't depend on which instrument the string is stretched across.
How do I find the width of a single fret gap instead of distance from the nut?
Calculate the distance from the nut for two adjacent fret numbers and subtract the smaller from the larger. Fret spacing naturally gets narrower as you move up the neck, since each fret's ratio is applied to a progressively shorter remaining string length.
What scale length should I use for a typical guitar?
It varies by instrument family and manufacturer — common examples run from roughly 24 inches on some short-scale electrics up past 34 inches on bass guitars, with classical and many steel-string acoustics landing around 25 to 26 inches. Check your instrument's specifications or measure nut-to-bridge-saddle directly for an exact figure.
Why is the formula based on the twelfth root of two specifically?
Because 12-tone equal temperament divides each octave (a 2:1 frequency ratio) into 12 equal semitone steps, and the number that, multiplied by itself 12 times, equals 2 is by definition the twelfth root of two. Each fret represents one of those 12 equal steps, so each one shortens the string by that same ratio.