Fret Calculator
Every fret on a guitar neck is one number wearing a position: the twelfth root of two, applied again and again down the scale. Enter your instrument’s scale length and this calculator returns the exact distance from the nut for the first twelve frets — the layout a luthier marks before a single slot is sawn.
The Twelfth Root of Two, Made Wood
Equal temperament divides the octave into twelve equal ratios, and a string’s pitch is inverse to its length — so the string remaining after fret n must be the scale divided by 2^(n/12), and the fret’s distance from the nut is what was taken away: d = L − L ÷ 2^(n/12). The constant 2^(1/12) ≈ 1.0594631 is the semitone itself, the most load-bearing irrational in music. Evaluate the formula and the neck’s entire geography falls out: fret 12 lands at exactly L/2 (the string halved, the octave sounded — the calculator’s note turns it into a built-in audit), fret 7 at about 0.3327 × L, fret 1 at about 0.0561 × L. Luthiers worked for centuries with the “rule of 18” — each fret placed 1/18 of the remaining string from the previous — an approximation (17.817 is the truer divisor) whose small error the modern formula retires; a neck laid out by 18s plays audibly sour by the octave, which is how you can still identify antique approximations by ear.
Worked Example: The Fender Scale
The defaults lay out the 25.5-inch scale of a Stratocaster. Fret 1 sits at 25.5 − 25.5 ÷ 1.0594631 = 1.431 inches from the nut; fret 5 at 6.397; fret 7 (the double-dot’s neighbour on many boards) at 8.481; and fret 12 at exactly 12.750 — half scale, the audit passing by construction. The remaining-string column shrinks geometrically (each fret takes 5.61% of what is left), which draws the neck’s familiar perspective: the first fret gap spans 1.43 inches of finger room while the eleventh-to-twelfth interval is under three quarters of an inch, and beyond the calculator’s dozen the crowding accelerates toward the dust region where only fingertips and conviction fit. Gibson’s 24.75 and the classical 650 mm run through the identical formula with a different L — scale length is a feel and tone decision (longer scales speak tenser and brighter), but the ratios never negotiate.
Compensation: Where the Formula Meets the String’s Physics
A real string is not the formula’s ideal line: pressing it to a fret stretches it slightly sharp, and its stiffness resists bending at the ends — so real bridges sit a hair beyond the nominal scale (saddle compensation, typically a millimetre or three, more for the thick wound strings), and nuts are sometimes shifted likewise. The calculator lays out the theoretical neck — the one the fret slots are actually cut to — because compensation belongs to the setup stage, per string, per gauge, per action height, adjusted at the saddle screws with a tuner as judge. Conflating the two stages is the beginner luthier’s classic error: frets placed by compensated measurements would bake one string set’s corrections into the wood for every future set. Cut the slots by the twelfth root of two; compensate at the hardware; the division of labour has been stable for a century because the mathematics insists on it.
Vincenzo’s Bones: The Formula Before the Formula
The fret problem predates logarithms in the workshop and follows them in the literature: Vincenzo Galilei — the lutenist father of the astronomer — published the 18-rule’s refinement in 1581, having noticed by ear that the semitone behaves like the ratio 18:17, and his experiments with string tension (pitch rising with the square root of tension, against Pythagorean doctrine) made the family name synonymous with measuring what tradition merely asserted. The twelfth root of two could not be computed conveniently until logarithms arrived (and Stevin and Mercator tabulated it within the century), so the rule of 18 is what an irrational constant looks like when a civilisation must build with it anyway: a fraction close enough to sing. Every modern neck is thus a small museum — geometry that waited two thousand years for its exact number, approximated brilliantly in the interim by men who trusted their ears over their arithmetic and were, within half a percent, right.
Temperament Is a Choice of Errors
The formula’s hidden premise deserves its paragraph: equal temperament itself is a negotiated settlement in which every interval except the octave is slightly out of tune (the fifth is narrowed by about 2 cents, the major third widened by about 14), so that all keys may be equally, tolerably wrong. The frets’ equal ratios are that settlement rendered in wire — a just-intonation neck, fretted for pure thirds, would need different fret positions per key and could not modulate, which is why luthiers of the meantone era literally moved tied gut frets between pieces. Modern experiments revisit the compromise (True Temperament’s wavy frets buy back some purity at the price of geometry per string), but the standard neck’s wager stands: distribute the error democratically, and no key shall be unusable. The calculator lays out the democratic neck — every fret a small act of distributive justice among the keys.
Related Tools
More from the Other Calculators hub: the Harmonic Series Calculator — the overtone series that makes this string worth fretting — and the Note Frequency Calculator for the pitches these positions produce.
Frequently Asked Questions
How do I measure my guitar’s scale length?
From the nut’s string-bearing edge to the bridge saddle’s contact point — conventionally doubled from the nut-to-12th-fret distance, which sidesteps the compensation built into the saddle’s actual position: measure nut to the centre of the 12th fret wire and double it. If that figure is 25.5 inches, you hold a Fender-scale neck; 24.75 says Gibson; classical guitars speak millimetres at 650. The calculator’s audit (fret 12 at exactly half your entered scale) turns the measurement back on itself: if your entered scale does not halve at your 12th fret, the measurement, not the maths, wants another look.
Why do the frets get closer together?
Because pitch ratios are multiplicative while the ruler is additive: each semitone must remove the same fraction (about 5.61%) of the string that remains, and equal fractions of a shrinking remainder are shrinking lengths. The geometry is identical to the harmonic series’ reciprocals wearing a different costume — and to compound interest run in reverse — which is why the upper neck’s geometry feels like a perspective drawing: it is one, drawn by an exponential. Players experience it as the high register’s fingertip precision; luthiers experience it as tolerances that must tighten as the frets ascend, since the same sawing error costs more pitch at fret 19 than at fret 2.
What is the rule of 18, and should I use it?
The pre-calculator luthier’s method: place each fret at 1/18 of the remaining string from the last. It approximates 2^(1/12)’s reciprocal geometry with the divisor 18 in place of the exact 17.817 — close enough to build centuries of playable instruments, far enough off to leave the octave measurably misplaced on careful work (a neck by 18s puts the 12th fret about 0.4% short of true half-scale, a discrepancy of roughly a tenth of an inch on a Fender scale). Use it to understand history and to rough out a cigar-box build by hand; use this page’s formula for anything whose intonation you will be judged by. The two agree to a hair at the first frets and argue by the twelfth, which is exactly how approximations behave.
Do basses and ukuleles use the same formula?
Identically — the twelfth root of two is no respecter of instruments. Bass scales (34 inches standard, 30 short) and ukulele scales (about 15 inches soprano) differ only in the L you enter, and in the practical footnotes: longer scales need the layout measured in correspondingly finer discipline, and the very short scales forgive sawing errors that a 34-inch neck would prosecute. Fanned-fret (multiscale) instruments run two scale lengths at once — the formula applied per string, the frets slanting to connect the two layouts — which this calculator supports one string at a time, the slant being a drawing exercise between the two runs.
What about the distance between frets rather than from the nut?
Subtract neighbours in the calculator’s table: the gap between fret n−1 and fret n is L ÷ 2^((n−1)/12) − L ÷ 2^(n/12), the remaining string’s 5.61% slice at that station. Templates and fret-slotting jigs are often specified in these step distances (or in cumulative-from-nut, as here — the safer convention, since cumulative measurement keeps any single marking error from propagating down the whole neck, a layout lesson every machinist will recognise). If your jig wants steps, differences of this page’s figures are exact; if it wants cumulative, they are printed as computed.
My 12th fret is not at half my measured scale. What is wrong?
One of three things, in order of likelihood: the scale was measured to the compensated saddle rather than the true scale (measure nut to 12th and double, as the first FAQ prescribes); the nut’s bearing edge is worn or the measurement started at the wrong face of the nut; or — on a handmade or vintage instrument — the neck was laid out by the rule of 18 or by a template with its own history, and its 12th fret genuinely sits a fraction off the modern ideal, which the saddle’s compensation has been quietly absorbing for decades. The formula is the court of appeal; instruments are permitted their biographies.