The equation and why it is exact
The distance from the nut to fret n is the scale length minus the scale length divided by 2 to the power n over 12. Written the other way round, the remaining vibrating length at fret n is the scale length divided by 2 to the power n over 12, which is the same twelfth-root-of-two ratio that governs equal temperament pitch. Frets are equal temperament made out of wire.
The consequence you can check with a ruler is that fret 12 sits at exactly half the scale length, because 2 to the power 12/12 is 2. On a 25.5 inch scale the twelfth fret is at 12.75 inches, and on a 24.75 inch scale it is at 12.375. Fret 24 sits at exactly three quarters of the scale. Those two checkpoints catch almost every layout error, and if the twelfth fret is not at half the scale, something upstream is wrong.
You may still see the rule of 18 in old shop books: divide the remaining length by 18 to get the next fret. The correct divisor is 17.817, which is 1 divided by (1 minus 2 to the power minus 1/12). The 18 version was a workshop simplification and it goes progressively sharp up the neck, ending noticeably wrong by the twelfth fret. There is no reason to use it now.
Measure from the nut every time
The table gives the fret-to-fret gap because it is useful for sanity-checking, not because you should measure that way. Laying out frets by stepping from one to the next accumulates every small error in the same direction. Two thousandths of an inch of drift per fret is a plausible marking error and it is invisible in isolation, but by fret 12 it has become 24 thousandths, which puts the octave measurably sharp.
Every distance in the first column is from the nut, and that is how the board should be marked. Professional layout goes further: a fret slotting template or a CNC-cut fixture references a single datum for all cuts, which removes marking error entirely.
The compensation question, which is not a calculation
Fret positions assume the string is an ideal flexible line whose vibrating length is exactly the distance between nut and saddle. Real strings are stiff, and pressing one to the fretboard stretches it slightly, raising its pitch above what the geometry predicts. The stretch is proportional to how far the string has to travel, so it grows with action height, and the pitch effect grows with how thick and stiff the string is.
The correction is to move the saddle backwards, lengthening the string so the fretted note comes back down to where the fret position says it should be. That extra length is compensation, and it is genuinely different for every string on the instrument. Wound low strings need the most, plain high strings the least, and on a set with a plain third the pattern usually breaks there, which is why guitar saddles are staggered rather than square.
How much is not something a page can tell you. Ordinary values run from a couple of thousandths on the treble E to well over an eighth of an inch on a low string, and they depend on the strings fitted, the action set, and how hard the player presses. Compensation is set by comparing the fretted twelfth-fret note with the twelfth-fret harmonic and moving the saddle until they agree, with the instrument in playing condition. The number the calculator gives you is a range so you know how much saddle travel to build in, not a target to cut to.
The nut can need compensating too, in the opposite direction, because the first few frets are where the stretch is proportionally largest. That is a finer adjustment and mostly matters on instruments already set up carefully.
Multiscale necks
A fanned-fret neck uses a longer scale on the bass side and a shorter one on the treble side, so the low strings gain tension and definition while the high strings stay easy to bend. The layout is not a special formula: each side gets its own scale length, each side is laid out with the ordinary equation, and the fret is a straight line joining the two positions.
What you have to decide first is the perpendicular fret, the one that sits square to the strings. Everything fans outward from it in both directions, so putting it around the seventh or ninth fret keeps the fan modest where the hand spends most of its time, while putting it at the nut throws the whole fan into the upper register. Fill in the second scale length above to see how far apart the two sides sit at each fret.
What the numbers do not cover
Fret positions are the easy part of a neck. Slot depth, tang size and the fit of the fret in the slot determine whether the board back-bows when the frets go in. Radius, fret height and levelling determine whether the instrument plays without buzzing. Neck relief interacts with all of it and moves with string tension, which is a separate calculation on the string tension calculator. Getting the layout right is necessary and nowhere near sufficient.
For the pitches those frets are producing, the note frequency calculator gives frequencies and cents at any reference. If the work is happening in a shop rather than on a bench, the workshop layout planner and the cut list optimizer handle the material side.
Questions people ask
What is the formula for fret positions?
Distance from the nut to fret n = S minus S divided by 2 to the power n over 12, where S is the scale length. Equivalently, the string length remaining above fret n is S divided by 2 to the power n over 12. On a 25.5 inch scale the first fret is 1.4312 inches from the nut, the twelfth is 12.75, and the twenty-fourth is 19.125. The twelfth fret at exactly half the scale is the check that tells you the layout is right.
Where exactly is the scale length measured from and to?
From the front edge of the nut to the theoretical bridge point, which is the position of the saddle before compensation. The reliable way to measure an existing instrument is to measure from the nut to the centre of the twelfth fret and double it, because that distance is exactly half the scale by construction. Measuring nut to saddle gives you a longer figure that includes compensation and varies string to string, so it is the wrong number to feed into a fret layout.
How much compensation should I add at the saddle?
It is set by measurement on the instrument, not by formula. The usual method is to compare the note fretted at the twelfth with the natural harmonic at the twelfth: if the fretted note is sharp, the string is too short and the saddle moves back, and if it is flat, the saddle moves forward. Do it with the strings you will use, at the action you will play, and repeat after any change to either. As orientation for how much saddle travel to build in, a couple of thousandths of an inch on a plain treble string up to an eighth of an inch or more on a wound low string covers most instruments. The exact figure belongs to the instrument.
Does a longer scale length make the frets further apart everywhere?
Yes, proportionally. Every fret position is the scale length times a fixed multiplier, so a 25.5 inch neck has all its frets 3 percent further apart than a 24.75 inch neck, at every fret. That is why the two feel different to fret across a stretch even though the neck profile and the string spacing may be identical. It is also why the scale length is the first thing to establish in any build: change it later and every single fret position changes with it.
Why is my instrument sharp up the neck even though the frets are in the right places?
Almost always compensation or action height rather than fret position. Pressing a string down stretches it, and the higher the action the further it has to travel and the sharper the fretted note goes. Check three things in order: the action at the twelfth fret, the intonation set by the twelfth-fret harmonic comparison, and the neck relief. If those are all right and it is still sharp only in the first few frets, the nut is likely cut too high, which stretches the string disproportionately down there. Frets that are genuinely mislocated show up as a pitch error that grows smoothly with fret number and does not respond to saddle adjustment at all.