Fret Position Calculator

Look down any guitar neck and the frets bunch up toward the body like fence posts seen from a train. It looks like a trick of perspective. It is not. The first fret space on a full-size guitar is nearly an inch and a half long, and the twentieth is under half an inch.

The reason fits in one sentence. Every fret shortens the string that is left by the same proportion, and the same proportion of a shorter string is a shorter distance. The rest of fret layout follows from it.

One ratio, applied again and again

A stretched string sounds a pitch set by three things: its vibrating length, its tension and its mass per unit length. The physics is old and simple. The fundamental frequency is the wave speed divided by twice the length, and the wave speed is the square root of tension over mass per length. Hold the tension and the string fixed, halve the length, and the frequency doubles. Pressing a string against a fret does exactly this kind of shortening, which is why stopping it on a higher fret gives a higher note.

Western instruments with fixed pitches divide the octave into twelve equal semitones. An octave is a frequency ratio of two to one, so each semitone must be the number that multiplies by itself twelve times to make two. That number is the twelfth root of two, approximately 1.059. To raise a note by one semitone, the vibrating length has to shrink by that factor, whatever the note was.

So the distance from the nut to fret n is the scale length times one minus two to the power of minus n over twelve. The scale length is the nominal sounding length of the open string, nut to saddle. Twelve steps of the ratio make exactly two, which puts the twelfth fret at exactly half the scale and the twenty-fourth at exactly three quarters. This is also why makers measure a scale length as twice the distance from the front of the nut to the middle of the twelfth fret: the octave fret is the one landmark that does not depend on where the saddle has been moved.

Take a 25.5 in scale, a common length for electric guitars. The first fret sits 1.4312 in from the nut. The second is 2.7821 in from the nut, only 1.3509 in beyond the first. By the twelfth fret, at 12.75 in, the spaces have shrunk to 0.7582 in, and the gap before the twenty-second fret is 0.4255 in. Each space is the one before it divided by the same 1.059. Put 25.5 in the tool above and those are the rows you will read.

Nothing in the formula cares that there are twelve notes. Replace the twelve with nineteen, twenty-four or thirty-one and you get the fret positions for those equal divisions, all of which have music written for them; the twenty-four-step version is the quarter-tone system used in Arab music. The octave fret simply moves to fret nineteen, twenty-four or thirty-one, still at exactly half the scale. More notes per octave means more frets, closer together, and less room for a fingertip.

The rule of 18, and what replaced it

Builders were placing frets long before anyone could conveniently raise two to a fractional power. Their method was the rule of 18: divide the scale length by 18 to find the first fret, then divide what remains between that fret and the bridge by 18 to find the next, and so on down the neck. One lutherie reference says plainly that the origins of the technique are unknown, and suggests that makers of early instruments with tied gut frets noticed the pattern on instruments that already played well.

The ratio was in use by the sixteenth century. Vincenzo Galilei, a lutenist and one of the first practical advocates of equal temperament, used the ratio 18 to 17 for fretting the lute and wrote dance suites on each of the twelve notes. In 1584 the Chinese scholar Zhu Zaiyu described his own calculation of equal temperament. Simon Stevin wrote about the twelfth root of two around 1605, in a manuscript that was not published until 1884. The craft and the mathematics ran side by side for a long time without quite touching.

They differ by less than you might guess. Dividing by 18 removes a slightly smaller share than the true semitone, so every fret lands a little closer to the nut than equal temperament wants. On a 25.5 in scale the rule of 18 puts the twelfth fret at 12.6573 in, which is 0.0927 in short of the true halfway point. By the twenty-fourth fret the shortfall is 0.0931 in, barely more, because each fret’s error is a share of the string still remaining, and less and less remains. On an ideal string the twelfth fret would sound about an eighth of a semitone flat.

The repair is to divide by a different number. Work out the true first-fret distance once, divide the scale length by it, and the answer is 17.817. Use 17.817 where the old rule used 18 and the same pencil-and-paper routine reproduces equal temperament, with the octave back at half the scale. The old rule nevertheless stayed in use well into the twentieth century, and the same reference gives the reason: with the right compensation at the bridge it produced instruments that played in tune. That is a clue about how much of the final result the fret formula really controls.

What scale length changes

Choose a longer scale and three things happen together. The frets spread out, which suits large hands and crowds small ones. The string must be pulled tighter to reach the same pitch. And the tone shifts: the maker behind the modern multiscale patent argued in a 1995 lecture to fellow luthiers that scale lengths have distinct harmonic voices.

The tension part is arithmetic. For a given string at a given pitch, tension rises with the square of the length. Going from 24.75 in to 25.5 in is a 3 percent increase in length and about 6 percent more tension. A bass at 34 in and a short-scale bass at 30 in differ by far more, about 28 percent for the same string at the same pitch. A five-string banjo maker lists 26 1/4 in from nut to bridge; a mandolin is a little under 14 in. Each instrument’s usual length is a settled bargain between the pitch wanted, the strings available and the reach of a hand.

It is worth being exact about the word tension. It is fixed completely by pitch, vibrating length and mass per unit length. The length of string behind the nut, the angle over the saddle and similar details do not change it, though they can change how stiff a string feels under the finger. Published scale lengths deserve the same care. One supplier warns that a famous nominal 24 3/4 in scale has rarely measured exactly that, having drifted with the factory’s tooling over the decades. A preset is a starting point. If you are copying an instrument, measure it.

A multiscale fingerboard refuses the bargain by giving each string its own length, longer on the bass side and shorter on the treble, with frets that fan out to connect them. The idea appeared on the Renaissance orpharion and was patented in a modern form in the United States in 1989, in a patent that has since expired. Each end of each fret is still placed by the same formula on its own scale. One fret is chosen to sit square to the neck, and that choice decides which way all the others lean.

What the formula leaves out

The formula describes an ideal string: perfectly flexible, and not stretched by the act of fretting it. Real strings fail both conditions, and both failures push fretted notes sharp. Pressing a string down to a fret stretches it and raises its tension. A stiff string’s overtones also run progressively sharp of true harmonics, more so as the sounding length gets shorter up the neck, and the ear hears the whole note as sharper.

Makers answer with compensation. The saddle is set a little farther from the nut than the scale length says, which lengthens every fretted note by a share that grows toward the high frets. Thick, stiff strings need more of it than thin flexible ones, steel more than nylon, which is why a saddle is so often set at a slant. Some builders also move the nut slightly toward the first fret. Physicists who modeled the problem and tested it on a monochord concluded that perfect intonation of every note can never be achieved, only an error small enough to live with.

This is the plain limit of any fret table, including the one above. It gives the uncompensated positions and draws the saddle at the theoretical end of the scale. It does not know your strings, your action height or how hard you press, so it cannot tell you where the saddle belongs. It also treats each slot as a line with no width: the numbers are slot centers, with no allowance for the saw kerf or for the shape of the fret crown that finally decides where the string stops.

How accurate a slot needs to be

Less accurate than the decimals suggest, and more careful than most first attempts. One widely used lutherie reference puts the working standard at calculations to three decimal places and saw placement within 0.01 in, and notes that wood moves with humidity and that crowns are filed by hand afterward. Finer arithmetic than that describes a fretboard nobody can build.

What ruins fretboards is not rounding but accumulation. Measure each fret from the one before and every small error is carried into all the frets after it; a supplier of fretting tools warns that marking with an accuracy of plus or minus 0.020 in this way can leave the twelfth fret off by as much as a quarter of an inch. Measure every slot from the nut instead and each error stays where it was made. The fret-to-fret column is there to check your marks, not to make them.

A printed template is a third way to transfer the numbers, and it carries its own trap, because printers quietly scale pages to fit. Print at 100 percent, then lay a rule you trust on the check ruler before any wood is cut. If 100 mm on the paper is not 100 mm, nothing else on the sheet is right either.

The pleasing thing about a fretboard is that its whole design is one ratio and a starting length. The humbling thing is that the ratio is only a first approximation, finished by ear at the saddle.