Calculating the positions of the frets requires the simplest bit of math. The (really) historical technique is called the rule of 18, and it involves successively dividing the scale length minus the offset to the previous fret by 18. This was easy enough to do using pencil and paper, although way back in the day they probably didn't even do that, instead relying on a mechanical device called a proportional divider to do the work. The invention of the pocket calculator made it possible to make use of a more accurate constant, and so these days we conventionally calculate fret positions for equal temperament by successively dividing the scale length minus the offset to the previous fret by 17.817. This series of calculations puts the 12th fret (octave) at exactly half the scale length, and we know from the physics of vibrating strings that halving the vibrating string length of a theoretically perfect string doubles the frequency. Those interested can read about the derivation of this constant below. Interestingly enough, the historical technique yielded good results for gut strings, as it included built in compensation, something that we do now by moving the bridge saddle so that the actual scale length is a bit longer than the nominal length. It is also interesting that the amount of compensation we add is empirically derived, as calculating it is a bit of a chore.
Some references specify a constant of 17.835, which is mathematically untenable but which will likely yield good results nonetheless. The reason that all these seemingly disparate systems can be used to good effect is that even the mathematically accurate equal temperament constant assumes a theoretically perfect string which both exhibits no inharmonicity in its vibration and doesn't need to be stretched to be fretted. Since real strings don't behave that way even the equal temperament system is just an approximation of the way real strings need to be fretted. See Gregory Byers nice paper on classic guitar intonation for more on this subject (see the Links section). Which constant should you use? Unless you are building or repairing historical instruments use the equal temperament constant of 17.817. It is the currently accepted standard. Want to know where this constant comes from? See the derivation section below.
Most folks that build instruments will never need to use this info, as the lutherie suppliers all provide slotted fret boards for all of the standard instruments. A number of suppliers use CNC machines to do the slotting and so can easily supply inexpensive slotted boards for any scale length and any number of frets.
Last updated: Tuesday, January 21, 2014
Let’s say the scale length of the instrument we want to calculate fret positions for is 25.5”. This is typical for Fender electric guitars and a lot of other guitars as well. To calculate the distance from the nut to the first fret, divide the scale length by 17.817:
ScaleLength = 25.5
NutToFret = ScaleLength / 17.817 = 1.431
There’s no need to maintain any more than three decimal places of accuracy (in fact two decimal places are plenty), since you won’t be able to saw the fret slots more accurately than a thousandth of an inch. Plus, even if you could, you couldn’t hear the difference anyway.
To calculate the distance from the nut to the second fret, subtract the distance from the nut to the first fret from the scale length. This yields the distance from the first fret to the bridge. Then divide that quantity by 17.817. This result is the distance from the first fret to the second fret. Add to this the distance from the nut to the first fret to get the distance from the nut to the second fret. The latter is the quantity we really want, so when you are sawing fret slots you are measuring each slot position from the nut. This way any errors in positioning your slots will not accumulate. So to calculate the distance from the nut to the second fret:
BridgeToFret = ScaleLength – NutToFret
NutToFret = (BridgeToFret / 17.817) + NutToFret
Repeat this process for each fret position on your fingerboard. If you consider that the nut is the same as a 0th fret, a generalized formulae for the calculation of the position of any fret n, given the scale length and the distance from the nut to the previous fret is:
BridgeToFret[n-1] = ScaleLength – NutToFret[n-1]
NutToFret[n] = (BridgeToFret[n-1] / 17.817) + NutToFret[n-1]
d = s – (s / (2 ^ (n / 12)))
Note that the root is expressed by exponentiation in the formula above – the twelfth root of 2 is the same as 2 raised to the 1/12th power.
If you calculate the offset for the first fret of any scale length and then divide the scale length by that offset, you get 17.817, the fret spacing constant.
One may well ask why the formula above is not more conventionally used – after all, it can be used to calculate the offset of any fret without having to go through the entire series, and doesn't accumulate errors. The problem is that it requires exponentiation, which is difficult to do with pencil and paper or a four function calculator. But for spreadsheets, scripts and CAD drawing software it is probably the cleanest way to go about the fret calculations.
Now, not everyone is really interested in how this works - some folks just want the answers. The following calculator will calculate fret offsets from the nut for all frets of an instrument, given the scale length.
The tabular results are useful for slotting fretboards by hand or making a fret spacing rule. Would you like a tool that will draw this (and a whole lot more) out for you so that you can import it directly into CAD or other drawing software? Try the "G" Thang Acoustic Guitar Design Tool.
Given that the derivation of the fret spacing constant is based on the offset to the first fret for a given scale length, it is possible to derive constants for each fret position. The first place I saw this approach mentioned in print was in Cumpiano and Natelson's book Guitarmaking Tradition and Technology. If the reciprocal of the constant for each fret offset is taken, you end up with a table of values that can be multiplied by the scale length to yield the offset from the nut to each fret. Here's the table:
A multiple scale length fretboard contains frets which are not perpendicular to the centerline of the fretboard. The picture here is of a bass version of the Colombian instrument called the bandola. It was built by Alberto Paredes and features a multiple scale length fretboard. First featured on the Renaissance instrument called the orpharion, multiple scale length fretboards can improve the intonation of the bass string(s), albeit with some degradation in playability of the instrument. They are found most often in wire strung instruments with extended range on the bass side, that is, with extra bass strings.
Figuring out fret placement for a multiple scale length instrument may seem difficult but the process is fairly straight forward. Fret placement for the treble side is calculated and then placement for the bass side is calculated. The relationship between the two sides is determined from the position of the one fret that is at the same horizontal position in both scales. That is, from the position of the fret that is not slanted, but is oriented straight across the fingerboard and perpendicular to the centerline.
Unlike with the calculators above that return results as offsets from the nut, this calculator returns offsets from the highest end of the nut, which is usually the bass side of the nut. Since both the nut and bridge saddle can be slanted on a multiple scale length instrument this calculator also provides offsets for each end of the nut and each end of the bridge saddle as well.
The best bet for making practical use of the output of the calculator is to start with a rectangular fingerboard blank and scribe a zero line perpendicular to one of the sides at one end of the face and also lines indicating the outside edges of the fingerboard. Scribe offset lines from the outside edges of the fingerboard to indicate placement of the two outermost strings. Then measure and mark the positions of the fret slot ends down both of these string lines, and connect the two end marks for each fret slot with a scribed line.
Some fretted basses and some guitars feature an extended range on the bass string(s) of the instrument. The way these usually work is they have some additional frets behind the nut on the bass string only, and on that string they have a regular fret right at what is the nut position for all the other strings. Then there is some sort of capo mechanism just for the bass string, so you can make use of the extended range if you want it, or capo the bass string at the normal nut position so you can use the instrument in the normal fashion.
The question is, how do you calculate the fret placement for those extra frets? And, if you are curious, what is the scale length of the extended range string? You can figure out both of these by using the very first calculator above, the one that only provides the fret offset from the nut for a single fret, given the scale length and fret number. Using this calculator you specify the scale length for the other strings, then if you specify the fret number as, say, -2 it will return the offset from the nut of a fret placed two fret positions behind the nut. So you can basically just use this calculator to figure out the fret positions of each of the frets for an extended range string. You can figure out the scale length of that string by simply adding the distance from the nut to the farthest fret behind the nut to the nominal scale length.
This is an interesting question for which there is no simple answer. In point of fact, there is no need to be any more accurate about fret placement and compensation than is necessary for the instrument to play in tune. But “play in tune” is not as easy to define as it might at first appear. Humans have very good hearing and very good pitch perception, but there are a large number of factors that affect perception of pitch. The relationship of the harmonics of a vibrating string to each other strongly affect the perception of the pitch of the fundamental. Pitch perception is strongly correlated with culture. What in one culture would be considered way out of tune would be considered right on the money in another. Pitch perception is affected by the instrument under consideration. For example, folks are generally much more tolerant of pitch variability from ideals when the instrument is a fretless instrument such as a violin. This can be tested easily enough by subjecting the individual notes of a solo violin recording of a player known for excellent intonation to pitch analysis. Pitch perception is also affected by the combination of instruments playing. It is even affected by the volume at which a piece is played. And last but not least, each individual has his/her own pitch sensitivity.
But another way to look at this is to ask how accurate you need to be with your calculations so that your results won't be any worse than all the other folks building instruments. Consider this. If you are calculating fret placement for actual use in sawing fret slots, you'll need to mark the calculated fret positions on the fingerboard and then saw the slots. If you are doing the measuring with a ruler and the sawing by hand, it is highly unlikely that you can do any better than 0.01” accuracy. If you are doing this on CNC machinery you might be able to expect better accuracy out of your equipment but do keep in mind that you're sawing wood and that wood moves around considerably with changes in temperature and humidity. For all practical purposes if you do your calculations to three decimal places and place your saw to within 0.01” you'll be at about the industry standard for this kind of work.
If you are building a standard instrument a suitable scale length for it will most likely be that of other instruments of its kind. Upright basses, classical guitars, mandolins, violins, etc. all use conventional scale lengths. An interesting exception to this rule is steel string guitars (and to some extent electric guitars) where a wide range of scale lengths are common. There are some engineering considerations associated with scale length. String tension increases with scale length, but for practical purposes this is not much of an issue as the range of guitar scale lengths is not that great and the designer has no control over the gage of the strings (also a factor in string tension – see the String Tension section) that the end user will use. Guitarists can’t seem to tell the difference when playing instruments of different scale lengths either, and Bob Benedetto has observed that players can’t detect differences in string tension or compliance associated with scale length differences as well. So for guitars probably the most useful approach to selecting a scale length is to just pick a common standard length and stick to it.
Choice of a scale length for non-standard instruments is most constrained by the gages (and to some extent lengths) of strings commonly available. A reasonable approach to this problem is to interpolate from existing instruments. When doing this it is a good idea to keep in mind the physical termination requirements of manufactured strings. Strings made for mandolins terminate in small wire loops, while guitars, electric basses, and upright basses all terminate in balls of different diameter. Termination on the tuning post should also be kept in mind for instruments in the bass range. Wound strings of much greater than 0.045” in diameter cannot have their wound part wound around a tuning post without breaking, and so have thin termination windings at the tuning machine end. This effectively constrains the scale lengths available for a given set of bass strings.