Calculating Fret Positions

This page is all about the calculation of the locations of the frets of a fretted instrument. For those that are interested, there is a bit of history about the way this has been historically calculated. There is also information for folks that want to do the calculations to figure out fret positions themselves. Not everyone has an academic interest in how all of this works, so this page includes some easy-to-use calculators that will perform all the calculations for you.

Information and calculators for the now old-fashioned Rule of 18 are included for those building reproductions of old instruments. Of course the info and calculators needed to calculate fret positions for modern instruments, using the 12^th root of 2 are here as well. There are also instructions and calculators for calculating the fret locations of multiscale (so called "fanned fret") fretboards, for microtonal instruments, and for instruments which feature extended range bass strings. Discussions about fret placement accuracy and choosing scale lengths are also included. See the Table of Contents for details.

The historical technique for calculating the location for each fret is called the Rule of 18, and it involves successively dividing the scale length minus the distance from the nut 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.

As mentioned in my book Building the Steel String Acoustic Guitar, the origins of this technique are unknown, but it is likely that builders of early instruments, like lutes with tied on gut frets, observed that the first fret was placed from the nut at approximately the scale length divided by 18; and that the second fret was placed from the second fret at approximately the distance from the bridge to the first fret divided by 18; and that this relationship was the same for all the rest of the frets all the way up the neck.

This fret placement technique was in use well into the 20^th century, even though a more mathematically accurate technique was generally available, possibly as early as the early 20^th century. The reason its use persisted is that, with appropriate compensation, it provides accurate intonation. The entire subject of compensation and its relationship with fret placement is quite complex. But as a general rule, modern instruments do not make use of the Rule of 18, so it is a wise idea to avoid its use unless a historical instrument is being built.

Calculating fret placement using the Rule of 18 can easily be done with pencil and paper or calculator. As described above, the calculations are performed sequentially, with the calculation for each fret making use of the location of the previous fret. These days it is unnecessary to perform such calculations manually. Special-purpose calculators such as the ones found below will perform the calculations accurately.

Here is a calculator that will calculate the fret position of a single fret, using the historical Rule of 18. You enter the scale length and the fret number, click the Calculate button, and the calculator provides the distance from the nut to that fret. Note that any units of measurement can be used (inches, mm, etc.). The value returned will be in the same units as you used for scale length.

Note that unless you are laying out frets for an antique instrument, this is probably not the calculator you want to use.

Note that when the Rule of 18 is used to calculate fret locations, the location of the 12^th fret is not half of the scale length.

Note to the mathematically inclined: Although it is possible to perform this calculation iteratively as described above, Rule of 18 fret position is accurately described by a simple exponentiation association function:

$\mathit{d}=\mathit{s}(1-\mathbf{exp}(-\mathit{kn}))$

where:

k = 5.71584144995393e-2.

Most of the time when folks make use of a fret position calculator they want the locations of all the frets. This Rule of 18 calculator takes the scale length and generates a table of the positions of 24 frets for that scale length.

These days we calculate fret positions using the 12^th root of 2 ( $\sqrt[12]{2}$ or $2^{1/12}$ ), which is the same constant used to calculate the frequencies of notes in equal temperament. The calculators below use this constant directly. In the olden days before computing power was readily available it was quite time-consuming to calculate this value over and over again. But if it can be calculated just once and used to determine the location of the first fret, then the scale length can be divided by the distance from nut to first fret. The result is a number, the approximate value of which is 17.817, which can be used in exactly the same way as the value 18 was used when sequentially calculating fret placement using the Rule of 18.

The invention of the pocket calculator made it possible to make use of this more accurate constant, and so in the recent past we conventionally calculated 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 12^th 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.

Some references specify a constant of 17.835, which is mathematically unrelated to the constant given above 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 bending stiffness and doesn't need to be stretched to be fretted. Since real strings don't behave that way, even the equal temperament based fret layout 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 if you want to sequentially calculate fret locations yourself? Unless you are building or repairing historical instruments use the equal temperament constant of 17.817. It is the currently accepted standard.

The conventional sequential calculation technique for calculating fret position is shown by example below. Use this to make your own fret rule if the only tools you have to do the math are a simple calculator or pencil and paper. I've also included information on the derivation of the fretting constant and an even more accurate formula you can use if you have spreadsheet or programming experience. There are Javascript calculators that use that formula below.

Most folks that build instruments will never need to use this info, as the lutherie suppliers all provide fret slotting templates as well as 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. And of course you can use the calculators below to do the calculation work for you.

In this section we'll discuss how to calculate fret positions, if what you have for tools are pencil and paper and maybe a four function calculator. If you don't want to go through all of this, skip down to the next section where you'll find a nice calculator that will do all the math for you.

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[1] = 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[1] = ScaleLength – NutToFret[1]

NutToFret[2] = (BridgeToFret[1] / 17.817) + NutToFret[1]

Repeat this process for each fret position on your fingerboard. If you consider that the nut is the same as a 0^th 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]

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. A much simpler way to end up with the same results is to use one of the calculators below with a scale length of 1. Here's the table:

Consider that the fundamental pitch of a vibrating perfect string goes up an octave if the length of the string is halved. Also consider that there are twelve tones in the equally tempered scale. From this it should be apparent that fret position offset will be a function of the twelfth root of two. The twelfth root of two is the number which, if multiplied by itself twelve times, would equal two. It is approximately 1.0594630943593. The offset from the nut can be calculated for any fret by using the following formula, presented in spreadsheet format and as a calculator applet:

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 1/12^th 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 mentioned in the previous section.

One may well ask why the formula above was 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.

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 definitiveness of tonality of the bass string(s), albeit with some degradation in playability of the instrument. They are found most often in steel-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 straightforward. Fret placement for the treble side is calculated and then placement for the bass side is calculated. The positional 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.

Note that there are other ways in which the positional relationship can be expressed. It is often expressed simply by indicating how much higher the nut is on the bass side than the treble side. But the reason it is expressed here as the number of the perpendicular fret is because this is the way the majority of musicians think about this, that is: where on the fretboard do the frets stop slanting toward the nut and start slanting toward the bridge. Note also that there is no need to specify that the transition point lands exactly on one fret. See the information about this below the calculator.

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. Note that compensation is not included in this calculator's results for placement of the bridge saddle.

By the way, you can do some special tricks with the non-slanted fret number parameter. Specifying a value of zero means the nut (i.e. the zeroth fret) will not be slanted. Want to make the bridge saddle not be slanted? Try a big number, like 1000, in this field. Don't want any of the frets to be non-slanted, but want the change from slanted one way to slanted the other way to be somewhere between two frets, say, between frets 7 and 8? Try a value like 7.5 in this field.

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 Modern (12^th Root of 2) Single Fret Position Calculator above, the one that 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 of the instrument, 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.

Building a microtonal instrument, that is, one with tones between the conventional scale tones, and want to figure out where to locate those additional frets? You can figure these out by using the Modern (12^th Root of 2) Single Fret Position Calculator above, the one that 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, then you specify the fret number as a decimal fraction. If you are building a quarter tone instrument for example, entering fret number 1.50 (i.e fret #1 plus 50 cents) will give you the location of the fret between frets 1 and 2. For an eighth tone instrument, entering fret numbers 1.25, 1.50, and 1.75 will give you the locations of the three frets between frets 1 and 2.

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 (compared to other animals), and musicians have very good (but hardly mathematically perfect) musical pitch perception (compared to other humans), but there are a large number of factors that affect perception of pitch. The relationship of the partials of a vibrating string to each other can 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. And don't forget that all of this locates the slots that the frets sit in, but it is the relative locations of the fret crowns that matter. Typically we simply file the crowns by hand with a crowning file which hardly locates them accurately relative to the fret slots. 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, all else being held constant, 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 gauge 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 subtly 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 gauges (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 danger of 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.

Retailers, do you want a calculator like the ones here for your commercial website? Contact me about custom software.