Fretting Calculations

Copyright © 2001 Howard Coleman and Doug Rickard (text, diagrams and software).
E-mail me at Howard<AT SIGN>acousticnotes.org.uk

Two methods are shown to calculate fret positions on stringed instruments: one known loosely as the Rule of 18 and the other as the 12th root of 2. They give a series of positions, using either a calculator or (better) a spreadsheet, measured from the nut and the bridge respectively. Adjustments to these positions are also discussed.

Calculation of Fret Positions from the Nut

This method started out as the "Rule of 18" and was improved in accuracy until it became what I call "The Rule of 17.817". It is mathematically equal to the vibrating length minus the 12th root of 2.  It gives perfect results for my guitars with low action and 0.012-0.053" strings. If fret positions need to be adjusted to get the 12th fret in the middle of the string length then the calculation has not not been carried out with sufficient decimal places. This has allowed a small error to creep in and accumulate. You need three decimal places in the calculation (not the measuring) to prevent this. The spreadsheet I use for calculating fret positions shows no discrepancy at the 12th fret. I can also use it to illustrates my discussion of an alternative fretting system. The Excel 97 and 5.0/95 versions have everything in one file. The Excel version 4.0 is split into three files. Please e-mail me if you need a different format.

• Excel 97
• Excel 5.0/95
• Excel 4.0 (Standard fretting method)
• Excel 4.0 (Nut adjusted fretting method)
• Excel 4.0 (Comparison of the two methods)

Please read my article on intonation as it covers related ground.

Calculation of Fret Positions from the Bridge

To some this method seems more respectable than the other as it has a mathematical pedigree rather than a history of continual refinement starting from the days of hobgoblins.  Note though that it gives exactly the same results. Instead of 17.817 as the magic number it uses  1.0594631 which is the 12th root of 2. I could give 17.817 some respectability by saying that it is the reciprocal of (1 - (reciprocal of the 12th root of 2)) but the hobgoblins and luthiers of old didn't know that so it still doesn't have the pedigree!

First, the theory behind the 12th root of 2.

The frequencies of the notes in the tempered musical scale are as follows.

A     440.00000 Hz
A#   466.16376 Hz
B     493.88330 Hz
C     523.25113 Hz
C#   554.36526 Hz
D     587.32953 Hz
D#   622.25396 Hz
E     659.25510 Hz
F     698.45645 Hz
F#   739.98883 Hz
G     783.99085 Hz
G#   830.60937 Hz
A     880.00000 Hz

We find that if we calculate the ratio between F sharp and F we get 739.98883/698.45645 = 1.059463. The ratio between F and E is now 698.45645/659.25510 = 1.059463. In fact we can take ANY pair of adjacent notes in the tempered scale and the ratio between them will be constant at approximately 1.059463. Now what is magic about this number? It turns out that in mathematical circles the musical scale is what is known as a "geometric progression" where every value in a scale is dependent upon the value just before it in the scale. In the case of the tempered music scale we have exactly twelve intervals in a region of the scale where the values double. That is, as we go from A (440 hertz) to A (880 Hertz), exactly one octave or double the frequency, we have exactly twelve divisions. The value of 1.059463 is in fact the "12th root of 2". This value is fundamental to the tempered scale in all Western music, and it affects the design of all musical instruments designed for the tempered scale. A more exact value for the 12th root of 2 is 1.059463094359... The value goes on for ever, it is unending, however for the purposes of this discussion, and when using a common 4 function calculator with about 8 digits, a value of 1.0594631 is sufficiently accurate for all practical purposes.

Now how does this magic value of the 12th root of 2 affect the average instrument maker. Simple - with all fretted instruments tuned to the tempered scale, the ratio between the length of a string from the bridge to the fret, and the length of the string from the bridge to the next fret, is exactly 12th root of 2, or 1.059463275.

Now since the dawn of invention of musical instruments luthiers have uses all sorts of geometric methods to derive various approximations of the 12th root of 2. Many of these methods have involved starting off at the nut end and making calculations as one progressed towards the bridge. However, just as luthiers over time have learned to use new epoxy resins, new varnishes, new tools, carbon composite materials, etc., it now behoves luthiers to move to a new method of fret calculation where the tool is a simple 4 function calculator, and the measurements are always made from the bridge end as they move towards the nut.

Let us take a very standard example of a guitar with a string length of 650 mm. An old style luthier might use his compass and protractors and work out that the first fret would go 36.5 mm from the nut. By this new method, the luthier would just divide the overall length of the string from the bridge to the nut, 650 mm, by 1.0594631 and get 613.51829 mm as the distance from the bridge to the fret. Mark the position for fret number 1. Now divide this new length, i.e. 613.51829 by 1.0594631 and we get 579.08415 mm as the distance from the bridge to the next fret. Now mark that fret position. This procedure is just repeated as many times as is required.

Let us create a table of values such as would be calculated for an instrument with a string length of 650 mm. The measurements in mm. from the bridge would be -

650.00000    Nut
613.51829    1st fret
579.08415    2nd fret
546.58265    3rd fret
515.90532    4th fret
486.94977    5th fret
459.61937    6th fret
433.82291    7th fret
409.47429    8th fret
386.49226    9th fret
364.80011    10th fret
344.32545    11th fret
324.99994    12th fret

Now note that theoretically the length from the bridge to the 12th fret should be exactly half of the full distance of 650 mm, i.e. it should really be 325.00000 mm. Instead, because of the approximation used for the 12th root of 2 (1.0594631), and errors in the calculator, we are out by 0.00006 mm. This is about one ten thousandth part of the thickness of a pencil line, and for most instrument making can be disregarded.

Now, how is all this put together in practice? Firstly, during the design stage of the instrument various decisions will have already been made. For example, what kind of instrument are we making? Is it a guitar, a dulcimer, a banjo, etc.? What is the scale of the instrument? Is it intended to be a tenor or bass instrument? All of these factors will influence the choice of the overall string length, and consequently the string gauge. If a lighter gauge string is used, a lesser tension will be required to reach the desired note. But a lesser tension means that the string will be somewhat like an elastic band and will not have a nice crisp note. On the other hand, if a heavier gauge string is chosen, a higher tension will be required in the string to reach the required note. Although a crisper note might be achieved, the overall increase in stress on the instrument with six or more strings might mean that there is just too much tension and the whole instrument will crumple up and self destruct. The amount of overall stress that the instrument can withstand is a function of the types of wood used, the method of construction, the glues used, etc. Only experience will teach the luthier what is the correct combination of string length and tension for the instruments they are building.

Once the overall string lengths have been determined, the next thing to determine is where the bridge and the nut will go. In the case of many guitar designs is it common for the 12th fret to go at the shoulder of the instrument. This means that as the 12th fret is exactly midway between the bridge and the nut, it automatically determines where the bridge and nut will go. On other instruments entirely different considerations determine where the bridge and nut will go.

Anyway, once the positions of the bridge and the nut have been determined, the fretting calculations can be commenced. Draw a pencil line down the centre of the finger board. Place a 1000mm steel rule along this line, so that the zero end of the rule is right at the bridge position, and one edge of the rule is along the centre line. Measure the exact distance from the bridge to the nut, let us say 650 mm. Divide this value by our magic number, 1.0594631 which gives us 613.51829 mm. You may round off this value to mark the fret position, BUT DO NOT USE THE ROUNDED OFF VALUE TO DO THE NEXT CALCULATION!!!. Any rounding errors accumulate and increase as you go along, so it is vitally important to maintain the full precision of the numbers for all calculations. Once this fret position has been marked, calculate the next fret position by dividing this value once again by our magic number giving 579.08415 mm. Once again measuring from the bridge end mark the position for the fret. The best method for marking the fret position is to use a small steel engineers square against the steel rule. Continuing on marking the positions in pencil of as many frets as is required. Do a double check and make sure that the 12th fret is exactly at the mid-point of the distance between the bridge and the nut.

Adjustments to the fret positions

The fret positions as marked are just the first part of the measurements involved. There are a number of other corrections that must be allowed for.

The first allowance is for the actual fret wire. The fret wire will usually have a small flat on the top where the string contacts. The front or leading edge of the fret is where the string takes off. In many cases this will be ahead of the tang of the fret wire, so the fret wire must be set back by a very small amount to allow for the distance between the tang and the leading edge. This obviously becomes more important for wide frets, for frets on small instruments or frets high up the neck of other instruments.
 


Another allowance is for the different gauges of strings used. Heavy strings need more compensation at the nut than light strings because they are stretched more by fretting.  This is due to their stiffness making a larger bending radius. This is compensated for by making the overall length of heavy gauge strings longer than light gauge strings. In many cases this is achieved by angling the bridge across the face of the instrument. To be theoretically exact, the bridge, the nut, and the frets should all be angled in different directions to achieve absolutely perfect fretting. However other factors come into play that mask these inaccuracies. For example, for the first few vibrations after a string has been plucked the string has been stretched slightly and the higher tension means that the first few cycles of the note are higher in pitch than a few milliseconds later. It therefore becomes a moot point as to which part of the sound wave should be considered the intended pitch.

Experience and common sense are the luthier's greatest assets in this area, and a good luthier will get to know what allowances can be made in his or her own particular methods of construction to accommodate this method of fretting.

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Last revised: June 24, 2007.