Calculating the Base Length

If one is making a guitar, and before one can place the bridge on the guitar, and translate the string spacing on the nut to the bridge and tailpiece, one has to know the base length of the guitar, the distance from the bridge side of the nut to the bridge line.  The bridge line is the average of all the distances from the nut to the bridge for each string, when that string has correct intonation.

The simplest method is to take the distance from the nut to fret 12 and double it.  But that is just a first estimate.  Because every string has slightly different nut to bridge length due to such effects as string diameter and stiffness, one should place a trial bridge on the guitar being made and determine the intonation and nut to bridge length for each string.  The standard tuning, EADGBE, with produce one base length.  Another tuning, like DGDGBD, with produce a slightly different base length.

If one wonders about the accuracy of the fret placement on the neck, there is a longer way to estimate base length, using the mathematical and musical relationship between the frets.  The frequency of the note on a string fretted at fret 12 is supposed to be twice that of the unfretted string, with the frets for notes in between spaced in a mathematical progression.  This can be expressed as:

[1]  F(12) = 2*F(0), where F(12) is the frequency of the note fretted at fret 12, and F(0) is the unfretted frequency, 0 being the fret designation for the nut.

Intonation effects aside, frequency is proportional to the distance from the fret being used to the bridge.  This can be expressed as:

[2]  L(12) = L(0)/2, where L(12) is the distance from fret 12 to the bridge, and L(0) is the distance from the nut to the bridge, or the base length.

The mathematical progression used to space the frets between the nut and bridge commonly uses either a factor of 17/18 (= 0.9444444…), or more exactly, 2 raised to the negative power of 1/12, or 2^(-1/12) (= 0.943874312).  This can be expressed as:

[3]  L(n+1) = 17*L(n)/18. where L(n+1) represents the fret one closer to the bridge than the fret at L(n) from the bridge, or

[4]  L(n) = L(0)*(17/18)^n, the base length time 17/18 raised to the n power,

or more exactly as:

[5]  L(n+1) = L(n)*2^(-1/12), or

[6]  L(n) = L(0)*2^(-n/12)  Thus, [2] becomes L(12) = L(0)*2^(-12/12) = L(0)/2.


Sheet 1, in 2015-08-25 base length and nut spacing calc.xls

2015-09-04 string spread calc Sheet 1

Sheet 1 in the spreadsheet, 2015-08-25 base length and nut spacing calc.xls, uses a version of equation [5] to fit the base length of a Golden Gate S-96FV neck to a number of measurements, made with a six-inch dial caliper (measuring to the nearest 0.001 inch) and a tape measure (measuring to the nearest 1/64 inch).  Because it uses successive multiplications of [5] instead of using [6], it may be subject to some small “least-significant-digit” computer math errors.  This may account for the deviation of the estimate of nut to fret 12 distance of 12.720” compared to the measured 12.703” with a tape measure.  Small changes in power calculations have much larger effects on answers  than other kinds of non-linear calculations.  On the other hand, the difference is about 1/64”, near the accuracy of the tape measure, which is only 0.13% of the predicted distance.

Sheet 1 also attempted to calculate the power of 2 as linear and quadratic functions of the fret number.  But this over-thought the problem, and produced strange results, likely due to the larger effects of small differences in power expressions.  For all practical purposes, equation [6] should be used.  The user can adjust Sheet 1 to make this so, setting cell B22 to (r =) 12, and solving for a minimum of cell D26 base only upon cell B21, the predicted base length.

Strictly speaking, in that case only one measurement, such as L(0) – L(21) = 17.875 (by tape measure), need be used.  But using more tends to average out measurement errors.  The smaller distances between frets came from the dial-caliper average of the measurements between two frets and the outsides of two frets.  Difference between such measurements and predictions give an idea of the accuracy of fret spacing.  The root-sum of squared errors, 0.026”, indicates the worst possible expected error in distance between frets, which for the average of distances measured, 4.42655”, is only 0.6%.

But this is a lot of trouble to go to.  If you have confidence in the quality of the neck, then multiplying the nut to fret 12 distance by two, and then checking the intonation by using a trial bridge, will likely be good enough.  This approach came from some confusion generated by different fret separation measurements, which produced calculations with a spread of different base lengths too wide to accommodate the designed and expected bridge intonation adjustments.  Hence an entirely over-thought approach.  It happens.