Formula for Computing a Fret Scale

Pythagoras was first to experiment with determining scalar intervals... and later, in the 16th century, Vicenzo Gallelei was credited with developing the "rule of 18". . . used for centuries by instrument makers to determine the fret scale of their instruments. For any given vibrating string length they would simply divide the length of the string by 18... yielding the distance from the nut to the first fret. By subtracting that figure from the original string length they arrived at a new shorter scale measurement which was then divided once again by 18 and resulted in the distance between the first and second frets. They continued in this manner until the entire scale was determined. Over the years several variations on this theme have been developed... The divisor has been refined, (based on a complex mathematical formula that utilizes the 12th root of 2) resulting in more accurate scales; other approaches have been tried as well. Here is the formula I use to determine my fret scales: it is a recurrence formula my brother Gary helped me refine.

vsl = vibrating string length (this is the basic scale length originating on the fretboard side of the nut and terminating at the leading edge of the saddle. It does not include a compensation factor.

c = the divisor -- I use the factor 17.817 as the constant in this formula, but others may use a different quantity.

i = 1... 30 or as many frets as you require.

L = the distance from the nut to the first fret position in millimeters.

L1 = the distance between the 1st fret and the nut in millimeters.

Li = the distance between the "i"th fret and the nut in millimeters.

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