Determining Scale Length Correctly

Guitarsmith Richard Stanley says:

When you start talking about scale length you start with the theoretical string length. If you devise a fret scale in a coherent way, the theoretical string length is twice the distance from the 12th fret to the theoretical fingerboard nut location, which, for example, on a 25.5" scale would be 12.75 inches from the fingerboard nut to the 12th fret.

The basic theory is related to the same math that is behind pitches in the equal tempered scale, which is the 12th root of 2. To determine the pitches of notes in equal temperament, you start at a given pitch, like A=440HZ, and you multiply 440 by the 12th root of 2, which is about 1.05946, and that gives you the Bb note (466.16HZ). Next, you multiply that by the 12th root of 2, and it gives you the B, and so forth for each succeeding pitch. Using the 12th root of two for calculating the 12 steps of the chromatic scale produces a 12th note whose pitch is exactly twice that of the first note (A at the first octave will =880HZ).

Since the relationship between pitch and string length is inversely proportional (one half the string length = twice the pitch) we use the inverse formula to determine the fret locations: Starting with the theoretical string length and using 12th root of 0.5 (one half) or ½ which is the inverse of two. The 12th root of 0.5 is approximately 0.943874 and to calculate the fret spacing we multiply the theoretical string length by the 12th root of 0.5 and subtract the result from the theoretical string length. The result of that subtraction is then multiplied by the 12th root of 0.5 to locate the second and succeeding frets. Notice that this approach locates the frets relative to the theoretical fingerboard nut location. Using this method will locate the 12th fret at exactly ½ of the theoretical string length (before making compensations at the nut for string stretch factor, and at the bridge for the string stiffness factor).

I spent two years on all this and went back to basic physics for a lot of this stuff when I found out some time in the mid 70s that fret scale problems were a significant issue. I don't know what the guitar companies use, but this has been my approach. And I call it a unified fret scale (i.e. one consistent theoretical string length applied to the entire fingerboard) — based on the 12th root of 2 and applied as the 12th root of .5.