Guitar Build 4: The mathematics of fretboards

Apparently, we have Pythagoras to thank for the musical scale. Using a device called the monochord, he was the first to notice the relationship between string length and octaves—namely, that halving the length of a string produces the same note one octave higher. An octave is twice the frequency of the original note (ratio 2:1).

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Fretboard before shaping. The cut at the bottom will be the nut, or the zero fret.

He also noticed that certain notes sound good together, a concept called consonance. For example, the note we call E sounds good with A. E is created from A (as its fifth) with the string ratio 3:2. A has frequency f=110 cycles per second (cps), so E has frequency \frac{3}{2}f =165 cps. D is created from A with ratio 4:3 (as its fourth) with frequency \frac{4}{3}f. And so on. Not all simple ratios produce consonance with A (5:4 does not), but using fourths and fifths around A, D, and E, we can produce a pentatonic scale.

 

At some point we decided 12 was the right number of notes, so the task was to produce a set of 11 frequencies between f and 2f that “sounded right.” The search eventually produced “the tempered scale” having the property f(n)=a f(n-1). This is a recursion, so f(2) = a f(1); f(3) = a f(2) = a^2f(1); f(4) = a f(3) = a^3f(1); and so on until we get f(12) = a^{12}f(1). But we also know f(12)=2f(1), so 2f(1) = a^{12}f(1) and our constant

a = \sqrt[\leftroot{-2}\uproot{3}12]{2} \approx 1.059463094359295264561825294946341700779204317494185628559.

(Much of the above comes from a nice article at Noyce Guitars.)

We are interested in the distance between frets on a fretboard, or to avoid accumulating errors, the distance from the nut to a particular fret. If the length of the string is s, the distance to the nth fret has the formula:

d[s,n]=s-\frac{s}{2^{n/12}}.

Notice that for the 12th fret,

2^{12/12}=2,

and we divide the string in half, as expected. See this nice article for a fret spacing calculator. So there we have it, a nice way to calculate the distance from the nut to each fret. Because cuts this precise would have taken so much time and because they are so critical to the sound, we ordered our fretboards from a guitar parts distributor.

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Maple top after glue up. Glue line is visible across the middle.

This week we glued up our maple tops in the same way we glued the two piece mahogany backs. Both faces had been planed, so we had only to joint one side of each piece and then glue and clamp.

Next we turned to the fretboard. After marking the width at the zero and 16th frets last week, we were ready to mark the cut lines and trim them on the band saw, leaving about 1/8th inch room to bring them down to exact shape with the bench plane. For the planing, we are using Ted’s “shooting board,” which is a jig that allows use of the plane as an edge jointer. I finished the night with my fretboard edges straightened, but I will have to remove more stock next week to bring it to the right width. We are checking the width with calipers, because width at the 16th fret is critical for the guitar’s cutaway to fair into the neck properly.

Production Notes

  • When cutting the fretboard on the bandsaw, leave the cut proud! You’ll take this off with the bench plane anyway.
  • Razor sharp and tuned plane is essential for every woodworking operation.
  • Need to build a shooting board for fretboard trimming.

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