In their book Guitarmaking authors Natelson and Cumpiano included a table containing pre-calculated fret spaces for a guitar with a scale length of 25.4 inches, which is the scale length I chose to use. It would have been perfectly fine for me to take this information without giving a second thought as to how it was obtained and go on my merry way with the layout of my fretboard. But I couldn’t do that, I wanted to understand how those numbers were being calculated, so I set off to try and figure it out for myself. Perhaps there is someone else that might find this not so rigorous derivation of interest.
I knew a little bit about music theory when I started, but I was by no means an expert (and I’m still not). I knew that in western music everything is referenced off of concert A, a note with its fundamental frequency component ringing at 440 Hz. I also knew that there were 12 steps or notes in western music and that they are arranged such that each note in a lower 12 note set will have half of the frequency of its partner in the 12 note set directly above it, and likewise each note in a higher 12 note set will have twice the frequency of its respective partner in the 12 note set directly below it. This table created by the Physics department of MTU helps me to visualize it, looking at the frequency of the A notes makes the pattern readily obvious: 440 Hz to 220 Hz to 110 Hz and so on.
So I knew I needed an equation that would give me an nth frequency, a f(n) = frequency as a function of n so to speak, where n represents some number of steps or notes away from the base or reference frequency, f0, like the equation below.
f(n) = f0*(some function of n)
So what’s that function of n? I tried for a while to figure it out by myself and finally gave up and looked around on the internet for some help and to see if I was on the right track. Again I found the Physics department at MTU to be quite helpful. It turns out I was on the right track, the base equation I got from MTU below is exactly what I was looking for, where a = 2 and n is replaced with n/12.
f(n) = f0*(a)n (1)
Doing a quick check shows that when n=0, f(0) simply equals f0 which makes sense, and when n=12 (meaning we are 12 steps higher than f0, which means we are starting to repeat the note cycle) f(12) equals twice the frequency of f0 which is exactly what was desired. Also when n=-12 (meaning we are 12 steps lower than f0, again we are repeating the note cycle but in the other direction) f(-12) equals half the frequency of f0 which is exactly what was desired. Now I had an equation to describe the relationship between a frequency n steps or notes away from a reference frequency which followed the rules of western music. The next step was to try and find a relationship between Equation 1 and the fret spacing on a guitar.
I had taken a couple physics courses in college and somewhere in there I had learned how the frequency of a vibrating string could be represented by string length, tension and some other factors, but all that information had since been forgotten. But it was no matter, I found out everything I needed to know and then some through good ol’ Wikipedia. The Vibrating String article provided the equation below (with a good derivation) for the frequency of a vibrating string, where L is the length of the string (the scale length in my case), T is the string tension and μ is the linear density of the string.
f = 1/(2L)*√(T/μ) (2)
It should be mentioned that Equation 2 relies on the assumption that vibrational movements normal to the axis of the string are small compared to the length of the string, which is the case for a guitar. Using Equation 2, Equation 1 can be written only in terms of string or scale length, L, so that the frequency terms in Equation 1 can be replaced as
f(n) = 1/(2L(n))*√(T/μ)
f0 = 1/(2L0)*√(T/μ)
and when inserted into Equation 2 produces
1/(2L(n))*√(T/μ) = [1/(2L0)*√(T/μ)]*(a)n
which simplifies to
L(n) = L0 /2n/12 (3)
Equation 3 gives the scale length of the nth fret number (distance from the nth fret to the saddle), where L0 is the guitars total scale length (distance from the face of the nut (zeroth fret) to the saddle). In terms of practicality Equation 3 isn’t very useful, it makes much more sense to measure the fret distances from the face of the nut, the top of the fretboard in my case, rather than the saddle. To do that L(n) in Equation 3 can be redefined as the distance from the face of the nut to the fret of interest, or in other words the result of Equation 3 must be subtracted from the guitars total scale length, as shown in Equation 4.
L(n) =L0 – L0 /2n/12 (4)
And that’s all there is to it. Plug in the desired scale length, L0, and the desired fret number, n, and you get the distance from the face of the nut to that fret. It works great in some sort of spreadsheet program. But before cutting fret slots I think its a really good idea to check your work against an online calculator (I like Stewmac’s) or an existing table.
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Planning and Design | Fret Spacing Calculation | Template and Workboard | Building the Neck | Making the Body Plates | Soundhole Rosette | Soundboard Bracing | Bending the Sides | Gluing Sides to the Soundboard | Creating and Installing the Back Plate | Binding the Body | Making the Dovetail Neck Joint | Carving the Neck | Making the Fretboard | Making the Bridge | Creating the Headstock Inlay | Finishing | The End Product
you are both eccelent luthiers. i’ve your book (guitarmaking tradiction and tecnology) and as far as i’m concerned is one of the best guitarbuilding book .i build guitars and mandolins just for pleasure i’m a totally self tout guy the only guide i ever hed was your book.well done.i leave in the south of italy,in calabria .every thing of the best,regards, Franco Rodi.
I found their book to be very complete reference also.