Spiral Math

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xylenz

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Jan 3, 2008
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I am an amateur guitar builder. I am trying to solve a problem for a hobby project but my math skills are not up to the task:

To calculate the locations for a number (N) of frets along a guitar string of length (L) we start at the nut and divide the remaining length of string by 17.817 (a number related to the 12th root of 2 = 1.059463094359). The frets get successively closer together as we move closer to the bridge.

The problem is how do you construct a spiral such that as that spiral is rolled along the string from the nut to the bridge it contacts a fret at every R degrees of rotation? R is a constant angle. My N=32, L=30", and R=36degrees but I would prefer a more general solution.

At first I thought that I could simply make concentric circles where each fret spacing subtends R degrees of each circle's circumference, then simply mark the edge of each segment and connect the dots. Unfortunately, this will not work because the axis changes height continually as it is rolled. It might be a logarithmic spiral of some sort. Although the logarithmic spiral formula on wikipedia baffles me (what is a & b?) I did find some more understandable java code that plots a log spiral. That doesnt really help me to solve this problem though. If I could understand the solution I might be able to edit the code to provide a general solution. I am a java engineer. Any clue?

Thanks,
Clark
 
Your REAL problem is that the spiral is not constant. It changes radius only as you approach completion of each lap. Once the change settles down (at the very beginning, for example), the radius is constant until you hit the next bump. Further, the abruptness of the changes decreases and the whole operation approaches the smooth, continuous change you are suggesting. You can approximate the real behavior with a smooth spiral, but you won't hit the right notes. Better to roll out the tape on a real guitar and copy the fret locations. You should be able to scale this as desired for larger or smaller models. Sometimes, the real solution doesn't quite synch up with the theoretical solution.
 
Think of it as a fretless guitar with "fret" lines. The spiral exists. I can prove it with a log spiral made of triangles using trig, just not with smooth curves. There IS a solution.
 
I would think this is a logarithmic spiral you are looking for. Afterall, the radius of a log spiral grows exponentially with the angle.

From the diagram, your angle alpha is 36 degrees and you know the length between the P points?.

If the string is the spiral, then P1 would be the nut, then P2 would be the first fret and so forth?.

You have 32 frets, so you go up to P33 with a distance of 30 inches.

I may be off base. If so, I am sorry, but this seems perhaps on the right track.

You know your angle is 36 degrees and the distance between the frets. What is it you want?. A general formula for this particular logarithmic spiral?. In that event, you need to find a and b.

I believe a=cot(36).

See here:

http://www.2dcurves.com/spiral/spirallo.html
 

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I think I did not adequately explain my position.

I have no doubt about the existence of a solution. It's easy enough to draw a pretty picture. Now find a material that actually does that. You will have the problems I described previously unless you select a very rigid material, then good luck bending, rolling, and unrolling it.
 
TkHunney, the materials are irrelevant. Its a thought problem that gives the solution to another undisclosed application.

Thanks Galactus! That is exactly what I needed. That is the only site ive seen that actually shows what a and b are supposed to represent!
I need to solve for b knowing a and the distance along the curve from P1 to P2.
 
Well, there I go dabbling in reality again. It's a bad habit. :shock:
 
xylenz said:
TkHunney, the materials are irrelevant. Its a thought problem....
Click to expand...
Then perhaps you should have presented this as being a "thought problem", rather than a real-world application (designing a guitar in a particular manner) in which practicalities such as materials would matter.

Eliz.
 
Ug. This has gone completely the wrong way. Let me clear it up. This IS a real problem and not just theory. tkhunney has assumed that he knew what i am doing with the spiral and then deduced that it is physically impossible. But his initial assumption about what I am doing with the spiral is incorrect. I never explained what it was that I was doing with the spiral. I only explained the problem. It has nothing to do with measuring the fret locations on a guitar... at all. Anyway, I have since modeled it in 3D CAD and found the solution by trial and error. There is another level of refinement that I still need to figure out. I may have to ask another question about it. Stay tuned...
 
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