GUITAR FRET BOARD: simple solution for layout out fret positions

[donsmac]

Here's a simple solution I worked out for laying out the fret positions on a guitar fret board. Once the geometry is sketched out roughly, a co-linear constraint is added to the Twelfth fret and center line of the box. Done!

[mstrater]

This felt wrong to me, and I spent a long time trying to prove it, but you're absolutely correct! I even dredged up my old algebra knowledge to verify it to myself geometrically. The trick is that the colinear constraint adjusts the angle of the slanted line to be precisely what's needed to fit the 12 line segments into half of the rectangles width. This angle value is arctan(1-2^(-1/12)) which is ~3.212deg. Very cool trick and thanks for the fun little math problem. 🙂

[jeff_strater]

@donsmac - that is...  amazing!  My son and I spent this evening proving to ourselves that it works, because it seemed too good to be true...

how did you come up with that?

[donsmac]

I could not find a satisfactory solution online so I decided to figure it out. After some unsuccessful attempts I laid down and thought about it and realized that the frets are positioned such that the overall length multiplied by the twelfth root of 0.5 will give the distance from the 'bridge' to first fret. Keep multiplying the result by the twelfth root of 0.5 (0.943874312681694) to get all the fret positions from the bridge. That math is easier than what I found online.

Once I realized it's simply a scaling factor, I sketched something where the lines were scaling by any factor, then forced the result by adding the co-linear constraint. 

Thanks guys for your responses.

[rishabh.bisht]

That's an awesome trick! Thanks for sharing!