This is a nice article about nut and saddle intonation on classical guitars. I have read it and had to skip most of the hard core mathematics in the middle, but it seems like there are rather simple and good enough formulas that can be used instead of my empiric method. You only need to make a gizmo to measure the elasticity of steel strings instead of the nylon strings in the article and put in the numbers. I might give it a try...
I learned that the saddle compensation is mainly needed because the pitch of the overtones increases with the stiffness of the strings. The nut compensation is needed mainly for the elasticity increase of the pitch when fretting. This coincides with my experience, the nut compensation mainly affects the upper part of the fretboard, the saddle compensation the lower part. The article clearly states that intonation will never be good over the whole fretboard with only a saddle compensation, a matching nut compensation is always needed for that.
Also, it's important to stretch the strings to it's final elasticity before doing the nut compensation. I'm lucky because I do the intonation last after three days of vibration and by that time the strings are stable.
Thanks for the post. If there is anything useful in the math, I'll translate and post. Integral calculus is used to compute changes-of-rate and it appears here to compute the outcome of changes in bridge and nut location.
Rates of change, i.e derivatives, are the subject of differential calculus. I don't see any integrals in that article.
Got a real world response. Per Hallgren, who is a well known classical guitar builder here in Sweden, read the paper 1995 and tried it out for real. He said it was a lot of work to use the concept and he stopped using it because nylon strings are not consistent. The well intonated guitar was not as well intonated when a new brand of strings was put on the guitar. He reverted to cutting the top of the fretboard 0.4 mm and using the standard 12th fret intonation.
Steel strings are much more stable.