The main use for nut compensation is to help correct intonation problems on the low frets. However, to do that, it has to be used correctly in conjunction with saddle compensation. Nut compensation used alone affects the frequency of notes on all frets equally. Think of it like this: Move the nut 1mm toward the first fret and the string will go ~3 cents sharp. That’s because the distance to the first fret is typically ~33mm and one fret distance is a semitone which equals 100 cents. So it’s simple proportion. However, the open string is normally tuned true, so the tension on the string is released to bring the open string tuning down by 3 cents to make it true, which makes the notes on all the other frets 3 cents flatter. So the net result of 1mm nut compensation towards the first fret is to flatten all the notes except the open string by ~3 cents.
String compensation at the saddle alters each fretted note differently. A change in position of the saddle has much more pitch effect on the high frets than it does on the low frets.hillguitar wrote: ↑Fri Feb 23, 2018 7:14 pmIndividual string compensation of the bridge can certainly make the 12th fret octaves perfect, all of which seems like a good thing. These refinements would affect intervals and harmonies all over the fingerboard to some varying extent, though maybe not as comprehensively...
Using these separate effects of nut and saddle compensation, together with a bit of scale length alteration (altering the string length compared to the scale length used to determine the fret positions) means that the typical errors on most guitars of the low frets playing sharp can be totally eliminated, so that accurate ET can be achieved. The fret positions do not have to move at all. Modelled results for a steel string guitar below. Classical strings give an even flatter chart. All the modeling, testing and maths are in the book.
Typical modeled intonation errors for a conventional saddle-only compensated steel string guitar:
Modeled results of full nut and saddle compensation, same guitar as above:
That’s the theory and in practice it works pretty well. What players think about this system has been volunteered in this comment on the effectiveness of nut and saddle compensation and also this one here.
However, the real world can be cruel to us. For example, most nylon strings will play sharp by ~ 3 cents just due to the heating effect caused by your finger on the string before you actually fret a note. Which means perhaps you should tune to fretted notes rather than open strings. Fluorocarbon strings are better behaved in this respect. Further, body resonances will also couple with the string vibrations to shift the frequency of the played note. This one can be serious with shifts of ~20 cents being commonplace for notes very close to body resonances. They are fixable, but not via fretboard geometry. That’s part of what modal tuning is about. Get all these various challenges sorted and you will have a guitar that plays accurate equal temperament. For a responsive guitar it takes more than just work on the fretboard geometry. It’s certainly worth it when it’s done right; like Fiji water compared to LA tap water, to paraphrase Kenny.