Some purists shun compensation at the peg head end of the string. Nobody disputes that adjustment at the bridge alone can adjust the total length of the string, and thus where the middle of the string is, and that the middle of the string should rest at the twelfth fret or slightly above it to compensate for fretted tension. Everyone agrees on this.
But the interval between the open string and the first fretted note is also crucial to overall intonation. Think of all of the first position chords with open strings juxtaposed to fretted notes. This area of the fret board more than any other exhibits troublesome intonation.
Adjusting the string end positions at both ends in the field / in reality is not a formulaic process. Variations in action and the concomitant difference in fretted-string-tension at any position on any neck produce uneven and varying outcomes. Variations exhibited on any particular instrument change with changes of string diameter and composition. Nut compensation is essentially changing the distance between the open string (recall that this is the note that you calibrate to your tuner determining open string tension) and all of the fretted notes. This is most evident in the that often pitchy move from open string to the first fret. Especially on shorter scale guitars given the shorter fret to fret distances, the pitch up effect of increasing the string tension as you depress / fret the string is a higher proportion of the pitch delta. Short scale instruments with high string tension can particularly benefit from individual nut position adjustments.
A combination of appropriate action and compensation at both ends of the string is the way to get as close to equal temperament as one can on any guitar with a given set of strings, tuned in a particular way. One aims to make as many areas of the fret board, in a given tuning, play as close as possible to in tune. Recall that each string behaves differently. This is a time consuming process sometimes, in the words of master luthier Mike McConnville, "a moving target". Sometimes compromises due to difficult to regulate or uneven action are necessary.
We can make your short scale guitar play in tune, even first position chords. Set up an appointment to get an estimate.
Any compensated nut is adjusted to make sense with a given bridge set up, neck relief, and a given guage of strings. You can't change string guages and expect the same set up to work, this is the case whether intonating just at the bridge or at both ends.
A compensated nut fitting always includes a set up and a full intonation. It is a multi stage process and takes significant time to get right. We start always with a raw bone or synthetic blank and make a custom nut from scratch.
Also, for acoustic guitars the normal range of saddle adjustment possibilities are limited by the narrow width of the saddle. We can make cantilevered bridge saddles that expand the compensations that can be achieved at the bridge on a regular acoustic guitar. Sometimes the bridge saddle as it is can work and just a compensated nut can make a big difference. Sometimes the compensated nut on its own doesn't really work over the whole fret board and some remedy at the bridge is required. Bridge slots can be filled and new saddles slots can be routed out in extreme cases, but a cantilevered nut can be a much faster and cheaper way to take a difficult to tune acoustic guitar with a poorly placed saddle and make it play in tune. Adding both a compensated nut and cantilevered bridge saddle can make minor and even significant saddle misplacement scenarios workable.