These diagrams will help you to better understand how guitar setup and guitar intonation work.
The above diagram illustrates the intonation problems associated with a traditional guitar setup, without any nut compensation and with the saddle intonated comparing notes at the 12th fret to the open strings.
The amount of error is different on each guitar, and is also affected by the type and gauge of the strings. Low action at the nut will minimize the intonation error.
This is the traditional intonation method, standard on most guitars, but it leaves a lot to be desired, especially for serious players. Anyone with an ear would readjust the tuning to make various chords sound better after tuning with a tuner, and would likely have to re-tune when changing keys or when placing a capo.
Note also that the errors are inverted at the 12th fret (sharp becomes flat) so any tweaking you do for the benefit of the lower frets will most likely make the upper frets sound worse.
This diagram assumes that the saddle has been compensated for each string individually - no problem on most electric guitars, but most acoustic guitars have a saddle with a single straight ridge. Let's see what that would look like in the next diagram.
In this diagram, a variation of Diagram 1, the saddle is straight across, instead of being compensated for each string. Let's assume that it is ideally positioned so that the 2nd and 5th strings are compensated perfectly. That would leave the low E and unwound G under-compensated (sharp) at the 12th fret, and High E and D over-compensated (a bit flat at the 12th fret). Notice that the pitch line for low E and plain G is more horizontal than on diagram 1 - that's a good thing.
Now imagine tuning the low E and unwound G strings downward - the sharp pitch line would drop down, making the E and plain G less sharp, compromising with the open string, which would become somewhat flat.
Here again, the player would adjust the tuning to get the best sound. The result of Diagram 1a may be an improvement over diagram 1, suggesting that it may help to intentionally intone strings that are sharp at the lower frets to also be sharp at the 12th (while open is in tune).
I hope this stimulates some creative thinking of how intonation adjustments work. There are lots of diagrams I can think of, and I hope you can to, but, to keep things simple, I'll continue with only the most important ones.
The above diagram illustrates the intonation improvement and shows the problems still remaining after moving the nut forward.
Notice that the 6th and unwound third are less sharp starting at the 1st fret, so the difference between open and fretted notes are less noticeable. Also, the notes above fret 12 are less flat. The 1st, unwound 3rd and 4th, however, will now be overcompensated, and so will play somewhat flat on the lower frets, with sharping above the 12th fret.
Although this intonation method reduces the maximum inharmony between open and fretted strings, as you can see from the diagram, the intonation error between the fretted strings has stayed the same.
Again, tweaking tuning by ear, or by offsets, is be expected - and the results will be somewhat improved over the traditional intonation method.
By now you may be asking the question: "Since the fretted strings are not in tune with the open strings (without nut compensation) why do we use the opens strings, along with the 12th fret, to intonate the bridge saddle?" That's a good question, and there are three answers: 1) It's the most convenient way to do it. 2) It's the quickest and cheapest way to do it! 3) Many people seem to be satisfied with it, or unaware that there may be a way to achieve better results.
In this diagram we illustrate the result of moving the nut forward, Intonating the saddle by comparing the pitch at fret 14 with that at fret 2, and then tuning at fret 2.
Intonating by comparing fret 14 to fret 2 and tuning at fret 2 causes most fretted notes to play in very good tune. Only the open notes are out of tune, but not as much as in Diagram 1, because of moving the nut forward. Intonation is greatly improved, even when playing with a capo!
For those who seldom play actual open strings, this setup is excellent, except for the inconvenience of having to tune to fretted notes.
Instead of having to tune to fretted notes, the cents-off of the open strings could be noted, and entered (with reverse sign) as tuning offsets into a strobe tuner; then you could tune the open strings and get the same effect.
It's very interesting to notice here that guitars could be built in this way at no extra per-unit cost over the traditional setup; only the cost of re-tooling.
For those that play open strings a lot, what's most important is fully compensating the nut, regardless of the method of saddle intonation.
This diagram shows the result of fully compensating the nut, compensating each string individually, after having set the fretboard intonation by comparing the pitch of two notes about an octave apart, on the fretboard.
Notes on the fretboard play in very good tune, and open strings are now in tune with the fretboard.
The intonation improvement of a guitar set up this way is amazing!
For an explanation of the slight sharping on the very high frets, see The Clothesline Effect.