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Perhaps someone can remove the mystery for me.

I understand how to adjust intonation, and I think I understand what proper intonation is, but what I don't get is why sometimes one string will be way off while everything else seems to be fine.

I've got a Martin acoustic in with a typical compensated saddle that to the eye looks fine, after checking the intonation every string is within 4 cents, except the A which is 15 cents sharp.

There was over .020 relief so I flattened the neck out just to see quickly if that would take care of things, it definitely changed the intonation but the A was still relatively just as far out. Perhaps the compensation is bad on that one string? Or maybe the string is just no good? He uses medium guage coated Martin SP's.

Always start with the strings. We are talking the cheapest part of the guitar.... literally a few cents worth of steel.

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"Act your age, not your shoe size" - Prince

The amount that a string plays sharp is a function of its stiffness, particularly at its ends. The stiffness is a function of core diameter, which explains why the B and D strings need more compensation than the G, it has a smaller core. You can actually calculate the amount that the string needs to be lengthened but the math is not trivial so most of us just do it by ear and mechanically move the break point. Your string might be sharp for a variety of reasons - that particular string, the break point of the saddle being in the wrong point, too high of action.

4 cents I wouldn't worry about, 15 is a lot. Try another string, get the action (and relief) to reasonable levels (0.020 is way too much), if that doesn't help make a new saddle with the break point pushed back. One little trick when you are making a saddle is to leave the top flat, get it down to near the action you want and put a little piece of wire on top under the strings (a piece of B string works nicely). Move it back and forth just like you would a saddle on a strat until you get the intonation you want, then mark and make that your break point. Gets really fun on a twelve string but is possible.

What is the guitar body's resonant frequency? I'm going to guess it's on A. In responsive guitars you can get weird intonation problems when the body resonance is on the scale tone.

That's an interesting thought and makes sense. But I doubt that was this guitars problem. It was one of the sustainable wood Martins, an okay guitar but not very resonant or lively, feels pretty overbuilt to me. Reminded me of a Taylor actually.

I put new strings on it, and the A went from being 14 or 15 cents sharp to 7, and I could drop it another 3 or 4 cents depending on how I fret the note at the 12th fret. The A saddle position surface was pretty flat so I filed some compensation into it and got it much closer to the rest of the strings.