So I figured out I can tune my G string just a little flat so all the frets sound in tune, except that, of course, leaves open G sounding like F#.
What should I do?
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Look at the nut. The string should make contact with the edge of the nut towards the fretboard. Here is a picture of what it should look like. You can fix it yourself or get it repaired at a luthier/guitar store.
This seems to be a nut/string saddle issue. If you're sure all the fretted notes are in tune when you de tune the third string, then it means that your open G string is too long because the string is not contacting the edge of the nut. The act of fretting the notes is enough to press the string down onto the correct contact point on the nut. You can buy a pocket-sized folding set of mig welder tip cleaning files for a few dollars. Select the appropriate file and ever so gently (think microns here) slope the nut slot so that it falls away on the headstock side. String up and re test after every file stroke: you do not want to take the slot too deep. It's possible that the intonation at the bridge will require a tweak too, depending on how the guitar was 'originally' set up. If your guitar has a bone/plastic saddle do not use the mig tip cleaning files; they should only be used on slots. If your guitar has adjustable saddles remember any adjustment will be tiny.
It is a nut slot issue but not a string hitting too far forward in the slot. It wouldn't matter where the string hits the nut because you are fretting a string and his fretted note is sharp, _ and that I believe is the original posters issue. So -just focusing on the G string here - So saying he gets his G to PERFECT pitch (open) but it's Sharp on the second fret(A) note - or even the first fret( G# )for that matter - this is his complaint I think . This is the most common issue (after you intonate everything of coarse)with guitar sounding out of tune (i.e. playing a Cowboy D). The problem is 2-fold.
It's because the G nut slot needs to be cut a little bit deeper. That eliminates (though ever so small) some of the downward stretch (travel) the string has to make to sound the A note(because the string will be closer to the fret when the nut slot is deeper )so it doesn't travel so far to make the string travel down to A as much (the further the string stretches down the sharper it becomes) It travels down as high as the nut slot-down to the fret.
Add a guitar with taller frets, it exasperates the issue that much. Taller frets even more travel .You have to get that G string slot as close as possible without fretting out and it will solve the issue. Problem is it's hard to get it that close without going TOO far .But it can be done and it will greatly if not fix totally depending on how good a job you do for this issue.
Worked as a guitar tech for Guitar Center and have done this so many times.It's a good idea to have an extra nut in case you do go to far but you can put a gap feeler gauge down on the fret board the exact height of the 1st fret and just take the slot down that far. If you don't have a gage ,in a pinch you could pile layers of duct tape on finger board as high as the 1st. fret but that's not too scientific .