I just finished (almost) my first 3 string cigar box guitar.  red oak neck.. 25 1/2 inch scale.. hard wood nut and movable (floating?) bridge.  Strung with ADG from standard string set and tuned up Gdg and re-tuned .. and re-tuned.. left for two days .. re-tuned.  String height is level at the bottoms and 1.5mm at first fret and 3 mm at 17th fret.

When I tune it (two different phone tuner apps) Gdg and just touch the string above the 12th fret it sounds good and the tuner apps say G2 is now G3.. D3 is now D4.. and G3 is now G4..... but if I FRET the 12th on any of the strings it is sharp.  This seems to me like string stretch making it sharp.  should I shave some off the bottom of the nut and bridge to make it even lower? or will this settle out in a couple days?

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I'd say strings too high at nut,for me, for slide or fretting note. I'd say too high,for me, at the 12th, (normally measured at 12 fret) for fretting but ok for slide. Make the note at the 12 fret sound the same as the open note, adjust string length to get this right. Move the bridge to increase or decrease compensation. Use finger on fret or slide on string over top of fret to check.
Cheers Taff
Sounds like your bridge compensation is off a tad. Is your bridge straight across the strings? You need to compensate for the difference in string diameters. The normal way to do this with a movable bridge is to angle it. First, make sure that your FRETTED notes at the 12th fret are spot on. Why? If you are touching the note DIRECTLY ABOVE the 12th fret with a slide, yes, they will be an octave above the open string. But if you fret the note, where's yer finger? Uh huh: BEHIND the fret, e.g., NOT on the fret. Your finger is depressing the string down, pulling the string across the top of the fret, rendering the note sharp. Sooo...you need to push your bridge just a little toward the tail. For a 3 stringer, get the middle string dialed in first at the 12th fret. Then, using that as a pivot point, angle the bridge just a little, so that the end nearest the thickest string goes back a bit toward the tail, while the end nearest the thinnest string moves toward the headstock. You will have to experiment a little to get them right, and even then, they won't be perfect, but can be very close to it. Couple cents off is no biggie, as that will tend to get canceled out by your fretting finger's positioning and vibrato.

OK .. maybe I wasn't clear with the OP so I will try again ... first with middle string only of open G tuning .. tuned to D3.  String touched just over the 12th fret with a slide and tuner shows D4. This SEEMS to tell me that the string is tuned properly, I havd set the bridge correctly if the note is exactly one octive higher at point above the 12th fret??  but if I FRET the 12th it goes sharp.  This SEEMS to tell me that the string is stretched(too high above the fret).  But if I fret the note at the 12th it is a little sharp.

Taffy:  You think the strings are too high.  what do you recommend for height at the 12th?  If you read does it seem like I still need to move the bridge or is it positioned correctly?

Ron:  If you read above does it seem like compensation at bridge is correct?  the bridge is not straight across BTW and in each case with the slide my tuner says tuned correct but if I press the string to the fret it is sharp.  Are you saying that if I move the bridge farther (make the strings slightly longer) it will compensate for the pitch increase when fretted and not make it out of tune otherwise?

Both Taff and Ron are more knowledgeable than me re intonation, but i suspect they may have missed some detail on your original post, to me, your scale length is in the ball park or better, if so, the only thing you have left is the amount of string stretch/distortion you get when you fret at 12, firstly, check how sharp you are when fretting the 1st fret, if excessive lower the nut to suit, all the time checking intonation at 12 and adjusting the bridge as need be, ideally you will end up with the strings, just clearing F1, and around 2-3 mm at 12, but even then, due to the fact your A string is a step down to get a G, it is near impossible to get perfect intonation at the same height from a set of matched strings, to improve from that point will require individual string height adjustment at the bridge and possibly the nut also, there is a degree of compromise in the set up generally and i've already spoken beyond my real knowledge, so i hope Ron or Taff can help you more

I might be repeating what Darryl said.. but...

First, you re not going to get the same perfect intonation with a slide and fretting... so adjust your intonation fretted...

Don't adjust the middle string... the high string should be about 3mm shorter than the low string. Anyway, adjust your intonation on the high string, then without changing the his string length, adjust the intonation on the low string... the result should be that the low string be a tad longer than the high string at the bridge.

Call it good, you're going to be as close as you can get...

If you can lower your action without buzzing while fretted, go for it... that will make it easier to play...

I don't play slide (never played any guitar before).  I am using a slide just to adjust the bridge.  My bridge was already slightly angled .. I did lower the bridge and nut by .5mm and I noticed that the amount it is sharp isn't so bad and the angle of the bridge changed a little bit.  Now very slight sharp on 1st fret and pretty darn close at the 12th fret so I will "Call it Good"  First (I know I will build more) is done.

Now I just need to learn how to play it.

Thanks for the comment Darryl. All I could add really is that with the use of frets on the fingerboard you will not get perfect intonation as you are battling against a few things that make it difficult. Fixed nut position(can be compensated) fixed saddle position (often compensated) fixed fret positions and different string diameters and mass. Also the finger pressure behind the fret will change intonation. For example a carefully played harmonic or open string may intonation perfectly when fretted slowly and carefully, but under playing conditions with fingers landing on strings from different angles and speeds that acracy is often lost. Strings can get pulled or pushed out of tune. I would think slide playing (if your ears work well) would be better at getting good/better pitched notes up and down the fingerboard.

It would be like having movable frets, notes a bit off at whatevever fret?....compensate by changing slide position slightly.

I heard someplace that a lot of slide guitar players in the old days were blind and couldn't see the positions on neck so the slide would be higher or lower and then move to what sounded right.

simple physics, pressing the string down to the fingerboard stretches it tighter.  You can't have both the fretted note in tune and the note from a slide perfectly above the center of the fret. 

I just checked a couple of my CBG's. Six string set up for slide is 3mm bass 3mm treble over the 12 fret, bridge moved until intonation is perfect. I do not finger frets that high up the neck as I play slide in that area. Does the slide sit above or behind the fret? it does not matter to me as the slide is always moving, making a vibrato effect. 

On electric guitars I would go for 1.75 to 2mm bass side [non slide], depending on customers preference.

I also set intonation on the first string first then do the Bass string # 3, 4 or six next , the others on a straight saddle, will be what they will be. On acoustic guitars I use a wide bone saddle and compensate individual strings according to their height and mass/gauge.

Cheers Taff

I found this video by Justin Johnson really useful, https://m.youtube.com/watch?v=IPTQmyNbwpY

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