My daughter's friend sold her an acoustic guitar. We could not tune it. A neighbor brought over a tuning meter. He said the strings looked fairly new but we could see each string caused the meter's needle to waver back and forth much more then he had ever seen before when each string was plucked. This confirmed what we were hearing and makes the guitar impossible to tune. Do we need new strings or a new guitar for her to learn on? Or maybe a new friend for my daughter.
It would be helpful to have a sound file to hear what's going on. But if the meter is wavering and can't focus in on the pitch, my guess is that the strings are bad. Cheap and/or old strings are sometimes not perfectly round, and thus have more than one fundamental frequency, which makes them sound sour and impossible to tune. But as others have intimated here, you need to have someone who knows their stuff look at it.
The pitch produced by a vibrating string depends on just three factors: the length of the vibrating portion, the tension the string is under, and the mass of the string. Any change to any of these three factors will change the pitch.
When a string won't hold pitch, it's usually related to tension. Either the string is physically stretching (common with new strings, especially nylon strings), or the tuning machine doesn't have enough friction to hold the tension steady. But both of those causes will only make the pitch drop - strings can't contract on their own, and a tuning machine can't increase the tension without any outside force tightening it.
From your description, the pitch is going up AND down while it's vibrating, and that means the length of the string must be changing - as the string gets longer the pitch goes up, and as it gets shorter the pitch goes down.
This means one (or even both) ends of the string aren't fixed in place. Since there are just two end points for the string, there are just two possible causes:
- The nut is moving, or
- The saddle is moving
Either of these can have more than one cause.
If the nut is moving it could be:
- The nut is not securely seated in the slot. It's possible that the nut slot is worn, or that there is some debris under the nut that's allowing it to rock back and forth as the string vibrates. Not likely, but possible.
- The peghead may be insecure. Because the peghead on most guitars is angled back to provide "string break" (which helps hold the string in the nut slot), this is a weak spot in the neck structure. Look for any tiny cracks on the back of the peghead.
- The neck joint is insecure, allowing the whole neck to swing back and forth. There are several different methods for neck attachment, but on most acoustic guitars the neck is glued into a dovetail joint in the "neck block" located inside the body. Look for any small breaks in the finish around the neck heel - that's the part where the neck gets thicker right where it joins the body.
If the saddle is moving it could be:
The saddle is not securely seated in the bridge slot. The underside of the saddle may not be flat, or there may be an imperfection in the slot that's allowing it to rock. Like nut problems, this is possible, but not likely.
The bridge may not be secure. Bridges are glued to the top, sometimes with additional pins through the bridge into the "bridge block" inside the body (if there are pins, they are rarely visible). In either case there will be some separation from the top at the back side of the bridge. If you can slip a business card under the back of the bridge at any point, this is the problem.
The top may be separating from the body. Inspect the edges of the top looking or any breaks in the finish, or any spots where the "purfling" (a decorative strip around the edge of some tops) shows any gaps.
In my experience, the most likely cause is a bridge joint failure, followed by a neck joint failure, followed by a peghead crack.
None of these are simple DIY repairs - try to identify the cause, and then take it to a luthier.
EDIT: In 40+ years of working on guitars I've only seen this problem about a dozen times. Eight hours after I posted the above, I see another one.
A half size guitar came into my shop because it won't hold tune. This one is caused by BOTH ends of the string moving - the bridge is coming loose from the body and the glue joint on the neck dovetail is failing
Between the two causes the pitch of any given string is shifting back and forth by 5-20 cents.