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I have a knock-off telecaster made by Stadium. The strings on it refuse to be in tune with themselves (the G and B strings especially). I hooked it up to a tuner and tested the intonation. The G string when open reads as G, at the 12th fret it also reads as G, The 12th fret harmonic reads G as well. When playing any other fret on that string however it is not an exact note. The notes are off by close to a quarter tone (50 cents) at some places, the 2nd and 3rd frets are particularly bad. I'm at a loss to what is wrong here.

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I don't have a true answer, but if it was my guitar I would probably try making a huge bridge saddle adjustment on the string and see if it improves or gets worse, eventually homing in on the best bridge adjustment. Telecasters typically have two-string saddles on the bridge IIRC, so you might need to compromise a little. –  horatio Dec 4 '13 at 19:43
    
To some extent, if the bridge is very high, the stretch of the string will cause detuning at lower frets when the 12th fret fingered matches the open harmonic. Even less helpful: switch to fretless like us bowed instrument players :-) –  Carl Witthoft Dec 4 '13 at 20:19
    
When did you last change strings? The older the string, the harder to tune. –  No'am Newman Dec 5 '13 at 4:20
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While the answers here are all valid, i would be inclined to first blame the strings as well. Do they look dull or corroded. Do they feel tacky or slow? Sorry, but this is kind of the "have you tried turning it off and on" test. –  Nathan Cooper Dec 5 '13 at 13:40
    
Old strings can have non-uniform weight distribution, which causes all kinds of problems. The intonation at the bridge isn't likely to have an effect at the 2nd and 3rd frets; but there can be intonation issues at the nut which do show up there. Are the strings not seating well in the nut? e.g, a wound G on a nut cut for an unwound may not sit properly. The string should be supported at the edge of the nut (closet to the bridge); if, for some reason, it's being supported at a point back from that edge, it can cause problems like this. –  greggo Dec 14 '13 at 3:00
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2 Answers

If it's an exceedingly low-quality product, it could just be that the fretboard is badly designed enough that the notes are just not in the right place, but realistically, all guitars exhibit tuning issues with fretted notes. Equal temperament is a compromise to begin with, and the guitar itself even more so.

Assuming it's not actually a manufacturing defect, guitarists have over the years tried to mitigate the problem. If I'm not mistaken, Eddie Van Halen liked to tune his B string slightly flat so that the major chords he played sounded better in tune.

(Relatedly, this is one reason power chords are so prevalent in guitar music -- aside from being easier to play, they also use few enough notes that tuning issues don't arise.)

The only real answer to this whole issue has arisen relatively recently thanks to advances in fretboard craftsmanship. The True Temperament system provides you with a neck built specifically for your instrument that moves the fret location for each note so that it is best in tune. Steve Vai in particular has taken a liking to them.

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I have never used a tuner across the fretboard like the OP, but his claim of "half step" off is probably farther than the sort of temperament compromises you mention. –  horatio Dec 4 '13 at 19:40
    
A half step is exactly between two notes. i.e: halfway between A and G#. –  Dude with a problem Dec 4 '13 at 21:05
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@Dudewithaproblem Actually, a “half step” is the distance between two consecutive notes—so G# to A is one half step. –  bdesham Dec 4 '13 at 22:22
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Whole tone = G-A. Half step (i.e. semitone) = G# - A. Quarter-tone = 1/2 of a semitone (or the distance from G# to halfway between G# and A). –  NReilingh Dec 5 '13 at 4:11
    
OP says '25 cents' which very specifically means 1/4 semitone (and 'quarter tone' which is presumably 1/2 a semitone). Either of which is pretty darn big for a fretboard error. –  greggo Dec 14 '13 at 3:04
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I don't know if this is the case here, but my first bass was an extremely cheap build, and I had something similar to this happening.

What I think may be happening is that you're fretting on your lower fret (Lets say 12 for sake of argument), and it may be playing the tone that should be heard at 13, or 14 or 15, etc... as if you were fretting that note. What causes that is often that there is a height difference between each fret, and the string is fretting itself on the higher of the two frets, this usually only affects a range of one to two frets before that fret, and on VERY rare occasions, 3 frets.

By (VERY CAREFULLY) filing the higher fret down slightly, you may be able to alleviate the problem without spending a ton of money on a (possibly) cheap guitar. If you're not comfortable doing this yourself, or you just plain don't know how to, find a instrument repair shop, and have them do it, you can try asking if you can sit in while the repair is done, but the answer will vary from shop to shop. Basically, it comes down to your skill level and budget.

Again, this may be completely off, but I had to make this modification myself when I first started, and your situation sounds similar to what I dealt with.

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This seems fairly extreme - you'd be able to tell for sure though - if your 3rd fret is so high that it comes into play when you use the 2nd fret, then you'd see no difference in the 2nd and 3rd fret notes. Also, when you use the 2nd fret you should be able to easily slide paper under the string at the 3rd; if not, it's likely a problem as you say. –  greggo Dec 14 '13 at 3:15
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