I read here that playing a string on the 12th fret will produce a note that is an octave above the note produced by playing the same string open (with no frets held down). For instance, if the bottom string is tuned to G, then playing the bottom string on the 12th fret should produce G also.
However, it seems that on my mandolin the octave is on the 13th fret. How should I fix it?
Playing a string on the 7th fret should produce the note on the next higher string (open), but it's almost a half-step off. I think this could be fixed by moving the bridge.
I didn't realize the bridge was a "floating" bridge, meaning it can be adjusted higher/lower with the screws. Could this be a simple matter of adjusting the bridge? I'm a little afraid to do so because I don't know which way to turn it. Also, there's a slight crack in the bridge right next to the screw: (I wonder if this is an indication that it's been "over-adjusted" before, leading to the intonation problems. That's just my guess though.)
Here's a closer shot of the crack in the bridge (I've also highlighted it):
The neck looks straight enough: