Not sure if this it but I had a weird overtone that did not sound a million miles from that on my Gibson LP style guitar recently.  It was on the High E string and sounded like there was a dissonant overtone. As I moved up the neck it become more prominent.

The fix ...?  Changed my strings.  I have had strings lose intonation on when a bit old but this one was really odd.   You can here it here --> http://jvmforum.com/phpBB3/viewtopic.php?f=8&t=6889&p=74320#p74320

I'd say the string cannot vibrate freely. Check this:

- Are the strings clean and new enough?

- Is the e-string set high enough? Set the e-string higher and test it.

- Do the frets have a clean shape? When the frets are worn the strings lie deeper and touch other frets. Then the frets have to be repaired and brought to the same level. Check if it always happen at the same fret.

I hear what you're talking about.  Typical beating aside, when you're playing the E string on the 11th and the B on the 12th simultaneously there's this weird droning noise that sounds like it's a wibbly wobbly C one octave below middle C.