You have a model 69 which is the only model with a tremolo. Perhaps that additional harmonic is coming from sympathetic vibrations from the springs?
Have you tried a few cotton balls at various places in that trem chamber or even on the bridge to see where the additional sound stops when suppressed?
Best of luck,
Hi was reading through the forum and saw this thread.
I just received my jtv59 and have the same issue with my low E. if I play the open E and let it ring out I get this type of sitar overtone. This is without the guitar plugged in. Driving my nuts.
Have tracked it down to the nut. If I place finger on where the string sits on the nut the issue goes away. Not sure exactly what's happening there but it's definately affecting the modelling.