"The 50/60 Hz problems you can have also with well grounded battery driven theremins. The reason: Over the pitch antenna and the high inductance serial coils one get up to some hundred of millivolts from electrical environment fog in houses. Fortunately this voltage is almost short circuited over the second resonance coil to ground. But not completely!"
The Phoenix uses a large (compared to the tank) coil to ground, something I thought of a while back but don't believe I mentioned here on TW. I ruled it out for the D-Lev because the coil has to be significantly larger than the series tank coil to not impart significant phase shift, I can deal with hum in software, and the hum imparts some dither to the input, which is kind of a good thing.
In my experiments I've found that a high value resistor from antenna to ground lowers hum/noise significantly without hurting the voltage swing too much. It forms an RC lowpass filter with the intrinsic C of the antenna & hand. So if you use 470k and have a 10pF antenna, you get -3dB at 34kHz, which at least kills some of the super higher harmonics and other environmental noise? Though it directly hurts Q, so I don't do this on the D-Lev (perhaps it is something I should look into though).
"The result is a modulation of the main signal and all overtones. You get additionally a more or less weakly plus/minus 50 Hz splitting of the main signal and all overtones. That sounds a bit scratchy and not clear especially in the near of zero."
I never thought of it from that angle (harmonic modulation) before (another reason to go digital) though I have thought of the influence of the mains hum field on the tank. Mains hum is coupled through the antenna intrinsic C, so adding a series C generally won't reduce it as the intrinsic C is so small. And the effect is FM/PM on the tank frequency. It acts like a hand waving near the antenna at 50/60Hz.