You may have a problem with conducted emissions or ground noise on one phase of your house power. If you want to investigate this go to your circuit breaker box and make a list of a few outlets in different rooms that are on each phase of the power (the left and right columns in the breaker box are different phases). Next move around to a few of those outlets to see if there is a correlation between where the problem appears and which phase the outlets are on.If you find that the problem outlets are pretty much limited to one phase (one column of circuit breakers) then you may have some electronic device that is introducing noise on that phase of power, and you can further isolate the problem by switching off breakers on that side of the box until the problem disappears. I have found carrier-current devices (things like X10 modules used for remotely controlling lights and appliances) to be likely culprits. Also consider light dimmers or LED lamps to be sources of noise that can be injected onto whichever phase of power they run on.In some cases the breaker box may have a phase coupler installed that will couple noise from one phase onto the other, but since you have some outlets that are better for the theremin than others this would seem unlikely.
Thanks for this analysis. I am not an electrician by any means, so this is very technical for me. There are so many things at play and my circuit breaker is different to what you describe. I am taking an easy way out and sticking to the "good" outlet.