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That sounds like a ground loop, actually. Do you know if the powered speakers are on the same AC ground line as the synth, or whether they're grounded at all? If not, that would explain quite a bit.

Probably the best thing you can do is to take the time (and hookup wire) necessary to set up a star-ground to get all of the equipment on the same ground plane. It's time-consuming, but very worth it in terms of cleaning up noise and hum, and might only be necessary for a few pieces of your gear. Have a look at https://www.sweetwater.com/insync/star-ground/ for a very basic explanation of what needs doing...and as I noted, if you can do this only with certain offending devices to clear up the ground loop issues, that works just fine as well.