What's the space between the pickup poles and the underside of the strings? If the pickup magnets are strong, and if they're too close to the strings, the magnets can affect string vibration and create overtones.
Normally, that's a sign of a bad string, or wear or foreign matter on the top of the changer fingers. Slack off the string and examine the "crown" (the very top surface) of the finger for dirt or a small groove. Wipe it clean, retune, and see if the noise is still there. If there's a groove, it will have to be polished out.
How long since the strings were changed ?
Some strings after being played a lot will Work Harden, At the bend over the Changer Finger. And lift up from the finger and cause weird overtones.
I notice the 4th string to my ear, Being Raised and Lowered going bad more quickly.
