In my basement bathroom, inside the wall next to the toilet stop valve, there seems to be some kind of flutter valve activated. I can hear air being released as the valve flutters 4-5 times in rapid succession, with about a 2 second pause between these cycles. I can also hear gurgling water as the valve is fluttering. What is the purpose of this valve? Use/non-use of the toilet and shutting off the stop value makes no difference. At this point I can't figure out anything that makes a difference. It comes and goes.

I know of no such flutter valve in residential plumbing. All I can imagine is that someone installed a AAV instead of a vent stack which would not be allowed by code. If everything is working properly I'd ignore it for now but if a problem develops you will need to open the wall to investigate. Do you know if this bathroom is properly vented through the roof line?

The bathroom in question is directly below the main floor bathroom, which is vented to the outside roof. I just assumed the vents were connected. I climbed onto the roof and checked the pipe for blockage as far down as I can see and it looks clear.

This may sound silly, but we did just get about 8-12 inches of rain over three days and my sump's been running nearly continuously for several days now. Could there be pressure in the system somehow related to all of that water in the ground?

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The bathroom in question is directly below the main floor bathroom, which is vented to the outside roof. I just assumed the vents were connected.

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Bob nailed it!!
You must run a separate vent for the downstairs bath. It can not use the same pipe as a vent,(vent stack) the upstairs bath uses as a discharge pipe,(stack vent) so instead of running up a separate vent and reventing in the attic the plumber installed a AAV, (Air admittance Vent) and tucked away it inside the wall. This is why every code going mandates a access panel so that when the AAV goes belly up like yours that it can be serviced. There's no doubt in my mind that your "flutter valve" is a AAV that's gone bad. Open up the wall, replace it and build a access panel in the wall so it can be serviced in the future. Good luck, Tom