Peter Able wrote:

his happens when nothing is happening in the house's
plumbing - not even a cistern refilling after a flush.

Could be started by a pressure wave coming in from the street, once it
sets up a oscillation e.g. involving a cistern float, it can then
continue by itself.

Can you set it off if you try?

Plus it can't be
relieved by opening any tap but the drinking water feed tap.Â* Nor by
flushing any cistern.Â* I've been down this paths.

You said two separate water feeds, is that literally two pipes coming in
from the street, or just that the drinking water tap is tee'ed off
before the piework continues to the softener, then splits to feed tge
rest of the house from there?

It is audible outside the house at, roughly, where the water supply -
both direct and indirect - enters the house.

As others have suggested, try sligtly closing down your stopcock(s).