Thank you all for your help. You have stimulated me to investigate a bit further, and as it's now too late here to get it in for service before next week, I'll impose a bit more on your patience.
The Sennheiser HD 555s are OK: nice inky black silence when connected to my puter or iRiver. They have a 1/4" plug, which goes naked into the Pioneer phone socket, so it's not adapter trouble. Rotating the plug in the socket produces no effect (except for making the woodpecker say "Lord bless my soul!", but that's in a very old song

).
The troublesome sound I now think sounds like AC hum. The interconnect was routed through a rat's nest of wires, but I've dragged it clear of all power cords, and the noise is still the same. It's also unaffected by the volume control, as I said, so I guess it's not something picked up through the input. (There's also a high pitched hiss that appears when the volume is turned up to about 7, but that's not what I'm bothered with, and I normally only have the knob halfway round the dial, anyway.) Even when I've cranked the volume up really high, I don't hear any of the noise coming from my speakers (which are KEF C Series bookshelf jobbies, in case it makes a difference).
My totally ignorant guess is that this is pointing to a leaking capacitor type of problem, which means it needs a trip to the hospital but should be fixable, yes? Or, no?
I'm going on a bit because there aren't too many audio repairers here, and the minimum charge is likely to be pretty close to the cost of a cmoy, so I'd like to have some sort of clue before I commit money.
With many thanks again
Michael