Why does my server play an E minor arpeggio through the PC speaker every five minutes?

Have you considered that someone might be playing a prank on you?
Back when I was working in an environment with lots of Sun workstations a co-worker of mine would follow people on the security cameras and make the machines scream or laugh as you walked past.

Assuming you can rule out pranks, as others have suggested perhaps it's a RAID card or remote-access card trying to tell you something. I assume the event log you've looked at is the Windows event log - have you checked the IPMI event log (methods to do this vary by server manufacturer)?

Have you seen http://support.microsoft.com/kb/261186?

I know nothing about music, so I don't know if either "Fur Elise" or "Its a small, small world" are an "E minor arpeggio", but even if not perhaps it's a different motherboard/bios manufacturer warning about something.

As it's coming from the PC speaker, rather than the sound card, I think you need to stop looking at the server side altogether because the server can't make sound come from the PC speaker.

I suggest either reinstalling or upgrading the remote desktop client or simply replace the client files with those from another PC that knows how to shut up. It's possible that there is some corruption in there somewhere which is causing the sound to be played.