LCD Monitor High-Pitched Whine

Check that all the wires are connected properly, try re-plugging every connection. If that fails, try a different cable and a different computer - that should give you some idea of where the problem is (screen DVI port, DVI cable, or computer DVI port).

From the information you provided, it seems unlikely that it's a problem with the computer's software or the screen's electronics.

I can sometimes detect with my naked eye if a monitor is using 60 Hz refresh rate. Rumor has it that 60 Hz is the frequency that somehow resonates with internal frequencies of the body, whatever that means.

In the past, with a CRT monitor, that was a real pain for my eyes, so I had to drop resolution in favor of refresh rate. With LCD screens, that's normaly not a problem but you may be more sensitive to that. Try to get hold of a large monitor with high refresh rate (85 Hz or higher), even if only for ten minutes, and see if it helps.