Monitors should not make any sound.
If the frequency of the tone changes with changing refresh-rates.You are hearing the windings of the deflection yoke. Many TV recievers do this as well. Some monitors will do it a bit more than others in the same model. It has to do with how tight the windings are when the yoke was made, and also how they were bonded down to the yoke core.
When there is no signal going into the monitor the scanning drives are freerunning. There is less noise because the drive pulses are not in constant phase allowing the yoke windings to go into resonance.
If it is not the yoke, There are a number of things that can cause a whiny high-pitched sound. It is almost always due to vibration of a component in the horizontal circuits, most often the flyback transformer or horizontal output transformer. I do not recommend you attempt to fix this yourself, unless you are trained and very comfortable working with extremely high voltages.
A service technician should be able to find the offending component and either replace it, remount it, or add some material which will stop the vibration.
Good Luck