Last time I experienced a mystery whine it ended up being my water pump.
A couple summers back, I start the engine and hear a funny whine. Hrmm. Pop the hood, start nosing around with the engine still running...then all of a sudden I'm covered in a mist of what appears to be really thin oil! IU was baffled, but it turns out my water pump insides had grenaded and was spraying water and picking up grime and crud from the engine bits on the way to my face so it looked dark in color. I stopped the engine and grabbed the fan and wiggled it, when I would pull it one way water would gush out from around the water pump shaft, when I would let go it would almost stop.
Also, for pinpointing mystery noises, I picked up a mechanics stethoscope, its basically similar to a medical one except the pickup is a long metal rod so you can touch it against various engine bits and really nail where a noise is coming from. I think I got it at Harbor Freight or some other cheapo tool store for peanuts, probably was only a few bucks at the most.