Posted by Psycher!

I don't know whether you would consider this bittersweet or not, but it's a beautiful piano piece and since you want it for sleeping apparently, it is nice and soft for such a thing. /bad explanation
[YOUTUBE]9FJmbei3x5Q&feature=relmfu[/YOUTUBE]

I think I found two that may convey the feeling you wish to get from a song.
[YOUTUBE]UrFhQKn-pLk&feature=relmfu[/YOUTUBE]
This one's more rock-ish.