It's not really tied to the stage(though I guess you could say it is in a way)...it's the Rival battle music you will hear. If you fight Ryu as your rival, you get Ryu's music during the fight. If Cammy or Rose is your rival, you will hear their music during the fight. Obviously, only certain people are the rivals for certain people though. The only way to get the music whenever you wanted would be the custom soundtrack option and use the Rival themes. Another reason for getting the soundtrack to the game. The Collector's Edition CD soundtrack also featured all the rival themes (the remix of the some of the characters and the specific 4 new people's themes). I ordered the full 2 CD soundtrack though anyways. The music is just too good here.
Cammy's and Rose's remix Rival themes are awesome, and Viper's theme music is epic.
Most of the stages aren't tied to any one character. Capcom mentioned this awhile back when they were originally going to do one for everyone, but it came down to quality over quantity. Time also probably played a huge factor.