A playoff mode so that you can challenge your friends to play a given song(s) in a head-to-head duel for bragging rights. This feature could even include an MMR style ranking with other features such as a record and leaderboards for different modes or game types. Competition against formidable opponents while playing appropriately difficult songs would really help users enter into the “flow state” and improve their learning experience.

For those who don’t find competition like that to be their cup of tea, you could add a cooperative mode in which you and your teammates have to beat a certain score etc.