There's a failure mode here.

For this incentive to work, you have to only send your partner songs that you have earned through accountability. Otherwise, you can not do the thing, but still send your partner a song.

For the carrot to work, you have not also give yourself unlimited access to carrots. Which means for the most part, it still functions sort of like a stick.

You have to deprive yourself of the privilege of sending your partner a song in order to motivate yourself to earn it back.

This is the general problem with these kinds of gamification schemes. They can work, but they do require the self-discipline to not let yourself acquire the goal without doing the thing.