When does the music change
During the early days in Animal Crossing: New Horizons, you simply won't have the option to change the music. Unlike previous games that had jet lag in the air, the main theme is pretty much everywhere this time around. And you can't change it right from the start of the game. Instead, the option unlocks after you upgrade Resident Services, which we explain how to do here.
The process will take a few days, even if you dedicate your time to checking the goal boxes involved. Once you've checked these boxes and Resident Services is upgraded, Isabelle will appear and you'll start getting different music in the background (this may actually be a bug, so watch for updates). Just be sure to save and quit to refresh the game. She is also the key to the next question of how to change the music in Animal Crossing: New Horizons, if you want to play with the Is; and Tune.
How to change the music
Once you've done the above, the music will automatically change every few hours, but if you want to play with your Island Tune, a jingle that plays intermittently while playing, you can also do that now. Just go to your recently improved resident services and talk to Isabelle.
She manages many functions on the island, including the "Island Tune". Choose the option to tell him about the island, then change the melody here. You have a few choices at this point, as it's pretty freeform. All you have to do is change the notes of Island Tune however you want, using question marks to rate a random entry. Modify it as you wish, then save it and your Is; and Tune will be changed.
Two separate things, but interrelated in an interesting way. These are the answers for when the music changes and how to change the music in Animal Crossing: New Horizons.