Pokémon Mystery Dungeon: Explorers of Sky was the first game to include a jukebox containing all of the music in the game and in this game, the music cycles through for 30 seconds on the menu screen. However, there is now also a feature within Pokemon Mystery Dungeon: Gates to Infinity that allows you to play all the music in the game.
Once you have defeated the game, Meloetta will start to appear within your Pokémon Paradise. She will mention that she wants to create a Musical Paradise, but needs special materials in order to open it, just like everywhere else in the Pokémon Paradise, and to make it worse, these materials are only found as rewards for the toughest missions.
Bring her the materials and she will be ecstatic and will instantly create the Musical Paradise. Here, you can listen to all 64 in-game tracks including the ones from various dungeons, boss fights and more. When you play these tunes, they will remain for as long as you are in the Pokémon Paradise, and you can deactivate them when there.
There is another perk to the Musical Paradise though. When you purchase and download the special Downloadable Content dungeons, you will also receive two tracks per dungeon. These tracks are not included normally and feature some classic Pokémon Mystery Dungeon tracks.
These tracks are also included in the cycle through on the title screen