According to the GPL FAQ, it depends on how tightly the programs are coupled.
- Tight coupling: This is what happens in the case of your game, as you almost certainly included the GPL music player within your binary or as a dynamically linked library. In those cases your software must be GPL, too.
- Loose coupling: If you have a generic program that happens to execute the music player as a separate program, there's no tight coupling going on. The GPL doesn't affect your code. You can choose any license you want, including a proprietary license. However, in that case you might want to add support for different music players, just to be sure. And you should refrain from shipping your application with the music player as a combined package, as that may give a wrong impression. On the other hand, a mere aggregation of multiple packages (as in software distributions) is fine.
Note that nobody forces you to distribute your application. If you don't distribute your application at all, not even in binary form, the license question disappears: There is nobody you could license your application to. The GPL and all other FLOSS licenses grant you the right of private modifications. The virality of copyleft licenses only kicks in as soon as you distribute your code.
Also note that tight/loose coupling is mostly a social term and not so much a technical term. Usually, linking to a GPL library counts as tight coupling, while executing a separate GPL programm counts as loose coupling. However, this is not an absolute. The line blurs as soon as you start playing nasty games, and people will recognize that.
For example, you could write a command-line wrapper around a GPL library, make only that wrapper GPL, and communicate with that wrapper using an application-specific RPC protocol. Even though your application starts the GPL code as a separate program, you would have a hard time explaining to FSF or a judge in a court that this is loose coupling. It would still count as tight coupling.