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If you are remixing other people's music, you need to get permission from their record companies if you want to do anything commercial with the resulting remix.
Obviously, if you make a mix tape for your own private use, you need not worry about getting any rights or permissions (after all, how would anyone know what you're doing if you're the only one to know about it?)
But if you want to sell your mix tape (or publicize it in any way, including giving it out for free), you will need to get permission from the artists (or the record companies- it's whoever holds the copyright for the recording of the song you are using) else you can find yourself in a rather unpleasant lawsuit.
Note that the key thing here is permission: if the artist requires you to pay them to get their permission, then that's where the money comes in. But if you work with unsigned artists (like young rappers who don't have a record deal yet), you may be able to get permission in exchange for the publicity they are getting by your mix tape release without having to pay them.
