A permit? Well, yes and no. It depends on whether the work involved is under copyright. (that is, if you are going to do it right). Indeed, this applies any time someone is going to perform a work under copyright in public for other than an educational purpose, such as a school or university recital (the educational fair use exemption).Anything other than that, one must contact the copyright holder and obtain permission from them -- and I would recommend in writing -- to perform the work. The copyright holder may or may not charge a fee; it depends a great deal on the holder and the nature of the performance. Now if you are adding to that recording the work and releasing the recording, the odds are that the copyright holder will charge a fee -- which will often be a percentage of the gross sales.And note that sometimes the copyright holder may say no, which they have the right to do.If the work in question is not under copyright, then you don't have to worry about any of that stuff. No permit is required.Of course, once you have done all that and released the recording, your performance of the work is now also copyright, and you have the right to limit its use or charge a fee for it, in the same way! I might note that enforcing that, so you can collect some income from your efforts, is almost impossible in this age of everyone pirating everything and downloading it, so good luck.I might add that a good agent should take care of all that stuff...
Copyright? What if the composer is dead?
I think in the U.S. the creator gets dibs on making the first recording. After that, if it's still under copyright, anyone can record it. They just have to pay royalties to the copyright holder.