Within the USA, If a musical composition has previously been released in recorded form, anyone may record their own version if they file a request for a compulsory license (which the copyright holder is required to grant), pay a certain fee per copy distributed (it used to be 8 cents for works up to 5 minutes, or 1.6 cents per minute for longer works, but has gone up somewhat), and abide by certain other restrictions. Most notably:
A compulsory license only authorizes a musician to distribute a work as a stand-alone recording, and not attach it to any sort of video or other imagery.
A compulsory license authorizes a musician only to perform a work without lyrics or other verbal content, or perform the work with the exact lyrics of the original. It does not authorize the use of any other lyrics or verbal content.
A compulsory license only authorizes a musician to perform the work essentially as written by the original composer; one may use different instrumentation from earlier recordings, and may adapt the work if necessary to accommodate that, but must preserve the essential character of the piece.
Any copyright interest in arrangements which are done to make the piece playable given the performer's instrumental or other limitations is automatically assigned to the copyright holder of the original work [to argue that such arrangements are themselves creative works of the performer would be to argue that one has changed the work beyond the extent permitted by the license]
There are some additional restrictions as well. If one doesn't like the standard terms, one may negotiate with a composer to seek better ones. Copyrights are often administered by agencies that impose minimum licensing requirements (e.g. they won't grant a license to distribute 5 copies for $0.50; instead it would be 500 for $50). The statute doesn't particularly authorize them to do that, but would impose enough other burdens on someone who wanted to license a piece without the cooperation of the copyright holder that it's easier to abide by such terms than try to fight them.