First recording rights refer to the ability/right of the copyright holder to decide who makes the first recording on their musical work. 

This is because, in the United States, copyright law states that the copyright owner of a musical work have complete control over recording rights of their works (i.e who can record and release their musical work) until they license the material for the first recording. After this first recording has been distributed to the public for consumption, the copyright owner becomes required by law to now license their work to any other person or entity who also wants to produce and distribute recordings of the copyrighted work in exchange for a fixed statutory royalty (often referred to as a compulsory mechanical license). 

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