Musical copyright is an enigma, and as a term is something of an omnibus. Lyrics are protected via literary copyright, the musical score is protected by musical copyright, and any accompanying actions or performance can be governed by dramatic copyright. And then there is sound copyright.
Of course, the copyrights can be bundled in ownership terms. Thus, in principle, a musical composition refers mostly to work of music as musical scores, often with accompanying lyrics, and it can be expressed in written form or in form of sound recordings.
However, the copyright for the sound recordings and for the musical score are totally distinct from each other. Sound recordings are the result of compilation of musical, spoken and other sounds in the form of a master recording. And that often leads to multiple stakeholders and different copyrights—but not always.
Jagger may have written the lyrics and Richards the music, but all copyrights, lyrics,
