A musical composition is the work behind a recording: its melody, harmony and rhythm, plus lyrics if there are any. Copyright law treats it separately from every sound recording of it. The US Copyright Office illustrates the point with the song “Respect” and Aretha Franklin’s recording of it, which are two distinct works.
Composition rights belong to the songwriters and, when they sign a deal, to their music publishers, who license and administer the work. Income from compositions, usually called publishing, includes performance royalties when music is played in public, broadcast or streamed, and mechanical royalties when it is reproduced. Much of it is collected by collective management organisations such as SIAE in Italy, PRS for Music in the UK or ASCAP and BMI in the US.
In instrumental jazz lo-fi, the composition is often the melody and harmonic writing a producer adds over the beat. If a track borrows a recognisable melody from an existing song, permission from its owners may be needed; for specific cases, a music lawyer or publisher is the right adviser.