Humans engage with music for various reasons that range from emotional regulation and relaxation to social bonding. While there are large inter-individual differences in how much humans enjoy music, little is known about the origins of those differences. Here, we disentangle the genetic factors underlying such variation. We collect data on several facets of music reward sensitivity, as measured by the Barcelona Music Reward Questionnaire, plus music perceptual abilities and general reward sensitivity from a large sample of Swedish twins (N = 9169; 2305 complete pairs). We estimate that genetic effects contribute up to 54% of the variability in music reward sensitivity, with 70% of these effects being independent of music perceptual abilities and general reward sensitivity. Furthermore, multivariate analyses show that genetic and environmental influences on the different facets of music reward sensitivity are partly distinct, uncovering distinct pathways to music enjoyment and different patterns of genetic associations with objectively assessed music perceptual abilities. These results paint a complex picture in which partially distinct sources of variation contribute to different aspects of musical enjoyment.