Music induces surprise and uncertainty in listeners as it unfolds. However, it remains unexamined whether it is also able to induce waxing and waning feelings of curiosity, how such feelings relate to the enjoyment of music, and what role music's information theoretic structure, on the one hand, and listeners' expertise and trait curiosity, on the other, may play. Here, we characterized melodies using a computational model and required participants to report on their experience of them as they unfolded. In a first experiment, listeners indicated, when cued, how curious they were as to how the melodies would continue. In a second experiment, a further set of participants completed two rating tasks -one in which they were cued to indicate felt curiosity, and another in which they were cued to indicate how much they were enjoying the melody -before completing a multidimensional assessment of curiosity. We found a positive association between curiosity and note information content (IC, surprisingness) that was more pronounced in low entropy (highly predictable) contexts. However, we found that curiosity ratings of listeners with no music-theory training (and little/no experience playing music) were less influenced by musical structure and more driven by judgments of stimulus valence. Finally, we showed that two subgroups of curious people, revealed using cluster analyses, did not differ in how well their curiosity ratings were explained by IC and entropy but differed in the extent to which their unfolding enjoyment of music changed as a function of IC. Taken together, our results demonstrate that musical structure interacts with musical background to influence the emergence of felt curiosity during music listening, while trait curiosity further influences how listening enjoyment emerges.