A small percentage of healthy individuals do not find music pleasurable, a condition known as specific musical anhedonia. These individuals have no impairment in music perception which might account for their anhedonia; their sensitivity to primary and secondary rewards is also preserved, and they do not show generalized depression. However, it is still unclear whether this condition is entirely specific to music, or rather reflects a more general deficit in experiencing pleasure, either from aesthetic rewards in general, or in response to other types of emotional sounds. The aim of this study is to determine whether individuals with specific musical anhedonia also show blunted emotional responses from other aesthetic rewards or emotional acoustic stimuli different than music. In two tasks designed to assess sensitivity to visual art and emotional sounds, we tested 13 individuals previously identified as specific musical anhedonics, together with two more groups with average (musical hedonic, HDN) and high (musical hyperhedonics, HHDN) sensitivity to experience reward from music. Differences among groups in skin conductance response and behavioral measures in response to pleasantness were analyzed in both tasks. Notably, specific musical anhedonics showed similar hedonic reactions, both behaviorally and physiologically, as the HDN control group in both tasks. These findings suggest that music hedonic sensitivity might be distinct from other human abstract reward processing and from an individual’s ability to experience emotion from emotional sounds. The present results highlight the possible existence of specific neural pathways involved in the capacity to experience reward in music-related activities.