Musicologists and sociologists have long been interested in patterns of music consumption and their relation to socioeconomic status. In particular, the Omnivore Thesis examines the relationship between these variables and the diversity of music a person consumes. Using data from social media users of Last.fm and Twitter, we design and evaluate a measure that reasonably captures diversity of musical tastes. We use that measure to explore associations between musical diversity and variables that capture socioeconomic status, demographics, and personal traits such as openness and degree of interest in music (into-ness). Our musical diversity measure can provide a useful means for studies of musical preferences and consumption. Also, our study of the Omnivore Thesis provides insights that extend previous survey and interview-based studies.