Alan Lomax's Cantometrics Project was arguably both the most ambitious and the most controversial undertaking in music and science that the world has known. Its flagship component, Lomax's "cantometric" analysis of approximately 1,800 songs from 148 worldwide populations using 36 classificatory features, sparked extensive debate. While Lomax responded to some criticisms, neither his final conclusions nor the evidence on which they were based were ever fully made clear. For decades, neither cantometrics nor Lomax's related projects involving dance, speech, popular music, digital humanities, pedagogy, and activism were widely adopted by other researchers, but there has been a resurgence of interest since Lomax's death in 2002. Here, I provide a comprehensive critical review of the Cantometrics Project, focusing on issues regarding the song sample, classification scheme, statistical analyses, interpretation, and ethnocentrism/reductionism. I identify misunderstandings, improvements that were made, and criticisms that remain to be addressed, and distil Lomax's sometimes-conflicting claims into diagrams summarizing his three primary results: (1) ten regional song-style types, (2) nine musical factors representing intra-musical correlations, and (3) correlations between these musical factors and five factors of social structure. Although Lomax's interpretations regarding correlations between song style and social structure appear weakly supported, his historical interpretations regarding connections ranging from colonial diaspora to ancient migrations provide a more promising starting point for both research and teaching about the global arts. While Lomax's attempts to correlate features of social structure such as gender, religion, politics, and economics with stylistic features of musical performance largely failed to gain acceptance, the Cantometrics Project can still provide both inspiration and cautionary lessons for future exploration of relationships between music and culture.