The digitization of music has changed how we consume, produce, and distribute music. In this paper, we explore the effects of digitization and streaming on the globalization of popular music. While some argue that digitization has led to more diverse cultural markets, others consider that the increasing accessibility to international music would result in a globalized market where a few artists garner all the attention. We tackle this debate by looking at how cross-country diversity in music charts has evolved over 4 years in 39 countries. We analyze two large-scale datasets from Spotify, the most popular streaming platform at the moment, and iTunes, one of the pioneers in digital music distribution. Our analysis reveals an upward trend in music consumption diversity that started in 2017 and spans across platforms. There are now significantly more songs, artists, and record labels populating the top charts than just a few years ago, making national charts more diverse from a global perspective. Furthermore, this process started at the peaks of countries charts, where diversity increased at a faster pace than at their bases. We characterize these changes as a process of Cultural Divergence, in which countries are increasingly distinct in terms of the music populating their music charts.