You can focus on how music affects society and culture for your argumentative essay. You might start with a statement such as, "The effects of violent lyrics on the actions of youth are overblown" and then discuss that assertion in the body of the essay, providing evidence from studies to support it. You could also go with a topic like, "Local governments should fund art endowments or scholarships," and discuss the importance of art on local culture and why it must be cultivated.
Argumentative essays can also examine specific genres. For example, you might choose a topic like, "Country music has abandoned its roots and evolved into a mishmash of multiple genres," and talk about influences of genres like pop music or hip hop on the country genre.
You can examine the actual performance of music, an area rich with possible topics. For example, a paper could talk about whether 18th century music should be performed on period instruments or if it's acceptable to use modern instruments. You might also argue that studios that excessively edit music threaten the integrity of music as an art form.
How music is taught is another important aspect that is ripe for an argumentative essay. Take a topic such as, "Music education should be integrated with other classes," and discuss how combining music with math and science might promote broader learning. Or, "School districts should develop their own music learning standards," an essay that could argue that schools should preserve a community's identity by pursuing a locally tailored education program independent of national curriculum.