"I do not listen to a lot of music."
Actually, no, because we're talking about music in general. When we use は, we can roughly get it as ‘As for music, I do not listen to lots of it.'
How do double negatives work in Japanese? If I were to translate this sentence directly I would write: おんがくはたくさん聞きません。 But the proper translation reads like ”does not listen to not a lot of music”