Translation:I don't know the name of the composer who wrote this music.
Both "that" and "who" are acceptable for a restrictive relative clause where the antecedent is human. Wikipedia notes there has been considerable debate on this kind of thing. See https://en.wikipedia.org/wiki/English_relative_clauses#Human_or_non-human_antecedents
"that" may have a bit of justification, but it sounds very jarring in English. The person who wrote the music is animate! One could say, "I don't know the name of that composer," but in the sentence given to us, the composer writes. "That" does not write. I don't think "that wrote this music" qualifies as a relative clause.
It doesn't sound great in English because relative pronouns normally apply to the closest noun. In this case, it sounds like you are saying that the name wrote the music. You'd likely be understood if you said that, but the natural way to say it would be "the name of the composer who wrote..."