But don’t I at least own my own arrangement of someone else’s song?

Unfortunately, no. Arrangements are copyrightable, but only as a derivative work. Derivative works are based on previously copyrighted material. In plain English: your arrangement could not exist without someone else’s song, so there’s a part of your arrangement that you do not own. You can’t copyright just your contribution, because it’s impossible to separate your contribution from the underlying source material.

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