Can I sell public domain music that I've recorded/produced?

I understand that many classical pieces are in the public domain (i.e. Beethoven, Bach, Mozart), but is it legal for me to produce my own tracks of these songs and sell them as royalty free music? Let's say I produce my own version of Fur Elise by Beethoven, do I legally own all the rights to sell this both... show more I understand that many classical pieces are in the public domain (i.e. Beethoven, Bach, Mozart), but is it legal for me to produce my own tracks of these songs and sell them as royalty free music? Let's say I produce my own version of Fur Elise by Beethoven, do I legally own all the rights to sell this both with royalties and royalty free?

if so, I'm assuming I should copyright a track before I sell/distribute it. Can I do that?
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