As I understand things (speaking from a museum perspective), there are several kinds of rights to a work and they do not necessarily go together. Among them are rights of display, performance, reproduction, alteration etc.

Initially, the creator of the work holds all the rights. When a work is sold some, but not all, of the rights are transferred to the new owner. Thus, if you buy a painting from an artist, transfer of the right to display it is implied in the sale: but if you want to make postcards to sell in your gift shop, or have it stabilized before it falls to pieces (a headache for modern art museum and "avant garde" works), etc. then you had better make sure that those rights are stipulated in your bill of sale.

With older works, by long-dead artists, the work is in the public domain and copyright cannot be re-established by the current owner. What the museum/owner controls is access to the image and the right to reproduce it and copyright on the reproduction.

In the case of your newly-discovered manuscript, as I understand it, the tune is in the public domain because the creator is long dead, but the owner can copyright the publication in which the tune appears.

However, this, like all else is subject to all sorts of legal shennanigans, so I would consult a good copyright attorney before doing anything rash.