What kind of reality does a piece of music have?

It has the same reality as a story does, or a theory. It belongs in the Ideosphere. Actually, the term I’ve heard used is Noosphere; but the early notions of noosphere that Wikipedia enumerates are kinda loony. But it’s a mental construct, the kind of thing that Dawkins actually had in mind when he first spoke of memes; and as such, the ideosphere is:

the “place” where thoughts, theories and ideas are thought to be created, evaluated and evolved. […] The ideosphere is not considered to be a physical place by most people. It is instead “inside the minds” of all the humans in the world.

The performance of the music isn’t the music; it’s an instantiation, and music still exists if is never performed, but is just written down on a score. The score of the music isn’t the music either. Think of the langue/parole distinction from Saussure: the music is the underlying code, the idealised mental construct, and it exists independent of instantiation.