Oh, that's not my experience at all! I downloaded this app to see how many distinct songs a mockingbird was singing one time (it was a lot) and the app just came back: "Mockingbird". I was both amazed and disappointed!
I have a Blue Jay that began imitating a Cooper's Hawk after one started showing up in the neighborhood. In its early attempts at imitation, it was still recognized as a Blue Jay, but the Blue Jay has improved and it now comes up as a Cooper's Hawk! I watch it sing, so I know this is the Blue Jay, not the hawk.
