It's a neat idea, but copyright protects against copying, if someone copies your work, even if your work is derived from the public domain, then you have reasonable grounds on which to sue. You'd have to have added something creative to the public domain melody, however.

In short copying from the public domain does not make one's whole work public domain.

This maybe helps against specious claims?

It may be useful to consider how Pantone's copyright on shades of colour relates; they don't get to sue people for using those colours (in general).