Walter:
Ah... This is a very good question ! At present time, there are no General Aviation gyrocopter pilot ratings LOL.
So I'm afraid that this once again depends on the country. As an example: in France, you would have to have on one hand a Pilot License (general avaiation), either fixed wing or helicopter, PLUS an ultralight gyrocopter license.
The first one because of general aviation regulations, radio and medicals, the second one because gyrocopter instructors are only found in the ultralight category, therefore you need to train with them :0)
A bit complicated, huh ? But, well, the one that wants to fly can indeed do it. Only, it will then be restricted to General Aviation regulations, e.g. only use GA airports... missibg the 600 ultralight airfields we have in that country :0) And it will have to be flown like a general aviation aircraft.
Note for our foreign friends: in France, radio is not mandatory for ultralights. Yes, many have it (and then should comply e.g. with proficiency English level 4), because it makes travelling easier, but you can as well, then avoiding the many controlled and military airsapces we have, fly without it.