That was my initial idea. But then there's no good way to divide them neatly into seasons, whether you want to have two, four, or six.

It does have the advantage that it comes very close to lining up with the lunar cycle, so that the lunar month would only slide very slowly against the calendar month (about half a lunar-month of phase shift per year). And since the definition of "season" is very mushy anyway, maybe thirteen months would be the way to go.