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February
By my logic, the first two meanings (lack of either a full or new moon in a month) can only occur in Februray, as the lunar cycle is 29.53 days, and the shortest non-Feburary month is 30 days.  Feburary, being 28 days (or 29 in a bissextile year), is the only month that is less than the full cycle, and that therefore can miss either a full or new moon in the month.  As the cycle from dark moon to dark moon is less than 29.53 days, it is also impossible to occur outside February.

As for the fourth meaning (two new moons in a month), obviously this cannot occur in February. —  talk "