Almost every Hebrew festival lands on a full moon. Rosh Hashanah is the exception: it falls on the first of Tishrei, which is the new moon itself.

The Talmud describes it as the festival on which the moon is covered, and reads the concealment as fitting for a day of judgement. It is also why Rosh Hashanah is two days long even in Israel — before a fixed calendar, nobody could be certain in advance which day the crescent would be confirmed.