Siman 599

The Tur wrote: For Arvit, one prays as on the first night. And if it falls on Shabbat, the Raviah wrote: I asked my teachers if one must say "Vatodieinu," because one says within it "and the festival-offering of the regal," and Rosh Hashanah is neither a "chag" nor a "regal." And they answered me that nevertheless one says "Vatodieinu," just as we say in the Havdalah of the conclusion of a Yom Tov "between the seventh day and the six days of labor," for we say: he is counting in the order of the Havdalot. Here too, he says the order of the established liturgy. Thus far his words.

And aside from this, behold, according to several of the early Geonim, Rosh Hashanah is called a "chag," for "at the hidden time for the day of our festival" refers to Rosh Hashanah, as I wrote in siman 597, and as the Tur himself wrote in siman 582 that it is called a "moed" according to several Geonim; see there. However, the Tur did not see fit to elaborate here, since in any case it is necessary to say so for the reason he wrote.