So here’s a question: why don’t we recite Hallel on Rosh Hashanah?
Hallel is a collection of psalms praising Hashem that are recited on holidays like Passover, Sukkot, etc. So as we celebrate our New Year, wouldn’t that be a great time for Hallel? Well, the rabbis pointed out the Rosh Hashanah is a time of very mixed emotions. On the one hand, it is indeed a celebration of the new year. On the other hand, it is a serious time of introspection and contrition. Instead of focusing on the celebration, through Avinu Malkeynu we emphasize the introspection.
There are numerous different versions of this prayer, of varying length. Among the many verses, there are a number that share a particular pattern: “Our Father, our King, write us in the book for….” But there is one interesting variation in this phrasing. On Rosh Hashanah, and during the 10 days of penitence between Rosh Hashanah and Yom Kippur, we recite the text as cited above. On Yom Kippur, though, when we recite Avinu Malkeynu for the last time in the Neilah service, the text reads, “seal us in the book for…,” based on the idea that the book in which Hashem describes our fate is written on Rosh Hashanah and sealed on Yom Kippur.
Avinu Malkeynu is also recited on fast days (like 10 Tevet or 17 Tammuz) because they, too, are days of introspection and contrition. But on those days, the text doesn’t read “write us…,” or “seal us…,” but instead “remember us…”. The idea here is that our fate has already been decided, and we ask Hashem to remember the (positive) fate that has been decreed for us.
One interesting quirk: there is one fast day when we don’t recite any version of Avinu Malkeynu—Tisha B’av, the anniversary of the destruction of Temple in Jerusalem. Why not? If ever there were a time for introspection and contrition, it would be then. But evidently the destruction of the Temple meant such a complete alienation from Hashem that it was (for the moment) futile to try. This can only add to the poignancy of the concluding verse in Avinu Malkeynu when we do have the opportunity to recite it: “Our Father our King, be gracious to us and answer us, for we are unworthy. Deal with us with charity and loving-kindness, and save us.”