The start of a new month is a joyful time in Judaism. The Torah speaks of Rosh Chodesh together with the major holidays: “And on your joyous occasions, your fixed festivals and new moon days…” (Numbers, 10:10).
One of the ways we are happy on Rosh Chodesh is by saying or, more commonly, singing Hallel, comprising Psalms 113-118 (albeit a shortened version). The Talmud enumerates 18 days when Hallel is said (Ta’anit 28b). They are times that mark a miracle which took place in Israel, going out of Egypt or physical salvation of the Jewish people.
Saying Hallel on Rosh Chodesh is not on the list. The custom began after the destruction of the Temple. The sage Rav thought to stop Babylonian Jews from saying it. But when he heard that they omitted part, he understood that they knew it was a tradition, not a biblical command and accepted their custom (Talmud Arachin 10a).
Rosh Chodesh is not be a celebration of redemption like Pesach or Succot. It is redemptive through its reminder that every new month can be a new beginning.