What additions does your local synagogue make to the morning service on Rosh Hodesh? What do you do if you are not praying in a synagogue? Today it is common practice to recite what is referred to as “half-Hallel” immediately after the Amida prayer. “Half-Hallel” consists of the normal recitation of psalms of thanksgiving (Tehillim 113–118), but skipping the first half of mizmorim 115 and 116. Generally speaking, the Ashekenazi community recites a blessing before this prayer, while the Sefardi community does not (see Shulhan Arukh, Orah Hayyim 422:2). In Chabad synagogues the tradition usually is for the hazzan to recite the blessing, and congregants respond to that berakha without saying it themselves (see the Siddur ha-Rav).
These varying traditions stem from the discussion of this topic in our Gemara and the lack of clarity in its conclusion.
The Gemara lists the days on which a full Hallel is said – eight days of Sukkot, eight days of Hanukkah, the first day of Pesah and on Shavu’ot (three days are added in Diaspora communities) – with no mention made of Hallel on Rosh Hodesh.
On this topic, the Gemara relates: Rav happened to come to Babylonia, where he saw that they were reciting Hallel on a New Moon. Unfamiliar with this practice, he thought to stop them, as he assumed that they were reciting Hallel unnecessarily. Once he saw that they were omitting portions, he said: I can learn from this that they are maintaining the custom of their forefathers, i.e., they know that it is a custom, not an obligation. It is taught in a baraita: An individual should not begin reciting Hallel on a New Moon, but if he has begun he should complete it.
Some understand that this story indicates that Hallel was not recited in Israel on Rosh Hodesh, perhaps because the closeness to the mikdash, where special sacrifices were brought to celebrate the day, was enough of an observance; in Bavel, on the other hand, they had a sense that there was a greater need to recognize and publicize the day.
The Ge’onim understood the concluding line of the Gemara as suggesting that individuals do not say Hallel at all on Rosh Hodesh. The Ri”f understands it to mean that an individual should not begin Hallel with a berakha, but if he did so, he should conclude with the appropriate blessing, as well.