Chatzi Hallel

There are basically 4 shitos on the issue of whether you make a bracha on chatzi hallel or not (for example - on Rosh Chodesh or the latter days of Pesach):

Shitas HaRambam: No bracha at all on chatzi hallel.

Shitas Rabbeinu Tam: One does make a bracha on chatzi hallel.

Shitas HaRaavad: On Rosh Chodesh you make a bracha and not on the latter days of Pesach.

Shitas HaRamban: On Rosh Chodesh you don't make a bracha and on the latter days of Pesach you do.

The Brisker Rav in Brachos 16:11 discusses this issue. Briefly, his basic premise is that one only makes a bracha on a cheftza shel mitzvah and the real machlokes is how exactly we define what is a cheftza shel mitzvah. Perhaps we will return to this issue in more depth another time.

The Machzor Vitri (page 192/siman 226) brings Rabbeinu Tam's shitah as well as some opposing views. Also, the Kol Bo in his halachos of Rosh Chodesh brings some shittos.

As far as why we say chatzi hallel on Rosh Chodesh, this comes from a gemara in Erchin 10b that no shira is needed if it isn't mekudash lechag. There is also a gemara in Taanis 28b that mentions that reciting hallel on Rosh Chodesh is a minhag. The gemara there relates this to the fact that the hallel is said bidilug.