By Rabbi Berach Steinfeld
The posuk in Bamidbar 28:11 says that a special korban was brought every Rosh Chodesh.
The Mishna in Shekalim 8:8 says that the limbs of a Korban Mussaf was placed on the mizbeach before it was burned. It was placed on the lower half of the ramp of the mizbeach. However, when it came to the Mussaf of Rosh Chodesh it was placed on the upper half of the ramp. The reason for the change in the Korban Mussaf of Rosh Chodesh is brought down in Meseches Sukkah 55a. It is to show the importance of Rosh Chodesh and it serves as a “pirsum” (advertisement) that people should know when Rosh Chodesh was set by Bais Din. Another indicator to show the chashivus of Rosh Chodesh is seen in Meseches Sukkah 54b when we are taught that when Rosh Chodesh fell out on Shabbos they sang the Shir Shel Yom of Rosh Chodesh before the Shir Shel Yom of Shabbos despite the fact that Shabbos comes more often than Rosh Chodesh. This shows the importance of Rosh Chodesh.
The gemara in Sanhedrin 41b tells us that Bais Din would let people know when it was Rosh Chodesh by blowing a shofar. The Aruch La’ner in Rosh Hashana 8b says we learn this out from a posuk in Tehillim 81:4 where it says “Tiku ba-chodesh shofar.” The Redak on Pirkei D’reb Eliezer explains that the blowing of the shofar was a pirsum for Rosh Chodesh.
The Rishonim write in Yereim 259 and Ohr Zarua, Hilchos Rosh Chodesh 552 and Shibalei Haleket 170 that the reason we bentch Rosh Chodesh the Shabbos before Rosh Chodesh is a takanas Chachamim to be mefarsem which day Rosh Chodesh will be. Since most people work during the week and are very busy, they established this announcement to take place on Shabbos when everyone is in Shul and not working. We could say that our bentching of Rosh Chodesh takes the place of the blowing of the shofar during the time of the Bais Hamikdash since it lets everyone know when Rosh Chodesh will take place.
The Minhag haolam is that we don’t bentch Rosh Chodesh Tishrei. The Levush 421 writes the reason for this is because since Rosh Hashana is the Yom Tov that is called  ”bakeseh,” it is covered and kept quiet. We don’t want to give a heads up for the satan to be mekatreg on us. This is also brought down in the Magen Avraham 417. We explained above that our bentching Rosh Chodesh takes the place of the blowing of the shofar to remind us when it is Rosh Chodesh. During the time of the Bais Hamikdash there was no need to blow the shofar for Rosh Chodesh Tishrei since everyone knew when it was Rosh Chodesh when they blew the shofar for Rosh Hashana. Therefore, since today we bentch Rosh Chodesh in place of the blowing of the shofar to remind us when Rosh Chodesh will be, and there was no blowing of the shofar for Rosh Chodesh Tishrei, we are consistent and do not bentch Rosh Chodesh.
May we all be zocheh to hear the shofar shel Moshiach.”