Eruvin 40B

אמר רבי זירא: שאני ראש חדש, מתוך שכולל לשחרית וערבית, כולל נמי במוספין.

R' Zeira said: Rosh Chodesh is different, for while it must indeed be mentioned according to Beit Shammai, it does not require a separate blessing. Since Rosh Chodesh is included in the regular morning and evening prayers without a separate blessing, it is included in the additional prayer as well without a separate blessing.

ומי אית להו לבית שמאי כולל? והתניא: ראש חדש שחל להיות בשבת, בית שמאי אומרים: מתפלל שמנה, ובית הלל אומרים: מתפלל שבע. קשיא.

The Talmud asks: And do Beit Shammai accept the view that one should include Rosh Chodesh in the regular prayer? Wasn’t it taught in a baraita that with regard to a New Moon that occurs on Shabbat, Beit Shammai say: One must pray an Amida that includes 8 blessings in the additional prayer, including a separate blessing for Rosh Chodesh; and Beit Hillel say: One must pray an Amida that includes 7 blessings, as Shabbat and Rosh Chodesh are mentioned in the same blessing? Therefore, according to Beit Shammai, we do not include Rosh Chodesh and other days in the same blessing, and the fact that Rosh Chodesh does not have its own blessing on Rosh HaShana is because one mention of remembrance counts for both Rosh HaShana and New Moon. The Talmud comments: Indeed, this is difficult.

וכולל עצמו ⁠— תנאי היא, דתניא: שבת שחל להיות בראש חדש או בחולו של מועד, ערבית שחרית ומנחה מתפלל כדרכו שבע, ואומר מעין המאורע בעבודה. רבי אליעזר אומר: בהודאה, ואם לא אמר מחזירין אותו.

The Talmud comments: The issue of whether or not one should include the mention of Rosh Chodesh in the blessing pertaining to the sanctity of the day of Shabbat is itself the subject of a dispute between the tanna’im, as it was taught in a baraita with regard to a Shabbat that occurs on a New Moon or on one of the intermediate days of a Festival: For the evening, morning, and afternoon prayers, one prays in his usual manner and recites 7 blessings, and says a passage pertaining to the event of the day, i.e. May there rise and come [ya’aleh veyavo], during the blessing of Temple service. R' Eliezer disagrees and says that this passage is said during the blessing of thanksgiving. And if he did not recite it, we require him to return to the beginning of the prayer and repeat it.