Greg Killian: A Rosh Chodesh (New Moon) - There are several other

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There are several other things which connect the festivals with Rosh Chodesh:

We rest on festivals; women rest on Rosh Chodesh[1]. Some have a minhag to reduce work on Rosh Chodesh.
We have a Musaf for festivals; we have a Musaf for Rosh Chodesh.
We have special additions (yaale v’yavo) to the Amida for festivals; we have special additions (yaale v’yavo) to the Amida for Rosh Chodesh.
We have special sacrifices on the festivals; we have special sacrifices on Rosh Chodesh[2] (Two oxen, a ram, seven lambs and their respective libations of wine, flour, and oil Numbers 28:11-15).
We have special Torah and Ashlamata (from Parshat Pinchas) for festivals; we have a special Torah and Ashlamata for Rosh Chodesh (from Parshat Pinchas). These special readings interrupt the triennial Torah cycle.
We have additional aliyot on festivals; we have an additional aliya on Rosh Chodesh.
We say Hallel on Festivals; we say half Hallel on Rosh Chodesh[3].
Fasting (mourning) is forbidden on the festivals; fasting (mourning) is forbidden on Rosh Chodesh[4].
We eat special meals in honor of the festivals; we eat a special meal or food in honor of Rosh Chodesh[5].
We wear special clothes for festivals; we wear special clothes for Rosh Chodesh.
Festivals are called Moedim; Rosh Chodesh is called moed[6].
We light candles for festivals; some Sephardim light a candle on Rosh Chodesh.
Sustenance for festivals is not fixed on Rosh HaShana; sustenance for Rosh Chodesh is not fixed on Rosh HaShana[7].

As to the minhag to recite Half Hallel on Rosh Chodesh, Rabbi J. B. Soloveitchik in his Yahrzeit Sheur on 4 Tevet, 5717, explained the statement in Arachin 10b, Rosh Chodesh, which is called moed, should require the recitation of Hallel. There is no prohibition to work on this day" as follows: A festival such as Passover possesses sanctity because of:

a) Musaf the additional-sacrifice offered in the Temple;

b) Prohibition of work;

c) Obligation to make the pilgrimage to Jerusalem;

d) Special mitzvot such as matza.

All festivals have one or more of these qualities. Rosh Chodesh, however, is distinguished only by musaf and is thus a festival in the Temple only, where Hallel was recited as din (law) not minhag. The minhag to recite Hallel on Rosh Chodesh consists in extending the sanctity from the Temple to Jewish communities outside its precincts.

The day before Rosh Chodesh is also known as Yom HaKippurim Katan. This means that the day before Rosh Chodesh is a minor Yom HaKippurim. The devout will fast on the last day of the month and reflect upon his actions on the first day of the month.

The Talmud[8] quotes an amazing comment of Rabbi Simeon ben Lakish that the he‑goat offered on Rosh Chodesh is called "a sin offering unto the Lord" because it is an atonement for HaShem Himself for having made the moon smaller than the sun. Arising out of this is the idea, expressed in the Rosh Chodesh liturgy, that Rosh Chodesh affords pardon for Israel's sins. Additionally, during the Musaf prayer service of Rosh Chodesh, we call Rosh Chodesh a "time of atonement". Thus in atonement, we see another connection between the festivals and Rosh Chodesh.

Shabbat Mevarchin

The Shabbat before Rosh Chodesh is known as Shabbat Mevarchin, which means "the Sabbath of blessing." We have an additional ashlamata (1 Shmuel 20:18-42) to prepare us for this special day. After the Torah reading in the Shabbat service, the prayer leader holds the Torah scroll, recites a blessing hoping for a good month, then announces the day of the upcoming week when the new month will begin and the name of the new month.

Shabbat Mevarchin is not observed during the month of Elul to announce the beginning of the month of Tishri, the month in which Rosh HaShana (the Jewish New Year) occurs.

Thus we announce Rosh Chodesh ahead of time, but we do not announce for festivals. This would seem to give special significance to Rosh Chodesh.

Finally, when the Sanhedrin sanctified the new moon it set the calendar date for the festivals of that month. The date of the festivals, therefore, depended on the new moon for their proper celebration. Rosh HaShana, which falls on Rosh Chodesh, is a festival which no one no the day or hour it begins because it’s beginning depends on the sanctification of the new moon. We have a midrash that makes this point: The angels ask HaShem, "When is Rosh HaShana?" "I do not know," HaShem responds. "Let us all go down to the bet din and see what they have decreed."