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One must recite Ya’aleh Ve-yavo in Birkat Ha-mazon as well. Even though one is not obligated to eat a festive meal on Rosh Ĥodesh, one must mention Rosh Ĥodesh when reciting Birkat Ha-mazon because of the importance of the day, on which the musaf offering was brought (Shabbat 24a; Tosafot ad loc.). Ya’aleh Ve-yavo is inserted in the berakha of Raĥem, because both are prayers of supplication.
If one forgot to recite Ya’aleh Ve-yavo in Birkat Ha-mazon, he need not repeat the prayer. This is because one who forgets to mention the sanctity of a particular day must repeat Birkat Ha-mazon only on days when there is an obligation to eat a meal with bread, like Shabbat and Yom Tov. On Rosh Ĥodesh and Ĥol Ha-mo’ed, however, one is not obligated to eat such a meal. Therefore, from the perspective of the sanctity of the day, one need not recite Birkat Ha-mazon at all. Consequently, one need not repeat it if he accidentally omitted Ya’aleh Ve-yavo (sa 424:1).[13]
One who began a meal on Rosh Ĥodesh, and managed to eat a kezayit (olive’s bulk) of bread before sunset, must recite Ya’aleh Ve-yavo in Birkat Ha-mazon, even if he continued eating long after tzeit ha-kokhavim (also called “tzeit”; meaning, “the emergence of stars,” i.e., nightfall), because the meal began on Rosh Ĥodesh (sa 188:10; see Kaf Ha-ĥayim 188:43 for the opposing view).
If one began to eat on the eve of Rosh Ĥodesh, and finished his meal after tzeit, he must recite Ya’aleh Ve-yavo, provided that he ate a kezayit of bread after Rosh Ĥodesh began (sa 271:6, mb 271:29).[14]
[13] If one realizes that he forgot Ya’aleh Ve-yavo before beginning the berakha of Ha-tov Ve-hametiv, the Sages decreed that he should say, “Blessed are You, Lord our God, King of the Universe, Who gave New Moons to His people Israel as a remembrance” (sa 188:7). According to bhl, one should recite the full berakha, invoking God’s name and kingship (shem u-malkhut), just as one does on Shabbat and Yom Tov. Kaf Ha-ĥayim 188:31, however, maintains that one should omit God’s name and kingship.
[14] If one did not eat a kezayit of bread after tzeit, sa rules that he must, nonetheless, recite Ya’aleh Ve-yavo, as is the law on Shabbat, as explained there. According to Rema, however, he should not recite it. The Aĥaronim also dispute the case of one who began se’uda shlishit on Shabbat and finished his meal on Motza’ei Shabbat (Saturday night), which is also the beginning of Rosh Ĥodesh. See Peninei Halakha: Laws of Shabbat, ch. 7 n. 7.