The fifth perek describes the offering of the korban pesach. The Mishnah (5:7) teaches that while the korban pesach was slaughtered, they would recite hallel. The Tosfot Yom Tov explains that it was an institution from the times of the neviim to recite hallel after having been redeemed from a difficult situation, which certainly applies to pesach. The question is, who recited hallel?
The Bartenura explains that the kitot (groups) recited hallel, implying it was those that brought the korban. This is all Rashi's position. The Tosfot Chadashim however cite the Tosfot and Rambam who explains that it was the leviim who recited the hallel while the korbanot were offered. The Tosfot (Pesachim 64a) cite a Tosfeta that records that this was indeed the case.
The Tosfot Chadashim however suggests that hallel was recited in a responsive manner lead by the leviiim, thereby reconciling the opinions of the Bartenura and Tosfot. This is also how the Tifferet Yisrael understands the Mishnah.
The Tosfot however ask that the Gemara (Erchin 11a) teaches that shira was only ever recited when a korban was offered along with wine libations (nesachim). The korban Pesach however was not brought with nesachim. The Tosfot explains that that principle however only applies to the [offering]{.underline} of the korban. Our Mishnah is referring to the time of the slaughter of the korban.
The Minchat Chinnuch (394) however asks that we find that the Gemara (Erchin 10a) questions that very law by noting that shira was recited when they brought the bikkurim. The Gemara there explains since the bikkurim were brought from grapes that were pressed for wine, that requirement was satisfied. We find therefore from that discussion that the principle is not just limited to the time of the offering of korban considering that the bikkurim was not "offered".
The Moadim Uzmanim (R' Shternbuch, 3:237) however explains that the reason why they recited hallel by the korban pesach is based on the Gemara (Pesach 95a): "is it possible that Yisrael would slaughter their korban pesach and not recite shira?" In other words, we find that there is a special din of simcha with the offering of the korban pesach. It is that simcha that obligates the recitation of hallel and not the korban itself. Since the people would be compelled to recited hallel, the leviim would join in the simcha.
Returning to the Minchat Chinnuch, with respect to the bikkurim, the obligation of shira is because it itself is a korban that requires waving and placement next to the mizbeach. Consequently, we find that the Gemara's question is sound when questioning how the requirement of wine is satisfied. In our case however, since obligation stems from simcha, there is no such requirement.1
1 The Moadim Uzmanim however also cites the Griz who explains Rashi as follows. There are two separate dinim in the recitation of hallel when slaughtering the korban pesach. One is from the perspective of the chabura. Just is they must recite hallel when consuming the korban, so too when it is slaughtered. That is what Rashi was referring to. The second din is the requirement of the leviim to recite shira when korbanot are offered. It is this second din that the Tosfefta was referring to. Rav Shternbuch however raises difficulties with this explanation and therefore maintains that explanation brought above. See inside for more details.
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