Hallel and the Challil

This post is a bit long and involved, but if you stick with it I think it's followable:

The Mishna in Erchin on 10a says that there are 12 days that during the time of the shiras haleviim (when the leviim sang in the bais hamikdash) they would also play the chalil (a certain musical instrument). The 12 days were:

1. the day of the shechitas korban pesach
2. the day of pesach sheni
3. the first day of pesach
4. the first day of shavuos
5. the 8 days of succos

Rashi on the mishna makes a somewhat perplexing statement that on the days of the shechitas korban pesach we say hallel. Where is there any mention of hallel in the mishna?

The Griz in Hilchos Korban Pesach writes that Rashi here is following the Yerushalmi that compares chalil to hallel. That is, any time the chalil is used hallel is recited.

Rashi, though, makes another interesting comment. The pashtus would've been that any time the chalil is used, individuals in their davening would say hallel. Rashi, however, says that the hallel is sung by the leviim in the mikdash instead of the shir shel yom. This Rashi needs to be explained.

Before we get into this issue, let's discuss precisely what Rashi is saying. Rashi in the beginning of Perek HaChalil says explicitly that the chalil was used at shacharis in the bais hamikdash. If so, Rashi here is saying that while they banged the chalil during shacharis, the leviim sang the hallel instead of the shir shel yom.

It's fascinating because the in Meseches Sukkah 55 says that on sukkos the tehillim of hu"mbi (an acronym) were recited. Rashi says this recitation took place during mussaf. The turei even in Rosh Hashana 30b says that Rashi must be saying this because at Shacharis the regular shir shel yom is recited. The Avnei Nezer 24:1 points out that the turei even in mistaken! Rashi says explicitly in Erchin that on sukkos the hallel replaces the shir shel yom. Even more interesting, I saw in the chasam sofer (Beitzah Daf 5) that on these days we really don't know what the leviim said in the Bais Hamikdash so one shouldn't say Hayom Yom... shebo hayu haleviim omrim.. because that would be dover shekarim!

Getting back now to our main issue, why does Rashi learn that hallel replaced the shir shel yom and was sung by the leviim? If you look at the gemara on the mishna, the gemara clearly says that these days corresponded to days where yechidim said hallel in their davening... but that the leviim said hallel as the shir shel yom seems to be from out of nowhere!?

Now, if you take a look at the gemara in Erchin, you will see that the gemara actually says that there were 18 days that full hallel was said. First of all, the gemara adds the days of chanuka. Rashi explains that although on Chanuka full hallel is said, still the chalil isn't beaten because chanuka has no korban mussaf. You may ask, if the chalil is beaten at shacharis, who cares if there is no korban mussaf. The Chikrei Lev says that, for whatever reason, korban mussaf and the challil are related. I thought to answer with what I saw in the Igros HaGrid, that without a Korban Mussaf there is no kedushas hayom. If that's the case then the chiddush may be that since chanuka has no kedushas hayom, so there is no challil.

All this answers for chanuka, but what about for the two days of shechitas korban pesach? On those days we do bang the challil, but the gemara doesn't mention that we say hallel on those days. Why not?

Tosafos answer that really on those days we do say hallel, but that is only in the times of the bais hamikdash. That is why the gemara doesn't mention those days.

Tosafos is therefore saying that in truth there is an obligation of hallel on these days of the shechitas hapesach. We will return to this shitah a little later.

There is another machlokes rishonim regarding chatzi hallel that I believe is very relevant. Tosafos there in Erchin (d"h Shemoneh Asar) says that any time one doesn't say a full hallel there is no chiyuv at all to say the hallel. Rather, it is only a minhag. This is explicit with regards to Rosh Chodesh in the gemara in Taanis 28b. The Ramban however (Shabbos dafei HaRif 11b see Ran there) argues that the minhag is only on Rosh Chodesh. On the second days of Pesach, however, there is a bona fide chiyuv to say a chatzi hallel. So the Ramban argues that there could be a chiyuv of chatzi hallel.

In a similar fashion, the Griz in Chanuka 3:4 points out that by the hallel by korban pesach the "third group" often didn't finish saying the whole hallel (see Pesachim 64b). The Griz asks, how is there a chiyuv of less than full hallel? The Griz asks the same question by the hallel of the seder and the hallel on nissim (see there). He answers based on a Rav Hai Gaon that there are two dinim in hallel. There is krias hallel, in which the whole hallel is recited, and there is a din of shiras hallel (where you just sing praise to God) in which it's a chiyuv, but you need not say the whole thing.

This distinction of the Griz may work in the Ramban as well. The chiyuv on the second days of Pesach is a chiyuv of shiras hallel and not krias hallel. Thus, the entire thing need not be recited.

Now, if we were to view the gemara in Erchin through the eyes of the Ramban, when the gemara says that there are 18 days when one must finish hallel, it probably means as opposed to the other days where there is only a chiyuv of chatzi hallel (or shiras hallel as the Griz may say).
If so, on the days of korban pesach there is no chiyuv of full hallel (in contrast to Tosafos) , only a chiyuv of shiras hallel. This is, as the Griz pointed out, explicit from Pesachim 64b that by korban pesach the third group never finished reciting the full hallel. The question of Tosafos would now swing back on the Ramban. If there is no chiyuv of full hallel on the days of korban pesach why is there banging of the chalil on those days?? (Remember Tosafos answered that there really is a chiyuv of full hallel on the days of korban pesach and the gemara only omitted it because it only applied during the bais hamikdash. We are suggesting that the Ramban disputes that based on Pesachim 64b.)

This is where Rashi steps in. Rashi in Taanis on 28b is mashma that he agrees with the Ramban's position that there is a chiyuv of chatzi hallel. Rashi here then is now stepping up and answering the question on the Ramban. If there is no chiyuv for a full hallel on days of korban pesach, why do we bang the challil? The answer is that the chiyuv of the challil is not dependent on the individuals recitation of full hallel during davening. Rather, the chiyuv depends upon the recitation of the leviim over the korbanos. The gemara in Pesachim 64b says explicitly that the leviim said hallel while the korban pesach was brought. This is the chiyuv of shiras hallel of the Griz. That chiyuv is mechayiv challil because the challil accompanies the leviim when they sing hallel in the mikdash.

The gemara in Erchin, however, was merely listing the days where individuals say full hallel in their davening. This does not apply on the days of korban pesach. However, on all days where this does apply there is a general rule that the leviim also sing the hallel on the korban tamid. Thus, on those days the challil bangs during the korban tamid.

The rule therefore is that the challil is used with a korban that the leviim say hallel on. When is this? Any time the individual says full hallel and additionally by the korban pesach as the gemara says in Pesachim 64b.