1) Your question can be answered with the help of a few brief words of the Rogatchover Gaon zt'l that are cited in Birkas Kohen to Erchin 10a (page 89) by Rav Baruch Shmuel Deutsch shlit'a, one of the Roshei Yeshivot of Kol Torah. The words of the Rogatchover appear in Tzafnas Pa'ane'ach, Mahadura Tinyana, on Rambam Hilchos Avodah Zarah 3:3. He writes as follows: "When the Mishnah in Erchin 10a states that the Chalil 'struck' in front of the Mizbe'ach, this did not take place on the Duchan, but it happened in front of the Mizbe'ach and was not considered as part of the singing of the Leviyim."

2) The Rogatchover seems to be saying that there is a big difference between the Mishnah on 10a and the Mishnah on 13b. The Mishnah on 10a discusses what music was played in front of the Mizbe'ach when the sacrifices were brought. This is not necessarily something unique to the Leviyim. In contrast, the Mishnah on 13b discusses the service in a different part of the Beis ha'Mikdash -- the Duchan -- and this was specifically the Avodah of the Leviyim.

3) In addition, the word "strike" in the Mishnah (10a) seems to indicate that the Chalil was used as a sort of percussion instrument. The Birkas Kohen writes that the Chalil was used merely for Simchah during the offering of the sacrifices, while the other instruments were an integral part of the Leviyim's service.

4) I saw in the Minchas Chinuch (Mitzvah 384:10) that if one did not sing and play the music properly, he has anulled a positive Mitzvah of the Torah. However, he cites the Sifri that says that even so, this does not disqualify the sacrifice that was brought and the Korban is valid. It thus seems difficult to make a distinction between l'Chatchilah and b'Di'eved.

Kol Tuv,

Dovid Bloom

Here are different answers to your question according to the Rishonim:

1) The Mesivta edition of the Gemara, on the Mishnah 13b (page 197 in Yalkut Bi'urim) cites the Chafetz Chaim in his Sefer Zevach Todah (end of Maseches Tamid) who learns that there is a dispute between Rashi and the Rambam concerning the Chalil. Rashi on the Mishnah (10a, DH v'Lo) writes that the reason why the maximum number of Chalilim is 12 is that this corresponds to the 12 days when the Chalil was played in front of the Mizbe'ach. The Chafetz Chaim learns from this that according to Rashi there were only 12 days in the year when the Chalil was played at all. The Chafetz Chaim writes that the Mishnah and Gemara on 13b is a proof for Rashi's opinion because no mention is made of the Chalil. This is because most days of the year the Chalil was not used at all, so the number of Leviyim at the Duchan would not be determined by the number of Chalilim.

2) In contrast, the Chafetz Chaim understands that according to the Rambam, the Chalil could be played any day of the year. We learn this from Rambam, Hilchos Klei ha'Mikdash 3:4, who writes, "With what instruments did they play? -With Nevalim and Chalilim, etc." This suggests that Chalilim were part of the fixed musical instruments all year round, not merely on the 12 days of Yom Tov. The Chafetz Chaim writes that the Sugya on 13b represents a difficulty to the position of the Rambam because the number of Leviyim at the Duchan should also have been decided by the number of Chalilim.

3) I would like to suggest a solution to the Rambam's Shitah, based on Rabeinu Gershom on 10a who writes that the reason why the maximum number of Chalilim is 12 is that this corresponds to the 12 Leviyim who stood at the Duchan, as the Mishnah states below (13b). This is a rather surprising thing for Rabeinu Gershom to write, because the Gemara on 13b gives a different reason for the number 12 -- namely, 9 Kinorot + 2 Nevalim + 1 cymbal. It appears to me that Rabeinu Gershom learns that the number 12 is derived as the Gemara on 13b says, but once we have that number we can then use it again to give us the number of Chalilim. At any rate, we can now suggest an answer to the Chafetz Chaim's question on the Rambam because we can say that the Rambam agrees with Rabeinu Gershom that there is indeed a connection between the number of the Leviyim at the Duchan and the number of Chalilim.

4) We have suggested that there is a hint to the Chalil on 13b, but we clearly have not solved the problem that if one includes the Chalil on 13b then 12 people are not enough to play all of the instruments. To solve this problem, I suggest that we look at the Tiferes Yisrael #47. He cites an opinion that the Leviyim mentioned on 13b not only played instruments but also sang. If so, this cannot include the Chalil, because it is not possible to sing while playing a flute.

5) To summarize, the answer to your question according to Rashi is that the Gemara on 13b does not mention Chalilim because the Gemara there is discussing what happened all year round, while Chalilim were used only on 12 special occasions in the year, as stated on 10a. According to the Rambam and Rabeinu Gershom, the answer to your question is that there is actually a hint on 13b to the Chalilim, because the number 12 is not only special to the Leviyim but is also shared by the Chalilim (as the maximum, not the minimum, number), and once we know the number of Leviyim we also know the number of Chalilim. In additon, we can say that 12 Leviyim are sufficient to play the musical instruments, because the Leviyim on the Duchan sung as well as played their musical instruments, so the Chalil cannot be included in this activity.

Kol Tuv,

Dovid Bloom

Donny, what you write is true according to the Rambam's opinion but the Tiferes Yisrael is discussing the opinion of Tosfos 10b DH Ein Pochasin MiY'b. It seems clear from Tosfos that the 12 Leviyim on the Duchan were the same ones that played the 2 nevalim 9 kinoros and 1 tzlatzal. Tiferes Yisrael #47 makes it clear that there is a dispute between the Rambam and Tosfos on this point

Wishing you a healthy summer

Dovid Bloom