Gemara

Mishnah: All Shofaros are Kesherim, except for that of a cow (or bull), for it is (called) Keren.

R. Yosi: All Shofaros are called Keren - "Bi'Mshoch b'Keren ha'Yovel"!

Gemara - Question: How do Chachamim answer R. Yosi?

Answer: All Shofaros are called Shofar, and also Keren. A cow's horn is called only Keren - "Bechor Shoro Hadar Lo v'Karnei Re'em Karnav".

R. Yosi says that a cow's horn is also called Shofar - "V'Sitav la'Shem mi'Shor Par";

Shor and Par both mean 'bull'. Surely, we expound the repetition and read 'Shofar'!

Chachamim: No. The verse alludes to a one-day old Shor that was a Par (a mature bull, i.e. one from the six days of creation), like Rav Masnah taught.

Ula: Chachamim do not allow a cow's horn because Ein Kategor Na'aseh Sanegor (the prosecutor, i.e. the species of the golden calf, cannot become the advocate).

Rav Chisda taught that this is why the Bigdei Kohen Gadol with gold cannot worn in the Kodesh ha'Kodoshim.

Ein Kategor Na'aseh Sanegor applies only inside (the Kodesh ha'Kodoshim; all other Avodah of the Kohen Gadol must be in the Bigdei Zahav). Since the Shofar is for Zikaron, it is like Avodah inside.

Question: In the Mishnah, Chachamim disqualify because it is a Keren!

Answer: They disqualify because Ein Kategor Na'aseh Sanegor, and also because it is a Keren.

R. Yosi does not consider the Shofar like Avodah inside.

Abaye: Chachamim disqualify a cow's horn because the Torah says to blow (one) Shofar. A cow's horn is layered, so it is like two or three Shofaros.

Abaye agrees that they also disqualify because it is called Keren, like the Mishnah says.

Chachamim say that since it is connected, it is considered one Shofar.

26b - Mishnah: The Shofar for Rosh Hashanah is straight, of a Ya'el (wild goat).

(Yom Kipur of) Yovel is like Rosh Hashanah regarding Teki'os;

R. Yehudah says, on Rosh Hashanah we blow (bent) Shofaros of males (rams). In Yovel we blow of a Ya'el.

Gemara - R. Levi: The Mitzvah on Rosh Hashanah and Yom Kipur is with a bent Shofar.

Question (Mishnah): The Shofar for Rosh Hashanah is straight, of a Ya'el.

Answer: R. Levi holds like the following Tana:

Beraisa - R. Yehudah: On Rosh Hashanah we blow bent Shofaros of males. In Yovel we blow of a Ya'el.

Question: If so, he should have said that the Halachah follows R. Yehudah!

Answer: Had he said so, one would have thought that he rules like R. Yehudah even regarding Yovel.

Question: What do they argue about?

Answer: R. Yehudah holds that on Rosh Hashanah it is best to be bent and submissive, and on Yom Kipur straightness is best;

Chachamim say that straightness is best also on Rosh Hashanah.

16a - R. Avahu: Hash-m commanded us to blow a ram's Shofar on Rosh Hashanah so that He will remember the Akeidah for us and consider it as if we bound ourselves in front of Him.

Rishonim

The Rif and Rosh (3:1) bring the Mishnayos, R. Levi and the Gemara's discussion of his ruling.

Ran (DH v'Chosvu): Tosfos says that we need not expound to disqualify horns that are not hollow. Surely, they are not called Shofar! 'Shofar' is related to Shefoferes (tube).

Question: Also a goat's horn is called Keren - "Veha'Tzafir Keren Chazus"!

Answer (Ran, ibid.): It is the same species as sheep - "Seh Kesavim v'Seh Izim". Alternatively, Daniel saw a horn without any hollow (alluding to the might of the kingdom), therefore it was called Keren.

Ran (DH u'Mihu Ika): Perhaps among Kosher animals only a cow's horn is Pasul, but the horn of any Tamei animal is Pasul. A Beraisa (Shabbos 28a) teaches that only Kosher animals are Kosher for Meleches Shomayim. This includes a Shofar, for it is like Avodas Panim. However, this is not so clear. The Gemara there did not learn about the Mishkan from this; it asked whether or not the Techashim were Tehorim.

Rav Hai Gaon (brought in Rashba Shabbos 28b DH Ela): The Gemara learned from Ohel ha'Mes that Behemos Temei'os could be used for the Mishkan. Therefore, it asked what we learn instead from the Beraisa.

Rambam (Hilchos Shofar 1:1): We blow a bent Shofar of a sheep, both on Rosh Hashanah and in Yovel. All other Shofaros are Pesulim.

Rebuttal #1 (Ra'avad): The Mitzvah l'Chatchilah is with bent Shofaros, but if one used a Shofar of a Ya'el he was Yotzei!

Rebuttal #2 (Ran DH u'Mihu Ika): R. Yehudah cannot say that what is Kosher for Rosh Hashanah is Pasul for Yovel and vice-versa, for he learns Rosh Hashanah from Yovel! Also, there is no source that R. Yehudah says 'ram' only to denote bent horns. Anything bent is l'Chatchilah.

Taz (OC 586:1): When the Rambam says 'sheep', this includes goats, and also a Ya'el, which is a kind of goat. The Rambam did not say that a ram's Shofar is best, for all of these are the species of rams.

Rosh (ibid.): We hold like R. Levi. Also R. Avahu (16a) holds like him. It seems that they argue only about l'Chatchilah, but all agree that all Shofaros are Kesherim except for a cow's, like the Mishnah says. If one cannot find a ram's Shofar, he is Yotzei with one of a Ya'el or goat. R. Levi himself said 'the Mitzvah...' (i.e. l'Chatchilah). Also, the verses do not mention straight or bent. Also, the argument of Chachamim and R. Yehudah appears to be independent of that of Chachamim and R. Yosi. If Chachamim and R. Yehudah argue even about b'Di'eved, it is all one large argument with R. Yosi and his Tana Kama! Tosfos asks, if bent is only l'Chatchilah because one should be submissive on Rosh Hashanah, how can we learn from a Gezeirah Shavah to Yovel, when this does not apply? This question does not override the solid proofs that I brought. Chachamim say that we can learn, even though the reason does not apply.

Defense (of Tosfos - Beis Yosef DH v'Da and Bedek ha'Bayis). The Mizrachi says that we learn from a Gezerah Shavah only things explicit in the verses. We must say that the Tana'im had a tradition regarding bent and straight Shofaros, based on the Kavanos we should have. However, he does not explain why a bent Shofar is better for Yovel. It should be straight, a symbol of freeing slaves!

Rosh (ibid.): Ya'elim were mentioned only because they are straight. All straight Shofaros are the same. Likewise, rams were mentioned only because they are bent. The Ramban says that all bent Shofaros are l'Chatchilah. R. Avahu explains that the custom to use a ram's is to recall the Akeidah. This is the ideal Mitzvah.

Ran (ibid.) The Aruch explains that Ya'el is a female sheep, because the Seifa discusses males. However, it appears that Rashi is correct, for the Targum of "v'Ako" (Devarim 14:5, one of the Kosher Chayos) is 'v'Ya'ala', and "Harim ha'Gevohim la'Y'elim" suggests that Ya'el is a Chayah.

Poskim

Shulchan Aruch (586:1): For Rosh Hashanah we use a bent Shofar of a ram.

Mishnah Berurah (2,4,5): 'Ram' includes female sheep. It is ideal to use a ram, for this is the best commemoration of the Akeidah. A goat is preferable to a Ya'el, for it is closer to a ram. One should spend an extra third for a ram's Shofar, for Hidur Mitzvah.

Shulchan Aruch (ibid.): B'Di'eved all Shofaros are Kesherim, whether straight or bent. It is a better Mitzvah if it is bent. A cow's horn is Pasul in any case. The horns of most Chayos, which are one bone without Zachrus (a protrusion from the animal's head into the hollow of the horn), are Pesulim.

Bach (DH Shofar): After R. Avahu taught that a ram is l'Chatchilah, anything else is b'Di'eved.

Kaf ha'Chayim (4): The Beis Yosef and Taz say that all bent Shofaros are l'Chatchilah; a ram's is best. However, the words of the Shulchan Aruch connote like the Bach. The Bach holds that a straight ram's Shofar is just as b'Di'eved as a bent Shofar of a Ya'el. If these are the only options, one should use the bent Shofar, which is l'Chatchilah according to the Taz.

Kaf ha'Chayim (11): A buffalo horn is Pasul, for it is Shor ha'Bar, which is a Behemah (hence it must be considered a bull).

Rema: Also horns of Tamei animals are Pesulim.

Mishnah Berurah (8): There is a Safek about this. If one cannot find a Shofar of a Kosher animal, he should blow one from a Tamei without a Berachah.