Gemara

Beraisa: The Teki'os of Yovel (and Rosh Hashanah - the Gra deletes this) override Shabbos in the Gevulin (outside the Mikdash or Yerushalayim), for a man and his household.

Question: What does 'a man and his household' mean?

Suggestion: It refers to a man and his wife.

Rejection: This is a Mitzvas Aseh sheha'Zman Grama (a positive Mitzvah that applies only at certain times). Women are exempt from all such Mitzvos!

33a - Mishnah: We do not stop children from blowing a Shofar (on Rosh Hashanah).

Inference: We stop women from blowing.

Contradiction (Beraisa): We do not stop women from blowing.

Answer (Abaye): The Mishnah is like R. Yehudah; the Beraisa is like R. Yosi and R. Shimon:

Beraisa - R. Yehudah: "Daber El Benei Yisrael..." - Benei Yisrael do Semichah (press on the neck of a Korban); Benos Yisrael do not.

R. Yosi and R. Shimon say, Semichah is optional for women.

Beraisa: Even on Shabbos we encourage children to blow, until they learn.

Shabbos 131b - Beraisa - R. Eliezer: Shofar and all its Machshirim override Shabbos.

He learns from "Yom Teru'ah Yihyeh Lachem" - on that day, even on Shabbos. Surely, the verse is not needed to permit blowing it!

Beraisa - Tana d'vei R. Yishmael: "Kol Meleches Avodah Lo Sa'asu" - this excludes blowing a Shofar and unsticking bread from the wall of the oven, for these are skills, not Melachos.

Rather, it permits preparations (detaching it from an animal).

Chachamim expound 'ba'Yom' - not at night.

Chulin 84b - Beraisa - R. Yosi: Bris Milah overrides Shabbos if we know that it is the eighth day. If we are not sure, it does not override even Yom Tov. Kisuy ha'Dam does not override Shabbos even for a definite Chayah, all the more so it does not override Yom Tov for a Safek Chayah!

Objection (Chachamim): Teki'as Shofar (outside the Mikdash) disproves this. We do not blow on Shabbos even for a definite Mitzvah, but we blow on Yom Tov for a Safek (male).

This is not difficult for R. Yosi, for he permits women to perform Teki'as Shofar (and all other Mitzvos Aseh sheha'Zman Grama), just like he permits Semichah.

Eruvin 96a - Beraisa: Michal bas Sha'ul used to wear Tefilin, and the wife of Yonah (the Navi) was Olah l'Regel. Chachamim did not protest against either of them.

Inference: They did not protest against Michal because they held that Tefilin is not Zman Grama!

Rejection: Perhaps the Tana holds like R. Yosi, who says that Semichah is optional for women.

Rishonim

Rif: We do not prevent children from blowing.

Ran (DH ul'Inyan Halachah): We learn from Michal and Yonah's wife that women may blow Shofar. We are not concerned for Shevus or Bal Tosif.

Rif (Shabbos 1b): Blowing a Shofar and Rediyas ha'Pas (peeling bread off the wall of the oven) are skills, not Melachos. Nevertheless, Rediyas ha'Pas is permitted only for a need of Shabbos.

Ran (DH Omar): Blowing a Shofar and Rediyas ha'Pas were not done in the Mishkan (Lechem ha'Panim was baked in molds. The dough was not stuck to the wall.) We do not need a verse to exempt for them. Rather, since "Meleches Avodah" connotes genuine Melachah, Chachamim did not forbid skills like Shevus. Rather, they are mere weekday actions, and they are permitted when there is a need. This is why we encourage children to blow. A proper Shevus is not permitted even for the Mitzvah itself! This is unlike the Ramban, who says that these are proper Shevusim.

Rambam (Hilchos Shofar 2:1): Women are exempt from hearing the Shofar.

Hagahos Maimoniyos: Just like women may do Semichah, even though it looks like Avodah with Kodshim, they may blow, even though it looks like Chilul Yom Tov. But if a man was already Yotzei, he may not sin to please others. The Ge'onim said like this, that a man may blow for them only before he was Yotzei. Maharam rules like this.

Rambam (2): If a woman blew a Shofar, one who heard was not Yotzei.

R. Mano'ach: We learn from this that women may blow without a Berachah. Similarly, Chachamim did not object to Michal and Yonah's wife for wearing Tefilin and being Olah l'Regel. The Rambam (Halachah 7, below) brings the Heter to allow children to blow a Shofar on Rosh Hashanah; he does not permit women. The Halachah follows our Stam Mishnah against an argument in a Beraisa (R. Yosi and R. Shimon). Also the Rif brings the Heter only for children, but not for women.

Note: R. Mano'ach retracted from his initial words. The proof from Michal and Yonah's wife are like R. Yosi, but R. Yehudah holds that Chachamim protested. However, the Rambam (Tzitzis 3:9, brought below) permits women to perform Mitzvas Aseh sheha'Zman Grama. Perhaps he holds that it is not clear like whom we rule, therefore we do not protest.

Rambam (7): We do not stop children below the age of Chinuch from blowing a Shofar on Shabbos that is not Rosh Hashanah, in order that they will learn. Adults may encourage children of any age to practice on Yom Tov, since Teki'ah is forbidden only mid'Rabanan.

Rambam (Hilchos Tzitzis 3:9): A woman is exempt from Tzitzis. If she wants to wear Tzitzis without a Berachah, she may. Similarly, if she performs any Mitzvah from which she is exempt without a Berachah, we do not protest.

Hagahos Maimoniyos (30): The Gemara (Eruvin 96a) says that Chachamim did not object to Michal and Yonah's wife for wearing Tefilin and being Olah l'Regel. The Yerushalmi says that Chachamim objected. I heard from a Gadol that we protest only where there is concern for transgression, such as Chulin ba'Azarah, Ervah (her hair might show), or blowing a Shofar on Rosh Hashanah, which is Shevus (if it is not a Mitzvah).

Note: The Ran (Rosh Hashanah 9b DH ul'Inyan Halachah): learns from Michal and Yonah's wife that women may blow Shofar; we are not concerned for Shevus or Bal Tosif. Seemingly, these are concerns also regarding Lulav (it is Muktzeh Machmas Chisaron Kis).

Rosh (Rosh Hashanah 4:7): R. Tam says that even though our Stam Tana is like R. Yehudah, the Halachah follows R. Yosi, for Nimuko Imo (he has great reasons to support himself). Also, Michal wore Tefilin and Yonah's wife was Olah l'Regel. Since actions were done, we follow them. Also, a Shelamim was brought to the Ezras Nashim for women to do Semichah on it to please them. Ba'al ha'Itur says that they may blow for themselves, but a man (who was already Yotzei) may not blow for them. In Ashkenaz, the custom is for a man to blow for women before they blow in the Beis ha'Keneses, so he himself is Yotzei then. Avi ha'Ezri says that since women may do proper Semichah, even though it looks like Avodah with Kodshim, similarly a man may blow for them, since Teki'as Shofar is a Chachmah, not a Melachah. This is even if the man was already Yotzei. One may take a Shofar through Reshus ha'Rabim to blow for women. This seems correct; women are no worse than children before Chinuch, whom we encourage to practice blowing, so they will learn. All the more so we encourage women to practice so they will learn, since they intend for a Mitzvah!

Rashba (Rosh Hashanah 33a DH R. Yosi): Our Mishnah is not a Stam Mishnah teaching that only children may blow, but not women. Rather, it permits children because all agree to this. It does not discuss women, for Tana'im argue about this. However, this is wrong. The Gemara learns from our Mishnah that R. Meir holds that Semichah is not Reshus for women!

Question: How could Chachamim (Chulin 84b) challenge R. Yosi from Teki'as Shofar, which is permitted on Yom Tov for a Safek? There, the reason it is forbidden on Shabbos, lest one carry a Shofar in Reshus ha'Rabim, does not apply on Yom Tov!

Answer (Tosfos Chulin 84b DH Teki'as): The reason does apply. One may carry in Reshus ha'Rabim on Yom Tov only for a need. If the Safek is truly a female, one may not carry a Shofar in Reshus ha'Rabim for her. Alternatively, Chachamim say that we forbid Teki'ah on Shabbos mid'Rabanan, lest one carry in Reshus ha'Rabim. Nevertheless it is permitted on Yom Tov for a Safek, even though normally Teki'ah is Asur mid'Rabanan!

Sha'arei Teshuvah (OC 589:2): The Sha'agas Aryeh (106) forbids to carry a Shofar in Reshus ha'Rabim for a female, but permits for a man who is unsure whether or not he was Yotzei. The Machazik Berachah (6) says that the Sha'agas Aryeh assumes that mid'Oraisa we are stringent about Safek mid'Oraisa. Rather, we are stringent about this mid'Rabanan, therefore it is forbidden. The Sha'agas Aryeh himself (108) forbids to carry in Reshus ha'Rabim for one who was already Yotzei mid'Oraisa and wants to hear the Teki'os of Musaf.

Poskim

Shulchan Aruch (589:6): Even though women are exempt, they may blow Shofar for themselves. A man may blow for them, even if he was already Yotzei.

Beis Yosef (589 DH v'Af and DH v'Chosav ha'Rav): The Ran and Rosh say that just like women may do Semichah and we are not concerned for Avodah with Kodshim, they may perform any Mitzvas Aseh sheha'Zman Grama and we are not concerned for Bal Tosif or Shevus of Teki'ah. R. Yeshayah says that if a woman blesses, she shows that it is a Chiyuv, and she transgresses Bal Tosif and a Berachah l'Vatalah. Regarding Bal Tosif, this is not true.

Gra (DH v'Chen): According to the second answer of Tosfos (Chulin 84b), a man who was already Yotzei may not blow for women. According to the first answer, he may. Avi ha'Ezri holds like the first answer, and rules like R. Yosi that one may carry a Shofar in Reshus ha'Rabim on Yom Tov for women. However, Tosfos (85a DH Nashim) says that R. Yosi permits only Isurim mid'Rabanan for women.

Mishnah Berurah (9). Even though it is Shevus to blow unnecessarily, it is somewhat of a Mitzvah for them to hear the Shofar.

Mishnah Berurah (11): Individuals should not blow in the first three hours of the day, for it is the time of judgment. It is better to blow for them after the Teki'os in the Beis ha'Keneses. However, then he may not bless for them unless he intended not to be Yotzei in the Beis ha'Keneses.

Kaf ha'Chayim (24): Teki'os are permitted to gratify women. Perhaps one should not blow for them more than once, like the Rema says (596) that men should not blow extra Teki'os after they were Yotzei.