When the Bais Hamikdash was standing, every person in Am Yisrael was instructed to give half a shekel. The money that was collected was used to purchase korbanos tzibbur, such as the Korban Tamid, Mussafim of Shabbos and moadim, and other things relating to the avodah of the korbanos (Rambam Shekalim 5 1).
The obligation of giving half a shekel was for men and not for women, as the passuk says “venasnu ish kofer nafsho” – ish and not an ishah (Shekalim 1 3; Rav Ovadiah Bartenura ibid). From this din, Rav Akiva Eiger proves that women did not have a part in the korbanos tzibbur, and therefore they are also exempt from davening Mussaf, which was instituted to correspond to the Korbanos Mussaf, which are a korban tzibbur.
However, Harav Yechezkel Heller, author of Amudei Ohr, differs from this view and claims that even though the Torah exempts women from giving machatzis hashekel, one should not derive from that they have no part in korbanos tzibbur. On the contrary, it is more likely to say that although they were exempt from payment for these korbanos, still, they receive atonement through them, and therefore, they are obligated min hadin in Tefillas Mussaf. And as we find regarding the par hachatas sacrificed by the Kohein Gadol on Yom Kippur, that even though the Kohein Gadol purchased it himself, it would atone for all his brethren the Kohanim (see Rambam Avodas Yom Hakippurim 5 13).
But halachah lema’aseh, the Amudei Ohr agrees that women are exempt from Tefillas Mussaf, because of the koach sevara that is mentioned in Tzelach (Brachos 26a): The Gemara (ibid 20b, based on Tosafos ibid ad loc. B’tefillah) explains that even though the mitzvah of tefillah is time-bound, because the essence of tefillah is asking for compassion and women also need that rachamim, therefore they are obligated in tefillah like men are, and are not exempt from it like the other time-bound mitzvos.
Indeed, this reason is correct regarding the daily tefillos which during the week include brachos with requests. But regarding Mussaf, which was instituted to correspond to the Korbanos Mussaf, and their essence is not rachamim, then even during the week they do not include requests, and this reason does not apply. If so, then the women should be exempt from it like the other mitzvos that are time-bound (see Rabbeinu Yona Brachos 13a in the Dapei Harif.)
But the sefer Magen Gibborim rejects this sevarah as well, because from the Yerushalmi (Brachos 4 6, and see Rosh Brachos 3 18) it seems that in earlier times, in Tefillas Mussaf that was said on weekdays, they would daven Shemoneh Esreh like in other tefillos and they would add a brachah to remember the Korban Mussaf. From this it emerges that when the tefillah of Mussaf was instituted, the women were also obligated in it, because it included the request for rachamim like in the other tefillos. Therefore, even though today we daven Mussaf with only seven brachos, their obligation has not expired.
Hagahos Rav Akiva Eiger Orach Chaim 106 2; Shu”t Amudei Ohr 7; Magen Gibborim Elef Hamagen 4
