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Who is obligated to hear Shofar: 
Women are not obligated to hear Shofar. Nevertheless if they desire to blow Shofar they may do so with a blessing. Practically women today have accepted the hearing of Shofar as an obligation and hence they must do so due to this custom.
May a man blow Shofar for women? It is permitted for a man to blow Shofar for women even if he has already fulfilled his obligation. It is even permitted for him to carry a Shofar through a public domain for the sake of women hearing Shofar. However if the man has already fulfilled his obligation of hearing Shofar [or plans to do so with a later blowing] then he may not say the blessing for a woman, and rather the blessing is to be said by one of the women. If a man desires to blow with a blessing for women then he is to blow for them before hearing Shofar in Shul and hence fulfill his obligation with this blowing. Alternatively he can intend not to fulfill his obligation with the blowing in Shul [of Meyushav] and then blow for women with a blessing [between Meyushav and Meumad] having intent to fulfill his obligation with this blowing.
May a woman blow Shofar for another man to fulfill his obligation?
No, as will be explained in H!
May a woman blow Shofar for herself or for another woman to fulfill the Mitzvah?
How many blows is a woman to hear?
A woman only needs to hear the 30 sounds required to fulfill the Mitzvah [as explained in Chapter 3]. Nevertheless it is proper for them to also hear the blows of Musaf and not speak until its conclusion.
Must a woman perform Hataras Nedarim if she is unable to hear Shofar one year for whatever reason?
Children are [Biblically] exempt from hearing Shofar. [However children who have reached the age of Chinuch are Rabbinically obligated to hear Shofar. It is thus a Mitzvah to bring them to Shul and educate them to hear Shofar.]
Children below the age of Chinuch: Due to the necessity for quiet during the blowing, children which are below the age of Chinuch are not to be brought to the [men’s side of] Shul for Shofar blowing in order so they do not disturb the listeners. However the women may have the children by their side as in any event they are not obligated in the Mitzvah.
At what age is a child considered to have reached the age of Chinuch? The age of education for positive commands depends on the sharpness and knowledge of that individual child in each particular Mitzvah, whether a Biblical or Rabbinical command. [Thus when the child understands the idea of the Mitzvah of Shofar he is obligated to hear Shofar. Some write that this is after age 6 or 7.]
Blowing for others: Even a child who has reached the age of Chinuch is unable to blow and fulfill the obligation of an adult which is Biblically obligated in the command.
May a child blow Shofar for another child to fulfill his obligation?
If the child has not reached the age of Chinuch then he cannot blow for the sake of the Mitzvah of a child that has reached the age of Chinuch. If the child has reached the age of Chinuch then some Poskim rule he may do so if he is being supervised and has intent to be Motzi the other child. Others rule that he cannot do so even if he has reached the age of Chinuch.
May a child blow Shofar for a woman to fulfill the Mitzvah?
Some Poskim rule that he cannot do so even if he has reached the age of Chinuch.
May a woman blow Shofar for a child to fulfill his obligation?
May one blow the Shofar on behalf of children above the age of Chinuch if he already fulfilled his obligation?
Yes. However in such a case the child is to say the blessing and not the Baal Tokeia.
May one blow the Shofar on behalf of children below the age of Chinuch?
See Halacha 22 in Q&A!
A mute which is able to hear is considered like a normal person and is obligated to hear Shofar. Likewise he may blow Shofar for others to fulfill their obligation. In such a case one of the listeners is to say the blessing and he is to blow.
A deaf person [is exempt from blowing Shofar]. Even if he is able to speak he is not allowed to blow for others as since he cannot hear he is exempt from the Mitzvah.
Does a person with a hearing aid have to hear Shofar? May he blow Shofar for others?
Electric hearing aid: One does not fulfill his obligation with hearing Shofar through an electric hearing aid. Therefore if one is able to hear without the hearing aid, he must remove it during the blowing in order for him to fulfill his obligation. If he already fulfilled his obligation he may blow for others even while wearing the hearing aid. If however he cannot hear without the hearing aid, then he is exempt from the Mitzvah and cannot blow for others.
A blind man is obligated in blowing Shofar and is able to blow for others to fulfill their obligation.
G. May one who is exempt from hearing Shofar blow for others?
(Although the main Mitzvah of hearing Shofar is merely to hear the sound nevertheless) if the blower is exempt from the Mitzvah then he cannot fulfill the obligation on behalf of others through blowing for them.
A child: Even a child who has reached the age of Chinuch is unable to blow and fulfill the obligation of an adult which is Biblically obligated in the command.
List of people exempt from Shofar:
If one is medically forbidden from blowing Shofar may he nevertheless choose to do so for himself and/or others?
If doing so can put his life in danger [such as a heart attack] then it is forbidden for him to blow the Shofar in such a case and if he does so it is questionable whether he or others fulfill their obligation.
Is an Onen obligated to hear Shofar? May he blow Shofar for others?
A. When is one defined as an Onen on R”H?
First day of Yom Tov/R”H: On the first day of Yom Tov one is never considered the status of an Onen unless he plans to bury the body on Yom Tov through gentiles. Practically however today the custom is not to bury the body on the first day of Yom Tov even through gentiles. He is thus liable for all the Mitzvos, including Shofar and can hence also blow Shofar on behalf of others.
Second day of Yom Tov/R”H: On the second day of Yom Tov and R”H one is considered the status of an Onen even if he does not plan to bury the corpse on that day unless this is due to incontrollable matters.
B. The law of an Onen:
Someone who is defined as an Onen, as stated above, is exempt from the Mitzvah of Shofar. Nevertheless some Poskim rule it is permitted for him to blow Shofar for others. Other Poskim however rule that he cannot blow Shofar for others and if he did so they do not fulfill their obligation. Practically he may not blow Shofar for others unless there is no other person available, in which case it may be done without a blessing.
 588/6-8; This follows the current printing of the Shulchan Aruch. However the Kuntrus Hashulchan writes that in truth this set of Halachas is the beginning of 589 and so is implied from 592/8.
 If one becomes Bar Mitzvah on R”H: If one was born on R”H and hence becomes Bar Mitzvah on R”H some say he is to be stringent to hear Shofar [again] after he reaches 13 years of age to the hour that he was born. Thus for example if he was born in the afternoon he is to hear Shofar [again] in the afternoon. See Piskeiy Teshuvos 589 footnote 15; Choshen Mishpat Shach 35/2; Smeh 35/1; Mefarshim there; Beis Shmuel 155/18; Kinyan Torah 3/53. Nevertheless this is not the custom. [Piskeiy Teshuvos ibid]
 See 595/1-2
 588/6; 589/2; Michaber 589/3; Kedushin 29
The reason: The reason for this is because hearing Shofar is a Mitzvas Aseh Shehazman Grama. [ibid] Regarding a Tumtum and Androgenus see 588/9.
 Admur 589/2; Michaber 589/6; R”H 33a as rules Rebbe Yossi and Rebbe Shimon; Rabbeinu Tam; Ran; Rosh there; Beis Yosef 589; See Kaf Hachaim 589/22
The reason it is allowed for women to blow Shofar on Rosh Hashanah despite it being a Shvus: Although blowing Shofar on Shabbos or Yom Tov is forbidden if it is not done for the sake of the Mitzvah and women are not obligated in the Mitzvah nevertheless the Sages allowed women to blow it if they intend to do so for the sake of the Mitzvah. The reason for this is because this specific prohibition of blowing a Shofar is not an actual complete Shvus as it is merely forbidden due to Uvdin Dechol, thus for the sake of appeasing the women the Sages permitted this light Rabbinical matter. [589/2]
 589/2; Rama 589/6; Rabbeinu Tam R”H 33a; Ran; Rosh in name of Mahritz Geios; Tur 589; Ritva Sukkah 2; Rashba 123; Maggid Mishneh Shofar 2 in name of Rashba; Shut Min Hashamayim 1; Birkeiy Yosef 654/2; Yosef Ometz 82; See Kaf Hachaim 589/23
The reason it is allowed for women to say a blessing despite them not being obligated in the command: Women are accustomed to say a blessing on even those commands that they are exempt from due to it being a Mitzvas Aseh Shehazman Grama. Thus in this case as well women may say the blessing before blowing Shofar. [ibid] In 17/3 Admur explains that women may say a blessing of “Asher Kidishana … Vitzivanu..” on commands that they are exempt from fulfilling being that the men are commanded in this Mitzvah. Furthermore although they do not receive as much reward as a man who is commanded in the Mitzvah, they do receive some reward and thus they too may say a blessing just like a man. [17/3]
Other Opinions-Sefaradim: Some Poskim rule that women may never recite a blessing over any Mitzvah that they are not obligated in. [Michaber 589/6; Beis Yosef 589; Rambam Hilchos Tzitzis; Igur in name of Reb Yeshaya] The Michaber ibid rules that they cannot say a blessing as Safek Brachos Lihakel. [Beis Yosef 589] The Birkeiy Yosef ibid states that after seeing the answer from heaven in Shut Min hashamayim he retracted his ruling like the Michaber and would tell women to recite a blessing. He concludes that if the Michaber would have seen the Shut Min Hashamayim certainly he too would have retracted his ruling.
 Admur 17/3; Shut Rav Akiva Eiger 1 Hashmatos; See Ashel Avraham Butchach 689 and Mahadurah Tinyana 592; so is implied also from Kitzur SHU”A 129/19; Ateres Zekeinim 689
 589/2; Michaber 589/6
 Admur ibid; Michaber ibid; Rama ibid; Ravaya brought in Rosh R”H 33; Tur; Beis Yosef; Levush; Erech Hashulchan 589/3; M”B 589/8
Other Poskim: Some Poskim rule it is forbidden for men to blow the Shofar on behalf of only women as it is forbidden to blow the Shofar for no need. [Baal Hitur brought in Tur 589; Peri Chadash; See M”E 589/12] The Kaf Hachaim 589/27 concludes that since there are Achronim that agree to the ruling of the Peri Chadash the custom therefore is that women come to Shul to hear Shofar and only in a case of need do we blow for them outside of Shul.
 Admur 589/2; Ravaya brought in Rosh R”H 33; Tur; Beis Yosef; Chida in Yosef Ometz 82; Kaf Hachaim 589/28; Igros Moshe 3/94; Az Nidbaru 13/38 [permitted even for female children]; Ketzei Hamateh 589/5 [“so is the custom”]; Piskeiy Teshuvos 518/4; 589/5
Other Opinions: Some Poskim rule one may not carry the Shofar through a public domain that does not contain an Eiruv simply for the sake of blowing for women. [Shaagas Aryeh 106 brought in Machazik Bracha 589/5; Shaareiy Teshuvah 589] The Chida in Yosef Ometz ibid negates this opinion and concludes like above. See Piskeiy Teshuvos ibid footnote 16 that perhaps today it is permitted according to all opinions as women have accepted upon themselves to hear Shofar and it is hence like a vow which is Biblical and one may hence carry on their behalf according to all.]
 The reason why a man may not say the blessing for women: Women are not obligated in the command of Shofar and hence are not commanded to say the blessing. Thus a man has no obligation to recite the blessing for them to fulfill this optional Mitzvah. In such a case if a man says the blessing for women it is considered a needless blessing and is hence defined as a blessing in vain. However women themselves can choose to say the blessing for reasons explained in previous footnotes. [ibid] Vetzaruch Iyun as to how to understand this reason. Why can’t men say this optional blessing for women just like women can say it for themselves? In other words if we permit an optional blessing for women why can’t we also permit it for men to say for the sake of women. Perhaps the reason is because since the blessing is optional and the man gains nothing for saying it [no even an exemption from Areivus] it is therefore considered an unnecessary blessing for him to say. However women may say it being that she gains the fulfillment of the Mitzvah and hence it is not unnecessary.
 This means that one is to leave Shul to blow for women between the first and second sets of blowing [between the blowing before Musaf and Musaf] and then return to Shul to hear the blowing of Musaf [Meumad]. [However one is not to have in mind not to fulfill his obligation and then only blow for women after Musaf as in such a case he has lost out on the Mitzvah of Tekios Meumad in Musaf which is only done with a Minyan.] Now although one makes an interval between his blessing and the blows of Musaf being that he says the blessing in the house of the women and returns to Shul for the blowing of Musaf, nevertheless he is not required to repeat the blessing before Tekios Meumad of Musaf as even if one talked in-between he is not required to repeat the blessing as explained in 592/6. [ibid]
 Admur ibid; See M”E 589/12 that says this is the best option in order to suspect for those that rule one may not blow on behalf of women. However see Alef Hamagen 589/11 that negates this as we do not ask of one to exclude himself from the community Mitzvah.
 M”E 589/8; Kaf Hachaim 589/15
 Shaagas Aryeh 105; Machazik Bracha 589/4; Shaareiy Teshuvah 589; Kaf Hachaim 589/26
 Piskeiy Teshuvos 589/2
 Piskeiy Teshuvos 589/4
 M”A 581/12; Degul Merivava Yoreh Deah 214/2; Yabia Omer 2/30
 Kaf Hachaim 568/46; 581/71; 589/34; Ben Ish Chaiy Netzavim 17; see Shach 214/2
 Admur 588/6; Michaber 589/2
 Michaber ibid; Admur ibid regarding that a Katan can’t be Motzi others; Vetzaruch Iyun why this exemption of a Katan was not explicitly written in Admur 588/6 and 589 and is only brought under the law of whether a Katan can be Motzi others.
 M”E 589/4; M”B 587/16; Kaf Hachaim 589/5; Gemara R”H 33B; Erechin 2b; Admur 343/2 regarding all Positive commands; See Admur 592/8 regarding children below Chinuch and 596/2 regarding educating children in how to blow Shofar; See Mikraie Kodesh 20 for why this law was not explicitly written in Shulchan Aruch here, as it is written regarding Shema; Tefila; Lulav; Megillah [Rama 596].
 592/8; Drashos Mahril R”H; Kneses Hagedola 589; M”A 589/1; Elya Raba 589/1; P”M 589; M”E 589/4; Kaf Hachaim 587/19; 589/5
 Admur 343/3; Alef Hamagen 589/5
 See Admur ibid that a child who understands the concept of Shabbos is obligated to hear Kiddush and Havdala. If the son knows how to garb himself in a Tallis, then the father is obligated to educate the child wearing Tzitzis. One cannot say that regarding Shofar it is when the child learns how to blow Shofar, as there are many adults that do not know how to blow. Thus seemingly it is when he understands the meaning of the Mitzvah.
 Birrur Halacha Tinyana in name of Rabbeinu Manoach on the Rambam Shofar 7/7 brought in Piskeiy Teshuvos 589 footnote 9
 Mikraeiy Kodesh 26; Piskeiy Teshuvos 589/2
 Tiferes Yisrael R”H 3/8
 Tiferes Yisrael R”H 3/8
 Halichos Beisa 15 footnote 91 in name of Rav SZ”A; Piskeiy Teshuvos 589/2
 Implication of M”E 589/8; Kaf Hachaim 589/15
 Peri Megadim 589 A”A 4; Chayeh Adam 140/7
 If the child knows how to say the blessing then this follows the same ruling as any time one is blowing for another in which case the person fulfilling the Mitzvah is to say the blessing . [See Halacha 9F]
If the child does not know how to say the blessing: The P”M ibid writes that the adult may say the blessing on behalf of the child since he is obligated in Mitzvos, however he concludes with a Tzaruch Iyun. The Piskeiy Teshuvos 589/2 writes based on this that the adult is not to say the blessing.
 Az Nidbaru 13/38
 Chanoch Lanaar 22/7 based on the Poskim that rule one may not blow for women if he already fulfilled his obligation. Hence by a child one is to be stringent.
 As the Mitzvah of blowing Shofar is to hear the sound and since he cannot hear the sound he is exempt. [ibid]
 See Piskeiy Teshuvos 589/3; Shraga Hameir 7/46; Likkutei Sichos 21/497; Shulchan Menachem 3/105
 M”B 589/4
 As he is obligated in the Mitzvah. However see Ketzei Hamateh 589/1 that this only applies if he is able to hear somewhat without the aids, otherwise he cannot blow for others.
 Piskeiy Teshuvos 589/3; See regarding Megillah and other Mitzvos: Rebbe in Shulchan Menachem 3/319; Daas Torah 689 based on Halachos Ketanos; Mahraiy Engel Brachos 25; Minchas Elazar 2/67; Minchas Shlomo 1/9; Eretz Tzvi 1/23; Minchas Yitzchak 1/37 and 3/38; Yechaveh Daas 2/68; 3/54; Mishneh Halachos 4/85; Kinyan Torah 1/75; Igros Moshe 2/108; 4/126; Sheivet Halevi 5/84; Piskeiy Teshuvos 689/3; Shearim hametzuyanim 129/25
The reason: As a microphone is not the actual sound of the person but an electrical generated sound and hence one does not fulfill his obligation of Shofar with hearing an electric sound imitation.
 As otherwise it is considered as if he is hearing the sound of a Shofar and the sound of an echo in which we rule he does not fulfill his obligation as rules in 587/1 [see Halacha 17]
 If the set blower became blind and the congregation decided to acquit him from the job due to that they suspect for the opinion that rules that a blind man is exempt from all Mitzvos then if he is the best Shofar blower available he is to be reinstated into his position. Nevertheless initially he may not be fired from his position due to becoming blind even if there is a better Shofar blower available being that we hold like the opinion that says a blind person is obligated in all the Mitzvos in the Torah. [ibid]
 588/6; Michaber 589/1
 The Kuntrus Hashulchan states that in truth this word should read “blowing Shofar”.
 Parentheses is in original.
 This refers to one who is exempt from the Mitzvah entirely and not to one who is obligated in the Mitzvah and has already fulfilled it, in which case he can still blow Shofar for others as will be explained in Halacha 9F. [Kaf Hachaim 589/1]
The reason: Since Hashem does not desire this person himself to perform the command, how can he fulfill the command on behalf of others. [Levush 589; Kaf Hachaim 589/2]
 588/6; Regarding one who is a half slave see 589/1
 See Minchas Yitzchak 4/102; Mahram Shick 260; Mishneh Halachos 4/77; Moadey Kodesh 8/121; Piskeiy Teshuvos 589/1
 Michaber Yoreh Deah 341/1 [“On Shabbos and Yom Tov… and he is obligated in all the Mitzvos… unless he plans to bury the body through gentiles”]; Admur 71/3; Mateh Efraim 585/6-8; Nitei Gavriel Aveilus 30/1; 31/3
 Nitei Gavriel Yom Tov 50/4 in name of: Rashal Beitza 1/10; Kneses Hagedola; Elya Raba; Chemed Moshe 2; Mor Uketzia; Olas Shabbos Chomas Adam 170/1; Kitzur SHU”A in Lechem Hapanim 200
 Michaber 341/1; Admur 71/3; See 548/5; Nitei Gavriel Yom Tov 47/5
 M”A 548/8; M”E 596/8; Nitei Gavriel Yom Tov 47/5
Such as there is a strike in the cemeteries etc. If however one is simply delaying the burial for the sake of relatives and the like then he is an Onen. [Nitei Gavriel ibid]
An out of town relative: Practically the accustomed Rabbinical directive given today is that if the relative will not be attending the funeral then he is not considered an Onen, and rather begins Shiva right away. [So rules Bach brought in Shach 341/5; 2nd opinion in Admur 71/1; Kitzur SHU”A 196/4; Yosef Daas 341; M”A 548/8; Imrei Yosher 1/51; Mahrsham 2/260; Sdei Chemed p. 3456; Nitei Gavriel 14/4; Shaareiy Halacha Uminhag 3/136 that this is what the Rebbe did in 1952; and so ruled to me Rav Asher Lemel Cohen and Rav Yosef Simcha Ginzberg. On the hand for Poskim that are stringent see: Michaber 341/1; 1st opinion in Admur 71/1 and that so is the custom] Aruch Hashulchan 341/10]
 Peri Megadim Pesicha Koleles 2/29; See Piskeiy Teshuvos 581 footnote 91
 Mateh Efraim 96/7; Tov Taam Veadaas Mahadurah Gimel 3/206
 See Piskeiy Teshuvos 589/1 Veztaruch Iyun Gadol on his advice that the listeners say the blessing, as many Poskim say they are not Yotzei and it is hence a blessing in vain! If the Onen heard Shofar prior to the burial and the burial took place on Yom Tov then he is to hear Shofar second time without a blessing. [Divrei Malkiel 22; Gesher Hachaim 18/2-10; Sdei Chemed Aninus 3]