Understanding the Mitzvah of Shofar: Why so many blasts?

Q: Is the mitzvah of shofar biblical or rabbinic?

Q: How do we know that the instrument to use on RH is a shofar (and not a trumpet)?

Q: Why do we blow during the day, and not at night?

Q: How do we know what sound(s) to blow? What is the difference - truah, shevarim, tekiah?

Q: How do we know how many of these sounds to blow? What is Biblical requirement? Rabbinic? Custom? 3, 9, 30, 100, or 101? Where do these numbers come from?

Q: Does the Torah provide clues as to the emotional message of a Tekiah vs a Teruah?

Q: How is shofar blowing structured for Ashkenazim vs Sefardim? Why don't we blow shofar during shacharit?

Q: Are women obligated to hear the shofar? Why? Why not?

Q: When the first day of R"H falls on Shabbat, do we fulfill our biblical obligation to hear shofar? Why/why not?

Q: If a man blows shofar for a group of women after he has fulfilled his own personal obligation, who says the blessing? Why? What is the halachic concept of "arvut"?

(כד) דַּבֵּ֛ר אֶל־בְּנֵ֥י יִשְׂרָאֵ֖ל לֵאמֹ֑ר בַּחֹ֨דֶשׁ הַשְּׁבִיעִ֜י בְּאֶחָ֣ד לַחֹ֗דֶשׁ יִהְיֶ֤ה לָכֶם֙ שַׁבָּת֔וֹן זִכְר֥וֹן תְּרוּעָ֖ה מִקְרָא־קֹֽדֶשׁ׃

(24) Speak to the Israelite people thus: In the seventh month, on the first day of the month, you shall observe complete rest, a sacred occasion commemorated with loud blasts.

(א) וּבַחֹ֨דֶשׁ הַשְּׁבִיעִ֜י בְּאֶחָ֣ד לַחֹ֗דֶשׁ מִֽקְרָא־קֹ֙דֶשׁ֙ יִהְיֶ֣ה לָכֶ֔ם כָּל־מְלֶ֥אכֶת עֲבֹדָ֖ה לֹ֣א תַעֲשׂ֑וּ י֥וֹם תְּרוּעָ֖ה יִהְיֶ֥ה לָכֶֽם׃

(1) In the seventh month, on the first day of the month, you shall observe a sacred occasion: you shall not work at your occupations. You shall observe it as a day when the horn is sounded.

(ט) וְהַֽעֲבַרְתָּ֞ שׁוֹפַ֤ר תְּרוּעָה֙ בַּחֹ֣דֶשׁ הַשְּׁבִעִ֔י בֶּעָשׂ֖וֹר לַחֹ֑דֶשׁ בְּיוֹם֙ הַכִּפֻּרִ֔ים תַּעֲבִ֥ירוּ שׁוֹפָ֖ר בְּכָל־אַרְצְכֶֽם׃

(9) Then you shall sound the horn loud; in the seventh month, on the tenth day of the month—the Day of Atonement—you shall have the horn sounded throughout your land.

From the verses below, do the tekiah and truah sounds seem to be used for different purposes? Which one is used for actions/activities of security/joy/gathering/holidays and which is used for war/dividing the nation/embarking into the unknown? Consider how this impacts our understanding of the shofar sounds on Rosh Hashanah.

(ט) סֵדֶר תְּקִיעוֹת, שָׁלשׁ, שֶׁל שָׁלשׁ שָׁלשׁ. שִׁעוּר תְּקִיעָה כְּשָׁלשׁ תְּרוּעוֹת. שִׁעוּר תְּרוּעָה כְּשָׁלשׁ יְבָבוֹת.

(9) The order of sounding the shofar is [that] three [sets] of three blows [are sounded] thrice; the required amount [of time] of the tekiah [long uninterrupted blow], is like [that of] three teruot [interrupted blows], and that of each tekiah is like [that of] three short blows [understood to be like sighs or cries].

(א) וַיְדַבֵּ֥ר ה' אֶל־מֹשֶׁ֥ה לֵּאמֹֽר׃ (ב) עֲשֵׂ֣ה לְךָ֗ שְׁתֵּי֙ חֲצֽוֹצְרֹ֣ת כֶּ֔סֶף מִקְשָׁ֖ה תַּעֲשֶׂ֣ה אֹתָ֑ם וְהָי֤וּ לְךָ֙ לְמִקְרָ֣א הָֽעֵדָ֔ה וּלְמַסַּ֖ע אֶת־הַֽמַּחֲנֽוֹת׃ (ג) וְתָקְע֖וּ בָּהֵ֑ן וְנֽוֹעֲד֤וּ אֵלֶ֙יךָ֙ כָּל־הָ֣עֵדָ֔ה אֶל־פֶּ֖תַח אֹ֥הֶל מוֹעֵֽד׃ (ד) וְאִם־בְּאַחַ֖ת יִתְקָ֑עוּ וְנוֹעֲד֤וּ אֵלֶ֙יךָ֙ הַנְּשִׂיאִ֔ים רָאשֵׁ֖י אַלְפֵ֥י יִשְׂרָאֵֽל׃ (ה) וּתְקַעְתֶּ֖ם תְּרוּעָ֑ה וְנָֽסְעוּ֙ הַֽמַּחֲנ֔וֹת הַחֹנִ֖ים קֵֽדְמָה׃ (ו) וּתְקַעְתֶּ֤ם תְּרוּעָה֙ שֵׁנִ֔ית וְנָֽסְעוּ֙ הַֽמַּחֲנ֔וֹת הַחֹנִ֖ים תֵּימָ֑נָה תְּרוּעָ֥ה יִתְקְע֖וּ לְמַסְעֵיהֶֽם׃ (ז) וּבְהַקְהִ֖יל אֶת־הַקָּהָ֑ל תִּתְקְע֖וּ וְלֹ֥א תָרִֽיעוּ׃ (ח) וּבְנֵ֤י אַהֲרֹן֙ הַכֹּ֣הֲנִ֔ים יִתְקְע֖וּ בַּֽחֲצֹצְר֑וֹת וְהָי֥וּ לָכֶ֛ם לְחֻקַּ֥ת עוֹלָ֖ם לְדֹרֹתֵיכֶֽם׃ (ט) וְכִֽי־תָבֹ֨אוּ מִלְחָמָ֜ה בְּאַרְצְכֶ֗ם עַל־הַצַּר֙ הַצֹּרֵ֣ר אֶתְכֶ֔ם וַהֲרֵעֹתֶ֖ם בַּחֲצֹצְר֑וֹת וֲנִזְכַּרְתֶּ֗ם לִפְנֵי֙ ה' אֱלֹֽקֵיכֶ֔ם וְנוֹשַׁעְתֶּ֖ם מֵאֹיְבֵיכֶֽם׃ (י) וּבְי֨וֹם שִׂמְחַתְכֶ֥ם וּֽבְמוֹעֲדֵיכֶם֮ וּבְרָאשֵׁ֣י חָדְשֵׁיכֶם֒ וּתְקַעְתֶּ֣ם בַּחֲצֹֽצְרֹ֗ת עַ֚ל עֹלֹ֣תֵיכֶ֔ם וְעַ֖ל זִבְחֵ֣י שַׁלְמֵיכֶ֑ם וְהָי֨וּ לָכֶ֤ם לְזִכָּרוֹן֙ לִפְנֵ֣י אֱלֹֽקֵיכֶ֔ם אֲנִ֖י ה' אֱלֹקֵיכֶֽם׃ (פ)

(1) The LORD spoke to Moses, saying: (2) Have two silver trumpets made; make them of hammered work. They shall serve you to summon the community and to set the divisions in motion. (3) When both are blown in long blasts (tekiah), the whole community shall assemble before you at the entrance of the Tent of Meeting; (4) and if only one (tekiah) is blown, the chieftains, heads of Israel’s contingents, shall assemble before you. (5) But when you sound short blasts (teruah), the divisions encamped on the east shall move forward; (6) and when you sound short blasts (teruah) a second time, those encamped on the south shall move forward. Thus short blasts shall be blown for setting them in motion, (7) while to convoke the congregation you shall blow long blasts (tekiah), not short ones. (8) The trumpets shall be blown by Aaron’s sons, the priests; they shall be for you an institution for all time throughout the ages. (9) When you are at war in your land against an aggressor who attacks you, you shall sound short blasts (teruah) on the trumpets, that you may be remembered before the LORD your God and be delivered from your enemies. (10) And on your joyous occasions—your fixed festivals and new moon days—you shall sound (tekiah) the trumpets over your burnt offerings and your sacrifices of well-being. They shall be a reminder of you before your God: I, the LORD, am your God.

(א) (א) ביובל ובראש השנה - דכתיב ביוה"כ של יובל והעברת שופר תרועה ובר"ה כתיב שבתון זכרון תרועה יום תרועה יהיה לכם:

(ב) (ב) פשוטה לפניה - דכתיב והעברת שופר תרועה משמע העברת קול אחד לפני התרועה [גמרא]:

1) On Yovel and Rosh Hashana - as it is written on Y"K of Yovel, "you shall pass a shofar balst". And on R'H it's written, a S

(ב) כָּֽל־הָ֭עַמִּים תִּקְעוּ־כָ֑ף הָרִ֥יעוּ לֵ֝אלֹקִ֗ים בְּק֣וֹל רִנָּֽה׃

(2) All you peoples, clap your hands, raise a joyous shout for God. (In Hebrew, clap has numerical value of 100 and follows word tekiah).

(כח) בְּעַד֩ הַחַלּ֨וֹן נִשְׁקְפָ֧ה וַתְּיַבֵּ֛ב אֵ֥ם סִֽיסְרָ֖א בְּעַ֣ד הָֽאֶשְׁנָ֑ב מַדּ֗וּעַ בֹּשֵׁ֤שׁ רִכְבּוֹ֙ לָב֔וֹא מַדּ֣וּעַ אֶֽחֱר֔וּ פַּעֲמֵ֖י מַרְכְּבוֹתָֽיו׃

(28) Through the window peered Sisera’s mother, Behind the lattice she whined: “Why is his chariot so long in coming? Why so late the clatter of his wheels?”

אף על פי שנשים פטורות יכולות לתקוע, וכן אחר שיצא כבר יכול לתקוע להוציאן. אבל אין מברכות ולא יברכו להן: הגה והמנהג שהנשים מברכות על מצות עשה שהזמן גרמא. ע"כ גם כאן מברכות לעצמן אבל אחרים לא יברכו להן אם כבר יצאו ואינן תוקעין רק לנשים אבל אם תוקעין לאיש המחוייב מברכין לו אע"פ שכבר יצאו כמו שנתבאר סי' תקפ"ה סעיף ב' בהג"ה א' (ד"ע):

Shulcahn Arukh Orech Chaim 589:6

Even though women and exempt, they are permitted to blow. So too, even after a man fulfilled his obligation, he is still able to blow for their obligation. However, [in these cases,] they should not make the blessing, nor should he make the blessing for them. Rem"a: And [our] custom is that women make blessings on positive time-bound mitzvot, so here too they would make the blessing for themselves. However, others should not bless for them if they have already fulfilled their obligation and are only blowing for the sake of women. However, if they are blowing [for women and] a man who is obligated, they make a blessing on his account, even though they already fulfilled their obligation, as has been explained in Chapter 585, subsection 2 in the first gloss

From Halachipedia.com, Rosh Hashana, footnote 50: Shulchan Aruch and Rama Orach Chaim 589:6. In general for the mitzvot that are time bound and so women are exempt but they may volunteer, there is a major dispute as to whether they can recite the brachot. Rambam (Hichot Tzitzit 3:9 and Hilchot Sukkah 6:13) holds that since women are exempt from the Mitzvah of Tzitzit they can't make a Bracha on it. However, the Raavad (Hilchot Tzitzit 3:9) and Tosfot (Eruvin 96a, Rosh Hashanah 33a, Kiddshin 31a s.v. lo mifkadana) quoting Rabbenu Tam argue that even if women are exempt from a mitzvah they may opt to recite the bracha if they want to do the mitzvah. The Maggid Mishna Hilhot Sukkah 6:13 explains the Rambam as saying that it is impossible to say VeTzivanu if a person is exempt from the mitzvah. Shulchan Aruch Orach Chaim 589:6 follows the Rambam, while the Rama Orach Chaim 17:2 accepts the Rabbenu Tam. What emerges from the halacha is that Ashkenazim hold that women may recite the bracha upon a mitzvah that they are volunteering to do, while according to Sephardim they may not. See Chida (Birkei Yosef 654:2) who opines that even Sephardim have what to rely upon to follow Rabbenu Tam and Kaf Hachaim Orach Chaim 17:4 who quotes this. Given the dozens of Poskim who rule that a Sephardic woman may recite the beracha and that that was the custom in their communities, Rav Mordechai Lebhar (Magen Avot, Orach Chaim 589:6) writes that women from those communities may continue with their traditions, but others may not, as the Shulchan Aruch rules stringently and we would say Safek Berachot Lehakel.

(א) סדר הראוי לתקיעת שופר ובו ט' סעיפים:
כמה תקיעות חייב אדם לשמוע בראש השנה תשע לפי שנאמר תרועה ביובל ובר"ה ג"פ וכל תרועה פשוטה לפניה ופשוטה לאחריה ומפי השמועה למדו שכל תרועות של חודש השביעי אחד הן בין בר"ה בין ביוה"כ של יובל תשע תקיעות תוקעין בכל אחד משניהם תר"ת תר"ת תר"ת:

(1) The Proper Order of the Shofar Blasts and in it 9 paragraphs<\b>How many Tekiahs is one obligated to hear on Rosh HaShana? Nine. (This is) because it says "Teruah" by Yovel and Rosh HaShana three times. And every Teruah has a straight (blast) before it and a straight (blast) after it. And from tradition (the Sages) learned that all Teruahs of the seventh month, whether on Rosh HaShana or Yom HaKippurim of Yovel, (are) nine Tekiahs. We blow on each of them TR"T, TR"T TR"T.