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Counting the Omer
(טו) וּסְפַרְתֶּ֤ם לָכֶם֙ מִמָּחֳרַ֣ת הַשַּׁבָּ֔ת מִיּוֹם֙ הֲבִ֣יאֲכֶ֔ם אֶת־עֹ֖מֶר הַתְּנוּפָ֑ה שֶׁ֥בַע שַׁבָּת֖וֹת תְּמִימֹ֥ת תִּהְיֶֽינָה׃ (טז) עַ֣ד מִֽמָּחֳרַ֤ת הַשַּׁבָּת֙ הַשְּׁבִיעִ֔ת תִּסְפְּר֖וּ חֲמִשִּׁ֣ים י֑וֹם וְהִקְרַבְתֶּ֛ם מִנְחָ֥ה חֲדָשָׁ֖ה לַה'׃
(15) And from the day on which you bring the sheaf of elevation offering—the day after the sabbath—you shall count off seven weeks. They must be complete: (16) you must count until the day after the seventh week—fifty days; then you shall bring an offering of new grain to the LORD.
(א) ממחרת השבת. מִמָּחֳרַת יוֹם טוֹב: (ב) תמימת תהיינה. מְלַמֵּד שֶׁמַּתְחִיל וּמוֹנֶה מִבָּעֶרֶב, שֶׁאִם לֹא כֵן אֵינָן תְּמִימוֹת (ספרא; מנחות ס"ו):

(1) ממחרת השבת FROM THE MORROW AFTER THE DAY OF REST — i. e. from the morrow after the first day of the Passover festival (Menachot 65b). (2) תמימת תהיינה [SEVEN WEEKS] SHALL BE COMPLETE — This teaches that one has to begin counting in the evening, for otherwise they would not be complete (Menachot 66a).

The Boethusians had a different idea about what "tomorrow after the day of rest" means.

The Boethusians were regarded by the Talmud as cynical and materialistic priests. They hired false witnesses to delude the Pharisees about the new moon (RH 22b; TJ, RH 57d; Tosef., RH 1:15). They maintained that the Omer (Men. 10:3) was to be offered on the first Sunday after Passover, and not on the morrow of the first day and, asPage 34 | Top of Articlea result, differed as to the date of Shavuot which according to them must always fall on a Sunday (Ḥog. 24). (EJ)

(ג) כֵּיצַד הָיוּ עוֹשִׂים. שְׁלוּחֵי בֵית דִּין יוֹצְאִים מֵעֶרֶב יוֹם טוֹב, וְעוֹשִׂים אוֹתוֹ כְרִיכוֹת בִּמְחֻבָּר לַקַּרְקַע, כְּדֵי שֶׁיְּהֵא נוֹחַ לִקְצֹר. וְכָל הָעֲיָרוֹת הַסְּמוּכוֹת לְשָׁם, מִתְכַּנְּסוֹת לְשָׁם, כְּדֵי שֶׁיְּהֵא נִקְצָר בְּעֵסֶק גָּדוֹל. כֵּיוָן שֶׁחֲשֵׁכָה, אוֹמֵר לָהֶם, בָּא הַשָּׁמֶשׁ, אוֹמְרִים, הֵן. בָּא הַשָּׁמֶשׁ, אוֹמְרִים הֵן. מַגָּל זוֹ, אוֹמְרִים הֵן. מַגָּל זוֹ, אוֹמְרִים הֵן. קֻפָּה זוֹ, אוֹמְרִים הֵן. קֻפָּה זוֹ, אוֹמְרִים הֵן. בְּשַׁבָּת אוֹמֵר לָהֶם, שַׁבָּת זוֹ, אוֹמְרִים הֵן. שַׁבָּת זוֹ, אוֹמְרִים הֵן. אֶקְצֹר, וְהֵם אוֹמְרִים לוֹ קְצֹר. אֶקְצֹר, וְהֵם אוֹמְרִים לוֹ קְצֹר. שָׁלשׁ פְּעָמִים עַל כָּל דָּבָר וְדָבָר, וְהֵם אוֹמְרִים לוֹ הֵן, הֵן, הֵן. וְכָל כָּךְ לָמָּה. מִפְּנֵי הַבַּיְתוֹסִים, שֶׁהָיוּ אוֹמְרִים, אֵין קְצִירַת הָעֹמֶר בְּמוֹצָאֵי יוֹם טוֹב:

(3) How would they perform the rite of the harvest of the omer? Emissaries of the court would emerge on the eve of the festival of Passover and fashion the stalks of barley into sheaves while the stalks were still attached to the ground, so that it would be convenient to reap them. The residents of all the towns adjacent to the site of the harvest would assemble there, so that it would be harvested with great fanfare. Once it grew dark, the court emissary says to those assembled: Did the sun set? The assembly says in response: Yes. The emissary repeats: Did the sun set? They again say: Yes. The court emissary next says to those assembled: Shall I reap the sheaves with this sickle? The assembly says in response: Yes. The emissary repeats: With this sickle? The assembly says: Yes. The court emissary then says to those assembled: Shall I place the gathered sheaves in this basket? The assembly says in response: Yes. The emissary repeats: In this basket? The assembly says: Yes. If the sixteenth of Nisan occurs on Shabbat, the court emissary says to the assembled: Shall I cut the sheaves on this Shabbat? The assembly says in response: Yes. The emissary repeats: On this Shabbat? The assembly says: Yes. The court emissary says to those assembled: Shall I cut the sheaves? And they say to him in response: Cut. The emissary repeats: Shall I cut the sheaves? And they say to him: Cut. The emissary asks three times with regard to each and every matter, and the assembly says to him: Yes, yes, yes. The mishna asks: Why do I need those involved to publicize each stage of the rite to that extent? The mishna answers: It is due to the Boethusians, as they deny the validity of the Oral Law and would say: There is no harvest of the omer at the conclusion of the first Festival day of Passover unless it occurs at the conclusion of Shabbat. The publicity was to underscore that the sixteenth of Nisan was the proper time for the omer harvest.

רבי יהודה בן בתירא אומר נאמר שבת למעלה ונאמר שבת למטה מה להלן רגל ותחלת רגל סמוך לה אף כאן רגל ותחלת רגל סמוך לה

Rabbi Yehuda ben Beteira says there is yet another refutation. It is stated “shabbat” above (Leviticus 23:15), with regard to starting the counting of the omer, and it is also stated “shabbat” below (Leviticus 23:16), with regard to the commencement of the festival of Shavuot. Just as there, with regard to the festival of Shavuot, it is stated: “Even until the morrow after the seventh week [hashabbat] you shall number fifty days,” and the word shabbat is referring to the beginning of the Festival and it immediately follows the end of the seventh week; so too here, with regard to the bringing of the omer, the word shabbat means Festival, so that the omer offering immediately follows the beginning of the Festival, on the second day of Passover. According to the Boethusians, the commencement of the counting could start well after the beginning of Passover. For example, if Passover occurs on a Sunday, the counting of the omer would start only the following Sunday.

(א) סדר תפל' ליל שני של פסח וספירת העומר. ובו י"ס:
בליל שני אחר תפל' ערבית מתחילין לספור העומר ואם שכח לספור בתחל' הלילה הולך וסופר כל הלילה ומצוה על כל אחד לספור לעצמו וצריך לספור מעומד ולברך תחל' וסופר הימים והשבועו' כיצד ביום הראשון אומר היום יום אחד (בעומר) עד שמגיע לשבעה ימים ואז יאמר היום שבעה ימים שהם שבוע אחד (בעומר) וביום שמיני אומר היום שמונה ימים שהם שבוע א' ויום א' (בעומר) וכן עד שיגיע לארבע' עשר יאמר היום ארבעה עשר ימים שהם שני שבועו' (בעומר) ועל דרך זה מונה והולך עד מ"ט יום:

(1) On the second night after the evening prayer, we begin to count the Omer. And if someone forgot to count, [that one may count] from the beginning of the evening onwards. He may account all night. It is a mitzvah for each person to count for themselves. He needs to count standing and bless before. He should count the days and weeks. How? On the first day he should say “today is day one of the Omer”, until he arrives to seven days. [At which point he should say], “they are one week of the Omer.” And on the eighth day he should say “today is eight days and they are one week and one day of the Omer.” And also, when he arrives to the 14th day, he should say “today is 14 days, they are two weeks of the Omer.” And in this way he should count and onwards until the 49th day.

ר"ע אומר למד תורה בילדותו ילמוד תורה בזקנותו היו לו תלמידים בילדותו יהיו לו תלמידים בזקנותו שנא' בבקר זרע את זרעך וגו' אמרו שנים עשר אלף זוגים תלמידים היו לו לרבי עקיבא מגבת עד אנטיפרס וכולן מתו בפרק אחד מפני שלא נהגו כבוד זה לזה והיה העולם שמם עד שבא ר"ע אצל רבותינו שבדרום ושנאה להם ר"מ ור' יהודה ור' יוסי ורבי שמעון ורבי אלעזר בן שמוע והם הם העמידו תורה אותה שעה תנא כולם מתו מפסח ועד עצרת אמר רב חמא בר אבא ואיתימא ר' חייא בר אבין כולם מתו מיתה רעה מאי היא א"ר נחמן אסכרה

Rabbi Akiva says that the verse should be understood as follows: If one studied Torah in his youth he should study more Torah in his old age; if he had students in his youth he should have additional students in his old age, as it is stated: “In the morning sow your seed, etc.” They said by way of example that Rabbi Akiva had twelve thousand pairs of students in an area of land that stretched from Gevat to Antipatris in Judea, and they all died in one period of time, because they did not treat each other with respect. And the world was desolate of Torah until Rabbi Akiva came to our Rabbis in the South and taught his Torah to them. This second group of disciples consisted of Rabbi Meir, Rabbi Yehuda, Rabbi Yosei, Rabbi Shimon, and Rabbi Elazar ben Shamua. And these are the very ones who upheld the study of Torah at that time. Although Rabbi Akiva’s earlier students did not survive, his later disciples were able to transmit the Torah to future generations. With regard to the twelve thousand pairs of Rabbi Akiva’s students, the Gemara adds: It is taught that all of them died in the period from Passover until Shavuot. Rav Ḥama bar Abba said, and some say it was Rabbi Ḥiyya bar Avin: They all died a bad death. The Gemara inquires: What is it that is called a bad death? Rav Naḥman said: Diphtheria.

Mourning customs during the Omer: refrain from haircuts, no weddings and other celebrations...

(א) דינים הנוהגים בימי העומר. ובו ד' סעיפים:
נוהגים שלא לישא אשה בין פסח לעצרת עד ל"ג בעומר מפני שבאותו זמן מתו תלמידי ר' עקיבא אבל לארס ולקדש שפיר דמי ונשואין נמי מי שקפץ וכנס אין עונשין אותו: הגה מיהו מל"ג בעומר ואילך הכל שרי (אבודרהם וב"י ומנהגים):

(ב) נוהגים שלא להסתפר עד ל"ג לעומר שאומרים שאז פסקו מלמות ואין להסתפר עד יום ל"ד בבוקר אלא אם כן חל יום ל"ג ערב שבת שאז מסתפרים בו מפני כבוד השבת: הגה ובמדינות אלו אין נוהגין כדבריו אלא מסתפרין ביום ל"ג ומרבים בו קצת שמחה ואין אומרים בו תחנון (מהרי"ל ומנהגים) ואין להסתפר עד ל"ג בעצמו ולא מבערב (מהרי"ל) מיהו אם חל ביום א' נוהגין להסתפר ביום ו' לכבוד שבת (מהרי"ל) מי שהוא בעל) ברית או מל בנו מותר להסתפר בספירה לכבוד המילה (הגהות מנהגים):

(1) It is customary not to get married between Pesach and Shavuot, until Lag BaOmer (the 33rd day), because during that time, the students of Rabbi Akiva died. However, to do "erusin" and "kiddushin" (engagement and betrothal) is OK. And even for "nisuin" (marriage), if someone did so, we do not punish him. Rema: however, from Lag Ba'Omer onwards, all this is permitted (Abudraham, Beit Yosef & Minhagim).

(2) It is customary not to cut one's hair until Lag BaOmer, since it is said that that is when they stopped dying. One should not cut one's hair until the 34th day, in the morning, unless the 33rd day falls on Friday, in which case one may cut one's hair then, in honor of the Sabbath ("kavod Shabbat"). Rema: But in these countries, we do not follow the custom he advocates; rather, we cut our hair on the 33rd day, and we rejoice a bit, and we do not say Tachanun (Maharil and Minhagim). And one should only cut one's hair on the 33rd day itself, but not on its evening. However, if it falls on Sunday, our custom is to cut our hair on Friday in honor of the Sabbath (Maharil). Someone who is performing a brit milah (i.e., the sandak, mohel and the infant's father), or circumcizing his son, is allowed to cut his hair during sefirah in honor of the circumcision (Hagahot Minhagim).

https://www.etzion.org.il/en/sefirat-ha-omer-mourning-practices-during-omer
During Which Days are the Mourning Customs Observed?
Different communities observe the minhagei aveilut during different parts of the Omerperiod. What is the basis for these customs? One can identify three different approaches, with different variations.
1- The entire Omer: The Sha’arei Teshuva (493:8) reports that the Ari z”l would not take a haircut for the entire period of the Omer, until Erev Shavuot. This, of course, is based upon the simple understanding of the Talmud’s description of the death of the students of R. Akiva, “from Pesachuntil Atzeret.” Similarly, the Mishna Berura (15) relates that some observe these minhagei aveilutfor the entire period of the Omer, excluding Rosh Chodesh Iyar, Lag Ba-Omer, and from Rosh Chodesh Sivan until Shavuot.
2- From Pesach until Lag Ba-Omer: Many are accustomed to observe only the first part of this period. Assuming that the minhagei aveilut were instituted in commemoration of the students of R. Akiva who died during this period, there are conflicting traditions regarding the days upon which they died. R. Yehoshua ibn Shu'ib (1280-1340), a student of the Rashba, cites two possibilities in his Derashot (Yom Rishon shel Pesach). According to one approach, based upon a midrash, the students of R. Akiva died until “peros ha-atzeret.” The term “peros ha-atzeret” is understood to refer to “half of a month," or at least fifteen days. If so, then peros ha-atzeret falls out on the 34thday of the Omer, fifteen days before the end of the count. One should therefore observe the customs of mourning for the first 34 days of the Omer. R. Ibn Shuib, however, writes that one may invoke the principle of “miktzat ha-yom ke-kulo” (part of a day counts as the entire day), which is applicable when the halakha demands that one count days, such as the “shiva neki’im” (seven clean days) of a zava gedola and the seven days of mourning (“shiva”). Therefore, just as a mourner finishes the shiva on the mourning of the seventh day, one may finish the mourning of the Omer period on the morning of the 34th day. The Shulchan Arukh (493:2) cites this opinion.
The Tur (495) mentions those who cut their hair on the 33rd day of the Omer, Lag Ba-Omer. Apparently, they understand that R. Akiva’s students died until the 33rd day of the Omer (Sefer Ha-Manhig, Hilkhot Eirusin Ve-Nisu’in; Meiri, Yevamot 62b; Mishna Berura 493:8). Therefore, using the same rationale, they invoke the principle of “miktzat ha-yom ke-kulo” and cease to observe the mourning practices on the morning of Lag Ba-Omer. The Gra explains that this is the basis for the opinion of the Rama, who writes that Ashkenazim do not observe mourning customs on Lag Ba-Omer, but rather take haircuts and “rejoice a bit.”
3- Thirty Three Days of Mourning: The Derashot Ibn Shu’ib cites the opinion of the Tosafot, which appears in the Maharil (Minhagim, Dinei Ha-Yamim She-Bein Pesach Le-Shavu’ot 7) as well, which claims that the students of R. Akiva did not die on days in which “techina” (tachanun) is not recited, i.e. on festive days. Therefore, if one subtracts the 16 days (seven days of Pesach, the three days of Rosh Chodesh (Iyar [2] and Sivan [1]), seven Shabbatot) from the 49 days of the Omer, we are left with 33 days upon which the students of R. Akiva died. Therefore, the custom fundamentally is to observe mourning practices for 33 days. According to this custom, when are these 33 days observed?
Some observe these 33 days from the beginning of the Omer, i.e. the second day of Pesach until Lag Ba-Omer. The Bach (493) explains that this is the reason behind the opinion of the Rama, cited above. Many communities, especially German communities, observed these 33 days during the “second half” of the Omer, as the Crusades occurred during the months of Iyar and Sivan. Some observe them from the second day of Rosh Chodesh Iyar until Erev Shavuot. Others begin from the first day of Rosh Chodesh (the 30th of Nisan) and observe until the 3rd of Sivan, leaving out the three days before Shavuot, known as the “shaloshet yemei hagbala.