The Mishnah (Ta'anit 4:8) records that Raban Shimon ben Gamaliel stated that "There were no happier days for the people of Israel than the fifteenth of Av (Tu B'av) and Yom Kippur (after the cohen gadol finished the avodah?) for on those days the daughters of Jerusalem used to go out….to dance in the vineyards. And what did they (the women) say, 'Young man lift up your eyes, and see what you will choose." This going out and dancing was that the men and women should meet which would explain why the days were so happy, but why did this dancing happen on Tu B'av and Yom Kippur?
The Talmud (Ta'anit 30a, 31b, and Baba Bathra 121a) suggests six reasons why Tu B'av was a special day. One, this was the day when people from different tribes were allowed to marry with each other. Two, on this day, the tribe of Binyamin was allowed to marry with the other tribes after the civil war between the all the tribes and Binyamin, Judges 20,21. Three, this was the day when the people stopped dying in the desert. (This reason is difficult since Bemidbar 33:38 records that Aharon died on the 1st of Av, and Bemidbar 20:29 records that the people cried for Aharon for a month, which implies that they did not travel for the entire month of Av. However, the people did not stop dying until they reached Nachal Zered, Devarim 2:13-16 and Bemidbar 21:12, which was after Aharon had died. Even if the people traveled when they were in mourning for Aharon it is doubtful they could have reached Nachal Zered by the 15th of Av since they had a war with Arad during this period. This chronology also calls into question the first explanation for Tu B'av since that explanation is based on the incident with the daughters of Tzelofhad, which appears to have occurred after the people had reached Nachal Zered.) Four, it was the day when Hoshea ben Ela removed the roadblocks that Yerovam ben Nabat had put up to stop the people from going to Jerusalem. Five, this was the day when the Romans allowed the Jews to bury their dead from the battle of Betar. (This reason cannot be the source for the uniqueness of the day since the women seemed to have danced when the Bet ha-Mikdash was standing and Betar fell 60 year after the Bet ha-Mikdash was destroyed.) Six, this was the last day that trees could be cut for the altar since after this period the wood was not dry because the sun was starting to get weaker and there could be worms in the wood. Megilat Ta'anit (chapter five) also relates Tu B'av to the wood donations to the Bet ha-Mikdash, but for almost exactly the opposite reason. Megilat Ta'anit states that this was the day when wood was donated to the Bet ha-Mikdash. (Mishnah Ta'anit 4:5 records that Tu B'av was one of nine days when wood was donated, though from the Mishnah it seems to be the most important of the nine.)
Maybe there is another reason related to the climate in Israel. The fifteenth of Av is a full moon, and also the midpoint in summer season, the months of Tammuz, Av and Elul. Strikingly, Tu B'av is six months apart from Tu be-Shevat, is also a full moon and is on the mid-point of the winter season, the months of Tevet, Shevat and Adar. Our designation of the seasons is based on the number of sunlight hours there are in the day. The summer begins when the day is the longest, and the winter begins when the day is the shortest. Yet, there exists a seasonal lag, that the climate does not exactly match the number of sunlight hours. For example, in Israel, on average, the hottest months are July and August and not June, while the coldest months are January and February. (For example, according to Table 1.3 in the 2010 Statistical Abstracts of Israel by Israel's Central Bureau of Statistics for the years 1981-2000, the average highest monthly temperatures at the Beit Dagan Meteorological Station was 29.2 in June, 30.8 in July, 31.2 in August and 30.4 in September, and the average lowest monthly temperatures were 8.6 in December, 7.2 in January, 7.1 in February and 8.8 in March. The corresponding readings for Jerusalem for the same period were 27.8 in June, 29 in July, 29.4 in August, 28.2 in September, 8.4 in December, 6.4 in January and February and 8.4 in March.)
Tu B'av occurs from the end of July to the middle of August, while Tu be-Shevat falls in the middle of January to the middle of February, which means that these dates roughly correspond to the peak changes in the temperatures in Israel. Furthermore, in Israel there are really only two seasons, the cold and rainy season and the dry and hot season. From the peak of the hottest temperature in Israel around Tu B'av, the temperature begins to fall, reaching its nadir around Tu be-Shevat, from which point it begins to rise reaching its peak around Tu B'av. Thus, Tu B'av and Tu- be-Shevat were a set of new years based on the climate. They were the full moons that were closest to the peaks in temperatures, and hence they signaled a change in the climate. Tu B'av would mark the end of the hot, dry season and the beginning of the cold and wet season, while Tu be-Shevat would mark the end of the cold and wet season, and the beginning of the hot and dry season. (The Meiri in his commentary on the first Mishnah of Rosh Hashanah notes that after Tu be-Shevat, it becomes less cold.)
If this idea is correct, there would be two sets of "new years," one based on climate, Tu B'av and Tu be-Shevat and one based on hours of sunlight, Rosh Hashanah (Rosh Chodesh Tishrei) and Rosh Chodesh Nisan, which roughly correspond to the two equinoxes. In the Torah, the beginning of the year is from Rosh Chodesh Nisan (Shemot 12:1), the system based on sunlight, but it very possible that the people also used the climate system to mark the beginning or changes in the year.
If this idea is true, then we can understand why the women went out to dance on Tu B'av, since it is normal for people to celebrate on a new year. Furthermore, this explanation could accord with the second explanation in the Talmud for Tu B'av, that it was the day that the tribe of Binyamin could rejoin the people. This explanation is based on the verse in Shoftim 21:16 that this event happened on some unidentified holiday, which is suggested to be Tu B'av. Maybe the holiday was Tu B'av that the people were celebrating the new year. In addition, this idea that the dancing on Tu B'av was derived from the celebrations of the new year, might also explain why the dancing was also on Yom Kippur. Rosh Hashanah was the start of the year, and hence we would expect that there would also be celebrations by Rosh Hashanah. However, Rosh Hashanah is a very somber time leading up to Yom Kippur, and hence it is possible that the celebrations were pushed off until the somber period ended either in the afternoon of Yom Kippur or the night after Yom Kippur.
Finally, these two systems of marking the year recall the first Mishnah in Rosh Hashanah, that there are four new years. The Mishnah lists, Rosh Hashanah, and Rosh Chodesh Nisan, and a disagreement whether the new year for the trees should have be Tu be-Shevat as maintained by Hillel or should be on Rosh Chodesh Shevat as maintained by Bet Shammai. The Mishnah also lists that Rosh Chodesh Elul is a new year, while R. Eliezer and R. Shimon argue that it is not a new year at all. It is not clear why Rosh Chodesh Elul should be a new year, as what does it mark? My guess is that the source for Rosh Chodesh Elul is Tu B'av. Chazal did not like the celebration of Tu B'av as the new year possibly because it was "competition" with Rosh Hashanah or because they wanted to emphasize that the dating of the year was to be based on the new moon and not the full moon. Thus, one idea was to minimize the day by moving "the new year" fifteen days ahead to the next new moon, Rosh Chodesh Elul, while R. Eliezer and R. Shimon wanted to abolish it altogether. It could be that this same idea is the basis for the argument between Bet Hillel and Bet Shammai with regard to Tu be-Shevat. Bet Hillel was willing to keep the people's celebration of the 15th of Shevat as a new year, while Bet Shammai wanted to move the marking of the "new year" to the new moon instead of the full moon. (I think I heard this idea or a variation of it concerning Tu be-Shevat from my brother-in-law Michael Segal.)
To summarize, I believe that there were two systems to mark the changes in the year, and each system had two festivals at the half-points of the year. One system was/ is based on the hours of sunlight, and the festivals are Rosh Hashanah and Rosh Chodesh Nisan, which roughly mark the equinoxes. The other system was based on change in climate and the two festivals were Tu B'av, which marked the peak of the dry and hot season, and Tu be-Shevat, which marked the peak of the wet and cold season.
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