Shekalim, Chapter 3, Mishnah 1. This mishnah opens with a discussion of the terumas halishkah: three times a year they would enter the chamber where all the half shekels had been deposited and take from it a sum of money designated for purchasing the public offerings. The mishnah lists three times in the year when the terumas halishkah, the withdrawal of the funds, was performed.
The three times of the terumas halishkah:
Half of the period before Pesach - in preparation for Pesach there is a thirty day period in which one begins to discuss and study the laws of the festival. Half of that period, that is, fifteen days before Pesach, is the first time of the terumas halishkah. This time falls exactly at the beginning of Nisan, when they would begin using the new half shekels that had been brought and bringing the offerings from these new funds.
Fifteen days before Shavuos.
Fifteen days before Sukkos.
The goren for maaser behemah:
From here the mishnah moves on to discuss maaser behemah, the animal tithe: one of every ten animals born in a given year, which must be brought as an offering. The transition to this topic is made because those very same times, the fifteen days before each of the three festivals, also serve as the goren for maaser behemah.
And what is a goren? With grain, the goren is the pile that is made from the produce. Once the pile has been completed, the produce becomes obligated in the separation of Terumah and tithes from that moment, and one may not eat from it even in a casual manner after this point. This term was borrowed for maaser behemah as well: there is a certain point on the timeline of the animals from which one is not permitted to slaughter any of the animals of the flock or the herd until their owner separates the animal tithe and sets it aside. In this sense it resembles the goren of grain.
The three times mentioned - the fifteen days before Pesach, Shavuos and Sukkos - are also the times at which, once they arrive, a person must separate maaser behemah before he can make use of the rest of his flock. This is the opinion of Rabbi Akiva, that the times for maaser behemah are identical to the times of the terumah.
The opinion of Ben Azzai:
The mishnah continues and brings various opinions regarding maaser behemah. "Ben Azzai omer" - Ben Azzai says that the three times are:
"B'esrim v'tishah b'Adar" - on the twenty ninth of Adar. This time is very similar to fifteen days before Pesach, and in truth Ben Azzai holds the same way. It is only that Rabbi Akiva did not state a specific date, because Adar is sometimes a month of twenty nine days and sometimes a month of thirty days, and therefore the fifteen days before Pesach sometimes fall on the twenty ninth of Adar and sometimes on the thirtieth. Ben Azzai holds that Adar is always a month of twenty nine days, and therefore fifteen days before Pesach is the twenty ninth of Adar.
"U'b'echad b'Sivan" - and on the first of Sivan: not fifteen days before Shavuos, but closer to the festival. The Gemara explains that in the stretch of time between fifteen days before Pesach and Shavuos not many animals were born, and therefore if one were to separate the animal tithe fifteen days before Shavuos, there is a concern that the owners would begin eating from many animals and there would be a shortage. Therefore only a short period is allowed before the festival of Shavuos.
"B'esrim v'tishah b'Av" - on the twenty ninth of Av. Instead of being closer to Sukkos, as Rabbi Akiva holds, Ben Azzai holds that the animals born during the month of Elul have their tithe separated separately. He therefore said that before Elul begins one separates the tithe from the earlier animals, while from the animals of Elul one separates their tithe at the end of Elul.
The opinion of Rabbi Eliezer and Rabbi Shimon:
Rabbi Eliezer and Rabbi Shimon say that the three times are: Rosh Chodesh Nisan - for they hold like the opinion of Rabban Shimon ben Gamliel, that the process of dealing with Pesach begins only fourteen days beforehand, and fourteen days before Pesach is Rosh Chodesh Nisan; the first of Sivan, like Ben Azzai, truly close to Shavuos; and the twenty ninth of Elul.
And why was the third time set on the twenty ninth of Elul and not on the first of Tishrei, which is fifteen days before Sukkos? The first of Tishrei is a festival, Rosh Hashanah, and one does not separate tithes on a festival. The Gemara explains two reasons why the first of Tishrei cannot be the time for separating maaser behemah:
A distinction must be made between the old and the new: Rosh Hashanah for maaser behemah separates between the animals born before Rosh Hashanah and those born after it. The years are not combined, and one does not separate maaser behemah from one group on behalf of the other, and therefore one must distinguish between them.
The way they marked which animal had been selected as maaser behemah was by smearing paint on it, and one may not do this sort of painting on a festival.
For these reasons they said that it must be done on the day before Rosh Hashanah, on the twenty ninth of Elul.
In summary: in this mishnah we learned the three times of the terumas halishkah - fifteen days before Pesach, before Shavuos and before Sukkos - and that they also serve as the goren for maaser behemah, from which point on one may not slaughter from the flock until he separates the tithe. We examined the opinion of Rabbi Akiva, the opinion of Ben Azzai (the twenty ninth of Adar, the first of Sivan and the twenty ninth of Av) and the opinion of Rabbi Eliezer and Rabbi Shimon, as well as the reasons why the time is not set on the first of Tishrei: the distinction between the old year and the new, and the prohibition against painting the animal on a festival.