Shekalim, Chapter Three, Mishnah One
Our mishnah teaches that three times a year they would take the gathered shekels out of the chamber in the Temple and appropriate them to buy public sacrifices. This time is connected to the period in which animals are tithed. In the remainder of the mishnah there are two other opinions as to when animals are tithed, and therefore as to when the shekels are appropriated.
1) At three periods of the year the appropriation is made [from the shekels] in the chamber: Half a month before Pesah, half a month before Shavuot, and half a month before Sukkot, and these are also the threshing floors [the seasons] for the tithe of beasts, the words of Rabbi Akiva.
2) Ben Azzai says: on the twenty-ninth of Adar, and on the first of Sivan, and on the twenty-ninth of Av.
3) Rabbi Elazar and Rabbi Shimon say: on the first of Nisan, on the first of Sivan, and on the twenty-ninth of Elul.
a) Why did they say, On the twenty-ninth of Elul and not on the first of Tishre?
b) Because the first of Tishre is a holy day, and it is not permitted to tithe on a festival, therefore they moved it up to the twenty-ninth of Elul.
Section one: According to Rabbi Akiva the shekels are appropriated at the same time of year that animals are tithed. I have translated A half-month before based on the Talmud. The mishnah itself says only Before, but uses a word which implies that this must occur at least a half-month before. The word threshing floors is borrowed from tithes for grain. The meaning is that in these three periods it becomes forbidden to eat any animal until every tenth animal has been tithed. The tithed animal still belongs to its owners, but its blood must be sprinkled on the altar and its meat eaten in Jerusalem. It seems likely to me that Rabbi Akiva said that animals were tithed before the holidays so that people would bring them to Jerusalem when they made their festival pilgrimages.
Section two: The remaining sages disagree as to when animals are tithed. Ben Azzai gives dates that are more precise and slightly differ from those given by Rabbi Akiva. The twenty-ninth of Adar is basically the same, as it is half a month before Pesah. The first of Sivan is only 7 days before Shavuot. He seems to give more time because there is not a lot of time between Pesah and Shavuot in which animals might be born. The third tithe is at the end of Av, because Ben Azai holds that animals born in Elul are tithed separately.
Section three: Rabbi Elazar and Rabbi Shimon that the animals are tithed on the first of Nissan, which is similar to both Rabbi Akiva and one day after Ben Azais time. The second tithing period is on the first of Sivan, the same as Ben Azai. The third tithing period is on the twenty-ninth of Elul. The mishnah explains that they would have said the first of Tishre but one cant tithe on the first of Tishre because it is a holiday, namely Rosh Hashanah. It seems that in this case he basically agrees with Rabbi Akiva who said half a month before Sukkot.