Midrash for Shekalim 3:1
בִּשְׁלשָׁה פְּרָקִים בַּשָּׁנָה, תּוֹרְמִין אֶת הַלִּשְׁכָּה, בִּפְרוֹס הַפֶּסַח, בִּפְרוֹס עֲצֶרֶת, בִּפְרוֹס הֶחָג, וְהֵן גְּרָנוֹת לְמַעְשַׂר בְּהֵמָה, דִּבְרֵי רַבִּי עֲקִיבָא. בֶּן עַזַּאי אוֹמֵר, בְּעֶשְׂרִים וְתִשְׁעָה בַּאֲדָר, וּבְאֶחָד בְּסִיוָן, וּבְעֶשְׂרִים וְתִשְׁעָה בְּאָב. רַבִּי אֶלְעָזָר וְרַבִּי שִׁמְעוֹן אוֹמְרִים, בְּאֶחָד בְּנִיסָן, בְּאֶחָד בְּסִיוָן, בְּעֶשְׂרִים וְתִשְׁעָה בֶּאֱלוּל. מִפְּנֵי מָה אָמְרוּ בְּעֶשְׂרִים וְתִשְׁעָה בֶּאֱלוּל, וְלֹא אָמְרוּ בְּאֶחָד בְּתִשְׁרֵי, מִפְּנֵי שֶׁהוּא יוֹם טוֹב, וְאִי אֶפְשָׁר לְעַשֵּׂר בְּיוֹם טוֹב, לְפִיכָךְ הִקְדִּימוּהוּ לְעֶשְׂרִים וְתִשְׁעָה בֶּאֱלוּל:
Three times a year they would remove (shekalim from) the lishkah. [They would deposit all the shekalim in one lishkah (compartment) in the Temple. And three times a year they would take from it and place (the shekalim) into three large baskets of three sa'ah for the purchase of communal offerings. The whole was not taken at one time for the needs of the entire year, for those in distant places had not yet brought all of their shekalim]: at the pross of Pesach [Fifteen days before the festival is referred to as the "pross" of the festival. For thirty days before the festival the halachoth of the festival are reviewed. ("pross," as in "prussah," i.e., half)], at the pross of Shavuoth, and at the pross of Succoth. And they (these time periods) are granoth ("threshing floors") for the beast-tithe. [These three times are three granoth for the beast-tithe, the sages having designated these times for the tithing of beasts that had been born. And just as (bringing the grain to) the threshing floor makes the grain subject to the tithe, so beasts that have been born may not be eaten after the arrival of these times until they are tithed. But before these times they may be eaten even if they have not been tithed. The sages designated these three times for the beast-tithe so that beasts be available for the festival pilgrims. For even though it is permitted to sell and slaughter and eat as long as the time of the "goren" has not yet arrived, still, people would not slaughter their beasts until they had tithed them. For one prefers doing a mitzvah with his property if he loses nothing thereby, as with the beast-tithe, the owner eating the tithed animal itself as a peace-offering. And if they did not tithe in these three periods, many would refrain from selling their beasts, not having tithed them, and beasts would not be available for the festival pilgrims.] These are the words of R. Akiva. Ben Azzai says: On the twenty-ninth of Adar, the first of Sivan, and the twenty-ninth of Av. R. Elazar and R. Shimon say: On the first of Nissan, the first of Sivan, and the twenty-ninth of Elul. [The reasons of all these tannaim, and (the bases of) their differences are explicated in the last chapter of Bechoroth.] Why did they say the twenty-ninth (of Elul) instead of the first of Tishrei? For it is a festival (Rosh Hashanah), and one cannot tithe on a festival, for which reason they moved it up to the twenty-ninth of Elul.