A short analysis of the reasons underlying the dates of Sukkot.
מָשְׁכֵ֖נִי אַחֲרֶ֣יךָ נָּר֑וּצָה הֱבִיאַ֨נִי הַמֶּ֜לֶךְ חֲדָרָ֗יו נָגִ֤ילָה וְנִשְׂמְחָה֙ בָּ֔ךְ נַזְכִּ֤ירָה דֹדֶ֙יךָ֙ מִיַּ֔יִן מֵישָׁרִ֖ים אֲהֵבֽוּךָ׃ (ס)
Draw me after you, let us run! The king has brought me to his chambers. Let us delight and rejoice in your love, Savoring it more than wine— Like new wine they love you!
According to the Vilna Gaon, the first three verses of Shir HaShirim correspond to the three festivals. The fourth verse corresponds to Shemini Atzeret and indicates particular closeness to Hashem. The Sages focus on the word בך, which has the gematria of 22, corresponding to the date of Shemini Atzeret itself, and also containing within it the idea of intimacy and closeness.
The Vilna Goan goes further, and asks: why do we celebrate Sukkot on 15 Tishri? Surely the clouds of glory were with the Jewish people throughout their sojourn in the desert, so Sukkot should be celebrated earlier on, closer to Pesach.
To answer this question, the Vilna Gaon explains that the reason the Jewish people lost the protection of the Clouds of Glory was due to the sin of the golden calf. Only after the building of the Mishkan, following the receiving of the second tablets, did the cloud return.
In order to bring these concepts together, the Vilna Goan makes reference to the following chronology of events:
All happened in the first year after coming out of Egypt, from Pesach through to Yom Kippur.
When did Moshe come down from Mount Sinai the second time, this time with the second luchot? By calculation of the days we can work out that this is Yom Kippur. In Parashat VaYakhel, Rashi says that this was the day after Yom Kippur. Moshe only tells the people on the 11 of Tishri that they now have to start building the Mishkan. For two days they collected donations (כל נדיב לב). On the 14th all of the instructions are given to Bezalel, and on the 16th of Tishri the work commenced.
Hence the 15th of Tishri was a key date for the Jewish people as it indicates that their teshuvah had been accepted. The work of the building of the Mishkan commenced and the Clouds of Glory returned. This is the origin of the festival of Sukkot.
Vilna Gaon writes this idea in his commentary to Shir haShirim, a relatively obscure place in Torah literature. Study of the ideas of the Vilna Gaon requires a broad knowledge of Torah sources.