According to tradition, we commemorate receiving the Torah on the holiday of Shavu’ot, which falls on the sixth day of the month of Sivan. In fact, there is a dispute in the Gemara regarding the day that the Torah was given.
On the sixth day of the month of Sivan, the Ten Commandments were given to the Jewish people. Rabbi Yosei says: On the seventh day of the month.
Rava said: Everyone agrees that the Jews came to the Sinai desert on the New Moon, as it is written here: “In the third month after the children of Israel were gone forth out of the land of Egypt, the same day came they into the wilderness of Sinai” (Shemot 19:1), without elaborating what day it was. And it is written there: “This month shall be to you the beginning of months; it shall be the first month of the year to you” (Shemot 12:2). Just as there, the term “this” is referring to the New Moon, so too, here the term is referring to the New Moon.
The word ḥodesh is understood throughout the Bible to mean month. Occasionally, it is a reference to the New Moon. Examples include: “Tomorrow is the ḥodesh” (I Samuel 20:18); “Its holiday, its ḥodesh, its” (Hoshea 2:13); and “The burnt-offering of the ḥodesh and its meal-offering” (Bamidbar 29:6). Therefore, the verse: “This ḥodesh shall be unto you the beginning of months” is understood as indicating that this New Moon is the first New Moon that the Jewish people are celebrating. In the verse describing their arrival in the desert, it is unclear whether ḥodesh refers to the month or the New Moon. It states: “In the third ḥodesh after the children of Israel went forth out of the land of Egypt” (Shemot 19:1). The emphasis on the words at the end of the verse, “on this day,” proves that it is referring to the day of the New Moon.
The Gemara continues:
And similarly, everyone agrees that the Torah was given to the Jewish people on, as it is written here in the Ten Commandments: “Remember the day to keep it holy” (Shemot 20:7), and it is written there: “And Moses said to the people: Remember this day, in which you came out from Egypt, out of the house of bondage, for by strength of hand the Lord brought you out from this place; there shall be no leaven eaten” (Shemot 13:3). Just as there, the mitzva of remembrance was commanded on the very day of the Exodus, so too, here the mitzva of remembrance was commanded on the very day of.
Where Rabbi Yosei and the Sages disagree is with regard to the determination of the month, meaning which day of the week was established as the New Moon. Rabbi Yosei held: The New Moon was established on the first day of the week, and on the first day of the week He did not say anything to them due to the weariness caused by the journey. On the second day of the week, He said to them: “And you shall be to Me a kingdom of priests and a holy nation; these are the words that you shall speak to the children of Israel” (Shemot 19:6).
The Gemara concludes that according to Rabbi Yosei, Moshe, on his own accord, added an extra day of separation prior to the giving of the Torah. Thus, according to Rabbi Yosei, the Torah was given only on the seventh day of Sivan.