The 3rd of Sivan begins the twelfth set of 22 days of the Jewish calendar, which parallels the letter Kaf Sofit, as well as the Trees of the Field and the Vine in Perek Shirah. This 22-day period runs from the Shlosha Yemei Hagbalah (three days of separation) and Shavuoth to close to the end of the month of Sivan.
The previous set of 22 days marked the last two letters of the Jewish alphabet, Shin and Tav. However, it is also common to include separately in the Aleph Bet the final letters: Kaf Sofit, Mem Sofit, Nun Sofit, Peh Sofit and Tzadik Sofit. These letters appear stand for the five parts of the mouth related to speech, which are connected to the "Five Gevurot" and the five primary vowels. (See Rabbi Aryeh Kaplan's translation and commentary to Sefer Yetzirah)
The twelfth cycle of 22 days appears related to the letter Kaf Sofit. As mentioned previously, the Kaf stands for Keter, which is the part of the soul that is associated with that which is above intellect. Kaf literally means the palm of the hand and/or a spoon, which is bent like a receptacle. Unlike the regular Kaf, the Kaf Sofit is not bent, but instead goes straight down.
The giving of the Torah came straight down in a way that penetrated the world, to the extent that on Mount Sinai the words of the Torah did not have an echo (ie. they did not bounce back from what they touched - instead they were absorbed).
The following is from Rabbi Michael Munk's, "The Wisdom of the Hebrew Alphabet:"
The Perek Shirah elements of this week are also related to Shavuot. Shavuot is known also as Chag HaBikurim, the Festival of the First Fruits, when the first fruits of the seven species with which the Land of Israel is blessed were harvested and brought to the Temple. It is therefore quite appropriate that in include the Trees of the Field in general as well as the Vine, which is the first (and arguably the most important) of the seven species. Shavuot is also called Chag HaKatzir, the Harvest Festival, and the third Chapter of Perek Shirah, which begins with this cycle, is all about the vegetable kingdom.
The verses of these two elements read as follows (translation from Rabbi Slifkin):
The Wild Trees are saying, "Then shall the trees of the forest sing out at the presence of G-d, because He comes to judge the earth." (Chronicles I 16:33)
The Vine is saying, "So says God: As the wine is found in the cluster, and one says: Do not destroy it, for a blessing is in it - so shall I do for the sake of my servants, so as not to destroy everything." (Isaiah 65:8)
The trees sing out in when they experience the presence of G-d, just as we celebrate our encounter with G-d on Mount Sinai.
Rabbi Slifkin explains that the Vine takes much labor to plant and to harvest, and to later produce its final outcome: wine. Yet, the greater amount of work brings about an even greater reward. ("Nature's Song, p. 170) Similarly, on Shavuot, we are repaid for all the hard work that took place during the Counting of the Omer.
The section of Pirkei Avot for this period describes the great reward for studying Torah, very much in line with the upcoming holiday.