"After the Saviour had miraculously healed the paralytic, the Jews, especially the Pharisees and Scribes, were moved with envy and persecuted Him, and sought to slay Him, using the excuse that He did not keep the Sabbath, since He worked miracles on that day. Jesus then departed to Galilee. About the middle of the Feast of Tabernacles, He went up again to the Temple and taught. The Jews, marvelling at the wisdom of His words, said, "How knoweth this man letters, having never learned?" But Christ first reproached their unbelief and lawlessness, then proved to them by the Law that they sought to slay Him unjustly, supposedly as a despiser of the Law, since He had healed the paralytic on the Sabbath. Therefore, since the things spoken by Christ in the middle of the Feast of Tabernacles are related to the Sunday of the Paralytic that is just passed, and since we have already reached the midpoint of the fifty days between Pascha and Pentecost, the Church has appointed this present feast as a bond between the two great feasts, thereby uniting, as it were, the two into one, and partaking of the grace of them both. Therefore today's feast is called Mid-Pentecost, and the Gospel Reading, "At Mid-feast"--though it refers to the Feast of Tabernacles--is used.
Finally, it is worth noting that the Great Church of Hagia Sophia (The Church of the Holy Wisdom) in Constantinople celebrated its feast on the feast of Mid-Pentecost. (http://www.johnsanidopoulos.com/2011/05/mid-pentecost-and-hagia-sophia.html)
Mid-way in the feast, refresh my thirsty soul with the flowing waters of piety. For You cried out to all, O Savior, "Let him who thirsts come to me and drink." You, O Christ our God, are the Fountain of Life, glory to You.