Midrasch zu Tehillim 119:148

Our Rabbis have taught: "The night has four watches," so says Rabbi. R. Nathan says "Three." What is R. Nathan's reason? It is written (Judges 6:19) "And Gidon, and the hundred men that were with him, came the edge of the camp in the beginning of the middle watch. And we are taught that Tichon (middle) cannot be used unless something precedes and something follows it. But Rabbi disputes this and says that "by the middle is meant one of the two middles." R. Nathan says: "Is it then written one of the middles? Behold! It is written the middle!" What is Rabbi's reason? R. Zerika in the name of R. Ami, who speaks in the name of R. Joshua ben Levi, said: "One passage says (Ps. 119:62) 'At midnight do I constantly rise to give thanks unto Thee,' etc., and another passage says (Ps. 119:148) 'My eyes are awake before the night watches. How is this possible? At midnight; because the night is divided into four watches, [midnight is therefore after two watches.]" But R. Nathan explains it by the statement of R. Joshua; as we are taught that R. Joshua says: "It is the custom of kings to rise on the third hour of the day. [i.e., six hours of the night and two hours of the day together make two night watches of four hours each]. R. Ashi said: "One watch and a half are also called watches (plural)."

“Arise, cry out at night, at the beginning of the watches, pour out your heart like water before the face of the Lord. Lift up your hands to Him for the life of your infants, who are faint with hunger at the head of every street” (Lamentations 2:19).
“Arise, cry out at night, at the beginning of the watches.” Rabbi [Yehuda HaNasi] says: There are four watches during the night and four watches during the day. The ona is one twenty-fourth of the et,¹²³The text should read: The ona is one twenty-fourth of an hour (Etz Yosef). and the et is one twenty-fourth of the ona. The rega is one twenty-fourth of the et. How long is a rega? Rabbi Berekhya said in the name of Rabbi Ḥelbo: As long as it takes to say it. The Rabbis say: Like the blink of an eye. Shmuel said: One fifty-six thousand five hundred and forty-eighth of an hour; that is a rega. Rabbi Natan said: There are three watches during the night.
Rabbi Zerika and Rabbi Ami said in the name of Rabbi Shimon ben Lakish: One verse says: “At midnight I rise to give You thanks” (Psalms 119:62), and one verse says: “My eyes precede the night watches” (Psalms 119:148). How can these two verses be reconciled? Rabbi Ḥizkiya, and some say Rabbi Zerika and Rabbi Ami: One said the source according to Rabbi [Yehuda HaNasi] and one says the source according to Rabbi Natan. The one who says the source according to Rabbi [Yehuda HaNasi], it works out well.¹²⁴There are four watches during the night, two before midnight and two after midnight. Thus, if one rises at midnight, that is still before two of the nightly watches. These verses can thus be a source for the opinion of Rabbi Yehuda HaNasi that the night is divided into four watches. The one who says the source according to Rabbi Natan: “At the beginning of the middle watch” (Judges 7:19).¹²⁵If there is a middle watch, there must be an odd number of watches. This can serve as a source for the opinion of Rabbi Natan that the night is divided into three watches.
But Rabbi Natan, how does he interpret “at midnight”? Rather, at times “at midnight,” and at times: “My eyes precede the night watches.”¹²⁶The two verses do not both mean the same thing, as according to Rabbi Yehuda HaNasi. Rather, King David, the author of Psalms, is stating that at times he awoke at midnight, and at times even earlier, before two of the three nightly watches. What would [David] do? When David would conduct his meal alone, he would eat until nine hours of the day, sleep until the beginning of the middle watch, and arise and engage in Torah study. When David would eat a feast of kings, he would eat until the evening, sleep until midnight, and arise and engage in Torah study from midnight onward.
In any case, dawn would not arrive with David asleep. That is what David said: “Awaken, my soul, awaken, harp and lyre; I will wake the dawn” (Psalms 57:9). Let my honor awaken before the honor of my Creator; my honor is nothing before the honor of my Creator. “I will wake the dawn,” I wake the dawn, the dawn does not awaken me.
Rabbi Pinḥas said in the name of Rabbi Elazar bar Menaḥem: A lyre was placed under his head and he would rise and play it at night. Rabbi Levi said: A lyre was suspended over David’s bed. When midnight arrived, the north wind would come and the lyre would play on its own. That is what is written: “It was as the instrument played” (II Kings 3:15). It is not written here “It was as he played the instrument,” but rather: “It was as the instrument played,” the instrument played on its own. When David would hear its sound, he would arise and engage in Torah study. [People] would say: If David king of Israel is engaging in Torah study, all the more so for us. They immediately would begin engaging in Torah study.
How does Rabbi interpret the verse of Rabbi Natan? Rabbi Huna said: The end of the second and the beginning of the third, which constitutes the midpoint [metavḥot] of the night. Rabbi Mani said: Had it said “middle [tikhonot],”¹²⁷Had the verse used the plural term for “middle” that would allow for it to be interpreted in accordance with Rabbi Yehua HaNasi, who holds that the night is divided into four watches. that would be correct. But does it not say “middle [tikhona]”?¹²⁸This term is singular. The first is not counted, as until then, the time has not yet arrived.¹²⁹The concept of dividing the night into periods of time called watches parallels when an earthly king would have soldiers standing guard. The first quarter of the night is not yet the time for guards, because people are still awake.