Ancient western Mesopotamian calendar using 7 periods of 50 days
The pentecontad calendar (from πεντηκοντάς pentēkontás) is an agricultural calendar system thought to be of Amorite origin in which the year is broken down into seven periods of fifty days (a total of 350 days), with an annual supplement of fifteen or sixteen days. Identified and reconstructed by Julius and Hildegaard Lewy in the 1940s, the calendar's use dates back to at least the 3rd millennium BCE in western Mesopotamia and surrounding areas. Used well into the modern age, forms of it have been found in Nestorianism and among the Fellahin of modern Palestine.
In Akkadian, the pentecontad calendar was known as hamšâtum and the period of fifteen days at the end of the year was known to Babylonians as shappatum.
Each fifty-day period was made up of seven weeks of seven days and seven Sabbaths, with an extra fiftieth day, known as the atzeret.
The liturgical calendar of the Essenes at Qumran was a pentecontad calendar, marked by festivals on the last day of each fifty-day period such as the Feast of New Wine, the Feast of Oil, and the Feast of New Wheat.
Philo expressly connected the "unequalled virtues" of the pentecontad calendar with the Pythagorean theorem, further describing the number fifty as the "perfect expression of the right-angled triangle, the supreme principle of production in the world, and the 'holiest' of numbers".
^Jonhatan Ben-Dov, The_History_of_Pentecontad_Time_Periods (I), in: A Teacher for All Generations. Essays in Honor of James C. VanderKam, (Gen. ed. E. Mason; JSJSup 153; Leiden: Brill, 2011), vol. I, pp. 93–111. This paper rebuts most of previous theories presented above.