Jewish Talmudic Calendar

The Jewish Talmudic Calendar is a lunisolar calendar using Tishri-years, observed by the Jewish people since Late Antiquity (300-700 CE). While it is based on Nisan-years, which began from the prebiblical Babylonian times (c. 2000 BCE), and the Tishri-years was formed in the time of David (c. 1000 BCE), the full formation of the Jewish Talmudic Calendar was during the time of the writing of Talmud (c. 300-600 CE), usually attributed to Hillel II.
Purpose.
The ancient Israelite calendar, whether Nisan-years or Tishri-years, was determined by the astronomical observation of the new moon and the agricultural observation of the growth of "Abib" (the spikes on the barley) in late winter and early spring. After the second (Roman) Diaspora, in order for the Jewish people away from the Land of Israel to be able to observe their spring and fall feasts in the same calendar, the Jewish Talmudic Calendar was formed, based on mathematical algorithms, and free from the required observations.
Accuracy.
The Jewish Talmudic Calendar, in agreement with the Almagest and the writings of Kidinnu, assumes that a month is uniformly of the length of an average synodic month, taken as exactly 29 days (about 29.530594 days, which is less than half a second from the modern scientific estimate); it also assumes that a tropical year is exactly 12 times that, i.e., about 365.2468 days. Thus it overestimates the length of the tropical year (365.2422 days) by 0.0046 days (about 7 minutes) per year, or about one day in 216 years. This error is less than the Julian years (365.2500 days) make (0.0078 days/year, or one day in 128 years), but much more than what the Gregorian years (365.2425 days/year) make (0.0003 days/year, or one day in 3333 years).
Days.
A day is defined to begin from the previous evening, at the time of sunset, when three medium-sized stars appear. Practically it begins from 6:00 p.m. (18:00) Jerusalem time (16:00 Universal Time). A day is divided into 24 hours, each with 1080 parts (halakim). Thus each minute is composed of 18 halakim; each halakim is 3.33 seconds.
Months.
There are twelve months in a common year, and thirteen months in a leap year. Rosh Hashana, the Jewish New Year, is the 1st day of Tishri. The months are 30 days and 29 days in turns. A regular common year is 354 days; while a regular leap year is 384 days (adding in a 30-day Adar I, and making the 29-day month Adar II).
Years.
There are times when a common year is 353 days, or a leap year is 383 days; these are called defective years. When a common year is 355 days, or a leap year is 385 days, these are called the excessive years. In defective years, the regularly 30-day month of Kislev is made 29 days; in excessive years, the regularly 29-day month of Heshvan is made 30 days.
Era.
The Jewish Talmudic calendar is coupled with the Rabbinical Chronology, taking Monday, (Julian) 7 October 3761 BCE. as the date of creation, and the beginning of the Era of the World (Anno Mundi). The reference junction of the Sun and the Moon (Molad 1) on the day of creation is considered to be at 5 hours and 204 halakim, or 11:11:20 p.m., in the evening of Sunday, 7 October 3761 BCE.
Cycles.
In every 19 years, the solar and lunar calendars basically synchronize, with only about 2 hours of difference. Thus each 19 years is called a Small Mahzor in the Jewish Talmudic calendar, which is equivalent to the Greek metonic cycle, although they do not start on the same year. The year of creation according to the Rabbinical Chronology (3761 BCE) is taken as year 1 in the first Small Mahzor. The Greek cycle begins from an arbitrarily year, usually from the beginning of the Common Era (Anno Domini).
If every 13 Small Mahzor is called an Iggul, because 12 times 2 hours is a day, and 30 days are a month, then in less than 30 Igguls a whole intercalary month should be removed.
Year patterns.
The position of the years in a small Mahzor is called the golden number. The pattern of the leap years change slightly in each Iggul, but the Jewish Talmudic calendar fixed the leap years in the year with golden numbers 3, 6, 8, 11, 14, 17, 19. If a Leap year marked L, and the Following year F, and the other common year as O, then
Because the Julian years are 365 and 1/4 days, in every 28 years the weekday pattern repeats. This is call the sun cycle. The beginning of this cycle is arbitrary.
Because every 50 years is a Jubilee year, there is a yovel cycle; Because every seven years is a sabbatical year, there is a seven-year release cycle. The placement of these cycles is controversial. Historically there are enough evidences to fix the sabbatical years in the Second Temple Period. But it may not match with the sabbatical cycle derived from the biblical period; and there is no consensus on whether or not the Jubilee year is the fiftieth year or the latter half of the forty ninth year.
Hebrew calendar.
A few suggestions were made to improve the Jewish Talmudic calendar, in order to reduce its accumulated error for the remote past and future. One is to delay the leap years gradually so that a whole intercalary month is taken out at the end of Iggul 21; another is to adopt the synodic month to be the more accurate 29.53058868 days, thus the length of the year would be (235*13*26-1)/(19*13*26) = 365.2426 days, very close to the actual 365.2422 days of the tropical year. The result is the "Hebrew Calendar" in the program CalMaster2000.