Israel's official calendar is the Hebrew one. According to Jewish counting, on September 24, 2014, we entered the Year 5775, that is - the supposed 5775th year since the world was created on Saturday night, October 6, 3761 BCE.
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This reckoning was instituted by Maimonides in the 12th century, in the stead of the previous system Jews had used before, which counted from the destruction of the Temple in 70 CE.
Moving onto today: Israel's official calendar is the Hebrew one. Under law, official Israeli documents must have the Hebrew date on them. Moreover, holidays in Israel are determined according to the Jewish calendar, not the Gregorian one. Thus a given fest – say, Rosh Hashanah – will happen on the same date each year according to Jewish reckoning, but on a different day each year according to the Gregorian calendar. That is because the Gregorian and Jewish calendars don't coincide.
Even civil holidays in Israel, such as Jerusalem Day, are based on the Jewish calendar.
Yet in their day to day lives, most Israelis are completely unaware of the Hebrew date and lead their lives according to the Gregorian calendar. That said, a not-insignificant religious minority still adhere to the Hebrew calendar of old.
The Hebrew calendar is very complicated, because it has to align the solar year (365 days, 5 hours, 48 minutes and 46-seconds) with the lunar year (12 months of 29 days, 12 hours, 44 minutes and 3 seconds).
This means that it must make allowance for the fact that 12 lunar months are about 12 days shorter than the solar year. This isn’t easy.
If you get it wrong, even by a little bit, over time the two would slowly drift apart and you’d have your spring holidays in the fall.
An ancient limestone calendar
Like most ancient peoples, at first the Jews followed a strictly lunar calendar. Our earliest record of this is a 10th century BCE calendar found in the Canaanite town of Gezer (midway between Jerusalem and Tel Aviv).
Regarding the "Israelite new year," this seems to have swung back and forth over the centuries. The Gezer calendar seems to place the "start" of the new year in the fall; by the time the Torah was redacted some four centuries later, the new year had moved to the spring. Today it's celebrated in the fall again. (It perhaps bears mention that the ancient Hebrews saw no need that we know of to mark the passing of an old year and start of a new one.)
Maybe calling the inscribed limestone tablet found at Gezer a "calendar" is an exaggeration. The etched stone, in either Phoenician or paleo-Hebrew (depending on who you ask) lists months (and month duos), which don't have proper names, just designated times of the year.
These months are referred to as “moon(s) of X” with X being the main agricultural activity of that month (or two months).
For example, the list begins in the fall with yarkhu asif, that is, “two moons of harvesting.”
As we said, some four centuries later when the Torah was being written, the year was considered to begin in the spring, with each month named by its ordinal number. Thus the spring month on which Passover fell was called the “first month,” the next month was the “second month” and so on.
The arrival of spring
The solar calendar is automatically in line with the seasons, but being some 12 days shorter, the lunar calendar is not.
The Bible doesn’t tell us how the lunar calendar was periodically adjusted to keep in line with the seasons. But it must have been, at least from the days of King Josiah and the centralization of the cult in Jerusalem (in the second half of the seventh century BCE). On Passover, Shavuot, and Sukkot – which have specific Hebrew dates - the people of Israel were required to bring agricultural produce to Jerusalem; and if there had been "seasonal creep," in at least some years, there would have been no produce to bring as tithes.
It is almost certain this is how they adjusted for seasonal creep: At the end of the 12th lunar month, priests in Jerusalem determined spring had arrived. If spring was indeed deemed to have arrived, great: the new moon was a sign that not only the first month had begun but also a new year, and messengers were sent out to announce that Passover was two weeks away.
But if the priests decided that it was still winter, an additional “leap month” or “embolismic month” as its called was added, and messengers would only be sent out a month later.
Thus the calendar was kept in line with the seasons. It is possible that this process was made more precise with the introduction of a sundial of sorts introduced by King Ahaz in the end of the eighth century BCE.
Jewish month - named for Babylonian god
It seems that before their banishment from the Holy Land and exile in Babylon, the ancient Hebrews hadn't given their months actual names. But after the Babylonian exile in the Second Temple period, the months began to be called by their Akkadian Babylonian names - which we use to this very day.
Some of these names are innocuous enough, such as Tishrei, meaning “beginning,” Marheshvan was shortened to Heshvan meaning “eighth month,” and so on. Other month names though are quite surprising: Tammuz for example originated with a pagan Babylonian god.
Another change that happened gradually in the Second Temple period was that the year was increasingly seen as beginning in the fall, as it is to this very day, and not in the spring as it was in biblical times.
The hand of imperfect men
At first the decision on when exactly one month ended and another began, and when a leap month was to be added, was in the hand of men. But in a gradual process spanning hundreds of years, a fixed system was devised, which we use to this day.
According to the Mishnah (redacted 200 CE), during the time of the Second Temple, witnesses would come to the Temple and report that they had seen the new moon. They would be grilled by priests, who would decide whether a new month had in fact begun.
If they decided it had, the priests would announce the new moon to the public.
Thus they would decide the length of each month and whether an embolismic month was to be added to the year. When the Temple was destroyed by Titus in 70 CE, these functions were taken up by the rabbinic council, the Great Sanhedrin, and more specifically by a subcommittee of three of the Sanhedrin members that would meet on the 29th of every month and receive witnesses and come to decisions regarding the calendar.
From Mishnaic times through to the ninth century CE at the latest, the Hebrew calendar gradually lost the errant human element and shifted to a fixed calendar based on predetermined rules and calculations.
Samuel the astronomer
A key figure in the process of formalizing the calendar from observation to calculation was the great Babylonian rabbi Samuel of Nehardea, who was a great astronomer. The Talmud quotes him as saying “I am as familiar with the courses of the stars as with the streets of Nehardea” (Berachot 58b).
Samuel developed the calculations necessary to determine the calendar without relying on messengers from Palestine, which, he explained, meant that Jews outside of Palestine no longer had to add an extra day to their holidays, though halakha still prescribes that these extra days should be kept just the same.
Throughout the leadership of Judah III (290-320 CE), the Great Sanhedrin continued to receive witnesses of the new moon, but only as a formality, and for the sake of preserving time-honored tradition. The decision was already made according to calculation.
Samuel's son and successor as Nasi of the Sanhedrin, Hillel II (330-365 CE) is credited with finalizing the fixed Hebrew calendar as we know it, though the rules we know today were only fully laid out in writing in the 9th century.
So how does it work? The basic Jewish year has 12 months with five months of 29 days, and five months of 30 days, which alternate. The two other months - Heshvan and Kislev - change from year to year, according to the rules elaborated below.
But not all Jewish years have 12 months. Seven out of every 19 years are "embolismic years," which means they have a 13th month, called "Adar Bet" or - a "second Adar".
Whether a year is embolismic or not is determined by its place on a 19-year cycle. Most years are “regular” 12-month years except for years 3, 6, 8, 11, 14, 17, and 19, which are embolismic.
Each year, whether embolismic or not, starts on Rosh Hashanah - the day the new moon of Tishri appears. But if one of four conditions called dekhiot (“postponements”) are met, the Rosh Hashanah holiday - and thus the beginning of the year - is postponed by a day or two. In practice, that happens in most years.
So what are these postponing rules? (1) If Rosh Hashanah falls on Sunday, Wednesday or Friday, it is postponed by a day. (2) If the new moon appears after noon, Rosh Hashanah is postponed by a day. If the new day is one of those in Rule 1, Rosh Hashanah gets postponed by two days. (3) In a regular 12-month year, that is one that isn’t embolismic, if the new moon of Tishri appears more than 20 seconds past 3:11 A.M. on a Tuesday, Rosh Hashanah is postponed by two days.
The last instance is an extremely rare one. (4) In years that follow an embolismic year, if the new moon appears on a Monday more than 43 seconds past 9:32 A.M. - then Rosh Hashanah is postponed by a day.
You might assume these one- or two-day postponements simply cause one or two days to be added to the last month of the year, Elul. Not so! In fact they were added six months in advance to the winter months of Heshvan and Kislev, which as we said, change in length from year to year.
Theoretically Heshvan and Kislev each have 29 days. But if the new year needs to be postponed by one day, the preceding Heshvan is expanded to 30 days and Kislev stays at 29.
If two days are to be added, then both the preceding Heshvan and Kislev have 30 days.
How is this calculated six months in advance? Because dekhiot can be and are calculated years in advance using moon charts and math. When calculating the calendar for the year to come, you can calculate what kind of year it will be well in advance, and monkey with the lengths of Heshvan and Kislev as needed.
That’s it. If you have the data on when the new moon will rise over Jerusalem for each year, information that is readily available online, you can calculate the Hebrew calendar yourself. Though, thankfully, you don’t have to. Shanah Tova!