You think your personal calendar is complex?

By Fred Reiss, Ed.D

WINCHESTER, California — Many people believe that the Jewish calendar is a lunar calendar because the months are determined by the motion of the moon, resulting in a year of twelve full-lunar cycles, or a little less than 354½ days. However, if the Jewish calendar were a lunar calendar, then each New Year would begin almost 11¼ days sooner than the previous year, allowing each holiday to retrogress one season in about nine years. Why? Because the motion of the earth around the sun, whose tropical or solar year, just short of 365¼ days, determines the seasons of the year.

However, we observe that year after year the Jewish holidays fall in their correct seasons, so the Jewish calendar cannot be a lunar calendar. Rather, it is a lunisolar calendar, a calendar in which the months begin with the new moon, but the length of the year approximates the length of a solar year, compensating for much of the 11¼ missing days.

From biblical times through the Second Temple period, the priesthood fixed each new month strictly through observations reported by eyewitnesses. From the Talmudic period through the time of Sanhedrin President Hillel II, who lived during the fourth century C.E., the Sanhedrin, or Jewish court, mandated the Jewish calendar through a combination of observations and computation. Witnesses came before a secret committee of the Sanhedrin, the Calendar Council, whose members were experts in astronomy and mathematics, testifying that they saw the faint crescent of the new moon. The members of the Calendar Council asked the witnesses such questions as: “Where did you see the moon in relation to the sun?” and “In what direction were the crescent points facing?” Based on the answers and their own expertise, the members determined whether or not a new month had begun. If no witnesses appeared because of, say, cloudy skies and if the calendar date were the thirty-first, that day automatically became the first of the new month, as no Jewish calendar month can have more than thirty days.

Additionally, whenever the committee determined by observation that spring had not arrived, as evidenced by such things as unripened barley and trees not yet in bloom, as well as by calculating that the vernal equinox fell after the sixteenth of the month of Nisan, they intercalated an extra month of thirty days before the month of Adar, known as Adar I, and in that year calling the month of Adar, Adar II. (The Hebrew Bible refers to Passover a spring holiday, and therefore, the First Day of Passover, 15 Nisan, must fall after the vernal equinox.) Hillel II, anticipating Roman Emperor Theodosius I’s abolishment of the Sanhedrin, abandoned the witness process and introduced a calculated Jewish calendar, a calendar determined strictly by mathematical rules, in 359 C.E., and this method has been utilized ever since.

The secular day is divided into 24 hours, each hour into 60 minutes, and each minute into 60 seconds. Jewish days are divided into 24 hours, each hour into 1080 parts (ḥalaqim), and each part into 76 moments (regaim). The time of the new month, or molad, is counted from 6 P.M., sunset, the start of the Jewish day.

Using the appropriate equations, for example, the molad of the new moon for Rosh Hashanah 5775 is 4d 14h 339p, which we read as: day 4, 14^th hour, 339 parts. This corresponds to Wednesday (day 4), 14 hours, 339 parts after 6 P.M. Fourteen hours after 6 P.M. is 8:00 A.M., the next day, and 339 parts computes to 18 minutes, 50 seconds. So the molad of Rosh Hashanah 5775 takes place on a Thursday at 8:18:50 A.M. (The actual date of Rosh Hashanah is determined by finding the number of days in a given calendar year and then counting forward from the previous Rosh Hashanah.)

In the secular calendar, the Gregorian calendar, common years have 365 days and leap years have 366 days. Leap years occur every four years, except on century years not divisible by 400. Regular or common Jewish years can have 353 (defective year), 354 (normal year), or 355 (abundant year) days. A Jewish leap year, a year in which an extra month of thirty days is added to the calendar, takes place at fixed intervals seven times (the 3^rd, 6^th, 8^th, 11^th, 14^th, 17^th, and 19^th years) within a nineteen-year cycle, known as the Metonic cycle, giving rise to leap years with 383 (defective year), 384 (normal year), or 385 (abundant year) days.

Unfortunately, the Jewish calendar is not really a lunisolar calendar either; it is rabbinic-Judaism’s mandated lunisolar calendar. Here’s why. Reason 1: if the molad of 1 Tishri (Rosh Hashanah) falls on a Wednesday or Friday, then Yom Kippur, ten days later, would fall either the day before or the day after Shabbat, a terrible, if not impossible inconvenience for observant Jews who cannot, among other things, prepare meals, as no work can be performed on either Shabbat or Yom Kippur. If the molad of 1 Tishri were to fall on a Sunday, then the holiday of Hoshana Rabbah (Seventh Day of Sukkot) would fall on the Sabbath, precluding the required Beating-of-the-Willows ceremony. Consequently, if the molad of 1 Tishri falls on a Sunday, Wednesday, or Friday, then Rosh Hashanah is postponed one day.

There are three other reasons for Rosh Hashanah being postponed for up to two days in any given year:

Reason 2: since the Jewish day begins at 6 P.M., the day is three-quarters over at noon the next day, and the rabbis consider it an “old moon.” If the molad of 1 Tishri occurs after 12 P.M. on any given day, then Rosh Hashanah is postponed one day. If, as a result of this postponement, Rosh Hashanah now falls on a Sunday, Wednesday, or Friday, the holiday is postponed one additional day.

Reason 3: if the molad of 1 Tishri falls in a common year on a Tuesday at or later than 9 hours (Wednesday, 3 A.M.) and 204 parts, then Rosh Hashanah is postponed two days, since it cannot fall on a Wednesday for reasons listed above. In addition, the rabbis understood that under this condition, the molad of 1 Tishri of the following year will fall after 12 P.M. on a Sabbath, thereby moving it to Sunday, which is not allowed and making the present common Hebrew year 356 days long, which is also not permitted.

Reason 4: if the molad of 1 Tishri falls in a common year following a leap year on a Monday at 15 hours (Tuesday, 9 A.M.), 589 parts or later, Rosh Hashanah is postponed one day for reasons similar to Reason 3; the preceding year will have only 382 days, which is unacceptable.

This Rosh Hashanah begins the year 5775, and in the next 5775 years, about 70 percent of all Rosh Hashanahs will be postponed one or more days. A postponement for Reason 1 applies nearly 57 percent of the time, whereas a postponement for Reason 2 takes place a little more than one-third of the time. Rosh Hashanah will be postponed for Reason 3 about eight percent of the time, with the next one occurring on Rosh Hashanah 5789 (2028 C.E.) A postponement for Reason 4 is the rarest, happening less than one percent of the time. Reason 4 will cause a postponement of Rosh Hashanah only 30 times over the next 5775 years, with the next one taking place in 6013 (2252 C.E.), followed by 6111 (2350 C.E.) A postponement for Reason 4 last occurred in 5766 (2005 C.E.)

These rabbinically-inspired postponements of Rosh Hashanah ripple through the year. For example, the first day of Passover can never fall on a Monday, Wednesday, or Friday, which of course means that the first Seder can never fall on a Sunday night, Tuesday night, or Thursday night. Since Lag Ba-Omer is the thirty-third day after Passover (18 Iyyar), it can never fall on a Monday, Tuesday, or Saturday, and the first day of Shavuot is the fiftieth day after Passover (6 Sivan), so it can never fall on a Tuesday, Thursday, or Saturday.

The Jewish calendar has a couple of built-in errors, resulting from unavoidable mathematical approximations. The first relates to the solar year. A solar year, the time it takes the earth to make one complete revolution around the sun, is 365 days, 5 hours, 48 minutes, and 46 seconds, or 365.2422 days. The average length of a Jewish year is 365.2468 days, so the Jewish calendar is in error by 4.6 days in a thousand years with respect to the solar year. Since the average length of the Jewish year is greater than the length of a solar year, the Jewish calendar is 4.6 days faster than it should be based on the length of a solar year after each thousand year period. Jews now celebrate the same Jewish holidays 4.6 days later than their ancestors did a thousand years ago.

The second error is between the Jewish and secular calendars. The secular or Gregorian calendar, which became the operating calendar in October 1582, also runs faster than the solar year by 0.3 days in a thousand years. The Gregorian calendar’s date on which Rosh Hashanah 5775 should fall and the observed date of Rosh Hashanah now differs by about 1.9 days. In essence, this upcoming Rosh Hashanah begins two days later than it should relative to the launch of the Gregorian calendar, some 433 years ago. The earliest that Rosh Hashanah can fall is September 5, which happened in 1899 and again in 2013. Rosh Hashanah will again fall on September 5, 2089, after which Rosh Hashanah will appear on the calendar no earlier than September 6.

Perhaps a clearer way of demonstrating this forward drift in the Jewish calendar is by observing the movement of the first day of Rosh Hashanah through the years. In the sixteenth century, the first day of Rosh Hashanah fell in October just 8 percent of the time, during the twenty-first century it will fall in October 14 percent of the time, and by the twenty-sixth century, as an example, Rosh Hashanah will fall in October 22 percent of the time.

Can the calendar’s drift problem be resolved? There are many ways to rectify and eliminate this error. But, the much more important question is who has the power to alter and then maintain a Jewish calendar without creating a great schism within the Jewish community? History shows that controlling the Jewish calendar is not without controversy. In the late Second Temple period, the Sadducees often intentionally lit new-moon fires on the wrong day to thwart the power of the Pharisees’ control over the calendar. The Pharisees eventually abandoned communication fires in favor of messengers. At the same time, the Qumran community, often associated with the Essenes, employed a solar calendar consisting of 12 months of thirty days each, plus one additional day at each of the solstices and equinoxes, for a total of 364 days. (While not as bad as a strictly lunar calendar, the error in the Qumran calendar is nearly 1¼ days per year.)

Whoever controls the Jewish calendar has enormous political power. As Samson Raphael Hirsch, an important nineteenth century rabbi noted, “The catechism of the Jew is his calendar.” During the Second Temple period, the high priest most likely became the final arbiter over the calendar. The victory of Roman general Vespasian and his son Titus over the Jews in 70 C.E. resulted in the destruction of the Temple and elimination of all priestly positions. Rabbi Joshua ben Ḥananiah, a student of the famous Rabbi Yoḥanan ben Zakki, founder of an academy at Yavne in the wake of the fall of Jerusalem and its Temple, challenged the authority of the Sanhedrin’s president, Rabbi Gamaliel, to set the Jewish calendar, fix the dates of the holidays, and intercalate months Fortunately, Rabbi Joshua backed down at the last moment, preventing a major schism (see Mishnah Rosh Hashanah 2:8-9.)

A second controversy broke out over the interpretation of postponement rule 3 between Aaron ben Meir, President of the Sanhedrin of the Palestinian Gaonate, the central academy and legal body of the Jewish community in Roman Palestine until its termination during the eleventh century, and the Babylonian exilarch (Head of the Diaspora) David ben Zakki in the period 921 to 923 C.E. ben Meir, representing the Jews of Palestine, changed the calendar calculations from the month of Tishri to Nisan, and used different rules for rounding-off time. As a result, ben Meir announced that Passover would fall two days earlier than with traditional calculations. Turmoil ensued, and only through the support of the famous Babylonian scholar Saadia Gaon did one side prevail, preventing a rupture among the Jews.

We know there is a pattern in the cycling of leap years in the Jewish calendar, are there any repetitive patterns in the cycling of defective, normal, and abundant years? The answer is no, but the Jewish calendar does have some rules. There are several ways of depicting the structure of the Jewish year. One method says that a complete description of a Jewish year’s configuration depends on exactly three things: (1) the day of the week on which Rosh Hashanah falls, (2) the type of year—deficient, normal, or abundant, and (3) the day of the week on which Passover falls in that Jewish year. Using this method, one finds that the calendar years are organized into precisely fourteen different useable structures, seven common years and seven leap years.

In the next 5775 years, the structure of the common year appearing most, about 18 percent of the time, is called a “5n7 year,” that is, a year in which Rosh Hashanah falls on a Thursday (day 5), Passover falls on a Saturday (day 7), and there are 354 days in the year (a normal year.) The least appearing common year, occurring only about 3.4 percent of the time, is a “5a1 year,” a year in which Rosh Hashanah falls on a Thursday, Passover falls on a Sunday, and the year contains 355 days (an abundant year). The leap year “7a5” occurs most, at just over 13.6 percent of the time, and a “5a1” year occurs least, where the “a” stands for 385 days, since it is a leap year, at 3.8 percent of the time. Additional rules say that leap years cannot follow leap years, normal years cannot follow normal years, deficient years cannot follow deficient years, but abundant common years may follow abundant leap years, and vise versa

Many different dates have been set aside for a national day of thanksgiving. President Lincoln established one with staying power to be held in November 1863, and all but a few states chose November 26 in that year. If we assume that after 1863 all Thanksgivings have been held on the fourth Thursday of November, then only in 1888 and 2013 did Thanksgiving and Chanukah coincide, and that will be it. In fact, after 2640 Chanukah will never fall in November again.

Beginning in 3031 C.E., there will be secular years without a Chanukah celebration! For example Chanukah will be celebrated on Monday, December 13, 3030, and not again until Sunday January 1, 3032. Of course, 3032 will have a second Chanukah celebration on Wednesday, December 19. The year 3032 is also the first year in which the first day of Chanukah falls in January. (On a side note: the first day of Chanukah last fell on the first day of Christmas in 1978, and will fall on the first day of Christmas only three more times this century, in the years 2016, 2027, and 2073.)

The Gregorian calendar is like a decimal fraction that repeats its structures after 400 digits, meaning years. How about the Jewish calendar? How often, if at all, do the Jewish calendar’s structures repeat? According to tradition the very first molad of 1 Tishri of the Jewish year 1 occurred on October 7, 3761 B.C.E., at 2d 5h 204p (Monday at 11:11:20 P.M.). Unfortunately, we will need to wait 98,496 years to get a complete repetition with respect to molad time. How often do both the day and time of the molad correspond with the day and time of the molad of the very first Rosh Hashanah? Since there are seven days in a week, the answer is that 7 x 98,496 = 689,472 years will pass before the structures of the Jewish calendar fully begin to repeat, and how often is there an exact correspondence between the Jewish and Gregorian calendar structures? Once every 14,390,140,400 years!

Now you know why people like Chassidic Rabbi Yanki Tauber say, “the Jew has what is probably the most complex calendar known to man,” and the late Wolfgang Alexander Schocken, mathematician, Jewish calendar expert, and concertmaster of the Israel Radio Orchestra, referred to the Jewish calendar as “the most sophisticated calendar system.” Happy Rosh Hashanah 5775!

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Dr. Fred Reiss is a retired public and Hebrew school teacher and administrator. He is the author of The Standard Guide to the Jewish and Civil Calendars;  Ancient Secrets of Creation: Sepher Yetzira, the Book that Started Kabbalah, Revealed; and a fiction book, Reclaiming the Messiah. The author can be reached  via fred.reiss@sdjewishworld.com.