The Jewish Calendar As Early and Late as it Gets Source Sheet (pdf)
The Jewish Calendar As Early and Late as it Gets (pdf)

purimMuch has been made of the fact that Pesach, last March 26, and Rosh Hashanah, on September 5 (2013), were the earliest on the Gregorian calendar since 1899 and the earliest these holidays will fall until 2089.  We can add to this the rare coincidence of Hanukkah and Thanksgiving (November 28) and even the recent Tu B’Shvat, on January 16, actually a day later than in 1900.  But while many articles, blogs and emails celebrate these facts, few, if any, really explain them.

The calendar in Judaism, as indeed in many cultures, is not simply an administrative tool; it can also be an expression of national identity.  In the past certain nations, e.g. France in 1793 and the Soviet Union in the 1920s and 1930s, asserted their sovereignty by instituting new, original calendars.  The children of Israel were instructed to adopt a new calendar, sovereignty in time, even before they began their departure from Egypt (Ex. 12:1-3).  Rashi’s first comment on the Torah is to explain why the Torah did not begin with the command of the calendar (Source 1).

Our secular (Gregorian) calendar is solar – with years of 365 or 366 days, corresponding to the earth’s revolution around the sun (365.25 days, to be more exact, though the astronomical data here are not precisely accurate), and months of 28, 29, 30 and 31 days, unrelated to astronomy.  Some cultures, such as Islam, use a purely lunar calendar, based on the 29.5 day circuit of the moon around the earth.  To paraphrase Seven Lonely Days (Georgia Gibbs 1953, Patsy Cline 1961), twelve lunar months make one 354-day year, 11.25 days shorter than the solar year, and Ramadan, for example, moves through the secular calendar at that pace.  Ramadan began on July 9 in 2013, will begin on June 28 this year (2014) and will fall close to July 9 again only in 2046.

The Jewish calendar is both lunar and solar – the months are lunar (Rashi on Ex 12:2, the first interpretation), either 29 or 30 days, rounding off for convenience.  But we are also obligated by the solar calendar because the Torah describes Pesach as Hag HaAviv – the Spring Festival (Deut 16:1) and Sukkot as Hag HaAsif – the Harvest Festival (Ex 34:22)  “at the turn of the year,” which lead the rabbis to require these holidays in their seasons and thus they are not “movable feasts” like Ramadan.  In fact Rashi defines shanim (“years”) in Gen 1:14 as solar – “l’sof 365 yamim – at the end of 365 days” (both Rashi’s in Source 2).

As a result we have to synthesize the lunar and solar calendars, which we do through shitat ha’ibur – the system of intercalating (inserting) an extra month roughly once in three years, or, more accurately, in 7 years out of 19.  The reason why the holidays have not been so early for 114 years and won’t be so early again for another 76 years lies in the fact that the “leap” years do not come regularly (“once in three years”).  It is no accident that both these numbers are multiples of 19.

We might note here that the Jewish calendar is not unique.  In fact the Chinese calendar, the one we associate with “the Year of the…” is almost identical.  Chinese months start with the new moon, the regular year is twelve lunar months, but over the course of 19 years the Chinese intercalate a “leap” month seven times. Confucius clearly knew his Rashi (Source 3).  Meton, a savvy Greek mathematician/astronomer back in the 5th century BCE, made the rather remarkable discovery that 235 lunar months = 19 solar years, which is the mathematical basis of both the Jewish and Chinese calendars.

A little math here, nothing complex – a regular Jewish year, namely one with 12 lunar months, averages 354 days, eleven days shorter than the solar year.  This happens 12 times in each 19 years.  A leap year, with the extra month, comes 7 times during the cycle and averages 384 days, 19 days longer.  So when we have a “regular” year, Rosh Hashanah (or Pesach or any date) will fall roughly 11 days earlier than the previous year – in 2012 Rosh Hashanah 5773 was on Sept 17.  And when we have a “leap” year, as the current year (5774) is, the next Rosh Hashanah will be roughly 19 days later than the prior year, in our case Sept 25.  When we have two regular years in a row, we are about 22 days earlier, and the leap month, roughly every third year, will compensate for only 19 of those 22 days.  So we’re still 3 days off.

Years 3, 6, 8, 11, 14, 17 and 19 are the leap years, but they don’t fall precisely every third year.  Notice – twice in the cycle, years 8 and 19, a leap year comes after only one regular year, instead of after two.  Each of these couplets, years 7-8 and 18-19 (354 + 384 days), leave a net excess of 8 days longer than the two solar years.

The impact of this becomes clear in the tables of the 19-year cycle (Sources 4a and 4b).  Year 1 of cycle #303 (divide 5774 by 19) began in 5758 (1997/98), but for understanding our point we start in year 17, 5755 (1994/95), and break the 19-year cycle into two segments: the 11 years from year 17 through year 8, and the 8 years from year 9 to year 16.  Leap years are bolded.

Viewed this way, the ebb and flow of the years is not at all evenly spaced over the cycle, and even the phrase “roughly one in three years” seems misleading. In the eleven years from year 17 through year 8 (Source 4a) there are 5 leap years, almost one year out of two (1:2.2, to be exact), meaning that in that period we accumulate some 29 days in excess of the secular years.  Thus the latest the holidays will come are in spring of year 8 and fall of year 9 (remember – the leap month, another Adar, is added “midyear,” just before Purim and Pesach).  For example Pesach and Rosh Hashanah in 2005 fell on April 25 and October 5 respectively.  Expect very “late” holidays again in 2024, 2043 and 2062.

In the eight years from year 9 to year 16 (Source 4b), on the other hand, there are only two leap years, one in four years, which leaves us roughly 28 days short vis-à-vis the secular calendar.  So the holidays at the end of this period (spring of year 16/fall of year 17) will be at their earliest vis-à-vis the secular calendar.  This is the point in the 19-year cycle we have just come through, thus the “earliest ever” Pesach and Rosh Hashanah.  The holidays will come very early again in 2032, 2051 and 2070.  The addition of Adar Alef from Feb 1, 2014, pushes Purim and Pesach a month later than would happen in a regular year, all to keep Pesach in the “spring” (namely after the spring equinox, March 21).

The variations in the Gregorian dates of the holidays (and of the coincidence of secular and Jewish birthdays) every 19 years reflect the little fractions here and there, the Gregorian leap year, and the dechi’ot, the occasions when we delay celebrating the new moon for one reason or another (such as lo b’A’D’U Rosh, that Rosh Hashanah not fall on Sunday, Wednesday or Friday, or Molad Zaken, when the new moon occurs at or after noon).  Thus most Rosh Hodeshes do not fall on the day of the actual lunar renewal, something that does not happen in the Chinese calendar.  The Chinese are evidently not worried if Yom Kippur falls on a Friday or a Sunday (lest we need to prepare for Shabbat on Yom Kippur or vice versa).

In sum the Jewish (Metonic) LuniSolar calendar plays against the secular/Gregorian solar calendar like the sea against the beach.  The years are like the waves, falling a little ahead or a little behind on an individual basis.  The 19-year cycle is like the tide, flowing slowly forward over eleven years and then ebbing backwards for the next eight.  We have just passed “low tide” and will reach “high tide” in 2024.