Why Jewish holidays "move"

What it is

The Jewish calendar is luni-solar, and since late antiquity it has been run by calculation rather than by watching the sky. ¹ The calendar keeps two clocks at once, and neither divides evenly into the other.

The month belongs to the moon. Jewish calendar arithmetic assigns one lunation a length of 29 days, 12 hours, and 793 parts. ² The measured value is 29 days, 12 hours, 44 minutes, and 2.841 seconds. ² For any purpose a calendar has, those are the same number.

The year belongs to the sun. The classical calculation uses a solar year of 365 days and six hours, though Maimonides records that the sages argued about the true figure and that some held it shorter. ⁶

Twelve lunar months come up short of that. In the numbers Maimonides sets out, the yearly shortfall is 10 days, 21 hours, and 204 parts, call it eleven days. ⁶ Leave it alone and Passover walks backward out of spring within a decade. Deuteronomy 16:1 does not permit this. The verse commands "Observe the month of Abib," and Abib is the spring. ⁷

So the calendar swallows the difference in one gulp. Seven years out of every nineteen get a thirteenth month. Maimonides fixes those leap years as the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th of the cycle, and states the purpose without ornament: it is so that Pesach always falls after the vernal equinox. ³ Twelve ordinary years plus seven leap years add up to nineteen solar years. ³ The month that gets added is an Adar. In a leap year the first Adar has 30 days and the second has 29. ²⁵

Everything else is nailed down. Tishrei always has 30 days and Tevet always 29, and from Shevat onward the months alternate: Shevat 30, Adar 29, Nisan 30, Iyar 29, Sivan 30, Tammuz 29, Av 30, Elul 29. ⁵ Two months are deliberately left loose. Marcheshvan and Kislev can both be full, both be short, or split it one each. ⁵ That slack is why a common year can run 353, 354, or 355 days, and a leap year 383, 384, or 385. ⁵

The slack exists to absorb the postponement rules. Rosh Chodesh Tishrei, which is to say Rosh Hashanah, is never set on a Sunday, a Wednesday, or a Friday. ⁴ Maimonides gives an astronomical reason: the arithmetic tracks the mean conjunction of sun and moon rather than the true one, and the postponements pull the date closer to the real event. ⁴ The commentary preserves the older practical reasons alongside it. Were Rosh Hashanah to fall on a Wednesday or Friday, Yom Kippur would land adjacent to Shabbat and produce two consecutive days on which the dead could not be buried; were it to fall on a Sunday, Hoshana Rabbah would fall on Shabbat. ⁴

By tradition the whole system was fixed by Hillel II in 359 CE, the year 4119 from creation. ¹² The Jewish Encyclopedia notes that some scholars have dated the fixing as late as 500 CE. ² The change was from a court in the land of Israel declaring each new month to a rule set that any Jew anywhere could run on his own. ¹² That calendar is today the official calendar of the State of Israel. ¹

Why it still matters

Passover begins on Nisan 15 every year, without exception. ¹ The Gregorian date underneath it changes every year.

The mechanism is arithmetic you can do at a kitchen table. A common Jewish year runs about eleven days shorter than a solar year, ⁶ so a holiday lands roughly eleven days earlier on the civil calendar than it did last year. Do that twice and you are three weeks early. Then a leap year arrives, a whole month goes in, and the holiday jumps back to about three weeks later than the year before. After nineteen years the pattern closes and begins again. ³

The holiday stays in its season, because keeping it there is the entire reason the leap month exists. ³ Pesach stays after the vernal equinox by design. ³ Rosh Hashanah stays in Tishrei, the month whose equinox Maimonides marks as the sun entering Libra. ⁶ Chanukah is exactly where it always is. December is what moves.

The postponement rules exist precisely because the mean conjunction is not the true one. ⁴ The solar year in the calculation is a round 365 days and six hours, over a dispute the sages themselves did not close. ⁶ This was a trade, and a deliberate one. The calendar gave up some astronomical precision in exchange for being portable and predictable.

That is why a Jewish date printed twenty years in advance is correct rather than provisional, and why the calculation returns the same answer wherever it is run. ¹

The words you'll hear

Luni-solar. A calendar whose months track the moon while the year is kept in step with the sun. The Jewish calendar has run this way by calculation, not observation, since late antiquity. ¹

Molad. The conjunction of sun and moon that begins a month in the calculation, reckoned at intervals of 29 days, 12 hours, and 793 parts. ²

The nineteen-year cycle. Twelve common years and seven leap years, which together come out to nineteen solar years. The leap years are the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th. ³

Adar I and Adar II. The doubled month in a leap year. The first has 30 days, the second has 29. ²⁵

Full and lacking months. A month of 30 days is full, one of 29 is lacking. Marcheshvan and Kislev are the two that can go either way, which is how a common year lands on 353, 354, or 355 days. ⁵

Lo ADU rosh. The rule that Rosh Hashanah is never set on a Sunday, Wednesday, or Friday, named for the Hebrew letters of those three days. ⁴

Tekufah. A season point in the solar year. Maimonides names four, at Nisan, Tammuz, Tishrei, and Tevet, spaced 91 days and seven and a half hours apart. ⁶

Anno Mundi. Counting from creation. The calendar's traditional fixing under Hillel II is dated to AM 4119, which is 359 CE. ¹²