The calendar that waits

Rosh Hashanah may never fall on a Sunday, a Wednesday or a Friday — and the moon itself may have to wait. A lunisolar machine with opinions about your week. ← Calendar Contraptions

Most calendars only have to agree with themselves. The Hebrew calendar has to satisfy the moon and the sun and the week — three clocks that share no common denominator — and it does it with the most opinionated rulebook in this whole corner. Months follow the moon (29 or 30 days), the year follows the sun (a thirteenth month seven years in nineteen), and then, just when the arithmetic is done, the calendar looks at the day the new year would land on and refuses to start: never a Sunday, never a Wednesday, never a Friday; not if the moon is "too old"; not if the year would come out impossible. Rosh Hashanah can start on time, wait one day, or wait two — and which of the three happens is decided by a lunar instant computed to a precision of three and a third seconds. Below: the working machine, the four rules of the wait, and the story of a calendar that used to wait for witnesses and barley before it learned to wait for arithmetic.

Read a date

The waiting machine

Every year, the molad — the mean conjunction of sun and moon — names a moment, and four rules decide whether the new year may start there. Type a Hebrew year and watch the walk.

From Hebrew back to Gregorian

Why the calendar waits

Twelve honest lunar months come to about 354.37 days; the solar year runs 365.24. The eleven-day gap is the whole problem, and Maimonides put it plainly in 1178: "By how much does the solar year exceed the lunar year? By approximately 11 days. Therefore, whenever this excess accumulates to about 30 days… one month is added and the particular year is made to consist of 13 months." Seven years in every nineteen take the extra month — years 3, 6, 8, 11, 14, 17 and 19 of the cycle, kept in the mnemonic GUCHADZaT — the same 19-year Metonic pattern the Christian computus uses for its own schematic moon. Rather than pay the eleven days somewhere hidden, this calendar pays them where everyone can see: in a month of its own, every second or third year.

But "waits" is the older half of the story. Before the calendar was fixed, the month itself waited for witnesses: two trustworthy people had to see the new crescent and testify before the court in the land of Israel could declare the month begun. And the year waited for the barley — the Tosefta lists three grounds for inserting a leap month: the ripeness of the aviv (spring barley), the fruit of the trees, and the equinox — "on two of these grounds it should be intercalated, but not on one of them alone." Passover had to fall in spring, because the Torah ties the festival month to the barley harvest. A calendar that waits for weather is a different machine from one that waits for arithmetic: this page runs the arithmetic, and the history of the older waits is at the bottom.

The arithmetic moon

Everything on this page descends from one number: the molad interval, the calendar's mean month, fixed at 29 days, 12 hours and 793 halakim — a helek being 1/1080 of an hour, exactly 3⅓ seconds. That is 29.5305941358 days. The modern value of the mean synodic month (at J2000) is 29.530588853 days — the Babylonians who set this number, around 300 BCE in their "System B", missed the truth by 0.46 seconds a month. Hipparchus adopted the value, Ptolemy printed it in the Almagest, and the rabbis built a calendar on it. In the Talmudic era, when the real mean month was slightly shorter than today's, the fit was even better — essentially perfect.

The epoch is named for its digits. The first molad — the molad tohu, "the new moon of chaos" — is reckoned as Day 2, 5 hours, 204 parts: Monday, 5 h 204 p after sunset that opened Sunday, which makes it 11:11:20 pm on Sunday, 6 October 3761 BCE in the proleptic Julian calendar (20:50:23 UTC; the "Jerusalem time" behind it is UTC+2 h 20 m 56.9 s). Tradition lands that moon about a year before creation: the world was made on 25 Elul of year 1, Adam — the story goes — saw the first crescent on the first Friday, and the count runs from there. Every molad since is that moment plus 765,433 parts per month, nothing more. A month's molad is still announced in synagogues on the Sabbath before it (except before Tishrei, which the Rosh Hashanah service covers) — a custom scholars trace only to about the 20th century. One warning survives from the practice: the announcement counts hours from sunset, and people who convert it to civil time by subtracting six get the weekday wrong a quarter of the time.

The four postponements

Now the part that makes this calendar different from every other lunar contraption: the week gets a veto. Four rules (deḥiyyot) stand between the molad and the first of Tishrei, and the machine above walks through all of them. Two are about holiness. Lo ADU — the new year may not open on Sunday, Wednesday or Friday, because that would put Yom Kippur next door to Shabbat: two days of full rest restrictions in a row with no way to cook in between (the Talmud puts it bluntly — the vegetables would wilt, and a corpse left unburied would putrefy), and Hoshana Rabbah's willow circuits cannot happen on Shabbat. Maimonides offers a second, more astronomical reason: the allowed days average out the difference between the mean and the true new moon. Either way the veto is absolute: across six thousand computed years in my check, Rosh Hashanah lands only on Monday, Tuesday, Thursday and Saturday, and the first day of Passover only on Saturday, Sunday, Tuesday and Thursday.

The other two rules are bookkeeping with teeth; they exist because a two-day postponement can strand a year at an impossible length. If a common year's molad falls on Tuesday at 9 h 204 p or later (GaTaRaD), the next new year would need a two-day delay too, making 356 days — so this start is pushed to Thursday at once. Symmetrically, if the molad after a leap year lands on Monday at 15 h 589 p or later (BeTUTeKaPoT), the leap year would shrink to 382 days — so the start moves to Tuesday. These are the rules that make 353-day and 385-day years exist at all. The result is a machine with a personality: about 39% of years start when the moon says, 47% wait one day, and 14% wait two — my own sweep of years 1–6000 gives 39.0 / 47.0 / 14.0, matching the classic figure. The medieval mind compressed all of this into a table of "four gates": the oldest surviving one was written by al-Khwarizmi in 824, and al-Biruni described the same machinery around 1000.

Years of 353 and 385

Take away the veto days and a Hebrew year would come in four natural lengths: 354 or 355 days common, 383 or 384 in a leap year. The postponements add the extremes — 353 and 385 — and every year gets a type name: chaserah (deficient, 353/383), kesidrah (regular, 354/384), shlemah (complete, 355/385). Only two months ever change length: Cheshvan can gain a day (30) and Kislev can lose one (29) — the tradition calls Cheshvan marcheshvan, read by some as "the month that marches out", waiting to see whether it fills up. Every combination of starting weekday, leap/common and type could in principle give 24 different years; only 14 survive the rules, and each has a code — the kevi'ah — like 5R7: Rosh Hashanah on day 5 (Thursday), a regular year, Passover opening on day 7 (Saturday). The commonest year is 5R7 (18.05% of years in the classical count; 18.07% in my 6,000-year sweep), the rarest 5C1 (3.31%). And the whole pattern nearly repeats every 247 years — the iggul of Rabbi Nahshon — because 13 Metonic cycles overshoot a whole number of weeks by just 905 parts, 50 minutes and 16⅔ seconds; in my check the near-repeat holds for about 96% of years. The exact repeat takes 689,472 years.

One practical casualty: the "Hebrew birthday every 19 years" is a folk approximation. A 19-year cycle runs 6,939 days 16 h 595 p, but actual cycles run 6,939 to 6,942 days depending on how the wait falls — so a birthday that returns to the same Gregorian date after 19 years can miss by a day or three, and the civil side has its own wobble (4 or 5 leap years per 19). The years themselves are written in letters: 5786 is תשפ״ו, the "minor era" that drops the thousands; the full form prefixes the millennium. One quirk of the notation: 15 and 16 are written ט״ו and ט״ז (9+6, 9+7) — never with the letters that spell two names of God.

The slow drift

The machine is exact against its own constants, and its constants are old. Nineteen Hebrew years take 6,939 days 16 h 33 m 03⅓ s; nineteen mean solar years take 6,939 days 14 h 26 m 15 s. The gap is just over two hours per cycle — about seven minutes a year, one day every 216 years. The assumed year (12 lunar months plus 7/19 of one, ≈ 365.2468 days) overestimates the tropical year (365.2422) by less than the Julian calendar did, and far more than the Gregorian's 26 seconds a year. Meanwhile the molad interval itself runs slow against today's moon by about half a second a month; the accumulated error since Talmudic times is around 97 minutes — already enough to change the computed Rosh Hashanah by a day in roughly 7% of years.

Where does the drift show? In Passover. When the calendar was fixed in the 4th century, the earliest Passover began with the first full moon after the spring equinox; that is still true in most years, but the leap years at positions 8, 11 and 19 of each cycle intercalate "early" by this standard, pushing their Passover a month late. The earliest 15 Nisan has drifted about eight days since the 4th century: over 2000–2100 the earliest first day of Passover is 26 March (in 2013 — re-derived from this page's own machine) and the latest is 25 April (2043). Left alone, the projection runs long: somewhere around year 16652 of the era (12892 CE) Passover would reach the summer solstice. Proposed rescues exist — Irv Bromberg's 353-year "rectified" cycle with 130 leap months, older proposals on 334- and 687-year cycles — and the Karaites, who never accepted the fixed calendar, still wait for barley and witnessed moons. The rabbinic calendar waits for nothing but its own arithmetic.

A brief history of waiting

In the Mishnah's world the month was declared by committee. Witnesses who had seen the crescent traveled to the court; the court tested them — by the time of Gamliel II (c. 100 CE) against a book of drawn crescents — and against calculations already known ("We calculate the new moon's birth," says the Talmud; "if it is born before midday, then certainly it will have been seen shortly before sunset"). Word went out by fire signals from mountaintop to mountaintop, until the Samaritans lit false fires and messengers took over. Communities the messengers couldn't reach in time kept festivals for two days — which is why diaspora communities still keep two days of each biblical festival. Even a lease, says the Mishnah, could not state in advance whether it ran twelve or thirteen months.

The Talmuds already know of calculators: Shmuel of Nehardea (c. 165–254) could fix festival dates by calculation, and a third-century ruling keeps Purim off Mondays and Shabbatot so Yom Kippur never touches the weekend — postponement rules in embryo. The tradition recorded by Hai Gaon (d. 1038) credits Hillel II with fixing the computed calendar "in the year 670 of the Seleucid era" — 358/359 CE — possibly because persecution was cutting off the patriarch's messengers; modern scholars doubt the story is the whole truth, noting that dates recorded in 506 and 776 don't fit the modern rules, so some arithmetic was still being settled in the Geonic era. The rules took their modern shape by the early 9th century (al-Khwarizmi describes them in 823), were contested in the famous 921/922 dispute between Aaron ben Meir and Saadia Gaon, were complete in al-Biruni's account by 1000, and were codified — with today's epoch — by Maimonides in 1178. The month names are Babylonian, adopted in exile: Nisanu, Ayaru, Simanu, Dumuzu, Abu, Ululu, Tashritu, Arakhsamna, Kislimu, Tebetu, Shabatu, Adaru. The Bible knows four older Canaanite names — Aviv, Ziv, Ethanim, Bul — and one deeper irony: Scripture numbers the months from Nisan ("this month is to you", Exodus 12:2) while the year number turns at Tishrei, the seventh month. The Mishnah lists four new years at once: 1 Nisan for kings and festivals, 1 Elul for the cattle tithe, 1 Tishrei for years and sabbaticals, 15 Shevat for trees. Which "new year" you mean depends on what you are counting — this calendar has always known that different things need different beginnings.

The arithmetic era has known real hardship: of the calendars kept by prisoners in Auschwitz, only two are known to survive, both made by women — a reminder that this machine has never been merely arithmetic to the people who keep it.

Days that start at sunset

"There was evening and there was morning" — the Hebrew day runs sunset to sunset, so every date on this page technically begins the evening before. The weekdays are simply numbered: Yom Rishon (day one, Sunday) through Yom Shishi, then Shabbat. Festival weekdays follow from the kevi'ah: a common mnemonic runs lo ADU rosh, velo BeDU pesach — Rosh Hashanah not on one-four-six (Sunday, Wednesday, Friday), Passover not on two-four-six (Monday, Wednesday, Friday). The months below carry their Babylonian originals and their lengths; Tishrei opens the year count, Nisan the festival count.

The numbers above are the civil count from Tishrei; in the biblical count Nisan is month one. In a leap year Adar I (30 days) is inserted and the original Adar follows as Adar II — Purim is kept in Adar II. Tammuz is named for a Babylonian deity; the fast of the 17th of Tammuz and the fast of the 9th of Av both mourn the Temple.

What this page doesn't do

Everything here is the fixed arithmetic — the machine that has run since roughly the Geonic era and was complete in Maimonides' day. Before about the 8th–9th century the historical calendar was observational, so this page's dates for those centuries are a projection backwards, not history. The machine can also differ by a day from modern observed practice (Karaites follow their own observations), and from the crescent any particular community actually saw. Dates are whole days with no location attached: "the day starts at sunset" is noted in the readings, but nothing here computes when the sun sets where you live. The Gregorian side is the proleptic machine used everywhere in this corner, and "today" means your device's own date. The year input reaches from 1 to 8000 — beyond that the letter-year notation runs out of patience before the arithmetic does.