Recently I got interested in the Jewish calendar. Somebody asked me about the way the Jewish holidays seem to bound around the Gregorian calendar, and after I had dazzled them with some mumbo-jumbo about the Metonic cycle, I realized that I didn’t really know much.

Wikipedia and Wikibooks have some marvelous detailed articles about the Jewish calendar, and I have some other calendrical references. I really got down into the algorithmic and numerical weeds, creating spreadsheets and writing programs to implement the rules of the Jewish calendar

So, now I know too much.

This is my attempt at the Goldilocks explanation, neither too little nor too much: the one I could start as we leave the building, and that conveys a useful amount of information by the time we reach our cars.

And then, having written it down, maybe I can stop thinking about it!

The Jewish calendar is a remarkable collision between the Sun and the Moon on the one hand – both going about their business with cosmic precision – and people on the other.

The Jewish calendar marks time by the moon, with each month
starting at the new moon. In
Hebrew this is the *molad*,
the moment when the sun, the earth, and the moon line up. Solar
eclipses take place at that moment.

In olden days, religious authorities would watch for the first sign of the new crescent moon and declare that a new month had begun. It didn’t take too many millenia for the deficiencies of that method to become obvious.

By the fourth century CE, the religious authorities switched to calculating the molad.

An arbitrary moment for the very first molad was established for
calculation purposes. Known as the *Molad Tohu*, it occurred, in the Gregorian calendar, at 11:11:33 and
⅓
seconds p.m. on Sunday, September 6, 3761 B.C.

At that time, an accurate value for the mean lunar month was known. Calculated from observations of solar eclipses, the value used for Jewish calendar calculations is within 0.6 seconds of the true value.

Our goal is to figure out the day of Rosh Hashanah, the Jewish
New Year. In the Jewish calendar, that day is the first day of the
month of Tishri. So, the absolute critical calculation, the one
from which the entire year follows, is the *Molad
of Tishri*.

Since we know the exact Molad Tohu, and because we know the exact length of the mean lunar month, we just need to know the number of months between the Molad Tohu and the month of Tishri of the year we are calculating.

In other words: given a year, how many months have elapsed since the Creation?

This is where the sun comes into the story.

The ancients knew of the Metonic cycle, an observation that 235 lunar months is almost exactly the same amount of time as 19 solar years. Nineteen years of 12 months is 228 months, which is 7 months short of the Metonic cycle’s 235.

To keep the Jewish calendar more-or-less lined up with the seasons of the sun, we add 7 additional months every Metonic cycle. An extra month is added to a year when you divide the Jewish year number by 19 and get a remainder of 0, 3, 6, 8, 11, 14, or 17.

The picture should be coming together: For a given year, use the Metonic cycle rules to come up with the number of months since the Creation. Multiply by the average length of the lunar month. Add that to the Molad Tohu; the result is the Molad of Tishri; the day of the molad is the day of Rosh Hashanah.

We now apply the first of four Rules of Postponement: If the molad is after noon, postpone Rosh Hashanah one day. (This keeps the evening of Rosh Hashanah close to the actual observed new moon.)

In a sensible world, we’d be done at this point. That calculation results in common years of 354 and 355 days, and leap years of 383 and 384 days. Easy peasy.

But, no. This is where the sun and the moon collide with people.

You see, if Rosh Hashanah fell on Wednesday or Friday, then Yom Kippur, nine days later, would fall on Friday or Sunday. Since no work can be done on Saturday, and likewise no work can be done on Yom Kippur, there would be a two-day period where no work can be done. This was regarded as too big a burden.

In addition, should Rosh Hashanah fall on Sunday, then Hoshana Rabbah, the seventh day of Sukkot, would fall on Saturday, and the tradition of carrying willow wands seven times around the synagogue would be forbidden.

This leads to the second Rule of Postponement: When the calculated day of Rosh Hashanah lands on Wednesday, Friday, or Sunday, postpone Rosh Hashanah to the following day.

The second rule leads to *eight* different year lengths: 353, 354, 355, 356, 382, 383, 384, and
385 days. The years of 356 and 382 days are rare, but they are
there.

This was regarded as too untidy.

The result is that there are two additional Rules of Postponement. When a year would have 382 days, it turns out that the following year would start on Monday. (That’s how the molad math works out.) So, that’s pushed off a day to Tuesday, resulting in a 383 day year.

And when a year would have 356 days, it turns out to start on Tuesday. So, that year gets shortened to 354 days by moving Rosh Hashanah from Tuesday to Thursday.

The postponement rules thus result in years of common years of
353, 354, 355 days, and leap years of 383, 384, 385 days. The
adjustment is made by modifying the number of days in the months
of Cheshvan and Kislev. In a *Regular*
year of 354/384 days, those months have 29 and 30 days
respectively. In a *Deficient* year of 353/383 days, Kislev gets shortened to 29 days. In
a *Complete* year of
355/385 days, Cheshvan gets bumped up to 30 days.

So, there you have it. It’s not the kind of thing you can do in your head. But I feel better now that I know why and how the leap months are inserted, how the Molad of Tishri is calculated, and why the varying number of days in a year comes from Rosh Hashanah having to start on Monday, Tuesday, Thursday, or Saturday.