The Jewish calendar

The Jewish calendar is lunisolar. It was originally an empirical calendar, based on careful observations of the moon. The problem with this is that precise moon observations depend on location! In the 4^th century Rabbi Hillel II established a calendar based on a theoretical moon, with a precisely defined period of lunation. As we shall see, the theoretical moon has a period very close to that of the actual moon.

To describe this, we need a new unit:

1 day = 24 hours

1 hour = 1080 halakim.

Notice that there are 1080/60=18 chalakim per minute.

The theoretical moon has a period of exactly 29 days, 12 hours and 793 halakim. In practice, we will write such numbers as 29, 12, 793. This period is measured from dark moon to dark moon; the theoretical moment that this occurs is called the molod. The decimal version is 29.5305941, which is very close to the modern average figure of 29.530589. Thus the Jewish calendar will be in synch with the moon!

This calendar is based on the Metonic cycle, a lunar calendar invented by the Greek astronomer Meton (ca. 432 BC). This is a 19 year cycle, in which there are 7 years with 13 months, called the embolismic years. They take place in the years 3, 6, 8, 11, 14, 17, 19. We then have (7)13+12(12)=235 months during the cycle.

We can compute 235 times 29, 12, 793 to figure the exact length of the cycle. We obtain 6939, 16, 595. In decimal form this is 6939.6896. If we divide this by the number 19 of years, we get 365.24682, a very accurate estimate for the year. Thus, the Jewish calendar will on average be in synch with the sun!

Here are the names of the months in the Jewish year:

Tishri	30 days
Heshvan	29 or 30 days
Kislev	30 or 29 days
Tevet	29 days
Shevat	30 days
Adar intercalated	30 days
Adar	29 days
Nisan	30 days
Iyyar	29 days
Sivan	30 days
Tammuz	29 days
Av	30 days
Elul	29 days

Now what is crucial for the Jewish calendar is that each year begins at an appropriate moment so that holy days occur on an acceptable day of the week. The holy days are specified by fixed dates (day and month), while the days of the week (the traditional seven) cycle in the usual way. Traditions regarding both the holy days and the Sabbath (Saturday) must be respected, and a given year is adjusted as to its length to ensure that this happens. In particular, the calendar should not require two days of fasting in a row. The traditional approach to meeting these requirements is a series of complicated rules called the postponements; we will instead describe this in terms of a diagram. These rules are called postponements, because we compute a theoretical start for the year, and then postpone a day or two if necessary.

To be more specific, the first day of the year is Tishri 1 – a day near the fall equinox. This is called Rosh Hashanah. It cannot occur on a Wednesday, Friday or Sunday. Consequently, a year cannot end on a Tuesday, Thursday or Saturday. We meet these requirements by adjusting the lengths of two of the months (as you can see in the table above, the months adjusted are Heshvan and Kislev). This leads to six possibilities: ordinary years of length 353, 354, 355, and embolismic years of length 383, 384, 385. We call the shorter years of a given type deficient, the longer years abundant, and the other years regular.

The Jewish calendar begins at the moment of creation, which according to this tradition occurs on Saturday, October 5, 3761 BC[1], at 6 p.m. Because the Jewish day begins at 6 p.m., the first day is really Sunday, October 6. The first molod takes place at C = 1, 5,204 (this would be about 11 p.m. on Sunday, October 6^th by our reckoning – or else 5 hours into Monday according to Jewish reckoning). The crucial calculation boils down to determining when the first molod in a given year occurs. By adjusting the length of the years, we can ensure that no forbidden configurations occur.

Let’s do the math! We will assume that we are working with Y, the Jewish calendar year, the number of years since creation. Notice that since the Jewish year begins in the fall, it will always begin in one Julian (or Gregorian) year, and end in another. The translation is easy though – we just add 3761. For example, in this Gregorian year 2006, the Jewish year beginning this coming fall will be 3761+2006=5767.

• We need first to determine whether or not year Y is an embolismic year. To do this we compute r, which is Y modulo 19. If r=3, 6, 8, 11, 14, 17, 19 then the year is embolismic; otherwise, it is regular. The regular years will turn out to have 353, 354, or 355 days; the embolismic years will have 383, 384, 385 days. We now have Y=19Q+r, where Q is a count of the completed 19 year Metonic cycles. Now some of the r-1 years in the current cycle will be embolismic (that's a) and the rest (that's b) will be the regular years. Note that r-1=a+b.

• We now compute N, the number of lunations since the first one. This is easy. It is N=235Q+13a+12b. This is true because each complete Metonic cycle has exactly 235 months in it; each embolismic year has 13 months, and each regular year has 12 months.

• We now compute T, the total time elapsed since creation, until the first dark moon of year Y. This is in principle easy: T=C+LN. That is, we take the moment of creation (T), and add to that the number of lunations (N) times how long each lunation lasts (L). In practice, the multiplication LN is quite painful arithmetic, because we have to change halakim into hours, and hours into days. Fortunately, I have provided you with a calculator in Excel (Rosh.xls) to perform this. You can find it on the course drive, alongside Euclid.

• We now take T and throw away weeks, because we are really only interested in the day of the week of the dark moon. In practice, reduce the number of days in T, modulo 7.

• We now take the result and look at the circular chart in the Ascher Handout. It will tell you exactly what the day of the week of Rosh Hashanah is, by passing to the inside circle. This chart encodes efficiently the rules of postponement that rabbinical scholars would have used. There is a basic shift of six hours (the nominal time between the molod and the appearance of the new moon), and then a shift of a day or two, to ensure that Rosh Hashanah falls on a permitted day of the week (in the current year, the next year and the previous year). There are some apparently bizarre special cases in the chart, but we can figure out when they apply, because we know whether or not Y and its neighbors are embolismic. We will encode the day D of the week of Rosh Hashanah as an integer modulo 7 (0 is Sunday, etc.)

• Now compute the day of the week for Rosh Hashanah in the next year. This is pretty easy. We take the time T for the dark moon for year Y, and then add either 13L if Y is embolismic, or else 12L if Y is regular. Throwing away weeks again, we obtain the time of the week for the first dark moon for Y+1. We call the day of the week D' for this next year.

• If Y is regular, then D'-D (modulo 7) will be 3 if Y has 353 days, 4 if Y has 354 days, and 5 if Y has 355 days. This determines which regular year to use. Similarly, if Y is embolismic, then D'-D (modulo 7) will be 5 if Y has 383 days, 6 if Y has 384 days, and 0 if Y has 385 days.

• This information regarding possibilities for D and D’ can actually be encoded efficiently in a chart below, which is essentially an addition table modulo 7. As we have observed earlier, Rosh Hashanah can only occur on Monday, Tuesday, Thursday and Saturday. That is, D must be congruent to 1, 2, 4, 6 modulo 7. But this must also be true for the next year. That, is D’ must also be congruent to one of these four cases, modulo 7. We can thus look at the movement from next year to the next, according to how many days the current year has. We get the following chart:

D	353≡3	354≡4	355≡5	383≡5	384≡6	385≡0
1	4	5	6	6	0	1
2	5	6	0	0	1	2
4	0	1	2	2	3	4
6	2	3	4	4	5	6

We have shaded the impossible cases. This leaves only 14 possible combinations!

• We have concluded exactly what kind of year Y is, and the day of the week on which it begins. This is all we need to know the full calendar for the Jewish year Y.

Here's the summary for the year Y=5765.

1. Q=303 and r=8; a=2 and b=5.
2. N=71291.
3. NL=2105265,14,83.
4. When we throw away weeks, we get 1,14,83.
5. When we add C we get 2,19,287. Consulting the chart gives us Thursday (the day of the week of Rosh Hashanah), and so D=4.
6. Now Y is embolismic, and so we compute 2,19,287+13L. This turns out to be 1,16,876. Consulting the chart gives Tuesday (the day of the week of Rosh Hashanah for 5766; D'=2.
7. Now D'-D=2-4=5, modulo 7. Among the possible embolismic years, only the 383 day year will work. Thus both months H and K will have 29 days. Alternatively, we could look at the table above. We have D=4, and we need an embolismic year. Notice that D’=2 appears in the column corresponding to the 383 day year.

Homework: Perform these calculations for the Jewish year 5426 – this is the year 1665 in our calendar. Then choose an AD year of your choice in the Gregorian calendar. Translate this to the appropriate Jewish year, and then compute the year type and day for Tishri 1.

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[1] This is of course a Julian date, translated back in time.