The Jewish Calendar

a study of its elements

Introduction

The Jewish calendar is a lunar calendar, adjusted so that: the first month of the year is always in the spring, the days begin and end at sunset, and the months follow the phases of the moon. The phases of the moon are a complete cycle of the apparitions of the moon, from waxing crescent to first quarter, to full moon, to last quarter and to waning crescent, as illustrated by the animated image on this page.

Between the waning cresent at the end of the month and the waxing wrescent at the beginning of the next month, there is a period of time in which the moon can not be seen. During this period of time a new lunar cycle begins all over again, and is called the time of the new moon, or the time when the cycle of the phases of the moon begins anew.

The determination of this time of the new moon is fraught with difficulty, because the moon is actually invisible during this period of time. As a result, there have been many different means employed to determine the time of the new moon. This determination is important, because it defines the beginning of the lunar month, the beginning of the cycle of the visible phases of the moon.

The New Moon

In ancient times, the new moon was determined by several different methods which became more reliable as knowledge of astronomy increased. By the time of the prophet Samuel, it was known that the lunar month was between 29 and 30 days long; so it became possible to predict within one day the time of the new moon. Visual observation was used around the predicted time of the new moon to decide if a month should end at 29 days or at 30 days. Each day was defined to begin and end at sunset.

For example, if at the end of the 29th day a faint waxing crescent could not be seen in the west just after sunset, then the end of the month was delayed (i.e., postponed) by one more day, making the month 30 days in length. Otherwise, if at the end of 29 days at sunset a faint crescent could be seen, then the current month was terminated at 29 days and the next day was declared to belong to the new month. That sighting of the faint crescent meant that a new moon had occured, because the cycle of lunar phases had started all over again.

This method of visual determination required fair weather. If the sky was cloudy, then it would not be possible to sight the new moon and decide if the passing month should be 29 or 30 days long. As a result, a special rule was formulated to deal with the problem of cloudy skies. This rule stipulated that there could be no more than eight 30-day months in a year. That way, if bad weather occured during the time of all the new moons in a year, then at least four of those months would have to be 29-day months. The reason for this was so that the average length of a calendar month would be close to 29.5 days.

If the average length of the calendar months deviated too much from 29.5 days, then some months might start several days after the waxing crescent. That would defeat the whole purpose of a lunar calendar, which is for the calendar months to follow the phases of the moon.

Because of the necessity to keep the months close the the cycle of the lunary phases, another rule was established similar to the first rule. This second rule stipulated that there could be no more than eight 29-day months in a year. That way, if unusually good weather and keen eyesight resulted in terminating eight months at 29 days, then no matter what happened in the year after that, all the months would have to be 30 days long, until the beginning of the next year. These two rules kept the average length of the calendar months close to 29.5 days.

Another way of summarizing the above two rules, is to say that in ancient times there could be no more than eight 29-day months, and no less than four 29-day months. All the remaining months of the year would be 30 days in length.

The Postponements

By the time of the 9th century C.E., these two rules were refined by the use of advanced astronomical knowledge. Although this resulted in a more complex calendar, the result has been a more accurate one. No other modifications have been made to the calendar since that time. The ancient rules were replaced with the new requirements that there can be no more than seven 29-day months in a year and no less than five 29-day months in a year. The increased precision with which these two rules have been stipulated is the result of advanced astronomical knowledge. The exact method by which it is determined if a year should have seven 29-day months, six 29-day months, or five 29-day months is stipulated by what are known as the postponement rules, which will be covered in a later section.

The Intercalations

One other aspect of the calendar in ancient times was harmonizing the calendar with the seasons, so that the first month always occurs in the spring. The ancient Babylonians determined that the actual average length of a lunar cycle is 29.53 days, so when it became necessary to adjust the the lunar calendar to the seasons, the solution was to add a month of 30 days to the end of the year, thus moving the start of the new year into the spring. The reason for this becomes appearent when you compare the length of 12 lunar months (354 days = six 30-day months + six 29-day months) to the length of the seasons (365.25 days). If a calendar had only 12 lunar months, in just three years the first month of the year would be in the winter, rather than the spring. By adding one extra 30-day month approximately every three years, it was possible to keep the first month in the spring.

A special rule was adopted in ancient times regarding when the 30-day leap month should be inserted into the calendar. It was decided to insert it just before the last month of the year, between the 11th month and the 12th month. This same ancient tradition continues today in the Jewish calendar.

By the time of the 9th century C.E., a set of rules were established to determine exactly when to insert a leap month; and it still is exactly as it was in ancient times -- always a 30-day month, and always inserted between the 11th and 12th months. The insertion of a leap month is called an intercalation (pronounced as in-turk-uhl-a-shun).

Thus, there are only two sets of rules used to adjust the calendar: a set of postponement rules (which determine how many 29-day months there will be in a year) and a set of intercalation rules (which determine when to insert a leap month of 30 days). I will take up both these topics in later sections.

The Calendar

Unless one or both of the two sets of rules mentioned above are invoked, the Jewish calendar simply has six 30-day months, and six 29-day months, as follows:

Note that the average length of all the calendar months is 29.5 days, which is very close to the actaul period of the lunar phases. Thus, if Nisan starts out in the spring within one day of the astronomical new moon, then all of the calendar months will follow the lunar phases to within a day. This is as close as a calendar based on a 24-hour day can get to the true lunar phases, because a month may have only 29 or 30 days, not an arbitrary fraction of days, which would be necessary if one were to track the lunar phases exactly. If a day were only 12 hours long, instead of 24 hours, then it would be possible to track the lunar phases more precisely, since an astronomical phase can occur at anytime of the day or night; but, a day can only begin and end at sunset.

Thus, there are three competing criteria used to determine the Jewish calendar: the day (defined from sunset to sunset), the month (defined by the cylce of lunar phases) and the year (defined from spring equinox to spring equinox). It is not possible to harmonize all of these 3 things exactly, but the Jewish calendar comes as close as possible to doing this.

The next two sections explain the adjustments that are necessary to keep the day, the month and the year harmonized.

Determination of the Leap Year

As mentioned above, 12 lunar months are about 11 days short of a solar year. A solar year is the cycle of the seasons, from spring to summer, to fall, to winter, and back to spring again. Just as the phases of the moon complete a full cycle every lunar month, so the seasons complete a full cycle every year. So, if we want to keep the first lunar month in the spring, we will need to add a leap month approximately every three years. This is exactly the reason why the Roman calendar has a leap day added every 4 years. It turns out that the cycle of the seasons, from spring to spring, is 365 and 1/4 days. Therefore, we need to add one day every four years to the Roman calendar in order to keep the start of the Roman calendar in the dead of winter. Otherwise, in just 240 years, the Roman calendar would be starting January 1 in the fall, rather than in the winter.

It turns out that the lunar cycle averages 29.53 days. If we divide 365.25 days by 29.53 days, we find out that there are almost exactly 12 and 7/19 months in a year. That means that in 19 years there will be 7 more months than the usual 12. Or, to put it another way, we need to add seven leap months in 19 years, in order to keep the first lunar month in the Spring.

Of course, we would have to spread these seven extra months evenly over the 19 years, in order to keep the first month of every year as close to spring as possible. This is a little different from the Roman calendar, where only one day has to be added every four years. But, with the lunar calendar, we need to add seven months every 19 years; so, we have to make a decision where to put those seven months. We know we need to keep them spread out evenly, otherwise we would have a wild variation between the first month and the Spring.

Since there are 4 seasons in a year, and 12 lunar months in an ordinary year, there are going to be 3 lunar months per season. But, in a leap year one of those seasons is going to have four months, since the leap year has 13 months. This turns out to be the key in determining where to place the seven extra months. We need to find out what years have four lunar months in the winter.

Since Passover and Unleavened Bread begin at the time of the full Moon in the spring, we need to determine which years have four full moons in the winter. Those years will be the years when we need to add a leap month, because we require the Passover (time of the full moon) to be in the spring.

Here is a table of 19 years, showing the number of full Moons in the winter season. Those years with four full moons during the winter season are the years where we need to put a leap month. Note that there are seven such years.

As can be seen from the above table, unless a full moon occurs within week or less of the winter solstice, there will be only three full moons during the winter season; but, if the full moon does occur within a week of the winter solstice, then that year will be a leap year. The basic pattern is such that there are four full moons squeezed into the winter season, rather than the usual three.

The patterns are nearly identical in adjacent 19-year cycles, so it doesn't matter which year you choose as your starting year, as long as you are consistent in repeating the same pattern of leap years in each 19-year cycle. In the above table, I started with 1981. In the 19-year cycle starting in 1981 there were seven leap years in years 3,5,8,11,14,16, and 19 of the cycle.

However, the Jewish calendar, for historical reasons, starts its 19-year cycles in the year 3761 B.C.E. That means that 1981 was the 4th year of a 19-year cycle, which explains the meaning of the last column above. From the above table, it should be clear that the years 3,6,8,11,14,17 and 19 in the Jewish 19-year cycle (which starts in 3761 B.C.E) are leap years.

Thus, to determine if a year is a leap year, one has to find the location of the year in the 19-year cycle. If that year is any one of the years 3,6,8,11,14,17 or 19, then that year is a leap year. In a leap year, one extra month of 30 days is inserted between the 11th and 12th months.

For example, 1994 was a leap year and here are the lengths of the months for 1994:

Note that the extra 30-day month has been inserted between Shebat and Adar II (which has 29 days and is the last month of the year.)

The next section further explains the means by which it is determined how many 30-day months and how many 29-day months the year will have.

The Postponement Rules

As mentioned above, in ancient times there could be as few as four 29-day months, and as many as eight 29-day months in a year. But, ever since the 9th Century, that variation has been reduced by one month, so that now there can be as few as five 29-day months and as many as seven 29-day months -- but, no more and no less. If it were not for the postponement rules, the calendar would vary much more in the number of 29-day months allowed.

There has been much debate about the postponement rules for many centuries because their history is obscure. Some have argued that they have only a religous basis and have nothing to do with astronomy. Others have argued that they have everything to do with astronomy and nothing to do with religion. However, one of the greatest Jewish scholars, Maimonides, believed that they had an astronomical basis. From the astronomical facts, it becomes clear that the postponement rules, whether they were motivated by astronomy or religion or both, clearly serve an astronomical purpose.

Before looking at the postponement rules in detail, it is necessary first to understand what they accomplish. As mentioned earlier, they determine if a year should have five, six or seven 29-day months. If no variation at all were allowed in the number of 29-day months, then each year would have six 29-day months and six 30-day months. A leap year would have six 29-day months and seven 30-day months.

That means that in every 19-year cycle, there would be 12 years of 354 days, and 7 years of 384 days. That is a total of 6936 days. However, 19 years, counting from spring to spring equinox, is actually 6939.6 days. That means that some years simply can't have six 29-day months and six 30-day months; otherwise, we would be off by nearly four days every 19 years, and in just 280 years Passover would be starting in winter, rather than the spring. Cleearly, there has to be some variation in the number of 29-day months, or the calendar will not follow the seasons. The calendar would be short by four days every 19-years, and it would be cumulative.

The first postponement rule attempts to make this correction for the missing four days. Since days begin and end at sunset, and since the astronomical elements of the Jewish calendar are based on ancient astronomy, which measured astronomical events from the mean sun at noon, there are 6 mean hours from mean noon to mean sunset. If a new moon is calculated to fall during that part of the day between noon and sunset, then the length of the lunar year is increased from 354 days to 355 days by adding 1 day to the 8th month (which ordinarily would have 29 days). This reduces the number of 29-day months from six down to five 29-day months. The change is made in the 8th month in order not to affect the Holy Days (annual Sabbaths in addition to the weekly Sabbath), which occur only during the first seven months of the year.

The first postponement rule occurs approximately once every four years, because a new moon can occur at any time of day or night, and since 6 hours is one quarter of a day, there is 1 chance in 4 that a new moon will fall in the interval between noon and sunset.

Since there is only one new moon that needs to be checked each year to determine the length of the year, there are two logical choices for the new moon to be selected for determining the length of the year. One choice is the first new moon of the year, at the very end of the 12th or 13th month. Another choice would be the last month of the holy day seasons. Since there are in fact a fixed number of days, always, in the first seven months of the year, the choice of which of those seven months to select for determining the length of the year is somewhat arbitrary. Since the new moon of the 7th month, the Feast of Trumpets, is the only new moon of those seven months that is a Holy Day (annual Sabbath), it is the 7th month that is selected to determine the length of the year.

Thus, the first postponement rule is: if the new moon of the 7th month falls between noon and sunset, then the 8th month of the previ ous year is increased to 30 days. Note that the extra day is added in the previous year, not in the month after Trumpets. This is important to notice, because the aim is to keep the new moon of Trumpets in one year from postponing the Trumpets for the following year. This very fact is often little understood by those who have argued against the postponement rules. They have argued that the postponement in one year causes a postponement in the next year; but they are wrong. The Jewish calendar has very strict rules to prevent this from ever happening!

However, as can be seen, adding one extra day to the length of the year approximately once in four years is too much of a correction. That means that there must also some years in which either the first rule is cancelled, or there must be some neighboring year shortened by one day. The remaining postponement rules accomplish just that.

As mentioned before, there has been much debate as to whether the remaining postponements needed to be specified for religious or for astronomical reasons, or if they are the only means for either cancelling the first postponement or shortening a neighboring year by 1 day, in order to reduce the over-corrections caused by the first rule. In any case, the remaining rules either cancel the first rule, or they cause the year to be shortened by 1 day. This is done by increasing the the number of 29-day months from six to seven 29-day months, by subtracting one day from the 9th month.

It can also happen that the remaining rules can lengthen a year, so two additional rules were also adopted to guarantee that the year can never vary by more than one day from the normal length of 354 days in an ordinary year, or 384 days in a leap year.