# Mathematics of the Jewish Calendar/Printable version

The current, editable version of this book is available in Wikibooks, the open-content textbooks collection, at
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# Basic principles

It is easy to understand the mathematical rules behind the Jewish calendar. Maimonides says that they are "such that even schoolchildren can learn and fully grasp them in three or four days".
The Jewish calendar is a vital part of Jewish law. The first command given to Moses, before even those relating to Pesach, was "This month shall be for you the beginning of the months", the command that underlies the calendar. Rashi, commenting on this, said that the Torah should have begun with this law because what went before is unimportant. For the unity of the Jews, there must be one and only one calendar, so there is never any dispute about when the festivals will occur.
The Jewish calendar is luni-solar, governed by both the moon and the sun. The months are regulated by the phases of the Moon, but the year always begins at roughly the same season as determined by the Sun. This differs both from the Gregorian calendar used in the UK, the USA and many other countries, where the months ignore the moon, and the Islamic calendar, where the years ignore the sun. However, there are other luni-solar calendars in use, such as the Chinese one.
G-d, when he created the Sun and Moon, said that they would be for signs, mo'adim, days and years. Mo'adim is usually translated here as seasons, but elsewhere it is used to mean festivals. Thus it was ordained at the creation that both the Sun and the Moon should be used to fix the calendar. Again, in the Psalms (104:19) it says "Who appoints the Moon for mo'adim; the Sun knows its setting", so it is the Moon that determines the exact dates of festivals.
The present Jewish calendar does not depend on observations, but is entirely based on calculations. The conversion to a calculated calendar was (according to a tradition recorded by Rabbi Hai Gaon in the 11th century CE) made by Rabbi Hillel II (4th century CE), and exactly the present rules were already regarded as old-established by Rabbi Saadia Gaon (10th century CE). As a result, anyone who knows the rules can calculate the calendar. The Talmud rules that except in dire emergency, all decisions about the calendar must be taken in Israel. However, the principle now adopted is that anyone performing these calculations is using the same rules as in Israel, hence will always get the same result (assuming there are no calculation errors). Thus, you are not making decisions, only finding out what decisions are being made in Israel by the people there who are using the same rules.
As the average interval between consecutive new moons is just over 29½ days, a month that keeps roughly in step with the Moon should have 29 or 30 days, and broadly we would expect 29 and 30 day months to alternate. A year of 12 months (an ordinary year) would have on average 354 days; this is 11 days too short, so we need a leap year of 13 months every two or three years to keep the calendar in step with the year determined by the sun, which currently averages about 365.2422 days.
The nominal aim is to ensure that the first day of the Festival of Pesach (Passover) falls on the first Full Moon after the vernal equinox, following the Biblical rule "Observe the month of Abib (Spring), and keep the Passover unto the Lord thy God". Whether this is achieved will be discussed later.
The day is supposed to last from 6pm to 6pm throughout the year. This differs from the usual practice in Jewish law, where the start of the day is related to the Sun (so Shabbat begins just before sunset on Friday and ends at the end of twilight on Saturday) so is earlier in Winter than in Summer and depends on each community's longitude and latitude. The day is divided into 24 hours, so hour 0 lasts from 6pm to 7pm and hour 23 from 5pm to 6pm. Hours 0 to 5 precede midnight so are regarded as the previous day in the Gregorian calendar.
When doing detailed calculations, it is customary to work with a unit of time called the chalak (plural chalakim). There are 1080 chalakim in an hour, so one chalak is 3600/1080 = 3⅓ seconds.
The assumed time of a new moon is called the molad. The interval between successive molads is assumed to be 29 days 12 hours 793 chalakim, or 29 days 12 hours 44 minutes 3⅓ seconds.

## References

* ↑ Gandz, Solomon (1956)
* ↑ Exodus, chap 12 verse 2
* ↑ Rashi's commentary ad loc.
* ↑ Genesis, chap 1
* ↑ Stern, Sacha (2001)
* ↑ Stern, Sacha (2001)
* ↑ Deuteronomy 16:1

# The 19 year cycle

In the Jewish calendar, it is usual to give the year as A.M. (Anno Mundi, Year of the World). This is calculated from the data given in the Tanach (Old Testament); the calculation is given in a book called the Seder Olam (Order of the World). The six days of creation described in Genesis chapter 1 have to be in some year, so they are assumed to be the last six days of Year 1. The first Shabbat (Saturday), described in Genesis chapter 2, was also the first day (Rosh Hashanah) of year 2AM.
An earlier tradition was that the first five days of creation were the last five days of Year 1 and the Friday was the first day of year 2AM. However, this is not in accordance with the present calendar rules, which forbid the year to start on Sunday, Wednesday or Friday.
According to the Jewish calculation, creation was in the autumn of 3761 BCE, slightly later than the traditional Christian figure of 4004 BCE. Thus the year 5701 began in 1940 in the Gregorian calendar and ended in 1941. Using that as a starting point, it is easy to convert years between the two calendars, remembering always that the new years do not coincide so a year in one calendar will always overlap with two years in the other.
A year is a leap year if the remainder (in AM) on division by 19 is 3, 6, 8, 11, 14, 17 or 0. Thus 5703, 5706, 5708, 5711, 5714, 5717 and 5719 were leap years with 13 months each, but not the years between, which only had 12 months. This rule is based on the Metonic cycle, which assumes that 19 years exactly equal 235 lunar months.
The 19 year cycle is quite accurate. Assuming 365.24219 days for the tropical year, to do better with a fixed cycle would require a cycle of 182 years containing 2,251 months. However, it is not perfect; see The long-term accuracy of the calendar.

## Length of a 19 year cycle

The total time between molads 19 years apart is 235 months, or 6939 days, 16 hours, and 595 chalakim. After the postponements are applied, however, a 19-year cycle may have 6939, 6940 or 6941 days. It is theoretically possible for it to have 6942 days, but this is very rare, as will be discussed later. However, 19 consecutive years, some in one cycle and some in the next, may have only 6938 days, for example 5719-5737 and 5790-5808, and 19 years of total length 6942 days are more common than exact cycles of this length.
Two days on the same date but exactly 19 years apart may thus be 6938, 6939, 6940, 6941 or 6942 days apart. In the Julian or Gregorian calendars, two days on the same date but exactly 19 years apart may be 6939 or 6940 days apart; they may be only 6938 days apart in the Gregorian calendar if the period includes say 1900 or 2100, which are not leap years. Thus, your 19th birthday by the two calendars is not necessarily on the same day, but may be one or two days different, and in rare cases the Jewish date may be three or possibly four days after the civil one.

# The lengths of the months

Although the New Year occurs at the start of the month of Tishri (that is the traditional spelling, although increasingly the spelling Tishrei is preferred), the Bible commands that Nissan should be regarded as the first month. ("This month shall be for you the beginning of months: it shall be the first month of the year for you." See Exodus Chapter 12 verse 2.)
As a general rule, the months are alternately 30 and 29 days. As will be explained later, two months (Cheshvan and Kislev) are of variable length. Further, in leap years, an extra month called Adar Rishon (First Adar) is inserted before the month of Adar, which is then renamed Adar Sheni (Second Adar). The months are as follows:
    Nisan, 30 days
    Iyar, 29 days
    Sivan, 30 days
    Tammuz, 29 days
    Av (or Menachem Av), 30 days
    Ellul, 29 days
    Tishri (or Tishrei), 30 days
    Cheshvan (or Marcheshvan), 29 days; sometimes 30 days
    Kislev, 30 days; sometimes 29 days
    Tevet, 29 days
    Shevat, 30 days
    [Adar Rishon (only in leap years), 30 days]
    Adar (called Adar Sheni in leap years), 29 days
In a regular (כסדרן) year, Cheshvan has 29 days and Kislev has 30 — consistent with the pattern of the lengths of the other months, and there are a total of 354 days (384 in a leap year). If Cheshvan and Kislev both have 30 days, the year is called complete (מלא) and there are 355 days (385 in a leap year). If Cheshvan and Kislev both have 29 days, the year is called deficient (חסר)and there are 353 days (383 in a leap year).
Thus an ordinary year can have 353, 354 or 355 days, and a leap year can have 383, 384 or 385 days. No year is allowed to fall outside these limits of length.
Note that 385 days is exactly 55 weeks, so that the Rosh Hashana of a year following an abundant leap year always falls on the same day of the week as the Rosh Hashana of that year. This does not happen with other year types.
It is possible for six consecutive years to contain all six lengths; for example, 5801-5806 does so.

# The calculation of the Molad
## The starting point

According to tradition, there was a Molad at exactly 14 hours (i.e. 8am) on the Friday of Creation, the time that Adam was created. Projecting the current rule back to that date, it follows, since Rosh Hashana cannot fall on a Friday (see later), that the following day was not only the first Shabbat but the first Rosh Hashana. This is regarded as Rosh Hashana of Year 2, since the six days of creation must belong to some year, which is designated as Year 1.
Working backwards, the previous Molad of Tishri was on a Monday, i.e. Day 2, at 5 hours and 204 chalakim. (With days from midnight to midnight, this is actually 11 hours 204 chalakim on Sunday pm.) Writing 2, 5, 200, 4 in Hebrew characters gives us BeHaRaD, so this is known as Molad BeHaRaD. It is also sometimes called Molad Tohu ("without form"), since it occurred while the world was still "without form and void" (Genesis ch.1). The date of this molad, and of Rosh Hashana for Year 1, would have been Monday 7 September 3761 BCE in the Gregorian calendar (7 October in the Julian one). (Note: when working with dates BCE, remember that there was no year 0; 1 BCE was immediately followed by 1 CE. Thus, the interval from say 1 June, 5 BCE to 1 June, 5 CE is only nine years, not ten.)

## Subsequent Molads

The date and time of each Molad, or time of New Moon, is computed by adding the standard interval 29 days, 12 hours, 793 chalakim to the date and time of the previous one.
If you are only interested in the day of the week and time of the molad, you can discard four weeks or 28 days and just add 1 day, 12 hours, 793 chalakim. If the result is 8 days or more, subtract a week.

## The Molad of Tishri

Only the Molad of Tishri is of importance in computing the calendar. Thus, for fixing the calendar, it suffices to start from the Molad of Tishri and add on 12 times the length of the standard interval (i.e. 354 days, 8 hours, 876 chalakim) if it is an ordinary year and 13 times the length of the standard interval (i.e. 383 days, 21 hours, 589 chalakim) if it is a leap year.
Again, if we are only concerned with days of the week and times, we can discard 50 and 54 weeks respectively to get 4 days, 8 hours, 876 chalakim and 5 days, 21 hours, 589 chalakim.
The calculation may begin from any known molad, and to save time if we want a date many years from the known molad, we can add the molad for a 19 year cycle, 6939 days, 16 hours, 595 chalakim or, discarding complete weeks, 2 days, 16 hours, 595 chalakim.

## Use of the other Molads

The Molads of the other months do have significance in determining whether the ceremony of blessing the New Moon may be performed. There are many rules and customs about this, but in particular it is not generally performed until at least 72 hours after the time of the Molad, to ensure that the Moon will be clearly visible. It must not be performed more than 14 days, 18 hours, 396 chalakim after the Molad, since after that time (half a chalak short of exactly half way from one Molad to the next), the Moon is regarded as waning and not waxing.

# The four postponements of the New Year

The first day of Tishri, which is Rosh Hashana or New Year (literally, "Head of the Year"), should be on the day on which the molad falls, 12 months (or if the previous year was a leap year, 13 months) after the molad for the previous Rosh Hashana. However, more often than not the first day of Tishri is postponed by one or two days, following four rules known as the dechiyot (singular dechiyah).
Once the date of Rosh Hashana for that year and the following year have been calculated, we know how many days there need to be in that year. If there should be 354 or 384, the year is regular, so as noted above, so Cheshvan has 29 days and Kislev has 30. If there should be 355 or 385, Cheshvan has an extra day, and if there should be 353 or 383, Kislev loses a day.
Rule 1
If the time of the molad is after noon, Rosh Hashana is postponed to the next day.
Rule 2
If Rosh Hashana would fall on a Sunday, Wednesday or Friday, then it is postponed to the next day. If it has already been postponed to Sunday, Wednesday or Friday by Rule 1, it is thus postponed for two days.
The reason for this rule is that if Rosh Hashana fell on a Wednesday or Friday, then Yom Kippur (the Day of Atonement), which is on 10th Tishri, would be a Friday or Sunday. Since both Yom Kippur and the Sabbath, which occurs every Saturday, are days when virtually all work is forbidden, it would be very inconvenient to have them on consecutive days. However, it is all right for Yom Kippur and the Sabbath to coincide, as then they are only one day.
If Rosh Hashana fell on a Sunday, then Hoshana Rabba, which is 21st Tishri, would be a Saturday, the Sabbath. It is traditional on that day to walk seven times round the Synagogue carrying a palm branch; this ritual could not be performed on the Sabbath.
This rule is often called Lo ADU Rosh. "Lo" is the Hebrew for "no". "ADU" represents the first, fourth and sixth letters of the Hebrew alphabet, and means that Rosh (Hashana) cannot fall on the first, fourth or sixth days of the week.
If these were the only postponement rules, it would be possible for an ordinary year to have 356 days or a leap year to have 382 days. To avoid this, there are two further rules.
Rule 3
If the calculated day of the New Moon is Tuesday, the calculated time is at least 9 hours 204 chalakim, and the year will be ordinary, the New Year is postponed; it cannot be on Wednesday by Rule 2 so it is moved to Thursday.
Molad Tishri of the following year will fall on Saturday at or after 18hr (noon), so by rules 1 and 2 the next Rosh Hashana would be postponed to Monday. Without this rule, the year would then have 356 days. This rule ensures that it has only 354.
A year postponed by this rule always becomes type 4, and the previous year becomes type 7 or 12, depending on whether it is an ordinary or a leap year.
Rule 4
If the calculated day of the New Moon is Monday, the calculated time is at least 15 hours 589 chalakim, and the year will be the year after a leap year, the New Year is postponed to Tuesday.
Molad Tishri of the previous leap year fell on or after Tuesday at 18hr (noon), so by rules 1 and 2 the previous Rosh Hashana was postponed to Thursday. Without this rule, that year would then have had only 382 days. This rule ensures that it has 383.
A year postponed by this rule always becomes type 3, and the previous year becomes type 11.

## How frequently is Rosh Hashana postponed?

If the Molad of Tishri falls on any of three days of the week, i.e. Sunday, Wednesday or Friday, Rosh Hashana is always postponed. Clearly, this happens on average in 3/7 of all years or about 43%.
If the Molad of Tishri falls on any of the other four days of the week, Rosh Hashana is only postponed if the Molad falls in the last quarter of the day. This happens on average in (4/7)/4 = 1/7 of all years or about 14%. Thus in total, from the first two rules, it is postponed in 4/7 or about 57% of all years.
If the Molad falls after noon on Saturday, Tuesday or Thursday, Rosh Hashana is postponed by two days. This happens on average in (3/7)/4 = 3/28 of all years or about 11%.
Rules 3 and 4 take effect more rarely. Rule 3 only affects ordinary years, i.e. 12 in a 19 year cycle. The range of Molads affected is 8 hours 876 chalakim or 5.24% of a week, so on average it applies to about 12/19 x 5.24% or 3.31% of years. Rule 4 only affects ordinary years that follow a leap year, i.e. 7 in a 19 year cycle. The range of Molads affected is 2 hours 491 chalakim or 1.46% of a week, so on average it applies to about 7/19 x 1.46% or 0.54% of years. (The last such year was 5766 and the next is 6013.) Thus only 3.85% of Rosh Hashanas are postponed for these reasons, so in total 61.0% of Rosh Hashanas are postponed.
Leap years are postponed less often than ordinary years, since rules 3 and 4 do not apply. Thus they are only postponed 57.1% of the time. Years immediately following leap years, where all rules apply, are postponed 63.8% of the time; other years, where rule 4 does not apply, are postponed 62.4% of the time.

## The "Four Gates"

Saadia Gaon (10th century CE) refers to the "Arbaah Shaarim" ("Four gates") as an ancient rule for regulating the calendar. It can be expressed as a table that shows, for each of four possible categories of years:
* Leap years
* Years preceding but not following a leap year, i.e. 2, 5, 10, 13, 16 in a cycle
* Years between two leap years, i.e. 7, 18 in a cycle
* Years following but not preceding a leap year, i.e. 1, 4, 9, 12, 15 in a cycle
the range of Molads that correspond with each possible year type. The rule is exactly equivalent to the four dechiyyot.

# The fourteen types of year

There are fourteen possible types of year. Two years are of the same type if they have the same number of days and every date in one year falls on the same weekday in the other year.
As explained above, a year may not start on Sunday, Wednesday or Friday. Thus it may start on any of four weekdays (Monday, Tuesday, Thursday or Saturday). It may be an ordinary year or a leap year, and it may be defective, regular or abundant. Thus there can at most be 4 x 2 x 3 = 24 year types.
Of these, nine are obviously impossible because they would cause the next Rosh Hashana to fall on a forbidden weekday. For example, if a leap year starts on Monday, it can be deficient or abundant, because then the next Rosh Hashana would fall on Saturday or Monday respectively, but it cannot be regular, because then the next Rosh Hashana would fall on Sunday.
Considering the possible Molad limits, one possible type of leap year (Rosh Hashana Tuesday, abundant) can never happen.
This leaves fourteen possible year types, and they all do occur.

## Nomenclature of year types

There are two systems used to denote year types. In the first, the weekday of Rosh Hashana is given by an integer based on its position in the week. Thus, Monday is 2, being the second day of the week, and similarly Tuesday, Thursday and Saturday are respectively 3, 5 and 7. There is then a letter to denote the year's length, lower case (d, r, a for deficient, regular and abundant) for ordinary years and capital (D, R, A) for leap years.
In the second, there are again a numeral and a letter, and then there is another digit giving the weekday of the first day of Pesach. This second digit enables a distinction between ordinary and leap years, so it does not matter whether the letter is a capital or lower case.
The fourteen types, in the two systems, are:
* 1 = 2d = 2d3
* 2 = 2a = 2a5
* 3 = 3r = 3r5
* 4 = 5r = 5r7
* 5 = 5a = 5a1
* 6 = 7d = 7d1
* 7 = 7a = 7a3
* 8 = 2D = 2D5
* 9 = 2A = 2A7
* 10 = 3R = 3R7
* 11 = 5D = 5D1
* 12 = 5A = 5A3
* 13 = 7D = 7D3
* 14 = 7A = 7A5
It follows that:
* A year beginning on Tuesday is always regular, whether it is an ordinary or leap year.
* A year beginning on Monday or Saturday is never regular, whether it is an ordinary or leap year.
* A year beginning on Thursday is never deficient if it is an ordinary year, and never regular, if it is a leap year.

## Possible sequences of two consecutive years

When considering which sequences of two consecutive years are possible, note that two consecutive years cannot both be leap years. Days of the week must be consistent: Rosh Hashana in the second year must always be two weekdays after Pesach in the first year. Thus in year type 1, Pesach is on Tuesday so the next year must be one where Rosh Hashana is Thursday.
To determine which pairs can actually occur, it is necessary to calculate the range of possible Molads that will produce the first year type. Then add on the shift in Molad due to an ordinary year or a leap year, as the case may be, and see what types of year are produced by the resultant range of possible Molads.
Thus only the following sequences can occur.
* Type 1 may be followed by types 4,5,12
* Type 2 may be followed by types 6,7,13,14
* Type 3 may be followed by types 7,14
* Type 4 may be followed by types 1,2,8,9
* Type 5 may be followed by types 3,10
* Type 6 may be followed by types 3,10
* Type 7 may be followed by types 4,11,12
* Type 8 may be followed by type 7
* Type 9 may be followed by types 1,2
* Type 10 may be followed by type 2
* Type 11 may be followed by type 3
* Type 12 may be followed by type 4
* Type 13 may be followed by types 4,5
* Type 14 may be followed by types 6,7
Two consecutive years cannot both be regular or both deficient, but they can both be abundant. Regular leap years are always followed by abundant years. Regular common years that begin on Tuesday are always followed by abundant common or leap years.

# The main festivals and fasts

The Jewish religion has many festivals (chagim, singular chag) and fasts (tzom or ta'anit). Their dates are of course determined by the Jewish calendar rather than the Gregorian one.
The Sabbath, festivals and fasts end just after the end of evening twilight. Festivals, and the fasts of Yom Kippur and Tisha B'Av, start just before sunset. Other fasts start just before morning twilight. These times must be calculated for each town's longitude and latitude. The definition of twilight may vary between different Jewish communities.
The day before the Sabbath or a festival or fast is known as Erev (literally, "evening", as in the English phrase "Christmas Eve"). Thus we have Erev Shabbat, Erev Succot, Erev Tisha B'Av, etc.
For historical reasons outside the scope of this book, some festivals that are observed for only one day in Israel are observed for two in the rest of the World by orthodox Jews. (Many non-orthodox movements have rejected this second day.) These are noted below in each case.

## Rosh Chodesh

Rosh Chodesh is the New Month (literally, the Head of the Month). It is the first day of each month. Additionally, if the previous month has 30 days, the 30th day of the previous month is also Rosh Chodesh. Thus for example, Rosh Chodesh Sivan is always just one day since Iyar has 29 days. However, since Sivan has 30 days, 30th Sivan is the first day of Rosh Chodesh Tammuz and 1st Tammuz is the second day.
Since Cheshvan and Kislev are of variable length, Rosh Chodesh Kislev and Tevet may each last either one or two days.
Since 1st Tishri is Rosh Hashana, it is rarely given the name Rosh Chodesh Tishri, although technically it is.
Consecutive Rosh Chodeshes (or, in Hebrew, Rashei Chodashim) fall on consecutive days of the week. This is because two Rosh Chodeshes are necessarily separated by exactly four weeks (2nd - 29th of the month), whether that month has 29 or 30 days. For example, if 1st Nisan is Sunday, we have
    1 Nisan: Rosh Chodesh Nisan: Sun
    30 Nisan: 1st day Rosh Chodesh Iyar: Mon
    1 Iyar: 2nd day Rosh Chodesh Iyar: Tue
    1 Sivan: Rosh Chodesh Sivan: Wed
    30 Sivan: 1st day Rosh Chodesh Tammuz: Thu
    1 Tammuz: 2nd day Rosh Chodesh Tammuz: Fri
    etc.
(Note when applying this rule that 1 Tishri must be counted.)

## Ten Days of Repentance

The Aseret Yeme Teshuva or Ten Days of Repentance are the first ten days of Tishri. The first two days are Rosh Hashana (even in Israel) and the tenth is Yom Kippur. There are seven days between these festivals; one and only one of these must be Sabbath and this is called Shabbat Shuvah (The Sabbath of Return or Repentance).
The day after Rosh Hashana is Tzom Gedalyahu (the Fast of Gedaliah). However, if this day is Shabbat Shuvah, the fast is postponed until Sunday, to avoid fasting on the Sabbath.
Although Yom Kippur is also a fast day, it may fall on Sabbath, since it is so holy that it overrides the normal prohibition of fasting on Sabbath.

## Succot

Succot or Tabernacles is also in Tishri. This is one of the festivals where the duration differs between Israel and the rest of the world.

### Israel

The festival lasts eight days, from 15 to 22 Tishri inclusive. The first and last days are full festivals. The other days, called Chol Hamoed or the Intermediate Days, are semi-festivals when most types of work are permitted. Unless the first and last days are Shabbat, one of the intermediate days will be Shabbat so is called Shabbat Chol Hamoed.
The eighth day is technically a different festival, called Shemini Atzeret (the eighth day of assembly). It is largely devoted to celebrating the reading of the Torah, so is also known as Simchat Torah or Rejoicing of the Law.
The day after the festival (23 Tishri) is known as Isru Chag (the binding of the festival) and has a somewhat joyous nature.

### Rest of the World

The festival lasts nine days, from 15 to 23 Tishri inclusive. The first two and last two days are full festivals. The other days, called Chol Hamoed or the Intermediate Days, are semi-festivals when most types of work are permitted. Unless the first and eighth days are Shabbat, one of the intermediate days will be Shabbat so is called Shabbat Chol Hamoed.
The eighth and ninth days technically form a different festival, called Shemini Atzeret (the eighth day of assembly). The ninth day is devoted to celebrating the reading of the Torah, so it is usually called Simchat Torah or Rejoicing of the Law, although it is the second day of Shemini Atzeret.
The day after the festival (24 Tishri) is known as Isru Chag (the binding of the festival) and has a somewhat joyous nature.

## Chanukah

The festival starts on 25 Kislev. It lasts for eight days (even in Israel), all days being of equal importance. It is 83 days after New Year except in an abundant year when Cheshvan has an extra day; in that case it is 84 days after New Year. This means that in an abundant year, the first day of Chanukah is on the same day of the week as the first day of Rosh Hashana. In other year types, it is one weekday earlier.
In a deficient year, when Kislev has 29 days, the sixth day of Chanukah is Rosh Chodesh Tevet and the last day of the festival is 3 Tevet. In a regular or abundant year, when Kislev has 30 days, the sixth and seventh days of Chanukah are both Rosh Chodesh Tevet and the last day of the festival is 2 Tevet.

## 10th Tevet

This is a fast day, generally just called Asara Betevet (which just means 10th Tevet). It is the only fast that can fall on Friday (but see Pesach); while fasting on Friday is not forbidden, it is discouraged so that you should not commence the Sabbath in a sad frame of mind.

## New Year for Trees

New Year for Trees (in Hebrew, Rosh Hashana L'Ilanot) is a minor festival that falls on 15th Shevat. It is also called Tu B'Shevat, Tu representing 15 in Hebrew letters.

## Purim

This celebrates the events described in the biblical Book of Esther. It is celebrated on 14th Adar (so is 30 days before Pesach). In cities that have been walled since the time of the biblical Book of Joshua (such as Jerusalem), it is celebrated on 15th Adar. Elsewhere, that day is called Shushan Purim and has a somewhat joyous character.
The day before Purim is the Fast of Esther (Ta'anit Esther). If this day is Saturday, fasting is forbidden. Normally, the fast would be postponed to Sunday; that cannot be done in this case, as Purim is a day of celebration and feasting. It could be moved back to Friday, but as noted above fasting on Friday is discouraged so the fast is moved back to Thursday 11th Adar.
In a leap year, all these days are observed in Adar Sheni. 14th and 15th Adar Rishon are called Purim Katon and Shushan Purim Katon respectively; katon means small.

## Pesach

Pesach is called "Passover" in English.
Erev Pesach is more important than most Erevs, for two reasons:
* On the morning of Erev Pesach, it is necessary to remove all trace of leavened bread from the home; no leavened bread is allowed throughout Pesach, so the festival in a sense begins on the morning of the day before.
* It is customary for first-born males to fast on that day, so it is called Ta'anit Bechorim (Fast of the Firstborn). If the first day of Pesach is a Saturday, the fast is on Friday, even though fasting on Friday is usually discouraged. If the first day of Pesach is a Sunday, the fast cannot be postponed to Monday, as that is still the festival of Pesach. It could be moved back to Friday, but since fasting on Friday is discouraged, the fast is moved back to Thursday 12th Nisan.
The first day is 15 Nisan, i.e. 163 days before the next New Year.

### Israel

The festival lasts seven days, from 15 to 21 Nisan inclusive. The first and last days are full festivals. The other days, called Chol Hamoed or the Intermediate Days, are semi-festivals when most types of work are permitted. Unless the first day is Shabbat, or Sunday so the seventh day is Shabbat, one of the intermediate days will be Shabbat so is called Shabbat Chol Hamoed.
The day after the festival (22 Nisan) is known as Isru Chag (the binding of the festival) and has a somewhat joyous nature.

### Rest of the World

The festival lasts eight days, from 15 to 22 Nisan inclusive. The first two and last two days are full festivals. The other days, called Chol Hamoed or the Intermediate Days, are semi-festivals when most types of work are permitted. Unless the first and eighth days are Shabbat, or the first day is Sunday so the seventh day is Shabbat, one of the intermediate days will be Shabbat so is called Shabbat Chol Hamoed.
The day after the festival (23 Nisan) is known as Isru Chag (the binding of the festival) and has a somewhat joyous nature.

### The Counting of the Omer

Starting on the second night of Pesach, every night is counted for seven weeks. Thus on the first night, "tonight is the first night of the Omer" is said, on the next night, "tonight is the second night of the Omer", and so on. From the seventh night, weeks and days are also counted, so on the last night, "tonight is the forty-ninth night of the Omer, making seven weeks" is said.
Parts of this period of seven weeks are times of mourning, when for example weddings should not be held. The precise dates when mourning is observed vary between communities.

### Pesach Sheni and Lag B'Omer

Two days in the Omer have special significance. Pesach Sheni ("Second Pesach") is on 14th Iyar, exactly a month after Erev Pesach. In the days of the Temple, it was an opportunity to make the special Pesach offering for those unable to make it on the correct date. Lag B'Omer, the 33rd day of the Omer or 18th Iyar, is regarded by all communities as a happy day when laws of mourning during the Omer cease or are suspended.

## Holocaust Memorial Day

In many communities, 27th Nisan is observed as Holocaust Memorial Day. If 27th Nisan falls on Friday, it is pushed back to Thursday (26th Nisan), whereas if 27th Nisan falls on Sunday, it is postponed to Monday (28th Nisan).

## Israel Independence Day

This is a new festival, created following the proclamation of the State of Israel in 1948. It is normally on 5th Iyar. The rules have changed slightly over the years, but are now that it is moved back to Thursday if it would be on Friday or Shabbat and postponed to Tuesday if it would be on Monday. Thus it can only fall on Tuesday, Wednesday or Thursday. Some Jews refuse to observe it because they do not regard the creation of the State of Israel as a miracle requiring a new festival. For those who do observe it, it is a joyous day that suspends Omer mourning.
The previous day, Yom HaZikaron ("Day of Remembrance"), is a day for recalling all those killed in battle, similar to Veterans' Day or Armistice Day in other countries.

## Jerusalem Day

This is an even newer festival, recalling the recapture of the Old City of Jerusalem from Jordanian occupation in 1967. It is on 28th Iyar.
As for Independence Day, some Jews refuse to observe it and for those who do observe it, it suspends Omer mourning.

## Shavuot

This is observed on the day following the completion of counting the Omer. This would be the 50th day of the Omer, or 50 days after the first day of Pesach, hence the English term "Pentecost". Before the present fixed calendar, it would have been possible for this to be the 5th, 6th or 7th of Sivan (depending on whether Nisan or Iyar had 29 and 30 days). However, now that Nisan always has 30 days and Iyar has 29, Shavuot is now always on 6th Sivan. Outside Israel, it is observed for two days, 6th and 7th Sivan. As for Succot and Pesach, the day afterwards is called Isru Chag.
Shavuot recalls the giving of the Ten Commandments at Mount Sinai. The three days before Shavuot are called the "Three Days of Bordering", recalling the time when the Israelites were forbidden to approach the mountain. (See Exodus Chapter 19.)

## The Fasts of Tammuz and Av

These fasts fall respectively on 17th Tammuz and 9th Av, so are known in Hebrew as Shiva Assar beTanmmuz and Tisha b'Av, Shiva Assar and Tisha meaining 17 and nine in Hebrew. They are 21 days or exactly three weeks apart, hence always fall on the same day of the week as each other. The period between them is known as "The Three Weeks" and is a period of mourning. Tisha b'Av is 62 days after the first day of Shavuot.
If they fall on Saturday, then to avoid fasting on Shabbat they are postponed to the next day.
It is customary to observe certain mourning practices on the day after Tisha b'Av. These are observed on the Monday if Tisha b'Av falls on Sunday and has not been postponed. However, if Tisha b'Av falls on Saturday and was postponed to Sunday, no mourning is observed on Monday.

# Minor fasts

The fasts listed in the previous chapter are accepted by all orthodox Jews. There are other fasts that are only observed by a minority of orthodox communities.

## Yom Kippur Katon

Yom Kippur Katon ("small" Yom Kippur) is observed on the 29th day of most months to atone for any sins committed in that month. It is normally on the day before Rosh Chodesh, i.e. 29th of the month, but is moved back to the previous Thursday if it would be on Shabbat, when fasting is forbidden, or on Friday, because it is not good to begin Shabbat while fasting; compare the Fast of Esther.
It is not observed in the months of Tishri (because it is a joyous month), Kislev (because it would fall during Chanukah), Nisan (because it is a joyous month) or Ellul (because we are about to have Rosh Hashana and Yom Kippur). In some communities, this fast is only kept in Av (to begin preparations for repentance in Ellul).

## Monday, Thursday, Monday

These are a consecutive Monday, Thursday and Monday to atone for any failings in observing Succot or Pesach. Since Tishri and Nisan are joyous months, they are postponed to Cheshvan and Iyar. The commonest custom is to fast on the Monday, Thursday and Monday following the first Shabbat after Rosh Chodesh Cheshvan and Iyar. There is no fast on Pesach Sheni (14 Iyar) if it is on Monday; instead, the last fast is on Thursday 17 Iyar. This happens if the first day of Pesach is Sunday.
The dates are as follows, depending on the day of the week of Rosh Hashana or Pesach:
Cheshvan:
* RH Mon: 6, 9, 13
* RH Tue: 5, 8, 12
* RH Thu: 10, 13, 17
* RH Sat: 8, 11, 15
Iyar:
* Pesach Sun: 7, 10, Thu 17
* Pesach Tue: 5, 8, 12
* Pesach Thu: 10, 13, 17
* Pesach Sat: 8, 11, 15
At one time, Israel Independence Day fell on Monday if the first day of Pesach was Tuesday; those who kept Israel Independence Day did not fast on that day, substituting Thursday 15th. Now it is postponed to Tuesday if it would fall on Monday, so this does not arise.
Some fast on the last Monday, Thursday and Monday in Cheshvan i.e. on or after 17th. The dates are as follows, depending on the day of the week of Rosh Hashana:
* RH Mon: 20, 23, 27; 29 is Yom Kippur Katon
* RH Tue: 19, 22, 26; 29 is Yom Kippur Katon
* RH Thu: 17, 20, 24; 27 is Yom Kippur Katon brought forward
* RH Sat: 22, 25, 29; 29 is also Yom Kippur Katon

## Shovavim Tat

This name is an acronym of the initial letters of the eight consecutive sidras Shemot, Vaera, Bo, Beshallach, Yitro, Mishpatim, Terumah and Tetsaveh (the initial vav of Vaera becoming a cholem vowel).
The original custom was that personal fasts were observed on both the Mondays and Thursdays of the weeks when these sidras are read. Nowadays, the more widespread custom is to observe them only on the Thursdays, and many only observe them in leap years. The fast is cancelled if it would coincide with Rosh Chodesh. (Fasting is also generally prohibited on Tu B'Shevat.)
These fasts may coincide with Yom Kippur Katon.
The fasts occur from mid-Tevet to mid-Adar (Adar I in leap years). The earliest date for a Monday fast is 13 Tevet and for a Thursday fast is 16 Tevet (year types 5 and 12). The latest date for a Thursday fast is 13 Adar (I) (year types 6 & 13); in a year type 6 this is also the Fast of Esther. The last fast also coincides with the Fast of Esther in year types 1 and 7, when the latter is moved back to Thursday to avoid falling on Shabbat. Thus the fasts always start after 10 Tevet and finish before Purim.

# Special Shabbats

There are various Shabbats that have special names; not all of them are within the scope of this book.

## Shabbat Rosh Chodesh

Shabbat Rosh Chodesh (SRC) is a Shabbat that falls on Rosh Chodesh. There are two such Shabbats in every year type, and three in types 4, 5, 9, 10,11 and 12..
In year types 6, 7, 13 and 14, Tishri 1 is also Shabbat, but that is Rosh Hashana, so does not count as SRC.

## Chanukah

The Shabbat during Chanukah is Shabbat Chanukah; it may be any day except the fifth (since the first day of Chanukah may be any weekday except Tuesday). If the first day is Shabbat, so is the last day (Shabbat Chanukah II). The day depends on the year type, as follows:
* Year types 7 & 14: 1st & 8th days
* Year types 6 & 13: 2nd day
* Year types 5 & 12: 3rd day
* Year types 4 & 11: 4th day
* Year types 2, 3, 9 &10: 6th day
* Year types 1 & 8: 7th day

## Lead-up to Pesach

There are five special Shabbats in the lead-up to Pesach:
* Shabbat Shekalim: this is on Rosh Chodesh Adar (Adar II in a leap year) if this is Shabbat, otherwise it is the Shabbat before.
* Shabbat Zachor: this is the Shabbat before Purim.
* Shabbat Parah: this is the Shabbat before Shabbat Hachodesh.
* Shabbat Hachodesh: this is on Rosh Chodesh Nisan if this is Shabbat, otherwise it is the Shabbat before.
* Shabbat Hagodol: this is the last Shabbat before Pesach.
These Shabbats cannot fall on five consecutive Shabbats. Shabbats that are gaps in the sequence are called hafsakahs, and there are two of these in every year. The sequence depends on the weekday of the first day of Pesach, as follows:
* Sunday: Shekalim, Zachor, Hafsakah, Parah, Hachodesh, Hafsakah, Hagodol
* Tuesday: Shekalim, Hafsakah, Zachor, Parah, Hachodesh, Hafsakah, Hagodol
* Thursday: Shekalim, Hafsakah, Zachor, Parah, Hachodesh, Hafsakah, Hagodol
* Saturday: Shekalim, Hafsakah, Zachor, Hafsakah, Parah, Hachodesh, Hagodol
In leap years, for Shevat read Adar I and for Adar read Adar II.

## Doubly Special Shabbats

Three of the Shabbats can be doubly special because they coincide with SRC, namely Chanukah, Shekalim and Hachodesh.
Chanukah: This coincides with SRC if day 6 is Shabbat. Day 7 can also be Rosh Chodesh, but not in a defective year, and Day 7 is only Shabbat in a defective year. Thus we get the double special in year types 2, 3, 9 and 10.
Shekalim: If the first day of Pesach is Sunday (year types 5, 6 & 11) it coincides with SRC.
Hachodesh: If the first day of Pesach is Saturday (year types 4, 9 & 10) it coincides with SRC.
Thus no double special occurs in year types 1, 7, 8, 12, 13 or 14; one occurs in year types 2, 3, 4, 5, 6 and 11; two occur in year types 9 and 10.
Another way is to consider the pre-Pesach period together with the following Chanukah, classified by year type of the later year. (The year type gives the weekday of Rosh Hashana, hence of the previous Pesach.) This currently corresponds to events in the same Gregorian year. This produces the following list of double specials:
* Year types 1, 8: Hachodesh
* Year types 2, 9: Hachodesh and Chanukah
* Year types 3, 10: Shekalim and Chanukah
Otherwise there are none.

# The Annual Cycle of Torah Readings

A section of the Torah is read every Shabbat morning. The universal custom nowadays among orthodox Jews is to read through the entire Torah once a year. The first section (Bereshit) is read on the Shabbat after Simchat Torah, and the last (Vezot Habracha) is read on Simchat Torah, the only time a regular section is read on a weekday. (In Israel, Simchat Torah can sometimes fall on a Shabbat; elsewhere, where it is a day later, it is always a weekday.)
There are special readings, outside the annual cycle, for the various festivals and fasts. If a festival (including Shabbat Chol Hamoed) falls on Shabbat, its reading supersedes the weekly one, which is postponed for a week (or possibly two if the first and eighth days of Pesach or Succot are both Shabbat).
Sometimes, there is a second Torah reading after the main one. On Shabbat and some festivals and fasts, there is a Haftara (reading from the Prophets) after the Torah reading. However, details of these are beyond the scope of this book.
The correct term for a weekly section is a parasha. A widely used alternative term is sidra, though strictly this is the name for one of the smaller sections used in the now obsolete custom of taking three years to read through the whole Torah.
Considering all of the 14 possible year types, there is a need for at least 48 and at most 54 sections in a year. The Torah is therefore divided into 54 sections, of which twelve are in six pairs, and when necessary a pair of sections (a double parasha or double sidra) is read on the same Shabbat.
If the eighth day of Pesach or the second day of Shavuot (outside Israel) fall on a Saturday, they will not be festival days in Israel. Thus, for a few weeks, there will be a difference between the sections read in Israel and elsewhere. (This does not apply to Succot, since the extra festival day outside Israel is always a weekday.)

## When are sections doubled?

The following rules decide which pairs are read on the same day and which are not:
* No sections are doubled before the month of Adar; this may date back to the time before the rules for determining leap years were fixed.
* In an ordinary year, we combine four pairs that are not combined in a leap year:
** Vayakhel/Pekude
** Tazria/Metsora
** Achare Mot/Kedoshim
** Behar/Bechukotai
Vayakhel/Pekude is not doubled in year type 5; in Israel, Behar/Bechukotai is not doubled in year type 4.
* The section Devarim must always be read on the Shabbat before the fast of Tisha b'Av. If 9th Av is a Shabbat, Devarim is read on that day and the fast is postponed. Two rules are necessary to ensure this.
** When 1st day Shavuot falls on Friday (i.e. 1st day Pesach on Thursday, next 1st day Rosh Hashana on Saturday), Chukkas/Balak are combined, otherwise they are separate.
** Normally, Naso is read on the Shabbat after Shavuot. However, in leap years, if the previous Rosh Hashana was on Thursday, Naso is read on the Shabbat before Shavuot. Thus in such years, Mattot and Masse are not doubled.
* The section Nitzavim is always read on the Shabbat before Rosh Hashana. Following that, there are only three more sections: Vayelech, Ha'azinu and Vezot Habracha. The last of these is read in Simchat Torah. If Rosh Hashana or Yom Kippur is Shabbat, their special readings supersede the annual cycle. There is always one Shabbat between Rosh Hashana and Yom Kippur when a section can be read. If the first day of Rosh Hashana is a Monday or Tuesday, there is a Shabbat between Yom Kippur and Succot; Ha'azinu is read then, and Vayelech before Yom Kippur. However, if the first day of Rosh Hashana is a Thursday or Saturday, there is no Shabbat between Yom Kippur and Succot; Ha'azinu must be read before Yom Kippur; Vayelech as well as Nitzavim must be read before Rosh Hashana.

## Rules in Israel

As noted above, the absence of second day Yom Tov in Israel means that sometimes the readings differ between Israel and elsewhere.
* If the 1st day of Pesach is Saturday, the next Saturday is not Yom Tov in Israel so there is an extra week.
** In an ordinary year, in Israel Behar and Bechukotai are not doubled.
** In a leap year, Naso is read before Shavuot in Israel and Mattot and Masse are not doubled.
* If the 1st day of Shavuot is Friday, the next day, Saturday, is not Yom Tov in Israel so again there is an extra week. The usual practice is that Chukkat and Balak are not doubled. This means that these sidras are never doubled in any year. There is an alternative custom not to double Mattot and Masse.

## All possible doublings and no doublings

Thus, outside Israel, all seven pairs of sidras are doubled in year types 2 and 3; none is doubled in year type 11. (The last year type 11 was 5768; the next is 5812.) A type 11 year is always followed by a type 3 year, which has all seven doubles.
In Israel, there cannot be more than six double sidras since Chukkas/Balak is never doubled. All six pairs of sidras are doubled in year types 1, 2, 3 and 6; none is doubled in year types 9, 10 and 11.
For reasons beyond the scope of this book, the haftara for Pinchas is only read when Mattot and Masse are not doubled (otherwise the haftara of Mattot is read with Pinchas). This only happens in year types 11 and 12 (except in those communities in Israel that divide mattot-Masse rather than Chukkat-Balak). There are runs of 23 consecutive years when it is not said, for example 5593-5615, 5924-5946.

# The Atbash

Once the day of the week of Rosh Hashana is known, so is the day of the week of every date from the previous 1st Adar (Adar Sheni in a leap year) until the next 29th Cheshvan. For dates before or after that, variations are possible depending on whether the previous year was a leap year and whether the current year is defective, regular or abundant (affecting the number of days in Cheshvan and Kislev).
A well-known mnemonic for calculating days of the week is the Calendar Atbash. An Atbash is a simple cypher where the first letter of the alphabet is replaced by the last, the second by the next to last, and so on. Thus Aleph is replaced by Tav, Beth by Shin and so on; this gives the acronym Atbash.
Applying the Atbash to the first seven days of Pesach, we get
* Aleph - Tav - Tisha B'Av
* Beth - Shin - Shavuot
* Gimel - Resh - Rosh Hashana
* Daled - Kuf - Keriat Hatorah, i.e. Simchat Torah, a day devoted to Keriat ("reading of") the Torah
* He - Tzadi - Yom Tzom Kippur, the Day of the Fast of Atonement
* Vav - Pe - Purim
* Zayin - Ayin - Yom ha-Atzmaut, Israel Independence Day
This is to be read "The first day of Pesach is on the same day of the week as the date beginning Tav, i.e. Tisha b'Av", etc. (The first line is spoilt if that day is Shabbat so that the fast has to be postponed to Sunday.) Israel Independence Day may also be moved. Note that the Atbash remained incomplete until the creation of the State of Israel meant that this new festival was created.
Since Rosh Hashana cannot fall on any of the three days Sunday, Wednesday or Friday, there are likewise three days of the week on which any given date from 1st Adar (Adar Sheni in a leap year) until 29th Cheshvan cannot fall. For dates before or after that, the situation is more complex; it is necessary to check the details of all fourteen possible types of year. Forbidden weekdays for some important dates are:
* Fast of Esther: Sunday, Tuesday, Friday; if it falls on Saturday (Shabbat), it is observed on Thursday instead. (It cannot be postponed until Sunday, as that day is Purim).
* Purim (14 Adar): Saturday, Monday, Wednesday (so Purim cannot fall on Shabbat except in places such as Jerusalem where it is observed a day late). Lag b'Omer is on the same day of the week as Purim.
* Pesach (1st day): Monday, Wednesday, Friday.
* Israel Independence Day (normally 5th Iyar, but movable): is subject to special rules, which have changed over the years; it can now only fall on Tuesday, Wednesday or Thursday.
* Shavuot (1st day): Tuesday, Thursday, Saturday. Next Hoshana Rabba is on the same day of the week as Shavuot.
* Fasts of Tammuz and Av: Monday, Wednesday, Friday; if they fall on Saturday (Shabbat) they are postponed to Sunday
* Rosh Hashana (1st day), 1st day Succot, Shemini Atzeret; Sunday, Wednesday, Friday.
* Fast of Gedaliah: Sunday, Tuesday, Friday, but if it falls on Saturday (Shabbat) it is postponed to Sunday
* Yom Kippur: Sunday, Tuesday, Friday.
* 1st day Chanukah: Tuesday.
* Fast of Tevet: Monday or Saturday; it can never be Wednesday in an ordinary year, or Thursday in a leap year. It is the only public fast that can fall on Friday.
* New Year for Trees: Sunday or Friday; it can never be Tuesday in an ordinary year, or Wednesday in a leap year.
* Purim Katon: Monday, Thursday, Saturday.

# The prayer for rain and the blessing of the sun

Two Jewish rituals appear to be fixed by the civil calendar rather than the Jewish calendar. These are the insertion of the words Tal umatar ("dew and rain") in the Amidah prayer recited three times every day, and the Blessing of the Sun made every 28 years. The former is done every day "from Ma'ariv (evening service) on 4th December (and in years preceding a civil leap year, 5th December)" up to and including Erev Pesach. The latter is done on 8th April. What is special about those rituals?
The reason is that they are fixed by reference to the tekufot (solstices and equinoxes). These are calculated from rules due to Rabbi Mar Samuel (3rd century CE). They assume that the length of the year is always 365¼ days, rather than varying between 12 and 13 lunar months, hence the fixed dates in the Gregorian calendar. Of course, 365¼ days is slightly too long (the same as the error in the Julian calendar). Thus the date for starting Tal umatar and for saying the Blessing of the Sun in the 20th and 21st centuries are both a day later than they were in the 19th century, and they will get later in the Gregorian calendar by three days every four centuries.

## The date of Tal umatar

In Israel, this prayer is said from 7th Cheshvan, following the Jewish calendar (following the ruling of Rabban Gamliel, Mishnah Ta'anit 1:3). However, the Gemara on that Mishnah (Ta'anit 10a) says that outside Israel, it should not be said until the 60th day from Tekufat Tishri (the autumnal equinox).

## The date of the Blessing of the Sun

Mar Samuel said that the world was created in Nisan. Every 28 years by his calculation, the Spring equinox falls on a Wednesday as it did at the time of creation. There is a custom to say a blessing on the Sun when this happens, "... who makes the work of creation". This contradicts the tradition of BeHaRaD, which assumes that the world was created in Tishri, or to be precise at the end of Ellul. Following this view, the Rosh Hashana prayers include the phrase: "the World was created today".
The same date is used both inside and outside Israel.

## References

* ↑ One who sees the sun at its turning point should say, "Blessed is He who reenacts the works of Creation." And when is this? Abaya said: every 28th year. Talmud Berachot 59b.

# The sixty-one types of cycle

Given the exact molad of the first year of a cycle, you can compute the molad of each year of the cycle hence their year types.
There are 61 possible sequences of years in a cycle.
This can be shown by considering every possible initial molad. Note that since the molad increases by 6939 days, 16 hours, 595 chalakim over a cycle, and 595 is of course a multiple of five, the molad at the start of any cycle must have a number of chalakim ending in either 4 or 9. Also, we may discard multiples of seven days since this makes no difference to the year type. Thus we can start with 7 days, 0 hours, 4 chalakim; by the postponement rules, this gives us Rosh Hashana on Monday and a defective year.
A full discussion and details of all 61 types are given in the book by Burnaby (see Mathematics of the Jewish Calendar/Further reading).
Most cycles have length 6,939, 6,940 or 6,941 days. Only type 61 (with an initial molad of 7 days, 16 hours, 689 chalakim to 7 days, 17 hours, 1079 chalakim) has 6,942 days. Such cycles are quite rare. If the present rules had always existed, the cycles starting in 2908 and 3155, i.e. 854 BCE and 607 BCE, would have been of this type. The next starts in 6765 (3004 CE).
Starts of cycles of length 6942 are separated by either 13 cycles (247 years), 190 cycles (3610 years) or 203 = 190+13 cycles (3857 years). As will be noted later, the calendar repeats approximately, although not exactly, after 13, 190 or 203 cycles.
When tabulating the Jewish calendar over a long period, it is much quicker to work with a whole cycle at a time than with individual years.

# Anniversaries

A problem arises if an event first occurred in Adar, Adar Rishon or Adar Sheni, or on 30th of Cheshvan or Kislev, as these days do not occur every year.
For example, a boy becomes Bar Mitzvah (legally an adult male) on his 13th birthday. If a boy is born in Adar and 13 years later it is a leap year, his Bar Mitzvah is in Adar Sheni. If a boy is born in Adar Rishon or Adar Sheni and 13 years later it is a leap year, his Bar Mitzvah is in Adar Rishon or Adar Sheni as the case may be. However, if a boy is born in Adar Rishon or Adar Sheni and 13 years later it is not a leap year, his Bar Mitzvah is in Adar. Take two boys born in 5746, which was a leap year, on 16th Adar Rishon and 2nd Adar Sheni. Both were Bar Mitzvah in 5759, not a leap year. If in both cases the ceremony was on the Shabbat after their 13th birthdays, the older one had his on 18th Adar and the younger one on 4th Adar, two weeks earlier. However, if he was born on 30th Adar Rishon, and 13 years later it is not a leap year, he becomes Bar Mitzvah on 1st Nisan.
If a boy is born on 30th of Cheshvan or Kislev and 13 years later there is not a 30th of the month, he becomes Bar Mitzvah on 1st Kislev or Tevet respectively.
The same rules apply to when a girl becomes Bat Mitzvah (legally an adult female), which is on her 12th birthday (or 13th in reform Jews).
Concerning the observance of Yahrzeit (literally, "year time", the anniversary of the death of a close relative), there are various customs and a rabbi should be consulted.

# Gauss' Formula for the Date of Pesach

In 1802, the mathematician Carl Friedrich Gauss published a formula to calculate the date of the first day of Pesach in any year. Albert Einstein is said to have remarked of the formula that not only could nobody but Gauss have produced it, but it would never have occurred to anyone but Gauss that such a formula was possible.
Let $\left[{\frac {m}{n}}\right]_{r}$ be the remainder when integer $m$ is divided by integer $n$.
Define:
    $A$ $=$ Jewish year
    $a=\left[{\frac {12A+17}{19}}\right]_{r}$
    $b=\left[{\frac {A}{4}}\right]_{r}$
    $M+m=32+{\frac {4343}{98496}}+\left(1+{\frac {272953}{492480}}\right)a+{\frac {b}{4}}-\left({\frac {313}{98496}}\right)A$
where $M$ is the integer part and $m$ the fractional part of the calculated number.
Then 15 Nisan is Mth March in the Julian calendar unless it is postponed for one of the three following reasons:
Define:
    $c=\left[{\frac {3A+5b+M+5}{7}}\right]_{r}$
    If $c$ = 2 or 4 or 6, then 15 Nisan is ($M$+1)th March.
    If $c$ = 1, $a$ > 6 and $m$ ≥ 1367/2160, then 15 Nisan is ($M$+2)th March.
    If $c$ = 0, $a$ > 11 and $m$ ≥ 23269/25920, then 15 Nisan is ($M$+1)th March.
It is trivial to adapt the formula to find the dates of the other main festivals; for example, Purim is always 30 days before Pesach and Rosh Hashana is 163 days after Pesach.
Note that the formula gives the date in the Julian calendar, so an adjustment to the Gregorian calendar is necessary. To find the difference, take the year and discard the last two digits; call the result $C$. Calculate
    $C-2-\left[{\frac {C}{4}}\right]$
where, in the division, just take the integer part and discard the remainder. Thus the adjustment is to add 11 days for 1 March 1700-28 February 1800, 12 days for 1 March 1800-28 February 1900, 13 days for 1 March 1900-28 February 2100 and 14 days for 1 March 2100- 28 February 2200. Centuries and therefore C in the above formula always start 1 March.
Note that in the very long term, due to calendar drift, it may be hard to know which year of the Julian calendar this Nisan 15th is in. (See Mathematics of the Jewish Calendar/The long-term accuracy of the calendar.)

## References

* "Berechnung des jüdischen Osterfestes", Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde, 5 (1802), 435-437; reprinted in: Carl Friedrich Gauss Werke (Königlichen Gesellschaft der Wissenschaften, Göttingen, 1874), vol. 6, pp. 80-81.
* Hamburger, M. (1896) Crelle's Journal, 116, p.90 gave the first rigorous proof.
* Paper by Zvi Har'El, Technion, Haifa, Israel

# Slonimsky's Formula for the Year Type

In 1844, Rabbi Chaim Zelig Slonimsky (1810-1904) published a modification of Gauss' formula that can be used to calculate the year type of any given year. The version below is his modified formula of 1852.
Let the Jewish year be A. Calculate the remainder r of the division
    (7A-6)/19
If r < 12, the year is ordinary, otherwise it is leap.
Calculate
    K = 0.178117457A + 0.777965458r + 0.2533747
and take the decimal part, discarding the integer part.
The year type may then be read from the following table:
    r < 5
    K
    ≥ 0.000000 ; 1
    ≥ 0.090410 ; 2
    ≥ 0.271103 ; 3
    ≥ 0.376121 ; 4
    ≥ 0.661835 ; 5
    ≥ 0.714282 ; 6
    ≥ 0.752248 ; 7
    r = 5, 6
    K
    ≥ 0.000000 ; 1
    ≥ 0.090410 ; 2
    ≥ 0.271103 ; 3
    ≥ 0.376121 ; 4
    ≥ 0.661835 ; 5
    ≥ 0.714282 ; 6
    ≥ 0.804693 ; 7
    r = 7 to 11
    K
    ≥ 0.000000 ; 1
    ≥ 0.090410 ; 2
    ≥ 0.285711 ; 3
    ≥ 0.376121 ; 4
    ≥ 0.661835 ; 5
    ≥ 0.714282 ; 6
    ≥ 0.804693 ; 7
    r > 11
    K
    ≥ 0.000000 ; 8
    ≥ 0.157466 ; 9
    ≥ 0.285711 ; 10
    ≥ 0.428570 ; 11
    ≥ 0.533590 ; 12
    ≥ 0.714282 ; 13
    ≥ 0.871750 ; 14
The formula calculates a function of the Molad. The lower limits are calculated as the same function of the lower bounds for the Molads of each year type. The function is as follows:
* Take the lower Molad limits for each year type.
* Add 0.25; if the result exceeds 8, subtract 7 from the result.
* Subtract 1 and divide by 7 to get a number in the range 0 to 1.

## Reference

* Ch. Z. Slonimsky (1844), "Eine allgemeineformel fur die gesammte judische Kalenderberechnung", Crelle's Journal, 28, p.179
* Ch. Z. Slonimsky (1852), Yesodei ha'ibbur, p. 26

# The recurrence period of the calendar

The recurrence period of a calendar is the period after which it will always repeat exactly, i.e. two days of the same date that many years apart will always be on the same day of the week. This means that the interval must be an exact number of weeks, or a multiple of seven days.
For the Julian calendar, the recurrence period is 28 years. This is also true for the Gregorian calendar unless the period covers the boundary between February and March in a century year which is not a leap year, such as 1900 or 2100. Even allowing for such years, the recurrence period is 400 years.
The recurrence period for the Jewish calendar is vastly longer: 689,472 years, or 36,288 cycles of 19 years. Any days of the same date this many years apart will always be 251,827,457 days or exactly 35,975,351 weeks apart, so any two dates separated by this number of years must fall on the same day of the week.

## Proof of the recurrence period

The time of the molad after a cycle of 19 years or 235 months exceeds a full week by 2 days, 16 hours, 595 chalakim, or 69,715 = 13,943 x 5 chalakim.
In a whole week there are 7 x 24 x 1080 = 36,288 x 5 chalakim.
It follows that the calendar cannot recur until the passage of 36,288 cycles of 19 years or 689,472 Jewish years.
Another way to express this is that the average year over a 19 year cycle has length 35975351/98496 days. Thus after 98,496 years the Molad will be at the same time of day as before, and hence after 98,496 x 7 = 689,472 years the Molad will be at the same time of day on the same day of the week as before.

## The 247 year cycle

It has often been claimed that the calendar always repeats itself after 247 years, or 13 cycles of 19 years. This is because after 247 years, the computed Molad of Tishri is only 905 chalakim (about 50 minutes) earlier than being at the same time on the same day of the week as before. This small difference rarely makes a difference to the year type, so corresponding dates in these years will nearly always be on the same weekday. This period is often called the Cycle of Nachshon Gaon, as it was attributed to Rabbi Nachshon, Gaon of Sura 871-9, by Abraham ibn Ezra (early 12th century).
However, "nearly always" does not mean always. Periods of 247 Hebrew years are usually 90216 = 12888 x 7 days long, an exact number of weeks, so two dates separated by 247 years are on the same day of the week. However, the period may last for 90215 or 90214 days, not an exact number of weeks, and the calendar does not recur.
The last time that Rosh Hashanah was not on the same day of the week as 247 years earlier was 5708 (1947CE), when it was Monday not Tuesday. The next time this happens will be 5848 (2087CE), when it will be Saturday not Monday.
Some printed copies of the Jewish law code known as the Arba Turim ("Four rows") give lists of the weekday of Rosh Hashana assuming the correctness of the 247 year cycle. The errors were pointed out by Rabbi Hezekiah di Silo (17th century CE) in his book P'ri Chadash ("New fruit").

## Other cycles

There are even closer (but still not exact) correspondences. For example:
* After 190 cycles or 190x19 = 3610 years there is an increase of 730 chalakim.
* After 190+13 = 203 cycles or 203x19 = 3857 years there is a decrease of 175 chalakim.
* After 1002 = 203x5-13 cycles or 1002x19 = 19038 years there is an increase of 30 chalakim.
* After 5213 = 1002x5+203 cycles or 5213x19 = 99047 years there is a decrease of 25 chalakim.
* After 1002+5213 = 6215 cycles or 6215x19 = 118085 years there is an increase of 5 chalakim; this is the best possible approximate recurrence, since the number of chalakim in a complete cycle is a multiple of 5 so a difference of less than 5 chalakim between starts of cycles is impossible.

## The Jewish-Gregorian calendar correspondence cycle

Thus the Jewish calendar repeats after 689,472 Jewish years, while the Gregorian calendar repeats after 400 Gregorian years.
400 Gregorian years are 146,097 days or 20,871 weeks. It follows that the correspondence cycle between the Hebrew and Gregorian calendars, the time interval after which any given Jewish date is guaranteed to fall on the same Gregorian date as before, is 20,871 of these 689,472 year cycles or 14,389,970,112 Jewish years. These amount to 5,255,890,855,047 days or 14,390,140,400 Gregorian years. This is approximately the current age of the universe according to the Big Bang model.
There are thus 170,288 more Gregorian years than Jewish ones in this period, reflecting the fact that the Jewish year is slightly longer (nearly 12 parts per million) than the Gregorian one. See the discussion later on calendar drift.

# Long-term data on the calendar

Since the calendar repeats exactly after 689,472 years, we can calculate the average properties of the calendar by working with data covering this time span. This may differ from the short-term results of analysing say a century of data.

## Number of years of each type

* Type 1: 39369
* Type 2: 81335
* Type 3: 43081
* Type 4: 124416
* Type 5: 22839
* Type 6: 29853
* Type 7: 94563
* Type 8: 40000
* Type 9: 32576
* Type 10: 36288
* Type 11: 26677
* Type 12: 45899
* Type 13: 40000
* Type 14: 32576
* Total: 689472
From this, we have for each possible weekday of:
Rosh Hashana:
* Mon: 193280
* Tue: 79369
* Thu: 219831
* Sat: 196992
Chanukah:
* Mon: 193280
* Wed: 151093
* Thu: 68738
* Fri: 69853
* Sat: 127139
* Sun: 79369
Fast of Tevet:
* Tue: 193280
* Wed: 26677
* Thu: 124416
* Fri: 138591
* Sun: 206508
New Year for Trees:
* Mon: 193280
* Tue: 26677
* Wed: 124416
* Thu: 138591
* Sat: 206508
And for each possible year length:
* 353 days: 69222
* 354 days: 167497
* 355 days: 198737
* 383 days: 106677
* 384 days: 36288
* 385 days: 111051

## Number of 19 year cycles

In the recurrence period, there are 36,288 19-year cycles. They may be tabulated by day of the week of the first day and by length.
* Mon: 9837
* Tue: 3811
* Thu: 12272
* Sat: 10368
* 6939 days: 17099
* 6940 days: 13648
* 6941 days: 5246
* 6942 days: 295
Note that fewer than 1% of cycles have 6942 days.

## Number of Molads

There are only 7 x 24 x 1080 = 181,440 possible Molads. Thus, over a full cycle, every Molad must occur three or four times. Molads with number of chalakim ending in 3 or 8 only occur three times; all the others occur four times. All 7 x 24 = 168 possible combinations of days and hours occur 4,104 times.
Two years with the same Molad Tishri, one ordinary and one leap, are of course always of different types. Two leap years with the same Molad Tishri are always of the same type. Two ordinary years with the same Molad Tishri are almost always of the same type; the only exception is where one year immediately follows a leap year and is subject to postponement rule 4 (Betuskapat), but the other year does not follow a leap year. For example, four years in the complete cycle have Molad 2d 15h 589ch: 88370, 205727, 396432 and 587137. 205727 is a leap year (type 9); 396432 is not postponed (type 2); the others are postponed (type 3).
The interval between two years with the same Molad is always either 117,357 or 190,705 years or the sum of these, 308,062 years. 689,472 = 3 x 190,705 + 117,357 = 2 x 190,705 + 308,062. If a molad occurs four times, then there are three gaps of 190,705 years and one of 117,357 years; if it occurs three times, then there are two gaps of 190,705 years and one of 308,062 years.
If the number of chalakim in the Molad has remainder 0, 1, 2, 3 and 4 respectively when divided by five, the number of ordinary years with that Molad are 2, 3, 2, 2 and 3; the number of leap years with that Molad are 2, 1, 2, 1 and 1.
Where there are three ordinary years per cycle with the same Molad, they are separated by gaps of 190705, 190705 and 308062 years. Where there are two years (ordinary or leap) with the same Molad, they are separated by gaps of 190705 and 498767 years. Where there is only one leap year with a given Molad, recurrence of course takes 689472 years.
Considering only the Molads of the first years of 19 year cycles, the number of chalakim always ends in 4 for odd cycles or 9 for even cycles; all possible Molads of these types occur exactly once over 689,472 years.
The first recurrence of the Molad for year 1 will be for year 117,358. It does not start a new cycle because it is the 14th, not the first, year of a 19 year cycle.

## Pairs of years of the same type

* It is impossible for two consecutive years to be of the same year type. This is easily seen: two consecutive years only begin on the same weekday if the first is an abundant leap year, and two consecutive years cannot both be leap years. (It is also impossible for pairs of leap years to be 4, 7, 12, 15, 18 years apart, or any interval a multiple of 19 greater than these.)
* It is also impossible for two years that are two, five, eight, 12, 15, 16, 22 or 25 years apart to be of the same type.
* Years three years apart can be of the same type, for all year types except 5. This is a rare occurrence for year types 1 and 6. For type 1 it would have happened in 3175/3178 if the present calendar had existed, and will next happen in 23130/23133. For type 6 it would have happened in 2990/2993 if the present calendar had existed, and will next happen in 55655/55658.
* Years four years apart can be of the same type, for all ordinary year types except 5; it is impossible for leap years, as pairs of leap years cannot be four years apart.
* Years six or 14 years apart can be of the same type, for year types 2, 4, 7 and 12. This is a rare occurrence for year type 12. It would have happened in 3173/3179 if the present calendar had existed, and will next happen in 35883/35889.
* Years seven or 20 years apart can be of the same type, for all ordinary year types (so seven is the smallest possible gap for year type 5); it is impossible for leap years, as pairs of leap years cannot be seven or 20 years apart.
* It is only possible for two years nine or 11 years apart to be of the same type if they are both of type 4.
* Years ten years apart can be of the same type, except for year types 1, 3, 5, 6 and 11.
* Years 13 and 18 years apart can be of the same type, for year types 4 and 7.
* Years 17 or 24 years apart can be of the same type, for all year types except 5.
* It is impossible for two years in the same position in consecutive 19 year cycles to be of the same type. The difference in Molad Tishri between such years is well over two days, wider than the Molad limits for any year type.
* Years 21 years apart can be of the same type, for year types 2, 4, 7, 8, 12 and 13.
* Years 23 years apart can be of the same type, for year types 2, 3, 4 and 7.
* Years 26 years apart can be of the same type, for year types 2, 4, 7.
* Years 27 years apart can be of the same type, for all year types; this is the smallest gap for which this is the case.
The longest possible gap between consecutive years of the same type is:
   * 1, 27 years
   * 2, 24 years
   * 3, 27 years
   * 4, 18 years
   * 5, 71 years
   * 6, 47 years
   * 7, 21 years
   * 8, 44 years
   * 9, 47 years
   * 10, 44 years
   * 11, 47 years
   * 12, 41 years
   * 13, 44 years
   * 14, 47 years
Gaps of 18 years between two consecutive type 4 years are rare; the first will be 42345/42363.
By far the longest possible gap is 71 years, for type 5; the last such gap was 5663/5734 and the next is 6255/6326.

## Possible year triplets

The following 52 sequences of year types in three consecutive years are possible:
* 1, 4, 9
* 1, 5, 10
* 1, 12, 4
* 2, 6, 10
* 2, 7, 11
* 2, 13, 4
* 2, 13, 5
* 2, 14, 6
* 3, 7, 11
* 3, 7, 12
* 3, 14, 6
* 3, 14, 7
* 4, 1, 12
* 4, 2, 13
* 4, 2, 14
* 4, 8, 7
* 4, 9, 1
* 4, 9, 2
* 5, 3, 14
* 5, 10, 2
* 6, 3, 14
* 6, 10, 2
* 7, 4, 8
* 7, 4, 9
* 7, 11, 3
* 7, 12, 4
* 8, 7, 4
* 8, 7, 11
* 8, 7, 12
* 9, 1, 4
* 9, 1, 5
* 9, 1, 12
* 9, 2, 6
* 9, 2, 13
* 10, 2, 6
* 10, 2, 7
* 10, 2, 13
* 10, 2, 14
* 11, 3, 7
* 11, 3, 14
* 12, 4, 1
* 12, 4, 2
* 12, 4, 8
* 12, 4, 9
* 13, 4, 2
* 13, 4, 9
* 13, 5, 3
* 13, 5, 10
* 14, 6, 3
* 14, 6, 10
* 14, 7, 4
* 14, 7, 11
Three consecutive years cannot all be ordinary; all sequences must be one of
* Ordinary, ordinary, leap
* Ordinary, leap, ordinary
* Leap, ordinary, ordinary
* Leap, ordinary, leap

## More on the 247 year recurrence

There are 24,073 times in a full cycle that there is a change in year type after a 247 year gap. It occurs 181 times for each year type in each possible position in the 19 year cycle; 24,073 = 181x7x19. (Each position in the cycle must be either always ordinary or always leap, so there are seven possible year types for each position.) Thus such a change occurs on average every 28 or 29 years. However, such changes are clustered. They only occur in 7,867 of the 19-year cycles and there may be up to seven changes in one cycle (e.g. cycles 233 and 436) though not six. Three consecutive years can have a change (e.g. 5521-3, 5933-5), but not four.
If there is only one change year in a 19 year cycle, it must be the 1st or 19th; often but not always, these form pairs, the 19th in one cycle and first in the next cycle. If there are two change years, they must be consecutive. If there are three, they must also be consecutive and not include the first year, unless they are the first, fourth and fifth year. If there are four, they form the following pattern: two consecutive, gap of two, two more consecutive. There is no simple pattern for five. For seven, they must be the 8th, 9th, 12th, 13th, 16th, 17th and 18th years of the cycle.
Put another way, if there are two change years in a 19 year cycle, any year may be involved, although year 1 is rare. If there are three, any year may be involved, although years 12 and 13 are rare. If there are four, any year but 17 may be involved, although years 9 and 13 are rare. If there are five, any year but 19 may be involved, although year 18 is rare.
The longest possible gap between "change years" is 183 years. The first such gap (had the calendar been in force then) was 3504-3687; the next is 7361-7544. Thus there is no full cycle of 247 years without a change. In fact, there will always be between two and 17 changes over any period of 247 years compared with the previous such period. The first period with 17 changes on the previous period is 11972-12218.

# The long-term accuracy of the calendar

The mathematics of the Jewish calendar with all their minutiae are accepted Jewish law and have undoubtedly been so for well over a thousand years. Thus they cannot be altered until a Sanhedrin can be convened, which is not possible at present. Thus it is interesting to ask how well the current rules met the two requirements in Jewish law: do the new months correspond with the first visibility of the New Moon, and will Pesach always fall in the Spring? There are also issues around the Prayer for Rain and the Blessing of the Sun.

## The Molad

The assumed average length of the month, the interval between two consecutive molads, is accurate to a fraction of a second according to the latest scientific knowledge. In making a long-term assessment of accuracy, it must be remembered that the motion of the Moon is very complex, and the average length of a month is varying over time. Further, the rotation of the Earth is slowing down; while the motion of the Moon is predicted using days of constant length, the Jewish calendar necessarily works with the day and night cycle, which is very slowly getting longer. Making the best possible forecasts of both effects, it seems that the formula for the Molad will not be in serious error in the next 3,000 years. Forecasts further ahead become very uncertain.
However, there are short-term irregularities in the Moon's motion, so the month constantly varies in length. As a result, the date and time of the Molad may be a few hours different from the true moment of New Moon. The main variation has an annual cycle, and since it is only the Molad of Tishri that determines the calendar, this annual cycle has little effect.
Agreement with the first visibility of the New Moon is more difficult to assess, since the visibility cannot be predicted with any certainty and will vary with latitude, longitude, altitude and other factors. However, there is no reason to believe that the mean interval between New Moon and first visibility is changing, so that as long as the Molad stays close to the true time of New Moon there should be no big problem.

## The date of Pesach

This is a much bigger problem than the Molad. The 19 year cycle is quite accurate, and far more accurate than any shorter cycle. However, it is not perfect. The average length of the Jewish year over a 19 year cycle is about 365 days, 5 hr, 55 min, 25.4 sec. This is 6 min, 40.2 sec longer than the current average value of the solar year (though that is changing very slowly) and 6 min, 13.4 sec longer than the average value of the year in the Gregorian calendar (which is itself slightly too long).
As a result, the average date of the first day of Pesach is getting later by one day in about 216 years compared to the Sun and 231 years compared to the Gregorian calendar. Thus it is already on average a few days later than at the time of Hillel II. In a 19 year cycle, Pesach currently occurs a month later than on the first Full Moon after the vernal equinox in the 8th, 11th and 19th year of each 19 year cycle, such as 2005, 2008 and 2016CE. This is still not inconsistent with Pesach falling in the Spring. However, the problem will get worse over the centuries. In 18,876 (15,115 CE), the first day of Pesach will be on 22nd June, clearly in the Summer, not the Spring.
In the Jewish year 25,963 (22,203 CE), Rosh Hashana will fall on 1 January. It will always fall on or after that date from 32,849 (29,088 CE). This reduces the difference between the Jewish and civil years at the date of Rosh Hashana by one. A further drift of the Jewish calendar through the Gregorian year will take roughly another 84,500 years, after which the difference will have decreased by one again.
Thus eventually the number of the Gregorian year will equal or exceed the number of the Jewish year, although this will take a very long time. The difference between these calendars averages 0.004322 days per year. So to remove a difference of 3761 years will take roughly 3761 x 365.2425/0.004322 or nearly 318 million years.

## The Tropical Year

The previous section refers to the Gregorian calendar. As noted above, the drift with respect to the true motion of the sun is slightly faster. Conversely, since the Julian calendar assumes an even greater value for the length of the year, the average date of the first day of Pesach is getting earlier with respect to that calendar.
Assuming 365.24219 days for the tropical year, to do better than the 19 year cycle with a fixed cycle would require a cycle of 182 years (67 leap, 115 ordinary) containing 2,251 months. Even this cycle would be out by one day in 256 years, compared with one day in 216 years for the 19 year cycle.
A much better approximation is a cycle of 334 years (123 leap, 211 ordinary), with an error of a day in about 47,000 years; this is in fact considerably more accurate than the Gregorian calendar, which has a 400 year cycle with an error of a day in about 3,200 years.

## The Prayer for Rain and the Blessing of the Sun

The problem with these is even worse than for Pesach (except for the prayer for rain in Israel), since the calculation of dates assumes a year of 365 days 6 hours, or 10 mins 48 seconds longer than the Gregorian year. Thus the dates get later (compared with the Gregorian calendar) by three days in four centuries. Eventually the starting date for the prayer for rain outside Israel will fall on or after the start of Pesach, which will cause a problem. It has been claimed that this happens for the first time in 37,258 CE.

## Do not look too far ahead

There is a widespread belief that the calendar should not be calculated after the year 6,000, i.e. 2240 CE, since it is assumed that the Prophet Elijah will have come by then to announce the end of the Jewish exile. This would allow a new Sanhedrin, which could alter the Jewish calendar. Thus the problems discussed above should not arise. (See w:Year 6000)

# Further reading
## Books

* Burnaby, Sherard B. (1901): Elements of the Jewish and Muhammadan Calendars, Geo. Bell & Sons
** Burnaby was a Church of England minister. While this book is very reliable on the mathematics of the calendar, it has many errors regarding the Jewish religion. The book may be downloaded here. There are a number of errors in the table on page 295 in the book, which details the 61 types of 19-year cycle.
* Gandz, Solomon (1956) Sanctification of the New Moon, Yale University Press
** An annotated edition of the laws of the Jewish calendar as given by Moses Maimonides.
* Greenfield, M. (1963) 150 Year Calendar
** Gives basic calendar information for 240-2240 and more detailed data for 1950-2100
* Guggenheimer, H. W (1998) Seder Olam: The Rabbinic View of Biblical Chronology
** This book is the basis for the calculation that according to the Bible, the world was created in 3761 BCE.
* Reingold, Edward M. and Dershowitz, Nachum (2008): Calendrical Calculations (3rd ed), Cambridge University Press, ISBN 978-0521702386
** Gives algorithms for calculating the Jewish and many other calendars.
* Richards, E. G. (1998): Mapping Time, The Calendar and its History, Oxford University Press, ISBN 0-19-850413-6
* Schamroth, Julian (1998): A Glimpse of Light, Feldheim, ISBN 1-56871-136-0
** Another annotated edition of the laws of the Jewish calendar as given by Moses Maimonides.
* Schocken, W. A. (1976): The Calculated Confusion of Calendars: Puzzles in Christian, Jewish and Moslem Calendars, Vantage Press
** Gives various formulae for computing these calendars
* Seidelmann, P. Kenneth (1992): Explanatory Supplement to the Astronomical Almanac, University Science Books, ISBN 0-935702-68-7: Chapter 12: Calendars (pages 575-608)
* Shneer, J. A. (2020): The Jewish Calendar and the Torah, 5th edition, Palos Verdes Historical Press. ((ISBN: 978-1-67800-545-0))
* Spier, A. (1986): The Comprehensive Hebrew Calendar (3rd ed.), Feldheim, ISBN 0-87306-398-8
** Gives parallel Jewish and Gregorian calendars for 1900-2100.
* Stern, Sacha (2001): Calendar and Community, a History of the Jewish Calendar, Oxford University Press, ISBN 0-19-827034-8

## Web sites

* http://www.hebcal.com/converter/ Converts between Jewish and Gregorian calendars
* http://hebrewcalendar.tripod.com/ The Hebrew Calendar Science and Myths by Remy Landau
* http://www.jewishencyclopedia.com/view.jsp?artid=44&letter=C&search=calendar The Calendar (Jewish Encyclopedia)
* http://www.jewishencyclopedia.com/view.jsp?artid=43&letter=C History of the Calendar (Jewish Encyclopedia)

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