Hebrew calendar

thumb|right|Jewish calendar, showing Adar II between 1927 and 1948

The Hebrew calendar (, ), also called the Jewish calendar, is a lunisolar calendar used today for Jewish religious observance and as an official calendar of Israel. It determines the dates of Jewish holidays and other rituals, such as , and the schedule of public Torah readings. In Israel, it is used for religious purposes, provides a time frame for agriculture, and is an official calendar for civil holidays alongside the Gregorian calendar.

Like other lunisolar calendars, the Hebrew calendar consists of months of 29 or 30 days that begin and end at approximately the time of the new moon. As 12 such months comprise a total of just 354 days, an extra lunar month is added every 2 or 3 years so that the long-term average year length closely approximates the actual length of the solar year.

Originally, the beginning of each month was determined based on physical observation of a new moon, while the decision of whether to add the leap month was based on observation of natural agriculture-related events in ancient Israel. Between the years 70 and 1178, these empirical criteria were gradually replaced with a set of mathematical rules. Month length now follows a fixed schedule that is adjusted based on the molad interval (a mathematical approximation of the mean time between new moons) and several other rules, while leap months are now added in 7 out of every 19 years according to the Metonic cycle.

Nowadays, Hebrew years are generally counted according to the system of (Latin: "in the year of the world"; , "from the creation of the world", abbreviated AM) according to traditional Jewish interpretation of the chronology of the Hebrew Bible. This system attempts to calculate the number of years since the creation of the world according to the Genesis creation narrative and subsequent Biblical stories. The current Hebrew year, AM , began at sunset on and will end at sunset on .

Components

Hours

Judaism uses multiple systems for dividing hours. In one system, the 24-hour day is divided into fixed hours equal to of a day, while each hour is divided into 1080 (parts, singular: ). A is seconds ( minute). The ultimate ancestor of the was a Babylonian unit of time called a barleycorn, equal to of a Babylonian time degree (1° of celestial rotation). These measures are not generally used for everyday purposes; their best-known use is for calculating and announcing the molad.

In another system, the daytime period is divided into 12 relative hours (, also sometimes called "halachic hours"). A relative hour is defined as of the time from sunrise to sunset, or dawn to dusk, as per the two opinions in this regard. Therefore, an hour can be less than 60 minutes in winter, and more than 60 minutes in summer; similarly, the 6th hour ends at solar noon, which generally differs from 12:00. Relative hours are used for the calculation of prayer times (zmanim); for example, the Shema must be recited in the first three relative hours of the day.

Neither system is commonly used in ordinary life; rather, the local civil clock is used. This is even the case for ritual times (e.g. "The latest time to recite Shema today is 9:38 AM").

Days

Based on the classic rabbinic interpretation of ("There was evening and there was morning, one day"), a day in the rabbinic Hebrew calendar runs from sunset (the start of "the evening") to the next sunset. Similarly, Yom Kippur, Passover, and Shabbat are described in the Bible as lasting "from evening to evening". The days are therefore figured locally.

Halachically, the exact time when days begin or end is uncertain: this time could be either sundown () or else nightfall (, "when the stars appear"). The time between sundown and nightfall () is of uncertain status. Thus (for example) observance of Shabbat begins before sundown on Friday and ends after nightfall on Saturday, to be sure that Shabbat is not violated no matter when the transition between days occurs.

Instead of the International Date Line convention, there are varying opinions as to where the day changes. (See International date line in Judaism.)

Weeks

The Hebrew week (, ) is a cycle of seven days, mirroring the seven-day period of the Book of Genesis in which the world is created.

The names for the days of the week are simply the day number within the week. The week begins with Day 1 (Sunday) and ends with Shabbat (Saturday). (More precisely, since days begin in the evening, weeks begin and end on Saturday evening. Day 1 lasts from Saturday evening to Sunday evening, while Shabbat lasts from Friday evening to Saturday evening.)

Since some calculations use division, a remainder of 0 signifies Saturday.

In Hebrew, these names may be abbreviated using the numerical value of the Hebrew letters, for example (Day 1, or ()):

{| class="wikitable"

! Hebrew name

! Abbreviation

! Translation

! English equivalent

| Yom Rishon (יום ראשון)

| First day

| Sunset on Saturday to sunset on Sunday

| Yom Sheni (יום שני)

| Second day

| Sunset on Sunday to sunset on Monday

| Yom Shlishi (יום שלישי)

| Third day

| Sunset on Monday to sunset on Tuesday

| Yom Revii (יום רביעי)

| Fourth day

| Sunset on Tuesday to sunset on Wednesday

| Yom Hamishi (יום חמישי)

| Fifth day

| Sunset on Wednesday to sunset on Thursday

| Yom Shishi (יום שישי)

| Sixth day

| Sunset on Thursday to sunset on Friday

| Yom Shabbat (יום שבת)

| Sabbath day

| Sunset on Friday to sunset on Saturday

The names of the days of the week are modeled on the seven days mentioned in the Genesis creation account. For example, Genesis 1:8 "... And there was evening and there was morning, a second day" corresponds to meaning "second day". (However, for days 1, 6, and 7, the modern name differs slightly from the version in Genesis.)

The seventh day, Shabbat, as its Hebrew name indicates, is a day of rest in Judaism. In Talmudic Hebrew, the word () can also mean "week", so that in ritual liturgy a phrase like "Yom Reviʻi beShabbat" means "the fourth day in the week".

Days of week of holidays

Jewish holidays can only fall on the weekdays shown in the following table:

{| class="wikitable"

|-style="background-color:#DDDDDD;text-align:center;"

!Purim

!Passover (first day)

!Shavuot (first day)

!17 Tammuz/ Tisha B'Av

!Rosh Hashanah/ Sukkot/ Shmini Atzeret (first day)

!Yom Kippur

!Chanukah (first day)

!10 Tevet

!Tu Bishvat

!Purim Katan (only in leap years)

|Thu||Sat||Sun||Sun*||Mon||Wed

|Sun or Mon

|Sun or Tue

|Sat or Mon

|Sun or Tue

|Fri||Sun||Mon||Sun||Tue||Thu||Mon||Tue||Mon||Tue

|Sun||Tue||Wed||Tue||Thu||Sat

|Wed or Thu

|Wed, Thu, or Fri

|Tue, Wed, or Thu

|Wed or Fri

|Tue||Thu||Fri||Thu||Sat||Mon

|Fri or Sat

|Fri or Sun

|Thu or Sat

|Fri or Sun

|colspan=10| *Postponed from Shabbat

The period from 1 Adar (or Adar II, in leap years) to 29 Marcheshvan contains all of the festivals specified in the Bible (Purim, Passover, Shavuot, Rosh Hashanah, Yom Kippur, Sukkot, and Shemini Atzeret). The lengths of months in this period are fixed, meaning that the day of week of Passover dictates the day of week of the other Biblical holidays. However, the lengths of the months of Marcheshvan and Kislev can each vary by a day (due to the Rosh Hashanah postponement rules, which are used to adjust the year length). As a result, the holidays falling after Marcheshvan (starting with Chanukah) can fall on multiple days for a given row of the table.

A common mnemonic is "", meaning: "Rosh HaShana cannot be on Sunday, Wednesday or Friday, and Passover cannot be on Monday, Wednesday or Friday" with each day's numerical equivalent, in gematria, is used, such that א' = 1 = Sunday, and so forth. From this rule, every other date can be calculated by adding weeks and days until that date's possible day of the week can be derived.

Months

The Hebrew calendar is a lunisolar calendar, meaning that months are based on lunar months, but years are based on solar years. The calendar year features twelve lunar months of 29 or 30 days, with an additional lunar month ("leap month") added periodically to synchronize the twelve lunar cycles with the longer solar year. These extra months are added in seven years (3, 6, 8, 11, 14, 17, and 19) out of a 19-year cycle, known as the Metonic cycle (See Leap months, below).

The beginning of each Jewish lunar month is based on the appearance of the new moon. Although originally the new lunar crescent had to be observed and certified by witnesses (as is still done in Karaite Judaism and Islam), nowadays Jewish months have generally fixed lengths that approximate the period between new moons. For these reasons, a given month does not always begin on the same day as its astronomical conjunction.

The mean period of the lunar month (precisely, the synodic month) is very close to 29.5 days. Accordingly, the basic Hebrew calendar year is one of twelve lunar months alternating between 29 and 30 days:

{| class="wikitable"

!colspan="2"| Month number

!rowspan="2"| Hebrew month

!rowspan="2"| Length

!colspan="2"| Range of possible Gregorian dates

! Biblical / after Exodus !! Civil / currently used

! First day !! Last day

| style="text-align:center;" | 1 || style="text-align:center;" | 7 || Nisan || style="text-align:center;" | 30 || style="text-align:center;" | 12 March to 11 April|| style="text-align:center;" | 10 April to 10 May

| style="text-align:center;" | 2 || style="text-align:center;" | 8 || Iyar || style="text-align:center;" | 29 || style="text-align:center;" | 11 April to 11 May|| style="text-align:center;" | 9 May to 8 June

| style="text-align:center;" | 3 || style="text-align:center;" | 9 || Sivan || style="text-align:center;" | 30 || style="text-align:center;" | 10 May to 9 June|| style="text-align:center;" | 8 June to 8 July

| style="text-align:center;" | 4 || style="text-align:center;" | 10 || Tammuz || style="text-align:center;" | 29 || style="text-align:center;" | 9 June to 9 July|| style="text-align:center;" | 7 July to 6 August

| style="text-align:center;" | 5 || style="text-align:center;" | 11 || Av || style="text-align:center;" | 30 || style="text-align:center;" | 8 July to 7 August|| style="text-align:center;" | 6 August to 5 September

| style="text-align:center;" | 6 || style="text-align:center;" | 12 || Elul || style="text-align:center;" | 29 || style="text-align:center;" | 7 August to 6 September|| style="text-align:center;" | 4 September to 4 October

| style="text-align:center;" | 7 || style="text-align:center;" | 1 || Tishrei || style="text-align:center;" | 30 || style="text-align:center;" | 5 September to 5 October|| style="text-align:center;" | 4 October to 3 November

| style="text-align:center;" | 8 || style="text-align:center;" | 2 || Cheshvan (or Marcheshvan) || style="text-align:center;" | 29 (or 30) || style="text-align:center;" | 5 October to 4 November|| style="text-align:center;" | 3 November to 2 December

| style="text-align:center;" | 9 || style="text-align:center;" | 3 || Kislev || style="text-align:center;" | 30 (or 29) || style="text-align:center;" | 4 November to 3 December|| style="text-align:center;" | 2 December to 31 December

| style="text-align:center;" | 10 || style="text-align:center;" | 4 || Tevet || style="text-align:center;" | 29 || style="text-align:center;" | 3 December to 1 January|| style="text-align:center;" | 1 January to 29 January

| style="text-align:center;" | 11 || style="text-align:center;" | 5 || Shevat || style="text-align:center;" | 30 || style="text-align:center;" | 1 January to 30 January|| style="text-align:center;" | 30 January to 28 February

| style="text-align:center;" | 12 || style="text-align:center;" | 6 || Adar I (this is added only in leap years)|| style="text-align:center;" | 30 || style="text-align:center;" | 31 January to 12 February|| style="text-align:center;" | 1 March to 12 March

| style="text-align:center;" | 12 || style="text-align:center;" | 6 || Adar (this is called Adar II in leap years)|| style="text-align:center;" | 29 || style="text-align:center;" | 11 February to 13 March|| style="text-align:center;" | 11 March to 10 April

!colspan=3| Total || 354 (or 353 or 355) 30 days more in leap years!! ||

Thus, the year normally contains twelve months with a total of 354 days. In such a year, the month of Marcheshvan has 29 days and Kislev has 30 days. However, due to the Rosh Hashanah postponement rules, in some years Kislev may lose a day to have 29 days, or Marcheshvan may acquire an additional day to have 30 days.

Normally the 12th month is named Adar. During leap years, the 12th and 13th months are named Adar I and Adar II (Hebrew: and —"first Adar" and "second adar"). Sources disagree as to which of these months is the "real" Adar, and which is the added leap month.

Justification for leap months

The Bible does not directly mention the addition of leap months (also known as "embolismic" or "intercalary" months). The insertion of the leap month is based on the requirement that Passover occur at the same time of year as the spring barley harvest (). (Since 12 lunar months make up less than a solar year, the date of Passover would gradually move throughout the solar year if leap months were not occasionally added.) According to the rabbinic calculation, this requirement means that Passover (or at least most of Passover) should fall after the March equinox.

Maimonides, discussing the calendrical rules in his Mishneh Torah (1178), notes:

By how much does the solar year exceed the lunar year? By approximately 11 days. Therefore, whenever this excess accumulates to about 30 days, or a little more or less, one month is added and the particular year is made to consist of 13 months, and this is the so-called embolismic (intercalated) year. For the year could not consist of twelve months plus so-and-so many days, since it is said: "throughout the months of the year", which implies that we should count the year by months and not by days.

</blockquote>

Years

New year

right|thumb|A made from a ram's horn is traditionally blown in observance of [[Rosh Hashanah, the beginning of the Jewish civic year.]]

The Hebrew calendar year conventionally begins on Rosh Hashanah, the first day of Tishrei. However, the Jewish calendar also defines several additional new years, used for different purposes. The use of multiple starting dates for a year is comparable to different starting dates for civil "calendar years", "tax or fiscal years", "academic years", and so on. The (c. 200 CE) identifies four new-year dates:

<blockquote>The 1st of Nisan is the new year for kings and festivals. The 1st of Elul is the new year for the cattle tithe, Rabbi Eliezer and Rabbi Shimon say on the first of Tishrei. The 1st of Tishri is the new year for years, of the Shmita and Jubilee years, for planting and for vegetables. The 1st of Shevat is the new year for trees—so the school of Shammai, but the school of Hillel say: On the 15th thereof.</blockquote>

Two of these dates are especially prominent:

1 Nisan is the ecclesiastical new year, i.e. the date from which months and festivals are counted. while Rosh Hashana (which begins on 1 Tishrei) is described as falling "in the seventh month".
1 Tishrei is the civil new year, and the date on which the year number advances. This date is known as Rosh Hashanah (). Tishrei marks the end of one agricultural year and the beginning of another,

As with (A.D. or AD), the words or abbreviation for (A.M. or AM) for the era should properly precede the date rather than follow it.

The reference junction of the Sun and the Moon (Molad 1) is considered to be at 5 hours and 204 halakim, or 11:11:20 p.m., on the evening of Sunday, 6 October 3761 BCE. According to rabbinic reckoning, this moment was Creation, but about one year "before" Creation, with the new moon of its first month (Tishrei) called (the mean new moon of chaos or nothing). It is about one year before the traditional Jewish date of Creation on 25 Elul AM 1, based upon the . Thus, adding 3760 before Rosh Hashanah or 3761 after to a Julian calendar year number starting from 1 CE will yield the Hebrew year. For earlier years there may be a discrepancy; see Missing years (Jewish calendar).

In Hebrew there are two common ways of writing the year number: with the thousands, called ("major era"), and without the thousands, called ("minor era"). Thus, the current year is written as &lrm;() using the "major era" and &lrm;() using the "minor era".

Cycles of years

Since the Jewish calendar has been fixed, leap months have been added according to the Metonic cycle of 19 years, of which 12 are common (non-leap) years of 12 months, and 7 are leap years of 13 months. This 19-year cycle is known in Hebrew as the ("small cycle").

Because the Julian years are days long, every 28 years the weekday pattern repeats. This is called the sun cycle, or the ("great cycle") in Hebrew. The beginning of this cycle is arbitrary. Its main use is for determining the time of Birkat Hachama.

Because every 50 years is a Jubilee year, there is a jubilee () cycle. Because every seven years is a sabbatical year, there is a seven-year release cycle. The placement of these cycles is debated. Historically, there is enough evidence to fix the sabbatical years in the Second Temple Period. But it may not match with the sabbatical cycle derived from the biblical period; and there is no consensus on whether the Jubilee year is the fiftieth year or the latter half of the forty ninth year.

Every 247 years, or 13 cycles of 19 years, form a period known as an , or the Iggul of Rabbi Nahshon. This period is notable in that the precise details of the calendar almost always (but not always) repeat over this period. This occurs because the interval (the average length of a Hebrew month) is 29.530594 days, which over 247 years results in a total of 90215.965 days. This is almost exactly 90216 days – a whole number and multiple of 7 (equalling the days of the week). So over 247 years, not only does the 19-year leap year cycle repeat itself, but the days of the week (and thus the days of Rosh Hashanah and the year length) typically repeat themselves.

Calculations

Leap year calculations

To determine whether a Jewish year is a leap year, one must find its position in the 19-year Metonic cycle. This position is calculated by dividing the Jewish year number by 19 and finding the remainder. (Since there is no year 0, a remainder of 0 indicates that the year is year 19 of the cycle.) For example, the Jewish year divided by 19 results in a remainder of , indicating that it is year of the Metonic cycle. The Jewish year used is the year, in which the year of creation according to the Rabbinical Chronology (3761 BCE) is taken as year 1. Years 3, 6, 8, 11, 14, 17, and 19 of the Metonic cycle are leap years. The Hebrew mnemonic GUCHADZaT refers to these years, while another memory aid refers to musical notation.

Whether a year is a leap year can also be determined by a simple calculation (which also gives the fraction of a month by which the calendar is behind the seasons, useful for agricultural purposes). To determine whether year n of the calendar is a leap year, find the remainder on dividing [(7 × n) + 1] by 19. If the remainder is 6 or less it is a leap year; if it is 7 or more it is not. For example, the The This works because as there are seven leap years in nineteen years the difference between the solar and lunar years increases by month per year. When the difference goes above month this signifies a leap year, and the difference is reduced by one month.

The Hebrew calendar assumes that a month is uniformly of the length of an average synodic month, taken as exactly days (about 29.530594 days, which is less than half a second from the modern scientific estimate); it also assumes that a tropical year is exactly times that, i.e., about 365.2468 days. Thus, it overestimates the length of the tropical year (365.2422 days) by 0.0046 days (about 7 minutes) per year, or about one day in 216 years. This error is less than the Julian years (365.2500 days) make (0.0078 days/year, or one day in 128 years), but much more than what the Gregorian years (365.2425 days/year) make (0.0003 days/year, or one day in 3333 years).

Rosh Hashanah postponement rules

Besides the adding of leap months, the year length is sometimes adjusted by adding one day to the month of Marcheshvan, or removing one day from the month of Kislev. Because each calendar year begins with Rosh Hashanah, adjusting the year length is equivalent to moving the day of the next Rosh Hashanah. Several rules are used to determine when this is performed.

To calculate the day on which Rosh Hashanah of a given year will fall, the expected molad (moment of lunar conjunction or new moon) of Tishrei in that year is calculated. The molad is calculated by multiplying the number of months that will have elapsed since some (preceding) molad (whose weekday is known) by the mean length of a (synodic) lunar month, which is 29 days, 12 hours, and 793 parts (there are 1080 "parts" in an hour, so that one part is equal to seconds). The very first molad, the molad tohu, fell on Sunday evening at 11:11:20 pm in the local time of Jerusalem, 6 October 3761 BCE (Proleptic Julian calendar) 20:50:23.1 UTC, or in Jewish terms Day 2, 5 hours, and 204 parts. The exact time of a molad in terms of days after midnight between 29 and 30 December 1899 (the form used by many spreadsheets for date and time) is

:−2067022 + (23 + 34/3/60)/24 + (29.5 + 793/1080/24)N

where N is the number of lunar months since the beginning. ( for the beginning of the 305th Machzor Katan on 1 October 2016.) Adding 0.25 to this converts it to the Jewish system in which the day begins at 6 pm.

In calculating the number of months since the known molad that one uses as the starting point, one must include any leap months that fall within the elapsed interval, according to the cycle of leap years. A 19-year cycle of 235 synodic months has 991 weeks 2 days 16 hours 595 parts, a common year of 12 synodic months has 50 weeks 4 days 8 hours 876 parts, while a leap year of 13 synodic months has 54 weeks 5 days 21 hours 589 parts.

Four conditions are considered to determine whether the date of Rosh Hashanah must be postponed. These are called the Rosh Hashanah postponement rules, or . The two most important conditions are:

If the molad occurs at or later than noon, Rosh Hashanah is postponed to the following day. This is called (, literally, "old birth", i.e., late new moon). This rule is mentioned in the Talmud, This ensures that the long-term average month length is 29.530594 days (equal to the molad interval), rather than the 29.5 days implied by the standard alternation between 29- and 30-day months.
If the molad occurs on a Sunday, Wednesday, or Friday, Rosh Hashanah is postponed to the following day. If the application of would place Rosh Hashanah on one of these days, then it is postponed a second day. This is called (), an acronym that means "not [weekday] one, four, or six".

This rule is applied for religious reasons, so that Yom Kippur does not fall on a Friday or Sunday, and Hoshana Rabbah does not fall on Shabbat. Since Shabbat restrictions also apply to Yom Kippur, if either day falls immediately before the other, it would not be possible to make necessary preparations for the second day (such as candle lighting). Additionally, the laws of Shabbat override those of Hoshana Rabbah, so that if Hoshana Rabbah were to fall on Shabbat, the Hoshana Rabbah ritual could not be performed.

Thus, Rosh Hashanah can only fall on Monday, Tuesday, Thursday, and Saturday. The uses the letters ה ,ג ,ב and ז (representing 2, 3, 5, and 7, for Monday, Tuesday, Thursday, and Saturday) to denote the starting day of Rosh Hashana and the year.

Two other rules are applied much less frequently and serve to prevent impermissible year lengths. Their names are Hebrew acronyms that refer to the ways they are calculated:

If the molad in a common year falls on a Tuesday, on or after 9 hours and 204 parts, Rosh Hashanah is postponed to Thursday. This is (, where the acronym stands for "3 [Tuesday], 9, 204").
If the molad following a leap year falls on a Monday, on or after 15 hours and 589 parts after the Hebrew day began (for calculation purposes, this is taken to be 6 pm Sunday), Rosh Hashanah is postponed to Tuesday. This is (), where the acronym stands for "2 [Monday], 15, 589".

Deficient, regular, and complete years

The rules of postponement of Rosh HaShanah make it that a Jewish common year will have 353, 354, or 355 days while a leap year (with the addition of Adar I which always has 30 days) has 383, 384, or 385 days.

Kevi'ah

{| class="wikitable" align="right" style="float; margin: 9px;"

! style="text-align:right" | Days in year →

| 353 || 354 || 355 || 383 || 384 || 385

! Day of Rosh HaShanah

! colspan="6" | English symbol

| Monday (2) || 2D3 || || 2C5 || 2D5 || || 2C7

| Tuesday (3) || || 3R5 || || || 3R7 ||

| Thursday (5) || || 5R7 || 5C1 || 5D1 || || 5C3

| Saturday (7) || 7D1 || || 7C3 || 7D3 || || 7C5

There are three qualities that distinguish one year from another: whether it is a leap year or a common year; on which of four permissible days of the week the year begins; and whether it is a deficient, regular, or complete year. Mathematically, there are 24 (2×4×3) possible combinations, but only 14 of them are valid.

Each of these patterns is known by a ( for 'a setting' or 'an established thing'), which is a code consisting of two numbers and a letter. In English, the code consists of the following:

The left number is the day of the week of , Rosh Hashanah
The letter indicates whether that year is deficient (D, "", from ), regular (R, "", from ), or complete (C, "", from )
The right number is the day of the week of , the first day of Passover or Pesach , within the same Hebrew year (next Julian/Gregorian year)

The in Hebrew letters is written right-to-left, so their days of the week are reversed, the right number for and the left for .

The also determines the Torah reading cycle (which are read together or separately.

The four gates

The , and thus the annual calendar, of a numbered Hebrew year can be determined by consulting the table of Four Gates, whose inputs are the year's position in the 19-year cycle and its molad Tishrei. In this table, the years of a 19-year cycle are organized into four groups (called "gates"): common years after a leap year but before a common year ; common years between two leap years ; common years after a common year but before a leap year ; and leap years .

This table

{| style="border-collapse: collapse;"

|+ Four gates or Table of Limits

|- style="background-color: #F4F4F4;"

! style="border: 1px solid #B0B0B0; text-align: center;" rowspan="2" | molad Tishrei ≥

! style="border: 1px solid #B0B0B0; text-align: center;" colspan="4" | Year of 19-year cycle

|- style="background-color: #F4F4F4;"

! style="border: 1px solid #B0B0B0; padding: 0px 10px;" | 1 4 9 12 15

! style="border: 1px solid #B0B0B0; padding: 0px 30px;" | 7 18

! style="border: 1px solid; border-color: #B0B0B0 #707070; padding: 0px 10px;" | 2 5 10 13 16

! style="border: 1px solid #B0B0B0; padding: 0px 10px;" | 3 6 8 11 14 17 19