View source for Hebrew calendar

{{Short description|Lunisolar calendar used for Jewish religious observances}}
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[[File:Jewish calendar, showing Adar II between 1927 and 1948.jpeg|thumb|right|Jewish calendar, showing Adar II between 1927 and 1948]]
{{Infobox calendar date today}}
{{Jewish culture}}
The '''Hebrew calendar''' ({{langx|he|הַלּוּחַ הָעִבְרִי|Lūaḥ hāʿĪḇrī}}), also called the '''Jewish calendar''', is a [[lunisolar calendar]] used today for [[Jew]]ish religious observance and as an official calendar of [[Israel]]. It determines the dates of [[Jewish holidays]] and other rituals, such as ''[[yahrzeit]]s'' and the schedule of [[Weekly Torah portion|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 which begin and end at approximately the time of the new moon. As [[Lunar calendar|12 such months]] comprise a total of just 354 days, an [[Intercalation (timekeeping)|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 [[History of ancient Israel and Judah|ancient Israel]].<ref name=ts22>[[Tosefta]] [https://www.sefaria.org.il/Tosefta_Sanhedrin.2.2?lang=bi&with=all&lang2=he Sanhedrin 2:2] "The year may be intercalated on three grounds: ''aviv'' [i.e.the ripeness of barley], fruits of trees, and the equinox. On two of these grounds it should be intercalated, but not on one of them alone."; also quoted in {{harvnb|Stern|2001|p=70}}; see also Talmud, [[Sanhedrin (tractate)|Sanhedrin]] 11b</ref> 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 which is adjusted based on the [[#Molad interval|molad interval]] (a mathematical approximation of the mean time between new moons) and [[#Rosh Hashanah postponement rules|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 {{lang|la|[[Anno Mundi]]}}  ([[Latin]]: "in the year of the world"; {{langx|he|{{Script/Hebrew|מבריאת העולם}}}}, "from the creation of the world", abbreviated AM) according to traditional Jewish interpretation of the [[Chronology of the Bible|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 {{#time:xjY}}, began at sunset on {{#time:j F Y|@{{Hebrew year/rhdatum}}-1 day}} and will end at sunset on {{#time:j F Y|@{{Hebrew year/rhdatum|{{#expr:{{#time:xjY}}+1}}}}-1 day}}.{{efn|This and certain other calculations in this article are now provided by a template ({{tl|Hebrew year/rhdatum}}). This template is mainly sourced from http://www.hebcal.com, though the information is widely available.}}

==Components==
===Days===
{{See also|Zmanim#Evening}}
Based on the classic rabbinic interpretation of {{Bibleverse|Genesis|1:5|HE}} ("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.<ref>{{cite book |url=https://books.google.com/books?id=t8VZga76bw4C&q=%22jewish+day+begins%22+evening&pg=PA169 |title=The Torah For Dummies |first=Arthur |last=Kurzweil |year= 2011 |publisher=John Wiley & Sons |isbn=9781118051832 |via=Google Books}}</ref> Similarly, [[Yom Kippur]], [[Passover]], and [[Shabbat]] are described in the Bible as lasting "from evening to evening".<ref>{{Bibleverse|Leviticus|23:32|HE}}; {{Bibleverse|Exodus|12:18|HE}}; regarding Shabbat ({{Bibleverse|Nehemiah|13:19|HE}}) only the beginning time is mentioned.</ref> The days are therefore figured locally.

[[Halakha|Halachically]], the exact time when days begin or end is uncertain: this time could be either sundown (''shekiah'') or else nightfall (''tzait ha'kochavim'', "when the stars appear"). The time between sundown and nightfall (''bein hashmashot'') is of uncertain status.<ref>{{cite web |title=Zmanim Briefly Defined and Explained |website= chabad.org |url=https://www.chabad.org/library/article_cdo/aid/134527/jewish/Zmanim-Briefly-Defined-and-Explained.htm}}</ref> 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.<ref>{{cite web |first=Willie |last=Roth |url=http://koltorah.org/ravj/The%20International%20Date%20Line%20and%20Halacha.htm |title=The International Date Line and ''Halacha''  |date=March 1, 2002  |url-status=dead |archive-date=July 18, 2011 |website=koltorah.org |archive-url=https://web.archive.org/web/20110718065425/http://koltorah.org/ravj/The%20International%20Date%20Line%20and%20Halacha.htm }}.</ref><ref>"Appendix II: Baal HaMaor's Interpretation of 20b and its Relevance to the Dateline" in ''Talmud Bavli'', Schottenstein Edition, Tractate ''Rosh HaShanah'', Mesorah Publications Ltd. ("ArtScroll") 1999, where "20b" refers to the 20th page 2nd folio of the tractate.</ref> (See [[International date line in Judaism]].)

===Hours===
{{See also|Zmanim#Relative hours|Relative hour}}
Judaism uses multiple systems for dividing hours. In [[Biblical and Talmudic units of measurement#Day|one system]], the 24-hour day is divided into fixed hours equal to {{frac|1|24}} of a day, while each hour is divided into 1080 ''halakim'' (parts, singular: ''[[helek]]''). A part is {{frac|3|1|3}} seconds ({{frac|1|18}} minute). The ultimate ancestor of the ''helek'' was a Babylonian time period called a ''barleycorn'', equal to {{frac|1|72}} of a Babylonian ''time degree'' (1° of celestial rotation).<ref name=Neugebauer1>{{cite journal|first=Otto |last=Neugebauer|title=The Astronomy of Maimonides and its Sources|journal=Hebrew Union College Annual|volume=23|date=1949|pages=321–363 |jstor=23506591}}</ref> 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 hour]]s (''sha'ah z'manit'', also sometimes called "halachic hours"). A relative hour is defined as {{frac|1|12}} 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.<ref>Mishna Berachot 1:2. Note that the mishna specifies that the Shema may be recited "until three hours"; this is understood to mean "until the end of the third hour".</ref>

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").<ref>See e.g. [https://www.myzmanim.com/day.aspx?askdefault=1&vars=27526341&q=jerusalem Zmanim: Jerusalem]</ref>

===Weeks===
{{Further|Week#Judaism}}
The Hebrew week ({{lang|he|שבוע}}, {{lang|he-latn|shavua}}) is a cycle of seven days, mirroring the [[creation week|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 [[Ordinal number (linguistics)|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 [[Hebrew numerals|numerical value]] of the Hebrew letters, for example {{lang|he|יום א׳}} (''Day 1'', or ''Yom Rishon'' ({{lang|he|יום ראשון}})):
{| class="wikitable" 
|-
! Hebrew name
! Abbreviation
! Translation
! English equivalent
|-
| Yom Rishon (יום ראשון)
| {{lang|he|יום א'}}
| First day
| [[Sunset]] on [[Saturday]] to sunset on [[Sunday]]
|-
| Yom Sheni (יום שני)
| {{lang|he|יום ב'}}
| Second day
| Sunset on [[Sunday]] to sunset on Monday
|-
| Yom Shlishi (יום שלישי)
| {{lang|he|יום ג'}}
| Third day
| Sunset on [[Monday]] to sunset on Tuesday
|-
| Yom Revii (יום רביעי)
| {{lang|he|יום ד'}}
| Fourth day
| Sunset on [[Tuesday]] to sunset on Wednesday
|-
| Yom Hamishi (יום חמישי)
| {{lang|he|יום ה'}}
| Fifth day
| Sunset on [[Wednesday]] to sunset on Thursday
|-
| Yom Shishi (יום שישי)
| {{lang|he|יום ו'}}
| Sixth day
| Sunset on [[Thursday]] to sunset on Friday
|-
| Yom Shabbat (יום שבת)
| {{lang|he|יום ש'}}
| 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 myth|Genesis creation account]].<ref>''Hebrew-English Bible,'' {{bibleverse||Genesis|1|HE}}</ref> For example, [[Genesis 1:8]] "... And there was evening and there was morning, a second day" corresponds to ''Yom Sheni'' 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 ''Shabbat'' ({{lang|he|שַׁבָּת}}) can also mean "week",<ref>[https://www.sefaria.org/Jastrow%2C_שַׁבָּת.1 Jastrow: שַׁבָּת]</ref> so that in ritual liturgy a phrase like "Yom Reviʻi beShabbat" means "the fourth day in the week".<ref>For example, when referring to the [https://www.sefaria.org/Siddur_Sefard%2C_Weekday_Shacharit%2C_Song_of_the_Day.15?lang=bi&with=all&lang2=en daily psalm recited in the morning prayer].</ref>

====Days of week of holidays====
{{Main|Days of week on Hebrew calendar}}

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

{| class="wikitable"
|-style="background-color:#DDDDDD;text-align:center;"
|-
![[Purim]]
![[Passover]]<br />(first day)
![[Shavuot]]<br />(first day)
![[Seventeenth of Tamuz|17 Tammuz]]/<br />[[Tisha B'Av]]
![[Rosh Hashanah]]/<br />[[Sukkot]]/<br />[[Shmini Atzeret]]<br />(first day)
![[Yom Kippur]]
![[Chanukah]]<br />(first day)
![[Tenth of Tevet|10 Tevet]]
![[Tu Bishvat]]
![[Purim|Purim Katan]]<br />(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| <small>*Postponed from Shabbat</small>
|}

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|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 "{{lang|he|לא אד"ו ראש, ולא בד"ו פסח}}", 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.<ref>{{Cite web |last=Posner |first=Menachem |title=On Which Days Do Jewish Holidays Begin? |url=https://www.chabad.org/library/article_cdo/aid/5504248/jewish/On-Which-Days-Do-Jewish-Holidays-Begin.htm |website=Chabad.org}}</ref>

===Months===
The Hebrew calendar is a [[lunisolar calendar]], meaning that months are based on [[lunar month]]s, but years are based on [[solar year]]s.{{efn|In contrast, the [[Gregorian calendar]] is a pure [[solar calendar]], while the [[Islamic calendar]] is a pure [[lunar calendar]].}} 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|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 [[Islamic calendar|Islam]]), nowadays Jewish months have generally fixed lengths which approximate the period between new moons. For these reasons, a given month does not always begin on the same day as its astronomical conjunction.<ref name="molad">{{cite web |last1=Bromberg |first1=Irv |date=August 5, 2010 |title=Moon and the Molad of the Hebrew Calendar |url=https://kalendis.free.nf/hebrew/molad.htm?i=1 |access-date=July 20, 2019 |publisher=utoronto.ca}}</ref>

The mean period of the lunar month (precisely, the [[Months#Synodic month|synodic month]]) is very close to 29.5&nbsp;days. Accordingly, the basic Hebrew calendar year is one of twelve lunar months alternating between 29 and 30 days:<ref name=companion>{{cite book|first1=Bonnie |last1=Blackburn |first2= Leofranc |last2=Holford-Strevens|title=The Oxford Companion to the Year: An Exploration of Calendar Customs and Time-reckoning|publisher= Oxford University Press|date= 2000|pages= 722–725|oclc=216353872}}</ref>

{| class="wikitable"
!colspan="2"| Month number*
!rowspan="2"| Hebrew month
!rowspan="2"| Length
!colspan="2"| Range of possible Gregorian dates{{efn|Valid at least for 1999-2050. In other years, the ranges for Kislev through Adar I may be a bit wider. After 2089 the earliest date for most months will be one day later, and from 2214 the last date will be one day later.}}
|-
! <small>Ecclesiastical/<br />biblical</small> !! Civil 
! 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 (Hebrew month)|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 (month)|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 (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]] (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) <br> 30 days more in leap years!! ||
|-style="background:#FFF;"
|colspan=6 style="text-align:center;"|<small>* – For the distinction between numbering systems, see {{slink||New year}} below.</small>
|}

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

Normally the 12th month is named [[Adar]]. During [[leap year]]s, the 12th and 13th months are named Adar I and Adar II (Hebrew: ''Adar [[Aleph (Hebrew)|Aleph]]'' and ''Adar Bet''—"first Adar" and "second adar"). Sources disagree as to which of these months is the "real" Adar, and which is the added leap month.<ref>[https://dinonline.org/2014/03/03/which-is-the-true-adar/ Which is the true Adar?]</ref>

====Justification for leap months====
The Bible does not directly mention the addition of leap months (also known as "embolismic" or "[[Intercalation (timekeeping)|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 (''aviv'').<ref>{{Bibleverse|Deuteronomy|16:1|HE}}, {{Bibleverse|Exodus|23:15|HE}}; see [https://www.daat.ac.il/he-il/hagim/luach_ivri/%D7%94%D7%9C%D7%95%D7%97-%D7%94%D7%A2%D7%91%D7%A8%D7%99/%D7%A0%D7%A1%D7%A4%D7%97%D7%99%D7%9D/aviv.htm למועד חודש האביב]</ref> (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]].<ref>Talmud, Rosh Hashana 21a; see [https://www.daat.ac.il/he-il/hagim/luach_ivri/%D7%94%D7%9C%D7%95%D7%97-%D7%94%D7%A2%D7%91%D7%A8%D7%99/%D7%A0%D7%A1%D7%A4%D7%97%D7%99%D7%9D/aviv.htm למועד חודש האביב] for elaboration.</ref> Similarly, the holidays of [[Shavuot]] and [[Sukkot]] are presumed by the Torah to fall in specific agricultural seasons.<ref>{{Bibleverse|Exodus|23:16|HE}}, {{Bibleverse-nb|Exodus|34:22|HE}}; {{Bibleverse|Leviticus|23:39|HE}}; {{Bibleverse|Deuteronomy|16:9,13|HE}}</ref>

[[Maimonides]], discussing the calendrical rules in his [[Mishneh Torah]] (1178), notes:
<blockquote>
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",<ref>''Hebrew-English Bible'', {{bibleverse||Num|28:14|HE}}.</ref> which implies that we should count the year by months and not by days.<ref name="tcjmwz">[[Mishneh Torah]], Sanctification of the New Moon 1:2; quoted in [https://personal.stevens.edu/~msenator/hand0.pdf Sanctification of the New Moon]. {{webarchive|url=https://web.archive.org/web/20100621084628/http://personal.stevens.edu/~msenator/hand0.pdf |date=2010-06-21 }}. Translated from the Hebrew by Solomon Gandz; supplemented, introduced, and edited by Julian Obermann; with an astronomical commentary by Otto Neugebauer. Yale Judaica Series, Volume 11, New Haven: Yale University Press, 1956.</ref>
</blockquote>

<div style="float:right;margin:10px 0 10px 10px;">{{Hebrew year}}</div>

===Years===
====New year====
[[File:Liten askenasisk sjofar 5380.jpg|right|thumb|A ''[[shofar]]'' 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 year]]s", "[[academic year]]s", and so on. The ''[[Mishnah]]'' (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 [[Rosh Hashanah|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 [[Tu Bishvat|15th thereof]].<ref>[[s:Translation:Mishnah/Seder Moed/Tractate Rosh Hashanah/Chapter 1/1|Rosh Hashanah 1:1]]</ref></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.<ref>''Hebrew-English Bible,'' {{bibleverse||Exodus|12:2|HE}} "This month [Aviv/[[Nisan]]] shall be unto you the beginning of months; it shall be the first month of the year to you."</ref> Thus [[Passover]] (which begins on 15 Nisan) is described in the Torah as falling "in the first month",<ref>''Hebrew-English Bible,'' {{bibleverse||Leviticus|23:5|HE}}</ref> while [[Rosh Hashana]] (which begins on 1 Tishrei) is described as falling "in the seventh month".<ref>''Hebrew-English Bible,'' {{bibleverse||Leviticus|23:24|HE}}</ref>
* 1 Tishrei is the ''civil new year'', and the date on which the year number advances. This date is known as [[Rosh Hashanah]] ({{lit|head of the year}}).  Tishrei marks the end of one ''agricultural'' year and the beginning of another,<ref>''Hebrew-English Bible,'' {{bibleverse||Exodus|23:16|HE}}, {{bibleverse-nb||Exodus|34:22|HE}}</ref> and thus 1 Tishrei is considered the new year for most agriculture-related commandments, including [[Shmita]], Yovel, [[Maaser Rishon]], [[Maaser Sheni]], and [[Maaser Ani]].

For the dates of the Jewish New Year see [[Jewish and Israeli holidays 2000–2050]].

====Anno Mundi====
{{Main|Anno Mundi}}
[[File:Creation of Light Detail 2.png|right|thumb|The Jewish calendar's reference point is traditionally held to be about one year ''before'' the [[Genesis creation myth|Creation]] of the world.]]

The Jewish year number is generally given by ''{{lang|la|Anno Mundi}}'' (from [[Latin]] "in the year of the world", often abbreviated ''AM'' or ''A.M.''). In this [[calendar era]], the year number equals the number of years that have passed since the [[Genesis creation narrative|creation of the world]], according to an interpretation of [[Biblical]] accounts of the creation and subsequent history. From the eleventh century, ''anno mundi'' dating became the dominant method of counting years throughout most of the world's Jewish communities, replacing earlier systems such as the [[Seleucid era]].<ref name=Jones/><ref>{{Cite book |url=https://books.google.com/books?id=0umDqPOf2L8C&pg=PA87 |title=The Easter Computus and the Origins of the Christian Era |author=Alden A. Mosshammer|year= 2008 |isbn=9780191562365|pages=87–89 |publisher=OUP Oxford }}</ref>
As with {{lang|la|[[Anno Domini]]}} (A.D. or AD), the words or abbreviation for {{lang|la|Anno Mundi}} (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&nbsp;p.m., on the evening of Sunday, 6 October 3761 BCE.<ref>Edgar Frank, ''Talmudic and Rabbinical Chronology: The System of Counting Years in Jewish Literature,'' (New York: Philip Feldheim, Publisher, 1956)</ref> According to rabbinic reckoning, this moment was ''not'' [[Creation according to Genesis|Creation]], but about one year "before" Creation, with the new moon of its first month (Tishrei) called ''molad tohu'' (the mean new moon of chaos or nothing). It is about one year ''before'' the traditional Jewish [[Dating Creation|date of Creation]] on 25 Elul AM 1,{{efn|The significance of 25 Elul derives from Adam and Eve being created on the sixth day of creation, 1 Tishrei AM 2. In this view, AM 2 is the actual first year of the world, while AM 1 is a "placeholder" year, so that calendar dates can be assigned to the days of creation.}} based upon the ''Seder Olam Rabbah''.{{efn|A minority opinion places Creation on 25 Adar AM 1, six months earlier, or six months after the modern epoch.}} 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)]]''.

{{anchor|writing}}In Hebrew there are two common ways of writing the year number: with the thousands, called {{lang|he|לפרט גדול}} ("major era"), and without the thousands, called {{lang|he|לפרט קטן}} ("minor era"). Thus, the current year is written as <span style="direction:rtl;">'''{{lang|he|{{#time:xhxjY}}}}'''</span> &lrm;({{#time:xjY}}) using the "major era" and <span style="direction:rtl;">'''{{lang|he|{{#invoke:string|sub|{{#time:xhxjY}}|3|-1}}}}'''</span> &lrm;({{#expr:{{#time:xjY}}mod1000}}) 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 ''Machzor Katan'' ("small cycle").

Because the Julian years are {{frac|365|1|4}} days long, every 28 years the weekday pattern repeats. This is called the sun cycle, or the ''Machzor Gadol'' ("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 (''yovel'') 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]].<ref>B. Zuckermann, ''A Treatise on the Sabbatical Cycle and the Jubilee'', trans. A.&nbsp;Löwy. New York: Hermon Press, 1974.</ref> 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 ''iggul'', or the ''Iggul of [[Nahshon ben Zadok|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 ''molad'' 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.<ref>Nadia Vidro, "The Origins of the 247-Year Calendar Cycle", ''Aleph'', '''17''' (2017), 95–137 [https://doi.org/10.2979/aleph.17.1.0095 doi link].</ref><ref>Dov Fischer, [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4364371 The Enduring Usefulness of the Tur’s 247-year Calendar Cycle (Iggul of Rabbi Nachshon)]</ref>

==Calculations==
===Leap year calculations===
{{See also|Golden number (time)}}
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 {{#time:xjY}} divided by 19 results in a remainder of {{#expr:{{#time:xjY}}mod 19}}, indicating that it is year {{#ifexpr:{{#time:xjY}}mod 19|{{#expr:{{#time:xjY}}mod 19}}|19}} of the Metonic cycle. The Jewish year used is the ''anno mundi'' 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 {{lang|he|גוחאדז״ט}} refers to these years,{{efn|In which the letters refer to [[Hebrew numerals]] equivalent to 3, 6, 8, 1, 4, 7, 9.}} while another memory aid refers to musical notation.{{efn|Intervals of the [[major scale]] follow the same pattern as do Jewish leap years, with ''do'' corresponding to year 19 (or 0): a [[whole step]] in the scale corresponds to two common years between consecutive leap years, and a [[half step]] to one common year between two leap years. This connection with the major scale is more plain in the context of [[19 equal temperament]]: counting the tonic as 0, the notes of the major scale in 19 equal temperament are numbers 0 (or 19), 3, 6, 8, 11, 14, 17, the same numbers as the leap years in the Hebrew calendar.}}

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&nbsp;×&nbsp;''n'')&nbsp;+&nbsp;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 {{Hebrew calendar/c|{{#time:xjY}}}} The {{Hebrew calendar/c|{{#expr:{{#time:xjY}}+1}}}}<ref>{{cite book |last1=Dershowitz |first1=Nachum |last2=Reingold |first2=Edward M. |title=Calendrical Calculations |title-link= Calendrical Calculations |date=2007 |edition=3rd |publisher=Cambridge University Press |page=91}}</ref> This works because as there are seven leap years in nineteen years the difference between the solar and lunar years increases by {{frac|7|19}} month per year. When the difference goes above {{frac|18|19}} 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 {{frac|29|13753|25920}} 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 {{frac|12|7|19}} 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 Calendar|Julian years]] (365.2500 days) make (0.0078 days/year, or one day in 128 years), but much more than what the [[Gregorian Calendar|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 {{frac|3|1|3}} seconds). The very first molad, the [[#Anno Mundi|molad tohu]], fell on Sunday evening at 11:11:20&nbsp;pm in the local time of [[Jerusalem]],<ref name=Tondering>{{cite web |url=https://www.tondering.dk/claus/cal/hebrew.php#newmoon |title=Calendar FAQ: the Hebrew calendar: New moon |first1=Trine |last1=Tøndering |first2= Claus |last2=Tøndering}}</ref>{{efn|UTC+02:20:56.9}} 6 October 3761 BCE ([[Proleptic Julian calendar]]) 20:50:23.1 [[Coordinated Universal Time|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. ({{nowrap|''N'' {{=}} 71440}} 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&nbsp;pm.

In calculating the number of months that will have passed since the known molad that one uses as the starting point, one must remember to include any leap months that falls 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 {{lang|he-Latn|deḥiyyot}}.<ref name=Ibbur>{{cite book|title=Sefer ha-Ibbur |volume= 2|chapter= 9,10 |author=R. Avraham bar Chiya ha-nasi |oclc=729982627 |date=1851 |language=he |location=London}}</ref><ref name=Tur>{{cite book|title=Tur, Orach Chaim ''(section 428)''|url=https://he.wikisource.org/wiki/%D7%98%D7%95%D7%A8_%D7%90%D7%95%D7%A8%D7%97_%D7%97%D7%99%D7%99%D7%9D_%D7%AA%D7%9B%D7%97}}</ref><ref name=HKC>{{cite book|author=Rambam|title=Hilchos Kiddush ha-Chodesh (chapters 6, 7, 8)|url=https://he.wikisource.org/wiki/%D7%A8%D7%9E%D7%91%22%D7%9D_%D7%94%D7%9C%D7%9B%D7%95%D7%AA_%D7%A7%D7%99%D7%93%D7%95%D7%A9_%D7%94%D7%97%D7%95%D7%93%D7%A9_%D7%95}}</ref><ref name=Feldman>{{cite book|author=W. M. Feldman|title=Rabbinical Mathematics and Astronomy|edition=2nd|publisher=Hermon Press|date=1965|chapter=Chapter 17: The Fixed Calendar}}</ref><ref name=Mandelbaum>{{cite book|author=Hugo Mandelbaum|chapter=Introduction: Elements of the Calendar Calculations|editor=Arthur Spier|title=The Comprehensive Hebrew Calendar|edition=3rd|date=1986}}</ref> 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 {{lang|he-Latn|deḥiyyat molad zaken}} ({{lang|he|דְחִיַּת מוֹלָד זָקֵן}}, literally, "old birth", i.e., late new moon). This rule is mentioned in the Talmud,<ref name="epstein"/> and is used nowadays to prevent the molad falling on the second day of the month.<ref>{{cite web|last=Landau|first=Remy|url=http://hebrewcalendar.tripod.com/#25|title=Hebrew Calendar Science and Myth: 'The Debatable Dehiyah Molad Zaquen'|access-date=7 February 2015}}</ref> 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 {{lang|he-Latn|deḥiyyah molad zaken}} would place Rosh Hashanah on one of these days, then it is postponed a second day. This is called {{lang|he-Latn|deḥiyyat lo ADU}} ({{lang|he|דְחִיַּת לֹא אד״ו}}), 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]].{{efn|This is the reason given by most [[halachic]] authorities, based on the [[Talmud]], Rosh Hashanah 20b and Sukkah 43b. [[Maimonides]] ([[Mishneh Torah]], Kiddush Hachodesh 7:7), however, writes that the arrangement was made (possible days alternating with impossible ones) in order to average out the difference between the mean and true [[lunar conjunction]]s.}} 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 [[Shabbat candles|candle lighting]]).{{efn|The Talmud (Rosh Hashanah 20b) puts it differently: over two consecutive days of full Shabbat restrictions, vegetables would wilt (since they can't be cooked), and unburied corpses would putrefy.}} Additionally, the laws of Shabbat override those of Hoshana Rabbah, so that if Hoshana Rabbah were to fall on Shabbat, the Hoshana Rabbah ''aravah'' ritual could not be performed.<ref>Yerushalmi, [https://www.sefaria.org/Jerusalem_Talmud_Sukkah.4.1.3 Sukkah 4:1] (18a, 54b)</ref>
:Thus Rosh Hashanah can only fall on Monday, Tuesday, Thursday, and Saturday. The ''kevi'ah'' 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.

Another two 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 {{lang|he-Latn|deḥiyyat GaTaRaD}} ({{lang|he|דְחִיַּת גטר״ד}}, 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&nbsp;pm Sunday), Rosh Hashanah is postponed to Tuesday. This is {{lang|he-Latn|deḥiyyat BeTUTeKaPoT}} ({{lang|he|דְחִיַּת בט״ו תקפ״ט}}), 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.<ref name=companion/>

*A {{transliteration|he|chaserah}} year (Hebrew for "deficient" or "incomplete") is 353 or 383 days long. Both Cheshvan and Kislev have 29 days.
*A {{transliteration|he|kesidrah}} year ("regular" or "in-order") is 354 or 384 days long. Cheshvan has 29 days while Kislev has 30 days.
*A {{transliteration|he|shlemah}} year ("complete" or "perfect", also "abundant") is 355 or 385 days long. Both Cheshvan and Kislev have 30 days.

Whether a year is deficient, regular, or complete is determined by the time between two adjacent Rosh Hashanah observances and the leap year.

A Metonic cycle equates to 235 lunar months in each 19-year cycle. This gives an average of 6,939 days, 16 hours, and 595 parts for each cycle.<ref name=weinberg/> But due to the Rosh Hashanah postponement rules (preceding section) a cycle of 19 Jewish years can be either 6,939, 6,940, 6,941, or 6,942 days in duration. For any given year in the Metonic cycle, the molad moves forward in the week by 2 days, 16 hours, and 595 parts every 19 years. The [[greatest common divisor]] of this and a week is 5 parts, so the Jewish calendar repeats exactly following a number of Metonic cycles equal to the number of parts in a week divided by 5, namely 7×24×216&nbsp;=&nbsp;36,288 Metonic cycles, or 689,472 Jewish years. There is a near-repetition every 247 years, except for an excess of 50 minutes {{frac|16|2|3}} seconds (905 parts).

Contrary to popular impression, one's Hebrew birthday does not necessarily fall on the same Gregorian date every 19 years, since the length of the Metonic cycle varies by several days (as does the length of a 19-year Gregorian period, depending whether it contains 4 or 5 leap years).<ref>[https://outorah.org/p/5696/ Tzarich Iyun: Your Hebrew Birthday]</ref>

===Keviah===
{| 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 ''Kevi'ah'' 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 [[combination]]s, but only 14 of them are valid.

Each of these patterns is known by a {{lang|he|kevi'ah}} ({{langx|he|קביעה}} 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 {{nowrap|1 Tishrei}}, Rosh Hashanah {{nowrap|(2 3 5 7; Hebrew: ב ג ה ז)}}
* The letter indicates whether that year is deficient (D, "ח", from {{Langx|he|חסרה|Chasera}}), regular (R, "כ", from {{Langx|he|כסדרה|Kesidra}}), or complete (C, "ש", from {{Langx|he|שלמה|Shlema}})
* The right number is the day of the week of {{nowrap|15 Nisan}}, the first day of Passover or Pesach {{nowrap|(1 3 5 7; Hebrew: א ג ה ז)}}, within the same Hebrew year (next Julian/Gregorian year)
The {{transliteration|he|kevi'ah}} in Hebrew letters is written right-to-left, so their days of the week are reversed, the right number for {{nowrap|1 Tishrei}} and the left for {{nowrap|15 Nisan}}.

The ''kevi'ah'' also determines the [[Torah reading]] cycle (which ''parshiyot'' are read together or separately.<ref name="Judaism 101">{{cite web|title=The Jewish Calendar: A Closer Look|url=http://www.jewfaq.org/calendr2.htm|publisher=Judaism 101|access-date=25 March 2011}}</ref>

=== The four gates ===
The ''keviah'', 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 [[#Rosh Hashanah postponement rules|molad Tishrei]].<ref name=biruni>{{citation |author=al-Biruni |title=The Chronology of Ancient Nations |url=https://archive.org/details/chronologyofanci00biru/page/150 |translator-last=Sachau |translator-first=C. Edward |year=1879 |orig-year=1000}}</ref>{{rp|150–152}}<ref>{{cite book |last=Bushwick |first=Nathan |year=1989 |title=Understanding the Jewish Calendar |location=New York/Jerusalem |publisher=Moznaim |isbn=0-940118-17-3 |pages=95–97}}</ref><ref>{{cite encyclopedia |last=Poznanski |first=Samuel |year=1910 |title=Calendar (Jewish) |encyclopedia=Encyclopædia of Religion and Ethics |editor-last=Hastings |editor-first=James |editor-link=James Hastings |title-link=Encyclopædia of Religion and Ethics |publisher=T. & T. Clark |location=Edinburgh |volume=3 |page=121 |url=https://archive.org/stream/encyclopaediaofr003hast#page/120/mode/2up |quote=limits, Qebi'oth [kevi'ot]}}</ref><ref>{{cite journal |last=Resnikoff |first=Louis A. |title=Jewish Calendar Calculations |page=276 |journal=Scripta Mathematica |volume=9 |year=1943}}</ref>{{efn|In the Four Gates sources ({{transliteration|he|kevi'ot}} cited here are in Hebrew in sources except al-Biruni): al-Biruni specified 5R (5 Intermediate) instead of 5D in leap years. Bushwick forgot to include 5D for leap years. Poznanski forgot to include 5D for a limit in his table although he did include it in his text as 5D1; for leap years he incorrectly listed 5C7 instead of the correct 5C3. Resnikoff's table is correct.}}<ref>{{cite web |first=Robert |last=Schram |date=1908  |title={{lang|de|Kalendariographische und Chronologische Tafeln|nocat=yes}} |url=https://archive.org/stream/kalendariograph00schrgoog#page/n231/mode/2up |pages= xxiii–xxvi, 190–238|publisher=Leipzig, J. C. Hinrichs }} Schram gives the type of Hebrew year for all years 1–6149 AM (−3760 to 2388 Julian/Gregorian) in a main table (3946+) and its adjunct (1+, 1742+) on pages 191–234 in the form 2d, 2a, 3r, 5r, 5a, 7d, 7a for common years and 2D, 2A, 3R, 5D, 5A, 7D, 7A for leap years. The type of year 1&nbsp;AM, 2a, is on page 200 at the far right.</ref> 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 {{nowrap|(1 4 9 12 15)}}; common years between two leap years {{nowrap|(7 18)}}; common years after a common year but before a leap year {{nowrap|(2 5 10 13 16)}}; and leap years {{nowrap|(3 6 8 11 14 17 19)}}.<ref>[https://hakirah.org/vol20AjdlerAppendices.pdf A Short History of the Jewish Fixed Calendar : Appendices].</ref>

This table<ref name="biruni" />{{rp|150}}<ref name=ajdler/>{{rp|183}} numbers the days of the week and hours for the limits of molad Tishrei in the Hebrew manner for calendrical calculations, that is, both begin at {{nowrap|6 pm}}, thus {{nowrap|7d 18h 0p}} is noon Saturday, with the week starting on {{nowrap|1d 0h 0p}} (Saturday 6pm, i.e. the beginning of Sunday reckoned in the Hebrew manner). The oldest surviving table of Four Gates was written by [[Muhammad ibn Musa al-Khwarizmi]] in [[824]].<ref>{{cite web | url=http://www.jphogendijk.nl/khwarizmi.html#JewCal | title=Muhammad ibn Musa (Al-)Khwarizmi (Or Kharazmi) (Ca. 780–850 CE) }}</ref>

{| 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 <br /> 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
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 7d 18h 0p
| style="border: 1px solid #707070; |
| style="border: 1px #707070; border-style: solid hidden; text-align: center;" | '''2D3''' &nbsp; {{resize|135%|בחג}}
| style="border: 1px solid #707070;" |
| style="border: 1px #B0B0B0; border-style: solid solid hidden; text-align: center;" | '''2D5''' &nbsp; {{resize|135%|בחה}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 1d 9h 204p
| style="border: 1px #707070; border-style: solid solid hidden;" colspan="3" |
| style="border: 1px solid; border-color: #707070 #B0B0B0;" | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 1d 20h 491p
| style="border: 1px solid #707070;" |
| style="border: 1px #707070; border-style: solid hidden; text-align: center;" | '''2C5''' &nbsp; {{resize|135%|בשה}} 
| style="border: 1px #707070; border-style: hidden solid;" |
| style="border: 1px #B0B0B0; border-style: solid solid hidden; text-align: center;" | '''2C7''' &nbsp; {{resize|135%|בשז}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 2d 15h 589p
| style="border: 1px #707070; border-style: solid solid hidden;" colspan="2" |
| style="border: 1px solid #707070;" |
| style="border: 1px solid; border-color: #707070 #B0B0B0;" | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 2d 18h 0p
| style="border: 1px solid #707070;" |
| style="border: 1px #707070; border-style: solid hidden; text-align: center;" | '''3R5''' &nbsp; {{resize|135%|גכה}}
| style="border: 1px solid #707070;" |
| style="border: 1px #B0B0B0; border-style: solid solid hidden; text-align: center;" | '''3R7''' &nbsp; {{resize|135%|גכז}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 3d 9h 204p
| style="border: 1px #707070; border-style: solid hidden solid solid;" rowspan="3" |
| style="border: 1px #707070; border-style: solid hidden; text-align: center;" rowspan="3" | '''5R7''' &nbsp; {{resize|135%|הכז}}
| style="border: 1px #707070; border-style: solid solid solid hidden;" rowspan="3" |
| style="border: 1px solid; border-color: #707070 #B0B0B0;" | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 3d 18h 0p
| style="border: 1px solid; border-color: #707070 #B0B0B0; text-align: center;" | '''5D1''' &nbsp; {{resize|135%|החא}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 4d 11h 695p
| style="border: 1px #B0B0B0; border-style: solid solid hidden;" | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 5d 9h 204p
| style="border: 1px solid #707070;" |
| style="border: 1px #707070; border-style: solid hidden; text-align: center;" | '''5C1''' &nbsp; {{resize|135%|השא}}
| style="border: 1px solid #707070;" |
| style="border: 1px solid; border-color: #707070 #B0B0B0; text-align: center;" | '''5C3''' &nbsp; {{resize|135%|השג}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 5d 18h 0p
| style="border: 1px solid #707070;" |
| style="border: 1px #707070; border-style: solid solid hidden hidden;" colspan="2" |
| style="border: 1px #B0B0B0; border-style: solid solid hidden; | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 6d 0h 408p
| style="border: 1px #707070; border-style: solid solid hidden;" |
| style="border: 1px #707070; border-style: hidden hidden solid solid; text-align: center;" | '''7D1''' &nbsp; {{resize|135%|זחא}}
| style="border: 1px solid #707070;" |
| style="border: 1px #B0B0B0; border-style: hidden solid; text-align: center;" | '''7D3''' &nbsp; {{resize|135%|זחג}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 6d 9h 204p
| style="border: 1px #B0B0B0; border-style: hidden hidden solid solid;" rowspan="2" |
| style="border: 1px #707070; border-style: solid solid hidden hidden;" colspan="2" |
| style="border: 1px solid; border-color: #707070 #B0B0B0;" | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding: 0px 10px;" | 6d 20h 491p
| style="border: 1px #B0B0B0; border-style: hidden hidden solid; text-align: center;" | '''7C3''' &nbsp; {{resize|135%|זשג}}
| style="border: 1px; border-style: hidden solid solid hidden; border-color: #A0A0A0 #707070;" |
| style="border: 1px solid #B0B0B0; text-align: center;" | '''7C5''' &nbsp; {{resize|135%|זשה}}
|}
====Incidence====
Comparing the days of the week of molad Tishrei with those in the {{transliteration|he|kevi'ah}} shows that during 39% of years {{nowrap|1 Tishrei}} is not postponed beyond the day of the week of its molad Tishrei, 47% are postponed one day, and 14% are postponed two days. This table also identifies the seven types of common years and seven types of leap years. Most are represented in any 19-year cycle, except one or two may be in neighboring cycles. The most likely type of year is 5R7 in 18.1% of years, whereas the least likely is 5C1 in 3.3% of years. The day of the week of {{nowrap|15 Nisan}} is later than that of {{nowrap|1 Tishrei}} by one, two or three days for common years and three, four or five days for leap years in deficient, regular or complete years, respectively.

{| class="wikitable"
|+Incidence (percentage)
! colspan="2" |common years
! colspan="2" |leap years
|-
|'''5R7'''
|18.05
|'''5C3'''
|6.66
|-
|'''7C3'''
|13.72
|'''7D3'''
|5.8
|-
|'''2C5'''
|11.8
|'''2D5'''
|5.8
|-
|'''3R5'''
|6.25
|'''3R7'''
|5.26
|-
|'''2D3'''
|5.71
|'''2C7'''
|4.72
|-
|'''7D1'''
|4.33
|'''7C5'''
|4.72
|-
|'''5C1'''
|3.31
|'''5D1'''
|3.87
|}

===Worked example===
Given the length of the year, the length of each month is fixed as described above, so the real problem in determining the calendar for a year is determining the number of days in the year. In the modern calendar, this is determined in the following manner.{{efn|The following description is based on the article "Calendar" in Encyclopaedia Judaica (Jerusalem: Ketter, 1972). It is an explanatory description, not a procedural one, in particular explaining what is going on with the third and fourth ''deḥiyyot''}}

The day of Rosh Hashanah and the length of the year are determined by the time and the day of the week of the Tishrei ''molad'', that is, the moment of the average conjunction. Given the Tishrei ''molad'' of a certain year, the length of the year is determined as follows:

First, one must determine whether each year is an ordinary or leap year by its position in the 19-year Metonic cycle. Years 3, 6, 8, 11, 14, 17, and 19 are leap years.

Secondly, one must determine the number of days between the starting Tishrei ''molad'' (TM1) and the Tishrei ''molad'' of the next year (TM2). For calendar descriptions in general the day begins at 6&nbsp;pm, but for the purpose of determining Rosh Hashanah, a ''molad'' occurring on or after noon is treated as belonging to the next day (the first ''deḥiyyah'').{{efn|So for example if the Tishrei molad is calculated as occurring from noon on Wednesday (the 18th hour of the fourth day) up until noon on Thursday, Rosh Hashanah falls on a Thursday, which starts Wednesday at sunset wherever one happens to be.}} All months are calculated as 29d, 12h, 44m, {{fraction|3|1|3}}s long (MonLen). Therefore, in an ordinary year TM2 occurs 12 × MonLen days after TM1. This is usually 354 calendar days after TM1, but if TM1 is on or after 3:11:20&nbsp;am and before noon, it will be 355 days. Similarly, in a leap year, TM2 occurs 13 × MonLen days after TM1. This is usually 384 days after TM1, but if TM1 is on or after noon and before {{fraction|2:27:16|2|3}}&nbsp;pm, TM2 will be only 383 days after TM1. In the same way, from TM2 one calculates TM3. Thus the four natural year lengths are 354, 355, 383, and 384 days.

However, because of the holiday rules, Rosh Hashanah cannot fall on a Sunday, Wednesday, or Friday, so if TM2 is one of those days, Rosh Hashanah in year 2 is postponed by adding one day to year 1 (the second ''deḥiyyah''). To compensate, one day is subtracted from year 2. It is to allow for these adjustments that the system allows 385-day years (long leap) and 353-day years (short ordinary) besides the four natural year lengths.

But how can year 1 be lengthened if it is already a long ordinary year of 355 days or year 2 be shortened if it is a short leap year of 383 days? That is why the third and fourth ''deḥiyyah''s are needed.

If year 1 is already a long ordinary year of 355 days, there will be a problem if TM1 is on a Tuesday,{{efn|This will happen if TM1 is on or after 3:11:20 am and before noon on a Tuesday. If TM1 is Monday, Thursday or Saturday, Rosh Hashanah in year 2 does not need to be postponed. If TM1 is Sunday, Wednesday or Friday, Rosh Hashanah in year 1 is postponed, so year 1 is not the maximum length.}} as that means TM2 falls on a Sunday and will have to be postponed, creating a 356-day year. In this case, Rosh Hashanah in year 1 is postponed from Tuesday (the third ''deḥiyyah''). As it cannot be postponed to Wednesday, it is postponed to Thursday, and year 1 ends up with 354 days.

On the other hand, if year 2 is already a short year of 383 days, there will be a problem if TM2 is on a Wednesday.{{efn|TM2 will be between noon and {{fraction|2:27:16|2|3}}&nbsp;pm on Tuesday, and TM3 will be between {{fraction|9:32:43|1|3}} and noon on Monday.}} because Rosh Hashanah in year 2 will have to be postponed from Wednesday to Thursday and this will cause year 2 to be only 382 days long. In this case, year 2 is extended by one day by postponing Rosh Hashanah in year 3 from Monday to Tuesday (the fourth ''deḥiyyah''), and year 2 will have 383 days.

=== Holidays ===
For calculated dates of Jewish holidays, see [[Jewish and Israeli holidays 2000–2050]].

==Accuracy==
===Molad interval===
A "[[new moon]]" (astronomically called a [[lunar conjunction]] and, in Hebrew, a [[molad]]) is the moment at which the sun and moon have the same [[Ecliptic coordinate system|ecliptic longitude]] (i.e. they are aligned horizontally with respect to a north–south line). The period between two new moons is a [[synodic month]]. The actual length of a synodic month varies from about 29 days 6 hours and 30 minutes (29.27 days) to about 29 days and 20 hours (29.83 days), a variation range of about 13 hours and 30 minutes. Accordingly, for convenience, the Hebrew calendar uses a long-term average month length, known as the '''molad interval''', which equals the [[Lunar month#Synodic month|mean synodic month]] of ancient times. The molad interval is 29 days, 12 hours, and 793 "parts" (1 "part" = <sup>1</sup>/<sub>18</sub> minute = 3<sup>1</sup>/<sub>3</sub> seconds) (i.e., 29.530594 days), and is the same value determined by the Babylonians in their [[Babylonian mathematics|System B]] about 300 BCE<ref name=Neugebauer2>Neugebauer, ''Astronomical cuneiform texts'', Vol 1, pp. 271–273</ref> and was adopted by [[Hipparchus]] (2nd century BCE) and by [[Ptolemy]] in the ''[[Almagest]]'' (2nd century CE). Its remarkable accuracy (less than one second from the current true value) is thought to have been achieved using records of lunar eclipses from the 8th to 5th centuries BCE.<ref>[[G. J. Toomer]], Hipparchus' Empirical Basis for his Lunar Mean Motions, ''Centaurus'', Vol 24, 1980, pp. 97–109</ref> In the Talmudic era, when the mean synodic month was slightly shorter than at present, the molad interval was even more accurate, being "essentially a perfect fit" for the mean synodic month at the time.<ref name=molad/>

Currently, the accumulated drift in the moladot since the Talmudic era has reached a total of approximately 97 minutes.<ref name=molad/> This means that the molad of Tishrei lands one day later than it ought to in (97 minutes) ÷ (1440 minutes per day) = nearly 7% of years. Therefore, the seemingly small drift of the moladot is already significant enough to affect the date of Rosh Hashanah, which then cascades to many other dates in the calendar year, and sometimes (due to the Rosh Hashanah postponement rules) also interacts with the dates of the prior or next year.

The rate of calendar drift is increasing with time, since the mean synodic month is progressively shortening due to gravitational [[tide|tidal]] effects. Measured on a strictly uniform time scale (such as that provided by an [[atomic clock]]) the mean synodic month is becoming gradually longer, but since the tides slow Earth's rotation rate even more, the mean synodic month is becoming gradually shorter in terms of mean solar time.<ref name=molad/>

===Metonic cycle drift===
A larger source of error is the inaccuracy of the Metonic cycle. Nineteen Jewish years average 6939d 16h 33m 03{{fraction|1|3}}s, compared to the 6939d 14h 26m 15s of nineteen mean solar years.<ref name=weinberg>Weinberg, I., [http://adsabs.harvard.edu/full/1956MNSSA..15...86W Astronomical Aspects of the Jewish Calendar], Monthly Notes of the Astronomical Society of South Africa, Vol. 15, p. 86.</ref> Thus, the Hebrew calendar drifts by just over 2 hours every 19 years, or approximately one day every 216 years.<ref name="richards">{{cite book| publisher = Oxford University Press| isbn = 978-0-19-286205-1| last = Richards| first = E. G| title = Mapping time: the calendar and its history| page = 224| url = https://archive.org/details/mappingtimecalen00rich| url-access = registration| date = 1998}}</ref><ref name=aviv>[https://www.daat.ac.il/he-il/hagim/luach_ivri/הלוח-העברי/נספחים/aviv.htm למועד חודש האביב]</ref> Due to accumulation of this discrepancy, the earliest date on which Passover can fall has drifted by roughly eight days since the 4th century, and the 15th of Nisan now falls only on or after 26 March (the date in 2013), five days after the actual equinox on 21 March. In the distant future, this drift is projected to move Passover much further in the year.<ref name=aviv/> If the calendar is not amended, then Passover will start to land on or after the summer solstice around approximately AM 16652 (12892 CE).{{efn|The exact year when this will begin to occur depends on uncertainties in the future tidal slowing of the Earth rotation rate, and on the accuracy of predictions of precession and Earth axial tilt.}}

===Implications for Jewish ritual===
When the calendar was fixed in the 4th century, the earliest Passover (in year 16 of the Metonic cycle) began on the first full moon after the [[March equinox]].{{efn|reference=That is to say, Passover began within a day or so of the full moon}} This is still the case in about 80% of years; but, in about 20% of years, Passover is a month late by this criterion.{{efn|reference=As it was in AM 5765, 5768 and 5776, the 8th, 11th and 19th years of the 19-year cycle = Gregorian 2005, 2008 and 2016 CE.}} Presently, this occurs after the "premature" insertion of a leap month in years 8, 11, and 19 of each 19-year cycle, which causes Passover to fall especially far after the March equinox in such years. Calendar drift also impacts the observance of [[Sukkot]], which will shift into Israel's winter rainy season, making dwelling in the [[sukkah]] less practical. It also affects the logic of the [[Shemini Atzeret]] prayer for rain, which will be more often recited once rains are already underway.

Modern scholars have debated at which point the drift could become ritually problematic, and proposed adjustments to the fixed calendar to keep Passover in its proper season.<ref name=aviv/> The seriousness of the calendar drift is discounted by many, on the grounds that Passover will remain in the spring season for many millennia, and the Torah is generally not interpreted as having specified tight calendrical limits. However, some writers and researchers have proposed "corrected" calendars (with modifications to the leap year cycle, molad interval, or both) which would compensate for these issues:

*Irv Bromberg has suggested a 353-year cycle of 4,366 months, which would include 130 leap months, along with use of a progressively shorter ''molad'' interval, which would keep an amended fixed arithmetic Hebrew calendar from drifting for more than seven millennia.<ref name=irv353>{{Cite web|first=Irv|last=Bromberg|title=The Rectified Hebrew Calendar.|url=http://individual.utoronto.ca/kalendis/hebrew/rect.htm|work= University of Toronto |access-date=13 May 2011 }}</ref> The 353 years would consist of 18 Metonic cycles, as well as an 11-year period in which the last 8 years of the Metonic cycle are omitted.<ref name=irv353/>
*Other authors have proposed to use cycles of 334 or 687 years.<ref name=aviv/>
*Another suggestion is to delay the [[leap years]] gradually so that a whole intercalary month is taken out at the end of Iggul 26; while also changing the [[synodic month]] to be the more accurate 29.53058868 days. Thus, the length of the year would be {{nowrap|(235 × 13 × 26 − 1)/(19 × 13 × 26) {{=}} 365.2422 days,}} very close to the actual [[tropical year]]. The result is the "Hebrew Calendar" in the program CalMaster2000.<ref>A. O. Scheffler and P. P. Scheffler, "Calmaster2000: Dates, Holidays, Astronomical Events". Pittsburgh, PA: Zephyr Services.</ref>

Religious questions abound about how such a system might be implemented and administered throughout the diverse aspects of the world Jewish community.<ref>{{cite web|url=http://www.thesanhedrin.org/en/index.php/Committee_concerning_the_fixing_of_the_Calendar|title=Committee concerning the fixing of the Calendar |work= The Sanhedrin }}</ref>

==Usage==
===In Auschwitz===
While imprisoned in [[Auschwitz concentration camp|Auschwitz]], Jews made every effort to preserve Jewish tradition in the camps, despite the monumental dangers in doing so. The Hebrew calendar, which is a tradition with great importance to Jewish practice and rituals was particularly dangerous since no tools of telling of time, such as watches and calendars, were permitted in the camps.<ref name=":0">{{Cite journal |title=Tracking Jewish time in Auschwitz |first=Alan |last=Rosen |year=2014 |journal=Yad Vashem Studies |volume=42 |issue=2 |page=41 |oclc=1029349665}}</ref> The keeping of a Hebrew calendar was a rarity amongst prisoners and there are only two known surviving calendars that were made in Auschwitz, both of which were made by women.<ref name=":0" /> Before this, the tradition of making a Hebrew calendar was greatly assumed to be the job of a man in Jewish society.<ref name=":0" />

===In contemporary Israel===
{{History of Israel}}
Early [[Zionist]] pioneers were impressed by the fact that the calendar preserved by Jews over many centuries in far-flung diasporas, as a matter of religious ritual, was geared to the climate of their original country: major Jewish holidays such as [[Sukkot]], [[Passover]], and [[Shavuot]] correspond to major points of the country's agricultural year such as planting and harvest. Accordingly, in the early 20th century the Hebrew calendar was re-interpreted as an agricultural rather than religious calendar.

After the creation of the [[State of Israel]], the Hebrew calendar became one of the official calendars of Israel, along with the [[Gregorian calendar]]. Holidays and commemorations not derived from previous Jewish tradition were to be fixed according to the Hebrew calendar date. For example, the Israeli Independence Day falls on 5 [[Iyar]], Jerusalem Reunification Day on 28 Iyar, [[Yom HaAliyah]] on 10 Nisan, and the Holocaust Commemoration Day on 27 [[Nisan]].

The Hebrew calendar is still widely acknowledged, appearing in public venues such as banks (where it is legal for use on cheques and other documents),<ref>[https://www.nevo.co.il/law_html/law01/p220m2_001.htm חוק השימוש בתאריך העברי, תשנ"ח-1998]</ref><ref>[https://www.aish.co.il/i/j/195842211.html צ'ק עם תאריך עברי?!]</ref> and on the mastheads of newspapers.<ref>{{cite web |url=http://www.israelnationalnews.com/ |title=Arutz Sheva }}; {{cite web |url=https://www.ynet.co.il |title=Yedioth Ahronoth }}; {{cite web |url=https://www.makorrishon.co.il/ |title=Makor Rishon }}; {{cite web |url=https://www.israelhayom.co.il/ |title=Israel HaYom }}; {{cite web |url=https://www.haaretz.co.il/ |title=Haaretz }}; {{cite web |url=https://www.themarker.com/ |title=The Marker }}; {{cite web |url=https://www.maariv.co.il/ |title=Maariv }}</ref>

The Jewish New Year (Rosh Hashanah) is a two-day public holiday in Israel. However, since the 1980s an increasing number of secular Israelis celebrate the Gregorian New Year (usually known as "[[Silvester]] Night"—{{lang|he|ליל סילבסטר}}) on the night between 31 December and 1 January. Prominent rabbis have on several occasions sharply denounced this practice, but with no noticeable effect on the secularist celebrants.<ref>{{cite news |url=http://www.israelnationalnews.com/News/News.aspx/163462#.Upe6pJuA2rY |title=Rabbinate: New Year's Eve Parties 'Not Kosher' |newspaper=[[Arutz Sheva]] |author=David Lev |date=23 December 2012 |access-date=30 November 2013}}</ref>

Wall calendars commonly used in Israel are hybrids. Most are organised according to Gregorian rather than Jewish months, but begin in September, when the Jewish New Year usually falls, and provide the Jewish date in small characters.

==History==
=== Early formation ===
Lunisolar calendars similar to the Hebrew calendar, consisting of twelve lunar months plus an occasional 13th [[Intercalation (timekeeping)|intercalary]] month to synchronize with the solar/agricultural cycle, were used in all ancient Middle Eastern civilizations except Egypt, and likely date to the 3rd millennium BCE.<ref>Britannica: [https://www.britannica.com/science/calendar/Ancient-and-religious-calendar-systems#ref313439 Calendar - Ancient, Religious, Systems]</ref> While there is no mention of this 13th month anywhere in the Hebrew Bible,<ref name="DeVaux">''Ancient Israel: Its Life and Institutions'' (1961) by Roland De Vaux, John McHugh, Publisher: McGraw–Hill, {{ISBN|978-0-8028-4278-7}}, p. 179</ref> still most Biblical scholars hold that the intercalation process was almost certainly a regularly occurring aspect of the early Hebrew calendar keeping process.<ref>[https://www.thetorah.com/article/what-is-the-bibles-calendar What Is the Bible’s Calendar?] The Torah.com. By Prof. Sacha Stern. Retrieved 2023-07-22.</ref>

===Month names===
[[File:Hebrew Calendar (Luach).jpg|thumb|Calendar for the year 1840/41. Printed by I. Lehrberger u. Comp., [[Rödelheim]]. In the collection of the [[Jewish Museum of Switzerland]].]]
Biblical references to the [[Babylonian captivity|pre-exilic]] calendar include ten of the twelve months identified by number rather than by name.

Prior to the [[Babylonian captivity]], the names of only four months are referred to in the [[Tanakh]]: ''[[Aviv]]'' (first month),<ref>{{bibleverse||Exodus|12:2|HE}}, {{bibleverse-nb||Exodus|13:4|HE}}, {{bibleverse-nb||Exodus|23:15|HE}}, {{bibleverse-nb||Exodus|34:18|HE}}, {{bibleverse||Deut.|16:1|HE}}</ref> ''[[Iyar|Ziv]]'' (second month),<ref>{{bibleverse|1|Kings|6:1|HE}}, {{bibleverse-nb|1|Kings|6:37|HE}}</ref> ''[[Ethanim]]'' (seventh month),<ref>{{bibleverse|1|Kings|8:2|HE}}</ref> and ''[[Cheshvan|Bul]]'' (eighth month).<ref>{{bibleverse|1|Kings|6:38|HE}}</ref> All of these are believed to be [[Canaanite languages|Canaanite names]].<ref>{{cite book|last=Hachlili|first=Rachel|title=Ancient Synagogues – Archaeology and Art: New Discoveries and Current Research|date=2013|publisher=Brill|isbn=978-9004257733|page=342|url=https://books.google.com/books?id=jRjhAQAAQBAJ&pg=PA342}}</ref> The last three of these names are only mentioned in connection with the building of the [[First Temple]] and Håkan Ulfgard suggests that the use of what are rarely used Canaanite (or in the case of Ethanim perhaps [[Northwest Semitic languages|Northwest Semitic]]) names indicates that "the author is consciously utilizing an archaizing terminology, thus giving the impression of an ancient story...".<ref>{{cite book|last=Ulfgard|first=Håkan|title=The Story of Sukkot : the Setting, Shaping and Sequel of the biblical Feast of Tabernacles|date=1998|publisher=Mohr Siebeck|isbn=3-16-147017-6|page=99|url=https://books.google.com/books?id=uxnXaYBj2wgC&pg=PA99}}</ref> Alternatively, these names may be attributed to the presence of Phoenician scribes in Solomon's court at the time of the building of the Temple.<ref>Seth L. Sanders, “Writing and Early Iron Age Israel: Before National Scripts, Beyond Nations and States,” in ''Literate Culture and Tenth-Century Canaan: The Tel Zayit Abecedary in Context'', ed. Ron E. Tappy and P. Kyle McCarter, (Winona Lake, IN, 2008), p. 101–102</ref>

During the Babylonian captivity, the Jewish people adopted the Babylonian names for the months. The [[Babylonian calendar]] descended directly from the Sumerian calendar.<ref>{{Cite web |url=http://www.ehebrew.org/articles/hebrew-calendar.php |title=Hebrew Calendar |archive-url=https://web.archive.org/web/20190721072923/http://www.ehebrew.org/articles/hebrew-calendar.php#.XTQUVo77SUk |archive-date=21 July 2019}}</ref> These Babylonian month-names (such as Nisan, Iyyar, Tammuz, Ab, Elul, Tishri and Adar) are shared with the modern [[Arabic names of Gregorian months#Syrian Calendar in Iraq, Syria, Lebanon, Palestine and Jordan|Levantine solar calendar]] (currently used in the [[Arabic]]-speaking countries of the [[Fertile Crescent]]) and the modern [[Assyrian calendar]], indicating a common origin.<ref name="DeVaux"/> The origin is thought to be the Babylonian calendar.<ref name="DeVaux"/>

{| class="wikitable"
|+Hebrew names of the months with their Babylonian analogs
! # 
! Hebrew
! [[Tiberian vocalization|Tiberian]]
! [[Academy of the Hebrew Language|Academy]]
! Common/<br>Other
! Length
! Babylonian analog
! Holidays/<br>Notable days
! Notes
|-
|  1 || style="text-align:right;" | {{Script/Hebrew|נִיסָן}} || Nīsān || [[Nisan]] || Nissan || {{nowrap|30 days}} || ''Nisanu'' || [[Passover]] || Called ''Abib''<ref>''Hebrew-English Bible'', {{bibleverse||Exodus|13:4|HE}}, {{bibleverse-nb||Exodus|23:15|HE}}, {{bibleverse-nb||Exodus|34:18|HE}}, {{bibleverse||Deut.|16:1|HE}}</ref> and Nisan<ref>''Hebrew-English Bible'', {{bibleverse||Esther|3:7|HE}}</ref> in the [[Tanakh]].
|-
|  2 || style="text-align:right;" | {{Script/Hebrew|אִיָּר / אִייָר}} || ʼIyyār || Iyyar || [[Iyar]] || 29 days || ''Ayaru'' || [[Pesach Sheni]]<br />[[Lag B'Omer]] || Called ''Ziv''<ref>''Hebrew-English Bible'', {{bibleverse|1|Kings|6:1|HE}}, {{bibleverse-nb|1|Kings|6:37|HE}}</ref>
|-
|  3 || style="text-align:right;" | {{Script/Hebrew|סִיוָן / סיוון}} || Sīwān || [[Sivan]] || Siwan || 30 days || ''Simanu'' || [[Shavuot]] || 
|-
|  4 || style="text-align:right;" | {{Script/Hebrew|תַּמּוּז }}|| Tammūz || [[Tammuz (Hebrew month)|Tammuz]] || Tamuz || 29 days || ''Dumuzu'' || [[Seventeenth of Tammuz]] || Named for the Babylonian god [[Tammuz (deity)|Dumuzi]]
|-
|  5 || style="text-align:right;" | {{Script/Hebrew|אָב }}|| ʼĀḇ || [[Av (month)|Av]] || Ab || 30 days || ''Abu'' || [[Tisha B'Av]]<br />[[Tu B'Av]] || 
|-
|  6 || style="text-align:right;" |{{Script/Hebrew| אֱלוּל }}|| ʼĔlūl || [[Elul]] ||  || 29 days || ''Ululu'' || ||
|-
|  7 || style="text-align:right;" |{{Script/Hebrew| תִּשְׁרֵי / תִּשְׁרִי}} || Tišrī || Tishri || [[Tishrei]] || 30 days || ''Tashritu'' || {{nowrap|[[Rosh Hashanah]]}}<br />[[Yom Kippur]]<br />[[Sukkot]]<br />[[Shemini Atzeret]]<br />[[Simchat Torah]] || Called ''Ethanim'' in Kings 8:2.<ref>''Hebrew-English Bible'', {{bibleverse|1|Kings|8:2|HE}}</ref> <br> First month of civil year. 
|-
|  8 || style="text-align:right;" |{{Script/Hebrew| מַרְחֶשְׁוָן /  מרחשוון }}|| Marḥešwān || Marẖeshvan || Marcheshvan <br>[[Cheshvan]] <br> Marẖeshwan || 29 or <br> 30 days || ''Arakhsamna'' ||  || Called ''Bul'' in Kings 6:38.<ref>''Hebrew-English Bible'', {{bibleverse|1|Kings|6:38|HE}}</ref>
|-
|  9 || style="text-align:right;" |{{Script/Hebrew| כִּסְלֵו / כסליו}}|| Kislēw || [[Kislev]] || Kislev <br> Chisleu <br> Chislev || 29 or <br> 30 days || ''Kislimu'' || [[Hanukkah]] ||
|-
| 10 || style="text-align:right;" |{{Script/Hebrew| טֵבֵת }}|| Ṭēḇēṯ || [[Tevet]] || Tebeth || 29 days || ''Tebetu'' || [[Tenth of Tevet]] ||  Called ''Tebeth'' in Esther 2:16.<ref>''Hebrew-English Bible'', {{bibleverse|Esther|2:16|HE}}</ref>
|-
| 11 || style="text-align:right;" |{{Script/Hebrew| שְׁבָט }}|| Šəḇāṭ || [[Shvat]] || Shevat <br> Shebat <br> Sebat || 30 days || ''Shabatu'' || [[Tu Bishvat]] ||
|-
| 12L<sup>*</sup> || style="text-align:right;" |{{Script/Hebrew| אֲדָר א׳ }}|| || Adar I<sup>*</sup> ||  || 30 days ||  || || rowspan="2"|<sup>*</sup>Only in Leap years.
|-
| 12 || style="text-align:right;" |{{nowrap|{{Script/Hebrew| אֲדָר / אֲדָר ב׳* }}}}|| ʼĂḏār || {{nowrap|[[Adar]] / Adar II<sup>*</sup>}} ||  || 29 days || ''Adaru'' || [[Purim]]
|}

===Past methods of dividing years===
According to some Christian and [[Karaite Judaism|Karaite]] sources, the tradition in ancient Israel was that 1 Nisan would not start until the barley is ripe, being the test for the onset of spring.{{efn|The barley had to be "eared out" (ripe) in order to have a wave-sheaf offering of the first fruits according to the Law.<ref>{{Cite book|title=Secrets of Time |last=Jones |first=Stephen |date=1996}}</ref>}} If the barley was not ripe, an intercalary month would be added before Nisan.

In the 1st century, [[Josephus]] stated that while –
<blockquote>Moses...appointed Nisan...as the first month for the festivals...the commencement of the year for everything relating to divine worship, but for selling and buying and other ordinary affairs he preserved the ancient order [i. e. the year beginning with Tishrei]."<ref name="Josephus, 1930">Josephus, ''Antiquities'' 1.81, Loeb Classical Library, 1930.</ref></blockquote>

[[Edwin Thiele]] concluded that the ancient northern [[Kingdom of Israel (Samaria)|Kingdom of Israel]] counted years using the ecclesiastical new year starting on 1 Aviv/Nisan ([[Nisan-years]]), while the southern [[Kingdom of Judah]] counted years using the civil new year starting on 1 Tishrei ([[Tishri-years]]).<ref name="Thiele">Edwin Thiele, ''[[The Mysterious Numbers of the Hebrew Kings]]'', (1st ed.; New York: Macmillan, 1951; 2d ed.; Grand Rapids: Eerdmans, 1965; 3rd ed.; Grand Rapids: Zondervan/Kregel, 1983). {{ISBN|0-8254-3825-X}}, 9780825438257</ref> The practice of the Kingdom of Israel was also that of [[Babylonian calendar|Babylon]],<ref>''The Chronology of the Old Testament'', 16th ed., Floyd Nolan Jones, {{ISBN|978-0-89051-416-0}}, pp. 118–123</ref> as well as other countries of the region.<ref name="DeVaux"/> The practice of Judah is continued in modern Judaism and is celebrated as [[Rosh Hashana]].

===Past methods of numbering years===
Before the adoption of the current {{lang|la|Anno Mundi}} year numbering system, other systems were used. In early times, the years were counted from some significant event such as [[the Exodus]].<ref>e.g., ''Hebrew-English Bible'', {{bibleverse|1|Kings|6:1|HE}}</ref> During the period of the monarchy, it was the widespread practice in western Asia to use era year numbers according to the accession year of the monarch of the country involved. This practice was followed by the united kingdom of Israel,<ref>e.g., ''Hebrew-English Bible'', {{bibleverse|1|Kings|14:25|HE}}</ref> kingdom of Judah,<ref>e.g., ''Hebrew-English Bible'', {{bibleverse|2|Kings|18:13|HE}}</ref> kingdom of Israel,<ref>e.g., ''Hebrew-English Bible'', {{bibleverse|2|Kings|17:6|HE}}</ref> Persia,<ref>(e.g., ''Hebrew-English Bible'', {{bibleverse||Nehemiah|2:1|HE}}</ref> and others. Besides, the author of [[Books of Kings|Kings]] coordinated dates in the two kingdoms by giving the accession year of a monarch in terms of the year of the monarch of the other kingdom,<ref>e.g., ''Hebrew-English Bible'', {{bibleverse|2|Kings|8:16|HE}}</ref> though some commentators note that these dates do not always synchronise.<ref name="Thiele" /> Other era dating systems have been used at other times. For example, Jewish communities in the Babylonian diaspora counted the years from the first deportation from Israel, that of [[Jehoiachin]] in 597 BCE.<ref>e.g., ''Hebrew-English Bible'', {{bibleverse||Ezekiel|1:1–2|HE}}</ref> The era year was then called "year of the captivity of Jehoiachin".<ref>e.g., ''Hebrew-English Bible'', {{bibleverse|2|Kings|25:27|HE}}</ref>

During the Hellenistic Maccabean period, [[Seleucid era]] counting was used, at least in the [[Land of Israel]] (under Greek influence at the time). The [[Books of the Maccabees]] used Seleucid era dating exclusively,<ref>e.g., ''Hebrew-English Bible'', {{bibleverse|1|Maccabees|1:54|NAB}}, {{bibleverse-nb|1|Maccabees|6:20|NAB}}, {{bibleverse-nb|1|Maccabees|7:1|NAB}}, {{bibleverse-nb|1|Maccabees|9:3|NAB}}, {{bibleverse-nb|1|Maccabees|10:1|NAB}}</ref> as did [[Josephus]] writing in the Roman period. From the 1st to 10th centuries, the center of world Judaism was in the Middle East (primarily [[Talmudic Academies in Babylonia|Iraq]] and [[Talmudic Academies in the Land of Israel|Palestine]]),{{efn|Ancient Iraq and [[Palestine (region)]], not the modern territories with those names}} and Jews in these regions also used Seleucid era dating, which they called the "Era of Contracts [or Documents]";<ref name=Jones>{{cite book |url=https://books.google.com/books?id=AvVPlyYjX7YC&pg=PA295 |title=Chronology of the Old Testament |author= Dr. Floyd Nolen Jones |date= 2005 |quote=When the center of Jewish life moved from Babylonia to Europe during the 8th and 9th centuries CE, calculations from the Seleucid era became meaningless. Over those centuries, it was replaced by that of the {{lang|la|anno mundi}} era of the {{lang|he-Latn|Seder Olam}}. From the 11th century, {{lang|la|anno mundi}} dating became dominant throughout most of the world's Jewish communities.|page=295|publisher=New Leaf Publishing |isbn=978-1-61458-210-6}}</ref> this counting is still sometimes used by [[Yemenite Jews]].<ref>Yitzhak Ratzabi, [https://www.maharitz.co.il/?CategoryID=345&ArticleID=511 The counting of years for Contracts], accessed on Maharitz on January 16, 2025.</ref> The [[Talmud]] states:

<blockquote>[[Rav]] [[Aha bar Jacob]] then put this question: How do we know that our Era [of Documents] is connected with the Kingdom of Greece at all? Why not say that it is reckoned from the Exodus from Egypt, omitting the first thousand years and giving the years of the next thousand? In that case, the document is really post-dated!<br />Said [[Rav Nachman|Rav Nahman]]: In the Diaspora the Greek Era alone is used.<br />He [Rav Aha] thought that Rav Nahman wanted to dispose of him anyhow, but when he went and studied it thoroughly he found that it is indeed taught [in a [[Baraita]]]: In the Diaspora the Greek Era alone is used.<ref>{{cite web|url=https://www.sefaria.org/Avodah_Zarah.10a.3?lang=bi&with=all&lang2=en|title=Babylonian Talmud: Avodah Zarah 10a|website=www.sefaria.org|publisher=[[Sefaria]]}}</ref></blockquote>

In the 8th and 9th centuries, as the center of Jewish life moved from Babylonia to Europe, counting using the Seleucid era "became meaningless", and thus was replaced by the {{lang|la|anno mundi}} system.<ref name=Jones /> The use of the Seleucid era continued till the 16th century in the East, and was employed even in the 19th century among [[Yemenite Jews]].<ref name=Zarah />

Occasionally in Talmudic writings, reference was made to other starting points for eras, such as destruction era dating, being the number of years since the 70 CE [[Siege of Jerusalem (70)|destruction of the Second Temple]].<ref name=Zarah>[https://halakhah.com/zarah/zarah_9.html Avodah Zarah 9a] Soncino edition, footnote 4: "The Eras in use among Jews in Talmudic Times are: (a) Era of Contracts  [H] dating from the year 380 before the Destruction of the Second Temple (312–1 BCE)... It is also termed Seleucid or Greek Era [H].... This Era... was generally in vogue in eastern countries till the 16th cent, and was employed even in the 19th cent, among the Jews of Yemen, in South Arabia... (b) The Era of the Destruction  (of the Second Temple) [H] the year 1 of which corresponds to 381 of the Seleucid Era, and 69–70 of the Christian Era. This Era was mainly employed by the Rabbis and was in use in Palestine for several centuries, and even in the later Middle Ages documents were dated by it."</ref>

===Leap months===
According to Rabbinic Judaism, {{Bibleverse|Exodus|12:1–2|HE}} requires that the months be determined by a proper court with the necessary authority to sanctify the months. Hence the court, not the astronomy, has the final decision.<ref>{{Cite book |title=The complete ArtScroll Machzor / [1.] Rosh Hashanah.|last=Scherman |first=Nosson |date =2005 |language=he |location=Brooklyn, NY |publisher= Mesorah Publ.|isbn=9780899066790}}</ref> When the observational form of the calendar was in use, whether a leap month was added depended on three factors: 'aviv [i.e., the ripeness of barley], fruits of trees, and the equinox. On two of these grounds it should be intercalated, but not on one of them alone.<ref name=ts22/> It may be noted that in the Bible the name of the first month, ''[[Aviv]]'', literally means "spring". Thus, if Adar was over and spring had not yet arrived, an additional month was observed.

===Determining the new month in the Mishnaic period===
[[File:To the trumpeting place.jpg|right|thumb|The [[Trumpeting Place inscription]], a stone (2.43×1 m) with [[Hebrew language|Hebrew]] inscription "To the Trumpeting Place" is believed to be a part of the Second Temple.]]
The [[Tanakh]] contains several [[613 Mitzvot|commandments]] related to the keeping of the calendar and the lunar cycle, and records changes that have taken place to the Hebrew calendar. Numbers 10:10 stresses the importance in Israelite religious observance of the new month (Hebrew: {{lang|he|ראש חודש}}, [[Rosh Chodesh]], "beginning of the month"): "... in your new moons, ye shall blow with the trumpets over your burnt-offerings..."<ref>''Hebrew-English Bible,'' {{bibleverse||Numbers|10:10|HE}}</ref> Similarly in Numbers 28:11.<ref>''Hebrew-English Bible,'' {{bibleverse||Numbers|28:11|HE}}</ref> "The beginning of the month" meant the appearance of a [[new moon]], and in Exodus 12:2.<ref>''Hebrew-English Bible,'' {{bibleverse||Exodus|12:2|HE}}</ref> "This month is to you".

According to the ''[[Mishnah]]'' and [[Tosefta]], in the Maccabean, Herodian, and Mishnaic periods, new months were determined by the sighting of a new crescent, with two eyewitnesses required to testify to the [[Sanhedrin]] to having seen the new lunar crescent at sunset.<ref>Mishnah ''Rosh Hashanah'' 1:7</ref> The practice in the time of [[Gamaliel II]] (c. 100 CE) was for witnesses to select the appearance of the moon from a collection of drawings that depicted the crescent in a variety of orientations, only a few of which could be valid in any given month.<ref>Mishnah ''Rosh Hashanah'' 2:6–8</ref> These observations were compared against calculations.<ref name="epstein">[[Babylonian Talmud]] Rosh Hashanah 20b: "This is what Abba the father of R. Simlai meant: 'We calculate the new moon's birth. If it is born before midday, then certainly it will have been seen shortly before sunset. If it was not born before midday, certainly it will not have been seen shortly before sunset.' What is the practical value of this remark? R. Ashi said: Confuting the witnesses." I. Epstein, Ed., ''The Babylonian Talmud Seder Mo'ed,'' Soncino Press, London, 1938, p. 85.</ref>

At first the beginning of each Jewish month was signaled to the communities of Israel and beyond by fires lit on mountaintops, but after the [[Samaritan]]s began to light false fires, messengers were sent.<ref>Mishnah ''Rosh Hashanah'' 2.2</ref> The inability of the messengers to reach communities outside Israel before mid-month High Holy Days ([[Succot]] and [[Passover]]) led outlying communities to celebrate scriptural festivals for two days rather than one, observing the second feast-day of the [[Jewish diaspora]] because of uncertainty of whether the previous month ended after 29 or 30 days.<ref>Babylonian Talmud ''Betzah'' 4b</ref>

==== Historicity ====
It has been noted that the procedures described in the Mishnah and Tosefta are all plausible procedures for regulating an empirical lunar calendar.{{sfn| Stern|2001|loc=pp. 162ff.}} Fire-signals, for example, or smoke-signals, are known from the pre-exilic Lachish ostraca.<ref>James B. Pritchard, ed., ''The Ancient Near East: An Anthology of Texts and Pictures'', Vol. 1, Princeton University Press, p. 213.</ref> Furthermore, the Mishnah contains laws that reflect the uncertainties of an empirical calendar. Mishnah Sanhedrin, for example, holds that when one witness holds that an event took place on a certain day of the month, and another that the same event took place on the following day, their testimony can be held to agree, since the length of the preceding month was uncertain.<ref>Mishnah ''Sanhedrin'' 5:3: "If one testifies, 'on the second of the month, and the other, 'on the third of the month:' their evidence is valid, for one may have been aware of the intercalation of the month and the other may not have been aware of it. But if one says, 'on the third', and the other 'on the fifth', their evidence is invalid."</ref> Another Mishnah takes it for granted that it cannot be known in advance whether a year's lease is for twelve or thirteen months.<ref>Mishnah ''Baba Metzia'' 8:8.</ref> Hence it is a reasonable conclusion that the Mishnaic calendar was actually used in the Mishnaic period.

The accuracy of the Mishnah's claim that the Mishnaic calendar was also used in the late [[Second Temple of Jerusalem|Second Temple]] period is less certain. One scholar has noted that there are no laws from Second Temple period sources that indicate any doubts about the length of a month or of a year. This led him to propose that the priests must have had some form of computed calendar or calendrical rules that allowed them to know in advance whether a month would have 30 or 29 days, and whether a year would have 12 or 13 months.<ref>Gandz, Solomon. "Studies in the Hebrew Calendar: II. The origin of the Two New Moon Days", ''Jewish Quarterly Review'' (New Series), 40(2), 1949–50. {{JSTOR|1452961}}. {{doi|10.2307/1452961}}. Reprinted in Shlomo Sternberg, ed., ''Studies in Hebrew Astronomy and Mathematics by Solomon Gandz'', KTAV, New York, 1970, pp. 72–73.</ref>

===The fixing of the calendar===
{{See also|Hillel II#Fixing of the calendar}}
Between 70 and 1178 CE, the observation-based calendar was gradually replaced by a mathematically calculated one.{{sfn|Stern|2001|}}

The Talmuds indicate at least the beginnings of a transition from a purely empirical to a computed calendar. [[Samuel of Nehardea]] (c. 165–254) stated that he could determine the dates of the holidays by calculation rather than observation.<ref>[https://www.sefaria.org.il/Rosh_Hashanah.20b.2?lang=bi&with=all&lang2=en Rosh Hashanah 20b]</ref> According to a statement attributed to Yose (late 3rd century), [[Purim]] could not fall on a Sabbath nor a Monday, lest [[Yom Kippur]] fall on a Friday or a Sunday.<ref>Yerushalmi ''Megillah'' 1:2, pp. 70b. Text:{{lang|he|א"ר יוסה לית כאן חל להיות בשני ולית כאן חל להיות בשבת, חל להיות בשני צומא רבא בחד בשובא, חל להיות בשבת צומא רבא בערובתא}}</ref> This indicates that, by the time of the redaction of the [[Jerusalem Talmud]] (c. 400 CE), there were a fixed number of days in all months from Adar to Elul, also implying that the extra month was already a second Adar added before the regular Adar. Elsewhere, [[Shimon ben Pazi]] is reported to have counseled "those who make the computations" not to set [[Rosh Hashana]] or [[Hoshana Rabbah]] on Shabbat.<ref>Yerushalmi ''Sukkah'' 54b. Text: {{lang|he|ר' סימון מפקד לאילין דמחשבין יהבון דעתכון דלא תעבדין לא תקיעתה בשבת ולא ערבתא בשבתא. ואין אדחקון עבדון תקיעתה ולא תעבדון ערבתא:}}</ref> This indicates that there was a group who "made computations" and controlled, to some extent, the day of the week on which Rosh Hashana would fall.

There is a tradition, first mentioned by [[Hai Gaon]] (died 1038 CE), that [[Hillel II]] was responsible for the new calculated calendar with a fixed intercalation cycle "in the year 670 of the Seleucid era" (i.e., 358–359 CE). Later writers, such as [[Nachmanides]], explained Hai Gaon's words to mean that the entire computed calendar was due to Hillel II in response to persecution of Jews. [[Maimonides]] (12th century) stated that the Mishnaic calendar was used "until the days of Abaye and Rava" (c. 320–350 CE), and that the change came when "the land of Israel was destroyed, and no permanent court was left." Taken together, these two traditions suggest that Hillel II (whom they identify with the mid-4th-century Jewish patriarch Ioulos, attested in a letter of the Emperor Julian,<ref>Julian, Letter 25, in John Duncombe, ''Select Works of the Emperor Julian and some Pieces of the Sophist Libanius'', Vol. 2, Cadell, London, 1784, pp. 57–62.</ref> and the Jewish patriarch Ellel, mentioned by Epiphanius<ref>Epiphanius, ''Adversus Haereses'' 30.4.1, in Frank Williams, trans., ''The Panarion of Epiphanius of Salamis Book I (Sections 1–46),'' Leiden, E. J.Brill, 1987, p. 122.</ref>) instituted the computed Hebrew calendar because of persecution. H. Graetz<ref>H. Graetz, Popular History of the Jews, (A. B. Rhine, trans.,) Hebrew Publishing Company, New York, 1919, Vol. II, pp. 410–411. Quoted in {{harvnb|Stern|2001|p=216}}</ref> linked the introduction of the computed calendar to a sharp repression following a failed Jewish insurrection that occurred during the rule of the Christian emperor [[Constantius II|Constantius]] and [[Constantius Gallus|Gallus]]. [[Saul Lieberman]] argued instead that the introduction of the fixed calendar was due to measures taken by Christian Roman authorities to prevent the Jewish patriarch from sending calendrical messengers.<ref>{{cite journal |last=Lieberman |first=S. |title=Palestine in the Third and Fourth Centuries |journal=Jewish Quarterly Review |pages=329–370 |date=1946 |volume=36 |issue=4 |jstor=1452134 |doi=10.2307/1452134}} Quoted in {{harvnb|Stern|2001|pp=216–217}}.</ref>

Both the tradition that Hillel II instituted the complete computed calendar, and the theory that the computed calendar was introduced due to repression or persecution, have been questioned.{{sfn|Stern|2001|loc= In particular section 5.1.1, discussion of the "Persecution theory."}}<ref>[[Samuel Poznanski|Poznanski, Samuel]], "Ben Meir and the Origin of the Jewish Calendar", ''Jewish Quarterly Review'', Original Series, Vol. 10, pp. 152–161 (1898). {{JSTOR|1450611}}. {{doi|10.2307/1450611}}.</ref><ref>"While it is not unreasonable to attribute to Hillel II the fixing of the regular order of intercalations, his full share in the present fixed calendar is doubtful." Entry "Calendar", ''Encyclopaedia Judaica'', Keter, Jerusalem, 1971.</ref> Furthermore, two Jewish dates during post-Talmudic times (specifically in 506 and 776) are impossible under the rules of the modern calendar, indicating that some of its arithmetic rules were established in Babylonia during the times of the [[Geonim]] (7th to 8th centuries).<ref>[[Samuel Poznanski]], "Calendar (Jewish)", [https://archive.org/details/encyclopaediaofr03hastuoft ''Encyclopaedia of Religion and Ethics'', vol. 3, p. 118].</ref> Most likely, the procedure established in 359 involved a fixed molad interval slightly different from the current one,{{efn|An interval of 29 days/12 hours/792 halakim, as opposed to the current interval of 29/12/793}} Rosh Hashana postponement rules similar but not identical to current rules,{{efn|Unlike in the current calendar, the first day of Rosh Hashana was permitted to fall on Sunday; otherwise the rules were about the same.}} and leap months were added based on when Passover preceded a fixed cutoff date rather than through a repeated 19-year cycle.<ref name=ajdler>[https://hakirah.org/vol20Ajdler.pdf A Short History of the Jewish Fixed Calendar: The Origin of the Molad]</ref> The Rosh Hashana rules apparently reached their modern form between 629 and 648, the modern molad interval was likely fixed in 776, while the fixed 19-year cycle also likely dates to the late 8th century.<ref name=ajdler/>

Except for the epoch year number (the fixed reference point at the beginning of year 1, which at that time was one year later than the epoch of the modern calendar), the calendar rules reached their current form by the beginning of the 9th century, as described by the [[Persia]]n [[Muslim]] astronomer [[Muhammad ibn Musa al-Khwarizmi]] in 823.<ref name=Kennedy/><ref name=Khwarizmi/> Al-Khwarizmi's study of the Jewish calendar describes the [[Metonic cycle|19-year intercalation cycle]],<ref>{{cite book |title={{transliteration|ar|Risāla fi istikhrāj ta’rīkh al-yahūd|nocat=yes}} ({{langx|ar|رسالة في إستخراج تأريخ اليهود|nocat=yes}}, "Extraction of the Jewish Era") |author=[[Muhammad ibn Musa al-Khwarizmi]] |date=823}} (date uncertain)</ref> the rules for determining on what day of the week the first day of the month [[Tishrei]] shall fall, the interval between the [[Anno Mundi|Jewish era]] (creation of Adam) and the [[Seleucid era]], and the rules for determining the mean longitude of the sun and the moon using the Jewish calendar.<ref name=Kennedy>E.S. Kennedy, "Al-Khwarizmi on the Jewish calendar", ''Scripta Mathematica'' '''27''' (1964) 55–59.</ref><ref name=Khwarizmi>"al-Khwarizmi", ''Dictionary of Scientific Biography'', VII: 362, 365.</ref> Not all the rules were in place by 835.{{sfn|Stern|2001}}

In 921, [[Aaron ben Meïr]] had a debate with [[Saadya Gaon]] about one of the rules of the calendar. This indicates that the rules of the modern calendar were not so clear and set.<ref>Haim Yehiel Bernstein, [https://hebrewbooks.org/46973 Mahloket Rav Sa'adya Gaon u-ben Me'ir], Warsaw 1904.</ref> In 1000, the Muslim chronologist [[al-Biruni]] described all of the modern rules of the Hebrew calendar, except that he specified three different epochs used by various Jewish communities being one, two, or three years later than the modern epoch.<ref name=biruni/>

In 1178, [[Maimonides]] included all the rules for the calculated calendar and their scriptural basis, including the modern epochal year, in his work ''[[Mishneh Torah]]''. He wrote<ref>[[Mishneh Torah]], ''Sanctification of the Moon'', 11:16</ref> that he had chosen the [[epoch]] from which calculations of all dates should be as "the third day of Nisan in this present year ... which is the year 4938 of the creation of the world" (22 March 1178).<ref>[[Solomon Gandz]] (1947–1948). "Date of the Composition of Maimonides' Code". ''Proceedings of the American Academy for Jewish Research'', Vol. 17, pp.&nbsp;1–7. {{doi|10.2307/3622160}}. {{JSTOR|3622160}}. Retrieved March 14, 2013.</ref> Today, these rules are generally used by Jewish communities throughout the world.

==Other calendars==
Outside of [[Rabbinic Judaism]], evidence shows a diversity of practice.

===Karaite calendar===
[[Karaite Judaism|Karaites]] use the lunar month and the solar year, but the Karaite calendar differs from the current Rabbinic calendar in a number of ways. The Karaite calendar is identical to the Rabbinic calendar used before the Sanhedrin changed the Rabbinic calendar from the lunar, observation based, calendar to the current, mathematically based, calendar used in Rabbinic Judaism today.

In the lunar Karaite calendar, the beginning of each month, the [[Rosh Chodesh]], can be calculated, but is confirmed by the observation in [[Israel]] of the first sightings of the new moon.<ref>{{cite web|url=http://www.karaite-korner.org/new_moon.shtml|title=Karaite Korner – New Moon and the Hebrew Month|website=www.karaite-korner.org}}</ref> This may result in an occasional variation of a maximum of one day, depending on the inability to observe the new moon. The day is usually "picked up" in the next month.

The addition of the leap month (Adar II) is determined by observing in Israel the ripening of barley at a specific stage (defined by Karaite tradition) (called [[aviv]]),<ref>{{cite web|url=http://www.karaite-korner.org/abib.shtml|title=Aviv Barley in the Biblical Calendar – Nehemia's Wall|date=24 February 2016}}</ref> rather than using the calculated and fixed calendar of [[rabbinic Judaism]]. Occasionally this results in Karaites being one month ahead of other Jews using the calculated rabbinic calendar. The "lost" month would be "picked up" in the next cycle when Karaites would observe a leap month while other Jews would not.

Furthermore, the seasonal drift of the rabbinic calendar is avoided, resulting in the years affected by the drift starting one month earlier in the Karaite calendar.

Also, the four rules of postponement of the rabbinic calendar are not applied, since they are not mentioned in the [[Tanakh]]. This can affect the dates observed for all the Jewish holidays in a particular year by one or two days.

In the Middle Ages many Karaite Jews outside Israel followed the calculated rabbinic calendar, because it was not possible to retrieve accurate aviv barley data from the land of Israel. However, since the establishment of the [[State of Israel]], and especially since the [[Six-Day War]], the Karaite Jews that have made ''[[aliyah]]'' can now again use the observational calendar.

===Samaritan calendar===
The [[Samaritan]] community's calendar also relies on lunar months and solar years. Calculation of the Samaritan calendar has historically been a secret reserved to the priestly family alone,<ref name="Sam1">{{cite web|title=The Samaritan Calendar|url=http://shomron0.tripod.com/articles/samaritancalendar.pdf|website=www.thesamaritanupdate.com|access-date=28 December 2017|date=2008}}</ref> and was based on observations of the new crescent moon. More recently, a 20th-century [[Samaritan High Priest]] transferred the calculation to a computer algorithm. The current High Priest confirms the results twice a year, and then distributes calendars to the community.<ref name="Sam2">{{cite web|last1=Benyamim|first1=Tzedaka|title=Calendar|url=https://www.israelite-samaritans.com/religion/calendar/|website=www.israelite-samaritans.com|access-date=28 December 2017}}</ref>

The epoch of the Samaritan calendar is year of the entry of the [[Children of Israel]] into the [[Land of Israel]] with [[Joshua]]. The month of Passover is the first month in the Samaritan calendar, but the year number increments in the sixth month. Like in the Rabbinic calendar, there are seven leap years within each 19-year cycle. However, the Rabbinic and Samaritan calendars' cycles are not synchronized, so Samaritan festivals—notionally the same as the Rabbinic festivals of Torah origin—are frequently one month off from the date according to the Rabbinic calendar. Additionally, as in the Karaite calendar, the Samaritan calendar does not apply the four rules of postponement, since they are not mentioned in the [[Tanakh]]. This can affect the dates observed for all the Jewish holidays in a particular year by one or two days.<ref name="Sam1" /><ref name="Sam2" />

===The Qumran calendar===
{{Main|Qumran calendrical texts}}
{{see also|Enoch calendar}}
Many of the [[Dead Sea Scrolls]] have references to a unique calendar, used by the people there, who are often assumed to have been [[Essenes]]. The year of this calendar used the ideal Mesopotamian calendar of twelve 30-day months, to which were added 4 days at the [[equinox]]es and [[solstice]]s (cardinal points), making a total of 364 days.<ref name=bendov>Jonathan Ben-Dov. ''Head of All Years: Astronomy and Calendars at Qumran in their Ancient Context''. Leiden: Brill, 2008, pp. 16–20</ref>

With only 364 days, the calendar would be very noticeably different from the actual seasons after a few years, but there is nothing to indicate what was done about this problem. Various scholars have suggested that nothing was done and the calendar was allowed to change with respect to the seasons, or that changes were made irregularly when the seasonal anomaly was too great to be ignored any longer.<ref name=bendov/>

===Other calendars used by ancient Jews===
Calendrical evidence for the postexilic Persian period is found in [[Elephantine papyri|papyri]] from the Jewish colony at [[Elephantine]], in Egypt. These documents show that the Jewish community of Elephantine used the [[Egyptian calendar|Egyptian]] and [[Babylonian calendar|Babylonian]] calendars.<ref>Sacha Stern, "The Babylonian Calendar at Elephantine", ''Zeitschrift für Papyrologie und Epigraphik'' 130, 159–171 (2000).</ref><ref>Lester L. Grabbe, ''A History of the Jews and Judaism in the Second Temple Period, Volume 1: Yehud: A History of the Persian Province of Judah'', T&T Clark, London, 2004, p. 186.</ref>

The [[Sardica paschal table]] shows that the Jewish community of some eastern city, possibly [[Antioch]], used a calendrical scheme that kept Nisan 14 within the limits of the Julian month of March.<ref>Eduard Schwartz, ''Christliche und jüdische Ostertafeln,'' (Abhandlungen der königlichen Gesellschaft der Wissenschaften zu Göttingen. Philologisch-Historische Klasse. Neue Folge, Band viii, Berlin, 1905 [https://archive.org/details/bub_gb_Sn8TAAAAYAAJ Internet Archive link].</ref> Some of the dates in the document are clearly corrupt, but they can be emended to make the sixteen years in the table consistent with a regular intercalation scheme. Peter, the bishop of Alexandria (early 4th century CE), mentions that the Jews of his city "hold their Passover according to the course of the moon in the month of [[Paremhat|Phamenoth]], or according to the intercalary month every third year in the month of [[Paremoude|Pharmuthi]]",<ref>Peter of Alexandria, quoted in the ''Chronicon Paschale''. ''Corpus Scriptorum Historiae Byzantinae, Chronicon Paschale'' Vol. 1, Weber, Bonn, 1832, p. 7</ref> suggesting a fairly consistent intercalation scheme that kept Nisan 14 approximately between Phamenoth 10 (6 March in the 4th century CE) and Pharmuthi 10 (5 April).

Jewish funerary inscriptions from [[Zoara|Zoar]] (south of the [[Dead Sea]]), dated from the 3rd to the 5th century, indicate that when years were intercalated, the intercalary month was at least sometimes a repeated month of Adar. The inscriptions, however, reveal no clear pattern of regular intercalations, nor do they indicate any consistent rule for determining the start of the lunar month.{{sfn| Stern|2001|loc=pp. 87–97, 146–153}}

==See also==
{{Portal|Judaism}}
* [[Biblical and Talmudic units of measurement]]
* [[Chronology of the Bible]]
* [[Gezer calendar]]
* [[Hebrew astronomy]]
* [[Jewish astrology]]
* [[Jewish and Israeli holidays 2000–2050]]
* [[List of observances set by the Hebrew calendar]]
<!-- Please do not add other calendars here as they are all in the {{calendars}} template below. Any that are particularly relevant will surely have been mentioned already in the body of the article. -->

==Notes==
{{notelist}}

==References==
{{Reflist|30em}}

==Bibliography==
{{refbegin|30em|indent=yes}}
* Ari Belenkiy. "A Unique Feature of the Jewish Calendar – ''Dehiyot''". ''Culture and Cosmos'' '''6''' (2002) 3–22.
* Sherrard Beaumont Burnaby. ''Elements of the Jewish and Muhammadan Calendars''. George Bell and Sons, London, 1901 – [https://archive.org/details/elementsofjewish00burnuoft Internet Archive link].
* Nathan Bushwick. ''Understanding the Jewish Calendar''. Moznaim, New York/Jerusalem, 1989. {{ISBN|0-940118-17-3}}
* William Moses Feldman. ''Rabbinical Mathematics and Astronomy'', 3rd ed., Sepher-Hermon Press, New York, 1978.
* ''The Code of Maimonides (Mishneh Torah), Book Three, Treatise Eight: Sanctification of the New Moon''. Translated by Solomon Gandz. Yale Judaica Series Volume XI, Yale University Press, New Haven, Conn., 1956.
* [[Edward M. Reingold]] and [[Nachum Dershowitz]]. ''Calendrical Calculations: The Millennium Edition''. Cambridge University Press; 2 edition (2001). {{ISBN|0-521-77752-6}} 723–730.
* Arthur Spier. ''The Comprehensive Hebrew Calendar: Twentieth to the Twenty-Second Century 5660–5860/1900–2100''. Feldheim Publishers, Jerusalem/New York, 1986.
* {{cite book |first=Sacha |last=Stern |title=Calendar and Community: A History of the Jewish Calendar 2nd Century BCE to 10th Century CE |publisher=Oxford University Press |date=2001 |isbn=978-0198270348}}
* Ernest Wiesenberg. "Appendix: Addenda and Corrigenda to Treatise VIII". ''The Code of Maimonides (Mishneh Torah), Book Three: The Book of Seasons''. Yale Judaica Series Volume XIV, Yale University Press, New Haven, Conn., 1961. pp.&nbsp;557–602.
* Francis Henry Woods. "Calendar (Hebrew)", ''[[Encyclopædia of Religion and Ethics]]''. T. & T. Clark, Edinburgh, 1910, vol.&nbsp;3, pp.&nbsp;108–109.
{{refend}}

==External links==
* [http://www.chabad.org/library/article_cdo/aid/55194/jewish/introduction.htm: Chabad.org: Introduction to the Jewish Calendar]
* [http://www.hebcal.com/ Hebcal.com: Jewish Holiday Calendars & Hebrew Date Converter]
* [http://www.aish.com/jewish-calendar/jewish_calendar.html Aish.com: Jewish Calendar]
* [http://hebrewcalendar.tripod.com/ Tripod.com: Hebrew Calendar Science and Myths]
* [http://www.yeshiva.co/calendar/ Yeshiva.co: Jewish Calendar with Halachic times date converter and daf yomi]
* [https://michoelr.github.io/Keviyus/FourGates/FourGates.htm Illustrating the "Four Gates"]

===Date converters===
* [http://www.torahcalc.com/molad/ TorahCalc.com: Molad Calculator]
* [http://www.kaluach.org/ Kaluach.org: Hebrew Date Converter]
* [http://www.hebcal.com/converter/ Hebcal Hebrew Date Converter]
* [http://www.chabad.org/calendar/converter_cdo/aid/6225/jewish/jewish-hebrew-date-converter.htm Chabad.org: Jewish/Hebrew Date Converter]
* [http://individual.utoronto.ca/kalendis/kalendis.htm University of Toronto: The "Kalendis" Calendar Calculator]
* [http://www.calendar-converter.com/jewishhebrew Calendar-Converter.com: Jewish/Hebrew Calendar Converter]
 
{{Calendars}}
{{Jews and Judaism}}
{{Jewish and Israeli holidays}}
{{Time in religion and mythology}}
{{Chronology}}
{{Authority control}}

[[Category:Hebrew calendar| ]]
[[Category:Lunisolar calendars]]
[[Category:Positive Mitzvoth]]