The Hebrew calendar (הלוח העברי ha'luach ha'ivri), or Jewish calendar, is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits (dates to commemorate the death of a relative), and daily Psalm readings, among many ceremonial uses. In Israel, it is an official calendar for civil purposes and provides a time frame for agriculture. The current year of the Jewish calendar (16 September 2012 to 4 September 2013) is 5773.
The Hebrew calendar has evolved over time. For example, until the Tannaitic period (approximately 10–220 CE) the months were set by observation of a new crescent moon, with an additional month added every two or three years to correct for the difference between twelve lunar months and the solar year, and therefore, to keep Passover in the spring. The addition of the extra month was also based on observation of natural events, namely the ripening of the barley crop, the age of the kids, lambs and doves, the ripeness of the fruit trees, and the relation to the Tekufah (seasons). Through the Amoraic period (200 to 500 CE) and into the Geonic period, this system was displaced by mathematical rules. The principles and rules appear to have been settled by the time Maimonides compiled the Mishneh Torah in the 12th century.
Because of the roughly eleven-day difference between twelve lunar months and one solar year, the length of the Hebrew calendar year varies in the repeating 19-year Metonic cycle of 235 lunar months, with the intercalary month added according to defined rules every two or three years, for a total of 7 times per 19 years. Seasonal references in the Hebrew calendar reflect its development in the region east of the Mediterranean and the times and climate of the Northern Hemisphere. The Hebrew calendar year is longer by about 6 minutes and 25+25/57 seconds than the present-day mean solar year, so that every 224 years, the Hebrew calendar will fall a full day behind the modern solar year, and about every 231 years it will fall a full day behind the Gregorian calendar year.
The present counting method for years use the Anno Mundi epoch (Latin for "in the year of the world", in Hebrew לבריאת העולם, "from the creation of the world"), abbreviated AM or A.M. and also referred to as the Hebrew era. Hebrew year 5772 began at sunset on 28 September 2011 and ended on 16 September 2012. Hebrew year 5773 began at sunset on 16 September 2012 and will end on 4 September 2013.
The Hebrew calendar is a lunisolar calendar, meaning that months are based on lunar months, but years are based on solar years. The calendar year features twelve lunar months of twenty-nine or thirty days, with an intercalary lunar month added periodically to synchronize the twelve lunar cycles with the longer solar year. (These extra months are added seven times every nineteen years. See Leap months, below.) The beginning of each Jewish lunar month is based on the appearance of the new moon. Although originally the new lunar crescent had to be observed and certified by witnesses, the moment of the new moon is now approximated arithmetically.
- Concurrently there is a weekly cycle of seven days, mirroring the seven-day period of the Book of Genesis in which the world is created. The names for the days of the week, like those in the Creation account, are simply the day number within the week, with Shabbat being the seventh day. Each day of the week runs from sunset to the following sunset and is figured locally.
The mean period of the lunar month (precisely, the synodic month) is very close to 29.5 days. Accordingly, the basic Hebrew calendar year is one of twelve lunar months alternating between 29 and 30 days: Tishrei (30 days), Marcheshvan (29 days), Kislev (30 days), Tevet (29 days), Shevat (30 days), Adar (29 days), Nisan (30 days), Iyar (29 days), Sivan (30 days), Tammuz (29 days), Av (30 days), and Elul (29 days), for a total of 354 days. In leap years (such as 5771) an additional month, Adar I (30 days) is added after Shevat, while the regular Adar is referred to as "Adar II."
- There are additional rules in the Hebrew calendar to prevent certain holidays from falling on certain days of the week. (See Rosh Hashanah postponement, below.) These rules are implemented by adding an extra day to Marcheshvan (making it 30 days long) or by removing one day from Kislev (making it 29 days long). Accordingly, a common Hebrew calendar year can have a length of 353, 354 or 355 days, while a leap Hebrew calendar year can have a length of 383, 384 or 385 days.
The insertion of the leap month mentioned above is based on the requirement that Passover—the festival celebrating the Exodus from Egypt, which took place in the spring—always occur in the [northern hemisphere's] spring season. Since the adoption of a fixed calendar, intercalations in the Hebrew calendar have been assigned to fixed points in a 19-year cycle. Prior to this, the intercalation was determined empirically:
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.
 Sources and history
- For smaller units of time, see Measurement of hours below.
The Jewish day is of no fixed length. The Jewish day is modeled on the reference to "...there was evening and there was morning..." in the Creation account in the first chapter of Genesis. Accordingly, it runs from sunset (start of "the evening") to the next sunset. At higher latitudes in summer, when the sun does not sink below the horizon, a day is counted from midday to midday, and in the winter, when the sun does not rise above the horizon, from midnight to midnight.
There is no clock in the Jewish scheme, so that a civil clock is used. Though the civil clock incorporates local adoptions of various conventions such as time zones, standard times and daylight saving, these have no place in the Jewish scheme. The civil clock is used only as a reference point - in expressions such as: "Shabbat starts at ...". The steady progression of sunset around the world and seasonal changes results in gradual civil time changes from one day to the next based on observable astronomical phenomena (the sunset) and not on man-made laws and conventions.
Instead of the international date line convention, there are varying opinions as to where the day changes. One opinion uses the antimeridian of Jerusalem. (Jerusalem is 35°13’ east of the prime meridian, so the antimeridian is at 144°47' W, passing through eastern Alaska.) Other opinions exist as well.
The Hebrew calendar follows a seven-day weekly cycle, which runs concurrently but independently of the monthly and annual cycles. The names for the days of the week are simply the day number within the week. In Hebrew, these names may be abbreviated using the numerical value of the Hebrew letters, for example יום א׳ (Day 1, or Yom Rishon (יום ראשון)):
- Yom Rishon - יום ראשון (abbreviated יום א׳) = "first day" = Sunday (starting at preceding sunset)
- Yom Sheni - יום שני (abbr. יום ב׳) = "second day" = Monday
- Yom Shlishi - יום שלישי (abbr. יום ג׳) = "third day" = Tuesday
- Yom Reviʻi - יום רביעי (abbr. יום ד׳) = "fourth day" = Wednesday
- Yom Chamishi - יום חמישי (abbr. יום ה׳) = "fifth day" = Thursday
- Yom Shishi - יום ששי (abbr. יום ו׳) = "sixth day" = Friday
- Yom Shabbat - יום שבת (abbr. יום ש׳) or more usually שבת - Shabbat = "Sabbath day (Rest day)" = Saturday.
The names of the days of the week are modeled on the seven days mentioned in the Creation story. For example, Genesis 1:5 "... And there was evening and there was morning, one day". One day also translates to first day or day one. Similarly, see Genesis 1:8, 1:13, 1:19, 1:23, 1:31 and 2.2.
 Importance of lunar months
From very early times, the Mesopotamian lunisolar calendar was in wide use by the countries of the western Asia region. The structure, which was also used by the Israelites, was based on lunar months with the intercalation of an additional month to bring the cycle closer to the solar cycle.
Num 10:10 stresses the importance in Israelite religious observance of the new month (Hebrew: ראש חודש, Rosh Chodesh, "beginning of the month"): "... in your new moons, ye shall blow with the trumpets over your burnt-offerings..." Similarly in Num 28:11. "The beginning of the month" meant the appearance of a new moon.
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 eye witnesses required to testify to the Sanhedrin to having seen the new lunar crescent at sunset. 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. These observations were compared against calculations. When thirty days elapsed since the last new moon, the witnesses were readily believed.
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 Samaritans began to light false fires, messengers were sent. 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.
In his work Mishneh Torah (1178), Maimonides included a chapter "Sanctification of the New Moon", in which he discusses the calendrical rules and their scriptural basis. He notes,
"By how much does the solar year exceed the lunar year? By approximately 11 days. Therefore, whenever this excess accumulates to about 30 days, or a little more or less, one month is added and the particular year is made to consist of 13 months, and this is the so-called embolismic (intercalated) year. For the year could not consist of twelve months plus so-and-so many days, since it is said: throughout the months of the year (Num 28:14), which implies that we should count the year by months and not by days."
 Names of months
Biblical references to the pre-Jewish calendar include ten months identified by number rather than by name. In parts of the Torah portion Noach ("Noah") (specifically, Gen 7:11, Gen 8:3-4, Gen 8:13-14) it is implied that the months are thirty days long. There is also indication that there were twelve months in the annual cycle (1Kin 4:7, 1Chr 27:1-15).
Many countries in the western Asian region used the Mesopotamian calendar from very early times, though the names of months varied. Prior to the Babylonian exile, the names of only four months are referred to in the Tanakh:
- Aviv - first month - literally "spring" (Exodus 12:2, 13:4, 23:15, 34:18, Deut. 16:1);
- Ziv - second month - literally "light" (1 Kings 6:1, 6:37);
- Ethanim - seventh month - literally "strong" in plural, perhaps referring to strong rains (1 Kings 8:2); and
- Bul - eighth month (1 Kings 6:38).
During the Babylonian exile, which started in 586 BCE, Babylonian month names were adopted, which are still in use. The Syrian calendar used in the Levant region shares many of the names for months as the Hebrew calendar, such as Nisan, Iyyar, Tammuz, Ab, Elul, Tishri, and Adar, indicating a common Babylonian origin.
Hebrew names and romanized transliteration may somewhat differ, as they do for חשוון / Marcheshvan or כסלו / Kislev: the Hebrew words shown here are those commonly indicated e.g. in newspapers.
|1||נִיסָן||Nīsān||Nisan||Nissan||30 days||Nisanu||Passover||Called Abib (Exodus 13:4, 23:15, 34:18, Deut. 16:1)|
and Nisan (Esther 3:7) in the Tanakh.
|2||אִיָּר / אייר||ʼIyyār||Iyyar||Iyar||29 days||Ayaru||Pesach Sheni
|Called Ziv in 1 Kings 6:1, 6:37.|
|3||סִיוָן / סיוון||Sīwān||Sivan||Siwan||30 days||Simanu||Shavuot|
|4||תַּמּוּז||Tammūz||Tammuz||Tamuz||29 days||Dumuzu||Seventeenth of Tammuz||Named for the Babylonian god Dumuzi|
|5||אָב||ʼĀḇ||Av||Ab||30 days||Abu||Tisha B'Av
|Called Ethanim in 1 Kings 8:2.|
First month of civil year.
|8||מַרְחֶשְׁוָן / מרחשוון||Marḥešwān||Marẖeshvan||Marcheshvan
|Arakhsamna||Called Bul in 1 Kings 6:38.|
|9||כִּסְלֵו / כסליו||Kislēw||Kislev||Kislev
|10||טֵבֵת||Ṭēḇēṯ||Tevet||Tebeth||29 days||Tebetu||Tenth of Tevet|
|30 days||Shabatu||Tu Bishvat|
|12L*||אֲדָר א׳||Adar I*||30 days||*Only in Leap years.|
|12||אֲדָר / אֲדָר ב׳||ʼĂḏār||Adar / Adar II*||29 days||Adaru||Purim|
In a regular (kesidran) year, Marcheshvan has 29 days and Kislev has 30 days. However, because of the Rosh Hashanah postponement rules (see below) Kislev may lose a day to have 29 days, and the year is called a short (chaser) year, or Marcheshvan may acquire an additional day to have 30 days, and the year is called a full (maleh) year. The calendar rules have been designed to ensure that Rosh Hashanah does not fall on a Sunday, Wednesday or Friday. This is to ensure that Yom Kippur does not directly precede or follow Shabbat, which would create practical difficulties, and that Hoshana Rabbah is not on a Shabbat, in which case certain ceremonies would be lost for a year.
 Leap months
The solar year is about eleven days longer than twelve lunar months. The Bible does not directly mention the addition of "embolismic" or intercalary months. However, without the insertion of embolismic months, Jewish festivals would gradually shift outside of the seasons required by the Torah. This has been ruled as implying a requirement for the insertion of embolismic months to reconcile the lunar cycles to the seasons, which are integral to solar yearly cycles.
When the observational form of the calendar was in use, whether or not an embolismic month was announced after the "last month" (Adar) depended on '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. 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. However, according to some traditions, the announcement of the month of Aviv could also be postponed depending on the condition of roads used by families to come to Jerusalem for Passover, adequate numbers of lambs to be sacrificed at the Temple, and on the ripeness of the barley that was needed for the first fruits ceremony.
Under the codified rules, the Jewish calendar is based on the Metonic cycle of 19 years, of which 12 are common years (12 months) and 7 leap years (13 months). The leap years are years 3, 6, 8, 11, 14, 17, and 19 of the Metonic cycle. Year 19 (there is no year 0) of the Metonic cycle is a year exactly divisible by 19 (when the Jewish year number, when divided by 19, has no remainder). In the same manner, the remainder of the division indicates the year in the Metonic cycle (years 1 to 18) the year is in.
During leap years Adar I (or Adar Aleph — "first Adar") is added before the regular Adar. Adar I is actually considered to be the extra month, and has 30 days. Adar II (or Adar Bet — "second Adar") is the "real" Adar, and has the usual 29 days. For this reason, holidays such as Purim are observed in Adar II, not Adar I.
 New year
- this month shall be unto you the beginning of months; it shall be the first month of the year to you.
Nisan 1 is referred to as the ecclesiastical new year.
In ancient Israel, the start of the ecclesiastical new year for the counting of months and festivals (i.e. Nisan) was determined by reference to Passover. Passover begins on 14 Nisan, (Leviticus 23:4-6) which corresponds to the full moon of Nisan. As Passover is a spring festival, 14 Nisan begins on the night of a full moon after the vernal equinox. According to normative Judaism, the verses in Exodus 12:1–2 require that the months be determined by a proper court with the necessary authority to sanctify the months.
According to some Christians and Karaites, 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. If the barley was not ripe an intercalary month would be added before Nisan.
The day most commonly referred to as the "New Year" is 1 Tishrei, which actually begins in the seventh month of the ecclesiastical year. On that day the formal New Year for the counting of years (such as Shmita and Yovel), Rosh Hashanah ("head of the year") is observed. (see Ezekiel 40:1, which uses the phrase "beginning of the year".) This is the civil new year, and the date on which the year number advances. Certain agricultural practices are also marked from this date.
Josephus, in the 1st century CE, states that while -
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]."
Edwin Thiele has concluded that the ancient northern Kingdom of Israel counted years using the ecclesiastical new year starting on 1 Aviv (Nisan), while the southern Kingdom of Judah counted years using the civil new year starting on 1 Tishrei. The practice of the Kingdom of Israel was also that of Babylon, as well as other countries of the region. The practice of Judah is still followed.
In fact the Jewish calendar has a multiplicity of new years for different purposes. The use of these dates has been in use for a long time. The use of multiple starting dates for a year is comparable to different starting dates for civil "calendar years", "tax or fiscal years", "academic years", "religious cycles", etc. By the time of the redaction of the Mishnah, Rosh Hashanah 1:1 (c. 200 CE), jurists had identified four new-year dates:
The 1st of Nisan is the new year for kings and feasts; the 1st of Elul is the new year for the tithe of cattle... the 1st of Tishri is the new year for years, of the years of release and jubilee years, for the planting and for vegetables; and the 1st of Shevat is the new year for trees-so the school of Shammai; and the school of Hillel say: On the 15th thereof.
 Anno Mundi (AM)
Since the codification by Maimonides in 1178CE, the Jewish calendar has used the Anno Mundi epoch (Latin for “in the year of the world,” abbreviated AM or A.M.; Hebrew לבריאת העולם), sometimes referred to as the “Hebrew era.”
During the Talmudic era, from the first to the tenth century CE, the center of world Judaism was in the Middle East, primarily in the Talmudic Academies of Iraq and Palestine. Jews in these regions used Seleucid Era dating (also known as the "Anno Graecorum (AG)" or the "Era of Contracts") as the primary method for calculating the calendar year. For example, the writings of Josephus and the Books of the Maccabees used Seleucid Era dating exclusively, and the Avodah Zarah states:
Rav Aha b. 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!
Said Rav Nahman: In the Diaspora the Greek Era alone is used. He [the questioner] 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.
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 70CE destruction of the Second Temple, and the number of years since the Creation year based on the calculation in the Seder Olam Rabbah of Rabbi Jose ben Halafta in about 160 AD. By his calculation, based on the Masoretic Text, Adam was created in 3760 BC, later confirmed by the Muslim chronologist al-Biruni as 3448 years before the Seleucid era. An example is the c.8th century CE Baraita of Samuel.
In the 8th and 9th centuries AD, the center of Jewish life moved from Babylonia to Europe, so calculations from the Seleucid era "became meaningless". In 1178CE, Maimonides wrote in the Misheh Torah, Sanctification of the Moon (11.16), 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" (March 22, 1178 CE). He included all the rules for the calculated calendar and their scriptural basis, including the modern epochal year in his work, and beginning formal usage of the anno mundi era. From the 11th century, anno mundi dating became dominant throughout most of the world's Jewish communities Today, the rules detailed in Maimonides' calendrical code are those generally used by Jewish communities throughout the world.
According to Rabbinic reckoning, the beginning of "year 1" is not Creation, but about one year before Creation, with the new moon of its first month (Tishrei) to be called molad tohu (the mean new moon of chaos or nothing). The Jewish calendar's epoch (reference date), 1 Tishrei 1 AM, is equivalent to Monday, 7 October 3761 BCE in the proleptic Julian calendar, the equivalent tabular date (same daylight period) and is about one year before the traditional Jewish date of Creation on 25 Elul AM 1, based upon the Seder Olam Rabbah of Rabbi Yossi ben Halafta, a 2nd century CE sage. Thus, adding 3760 before Rosh Hashanah or 3761 after to a Julian or Gregorian year number after 1 CE will yield the Hebrew year. For earlier years there may be a discrepancy (see: Missing years (Jewish calendar)). The Jewish year 5771 AM began on 9 September 2010 (1 Tishrei or Rosh Hashanah) and, since it is a 13-month year, ended on 28 September 2011 (29 Elul).
In Hebrew there are two common ways of writing the year number: with the thousands, called לפרט גדול ("major era"), and without the thousands, called לפרט קטן ("minor era").
 Other systems
Before the adoption of the current year numbering system, other systems were in use. In early times, the years were counted from some significant historic event. (e.g. 1 Kings 6:1) 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 also followed by the united kingdom of Israel (e.g. 1 Kings 14:25), kingdom of Judah (e.g. 2 Kings 18:13), kingdom of Israel (e.g. 2 Kings 17:6), Persia (e.g. Nehemiah 2:1) and others. Besides, the author of 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, (e.g. 2 Kings 8:16) though some commentators note that these dates do not synchronise. Other era dating systems have been used at other times. For example, Jewish communities in Babylonian captivity counted the years from the first deportation, that of Jehoiachin, (e.g. Ezekiel 1:1-2). The era year was then called "year of the captivity of Jehoiachin". (e.g. 2 Kings 25:27) During the Greek period, Seleucid era counting was used, (e.g. 1 Maccabees 1:54) at least in the Greek-influenced area of Israel.
The Seder Olam Rabbah also recognized the importance of the Jubilee and Sabbatical cycles as a long-term calendrical system, and attempted at various places to fit the Sabbatical and Jubilee years into its chronological scheme.
 Karaite calendar
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 Rabbinical 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. 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), 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, as they are not mentioned in the Tanakh. This can affect the dates observed for all the Jewish holidays in a particular year by one day.
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.
 Dates of Jewish New Year
|5762||18 September 2001||354||5782||7 September 2021||384*|
|5763||7 September 2002||385*||5783||26 September 2022||355|
|5764||27 September 2003||355||5784||16 September 2023||383*|
|5765||16 September 2004||383*||5785||3 October 2024||355|
|5766||4 October 2005||354||5786||23 September 2025||354|
|5767||23 September 2006||355||5787||12 September 2026||385*|
|5768||13 September 2007||383*||5788||2 October 2027||355|
|5769||30 September 2008||354||5789||21 September 2028||354|
|5770||19 September 2009||355||5790||10 September 2029||383*|
|5771||9 September 2010||385*||5791||28 September 2030||355|
|5772||29 September 2011||354||5792||18 September 2031||354|
|5773||17 September 2012||353||5793||6 September 2032||383*|
|5774||5 September 2013||385*||5794||24 September 2033||355|
|5775||25 September 2014||354||5795||14 September 2034||385*|
|5776||14 September 2015||385*||5796||4 October 2035||354|
|5777||3 October 2016||353||5797||22 September 2036||353|
|5778||21 September 2017||354||5798||10 September 2037||385*|
|5779||10 September 2018||385*||5799||30 September 2038||354|
|5780||30 September 2019||355||5800||19 September 2039||355|
|5781||19 September 2020||353||5801||8 September 2040||383*|
* Year of 13 months.
 Change to a calculated calendar
 Observational principles
 Evaluation of the Mishnaic evidence
It has been noted that the procedures described in the Mishnah and Tosefta are all plausible procedures for regulating an empirical lunar calendar. Fire-signals, for example, or smoke-signals, are known from the pre-exilic Lachish ostraca. 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. Another Mishnah takes it for granted that it cannot be known in advance whether a year's lease is for twelve or thirteen months. 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 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.
 Modern calendar
Between 70 CE and 1178 CE, the observation-based calendar was gradually replaced by a mathematically calculated one. Except for the epoch year number, the calendar rules reached their current form by the beginning of the 9th century, as described by the Muslim astronomer al-Khwarizmi (c. 780–850 CE) in 823.
One notable difference between the calendar of that era and the modern form was the date of the epoch (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.
Most of the present rules of the calendar were in place by 823, according to a treatise by al-Khwarizmi. Al-Khwarizmi's study of the Jewish calendar, Risāla fi istikhrāj taʾrīkh al-yahūd "Extraction of the Jewish Era" describes the 19-year intercalation cycle, the rules for determining on what day of the week the first day of the month Tishrī shall fall, the interval between the 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.
In 921, Aaron ben Meïr proposed changes to the calendar. Though the proposals were rejected, they indicate that all of the rules of the modern calendar (except for the epoch) were in place before that date. 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.
There is a tradition, first mentioned by Hai Gaon (died 1038 CE), that Hillel b. R. Yehuda "in the year 670 of the Seleucid era" (i.e., 358–359 CE) was responsible for the new calculated calendar with a fixed intercalation cycle. Later writers, such as Nachmanides, explained Hai Gaon's words to mean that the entire computed calendar was due to Hillel b. Yehuda in response to persecution of Jews. Maimonides, in the 12th century, stated that the Mishnaic calendar was used "until the days of Abaye and Rava", who flourished 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 b. Yehuda (whom they identify with the mid-4th century Jewish patriarch Ioulos, attested in a letter of the Emperor Julian, and the Jewish patriarch Ellel, mentioned by Epiphanius) instituted the computed Hebrew Calendar because of persecution. H. Graetz  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 and Gallus. A later writer, S. 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.
Both the tradition that Hillel b. Yehuda instituted the complete computed calendar, and the theory that the computed calendar was introduced due to repression or persecution, have been questioned. Furthermore, two Jewish dates during post-Talmudic times (specifically in 506 and 776) are impossible under the rules of the modern calendar, indicating that its arithmetic rules were developed in Babylonia during the times of the Geonim (7th to 8th centuries). The Babylonian rules required the delay of the first day of Tishrei when the new moon occurred after noon.
The Talmuds do, however, indicate at least the beginnings of a transition from a purely empirical to a computed calendar. According to a statement attributed to Yose, an Amora who lived during the second half of the 3rd century, the feast of Purim, 14 Adar, could not fall on a Sabbath nor a Monday, lest 10 Tishrei (Yom Kippur) fall on a Friday or a Sunday. 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. In another passage, a sage is reported to have counseled "those who make the computations" not to set the first day of Tishrei or the Day of the Willow on the sabbath. This indicates that there was a group who "made computations" and were in a position to control, to some extent, the day of the week on which Rosh Hashanah would fall.
 Other practices
Outside of Rabbinic circles, evidence shows a diversity of Jewish practice.
 The Qumran calendar
Many of the Dead Sea (Qumran) Scrolls have references to a unique calendar, used by the people there, who are often assumed to be Essenes.
The year of this calendar used the ideal Mesopotamian calendar of twelve 30-day months, to which were added 4 days at the equinoxes and solstices (cardinal points), making a total of 364 days.
There was some ambiguity as to whether the cardinal days were at the beginning of the months or at the end, but the clearest calendar attestations give a year of four seasons, each having three months of 30, 30, and 31 days with the cardinal day the extra day at the end, for a total of 91 days, or exactly 13 weeks. Each season started on the 4th day of the week (Wednesday), every year. (Ben-Dov, Head of All Years, p. 16-17)
With only 364 days, it is clear that the calendar would after a few years be very noticeably different from the actual seasons, but there is nothing to indicate what was done about this problem. Various suggestions have been made by scholars. One is that nothing was done and the calendar was allowed to change with respect to the seasons. Another suggestion is that changes were made irregularly, only when the seasonal anomaly was too great to be ignored any longer. (Ben-Dov, Head of All Years, p. 19-20)
The writings often discuss the moon, but the calendar was not based on the movement of the moon any more than indications of the phases of the moon on a modern western calendar indicate that that is a lunar calendar.
 Persian civil calendar
Calendrical evidence for the postexilic Persian period is found in papyri from the Jewish colony at Elephantine, in Egypt. These documents show that the Jewish community of Elephantine used the Egyptian and Babylonian calendars.
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. 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 Phamenoth, or according to the intercalary month every third year in the month of Pharmuthi", suggesting a fairly consistent intercalation scheme that kept Nisan 14 approximately between the Phamenoth 10 (March 6 in the 4th century CE) and Pharmuthi 10 (April 5). Jewish funerary inscriptions from Zoar, south of the Dead Sea, dated from the 3rd to the 5th century CE, indicate that when years were intercalated, the intercalary month was at least sometimes a repeated month of Adar. But the inscriptions reveal no clear pattern of regular intercalations, nor do they indicate any consistent rule for determining the start of the lunar month.
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. Today, the rules detailed in Maimonides' code are those generally used by Jewish communities throughout the world.
 Usage in contemporary 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: the Jewish New Year marks the transition from the dry season to the rainy one, and 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. The Kibbutz movement was especially inventive in creating new rituals fitting this interpretation.
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, and the Holocaust Commemoration Day on 27 Nisan.
Nevertheless, since the 1950s usage of the Hebrew calendar has steadily declined, in favor of the Gregorian calendar. At present, Israelis — except for a minority of the religiously observant — conduct their private and public life according to the Gregorian calendar, although 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, though only rarely do people make use of this option) and on the mastheads of newspapers.
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" — "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. 
The disparity between the two calendars is especially noticeable with regard to commemoration of the assassinated Prime Minister Yitzchak Rabin. The official Day of Commemoration, instituted by a special Knesset law, is marked according to the Hebrew calendar - on 12 Marcheshvan. However, left-leaning Israelis, who revere Rabin as a martyr for the cause of peace and who are predominantly secular, hold their commemoration on 4 November. In some years the two competing Rabin Memorial Days are separated by as much as two weeks.
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.
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 (2×4x3) possible combinations, but only 14 of them are valid. Each of these patterns is called a keviyah (Hebrew קביעה for "a setting" or "an established thing"), and is encoded as a series of three Hebrew letters.
 Leap years
The Jewish calendar is based on 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. 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. For example, Jewish year 5771 divided by 19 results in a remainder of 14, indicating that it is year 14 of the Metonic cycle. Since there is no year 0, a remainder of 0 indicates that the year is year 19 of the cycle. (See also Golden number (time).)
Years 3, 6, 8, 11, 14, 17, and 19 the Metonic cycle are leap years. To assist in remembering this sequence, some people use the mnemonic Hebrew word GUCHADZaT "גוחאדז"ט", where the Hebrew letters gimel-vav-het aleph-dalet-zayin-tet are used as Hebrew numerals equivalent to 3, 6, 8, 1, 4, 7, 9. The keviyah merely records whether the year is leap or common; פ for p'shutah, meaning simple and indicating a common year, and מ indicating a leap year.
Another memory aid notes that 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 remarkable in the context of 19 equal temperament.
A mathematical way to determine leap year is to calculate the remainder using the following calculation: (7 × the Jewish year number + 1) / 19; if the remainder is less than 7, the year is a leap year. Also, rounding the result of (7 × the Jewish year number + 1) / 13 to the nearest whole number calculates a 0 for leap years and 1 for common years.
 Rosh Hashanah postponement
|Day of week||Number of days|
To calculate the day on which Rosh Hashanah falls, it is necessary to first calculate the molad (lunar conjunction or new moon) of Tishrei, and then determine whether the start of the year must be postponed. The molad can be calculated by multiplying the mean length of a (synodic) lunar month (29 days, 12 hours, and 793 parts) by the elapsed time since another molad whose weekday is known. (There are 1080 "parts" in an hour, making one part equal to 31/3 seconds.) The molad tohu began 2 days, 5 hours, and 204 parts after the beginning of the week.
The rules are complicated by the fact that the months subject to adjustment, Marcheshvan and Kislev, are the eighth and ninth months of the ecclesiastical year while Tishrei is the seventh month. This means that adjustments must be made in one year in anticipation of the day of the week on which Rosh Hashanah will fall in the next year, which may itself be affected by the day on which it will fall in the third year, and so on. The process is further complicated by the need to insert leap months in accordance with their own cycle.
Nevertheless, only four possible adjustments are needed. These are called the Rosh Hashanah postponement rules, or deḥiyyot:
- If the molad occurs at or later than 18 hours, Rosh Hashanah is postponed a day. This is called deḥiyyah molad zaken, meaning an "old conjunction."
- If the molad occurs on a Sunday, Wednesday, or Friday, Rosh Hashanah is postponed a day. If deḥiyyah molad zaken places Rosh Hashanah on one of these days, it is postponed a second day. This is called deḥiyyah lo ADU, an acronym meaning "not one, four, or six."
The first of these (deḥiyyah molad zaken) is thought to be a relic of when the calendar was established empirically (although there is some doubt); the second (deḥiyyah lo ADU) is applied for religious reasons.
The last two rules are applied much less frequently and exist to prevent illegal year lengths. Because they apply to the molad, they are never used if another postponement is made. Their names are Hebrew acronyms for the way they are calculated:
- If the molad in a common year falls on a Tuesday after 9 hours and 204 parts, Rosh Hashanah is postponed to Thursday. This is deḥiyyah GaTaRaD, an acronym meaning "3 (Tuesday), 9, 204."
- If the molad following a leap year falls on a Monday after 15 hours and 589 parts, Rosh Hashanah is postponed to Tuesday. This is deḥiyyah BeTUTeKaPoT, and acronym for "2 (Monday), 15, 589."
At the innovation of the rabbis, the mathematical calendar has been arranged to ensure that Yom Kippur does not fall on a Friday or Sunday, and Hoshana Rabbah does not fall on Shabbat. These rules have been instituted because Shabbat restrictions also apply to Yom Kippur, so that if Yom Kippur were to fall on Friday, it would not be possible to make necessary preparations for Shabbat (such as candle lighting). Similarly, if Yom Kippur fell on a Sunday, it would not be possible to make preparations for Yom Kippur because the preceding day is Shabbat. Additionally, the laws of Shabbat override those of Hoshana Rabbah, so that if Hoshana Rabbah were to fall on Shabbat certain rituals that are a part of the Hoshana Rabbah service (such as carrying willows, which is a form of work) could not be performed.
To prevent Yom Kippur (10 Tishrei) from falling on a Friday or Sunday, Rosh Hashanah (1 Tishrei) cannot be a Wednesday or Friday. Likewise, to prevent Hoshana Rabbah (21 Tishrei) from falling on a Saturday, Rosh Hashanah cannot be a Sunday. This leaves only four days on which Rosh Hashanah can fall: Monday, Tuesday, Thursday, and Saturday, which are referred as the "four gates." Each day is associated with a number (its order in the week, starting with Sunday as 1), and these numbers are associated with Hebrew letters. Therefore the keviyah 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 the year.
 Deficient, regular, and complete years
The postponement of the year is compensated for by adding a day to the second month or removing one from the third month. A Jewish common year can only have 353, 354, or 355 days. A leap year is always 30 days longer, and so can have 383, 384, or 385 days.
- A chaserah year (Hebrew for "deficient" or "incomplete") is 353 or 383 days long. Both Marcheshvan and Kislev have 29 days. The Hebrew letter ח "het" is used in the keviyah.
- A kesidrah year ("regular" or "in-order") is 354 or 384 days long. Marcheshvan has 29 days while Kislev has 30 days. The Hebrew letter כ "kaf" is used in the keviyah.
- A shlemah year ("complete" or "perfect") is 355 or 385 days long. Both Marcheshvan and Kislev have 30 days. The Hebrew letter ש "shin" is used in the keviyah.
Whether a year is deficient, regular, or complete is determined by the time between two adjacent Rosh Hashanah observances and the leap year. While the keviyah is sufficient to describe a year, a variant specifies the day of the week for the first day of Pesach (Passover) in lieu of the year length.
A Metonic cycle equates to 235 lunar months in each 19-year cycle. This gives an average of 6939 days, 16 hours, and 595 parts for each cycle. But due to the Rosh Hashanah postponement rules (preceding section) a cycle of 19 Jewish years can be either 6939, 6940, 6941, or 6942 days in duration. Since none of these values is evenly divisible by seven, the Jewish calendar repeats exactly only following 36,288 Metonic cycles, or 689,472 Jewish years. There is a near-repetition every 247 years, except for an excess of 50 minutes (905 parts).
 Days of week of holidays
The period from 1 Adar (or Adar II, in leap years) to 29 Heshvan contains all of the festivals specified in the Bible - Purim (14 Adar), Pesach (15 Nisan), Shavuot (6 Sivan), Rosh Hashanah (1 Tishrei), Yom Kippur (10 Tishrei), Sukkot (15 Tishrei), and Shemini Atzeret (22 Tishrei). This period is fixed, during which no adjustments are made.
|10 Tevet||Tu Bishvat|
|Thu||Sat||Sun||Sun*||Mon||Wed||Sun or Mon||Sun or Tue||Sat or Mon|
|Sun||Tue||Wed||Tue||Thu||Sat||Wed or Thu||Wed, Thu, or Fri||Tue, Wed, or Thu|
|Tue||Thu||Fri||Thu||Sat||Mon||Fri or Sat||Fri or Sun||Thu or Sat|
|*Postponed from Shabbat|
 Measurement of hours
Every hour is divided into 1080 halakim or parts. A part is 3⅓ seconds or 1/18 minute. The ultimate ancestor of the helek was a small Babylonian time period called a barleycorn, itself equal to 1/72 of a Babylonian time degree (1° of celestial rotation). Actually, the barleycorn or she was the name applied to the smallest units of all Babylonian measurements, whether of length, area, volume, weight, angle, or time.
The weekdays start with Sunday (day 1) and proceed to Saturday (day 7). Since some calculations use division, a remainder of 0 signifies Saturday.
While calculations of days, months and years are based on fixed hours equal to 1/24 of a day, the beginning of each halachic day is based on the local time of sunset. The end of the Shabbat and other Jewish holidays is based on nightfall (Tzeth haKochabim) which occurs some amount of time, typically 42 to 72 minutes, after sunset. According to Maimonides, nightfall occurs when three medium-sized stars become visible after sunset. By the 17th century this had become three second-magnitude stars. The modern definition is when the center of the sun is 7° below the geometric (airless) horizon, somewhat later than civil twilight at 6°. The beginning of the daytime portion of each day is determined both by dawn and sunrise. Most halachic times are based on some combination of these four times and vary from day to day throughout the year and also vary significantly depending on location. The daytime hours are often divided into Sha`oth Zemaniyoth or "Halachic hours" by taking the time between sunrise and sunset or between dawn and nightfall and dividing it into 12 equal hours. The nighttime hours are similarly divided into 12 equal portions, albeit a different amount of time than the "hours" of the daytime. The earliest and latest times for Jewish services, the latest time to eat Chametz on the day before Passover and many other rules are based on Sha`oth Zemaniyoth. For convenience, the modern day using Sha`oth Zemaniyoth is often discussed as if sunset were at 6:00pm, sunrise at 6:00am and each hour were equal to a fixed hour. For example, halachic noon may be after 1:00pm in some areas during daylight saving time. Within the Mishnah, however, the numbering of the hours starts with the "first" hour after the start of the day.
 Worked example
||This section uses abbreviations that may be confusing or ambiguous. (July 2011)|
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.
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 p.m., but for the purpose of determining Rosh Hashanah, a molad occurring on or after noon is treated as belonging to the next day (the second deḥiyyah). All months are calculated as 29d, 12h, 44m, 3⅓s 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 a.m. 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 2:27:16⅔ p.m., 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 first deḥiyyah). To compensate, one day is subtracted from year 2. It is to allow 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 deḥiyyahs 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, 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. 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.
 Astronomic calculations
 Synodic month - the molad interval
A "new moon" (astronomically called a lunar conjunction and in Hebrew called a molad) is the moment at which the sun and moon are aligned horizontally with respect to a north-south line (technically, they have the same ecliptical longitude). 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, a long-term average length called the mean synodic month (also called the molad interval) is used. The mean synodic month is days, or 29 days, 12 hours, and 793 parts (44+1/18 minutes) (i.e. 29.530594 days), and is the same value determined by the Babylonians in the System B in about 300 BCE and was adopted by the Greek astronomer Hipparchus in the 2nd century BCE and by the Alexandrian astronomer Ptolemy in Almagest in the 2nd century CE (who cited Hipparchus as his source). Its remarkable accuracy (less than one second from the true value) is thought to have been achieved using records of lunar eclipses from the 8th to 5th centuries BCE.
This value is as close to the correct value of 29.530589 days as it is possible for a value to come that is rounded off to whole parts (1/18 minute). The discrepancy makes the molad interval about 0.6 seconds too long. Put another way, if the molad is taken as the time of mean conjunction at some reference meridian, then this reference meridian is drifting slowly eastward. If this drift of the reference meridian is traced back to the mid-4th century CE, the traditional date of the introduction of the fixed calendar, then it is found to correspond to a longitude midway between the Nile and the end of the Euphrates. The modern molad moments match the mean solar times of the lunar conjunction moments near the meridian of Kandahar, Afghanistan, more than 30° east of Jerusalem. Furthermore, due to the eccentricity of Earth's orbit, series of shorter lunations alternate with series of longer lunations. Consequently the actual lunar conjunction moments can range from 12 hours earlier than to 16 hours later than the molad moment, in terms of Jerusalem mean solar time.
Furthermore, the discrepancy between the molad interval and the mean synodic month is accumulating at an accelerating rate, since the mean synodic month is progressively shortening due to gravitational 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.
 Seasonal drift
The mean Hebrew calendar year is 365 days 5 hours 55 minutes and 25+25/57 seconds (365.2468 days) - computed as the molad/monthly interval of 29.530594 days × 235 months in a 19-year metonic cycle ÷ 19 years per cycle. As the present-era mean northward equinoctal year is 365 days 5 hours 49 minutes 1 second long (365.2424 days), the Hebrew calendar has a "seasonal drift" in relation to the tropical year of about a day every 224 years. In relation to the Gregorian calendar, the mean Gregorian calendar year is 365 days 5 hours 49 minutes and 12 seconds (365.2425 days), and the drift of the Hebrew calendar in relation to it is about a day every 231 years.
 Implications for Jewish ritual
Although the molad of Tishrei is the only molad moment that is not ritually announced, it is actually the only one that is relevant to the Hebrew calendar, for it determines the provisional date of Rosh Hashanah, subject to the Rosh Hashanah postponement rules. The other monthly molad moments are announced for mystical reasons. With the moladot on average almost 100 minutes late, this means that the molad of Tishrei lands one day later than it ought to in (100 minutes) ÷ (1440 minutes per day) = 5 of 72 years or 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 molad drift could be corrected by using a progressively shorter molad interval that corresponds to the actual mean lunar conjunction interval at the original molad reference meridian. Furthermore, the molad interval determines the calendar mean year, so using a progressively shorter molad interval would help correct the excessive length of the Hebrew calendar mean year, as well as helping it to "hold onto" the northward equinox for the maximum duration.
When the 19 - year intercalary cycle was finalised in the 4th century CE the earliest Passover (in year 16 of the cycle) coincided with the northward equinox, which means that Passover fell near the first full moon after the northward equinox, or that the northward equinox landed within one lunation before 16 days after the molad of Nisan. This is still the case in about 80% of years, but in about 20% of years Passover is a month late by these criteria (as it was in Hebrew years 5765 and 5768, the 8th and 11th years of the 19-year cycle = Gregorian 2005 and 2008 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 the northward equinox to land on exceptionally early Hebrew dates in such years. This problem will get worse over time, and so beginning in Hebrew year 5817 (2057 CE), year 3 of each 19-year cycle will also be a month late. Furthermore, the drift will accelerate in the future as perihelion approaches and then passes the northward equinox, and if the calendar is not amended then Passover will start to land on or after the summer solstice around Hebrew year 16652 (12892 CE). (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.)
The seriousness of the spring equinox drift is widely discounted on the grounds that Passover will remain in the spring season for many millennia, and the text of the Torah is generally not interpreted as having specified tight calendrical limits. On the other hand, the mean southward equinoctial year length is considerably shorter, so the Hebrew calendar has been drifting faster with respect to the autumn equinox, and at least part of the harvest festival of Sukkot is already more than a month after the equinox in years 1, 9, and 12 of each 19-year cycle; beginning in Hebrew year 5818 (2057 CE), this will also be the case in year 4. (These are the same year numbers as were mentioned for the spring season in the previous paragraph, except that they get incremented at Rosh Hashanah.) This progressively increases the probability that Sukkot will be cold and wet, making it uncomfortable or impractical to dwell in the traditional succah during Sukkot. The first winter seasonal prayer for rain is not recited until Shemini Atzeret, after the end of Sukkot, yet it is becoming increasingly likely that the rainy season in Israel will start before the end of Sukkot.
No equinox or solstice will ever be more than a day or so away from its mean date according to the solar calendar, while nineteen Jewish years average 6939d 16h 33m 03⅓s compared to the 6939d 14h 26m 15s of nineteen mean tropical years. This discrepancy has mounted up to six days, which is why the earliest Passover currently falls around 27 March.
 "Rectifying" the Hebrew calendar
It has been argued by some[who?] that, as the fixed arithmetic Hebrew calendar was established on the authority of Hillel II, President of the Sanhedrin in Hebrew year 4119 (358 CE), only an equal authority (the modern Sanhedrin) can either amend it or reinstate the observational Hebrew calendar. The attribution of the fixed arithmetic Hebrew calendar solely to Hillel II has, however, been questioned by a few authors, such as Sasha Stern, who claim that the calendar rules developed gradually over several centuries.
Given the importance in Jewish ritual of establishing the accurate timing of monthly and annual times, some futurist writers and researchers have considered whether a "corrected" system of establishing the Hebrew date is required. The Hebrew calendar mean year is more than 6 minutes and 25 seconds in excess of the northern hemisphere spring equinox year, and has "drifted" an average of 7–8 days late relative to the equinox relationship that it originally had. It is not possible, however, for any individual Hebrew date to be a week or more "late", because Hebrew months always begin within a day or two of the molad moment. What happens instead is that the traditional Hebrew calendar "prematurely" inserts a leap month one year before it "should have been" inserted, where "prematurely" means that the insertion causes the spring equinox to land more than 30 days before the latest acceptable moment, thus causing the calendar to run "one month late" until the time when the leap month "should have been" inserted prior to the following spring. This presently happens in 4 years out of every 19-year cycle (years 3, 8, 11, and 19), implying that the Hebrew calendar currently runs "one month late" more than 21% of the time. To a minor degree the tardiness of the calendar is also due to not correcting for the progressively shorter mean astronomical lunation interval — although presently this only accounts for a little over six seconds of the yearly equinox drift, it more importantly accounts for nearly two hours of molad drift relative to actual mean lunar conjunctions, which is enough to cause Rosh HaShanah to start on the "wrong" date in an appreciable number of years.
Dr. Irv Bromberg has proposed a 353-year cycle of 4366 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. It takes about 31⁄2 centuries for the spring equinox to drift an average of 1⁄19th of a molad interval earlier in the Hebrew calendar. That is a very important time unit, because it can be cancelled by simply truncating a 19-year cycle to 11 years, omitting 8 years including three leap years from the sequence. That is the essential feature of the 353-year leap cycle ((9 × 19) + 11 + (9 × 19) = 353 years).
Religious questions abound about how such a system might be implemented and administered throughout the diverse aspects of the world Jewish community.
 See also
- Babylonian calendar
- Biblical and Talmudic units of measurement
- Chronology of the Bible
- Islamic calendar
- Jewish holidays 2000–2050
- Talmud, Sanhedrin 11b
- In contrast, the Gregorian calendar is a pure solar calendar, while the Islamic calendar is a pure lunar calendar.
- Under the fixed, calculated calendar, this is only loosely true. Because the calculations are based on mean lunar months, not observed ones–and because of the Rosh Hashanah postponement rules—a given month may not begin on the same day as its astronomical conjunction. See Bromberg, Dr. Irv (August 5, 2010). "Moon and the Molad of the Hebrew Calendar". utoronto.ca. http://www.sym454.org/hebrew/molad.htm. Retrieved December 16, 2012.
- This practice continues to be used in Karaite Judaism as well as in the Islamic calendar.
- Tosefta Sanhedrin 2.2, Herbert Danby, Trans., Tractate Sanhedrin Mishnah and Tosefta, Society for Promoting Christian Knowledge, London and New York, 1919, p. 31. Also quoted in Sacha Stern, Calendar and Community: A History of the Jewish Calendar Second Century BCE-Tenth Century CE, Oxford University Press, 2001, p. 70.
- Gen 1:5, Gen 1:8, Gen 1:13, Gen 1:19, Gen 1:23, Gen 1:31 and Gen 2.2.
- "CALENDAR - JewishEncyclopedia.com". http://www.jewishencyclopedia.com/articles/3921-calendar. Retrieved 2012-04-24.
- See Willie Roth's essay The International Date Line and Halacha.
- "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.
- For example, according to Morfix מילון מורפיקס, Morfix Dictionary, which is based upon Prof. Yaakov Choeka's Rav Milim dictionary. But the word meaning a non-Talmudic week is שָׁבוּע (shavuʻa), according to the same "מילון מורפיקס".
- For example, when referring to the daily psalm recited in the morning prayer (Shacharit).
- Ancient Israel: Its Life and Institutions (1961) by Roland De Vaux, John McHugh, Publisher: McGraw-Hill, ISBN 978-0-8028-4278-7, p.179
- M. Rosh Hashanah 1.7
- M. Rosh Hashanah 2.6-8
- b. 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.
- M. Rosh Hashanah 2.2
- b. Betzah 4b
- Sanctification of the New Moon. 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
- Gen 7:11 says "... on the seventeenth day of the second month—on that day all the springs of the great deep burst forth..." and Gen 8:3-4 say "...At the end of the hundred and fifty days the water had gone down, (4) and on the seventeenth day of the seventh month the ark came to rest on the mountains of Ararat..." There is an interval of 5 months and 150 days, making each month 30 days long.
- Scherman, Nosson (2005). Artscroll Chumash.
- The barley had to be "eared out" (ripe) in order to have a wave-sheaf offering of the first fruits according to the Law. Jones, Stephen (1996). Secrets of Time.
- See Maaser Rishon, Maaser Sheni, Maaser Ani.
- Josephus, Antiquities 1.81, Loeb Classical Library, 1930.
- 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
- The Chronology of the Old Testament, 16th ed., Floyd Nolan Jones, ISBN 978-0-89051-416-0, p. 118-123
- M. Rosh Hashanah 1, in Herbert Danby, trans., The Mishnah, Oxford University Press, 1933, p. 188.
- Chronology of the Old Testament, Dr. Floyd Nolen Jones "When the center of Jewish life moved from Babylonia to Europe during the 8th and 9th centuries AD, calculations from the Seleucid era became meaningless. Over those centuries, it was replaced by that of the anno mundi era of the Seder Olam. From the 11th century, anno mundi dating became dominant throughout most of the world's Jewish communities."
- Avodah Zarah, tractate 10
- Avodah Zarah, tractate 9 Footnote: "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 B.C.E.) when, at the Battle of Gaza, Seleucus Nicator, one of the followers of Alexander the Great, gained dominion over Palestine. It is also termed Seleucid or Greek Era [H]. Its designation as Alexandrian Era connecting it with Alexander the Great (Maim. Yad, Gerushin 1, 27) is an anachronism, since Alexander died in 323 B.C.E. — eleven years before this Era began (v. E. Mahler, Handbuch der judischen Chronologie, p. 145). This Era, which is first mentioned in Mac. I, 10, and was used by notaries or scribes for dating all civil contracts, 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 (Eben Saphir, Lyck, 1866, p. 62b). (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. One of the recently discovered Genizah documents bears the date 13 Tammuz 987 after the Destruction of the Temple — i.e. 917 C.E. — (Op. cit. p. 152, also Marmorstein ZDMG, Vol. VI, p. 640). The difference between the two Eras as far as the tens and units are concerned is thus 20. If therefore a Tanna, say in the year 156 Era of Dest. (225 C.E.), while remembering, naturally, the century, is uncertain about the tens and units, he should ask the notary what year it is according to his — Seleucid — era. He will get the answer 536 (156 + 380), on adding 20 to which he would get 556, the last two figures giving him the year  56 of the Era of Destruction."
- p.107, Kantor
- See The Remaining Signs of Past Centuries.
- The Easter Computus and the Origins of the Christian Era, Alden A. Mosshammer
- A minority opinion places Creation on 25 Adar 1 AM, six months earlier, or six months after the modern epoch.
- The Karaite Korner: The New Moon in the Hebrew Bible
- The Karaite Korner: Aviv (Barley)
- Sacha Stern, Calendar and Community, Oxford University Press, 2001, pp. 162ff.
- James B. Pritchard, ed., The Ancient Near East: An Anthology of Texts and Pictures, Vol. 1, Princeton University Press, p. 213.
- M. 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."
- M. Baba Metzia 8.8.
- Solomon Gandz, "The origin of the Two New Moon Days", Jewish Quarterly Review (New Series), v. 40, 1949-50. Reprinted in Shlomo Sternberg, ed., Studies in Hebrew Astronomy and Mathematics by Solomon Gandz, KTAV, New York, 1970, pp. 72–73.
- Sacha Stern, Calendar and Community.
- E.S. Kennedy, "Al-Khwarizmi on the Jewish calendar", Scripta Mathematica 27 (1964) 55–59.
- "al-Khwarizmi", Dictionary of Scientific Biography, VII: 362, 365.
- See The Remaining Signs of Past Centuries.
- 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.
- Epiphanius, Adversus Haereses 30.4.1, in Frank Williams, trans., The Panarion of Epiphanius of Salamis Book I (Sects 1-46), Leiden, E. J.Brill, 1987, p. 122.
- H. Graetz, Popular History of the Jews, (A. B. Rhine, trans.,) Hebrew Publishing Company, New York, 1919, Vol. II, pp. 410–411. Quoted in Sacha Stern, Calendar and Community, p. 216.
- S Lieberman, "Palestine in the 3rd and 4th Centuries", Jewish Quarterly Review, New Series 36, pp. 329–370(1946). Quoted in Sacha Stern, Calendar and Community, pp. 216–217.
- Sacha Stern, Calendar and Community: A History of the Jewish Calendar Second Century BCE-Tenth Century CE, Oxford University Press, 2001. In particular section 5.1.1, discussion of the "Persecution theory."
- Samuel Poznanski, "Ben Meir and the Origin of the Jewish Calendar", Jewish Quarterly Review, Original Series, Vol. 10, pp. 152–161(1898).
- "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", Encyclopedia Judaica, Keter, Jerusalem, 1971.
- Samuel Poznanski, "Calendar (Jewish)", Encyclopaedia of Religion and Ethics, vol. 3.
- Yerushalmi Megillah 70b.
- Yerushalmi Sukkah 54b.
- Sacha Stern, "The Babylonian Calendar at Elephantine", Zeitschrift für Papyrologie und Epigraphik 130, 159–171(2000).
- 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.
- Eduard Schwartz, Christliche und jüdische Ostertafeln, (Abhandlungen der königlichen Gesellschaft der Wissenschaften zu Göttingen. Philologisch-Historische Klasse. Neue Folge, Band viii, Berlin, 1905.
- Peter of Alexandria, quoted in the Chronicon Paschale. Corpus Scriptorum Historiae Byzantinae, Chronicon Paschale Vol. 1, Weber, Bonn, 1832, p. 7
- Sacha Stern, Calendar and Community, pp. 87–97, 146-153.
- "The Jewish Calendar: A Closer Look". Judaism 101. http://www.jewfaq.org/calendr2.htm. Retrieved 25 March 2011.
- "Chelm.org's explaination of the Jewish calendar". 1 August 1999. http://www.chelm.org/jewish/calendar/explain.html. Retrieved 25 March 2011.
- Landau, Remy. "Hebrew Calendar Science and Myths". http://hebrewcalendar.tripod.com/. Retrieved 25 March 2011.
- 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 conjunctions.
- 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.
- Yerushalmi, Sukkah 54b.
- Otto Neugebauer, "The astronomy of Maimonides and its sources", Hebrew Union College Annual 23 (1949) 322–363.
- See, for example, Berachot chapter 1, Mishnah 2.
- 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
- 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 of course starts Wednesday at sunset wherever one happens to be.
- This will happen if TM1 is on or after 3:11:20 a.m. 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.
- TM2 will be between noon and 2:27:16⅔ p.m. on Tuesday, and TM3 will be between 9:32:43⅓ and noon on Monday.
- G. J. Toomer, Hipparcus' Empirical Basis for his Lunar Mean Motions, Centaurus, Vol 24, 1980, pp. 97–109
- Weinberg, I., Astronomical Aspects of the Jewish Calendar, Monthly Notes of the Astronomical Society of South Africa, Vol. 15, p. 86; available at 
- Bromberg, Irv. "The Rectified Hebrew Calendar.". http://individual.utoronto.ca/kalendis/hebrew/rect.htm. Retrieved 2011-05-13.
- Committee concerning the fixing of the Calendar at TheSanhedrin.org
- al-Biruni. The Chronology of Ancient Nations, Chapter VII. tr. C. Edward Sachau. London, 1879.
- Ari Belenkiy. "A Unique Feature of the Jewish Calendar — Dehiyot". Culture and Cosmos 6 (2002) 3-22.
- Jonathan Ben-Dov. Head of All Years: Astronomy and Calendars at Qumran in their Ancient Context. Leiden: Brill, 2008.
- Bonnie Blackburn and Leofranc Holford-Strevens. The Oxford Companion to the Year: An Exploration of Calendar Customs and Time-reckoning. Oxford University Press; USA, 2000.
- Sherrard Beaumont Burnaby. Elements of the Jewish and Muhammadan Calendars. George Bell and Sons, London, 1901.
- Nathan Bushwick. Understanding the Jewish Calendar. Moznaim, New York/Jerusalem, 1989. ISBN 0-940118-17-3
- William Moses Feldman. Rabbinical Mathematics and Astronomy, 3rd edition, Sepher-Hermon Press, New York, 1978.
- Eduard Mahler, Handbuch der jüdischen Chronologie. Buchhandlung Gustav Fock, Leipzig, 1916.
- Otto Neugebauer. Ethiopic astronomy and computus. Österreichische Akademie der Wissenschaften, philosophisch-historische Klasse, Sitzungsberichte 347. Vienna, 1979.
- 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.
- Samuel Poznanski. "Calendar (Jewish)". Encyclopædia of Religion and Ethics. T. & T. Clark, Edinburgh, 1910, vol. 3, pp. 117–124.
- Edward M. Reingold and Nachum Dershowitz. Calendrical Calculations: The Millennium Edition. Cambridge University Press; 2 edition (2001). ISBN 0-521-77752-6
- Louis A. Resnikoff. "Jewish Calendar Calculations", Scripta Mathematica 9 (1943) 191-195, 274-277.
- Eduard Schwartz, Christliche und jüdische Ostertafeln (Abhandlungen der königlichen Gesellschaft der Wissenschaften zu Göttingen. Philologisch-Historische Klasse. Neue Folge, Band viii), Berlin, 1905.
- Arthur Spier. The Comprehensive Hebrew Calendar: Twentieth to the Twenty-Second Century 5660-5860/1900-2100. Feldheim Publishers, Jerusalem/New York, 1986.
- Sacha Stern, Calendar and Community: A History of the Jewish Calendar Second Century BCE-Tenth Century CE. Oxford University Press, 2001.
- 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. 557–602.
- Francis Henry Woods. "Calendar (Hebrew)", Encyclopædia of Religion and Ethics. T. & T. Clark, Edinburgh, 1910, vol. 3, pp. 108–109.
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