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Hebrew calendar

2007 Schools Wikipedia Selection. Related subjects: Ancient History,
Classical History and Mythology

   The Hebrew calendar (Hebrew: הלוח העברי) or Jewish calendar is the
   annual calendar used in Judaism. It determines the dates of the Jewish
   holidays, the appropriate Torah portions for public reading, Yahrzeits
   (the date to commemorate the death of a relative), and the specific
   daily Psalms which some customarily read. Two major forms of the
   calendar have been used: an observational form used prior to the
   destruction of the Second Temple in 70 CE, and based on witnesses
   observing the phase of the moon, and a rule-based form first fully
   described by Maimonides in 1178 CE, which was adopted over a transition
   period between 70 and 1178.

   The "modern" form is a rule-based lunisolar calendar, akin to the
   Chinese calendar, measuring months defined in lunar cycles as well as
   years measured in solar cycles, and distinct from the purely lunar
   Islamic calendar and the almost entirely solar Gregorian calendar.
   Because of the roughly 11 day difference between twelve lunar months
   and one solar year, the calendar repeats in a Metonic 19-year cycle of
   235 lunar months, with an extra lunar month added once every two or
   three years, for a total of seven times every nineteen years. As the
   Hebrew calendar was developed in the region east of the Mediterranean
   Sea, references to seasons reflect the times and climate of the
   Northern Hemisphere.

History

Biblical period

   Mosaic pavement of a zodiac in the 6th century synagogue at Beit Alpha,
   Israel.
   Enlarge
   Mosaic pavement of a zodiac in the 6th century synagogue at Beit Alpha,
   Israel.
   This figure, in a detail of a medieval Hebrew calendar, reminded Jews
   of the palm branch (Lulav), the myrtle twigs, the willow branches, and
   the citron (Etrog) to be held in the hand and to be brought to the
   synagogue during the holiday of sukkot, near the end of the autumn
   holiday season.
   Enlarge
   This figure, in a detail of a medieval Hebrew calendar, reminded Jews
   of the palm branch ( Lulav), the myrtle twigs, the willow branches, and
   the citron ( Etrog) to be held in the hand and to be brought to the
   synagogue during the holiday of sukkot, near the end of the autumn
   holiday season.

   Jews have been using a lunisolar calendar since Biblical times. The
   first commandment the Jewish People received as a nation was the
   commandment to determine the New Moon. The beginning of Exodus Chapter
   12 says "This month (Nissan) is for you the first of months.". The
   months were originally referred to in the Bible by number rather than
   name. Only four pre-exilic month names appear in the Tanakh (the Hebrew
   Bible): Aviv (first; literally "Spring", but originally probably meant
   the ripening of barley), Ziv (second; literally "Light"), Ethanim
   (seventh; literally "Strong" in plural, perhaps referring to strong
   rains), and Bul (eighth), and all are Canaanite names, and at least two
   are Phoenician (Northern Canaanite). It is possible that all of the
   months were initially identifiable by native Jewish numbers or foreign
   Canaanite/Phoenician names, but other names do not appear in the Bible.

   Furthermore, because solar years cannot be divided evenly into lunar
   months, an extra embolismic or intercalary month must be added to
   prevent the starting date of the lunar cycles from "drifting" away from
   the Spring, although there is no direct mention of this in the Bible.
   There are hints, however, that the first month (today's Nissan) had
   always started only following the ripening of barley; according to some
   traditions, in case the barley had not ripened yet, a second last month
   would have been added. Only much later was a systematic method for
   adding a second last month, today's Adar I, adopted.

Babylonian exile

   During the Babylonian exile, immediately after 586 BCE, Jews adopted
   Babylonian names for the months, and some sects, such as the Essenes,
   used a solar calendar during the last two centuries BCE. The Babylonian
   calendar was the direct descendant of the Sumerian calendar.

Names and lengths of the months

   CAPTION: Hebrew names of the months with their Babylonian analogs

   Number Hebrew name Length Babylonian analog Notes
   1 Nisan / Nissan 30 days Nisanu called Aviv in the Tanakh
   2 Iyar 29 days Ayaru called Ziv in the Tanakh
   3 Sivan 30 days Simanu
   4 Tammuz 29 days Du`uzu
   5 Av 30 days Abu
   6 Elul 29 days Ululu
   7 Tishrei 30 days Tashritu called Eitanim in the Tanakh
   8 Cheshvan 29 or 30 days Arakhsamna also spelled Heshvan or Marheshvan;
   called Bul in the Tanakh
   9 Kislev 30 or 29 days Kislimu also spelled Chislev
   10 Tevet 29 days Tebetu
   11 Shevat 30 days Shabatu
   12 Adar I 30 days Adaru Only in leap years
   13 Adar / Adar II 29 days Adaru

   During leap years Adar I (or Adar Aleph — "first Adar") is considered
   to be the extra month, and has 30 days. Adar II (or Adar Bet — "second
   Adar") is the "real" Adar, and has 29 days as usual. For example, in a
   leap year, the holiday of Purim is in Adar II, not Adar I.

Second Temple era

   In Second Temple times, the beginning of each lunar month was decided
   by two eyewitnesses testifying to having seen the new crescent moon.
   Patriarch Gamaliel II (c. 100) compared these accounts to drawings of
   the lunar phases. According to tradition, these observations were
   compared against calculations made by the main Jewish court, the
   Sanhedrin. Whether or not an embolismic month (a second Adar) was
   needed depended on the condition of roads used by families to come to
   Jerusalem for Passover, on an adequate number of lambs which were to be
   sacrificed at the Temple, and on the earing of barley needed for first
   fruits.

   Once decided, the beginning of each Hebrew month was first announced to
   other communities by signal fires lit on mountaintops, but after the
   Samaritans and Boethusaeans began to light false fires, a shaliach was
   sent. The inability of the shaliach to reach communities outside Israel
   within one day, led outlying communities to celebrate scriptural
   festivals for two days rather than for one, observing the second
   feast-day of the Jewish diaspora because of uncertainty of whether the
   previous month was 29 or 30 days.

   From the times of the Amoraim (third to fifth centuries), calculations
   were increasingly used, for example by Samuel the astronomer, who
   stated during the first half of the third century that the year
   contained 365 ¼ days, and by "calculators of the calendar" circa 300.
   Jose, an Amora who lived during the second half of the fourth century,
   stated that the feast of Purim, 14 Adar, could not fall on a Sabbath
   nor a Monday, lest 10 Tishri ( Yom Kippur) fall on a Friday or a
   Sunday. This indicates 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.

Roman Era

   The Jewish-Roman wars of 66–73, 115–117, and 132–135 caused major
   disruptions in Jewish life, also disrupting the calendar. During the
   third and fourth centuries, Christian sources describe the use of
   eight, nineteen, and 84 year lunisolar cycles by Jews, all linked to
   the civil calendars used by various communities of Diaspora Jews, which
   were effectively isolated from Levant Jews and their calendar. Some
   assigned major Jewish festivals to fixed solar calendar dates, whereas
   others used epacts to specify how many days before major civil solar
   dates Jewish lunar months were to begin.

Alexandrian Jewish calendar

   The Ethiopic Christian computus (used to calculate Easter) describes in
   detail a Jewish calendar which must have been used by Alexandrian Jews
   near the end of the third century. These Jews formed a relatively new
   community in the aftermath of the annihilation (by murder or
   enslavement) of all Alexandrian Jews by Emperor Trajan at the end of
   the 115–117 Kitos War. Their calendar used the same epacts in nineteen
   year cycles that were to become canonical in the Easter computus used
   by almost all medieval Christians, both those in the Latin West and the
   Hellenist East. Only those churches beyond the eastern border of the
   Byzantine Empire differed, changing one epact every nineteen years,
   causing four Easters every 532 years to differ.

Transition period

   The period between 70 and 1178 was a transition period between the two
   forms, with the gradual adoption of more and more of the rules
   characteristic of the modern form. Except for the modern year number,
   the modern rules reached their final form before 820 or 921, with some
   uncertainty regarding when. The modern Hebrew calendar cannot be used
   to calculate Biblical dates because new moon dates may be in error by
   up to four days, and months may be in error by up to four months. The
   latter accounts for the irregular intercalation (adding of extra
   months) that was performed in three successive years in the early
   second century, according to the Talmud.

Evidence for adoption of the modern rules

   A popular tradition, first mentioned by Hai Gaon (d.1038), holds that
   the modern continuous calendar was formerly a secret known only to a
   council of sages or "calendar committee," and that Patriarch Hillel II
   revealed it in 359 due to Christian persecution. However, the Talmud,
   which did not reach its final form until c. 500, does not mention the
   continuous calendar or even anything as mundane as either the
   nineteen-year cycle or the length of any month, despite discussing the
   characteristics of earlier calendars.

   Furthermore, Jewish dates during post-Talmudic times (specifically in
   506 and 776) are impossible using modern rules, and all evidence points
   to the development of the arithmetic rules of the modern calendar in
   Babylonia during the times of the Geonim (seventh to eighth centuries),
   with most of the modern rules in place by about 820, according to the
   Muslim astronomer Muḥammad ibn Mūsā al-Ḵwārizmī. One notable difference
   was the date of the epoch (the fixed reference point at the beginning
   of year 1), which at that time was identified as one year later than
   the epoch of the modern calendar.

Controversy over the Passover of 4682 AM

   The Babylonian rules required the delay of the first day of Tishri when
   the new moon occurred after noon.

   In 921, Aaron ben Meir, a person otherwise unknown, sought to return
   the authority for the calendar to the Land of Israel by asserting that
   the first day of Tishri should be the day of the new moon unless the
   new moon occurred more than 642 parts (35 2/3 minutes, where a "part"
   is 1/1080 of an hour) after noon, when it should be delayed by one or
   two days. He may have been asserting that the calendar should be run
   according to Jerusalem time, not Babylonian. Local time on the
   Babylonian meridian was presumably 642 parts later than on the meridian
   of Jerusalem.

   An alternative explanation for the 642 parts is that if Creation
   occurred in the Autumn, to coincide with the observance of Rosh Hashana
   (which marks the changing of the calendar year), the calculated time of
   New Moon during the six days of creation was on Friday at 14 hours
   exactly (counting from the day starting at 6pm the previous evening).
   However, if Creation actually occurred six months earlier, in the
   Spring, the new moon would have occurred at 9 hours and 642 parts on
   Wednesday. Ben Meir may thus have believed, along with many earlier
   Jewish scholars, that creation occurred in Spring and the calendar
   rules had been adjusted by 642 parts to fit in with an Autumn date;

   In any event he was opposed by Saadiah Gaon. Only a few Jewish
   communities accepted ben Meir's opinion, and even these soon rejected
   it. Accounts of the controversy show that all of the rules of the
   modern calendar (except for the epoch) were in place before 921.

   In 1000, the Muslim chronologist al-Biruni also described all of the
   modern rules except that he specified three different epochs used by
   various Jewish communities being one, two, or three years later than
   the modern epoch. Finally, in 1178 Maimonides described all of the
   modern rules, including the modern epochal year.

When does the year begin?

   According to the Mishnah (Rosh Hashanah 1:1), there are four days which
   mark the beginning of the year, for different purposes:
     * Months are numbered from Nisan, reflecting the injunction in Exodus
       12:2, "This month shall be to you the beginning of months," and
       Nisan marks the new year for civil purposes.
     * The day which is most often referred to as the "New Year" is
       observed on the first of Tishri, when the year number increases by
       1 and the formal new year festival Rosh Hashana is celebrated. It
       also marks the new year for certain agricultural laws.
     * The month of Elul is the New Year for certain matters connected
       with animals.
     * Tu Bishvat ("the 15th of Shevat (ט"ו בשבט),") marks the new year
       for trees.

   There may be an echo here of a controversy in the Talmud about whether
   the world was created in Tishri or Nisan; it was decided that the
   answer is Tishri.

Modern calendar

Epoch

   The epoch of the modern Hebrew calendar is 1 Tishri AM 1 (AM = anno
   mundi = in the year of the world), which in the proleptic Julian
   calendar is Monday, October 7, 3761 BCE, the equivalent tabular date
   (same daylight period). This date is about one year before the
   traditional Jewish date of Creation on 25 Elul AM 1. (A minority
   opinion places Creation on 25 Adar AM 1, six months earlier, or six
   months after the modern epoch.) Thus, adding 3760 to any
   Julian/Gregorian year number after 1 CE will yield the Hebrew year
   which roughly coincides with that English year, ending that autumn.
   (Add 3761 for the year beginning in autumn). Due to the slow drift of
   the modern Jewish calendar relative to the Gregorian calendar, this
   will be true for about another 20,000 years.

   The traditional Hebrew date for the destruction of the First Temple
   (3338 AM) differs from the modern scientific date, which is usually
   expressed using the Gregorian calendar (586 BCE). The scientific date
   takes into account evidence from the ancient Babylonian calendar and
   its astronomical observations. In this and related cases, a difference
   between the traditional Hebrew year and a scientific date in a
   Gregorian year results from a disagreement about when the event
   happened — and not simply a difference between the Hebrew and Gregorian
   calendars. See the "Missing Years" in the Hebrew Calendar.

Measurement of the month

   The Hebrew month is tied to an excellent measurement of the average
   time taken by the Moon to cycle from lunar conjunction to lunar
   conjunction. Twelve lunar months are about 354 days while the solar
   year is about 365 days so an extra lunar month is added every two or
   three years in accordance with a 19-year cycle of 235 lunar months (12
   regular months every year plus 7 extra or embolismic months every 19
   years). The average Hebrew year length is about 365.2468 days, about 7
   minutes longer than the average tropical solar year which is about
   365.2422 days. Approximately every 216 years, those minutes add up so
   that the modern fixed year is "slower" than the average solar year by a
   full day. Because the average Gregorian year is 365.2425 days, the
   average Hebrew year is slower by a day every 231 Gregorian years.
   During the last century a number of Jewish scholars suggested that the
   chief rabbinate in Jerusalem consider modifying this rule to avoid this
   effect.

Pattern of calendar years

   There are exactly 14 different patterns that Hebrew calendar years may
   take. Each of these patterns is called a "keviyah" (Hebrew for "a
   setting" or "an established thing"), and is distinguished by the day of
   the week for Rosh Hashanah of that particular year and by that
   particular year's length.
     * A chaserah year (Hebrew for "deficient" or "incomplete") is 353 or
       383 days long because a day is taken away from the month of Kislev.
       The Hebrew letter ח "het", and the letter for the weekday denotes
       this pattern.
     * A kesidrah year ("regular" or "in-order") is 354 or 384 days long.
       The Hebrew letter כ "kaf", and the letter for the week-day denotes
       this pattern.
     * A shlemah year ("abundant" or "complete") is 355 or 385 days long
       because a day is added to the month of Heshvan. The Hebrew letter ש
       "shin", and the letter for the week-day denotes this pattern.

   A variant of this pattern of naming includes another letter which
   specifies the day of the week for the first day of Pesach (Passover) in
   the year.

Measurement of hours

   Every hour is divided into 1080 halakim or parts. A part is 3^1/[3]
   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. But by the twelfth century that source had been
   forgotten, causing Maimonides to speculate that there were 1080 parts
   in an hour because that number was evenly divisible by all numbers from
   1 to 10 except 7. But the same statement can be made regarding 360. 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 (Tzeis Hacochavim) 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 seventeenth century this had become three
   second-magnitude stars. The modern definition is when the centre 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 Shaos Zemaniyos or Halachic
   hours by taking the time between sunrise and sunset or between dawn and
   nightfall and diving into 12 equal hours. 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 Shaos Zemaniyos. For
   convenience, the day using Shaos Zemaniyos is often discussed as if
   sunset were at 6:00pm, sunrise at 6:00am and each hour were equal to a
   fixed hour. However, for example, halachic noon may be after 1:00pm in
   some areas during daylight savings time.

Measurement of "molads" (lunar conjunctions)

   The calendar is based on mean lunar conjunctions called "molads" spaced
   precisely 29 days, 12 hours, and 793 parts apart. Actual conjunctions
   vary from the molads by up to 7 hours in each direction due to the
   nonuniform velocity of the moon. This value for the interval between
   molads (the mean synodic month) was measured by Babylonians before 300
   BCE and was adopted by the Greek astronomer Hipparchus and the
   Alexandrian astronomer Ptolemy. Its remarkable accuracy was achieved
   using records of lunar eclipses from the eighth to fifth centuries BCE.
   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 rotation of the earth is slowing even more the mean synodic
   month is becoming gradually shorter in terms of the day-night cycle.
   The value 29-12-793 was almost exactly correct at the time of Hillell
   II and is now about 0.6 s per month too great. However it is still the
   most correct value possible as long as only whole numbers of parts are
   used. Especially, it is far more accurate than the average solar year
   due to the 19-years-235-months equality described above — the total
   accumulated error of 29-12-793 from its Babylonian measurement until
   the present amounts to only about five hours.

Metonic cycle

   The 19 year cycle has 12 common and 7 leap years. There are 235 lunar
   months in each cycle. This gives a total of 6939 days, 16 hours and 595
   parts for each cycle. Due to the vagaries of the Hebrew calendar, a
   cycle of 19 Hebrew years can be either 6939, 6940, 6941, or 6942 days
   in duration. To start on the same day of the week, the days in the
   cycle must be divisible by 7, but none of these values can be so
   divided. This keeps the Hebrew calendar from repeating itself too
   often. The calendar almost repeats every 247 years, except for an
   excess of 50 minutes (905 parts). So the calendar actually repeats
   every 36,288 cycles (every 689,472 Hebrew years).

   Leap years of 13 months are the 3rd, 6th, 8th, 11th, 14th, 17th, and
   the 19th years beginning at the epoch of the modern calendar. Dividing
   the Hebrew year number by 19, and looking at the remainder will tell
   you if the year is a leap year (for the 19th year, the remainder is
   zero). A Hebrew leap year is one that has 13 months in it, a common
   year has 12 months. A mnemonic word in Hebrew is GUCHADZaT "גוחאדז"ט"
   (the Hebrew letters gimel-vav-het aleph-dalet-zayin-tet, i.e. 3, 6, 8,
   1, 4, 7, 9. See Hebrew numerals). Another mnemonic is that the
   intervals of the major scale follow the same pattern as do Hebrew leap
   years: a whole step in the scale corresponds to two common years
   between consecutive leap years, and a half step to one common between
   two leap years.

   A Hebrew common year will only have 353, 354, or 355 days. A leap year
   will have 383, 384, or 385 days.

Special holiday rules

   Although simple math would calculate 21 patterns for calendar years,
   there are other limitations which mean that Rosh Hashanah may only
   occur on Mondays, Tuesdays, Thursdays, and Saturdays (the "four
   gates"), according to the following table:

                                              Day of Week   Number of Days
                                                   Monday 353 355 383  385
                                                  Tuesday 354          384
                                                 Thursday 354 355 383  385
                                                 Saturday 353 355 383  385

   The lengths are described in the section Names and lengths of the
   months.

   In leap years, a 30 day month called Adar I is inserted immediately
   after the month of Shevat, and the regular 29 day month of Adar is
   called Adar II. This is done to ensure that the months of the Jewish
   calendar always fall in roughly the same seasons of the solar year, and
   in particular that Nisan is always in spring. Whether either Chesvan or
   Kislev both have 29 days, or both have 30 days, or one has 29 days and
   the other 30 days depends upon the number of days needed in each year.
   Thus a leap year of 13 months has an average length of 383½ days, so
   for this reason alone sometimes a leap year needs 383 and sometimes 384
   days. Additionally, adjustments are needed to ensure certain holy days
   and festivals do or do not fall on certain days of the week in the
   coming year. For example, Yom Kippur, on which no work can be done, can
   never fall on Friday (the day prior to the Sabbath), to avoid having
   two consecutive days on which no work can be done. Thus some
   flexibility has been built in.

   The 265 days from the first day of the 29 day month of Adar (i.e. the
   twelfth month, but the thirteenth month, Adar II, in leap years) and
   ending with the 29th day of Heshvan forms a fixed length period that
   has all of the festivals specified in the Bible, such as Pesach (Nisan
   15), Shavuot (Sivan 6), Rosh Hashana (Tishri 1), Yom Kippur (Tishri
   10), Sukkot (Tishri 15), and Shemini Atzeret (Tishri 22).

   The festival period from Pesach up to and including Shemini Atzeret is
   exactly 185 days long. The time from the traditional day of the vernal
   equinox up to and including the traditional day of the autumnal equinox
   is also exactly 185 days long. This has caused some unfounded
   speculation that Pesach should be March 21, and Shemini Atzeret should
   be September 21, which are the traditional days for the equinoxes. Just
   as the Hebrew day starts at sunset, the Hebrew year starts in the
   Autumn (Rosh Hashanah), although the mismatch of solar and lunar years
   will eventually move it to another season if the modern fixed calendar
   isn't moved back to its original form of being judged by the Sanhedrin
   (which requires the Beit Hamikdash)

Karaite interpretation

   Karaites use the lunar month and the solar year, but determine when to
   add a leap month by observing the ripening of barley (called abib) in
   Israel, rather than the calculated and fixed calendar of Rabbinic
   Judaism. This puts them in sync with the Written Torah, while other
   Jews are often a month later. (For several centuries, many Karaites,
   especially outside Israel, have just followed the calculated dates of
   the Oral Law (the Mishnah and the Talmud) with other Jews for the sake
   of simplicity. However, in recent years most Karaites have chosen to
   again follow the Written Torah practice.)

Accuracy

   The average length of the month assumed by the calendar is correct
   within a fraction of a second (although individual months may be a few
   hours longer or shorter than average). There will thus be no
   significant errors from this source for a very long time. However, the
   assumption that 19 tropical years exactly equal 235 months is wrong, so
   the average length of a 19 year cycle is too long (compared with 19
   tropical years) by about 0.088 days or just over 2 hours. Thus on
   average the calendar gets further out of step with the tropical year by
   roughly one day in 216 years. If the intention of the calendar is that
   Pesach should fall on the first full moon after the vernal equinox,
   this is still the case in most years. However, at present three times
   in 19 years Pesach is a month late by this criterion (as in 2005).
   Clearly, this problem will get worse over time and if the calendar is
   not amended, Pesach and the other festivals will progress through a
   complete cycle of seasons in about 79,000 years.

   As the 19 year cycle (and indeed all aspects of the calendar) is part
   of codified Jewish law, it would only be possible to amend it if a
   Sanhedrin could be convened. It is traditionally assumed that this will
   take place upon the coming of the Messiah, which will mark the
   beginning of the era of redemption according to Jewish belief.
   Theoretically, if Jewish law could be modified, one solution would be
   to replace the 19-year cycle with a 334-year cycle of 4131 lunations.
   This cycle has an error of only one day in about 11,500 years. However,
   this would be impossibly cumbersome in practice. Further, no such
   mathematically fixed rule could be valid in perpetuity, because the
   lengths of both the month and tropical year are slowly changing.
   Another possibility would be to calculate the approximate time of the
   vernal equinox and have a leap year if and only if Pesach would
   otherwise start before the vernal equinox. Similar ideas are used in
   the Chinese calendar and some Indian calendars.

Programmer's guide

   The audience for this summary of the mechanics of the Hebrew calendar
   is computer programmers who wish to design software that accurately
   computes dates in the Hebrew calendar. The following details are
   sufficient to generate such software.

   1) The Hebrew calendar is computed by lunations. One mean lunation is
   reckoned at 29 days, 12 hours, 44 minutes, 3⅓ seconds, or equivalently
   765433 parts = 29 days, 13753 parts, where 1 minute = 18 parts (halakim
   plural, helek singular).

   2) A common year must be either 353, 354, or 355 days; a leap year must
   be 383, 384, or 385 days. A 353 or 383 day year is called haserah. A
   354 or 384 day year is kesidrah. A 355 or 385 day year is shlemah.

   3) Leap years follow a 19 year schedule in which years 3, 6, 8, 11, 14,
   17, and 19 are leap years. The Hebrew year 5758 (which starts in
   Gregorian year 1997) is the first year of a cycle.

   4) 19 years is the same as 235 lunations.

   5) The months are Tishri, Cheshvan, Kislev, Tevet, Shevat, Adar, Nisan,
   Iyar, Sivan, Tammuz, Av, and Elul. In a leap year, Adar is replaced by
   Adar II (also called Adar Sheni or Veadar) and an extra month, Adar I
   (also called Adar Rishon), is inserted before Adar II.

   6) Each month has either 29 or 30 days. A 30 day month is full (male,
   maley, or malei), whereas a 29 day month is defective (haser or
   chaser).

          Nisan, Sivan, Av, Tishri, and Shevat are always full.

          Iyar, Tammuz, Elul, Tevet, and Adar (Adar II in leap years) are
          always defective.

          Adar I, added in leap years before Adar II, is full.

          Cheshvan and Kislev vary. There are three possible combinations:
          both defective, both full, Cheshvan defective and Kislev full.

   7) Tishri 1 (Rosh Hashana) is the day during which a molad (instant of
   the mean lunar conjunction) occurs unless that conflicts with certain
   postponements (dehiyyot plural; dehiyyah singular). Note that for
   calendar computations, the Jewish date begins at 6 pm or six fixed
   hours before midnight when the date changes in the Gregorian calendar,
   not at nightfall or sunset when the observed Hebrew date begins.

          Postponement A is required whenever Tishri 10 (Yom Kippur) would
          fall on a Friday or a Sunday, or if Tishri 21 (7th day of
          Sukkot) would fall on a Saturday. This is equivalent to the
          molad being on Sunday, Wednesday, or Friday. Whenever this
          happens, Tishri 1 is delayed by one day.

          Postponement B is required whenever the molad occurs at or after
          noon. When this postponement exists, Tishri 1 is delayed by one
          day. If this conflicts with postponement A then Tishri 1 is
          delayed an additional day.

          Postponement C: If the year is to be a common year and the molad
          falls on a Tuesday at or after 3:11:20 am (3 hours 204 parts),
          Tishri 1 is delayed by two days—if it weren't delayed, the
          resulting year would be 356 days long.

          Postponement D: If the new year follows a leap year and the
          molad is on a Monday at or after 9:32:43⅓ am (9 hours 589
          parts), Tishri 1 is delayed one day—if it weren't, the preceding
          year would have only 382 days.

   8) Postponements are implemented by adding a day to Kislev of the
   preceding year, making it full. If Kislev is already full, the day is
   added to Cheshvan of the preceding year, making it full also. If a
   delay of two days is called for, both Cheshvan and Kislev of the
   preceding year become full.

   9) Because all postponements are defined in terms of a seven-day week,
   whole weeks between the epoch and the molad of the current year can be
   eliminated, leaving only a partial week with a few days, hours and
   parts.

          A nineteen-year cycle has 235 months of 29d 12h 793p each or
          6939d 16h 595p. Eliminating 991 weeks leaves a partial week of
          2d 16h 595p or 69715p.

          A common year has 12 months of 29d 12h 793p each or 354d 8h
          876p. Eliminating 50 weeks leaves a partial week of 4d 8h 876p
          or 113196p.

          A leap year has 13 months of 29d 12h 793p or 383d 21h 589p.
          Eliminating 54 weeks leaves a partial week of 5d 21h 589p or
          152869p.

   10) Postponement B requiring a delay until the next day (beginning at 6
   pm) if a molad occurs at or after noon effectively means that the week
   begins at noon Saturday for computational purposes.

   11) Calculate the partial week between the molad of the desired Hebrew
   year and the preceding noon Saturday considering the partial week
   before molad Tishri of AM 1 (or the first year of a more recent
   nineteen-year cycle) and the partial weeks from the intervening cycles
   and years within the current cycle, eliminating whole weeks via mod
   181440, the number of parts in one week.

   Thus molad Tishri AM 1, which is 1d 5h 204p after 6 pm Saturday, is
   increased by 6 hours to 1d 11h 204p or 38004p. This is 5h 204p after
   the beginning (6 pm) of the second day of the week. In Western terms,
   this is 23:11:20 on Sunday (because it is before midnight), 6 October
   3761 BCE in the proleptic Julian calendar. Although the Julian day
   number begins at noon, it can be advanced twelve hours earlier for
   programming purposes, so this date can be regarded as the midnight
   Julian day number 347997. Consulting the Table of Limits below, 1
   Tishri is the second day of the week, equivalent to the tabular Western
   day of Monday (same daylight period as the Hebrew day), which is 7
   October 3761 BCE. This means no postponement was needed (both the molad
   Tishri and 1 Tishri were on the second day of the week).

   Alternatively, the molad of a more recent Hebrew year may be selected
   as the epoch if it is the first year of a nineteen-year cycle, such as
   5758, which is 303 nineteen-year cycles after molad Tishri AM 1. Thus
   molad Tishri 5758 is (38004 + 303×69715) mod 181440 = 114609 parts
   after noon Saturday, or 4d 10h 129p, which is 4h 129p after the
   beginning (6 pm) of the fifth day of the week. In Western terms, this
   is before midnight at 22:07:10 on Wednesday, 1 October 1997
   (Gregorian), midnight Julian day number 2450723. Consulting the Table
   of Limits, 1 Tishri is the fifth day of the week, or tabular Thursday 2
   October 1997 (Gregorian), again no postponement was needed.

   By applying the postponements to the moladim Tishri at the beginning
   and end of two successive Hebrew years, a table of limits can be
   developed which uniquely identifies which of the fourteen types the
   second year is (the day of the week of 1 Tishri, the number of days in
   Cheshvan and Kislev, and whether common or leap (embolismic)). The
   first table of limits was developed by Saadiah Gaon (d. 1038). In the
   following table, the years of a nineteen-year cycle are listed in the
   first row, organized into four groups: a common year after an leap year
   but before a common year (1 4 9 12 15), a common year between two leap
   years (7 18), a common year after a common year but before a leap year
   (2 5 10 13 16), or an leap year between two common years (3 6 8 11 14
   17 19). The week since noon Saturday in the first column is partitioned
   by a set of limits between which the molad Tishri of the Hebrew year
   can be found. The resulting type of year indicates the day of the
   Hebrew week of 1 Tishri (2, 3, 5, or 7 due to postponement A) and
   whether the year is deficient (−1), regular (0), or abundant (+1).

   CAPTION: Table of Limits

                            1 4 9 12 15 7 18 2 5 10 13 16 3 6 8 11 14 17 19
   0 <= molad <      16404                   2 , −1
   16404 <= molad <  28571
   28571 <= molad <  49189                   2 , +1
   49189 <= molad <  51840
   51840 <= molad <  68244                   3 , 0
   68244 <= molad <  77760
   77760 <= molad <  96815                   5 , 0        5 , −1
   96815 <= molad <  120084
   120084 <= molad < 129600                  5 , +1
   129600 <= molad < 136488
   136488 <= molad < 146004                  7 , −1
   146004 <= molad < 158171
   158171 <= molad < 181440                  7 , +1
   Retrieved from " http://en.wikipedia.org/wiki/Hebrew_calendar"
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   with only minor checks and changes (see www.wikipedia.org for details
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