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

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

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 3¹/₃ 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).

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