Hebrew calendar

The Hebrew calendar (Template:Lang-he) or Jewish calendar is the calendar used by Jews for religious purposes. Nowadays it determines dates for 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.

In ancient days the Hebrew calendar was used by Jews for all day to day purposes, but by the times of the Romans they were compelled to follow the imperial, civil calendar for all civil matters, such as the payment of taxes and dealings with government offices and officials.

Two major forms of the calendar have been used. Before the destruction of the Second Temple in 70 CE, the start of each month was based on the observation of a new crescent. Witnesses would be required to testify that they had seen the new crescent, which would signify the start of the new month. Between 70 and 1178 CE a rule-based system was adopted. The principles were fully described by Maimonides in 1178 CE.

The "modern" form is a fixed arithmetic lunisolar 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 7 times per 19 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.

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 months were originally referred to in the Bible by number rather than name. For example, in the story of Noah and the Flood (Genesis, 7:11, 8:4, :5, :13, :14), the months are referred to by a number. One should note that there is a hint that the months are of 30 days duration. There is no indication as to how many months are in the annual cycle, but there are least ten mentioned. One should also note that this is a pre-Jewish calendar, and no indication that the same cycles were in use at the time of Sinai.

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.". 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). All of these 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.

There is no direct mention in the Bible of an extra embolismic or intercalary month, required to prevent the starting date of the lunar cycles from "drifting" away from the Spring. There are hints, however, that the first month (today's Nissan) had always started only following the ripening of barley. According to some traditions, if the barley had not yet ripened, the start of the first month would have to wait for another monthly cycle to be completed. Only much later was a systematic method for inserting a second-last month, today's Adar I, adopted. The intercalary month is the second-last of the annual cycle. The logic is that at the end of Adar, it is discovered that the year has not yet concluded, so the Adar that had just finished was a false Adar (ie the last month), so the next month, Adar II, is the correct one.

Babylonian exile

The Babylonians also had a lunisolar calendar, which was the direct descendant of the Sumerian calendar. During the Babylonian exile, which started in 586 BCE, Jews adopted Babylonian names for the months, which have continued to be used to this day.

Names and lengths of the months

Hebrew names of the months with their Babylonian analogs

Number	Hebrew	Hebrew name	Length	Babylonian analog	Notes
1	ניסן	Nisan / Nissan	30 days	Nisanu	called Aviv and Nisan in the Tanakh
2	איר / אייר	Iyar	29 days	Ayaru	called Ziv in the Tanakh
3	סיוון	Sivan	30 days	Simanu
4	תמוז	Tammuz	29 days	Du'uzu
5	אב	Av	30 days	Abu
6	אלול	Elul	29 days	Ululu
7	תשרי	Tishrei	30 days	Tashritu	called Eitanim in the Tanakh. Modern first month, Rosh Hashana is celebrated in Tishrei.
8	חשוון	Marcheshvan	29 or 30 days	Arakhsamna	often shortened to Cheshvan; called Bul in the Tanakh
9	כסלו	Kislev	30 or 29 days	Kislimu	also spelled Chislev
10	טבת	Tevet	29 days	Tebetu
11	שבט	Shevat	30 days	Shabatu
12*	אדר א׳	Adar I*	30 days	Adaru	*Only in leap years
12 / 13*	אדר / אדר ב׳	Adar / Adar II*	29 days

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.

Names of the weekdays

The Hebrew calendar follows the common seven-day weekly cycle. The Hebrew names for the weekdays are simply the day number within the week, in Hebrew, sometimes (noticeably in the newspapers) abbreviated as יום א׳ (Day 1 = Sunday) and so on, using the numerical value of the Hebrew letters:

Yom Rishon (Hebrew: יום ראשון), abbreviated יום א׳ = "first day" = Sunday
Yom Sheni (יום שני), abbr. יום ב׳ = "second day" = Monday
Yom Shlishi (יום שלישי), abbr. יום ג׳ = "third day" = Tuesday
Yom Reviʻi (יום רבעי), abbr. יום ד׳ = "fourth day" = Wednesday
Yom Ḥamishi (יום חמישי), abbr. יום ה׳ = "fifth day" = Thursday
Yom Shishi (יום ששי), abbr. יום ו׳ = "sixth day" = Friday
Yom Shabbat (יום שבת or more usually שבת - Shabbat), abbr. יום ש׳ = "Sabbath day (Rest day)" = Saturday

In Hebrew, the word "Shabbat" (שַׁבָּת) can also mean "(Talmudic) week", so that in ritual liturgy a phrase like "Yom Reviʻi bəShabbat" means "the fourth day in the week".

Second Temple era

In Second Temple times, the beginning of each lunar month was decided on the basis of two eyewitnesses testifying to have seen the new lunar crescent at sunset. Patriarch Gamaliel II (c. 100) asked the 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. According to tradition, these observations were compared against calculations made by the supreme Jewish court, the Sanhedrin. Whether or not an embolismic month was to be inserted depended on the calculated estimate of the spring equinox moment, 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.

If one back-calculates the moments of the traditional moladot using modern astronomical calculations then the closest that their reference meridian of longitude ever got to Israel was midway between the Nile River and the end of the Euphrates River (about 4° east of Jerusalem), and that was in the era of the Second Temple.

At first the beginning of each Hebrew month was signaled to the communities of Israel and beyond by fires lit on mountaintops, but after the Samaritans and Boethusians 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, 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.

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 Tishrei (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

By the Roman era, some sects, such as the Essenes, used a solar calendar during the last two centuries BCE.

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 ±2 days, and months may be in error by ±2 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 Tishrei 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 Tishrei should be the day of the new moon unless the new moon occurred more than 642 parts (35⅔ minutes, where a "part" is ¹/₁₀₈₀ of an hour or ¹/₁₈ of a minute or 3⅓ seconds) 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 indeed about 642 parts (35 minutes and 40 seconds) later than (ahead of) the meridian of Jerusalem, corresponding to a longitude difference of 8° 55'.

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".
• The day which is most often referred to as the "New Year" is observed on the first of Tishrei, when the year number increases by 1 and the formal new year festival Rosh Hashanah is celebrated. It also marks the new year for certain agricultural laws such as tithes (See Maaser Rishon, Maaser Sheni, Maaser Ani).
• The month of Elul is the new year for counting animal tithes (ma'aser).
• Tu Bishvat ("the 15th of Shevat") marks the new year for trees (and agricultural tithes).

There may be an echo here of a controversy in the Talmud about whether the world was created in Tishrei or Nisan; it was decided that the answer is Tishrei, and this is now reflected in the prayers on Rosh Hashanah.

Modern calendar

Epoch

The Hebrew calendar uses an era anno mundi ("in the year of the world"), abbreviated AM. Ironically, the beginning of its "year 1" is not Creation, but about one year before Creation. This caused the new moon of its first month (Tishrei) to be called molad tohu (the mean new moon of chaos or nothing).

Its epoch 1 Tishrei 1 AM, is equivalent to Monday, October 7, 3761 BCE in the proleptic Julian calendar, 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, based upon the Seder Olam of Rabbi Yossi ben Halafta, a second century CE sage. (A minority opinion places Creation on 25 Adar AM 1, six months earlier, or six months after the modern epoch.) Thus, adding 3760 (from September-October through December, 3761) to any Julian/Gregorian year number after 1 CE will yield the Hebrew year, ending in September-October, which roughly coincides with that Julian/Gregorian year. Owing 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 = 423 BCE) 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

Traditionally new moon refers to the first visible crescent of the moon. If one uses the "observation" method of determining a monthly cycle, one would wait for the new crescent to be visible, somewhere in the sky, to indicate the start of the new month. In this manner, the lunar month may be either 29 or 30 days in length.

However, that is not a scientific definition of a lunar cycle, which is called a lunar conjunction. The period of time between two lunar conjunctions is the synodic month (known in that context as the molad in Hebrew). The adopted mean lunar month length (molad interval) equals exactly ⁷⁶⁵⁴³³/₂₅₉₂₀ days, or 29 days 12 hours and 44+¹/₁₈ minutes (or 29.5306 days).

Combining the observation method with the scientific lunar month length works as follows: assuming we start at a particular new month (which we'll call "the base date"), that month will be 29 days long, with 12 hours, 44+¹/₁₈ minutes left over ("the carry forward amount"). Adding that carry forward amount to the next month will make it equal 30 days, 1 hour and 24+²/₁₈ minutes (30.0612 days). So the second month would be 30 days long, and 1 hour 24+²/₁₈ minutes (2 x carry forward amount) would be carried forward to be added to the next cycle, and so on. Then every 17 lunar cycles the carry forward amounts are over 24 hours, which would require an additional day to be added to whatever length that month would have been. In summary, the progression becomes: year 1 | 29 - 30 - 29 -30 - 29 - 30 - 29 - 30 - 29 - 30 - 29 - 30 | year 2 | 29 - 30 - 29 - 30 - 30 - 29 - etc.

The other issue is the number of months in a year. Twelve lunar months are about 354 or 355 days (see above) 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).

Calendar mean year

The mean Hebrew calendar year length is exactly equal to the molad interval × 235 months per cycle ÷ 19 years per cycle = 365 days, 5 hours 55 minutes, and 25+²⁵/₅₇ seconds. The present era mean northward equinoctial year is about 365 days 5 hours 49 minutes and zero seconds, so the Hebrew calendar mean year is about 6 minutes and 25+²⁵/₅₇ seconds too long per year. Approximately every 224 years, those minutes add up so that the modern fixed year is "slower" than the average solar year by a full day. Because the mean Gregorian calendar year is 365.2425 days (exactly 365 days 5 hours 49 minutes and 12 seconds) and the mean Hebrew calender year is 365.2468 days, the Hebrew calendar falls further behind the Gregorian calendar by about a day about every 231 years.

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 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. 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 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 Shaos Zemaniyos or Halachic hours by taking the time between sunrise and sunset or between dawn and nightfall and dividing it 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 saving time.

Measurement of "molads" (lunar conjunctions)

The calendar is based on estimated mean lunar conjunctions called moladot spaced at intervals of exactly 29 days, 12 hours, and 793 parts (44+¹/₁₈ minutes). In the present era actual lunar conjunction intervals can be as short as 29 days 6 hours and 30 minutes to as long as 29 days and 20 hours, an astonishing variation range of about 13 hours and 30 minutes. 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 (make the conjunction moments 16 minutes earlier if referred to the original molad reference meridian midway between the Nile River and the end of the Euphrates River, about 4° east of Jerusalem). Today, in terms of the mean solar time at the meridian of Qandahar, Afghanistan the actual lunar conjunctions vary ±14 hours relative to the traditional moladot.

The traditional molad interval matches the mean synodic month as determined by the Babylonians before 300 BCE and as adopted by the Greek astronomer Hipparchus and the Alexandrian astronomer Ptolemy. Its remarkable accuracy is thought to have been 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 due to the tides the Earth rotation rate slowing even more the mean synodic month is becoming gradually shorter in terms of mean solar time. The value 29-12-793 was almost exactly correct at the time of Hillell II and is now about 0.6 seconds per month too long. However, it is still the most correct value possible as long as only whole parts (¹/₁₈ minute) are used.

Metonic cycle

Main article: 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 Rosh HaShanah postponement rules 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). The calendar actually repeats only after 36,288 cycles = 689,472 Hebrew years!

A Hebrew leap year is one that has 13 months, a common year has 12 months. 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). Alternatively, the following expression yields the leap status of the year:

hYear is a leap year if the remainder of ( 7 x hYear + 1 ) / 19 is less than 7, where hYear is the Hebrew year number.

With 7 leap years per 19-year cycle, the average interval between leap years = ¹⁹/₇ = 2+⁵/₇ years, which means that 3-year intervals are more common that 2-year intervals.

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 non-leap year can only have 353, 354, or 355 days. A leap year can have 383, 384, or 385 days (always 30 days longer than the non-leap length).

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 (Tishrei 1), Yom Kippur (Tishrei 10), Sukkot (Tishrei 15), and Shemini Atzeret (Tishrei 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 the Karaite calendar differs from the Rabbinical calendar in a few ways: Determination of the first month of the year - (called aviv), which is the month Passover falls out and determination of the first day of each month (Rosh Chodesh).

The 4 rules of postponement are not applied, as they are not found in the Tanakh. It is determined when to add a 13th month by observing the ripening of barley (called abib) in Israel, rather than the calculated and fixed calendar of Rabbinic Judaism. This puts Karaites in sync with the Written Torah, while other Jews are often a month later.

The beginning of each month is determined by the Rosh Chodesh - which can be calculated, but is confirmed by observation of the first sightings of the new moon in Israel.

For several centuries, many Karaites, especially those 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 molad interval is currently about 0.6 seconds too long, and the discrepancy is accumulating at an accelerating rate as the mean lunation interval is getting progressively shorter, due to Earth-Moon gravitational tidal effects. The accumulated "error" since the era of Hillel II is such that the molad moments are now almost 1 hour and 40 minutes late, relative to the mean lunar conjunctions at the original reference meridian that was midway between the Nile River and the end of the Euphrates River. Today the molad moments match the mean lunar conjunction moments in terms of the mean solar time near the meridian of Qandahar, Afghanistan, more than 30° east of Jerusalem!

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.

If the intention of the calendar is that Passover should fall near the first full moon after the northward equinox, or that the northward equinox should land within one lunation before 16 days after the molad of Nisan, then 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 year 5765, an 8th year of the 19-year cycle = Gregorian 2005 AD). Presently this occurs after the "premature" insertion of a leap month in years 8, 19, and 11 of each 19-year cycle, which causes the northward equinox to land at exceptionally early moments in such years. This problem will get worse over time, and so beginning in Hebrew year 5817 the 3rd year 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, or about 10885 years from the present. (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 9, 1, 12 and 4 of each 19-year cycle (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.

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. [This paragraph is in conflict with the historical gradual evolution of the calendar rules that was outlined above. If the calendar development was indeed gradual and did not reach its final form until Maimonides, who published the first complete and unambiguous codification of both the observational and fixed-arithmetic Hebrew calendars, then a Sanhedrin is not required to change it. If the calendar rules were set by the Sanhedrin of Hillel II, then the gradual history outlined above is wrong and only the present or future Sanhedrin can change them.]

An excellent solution would be to replace the 19-year cycle with a 353-year cycle of 4366 lunations, including 130 leap months. It is predicted that this cycle, together with use of a progressively shorter molad interval, will keep the amended calendar from drifting for more than 7 millennia (deduct 3 millennia if the traditional molad interval is retained). The calendar arithmetic to do this is straightforward and is documented in the public domain (see the external link to the Rectified Hebrew Calendar).

Another possibility would be to calculate the astronomical moment of the actual northward equinox and declare a leap year if and only if Pesach would otherwise start before the equinox. Similar ideas are used in the Chinese calendar and some Indian calendars. This would be very accurate, but would require a central authority to be responsible for the official calculations, because there are small differences between astronomical algorithms, depending on the methods employed.

Adopting an astronomical calendar would require more explicit definition of the calendar rules. Should the calculated equinox moment be the actual astronomical equinox, or the mean astronomical equinox, and which meridian of longitude should the moment be referred to? (The traditional equinox moments of Tekufat Shmuel drift at the same rate as the Julian calendar, and those of Tekufat Adda drift at the same rate as the fixed arithmetic Hebrew calendar, so neither can be used.) Should the leap month be inserted if the equinox would otherwise land after the end of the first day of Passover (as Maimonides suggested), or should the cutoff be the moment of the Korban Pesach sacrifice 30 minutes after noon on the 14th of Nissan (most compatible with the Torah command in Deuteronomy 16:1), or should the average equinox moment align with the average moment when the month of Nissan starts (calendrically most sensible)?

Should a progressive molad be used, or the actual lunar conjunction, or a prediction of new lunar crescent visibility (a reliable way to do that still doesn't exist), and which meridian of longitude should the moment be referred to? Should month lengths vary such that any month can have 29 or 30 days, or should the present rules for fixed month lengths be continued? (In particular, should the length of Elul be fixed at 29 days, which was mentioned in many places in the Talmud?) Should there be any offset between the "molad" moment (however determined) and the start of months (one day yields good agreement with the performance of the fixed arithmetic calendar)? Should Rosh HaShanah postponement rules be continued, or advance/postpone used instead (arithmetically much simpler)?

The compatibility of the selected astronomical rules with the dates of High Holy Days and other events, and with the weekly Torah portions, needs to be evaluated and confirmed as acceptable.

Programmers' guide

The intended 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 may prove useful for validating such software. Note, however, that published Hebrew calendar algorithms are much simpler than the details listed below, and there is no need to employ tables in computer implemention of Hebrew calendar arithmetic. As usual, tables are useful shortcuts for humans carrying out the calculations manually.

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 Tishrei, 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 (מלא pronounced: maleh, maley, or malei), whereas a 29 day month is defective (חסר pronounced: ħaser or khaser).
   • Nisan, Sivan, Av, Tishrei, 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. Tishrei 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 Tishrei 10 (Yom Kippur) would fall on a Friday or a Sunday, or if Tishrei 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, Tishrei 1 is delayed by one day.
   • Postponement B is required whenever the molad occurs at or after noon. When this postponement exists, Tishrei 1 is delayed by one day. If this conflicts with postponement A then Tishrei 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), Tishrei 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), Tishrei 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. A reference epoch in modern times is molad Tishrei for Hebrew year 5758, which is at 22:07:10 on Wednesday, 1 October 1997 (Gregorian), or equivalently midnight-referenced Julian day number 2450723 plus 23889 parts. This epoch also marks the beginning of a cycle. Note: Although the Julian day number begins at noon, it can be reckoned twelve hours earlier for programming purposes, which is what is meant here by the phrase, "midnight-referenced."

Calculation by use of partial weeks

There are a number or approaches that can be taken in calculating Hebrew dates. One that is widely documented uses partial weeks and a table of limits. This method relies on all postponements being defined in terms of a seven-day week. That means that 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.

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.

Calculate the partial week between the molad of the desired Hebrew year and the preceding noon Saturday considering the partial week before molad Tishrei 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 Tishrei 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. This date is midnight-referenced Julian day number 347997. Consulting the Table of Limits below, 1 Tishrei 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 Tishrei and 1 Tishrei 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 (used in rule 9), which is 303 nineteen-year cycles after molad Tishrei AM 1. Thus molad Tishrei 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, which yields the date and time indicated in rule 9. Consulting the Table of Limits, 1 Tishrei 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 moladot Tishrei at the beginning and end of any Hebrew year, a table of four gates (Hebrew: arba'ah sha'arim), which is also a table of limits, can be developed which uniquely identifies which of the fourteen types the year is (the day of the week of 1 Tishrei, the number of days in Cheshvan and Kislev, and whether common or leap (embolismic)). "Four gates" refers to the four allowable days of the week with which the year can begin. The first table of four gates was developed by Saadiah Gaon (892–942). In the following table, the years of a nineteen-year cycle are listed in the top row, organized into four groups: a common year after a leap year but before a common year (LCC, 1 4 9 12 15), a common year between two leap years (LCL, 7 18), a common year after a common year but before a leap year (CCL, 2 5 10 13 16), or a leap year between two common years (CLC, 3 6 8 11 14 17 19). The week since noon Saturday on the left is partitioned by a set of limits between which the molad Tishrei of the Hebrew year can be found. The resulting type of year in the body of the table indicates the day of the Hebrew week of 1 Tishrei (2, 3, 5, or 7), the four gates, and whether the year is deficient (−1), regular (0), or abundant (+1).

Table of four gates

		LCC 1 4 9 12 15	LCL 7 18	CCL 2 5 10 13 16	CLC 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

Hebrew Calendar and Gregorian Calendar 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 moment of transition from the Dry Season to the Rainy one, and major Jewish Holidays such as Sukkot, Passover, or 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.

With the creation of the State of Israel the Hebrew Calendar was made its official calendar. New holidays and commemorations not derived from previous Jewish tradition invariably were to be defined according to their Hebrew dates - notably the Israeli Independence Day on Iyar 5 and the Holocaust Commemoration Day on Nisan 27 (Hebrew date of the start of the Warsaw Ghetto Uprising).

Nevertheless, since the 1950s the Hebrew calendar steadily declined in importance in Israeli daily life, in favor of the worldwide Gregorian Calendar. At present, Israelis - except for the minority of religiously observant - conduct their private and public life according to the Gregorian Calendar, and an average Israeli would not know what the Hebrew date is without specifically looking it up, and questions such as "On what date does Hanukkah start this year?" are not uncommon.

The Jewish New Year (Rosh Hashana) is a two-day public holiday in Israel. However, since the 1980s an increasing number of secularist Israelis had taken up the habit of celebrating the Gregorian New Year (usually known as "Sylvester Night" - "ליל סילבסטר") by holding all-night parties on the night between December 31 and January 1. 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 Heshvan 12. However, left-leaning Israelis, who revere Rabin as a martyr for the cause of peace and who are predominantly secularist, prefer to hold their own mass memorial rallies on November 4. In some years, the two competing Rabin Memorial Days are separated by as much as two weeks.

The wall Calendars commonly used in Israel are hybrids - organised according to Gregorian rather than Jewish months, but beginning in September, where the Jewish New Year usually falls, and providing the Jewish date in small characters.

References

• 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.
• 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.
• Samuel Poznanski. "Calendar (Jewish)". Encylopædia of Religion and Ethics, 1911.
• F.H. Woods. "Calendar (Hebrew)", Encylopædia of Religion and Ethics, 1911.
• Sherrard Beaumont Burnaby. Elements of the Jewish and Muhammadan Calendars. George Bell and Sons, London, 1901.
• W.H. Feldman. Rabbinical Mathematics and Astronomy,3rd edition, Sepher-Hermon Press, 1978.
• Otto Neugebauer. Ethiopic astronomy and computus. Österreichische Akademie der Wissenschaften, philosophisch-historische klasse, sitzungsberichte 347. Vienna, 1979.
• Ari Belenkiy. "A Unique Feature of the Jewish Calendar — Dehiyot". Culture and Cosmos 6 (2002) 3-22.
• Arthur Spier. The Comprehensive Hebrew Calendar. Feldheim, 1986.
• Nathan Bushwick. Understanding the Jewish Calendar. Moznaim, 1989. ISBN 0940118173
• L.A. Resnikoff. "Jewish calendar calculations", Scripta Mathematica 9 (1943) 191-195, 274-277.
• Edward M. Reingold and Nachum Dershowitz. Calendrical Calculations: The Millennium Edition. Cambridge University Press; 2 edition (2001). ISBN 0-521-77752-6
• 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. pp 723-730.

See also

• Jewish holidays 2000-2050
• molad

External links

• Ancient Jewish Text calculates Lunar Month
• Perpetual Hebrew / Civil Calendar
• Jewish Calendar Details various Jewish points-of-view about the history of the Jewish calendar/Hebrew calendar. Includes several charts.
• Hebrew Calendar Science and Myth gives complete rules of the Hebrew calendar and a lot more.
• The Jewish Controversy about Calendar Postponements
• Jewish Calendar with Zmanim - Halachic times and date converter chabad.org
• Jewish calendar scientific explanation at the NASA web site
• Jewish Encyclopedia: Calendar
• Calendar Hebrew for Christians website
• Karaite Holidays Karaite website
• Hebrew CalendarDates and Holydays (Diaspora or Israel)
• The Lengths of the Seasons (numerical integration analysis)
• The Hebrew Calendar (astronomical analyses)
• The Molad of the Hebrew Calendar (astronomical analysis)
• The Rectified Hebrew Calendar (calendar reform proposal, includes full arithmetic algorithms for both the Traditional and the Rectified calendars)
• Hebrew - Gregorian Calendar with options (Multilingual)

Date converters

• Jewish Calendar for Outlook - Incorporate Jewish dates and holidays into Microsoft Office Outlook.
• Molad - Jewish Calendar with Zmanim and holidays for Mobiles.
• Kaluach - Hebrew/civil calendars
• Hebcal Hebrew Date Converter
• Jewish/Gregorian/Julian Perpetual Calendar Converter - Also contains a full year view for the Hebrew Calendar.
• Sample VB.Net and Javascript code to convert the Hebrew Date to the Gregorian Date
• Jewish / Civil Date Converter
• Gregorian-Mayan-Julian-Islamic-Persian-Hebrew Calendar Converter
• Kalendis Calendar Calculator
• Gregorian to Hebrew date with weekly Parshat HaShavua.
• Full Jewish Year in Hebrew Characters
• Full Jewish Year in Western Characters