The Hebrew or Jewish calendar (הַלּוּחַ הָעִבְרִי, Ha-Luah ha-Ivri) is a lunisolar calendar used today predominantly for Jewish religious observances. It determines the dates for Jewish holidays and the appropriate public reading of Torah portions, yahrzeits (dates to commemorate the death of a relative), and daily Psalm readings, among many ceremonial uses. In Israel, it is used for religious purposes, provides a time frame for agriculture and is an official calendar for civil purposes, although the latter usage has been steadily declining in favor of the Gregorian calendar.
The present Hebrew calendar is the product of evolution, including a Babylonian influence. Until the Tannaitic period (approximately 10–220 CE), the calendar employed a new crescent moon, with an additional month normally added every two or three years to correct for the difference between twelve lunar months and the solar year. The year in which it was added was based on observation of natural agriculture-related events in ancient Israel. Through the Amoraic period (200–500 CE) and into the Geonic period, this system was gradually displaced by the mathematical rules used today. The principles and rules were fully codified by Maimonides in the Mishneh Torah in the 12th century. Maimonides' work also replaced counting "years since the destruction of the Temple" with the modern creation-era Anno Mundi.
The Hebrew lunar year is about eleven days shorter than the solar year and uses the 19-year Metonic cycle to bring it into line with the solar year, with the addition of an intercalary month every two or three years, for a total of seven times per 19 years. Even with this intercalation, the average Hebrew calendar year is longer by about 6 minutes and 40 seconds than the current mean tropical year, so that every 217 years the Hebrew calendar will fall a day behind the current mean tropical year; and about every 238 years it will fall a day behind the mean Gregorian calendar year.
The era used since the Middle Ages is the Anno Mundiepoch (Latin for "in the year of the world"; Hebrew: לבריאת העולם, "from the creation of the world"). As with Anno Domini (A.D. or AD), the words or abbreviation for Anno Mundi (A.M. or AM) for the era should properly precede the date rather than follow it.
AM 5779 began at sunset on 9 September 2018 and will end at sunset on 29 September 2019.
The Jewish day is of no fixed length. The Jewish day is modeled on the reference to "...there was evening and there was morning..." in the creation account in the first chapter of Genesis. Based on the classic rabbinic interpretation of this text, a day in the rabbinic Hebrew calendar runs from sunset (start of "the evening") to the next sunset. Halachically, a day ends and a new one starts when three stars are visible in the sky. The time between true sunset and the time when the three stars are visible (known as 'tzait ha'kochavim') is known as 'bein hashmashot', and there are differences of opinion as to which day it falls into for some uses. This may be relevant, for example, in determining the date of birth of a child born during that gap.
There is no clock in the Jewish scheme, so that the local civil clock is used. Though the civil clock, including the one in use in Israel, incorporates local adoptions of various conventions such as time zones, standard times and daylight saving, these have no place in the Jewish scheme. The civil clock is used only as a reference point – in expressions such as: "Shabbat starts at ...". The steady progression of sunset around the world and seasonal changes results in gradual civil time changes from one day to the next based on observable astronomical phenomena (the sunset) and not on man-made laws and conventions.
In Judaism, an hour is defined as 1/12 of the time from sunrise to sunset, so, during the winter, an hour can be much less than 60 minutes, and during the summer, it can be much more than 60 minutes. This proportional hour is known as a sha'ah z'manit (lit. a timely hour). A Jewish hour is divided into 1080 halakim (singular: helek) 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). These measures are not generally used for everyday purposes.
The weekdays start with Sunday (day 1, or Yom Rishon) and proceed to Saturday (day 7), Shabbat. Since some calculations use division, a remainder of 0 signifies Saturday.
While calculations of days, months and years are based on fixed hours equal to 1/24 of a day, the beginning of each halachic day is based on the local time of sunset. The end of the Shabbat and other Jewish holidays is based on nightfall (Tzeth haKochabim) which occurs some amount of time, typically 42 to 72 minutes, after sunset. According to Maimonides, nightfall occurs when three medium-sized stars become visible after sunset. By the 17th century, this had become three-second-magnitude stars. The modern definition is when the center of the sun is 7° below the geometric (airless) horizon, somewhat later than civil twilight at 6°. The beginning of the daytime portion of each day is determined both by dawn and sunrise. Most halachic times are based on some combination of these four times and vary from day to day throughout the year and also vary significantly depending on location. The daytime hours are often divided into Sha'oth Zemaniyoth or "Halachic hours" by taking the time between sunrise and sunset or between dawn and nightfall and dividing it into 12 equal hours. The nighttime hours are similarly divided into 12 equal portions, albeit a different amount of time than the "hours" of the daytime. The earliest and latest times for Jewish services, the latest time to eat chametz on the day before Passover and many other rules are based on Sha'oth Zemaniyoth. For convenience, the modern day using Sha'oth Zemaniyoth is often discussed as if sunset were at 6:00 pm, sunrise at 6:00 am and each hour were equal to a fixed hour. For example, halachic noon may be after 1:00 pm in some areas during daylight saving time. Within the Mishnah, however, the numbering of the hours starts with the "first" hour after the start of the day.
Shavua [שבוע] is a weekly cycle of seven days, mirroring the seven-day period of the Book of Genesis in which the world is created. The names for the days of the week, like those in the creation account, are simply the day number within the week, with Shabbat being the seventh day. Each day of the week runs from sunset to the following sunset and is figured locally.
The Hebrew calendar follows a seven-day weekly cycle, which runs concurrently with but independently of the monthly and annual cycles. The names for the days of the week are simply the day number within the week. In Hebrew, these names may be abbreviated using the numerical value of the Hebrew letters, for example יום א׳ (Day 1, or Yom Rishon (יום ראשון)):
Yom Rishon – יום ראשון (abbreviated יום א׳), meaning "first day" [corresponds to Sunday] (starting at preceding sunset of Saturday)
Yom Sheni – יום שני (abbr. יום ב׳) meaning "second day" [corresponds to Monday]
Yom Shlishi – יום שלישי (abbr. יום ג׳) meaning "third day" [corresponds to Tuesday]
Yom Reviʻi – יום רביעי (abbr. יום ד׳) meaning "fourth day" [corresponds to Wednesday]
Yom Chamishi – יום חמישי (abbr. יום ה׳) = "fifth day" [corresponds to Thursday]
Yom Shishi – יום שישי (abbr. יום ו׳) meaning "sixth day" [corresponds to Friday]
Yom Shabbat – יום שבת (abbr. יום ש׳), or more usually, simply Shabbat – שבת meaning "rest day" [corresponds to Saturday]
Yom Shabbat (יום שבת) is also known as Yom Shabbat Kodesh – יום שבת קודש meaning "holy rest day."
The names of the days of the week are modeled on the seven days mentioned in the Genesis 1:5 "... And there was evening and there was morning, one day". One day (יוֹם אֶחָד) in Genesis 1:5 is translated in JPS as first day, and in some other contexts (including KJV) as day one. In subsequent verses, the Hebrew refers to the days using ordinal numbers, e.g., 'second day', 'third day', and so forth, but with the sixth and seventh days the Hebrew includes the definite article ("the").
The rest day, Shabbat, has a special role in the Jewish weekly cycle as being a special and set apart day, where no work is done. There are many special rules that relate to Shabbat, discussed more fully in the Talmudic tractate Shabbat.
In (Talmudic) Hebrew, the word Shabbat (שַׁבָּת) can also mean "week", so that in ritual liturgy a phrase like "Yom Reviʻi bəShabbat" means "the fourth day in the week".
The period from 1 Adar (or Adar II, in leap years) to 29 Marcheshvan contains all of the festivals specified in the Bible – Pesach (15 Nisan), Shavuot (6 Sivan), Rosh Hashanah (1 Tishrei), Yom Kippur (10 Tishrei), Sukkot (15 Tishrei), and Shemini Atzeret (22 Tishrei). This period is fixed, during which no adjustments are made.
There are additional rules in the Hebrew calendar to prevent certain holidays from falling on certain days of the week. (See Rosh Hashanah postponement rules, below.) These rules are implemented by adding an extra day to Marcheshvan (making it 30 days long) or by removing one day from Kislev (making it 29 days long). Accordingly, a common Hebrew calendar year can have a length of 353, 354 or 355 days, while a leap Hebrew calendar year can have a length of 383, 384 or 385 days.
The Hebrew calendar is a lunisolar calendar, meaning that months are based on lunar months, but years are based on solar years. The calendar year features twelve lunar months of twenty-nine or thirty days, with an intercalary lunar month added periodically to synchronize the twelve lunar cycles with the longer solar year. (These extra months are added seven times every nineteen years. See Leap months, below.) The beginning of each Jewish lunar month is based on the appearance of the new moon. Although originally the new lunar crescent had to be observed and certified by witnesses, the moment of the true new moon is now approximated arithmetically as the molad, which is the mean new moon to a precision of one part.
The mean period of the lunar month (precisely, the synodic month) is very close to 29.5 days. Accordingly, the basic Hebrew calendar year is one of twelve lunar months alternating between 29 and 30 days:
* – For the distinction between numbering systems, see § New year below.
In leap years (such as 5779) an additional month, Adar I (30 days) is added after Shevat, while the regular Adar is referred to as "Adar II."
The insertion of the leap month mentioned above is based on the requirement that Passover—the festival celebrating the Exodus from Egypt, which took place in the spring—always occurs in the [northern hemisphere's] spring season. Since the adoption of a fixed calendar, intercalations in the Hebrew calendar have been assigned to fixed points in a 19-year cycle. Prior to this, the intercalation was determined empirically:
The year may be intercalated on three grounds: 'aviv [i.e.the ripeness of barley], fruits of trees, and the equinox. On two of these grounds it should be intercalated, but not on one of them alone.
Importance of lunar months
From very early times, the Mesopotamianlunisolar calendar was in wide use by the countries of the western Asia region. The structure, which was also used by the Israelites, was based on lunar months with the intercalation of an additional month to bring the cycle closer to the solar cycle, although there is no evidence of a thirteenth month mentioned anywhere in the Hebrew Bible.
Num 10:10 stresses the importance in Israelite religious observance of the new month (Hebrew: ראש חודש, Num 28:11. "The beginning of the month" meant the appearance of a Exod 12:2. "This month is to you".
According to the Mishnah and Tosefta, in the Maccabean, Herodian, and Mishnaic periods, new months were determined by the sighting of a new crescent, with two eyewitnesses required to testify to the Sanhedrin to having seen the new lunar crescent at sunset. The practice in the time of Gamaliel II (c. 100 CE) was for witnesses to select the appearance of the moon from a collection of drawings that depicted the crescent in a variety of orientations, only a few of which could be valid in any given month. These observations were compared against calculations.
At first the beginning of each Jewish month was signaled to the communities of Israel and beyond by fires lit on mountaintops, but after the Samaritans began to light false fires, messengers were sent. The inability of the messengers to reach communities outside Israel before mid-month High Holy Days (Succot and Passover) led outlying communities to celebrate scriptural festivals for two days rather than one, observing the second feast-day of the Jewish diaspora because of uncertainty of whether the previous month ended after 29 or 30 days.
In his work Num 28:14), which implies that we should count the year by months and not by days."
Names of months
Both the Syrian calendar, currently used in the Arabic-speaking countries of the Fertile crescent, and the modern Assyrian calendar share many of the names for months with the Hebrew calendar, such as Nisan, Iyyar, Tammuz, Ab, Elul, Tishri and Adar, indicating a common origin. The origin is thought to be the Babylonian calendar. The modern Turkish calendar includes the names Şubat (February), Nisan (April), Temmuz (July) and Eylul (September). The former name for October was Tesrin.
The Jewish people left Babylon and returned to live in Judea around 586 BCE. At this time they adopted the Babylonian names for the months. The Babylonian calendar descended directly from the Sumerian calendar.
Biblical references to the pre-exilic calendar include ten months identified by number rather than by name. In parts of the 8:13–14) it is implied that the months are thirty days long. There is also an indication that there were twelve months in the annual cycle (1 Kings 4:7, 1 Chronicles 27:1–15). Prior to the Babylonian exile, the names of only four months are referred to in the Tanakh:
All of these are believed to be Canaanite names. These names are only mentioned in connection with the building of the First Temple. Håkan Ulfgard suggests that the use of what are rarely used Canaanite (or in the case of Ethanim perhaps Northwest-semitic) names indicates that "the author is consciously utilizing an archaizing terminology, thus giving the impression of an ancient story...".
In a regular (kesidran) year, Marcheshvan has 29 days and Kislev has 30 days. However, because of the Rosh Hashanah postponement rules (see below) Kislev may lose a day to have 29 days, and the year is called a short (chaser) year, or Marcheshvan may acquire an additional day to have 30 days, and the year is called a full (maleh) year. The calendar rules have been designed to ensure that Rosh Hashanah does not fall on a Sunday, Wednesday or Friday. This is to ensure that Yom Kippur does not directly precede or follow Shabbat, which would create practical difficulties, and that Hoshana Rabbah is not on a Shabbat, in which case certain ceremonies would be lost for a year.
Hebrew names of the months with their Babylonian analogs
The solar year is about eleven days longer than twelve lunar months. The Bible does not directly mention the addition of "embolismic" or intercalary months. However, without the insertion of embolismic months, Jewish festivals would gradually shift outside of the seasons required by the Torah. This has been ruled as implying a requirement for the insertion of embolismic months to reconcile the lunar cycles to the seasons, which are integral to solar yearly cycles.
When the observational form of the calendar was in use, whether or not an embolismic month was announced after the "last month" (Adar) depended on 'aviv [i.e., the ripeness of barley], fruits of trees, and the equinox. On two of these grounds it should be intercalated, but not on one of them alone. It may be noted that in the Bible the name of the first month, Aviv, literally means "spring". Thus, if Adar was over and spring had not yet arrived, an additional month was observed.
During leap years Adar I (or Adar Aleph – "first Adar") is added before the regular Adar. Adar I is actually considered to be the extra month, and has 30 days. Adar II (or Adar Bet – "second Adar") is the "real" Adar, and has the usual 29 days. For this reason, holidays such as Purim are observed in Adar II, not Adar I.
Chronology was a chief consideration in the study of astronomy among the Jews; sacred time was based upon the cycles of the Sun and the Moon. The Talmud identified the twelve constellations of the zodiac with the twelve months of the Hebrew calendar. The correspondence of the constellations with their names in Hebrew and the months is as follows:
Some scholars identified the 12 signs of the zodiac with the 12 sons of Jacob/twelve tribes of Israel.
It should be noted that the 12 lunar months of the Hebrew calendar are the normal months from new moon to new moon: the year normally contains twelve months averaging 29.52 days each. The discrepancy compared to the mean synodic month of 29.53 days is due to Adar I in a leap year always having thirty days. This means that the calendar year normally contains 354 days.
The Hebrew calendar year conventionally begins on Rosh Hashanah. However, other dates serve as the beginning of the year for different religious purposes.
There are three qualities that distinguish one year from another: whether it is a leap year or a common year, on which of four permissible days of the week the year begins, and whether it is a deficient, regular, or complete year. Mathematically, there are 24 (2×4×3) possible combinations, but only 14 of them are valid. Each of these patterns is called a keviyah (Hebrew קביעה for "a setting" or "an established thing"), and is encoded as a series of two or three Hebrew letters. See Four gates.
In Hebrew there are two common ways of writing the year number: with the thousands, called לפרט גדול ("major era"), and without the thousands, called לפרט קטן ("minor era"). Thus, the current year is written as ה'תשע"ט (5779) using the "major era" and תשע"ט (779) using the "minor era".
The Jewish calendar's reference point is traditionally held to be about one year before the Creation of the world.
In 1178 CE, Maimonides wrote in the Mishneh Torah, Sanctification of the Moon (11.16), that he had chosen the epoch from which calculations of all dates should be as "the third day of Nisan in this present year ... which is the year 4938 of the creation of the world" (22 March 1178). He included all the rules for the calculated calendar and their scriptural basis, including the modern epochal year in his work, and beginning formal usage of the anno mundi era. From the eleventh century, anno mundi dating became dominant throughout most of the world's Jewish communities.[page needed] Today, the rules detailed in Maimonides' calendrical code are those generally used by Jewish communities throughout the world.
Since the codification by Maimonides in 1178, the Jewish calendar has used the Anno Mundiepoch (Latin for "in the year of the world," abbreviated AM or A.M., Hebrew לבריאת העולם), sometimes referred to as the "Hebrew era", to distinguish it from other systems based on some computation of creation, such as the Byzantine calendar.
According to rabbinic reckoning, the beginning of "year 1" is notCreation, but about one year before Creation, with the new moon of its first month (Tishrei) to be called molad tohu (the mean new moon of chaos or nothing). The Jewish calendar's epoch (reference date), 1 Tishrei AM 1, is equivalent to Monday, 7 October 3761 BCE in the proleptic Julian calendar, the equivalent tabular date (same daylight period) and is about one year before the traditional Jewish date of Creation on 25 Elul AM 1, based upon the Seder Olam Rabbah. Thus, adding 3760 before Rosh Hashanah or 3761 after to a Julian year number starting from 1 CE will yield the Hebrew year. For earlier years there may be a discrepancy [see: Missing years (Jewish calendar)].
The Seder Olam Rabbah also recognized the importance of the Jubilee and Sabbatical cycles as a long-term calendrical system, and attempted at various places to fit the Sabbatical and Jubilee years into its chronological scheme.
Occasionally, Anno Mundi is styled as Anno Hebraico (AH), though this is subject to confusion with notation for the Islamic Hijri year.
Before the adoption of the current AM year numbering system, other systems were in use. In early times, the years were counted from some significant historic event. (e.g., 1 Kings 6:1) During the period of the monarchy, it was the widespread practice in western Asia to use era year numbers according to the accession year of the monarch of the country involved. This practice was also followed by the united kingdom of Israel (e.g., 1 Kings 14:25), kingdom of Judah (e.g., 2 Kings 18:13), kingdom of Israel (e.g., 2 Kings 17:6), Persia (e.g., Nehemiah 2:1) and others. Besides, the author of Kings coordinated dates in the two kingdoms by giving the accession year of a monarch in terms of the year of the monarch of the other kingdom, (e.g., 2 Kings 8:16) though some commentators note that these dates do not always synchronise. Other era dating systems have been used at other times. For example, Jewish communities in the Babylonian diaspora counted the years from the first deportation from Israel, that of 2 Kings 25:27)
During the Hellenistic Maccabean period, Seleucid era counting was used, at least in the Greek-influenced area of Israel. The 10:1). Josephus writing in the Roman period also used Seleucid era dating exclusively. During the Talmudic era, from the 1st to the 10th century, the center of world Judaism was in the Middle East, primarily in the Talmudic Academies of Iraq and Palestine. Jews in these regions used Seleucid era dating (also known as the "Era of Contracts"). Tractate Avodah Zarah states:
RavAha b. Jacob then put this question: How do we know that our Era [of Documents] is connected with the Kingdom of Greece at all? Why not say that it is reckoned from the Exodus from Egypt, omitting the first thousand years and giving the years of the next thousand? In that case, the document is really post-dated! Said Rav Nahman: In the Diaspora the Greek Era alone is used. He [the questioner] thought that Rav Nahman wanted to dispose of him anyhow, but when he went and studied it thoroughly he found that it is indeed taught [in a Baraita]: In the Diaspora the Greek Era alone is used.
The use of the era of documents (i.e., Seleucid era) continued till the 16th century in the East, and was employed even in the 19th century among the Jews of Yemen.
Occasionally in Talmudic writings, reference was made to other starting points for eras, such as destruction era dating, being the number of years since the 70 CE destruction of the Second Temple. In the 8th and 9th centuries, as the center of Jewish life moved from Babylonia to Europe, counting using the Seleucid era "became meaningless". There is indication that Jews of the Rhineland in the early Middle Ages used the "years after the destruction of the Temple" (e.g., Mainz Anonymous).
A shofar made from a ram's horn is traditionally blown in observance of Rosh Hashanah, the beginning of the Jewish civic year.
this month shall be unto you the beginning of months; it shall be the first month of the year to you.
Nisan 1 is referred to as the ecclesiastical new year.
In ancient Israel, the start of the ecclesiastical new year for the counting of months and festivals (i.e., Nisan) was determined by reference to Leviticus 23:4–6) which corresponds to the full moon of Nisan. As Passover is a spring festival, it should fall on a full moon day around, and normally just after, the Exodus 12:1–2 require that the months be determined by a proper court with the necessary authority to sanctify the months. Hence the court, not the astronomy, has the final decision.
According to some Christian and Karaite sources, the tradition in ancient Israel was that 1 Nisan would not start until the barley is ripe, being the test for the onset of spring. If the barley was not ripe, an intercalary month would be added before Nisan.
The day most commonly referred to as the "New Year" is 1 Tishrei, which actually begins in the seventh month of the ecclesiastical year. On that day the formal New Year for the counting of years (such as Shmita and Yovel), Ezekiel 40:1, which uses the phrase "beginning of the year".) This is the civil new year, and the date on which the year number advances. Certain agricultural practices are also marked from this date.
Moses...appointed Nisan...as the first month for the festivals...the commencement of the year for everything relating to divine worship, but for selling and buying and other ordinary affairs he preserved the ancient order [i. e. the year beginning with Tishrei]."
Edwin Thiele has concluded that the ancient northern Kingdom of Israel counted years using the ecclesiastical new year starting on 1 Aviv (Nisan), while the southern Kingdom of Judah counted years using the civil new year starting on 1 Tishrei. The practice of the Kingdom of Israel was also that of Babylon, as well as other countries of the region. The practice of Judah is still followed.
In fact the Jewish calendar has a multiplicity of new years for different purposes. The use of these dates has been in use for a long time. The use of multiple starting dates for a year is comparable to different starting dates for civil "calendar years", "tax or fiscal years", "academic years", "religious cycles", etc. By the time of the redaction of the Mishnah, Rosh Hashanah 1:1 (c. 200 CE), jurists had identified four new-year dates:
The 1st of Nisan is the new year for kings and feasts; the 1st of Elul is the new year for the tithe of cattle... the 1st of Tishri is the new year for years, of the years of release and jubilee years, for the planting and for vegetables; and the 1st of Shevat is the new year for trees—so the school of Shammai; and the school of Hillel say: On the 15th thereof.
The month of Elul is the new year for counting animal tithes (ma'aser behemah). Tu Bishvat ("the 15th of Shevat") marks the new year for trees (and agricultural tithes).
The Jewish calendar is based on the Metonic cycle of 19 years, of which 12 are common (non-leap) years of 12 months and 7 are leap years of 13 months. To determine whether a Jewish year is a leap year, one must find its position in the 19-year Metonic cycle. This position is calculated by dividing the Jewish year number by 19 and finding the remainder. (Since there is no year 0, a remainder of 0 indicates that the year is year 19 of the cycle.) For example, the Jewish year 5779 divided by 19 results in a remainder of 3, indicating that it is year 3 of the Metonic cycle. 
Years 3, 6, 8, 11, 14, 17, and 19 of the Metonic cycle are leap years. To assist in remembering this sequence, some people use the mnemonic Hebrew word GUCHADZaT "גוחאדז"ט", where the Hebrew letters gimel-vav-het aleph-dalet-zayin-tet are used as Hebrew numerals equivalent to 3, 6, 8, 1, 4, 7, 9. The keviyah records whether the year is leap or common: פ for peshuta (פשוטה), meaning simple and indicating a common year, and מ indicating a leap year (me'uberet, מעוברת).
Another memory aid notes that intervals of the major scale follow the same pattern as do Jewish leap years, with do corresponding to year 19 (or 0): a whole step in the scale corresponds to two common years between consecutive leap years, and a half step to one common year between two leap years. This connection with the major scale is more plain in the context of 19 equal temperament: counting the tonic as 0, the notes of the major scale in 19 equal temperament are numbers 0 (or 19), 3, 6, 8, 11, 14, 17, the same numbers as the leap years in the Hebrew calendar.
A simple rule for determining whether a year is a leap year has been given above. However, there is another rule which not only tells whether the year is leap but also gives the fraction of a month by which the calendar is behind the seasons, useful for agricultural purposes. To determine whether year n of the calendar is a leap year, find the remainder on dividing [(7 × n) + 1] by 19. If the remainder is 6 or less it is a leap year; if it is 7 or more it is not. For example, the remainder on dividing [(7 × 5779) + 1] by 19 is 3, so the year 5779 is a leap year. The remainder on dividing [(7 × 5780) + 1] by 19 is 10, so the year 5780 is not a leap year. This works because as there are seven leap years in nineteen years the difference between the solar and lunar years increases by 7/19-month per year. When the difference goes above 18/19-month this signifies a leap year, and the difference is reduced by one month.
Rosh Hashanah postponement rules
Day of week
Number of days
To calculate the day on which Rosh Hashanah of a given year will fall, it is necessary first to calculate the expected molad (moment of lunar conjunction or new moon) of Tishrei in that year, and then to apply a set of rules to determine whether the first day of the year must be postponed. The molad can be calculated by multiplying the number of months that will have elapsed since some (preceding) molad whose weekday is known by the mean length of a (synodic) lunar month, which is 29 days, 12 hours, and 793 parts (there are 1080 "parts" in an hour, so that one part is equal to 31⁄3 seconds). The very first molad, the molad tohu, fell on Sunday evening at 11.111⁄3, or in Jewish terms Day 2, 5 hours, and 204 parts.
In calculating the number of months that will have passed since the known molad that one uses as the starting point, one must remember to include any leap months that falls within the elapsed interval, according to the cycle of leap years. A 19-year cycle of 235 synodic months has 991 weeks 2 days 16 hours 595 parts, a common year of 12 synodic months has 50 weeks 4 days 8 hours 876 parts, while a leap year of 13 synodic months has 54 weeks 5 days 21 hours 589 parts.
The two months whose numbers of days may be adjusted, Marcheshvan and Kislev, are the eighth and ninth months of the Hebrew year, whereas Tishrei is the seventh month (in the traditional counting of the months, even though it is the first month of a new calendar year). Any adjustments needed to postpone Rosh Hashanah must be made to the adjustable months in the year that precedes the year of which the Rosh Hashanah will be the first day.
Just four potential conditions are considered to determine whether the date of Rosh Hashanah must be postponed. These are called the Rosh Hashanah postponement rules, or deḥiyyot:
If the molad occurs at or later than noon, Rosh Hashanah is postponed a day. This is called deḥiyyat molad zaken (דְחִיַּת מוֹלָד זָקֵן, literally, "old birth", i.e., late new moon).
If the molad occurs on a Sunday, Wednesday, or Friday, Rosh Hashanah is postponed a day. If the application of deḥiyyah molad zaken would place Rosh Hashanah on one of these days, then it must be postponed a second day. This is called deḥiyyat lo ADU (דְחִיַּת לֹא אד”ו), an acronym that means "not [weekday] one, four, or six."
The first of these rules (deḥiyyat molad zaken) is referred to in the Talmud. Nowadays, molad zaken is used as a device to prevent the molad falling on the second day of the month. The second rule, (deḥiyyat lo ADU), is applied for religious reasons.
Another two rules are applied much less frequently and serve to prevent impermissible year lengths. Their names are Hebrew acronyms that refer to the ways they are calculated:
If the molad in a common year falls on a Tuesday after 9 hours and 204 parts, Rosh Hashanah is postponed to Thursday. This is deḥiyyat GaTaRaD (דְחִיַּת גטר”ד, where the acronym stands for "3 [Tuesday], 9, 204."
If the molad following a leap year falls on a Monday, more than 15 hours and 589 parts after the Hebrew day began (for calculation purposes, this is taken to be 6 pm Sunday), Rosh Hashanah is postponed to Tuesday. This is deḥiyyat BeTUTeKaPoT (דְחִיַּת בט”ו תקפ”ט), where the acronym stands for "2 [Monday], 15, 589."
At the innovation of the sages, the calendar was arranged to ensure that Yom Kippur would not fall on a Friday or Sunday, and Hoshana Rabbah would not fall on Shabbat. These rules have been instituted because Shabbat restrictions also apply to Yom Kippur, so that if Yom Kippur were to fall on Friday, it would not be possible to make necessary preparations for Shabbat (such as candle lighting). Similarly, if Yom Kippur fell on a Sunday, it would not be possible to make preparations for Yom Kippur because the preceding day is Shabbat. Additionally, the laws of Shabbat override those of Hoshana Rabbah, so that if Hoshana Rabbah were to fall on Shabbat certain rituals that are a part of the Hoshana Rabbah service (such as carrying willows, which is a form of work) could not be performed.
To prevent Yom Kippur (10 Tishrei) from falling on a Friday or Sunday, Rosh Hashanah (1 Tishrei) cannot fall on Wednesday or Friday. Likewise, to prevent Hoshana Rabbah (21 Tishrei) from falling on a Saturday, Rosh Hashanah cannot fall on a Sunday. This leaves only four days on which Rosh Hashanah can fall: Monday, Tuesday, Thursday, and Saturday, which are referred to as the "four gates". Each day is associated with a number (its order in the week, beginning with Sunday as day 1). Numbers in Hebrew have been traditionally denominated by Hebrew letters. Thus the keviyah uses the letters ה ,ג ,ב and ז (representing 2, 3, 5, and 7, for Monday, Tuesday, Thursday, and Saturday) to denote the starting day of the year.
Deficient, regular, and complete years
The postponement of the year is compensated for by adding a day to the second month or removing one from the third month. A Jewish common year can only have 353, 354, or 355 days. A leap year is always 30 days longer, and so can have 383, 384, or 385 days.
A chaserah year (Hebrew for "deficient" or "incomplete") is 353 or 383 days long. Both Cheshvan and Kislev have 29 days. The Hebrew letter ח "het" is used in the keviyah.
A kesidrah year ("regular" or "in-order") is 354 or 384 days long. Cheshvan has 29 days while Kislev has 30 days. The Hebrew letter כ "kaf" is used in the keviyah.
A shlemah year ("complete" or "perfect", also "abundant") is 355 or 385 days long. Both Cheshvan and Kislev have 30 days. The Hebrew letter ש "shin" is used in the keviyah.
Whether a year is deficient, regular, or complete is determined by the time between two adjacent Rosh Hashanah observances and the leap year. While the keviyah is sufficient to describe a year, a variant specifies the day of the week for the first day of Pesach (Passover) in lieu of the year length.
A Metonic cycle equates to 235 lunar months in each 19-year cycle. This gives an average of 6939 days, 16 hours, and 595 parts for each cycle. But due to the Rosh Hashanah postponement rules (preceding section) a cycle of 19 Jewish years can be either 6939, 6940, 6941, or 6942 days in duration. Since none of these values is evenly divisible by seven, the Jewish calendar repeats exactly only following 36,288 Metonic cycles, or 689,472 Jewish years. There is a near-repetition every 247 years, except for an excess of about 50 minutes (905 parts).
The annual calendar of a numbered Hebrew year, displayed as 12 or 13 months partitioned into weeks, can be determined by consulting the table of Four gates, whose inputs are the year's position in the 19-year cycle and its molad Tishrei. The resulting type (keviyah) of the desired year in the body of the table is a triple consisting of two numbers and a letter (written left-to-right in English). The left number of each triple is the day of the week of 1 Tishrei, Rosh Hashanah (2 3 5 7); the letter indicates whether that year is deficient (D), regular (R), or complete (C), the number of days in Chesvan and Kislev; while the right number of each triple is the day of the week of 15 Nisan, the first day of Passover or Pesach (1 3 5 7), within the same Hebrew year (next Julian/Gregorian year). The keviyah in Hebrew letters are written right-to-left, so their days of the week are reversed, the right number for 1 Tishrei and the left for 15 Nisan. The year within the 19-year cycle alone determines whether that year has one or two Adars.
This table numbers the days of the week and hours for the limits of molad Tishrei in the Hebrew manner for calendrical calculations, that is, both begin at 6 pm, thus 7d 18h 0p is noon Saturday. The years of a 19-year cycle are organized into four groups: common years after a leap year but before a common year (1 4 9 12 15); common years between two leap years (7 18); common years after a common year but before a leap year (2 5 10 13 16); and leap years (3 6 8 11 14 17 19), all between common years. The oldest surviving table of Four gates was written by Saadia Gaon (892–942). It is so named because it identifies the four allowable days of the week on which 1 Tishrei can occur.
Comparing the days of the week of molad Tishrei with those in the keviyah shows that during 39% of years 1 Tishrei is not postponed beyond the day of the week of its molad Tishrei, 47% are postponed one day, and 14% are postponed two days. This table also identifies the seven types of common years and seven types of leap years. Most are represented in any 19-year cycle, except one or two may be in neighboring cycles. The most likely type of year is 5R7 in 18.1% of years, whereas the least likely is 5C1 in 3.3% of years. The day of the week of 15 Nisan is later than that of 1 Tishrei by one, two or three days for common years and three, four or five days for leap years in deficient, regular or complete years, respectively.