The Hebrew calendar is the annual calendar used in Judaism. Like the Chinese calendar, it is also a lunisolar calendar, based upon both lunar months and a solar cycle (which defines its years). This is in contrast to the Gregorian calendar, which is based solely upon a solar cycle, or the Islamic calendar, which is purely lunar.
Jews use this calendar to determine when the new Hebrew months start; this calendar determines the Jewish holidays, which Torah portions to read, Jahrzeits, and which set of Psalms should be read each day.
Jews have been using a lunisolar calendar since Biblical times, but originally referred to the months by number rather than name. During the Babylonian exile, they adopted Babylonian names for the months. Some sects, such as the Essenes, used a solar calendar.
The beginning of each Hebrew year used to be decided by observing the growth of barley, and the beginning of each lunar month used to be decided by observing the moon (in case of cloudy skies, by guessing); the latter method is still in use by a minority of Muslim nations even today, for the Islamic calendar. The beginning of each Hebrew month had been decided in Israel and the information was transfered by torches to the main diaspora areas in Babylonia (now Iraq) and Egypt.
The observation-independent Hebrew calendar described hereafter was set in Babylonia sometime in the 4th century. The change was probably performed by a council of sages headed by Shmuel hakatan.
The epoch of the modern Hebrew calendar began 1 Tishri AM 1 (AM = Anno Mundi = in the year of the world) with a tabular date (same daylight period) in the proleptic Julian calendar of Monday, October 7, 3761 BC. This date is about one year before the traditional Jewish date of Creation on 25 Elul AM 1! A minority place Creation on 25 Adar AM 1, about six months after the modern epoch. Thus adding 3761 to a Gregorian year number will yield the Hebrew year number beginning in autumn (add 3760 for that ending in autumn). This holds until the Gregorian year 1 BC. After that (due to the lack of year 0), adding 3760 to the Gregorian year yields the Hebrew year beginning in autumn (3759 for that ending in autumn). Because the Hebrew year drifts relative to the Gregorian year, this actually only works until the year 22,203, but it is a fairly good rule of thumb.
The Hebrew month is tied to an estimate 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 Hebrew 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. A number of Jewish scientists have recently suggested that the chief rabbinate in Jerusalem consider modifying this rule to avoid this effect.
There are exactly 14 different patterns that Hebrew calendar years may take. Each of these patterns is called a "keviyah" (Hebrew for "species"), and is distinguished by the day of the week for Rosh Hashanah of that particular year and by that particular year's length.
A variant of this pattern naming includes another letter which specifies the day of the week for the first day of Pesach (Passover) in the year.
Every hour is divided into 1080 parts. A part (31/3 seconds or 1/18 minute) equals a small Babylonian time period called a barleycorn, itself equal to 1/72 of a Babylonian time degree (1° of celestial rotation). The weekdays start with Sunday (day 1) and proceed to Saturday (day 7). Since some calculations use division, a remainder of 0 signifies Saturday.
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 13 hours in each direction due to the nonuniform velocity of the moon. This value for the interval between molads (the mean synodic month) was known to the Babylonians by about 250 BC and was later used by the Greek astronomer Hipparchus and the Alexandrian astronomer Ptolemy. Its remarkable accuracy was achieved using records of lunar eclipses over several centuries. Measured using an absolute scale, such as 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 in AD 1 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 19-years-235-months equality described above; the total accumulated error of the 29-12-793 calculation since the start of the Hebrew calendar amounts to only about five hours until now.
The 19 year cycle has 12 non-leap 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, 19 Hebrew years can be either 6939, 6940, 6941, or 6942 days each. 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).
The leap years of 13 months are the 3rd, 6th, 8th, 11th, 14th, 17th, and the 19th years. 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 non-leap year has 12 months. A mnemonic word in Hebrew is GUCHADZaT (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 non-leap years between consecutive leap years, and a half step to one non-leap between two leap years.
A Hebrew non leap-year will only have 353, 354, or 355 days. A leap year will have 383, 384, or 385 days.
Although simple math would calculate 21 patterns for the calendar years, there are other limitations which means that Rosh Hashanah may only occur on Mondays, Tuesdays, Thursdays, and Saturdays, according to the following table:
For leap years, a 30 day month of Adar 1 is added immediately after the month of Shevat, and the 29 day Adar is called Adar 2. This is to ensure that the months remain at the same season rather than continuing to drift earlier by about 11 days per year.
The 265 days from the first day of the 29 day month of Adar (the last one of the year) 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 Hashannah (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 calendar isn't reformed (this will not happen for thousands of years).
Karaites use the lunar month and the solar year, but determine when to add a leap month by observing barley, rather than a fixed calendar. This occasionally puts them a month out of sync with the rest of the Jews.