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The Jews have been using the lunar calendar since Biblical times, but usually referred to months by number rather than name. During the Babylonian exile[?], they adopted Babylonian names for months and possibly a regular pattern of intercalating the 13th month. Some sects, such as the Essenes, used a solar calendar.
The Hebrew year 1 started on Monday, September 7, 3761 B.C.E.. This means that adding 3761 to a Gregorian year number will yield the Hebrew year number (within one year). This actually only works until the Gregorian year 22,203, but it's a fairly good rule of thumb.
The Hebrew month is tied to the average time taken by the Moon to cycle from lunar conjunction to lunar conjunction. Twelve lunar months are approx. 354 days while while the solar year is approx. 365 days so an extra lunar month must be added every two or three years.
The calendar is thus also tied to a 19year cycle of 235 lunar months. The average Hebrew year length is 365.2468 days, in contrast to the average tropical solar year which is measured at roughly 365.2422 days. Approximately every 216 years, the Hebrew year is "slower" than the average solar year by a full day. Since the average Gregorian year is 365.2425 days and repeats every 400 years, the average Hebrew year is slower by a day every 231 Gregorian years.
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 Hashannah 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.
Hebrew time measurement is governed by rabbinic law[?], which divides the hours up into 1080 parts, (a part lasts 3 and 1/3 seconds and each minute has 18 parts). This simplifies calculations, as only days, hours and parts are required. 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 virtual 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 BCE and later verified by the Greek astronomer Hipparcus. Its remarkable accuracy was achieved using records of eclipses over long periods. 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 daynight cycle. The value 2912793 was almost exactly correct in 1 CE and is now about 0.6 sec per month too great.
The 19 year cycle has 12 nonleap 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 nonleap year has 12 months. A mnemonic word in Hebrew is GUCHADZaT (the Hebrew letters gimelvavhet alephdaletzayintet, i.e. 3, 6, 9, 1, 4, 7, 9. See Hebrew numerals).
A Hebrew non leapyear 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 Hashannah may only occur on Mondays, Tuesdays, Thursdays, and Saturdays, 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 
Basically, the Hebrew months alternate between a short month and a long month, for example: Tishrei[?] (30 days), Cheshvan[?] (also spelled Heshvan) (29 days), Kislev (30 days), Tevet (29 days), Shevat (30 days), Adar (29 days), Nisan (30 days), Iyar (29 days), Sivan[?] (30 days), Tammuz (29 days), Av (30 days), Elul[?] (29 days).
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 because the 11.25 days between 12 lunar months and one solar year in just three years adds up to more than a month.
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 (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 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 21st, 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.
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.
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