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Adar I is added in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of the cycle.The current cycle began in Jewish year 5758 (the year that began October 2, 1997).
The civil calendar used by most of the world has abandoned any correlation between the moon cycles and the month, arbitrarily setting the length of months to 28, 30 or 31 days.
The Jewish calendar, however, coordinates all three of these astronomical phenomena.
Months are either 29 or 30 days, corresponding to the 29½-day lunar cycle.
After many years of blissful ignorance, I finally sat down and worked out the mathematics involved, and I have added a page on The Jewish Calendar: A Closer Look, which may be of interest to those who want a deeper understanding or who want to write a Jewish calendar computer program.
For the rest of us, there are plenty of easily accessible computer programs that will calculate the Jewish calendar for more than a millennium to come. Note that the number of days between Nissan and Tishri is always the same.
These three phenomena are independent of each other, so there is no direct correlation between them.
On average, the moon revolves around the Earth in about 29½ days.In addition, Yom Kippur should not fall adjacent to Shabbat, because this would cause difficulties in coordinating the fast with Shabbat, and Hoshanah Rabbah should not fall on Saturday because it would interfere with the holiday's observances. (Before the Common Era), which are commonly used by scholars today.A day is added to the month of Cheshvan or subtracted from the month of Kislev of the previous year to prevent these things from happening. The "first month" of the Jewish calendar is the month of Nissan, in the spring, when Passover occurs.In ancient times, this month was added by observation: the Sanhedrin observed the conditions of the weather, the crops and the livestock, and if these were not sufficiently advanced to be considered "spring," then the Sanhedrin inserted an additional month into the calendar to make sure that Pesach (Passover) would occur in the spring (it is, after all, referred to in the Torah as Chag he-Aviv, the Festival of Spring! A year with 13 months is referred to in Hebrew as Shanah Me'uberet (pronounced shah-NAH meh-oo-BEH-reht), literally: a pregnant year. The additional month is known as Adar I, Adar Rishon (first Adar) or Adar Alef (the Hebrew letter Alef being the numeral "1" in Hebrew).The extra month is inserted before the regular month of Adar (known in such years as Adar II, Adar Sheini or Adar Beit).When the Sanhedrin heard testimony from two independent, reliable eyewitnesses that the new moon occurred on a certain date, they would declare the rosh chodesh (first of the month) and send out messengers to tell people when the month began.