Mathematics of the Jewish Calendar/Basic principles
It is easy to understand the mathematical rules behind the Jewish calendar. Maimonides says that they are "such that even schoolchildren can learn and fully grasp them in three or four days".[1]
The Jewish calendar is a vital part of Jewish law. The first command given to Moses, before even those relating to Pesach, was "This month shall be for you the beginning of the months"[2], the command that underlies the calendar. Rashi, commenting on this,[3] said that the Torah should have begun with this law because what went before is unimportant. For the unity of the Jews, there must be one and only one calendar, so there is never any dispute about when the festivals will occur.
The Jewish calendar is luni-solar, governed by both the moon and the sun. The months are regulated by the phases of the Moon, but the year always begins at roughly the same season as determined by the Sun. This differs both from the Gregorian calendar used in the UK, the USA and many other countries, where the months ignore the moon, and the Islamic calendar, where the years ignore the sun. However, there are other luni-solar calendars in use, such as the Chinese one.
G-d, when he created the Sun and Moon, said that they would be for signs, mo'adim, days and years.[4] Mo'adim is usually translated here as seasons, but elsewhere it is used to mean festivals. Thus it was ordained at the creation that both the Sun and the Moon should be used to fix the calendar. Again, in the Psalms (104:19) it says "Who appoints the Moon for mo'adim; the Sun knows its setting", so it is the Moon that determines the exact dates of festivals.
The present Jewish calendar does not depend on observations, but is entirely based on calculations. The conversion to a calculated calendar was (according to a tradition recorded by Rabbi Hai Gaon in the 11th century CE) made by Rabbi Hillel II (4th century CE), and exactly the present rules were already regarded as old-established by Rabbi Saadia Gaon (10th century CE).[5] As a result, anyone who knows the rules can calculate the calendar. The Talmud rules that except in dire emergency, all decisions about the calendar must be taken in Israel.[6] However, the principle now adopted is that anyone performing these calculations is using the same rules as in Israel, hence will always get the same result (assuming there are no calculation errors). Thus, you are not making decisions, only finding out what decisions are being made in Israel by the people there who are using the same rules.
As the average interval between consecutive new moons is just over 29½ days, a month that keeps roughly in step with the Moon should have 29 or 30 days, and broadly we would expect 29 and 30 day months to alternate. A year of 12 months (an ordinary year) would have on average 354 days; this is 11 days too short, so we need a leap year of 13 months every two or three years to keep the calendar in step with the year determined by the sun, which currently averages about 365.2422 days.
The nominal aim is to ensure that the first day of the Festival of Pesach (Passover) falls on the first Full Moon after the vernal equinox, following the Biblical rule "Observe the month of Abib (Spring), and keep the Passover unto the Lord thy God" [7]. Whether this is achieved will be discussed later.
The day is supposed to last from 6pm to 6pm throughout the year. This differs from the usual practice in Jewish law, where the start of the day is related to the Sun (so Shabbat begins just before sunset on Friday and ends at the end of twilight on Saturday) so is earlier in Winter than in Summer and depends on each community's longitude and latitude. The day is divided into 24 hours, so hour 0 lasts from 6pm to 7pm and hour 23 from 5pm to 6pm. Hours 0 to 5 precede midnight so are regarded as the previous day in the Gregorian calendar.
When doing detailed calculations, it is customary to work with a unit of time called the chalak (plural chalakim). There are 1080 chalakim in an hour, so one chalak is 3600/1080 = 3⅓ seconds.
The assumed time of a new moon is called the molad. The interval between successive molads is assumed to be 29 days 12 hours 793 chalakim, or 29 days 12 hours 44 minutes 3⅓ seconds.