About the Hebrew Calendar
The Hebrew calendar is a lunar calendar based on 19-year cycles. The following sections explain the principles of this calendar. The relationship of the Hebrew year to the Civil year is illustrated by this chart.
Units of Time
The calculations in the subsequent sections are based on an ancient form of measuring time. Each day is divided into 24 hours, which are each divided into 1080 parts (or "halakhim").
A month is defined as the period from one new moon to the next, corresponding to the astronomical definition of a siderial month. For these calculations the length of a month is taken to be 29 days, 12 hours, and 793 halakhim.
The Biblical date of creation is used as the basis for the Hebrew calendar. Reverse calculation gives the time of the first new moon of Year 1 as 5 hours and 204 halakhim from the beginning of the second day of the week (Monday). This date corresponds to October 7, 3761 BCE, using the Julian (not Gregorian) calendar. For later arithmetic convenience, we will define the preceding Sabbath as day number 0, so that this date, 1 Tishrei of Year 1, is day number 2.
Types of Years
A year in the Hebrew calendar is normally twelve months: Tishrei, Cheshvan, Kislev, Tevet, Sh'vat, Adar, Nisan, Iyar, Sivan, Tammuz, Av, and Elul. An extra month (Adar is replaced with Adar I and Adar II) is intercalated in seven of every nineteen years (years 3, 6, 8, 11, 14, 17, and 19). Every 19-year cycle thus contains 235 months. These two cycles of years and months are so close that they diverge at the rate of only 4.5 days every 1000 years.
Because a lunar month is not a whole number of days, the length of the months can be either 29 or 30 days. A typical year alternates 30- and 29-day months, beginning with Tishrei, which is 30 days. In leap years, Adar I is 30 days and Adar II is 29 days. To further bring the calendar into alignment with the lunar cycle, a year may need to be shortened or lengthened by one day: Kislev may be shortened to 29 days, or Cheshvan may be lengthened to 30 days. Thus, there are six types of years:
- Defective common years, with 353 days
- Regular common years, with 354 days
- Excessive common years, with 355 days
- Defective leap years, with 383 days
- Regular leap years, with 384 days
- Excessive leap years, with 385 days
Finding the Molad Tishrei
In the ancient Hebrew calendar, each month began with the first observation of the first sliver of moon after a new moon. The beginning of each month is now determined solely by calculation. The first step is to determine the date and time of the first new moon of the year, the Molad Tishrei.
This, and the Molad Tishrei for the following year, determine whether the year needs to be defective, regular, or excessive, and the year's position in the 19-year cycle determines whether it is a leap year or a common year.
To find the Molad Tishrei for a given year, simply count from year 1 (starting with 2 days, 5 hours, and 204 halakhim as the Molad Tishrei for that year), counting 29 days, 12 hours, and 793 halakhim for each month. This can be done efficiently by first calculating the number of days from Day 0 until the beginning of the current 19-year cycle, counting 235 months (991 weeks, 2 days, 16 hours, and 595 halakhim) for each cycle. Then, for each year from there to the current year, 12 or 13 months should be added. The total is best expressed in weeks, days, hours, and halakhim.
Determining the Date of Rosh HaShanah
Once the date of the Molad Tishrei has been calculated, some additional considerations must be taken into account to determine the actual date of Rosh HaShanah. These considerations may cause the actual date of 1 Tishrei to be postponed from the date of the Molad Tishrei. There are four such postponements:
First, if the time of the Molad Tishrei is later than 18 hours from the beginning of the day, Rosh HaShanah is postponed to the next day. This probably accounts for the fact that the young moon could not have been observed until the next day.
Second, for common years only, if the Molad Tishrei falls on a Tuesday, and is later than 9 hours and 204 halakhim from the start of the day, Rosh HaShanah is postponed to the next day. This rule prevents a situation in which the postponements for the next year would require the year to be 356 days long.
Third, for years following leap years only, if the Molad Tishrei falls on a Monday, and is later than 15 hours and 589 halakhim from the start of the day, Rosh HaShanah is postponed to the next day. This rule prevents a situation that would require the previous year to be only 382 days long.
Finally, if Rosh HaShanah would fall on Sunday, Wednesday, or Friday, it is postponed to the next day. In combination with one of the above postponements, Rosh HaShanah could be postponed by as much as two days. This postponement prevents certain holidays from falling on the Sabbath.
Converting to the Civil Calendar
Since we know the Civil date corresponding to Day 0 in the Hebrew calendar (October 5, 3761 BCE -- two days before the Molad Tishrei for Year 1), it is a straightforward matter to add the weeks and days to this date to arrive at the corresponding date for any date in the Hebrew calendar.
This conversion is usually done using the astronomical convention of Julian day numbers, where the Hebrew calendar's Day 0 corresponds to Julian Day 347,996. Given a number of weeks and days from Day 0, we can easily calculate the corresponding Julian Day number. For Rosh HaShanah 5757, we add 300,334 weeks and 7 days (or 2,102,345 days), arriving at Julian Day 2,450,341.
Given a Julian day number, it can be converted to the civil (Gregorian) calendar as follows.
Step 1. Convert it to the number of days since March 3 of Civil Year 0 (commonly written 1 BCE), by subtracting 1,721,120, which is the Julian day number for that day (using the Gregorian calendar). March 3, 1BCE, in the Julian calendar would have been March 1 had the Gregorian calendar been in use at the time.
Step 2. Compute the number of whole centuries since Civil Year 0 by dividing the result of Step 1 by 36,524.25, the average number of days per century. Express the result as a whole number of centuries, then calculate the whole number of days left over.
Step 3. Compute the number of whole years since the beginning of the current century by dividing the remainder from Step 2 by 365.25, the average number of days per year in a century. Express the result as a whole number of years, then calculate the whole number of days left over.
Step 4. Combine the count of centuries from Step 2 with the count of years from Step 3 to find the Civil Year, and use the remainder from Step 3 to find the month and day of the year (count from March 1). If the month is January or February, the date actually in the following year (this won't be a concern in Rosh HaShanah calculations). September 1 is 184 days after March 1; October 1 is 214 days after March 1.