# Mean Hebrew Year Length

I am researching the kings of Judah and Israel and realized that the Hebrew year and the Gregorian year may be different. I know the mean length of the Gregorian year (365.2425 days), but I do not know the length of the Hebrew year. So I come here to find out. What is the mean length (in days) of a Hebrew year, to seven significant digits, like I've shown for the Gregorian year? (In case you don't know, when I say "mean" I am saying "average").

• I know that what I am going to say now might sound trivial, but the best way to determine an average is to simply compute it. Using this particularly helpful tool, evaluate the Julian day value of the same Hebrew calendar date (say, Nisan 1 or Tishri 1), but situated 19 x N centuries apart (since a Metonic cycle consists of 19 years), subtract the two results, and then divide their difference by the number of years contained within that specific time span. – Lucian Sep 27 '18 at 22:38

## 2 Answers

A mean Hebrew month is 29 days + 12 hours + 793/1080 of an hour.

Back then they would add a 13th month into the year whenever the drift appeared to be getting too big, but not on a regular schedule, so it's hard to give an exact estimate of mean year length.

Currently we use a regular schedule of 7 leap years every 19 years, for 235 months in 19 years. Do your math to find a mean year length of 365.2468 days. (No year is actually that long. They are all on the order of ~354 or ~384.)

(In the current fixed calendar, there is also some variance in the lengths of certain months for other reasons (putting holidays on certain days of the week), so the full cycle of the calendar is actually 36288 cycles of 19 year cycles, for a total of 689472 years.)

• Interesting math calculations. Can you link in a source or two to support, esp. the info in the last paragraph? I'm curious how they determined those numbers? – DanF Dec 22 '16 at 15:56
• @DanF Just find the lcm for 19 and the number of 19 year cycles to get back to where you started. A 19 year cycle pushes the molad 69715 chalakim and there's 181440 in a week, so solve 69715*x=0 mod 181440. Since 19 is prime (and of different parity) we can find the lcm directly: 19*36288=689472. – Double AA Dec 22 '16 at 16:08
• @DoubleAA sorry I don't see how you are doing to convert the 235 months on days – kouty Dec 22 '16 at 17:17
• @kouty Take the 235 months modulo 1 week. The goal is to find two molads of the same month in the same year of the 19 year cycle at the exact same time of day on the same day of the week. – Double AA Dec 22 '16 at 17:19
• – kouty Dec 22 '16 at 17:35

The Jewish year varies in length, with an extra month some years and with some years having one more or one less day than usual. Ultimately, the main reason for the variation is to keep it from drifting too much from the solar year, such that Passover stays in the Spring time. Therefore, the mean length of the Hebrew year, after all of the variations, ends up, by design, very close to the mean length of the solar year.

According to this WP article, the mean length of the Jewish year is 365.2468 days, or just a smidge longer than the Gregorian year.