# How Do I Know If it's a Hebrew Leap Year?

How can I tell if a given year is a Hebrew Leap Year? The simpler, the better.

[Assuming we are using a fixed calendar, not like the "olden days".]

If you have a piano keyboard handy, you can use it as a mnemonic aid:

``````              Y mod 19 =

│  ▐██▌  ▐██▌  │  ▐██▌  ▐██▌  ▐██▌  │
│  ▐██▌  ▐██▌  │  ▐██▌  ▐██▌  ▐██▌  │
│  ▐██▌  ▐██▌  │  ▐██▌  ▐██▌  ▐██▌  │
│  ▐██▌  ▐██▌  │  ▐██▌  ▐██▌  ▐██▌  │
│  ▐██▌  ▐██▌  │  ▐██▌  ▐██▌  ▐██▌  │
│    │    │    │    │     │    │    │
│    │    │    │    │     │    │    │
│  0 │  3 │  6 │  8 │  11 │ 14 │ 17 │
└────┴────┴────┴────┴─────┴────┴────┘
``````
• half step between white keys (no intervening black key) = leap years 2 years apart
• whole step between white keys (black key between them) = leap years 3 years apart
• That's pretty cool! – Dave Mar 23 '11 at 6:59
• Daniel ben Noach, Welcome to mi.yodeya, and thanks very much for the awesome mnemonic and accompanying ASCII art! I look forward to seeing you around. – Isaac Moses Mar 23 '11 at 13:36
• So, is there a common consideration that causes this coincidence? Is it just that you want to put your two short intervals as far apart as possible in both cases? I don't know enough about music to understand what drove the design of scales and therefore pianos. – Isaac Moses Mar 23 '11 at 17:18
• According to en.wikipedia.org/wiki/…, it's just a coincidence. – Daniel ben Noach Mar 24 '11 at 0:06
• @Monica sorry to bug you about this but I think I remember you have musical training. How does this picture make sense? Forget the leap year aspect, there are lines here I don't understand and an apparent misslabeling of whole-half intervals. Am I missing something? The line between 3 and six for example whether meant to represent a separation of keys messes things up. There is only one key there. – user6591 Jan 17 '16 at 0:34

A Hebrew Leap Year (in modern times) depends on which year of the 19-year cycle we are in. The years of the 19 year cycle that are leap years are: 3, 6, 8, 11, 14, 17, and 19. (to figure out which year of the cycle you're in, find the remainder of the current year when divided by 19).

It turns out there's a simple equation you can use:

``````(7y+1) mod 19 < 7
``````

[read, 7 times year, plus 1, divided by 19 & find remainder].

if the remainder is less than 7, it's a leap year. If it's 7 or greater, it's a regular year.

For example, for 5771 (this year):

``````7 * 5771 = 40397
40397 + 1 = 40398
40398 / 19 = 2126 remainder 4
4 is less than 7, so it's a leap year!
``````

OR, you can use the "old-fashioned," non-mathy method of memorizing גוחאדז"ט - which comes out to: 3, 6, 8, 1, 4, 7, 9. Adding ten to the ones after 8 gets you: 3, 6, 8, 11, 14, 17, and 19 - like we found above.

• I once came across a different equation in an old book about the Jewish calendar: (12y+17) mod 19 > 11. – Alex Mar 17 '11 at 22:22
• Note that both R'yydl's and R'Alex's formulas require you to assume the numbers modulo 19 are 0, ..., 18, not 1, ..., 19 (or anything else). In fact, they're the same formula reworded, believe it or not: Because 7=-12 and 1=-18 (mod 19), we have 7y+1=-(12y+18)=-(12y+17)-1, so when 7y+1<7, 12y+17+1>-7=12 i.e. 12y+17>11. – msh210 Mar 18 '11 at 2:40
• A little shortcut: Since 5700 is evenly divisible by 19, you can just use the last two digits for recent years (e.g. 71 in 5771). – Dave Mar 18 '11 at 4:25
• @Dave, years 3, 6, 8, 11, 14, 17, and 19 of each 19-year cycle are leap years. It happens to be that if you multiply each of those by 7 and add 1 you get a number whose remainder mod 19 is 0, 1, 2, 3, 4, 5, or 6, and that the same is false of the other years. I don't think there's anything deeper than that to it. I can't think of any way to discover the formula other than trial and error, but perhaps there is a way... likely someone at Mathematics will know. – msh210 Jun 22 '11 at 15:24
• @msh210 - Good Idea! See here: math.stackexchange.com/questions/46964/… – Dave Jun 22 '11 at 19:37

Simpler is 5771 / 19 = 303 remainder 14 so it is a leap year.

• Yep. But that requires memorizing the 7 years that are leap years. – yydl Mar 17 '11 at 22:56

Here's a version that can be done mentally. It works for all years between 5700-5799 (1940-2039):

1) Take the last two digits of the year, and add the second digit to half of the first. If there is a remainder, add 10. If this brings the total to 19 or more, subtract 19. You now know which year in the cycle it is. [Year 19 = 0.]

2) For cycle numbers greater than 7, add 1. Then check if it's evenly divisible by 3 -- if so, it is a leap year.

Example:

The last two digits of 5771 are 71.

1 + 3 (half of 7) = 4

Since there is a remainder: 4 + 10 = 14. --> [It is year #14 in the cycle.]

14 is more than 7, so we add 1, raising it to 15.

Since 15 is divisible by 3, it is a leap year.

After dividing the year by 19, if the remainder's digits have only curves or only straight lines, it's a leap year. Otherwise, it's not.

``````00 - all curved
03 - all curved
06 - all curved
08 - all curved
11 - all straight
14 - all straight
17 - all straight
``````

Of course, this depends on the font you use. We're assuming that 0,3,6,8 are curved; 1,4,7 are straight, and 2,5,9 are mixed.

Using the anno mundi system started by Maimonides , devide the Hebrew year by 19 . This year is 5776. It is divisible evenly by 19 , and yielding 304. In other words, there have been 304 cycles since it started. Next, the next leap years are in a sequence of 3-3-2, and 3-3-3-2, giving 7 leap years out of every 19 year cycle . The next leap year will be 5779 and so on . As a coincidence, the 3323332 sequence resembles the sequence of alternating black and white keys of a piano

• Thanks for the answer post. But everything in it is already in preexisting answers to this question, so this will likely get deleted. – msh210 Apr 28 '16 at 14:17
• I am surprising that to know piano is so helpfull in hilchot kidush Hachodesh. – kouty Apr 28 '16 at 14:31
• @kouty see the other answers to this question. – sabbahillel Apr 28 '16 at 16:33