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The Math of the Jewish Calendar

NOTE: Be critical of the results and compare them with other sources.

NOTE: All non-Jewish dates are Gregorian calendar dates. Other sources may use Julian dates on years before the 18th century. This may cause the dates to be a few days different when comparing results.

Glossary

There is a glossary at the end of this page for a few terms you will need to be familiar with.

Date and time formatting

You will see time formatting like this, 29d 12h 793p. This is the same as 29 days, 12 hours and 793 parts.

Overview of the Steps

  1. Choose a Jewish Year
  2. Count the number of months since the first molad to the molad of Tishri in the selected year.
  3. Multiply the number of months by the length of the molad 29d 12h 793p and add 1st molad (molad tohu)
  4. Determine the day of the week of the molad (before postponements are applied)
  5. Apply the rules of postponement to determine the date of Rosh Hashanah (Tishri 1) for the year .
  6. To get the secular date, add the number of days calculated above to the secular starting date.

Getting Started

Enter a Jewish year below (the current year is entered by default) and the following calculations will be presented on this page. This will help you learn how the Jewish calendar works, so you can replicate this process by hand.

Choose a Jewish Year

To determine the Jewish year for Tishri 1, take the Gregorian year and add 3761. For example, the year 2016 is the Jewish years 5777 (2016 + 3761 = 5777).

Jewish Year:

(Gregorian year )

Choose a year between 3761 and 5811 (This is years 1 to 2050 in Gregorian years).

Skip to the final dates

Count the number of months since the first molad to the selected year ()

The Jewish calendar works by totaling the number of months since the very first molad of Tishri 1 and multiplying this number by the average length of a month.

The Jewish calendar uses the 19 year Metonic cycle, which contains 7 leap years. Each of these leap years has 13 months instead of 12. Using the 19 year cycle to count the months helps simplify the counting since there are 235 months in a 19 year cycle, which takes into account the 7 extra months from the 7 leap years in the cycle.

This means that 19 years times 12 months plus 7 extra months for every leap year equals 235 months every 19 years. (19 × 12) + 7 = 235

Determine the number of 19 year cycles:

( - 1) / 19 = remainder of y.

There are "19 year" cycles. Let's convert this number into the number of months.

Determine the number of months:

× 235m = m

We need to add the months from the remainder of years (there are/is years).

This is complicated because we need to take into account when the Jewish calendar has leap years during the 19 year cycle. The remainder value is the count of the year into the last 19 year cycle we have counted.

Let's add 12 for all the remainder years, and then determine if there are any leap years in the remainder, and then add extra months for those.

Determine the number of months in the remainder years:

y × 12m = m

Let's add up all the extra leap months.

Years 3,6,8,11,14 and 17 are all leap years.

Add 1 month for every leap year left in the cycle: (3,6,8,11,14,17 and 19 are leap years)

leap months + months + = total months

Our final count of months from the very first molad of the first molad of Tishri 1 (Sunday, September 6, 3761 BCE at 11:11pm) till the molad of Tishri 1 of the year is is months.

Multiply the number of months by the length of the molad 29d 12h 793p and add 1st molad (molad tohu)

We multiply the number of months by the molad 29d 12h 793p and then add the first molad (molad tohu), which is 2d 5h 204p.

Because we will be adding a number of remainders together, it's most simple to add the molad tohu (the first molad) at the same time as we are adding all the molads together.

Molad is 29d 12h 793p, now we multiply it by m

Parts

m × 793p + 204p (molad tohu) = p

Reduce parts to hours (1080 parts in a hour), and keep the remainder for the final time

p / 1080 = h remainder p

Hours

m × 12h + 5h (molad tohu) + h (hours from parts) = h

Reduce hours to days, and keep the remainder for the final time

h / 24 = d remainder h

Days

m × 29d + 2d (molad tohu) + d (days from hours) = d

Final time for Molad in year

d h p

Determine the day of the week of the molad (before postponements are applied)

We need to determine the day of the week the molad is on. We use this to determine how to apply the rules of postponement.

It is assumed that the weeks have been 7 days consistently since the beginning without interuption, so we can just divide by seven, and use the remainder.

Day 1 is Sunday, 2 is Monday, 3 is Tuesday, 4 is Wednesday, 5 is Thursday, 6 is Friday 0 is Saturday.

Determine the day of the week for the molad

/ 7 = remainder ()

The remainder is therefore the day of the week for the molad is

Apply the rules of postponement to determine the date of Rosh Hashanah (Tishri 1) for the year .

This is the most complicated part of the process, it's mostly logic that only alters Tishri 1 by pushing it back (postponing) it by one or two days at most. These rules alter the date of Rosh Hashanah (Tishri 1), but do not alter the time of the molad.

Why do we apply rules of postponement?

The rules of postponement have complicated and long explanations that relate most commonly to relgious issues the Jews have with keeping sabbaths on certain days of the week, but also ensuring the year lengths fit within certain parameters so that the math of the calendar functions consistently.

You can read all the details about the postponents at the end of this document here: Explanation of the Rules of Postponements

Order of application

Because of the logic surrounding how these rules work, we will alter the order in which they are checked and applied.

Here's the order in the way they are checked, and explanations on why this order is chosen.

  1. Rule #3 is checked first. If applied, none of the other rules are applied. Also, if it is applied it moves Rosh Hoshanna (Tishri 1) back a day. And in doing this, it automatically triggers rule #2, which just postpones for another day. So instead of checking #2, we just add 2 days instead of one.
  2. Rule #4 is checked second. This is because if it's applied, none of the other rules are applied.
  3. Rule #1 is checked third. If applied, it may trigger #2 or it may cause #2 to not be applied. This rule and rule #2 can both be triggered.
  4. Rule #2 is checked last, as it won't trigger or cause other rules to not be checked.

What are the effects of the rules of postponement?

All of the rules postpone Rosh Hoshanna 1 day. But since rule #3 moves Rosh Hoshanna from a Tuesday to a Wednesday, it then also triggers rule #2. So for rule #3, we simply postpone Rosh Hoshanna 2 days, instead of 1 day twice for both rules.

NOTE: We will run through all the LOGIC of the rules, and then after that, we will apply the MATH of the rules.

Rule #3 Gatarad
Ask 3 questions, if all 3 are YES, then we apply this rule. If any are NO, we skip this rule.

Question #1: Is day of the week of the molad on Tuesday (day #3)?

Answer: The day of the week for the molad of year is (), so the answer is .


Question #2: Is the molad after 9h 204p?

Answer: The molad of year is at h p, so the answer is .


Question #3: Is the year a normal year (ie, NOT a leap year)?

Answer: The leap years in a 19 year cycle are years 3,6,8,11,14,17 and 19. (If the remainder is 0, then it is year 19)

If the remainder of the year divided by 19 is one of these numbers, then it's a leap year, otherwise it's a normal year.

/ 19 = remainder of

The year is year # in the last 19 year cycle, so the answer is .

Do we apply postponement rule #3 Gatarad?
Are ALL of the answers "YES" to the three questions above?

Rule #4 Betutkafot
Ask 3 questions, if all 3 are YES, then we apply this rule. If any are NO, we skip this rule.

Question #1: Is day of the week of the molad on Monday (day #2)?

Answer: The day of the week for the molad of year is (), so the answer is .


Question #2: Is the molad after 9h 204p?

Answer: The molad of year is at h p, so the answer is .


Question #3: Is the PREVIOUS year () a LEAP year?

Answer: The leap years in a 19 year cycle are years 3,6,8,11,14,17 and 19. (If the remainder is 0, then it is year 19)

If the remainder of the year divided by 19 is one of these numbers, then it's a leap year, otherwise it's a normal year.

/ 19 = remainder of

The year is the PREVIOUS year, and is year # in the last 19 year cycle. so the answer is .

Do we apply postponement rule #4 Betutkafot?
Are ALL of the answers "YES" to the three questions above?

Rule #1 Molad Zaken
Ask 2 questions, if both are YES, then we apply this rule. If any are NO, we skip this rule.

Question #1: Did we skip both rules #3 and #4? (ie, we did not apply either of them)

Answer:


Question #2: Is the molad at or after 18h?

Answer: The hour for molad for year is h so the answer is

Do we apply postponement rule #1 Molad Zaken?
Are ALL of the answers "YES" to the two questions above?

Rule #2 Lo ADU Rosh
Ask 2 questions, if both are YES, then we apply this rule. If any are NO, we skip this rule.

Question #1: Did we skip both rules #3 and #4? (ie, we did not apply either of them)

Answer:


Question #2: Is the day of the week either 1, 4 or 6? (Sunday, Wednesday or Friday?)

Answer: The day of the week for the molad of year is (), so the answer is .

Do we apply postponement rule #2 Lo ADU Rosh?
Are ALL of the answers "YES" to the two questions above?

Results after Postponements applied

Day of the week is: ()

Number of days:

To get the secular date, add the number of days calculated above to the secular starting date.

The number of days from the molad tohu (the first molad in the Jewish calendar) is .

Sunday, September 6, 3761 BCE at 11:11pm

To convert this to a Gregorian date, we need to count the number of days from September 6th, 3761 BCE till the year.

Since there is a perfectly repeatable count of 146,097 days for every 400 Gregorian years, and this includes a perfect counting of weeks, we can use this number to help us count years, including the extra days in leap years, just like the 19 year cycle has 235 months, including the extra months in leap years in the Jewish calendar.

As a reference point, 14 × 146,097 = 2,045,358 days have elapsed from the molad tohu until 7 September 1840.

Determine the date for Tishri 1 in Jewish year / Gregorian year

The Final Dates

The calculations are complete for the Jewish year . Here's the dates you get from the Jewish calendar.

The Feast of Trumpets is on Tishri 1, all other holy days are simple addition/subtractions from this date.

Feast of Trumpets (Tishri 1, Rosh Hoshanna, Jewish New Year)

Passover/Unleavened Bread (15th of Nissan) (minus 163 days) ( is the 14th of Nissan, which is minus 164 days)

Pentecost (minus 113 days)

Atonement (plus 9 days)

Feast of Tabernacles (plus 14 days)


Appendix

Explanation of the Rules of Postponements

This is a simplified explanation of the postponements.

The Hebrew calendar makes four exceptions where we push off Rosh Hashanah one or two days. This is done to prevent three situation. Without explaining why, the three situations are:
  1. We don't want Rosh Hashanah to come out on Sunday, Wednesday or Friday (Postponement #2)
  2. We don't want Rosh Hashanah to be on the day of the molad if the molad occurs after the beginning of 18th hour. (Postponement #1)
  3. We want to limit years to specific lengths. For non-leap years, we limit it to either 353, 354 or 355 days.
  4. For leap years, we limit it to either 383, 384 or 385 days. If setting Rosh Hashanah to the day of the molad will cause this year, or the previous yearto fall outside these lengths, we push off Rosh Hashanah to get the year back to a valid length. (Postponements #3 and #4)
http://dafaweek.com/HebCal/HebCalSampleSource.php

Rule #1: Molad Zaken

This rule states that if the molad (the mathematically calculated conjunction of the new moon, not the real conjunction) is at or after noon, then postpone Rosh Hoshanna one day.

The modern explanation for this rule is that it's related to when a new moon would be declared by the Sanhedrin based on witnesses.

Rule #2: Lo ADU Rosh

This prevents Rosh Hoshanna from being on Sunday, Wednesday or Friday. The reason for this rule is so that there are not two sabbaths in a row for Atonement (this is why Wednesday and Friday are forbidden) and that Hoshana Rabbah doesn't fall on a Saturday (which is why Sunday is forbidden).

Rule #3: Gatarad

This prevents the year from being 356 days. We have to push Rosh Hashanah off two days because if we pushed it off only one day, Rosh Hashanah would comes out on a Wednesday. Check the Hebrew year 5745 for an example. http://dafaweek.com/HebCal/HebCalSampleSource.php

Rule #4: Betutkafot

This prevents the previous year from being 382 days. Check the Hebrew Year 5766 for an example. If Rosh Hashanah was not pushed off a day then 5765 would be 382 days http://dafaweek.com/HebCal/HebCalSampleSource.php

Glossary of terms

It's important to know the meaning of a few terms when calculating of the Jewish calendar.

Tishri: A Babylonian name for the 7th month of the religious year, it is the first month in the Jewish civil year. Adopted by Jews during their captivity in Babylon.

Tishri 1: The foundation of the Jewish calendar is the 1st day of Tishri. All the holy days in the Jewish calendar are counted from Tishri 1. Also called Rosh Hoshanna or the Feast of Trumpets.

Halakim / parts: A measurement of time that is equal to 1080 parts per hour, or 3 1/3 seconds per part.

Synodic month: The time it takes the moon to make a complete orbit around the earth.

Conjunction: The astronomical event when the sun, moon and earth are in alignment, generally referred to as the "new moon". This is the beginning and ending of a synodic month.

Leap year: A leap year in the Jewish calendar has 13 months, instead of 12 months.

Metonic cycle: There are 235 synodic months every 19 years (Though it is about 2 hours short). This includes 7 leap years and 12 normal years. This phenomena is named after a Greek astronomer called Meton. Years 3,6,8,11,14,17 and 19 of the 19 year cycle are leap years in the Jewish calendar.

Molad: A Jewish term that generally references the conjunction of the new moon but also the "average time" from one conjuction to the next. The length of time of the molad is 29 days 12 hours and 793 parts. (NASA's measurements of the average time between new moons is about a half second shorter than this time) Literally, the word means "birth".

Molad tohu: According to Jewish tradition this is the first molad, before the world was created and was without form. (tohu) The date for the Molad tohu is Monday 7 September 3761 BCE. In the Jewish calendar this is also called BaHaRaD, which means 2d 5h 204p (2 days 5 hours 204 parts).

Postponement: The Jewish calendar has 4 rules which, when applied, can alter the date of Tishri 1, (aka Rosh Hoshanna/Jewish New Year).


Calculated dates from the code on this page: (Compare to the dates from http://www.timeanddate.com/, https://www.hebcal.com/ and http://www.math.harvard.edu/computing/javascript/Calendar/)

The original source code used here for the calculations has been copied and modified from these sources:

  • http://dafaweek.com/HebCal/HebCalSampleSource.php
  • http://www.jewfaq.org/calendr2.htm

Source for converting the count of days from the molad tohu to a Gregorian date is thanks to this source:

http://hebrewcalendar.tripod.com/gregor.html

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