A month is calculated as 29days, 12 hours, and 793"parts" Leap years occur in years 3,6, 8, 11, 14, 17 and 19 of a 19-year cycle Adjustments (dechiyot)prevent round off the datecalculated Dechiyot prevent oddities inthe length of the year Dechiyot prevent holidaysfrom falling on the wrong dayof the week Some months have variablelengths There are 14 possible formatsof year, identified by codes The calendar is not perfect,but it is very accurate

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Calendar: Closer Look

Calendar EssentialsCalculating the Calendar

Rosh HashanahOther Days

Encoding the YearAccuracy of the CalendarFurther Reading

See also:Current CalendarJewish Calendar

The JewishCalendar: ACloser LookLevel: Intermediate

The basics of the Jewish calendar were explained on theprevious page, and will be mentioned only in passing here.This page is intended for those who are interested in adeeper understanding of the workings of Rabbi Hillel II'sfixed calendar, or those who want to be able to build theirown Jewish calendar computer programs.Although this page will focus primarily on calendarcalculations, I encourage you not to dismiss this as purely amathematical exercise devoid of spiritual value. The sagesemphasized the value of studying astronomy as a way ofappreciating the greatness of the Creator's work. This pagedoes focus on some arcane mathematics, but do not beintimidated by it: the Jewish scholar Rambam wrote that,"the method of the fixed calendar is one which an averageschool child can master in 3 or 4 days." (Hilkhot QiddushHaHodesh 11:4). A lot of the confusion people experiencestems from variations in the way different sources say thesame thing, and the way some sources use familiar terms tomean unfamiliar things. I will do my best to keep thesevariations straight for you.This page includes JavaScript that performs all of thecalculations described, in the order that the concepts arepresented here and with detailed comments. Those who arecomfortable with programming languages may find it faster

and easier to understand the math by looking at the code.This code is not necessarily the best or most efficient scriptpossible, but it's not intended to be; it's intended to illustratehow the calendar is calculated. Despite the inefficiency ofthis code, I have no doubt that it will be appearing on otherwebsites in short order. Would it kill you to give me creditand a link back?

Calendar EssentialsThe Jewish calendar is based on three astronomicalphenomena: the rotation of the Earth about its axis (a day);the revolution of the moon about the Earth (a month); andthe revolution of the Earth about the sun (a year). Thesethree phenomena are independent of each other, so there isno direct correlation between them. On average, the moonrevolves around the Earth in about 29 days. The Earthrevolves around the sun in about 365 days, that is, about12 lunar months and 11 days.To coordinate these three phenomena, and to accommodatecertain ritual requirements, the Jewish calendar consists of12 or 13 months of 29 or 30 days, and can be 353, 354, 355,383, 384 or 385 days long. The keystone of the calendar isthe new moon, referred to in Hebrew as the molad.A new month on the Jewish calendar begins with the molad,(pronounced moh-LAHD). Molad is a Hebrew wordmeaning "birth," and refers to what we call the "new moon"in English. The molad for the month of Tishri (the monththat starts with Rosh Hashanah) is the most important onefor calendar calculations, and is referred to as Molad Tishri.Note that the calculated molad does not necessarilycorrespond precisely to the astronomical new moon. Thelength of time from one astronomical new moon to the nextvaries somewhat because of the eccentric orbits of the Earthand Moon; however, the moladot of Rabbi Hillel's calendarare set using a fixed average length of time: 29 days, 12hours, and 793 "parts" (or in Hebrew, chalakim). Theamount of time is commonly written in an abbreviated form:29d 12h 793p.A "part" (or in Hebrew, cheilek) is a unit of time used in theJewish calendar, equal to 3-1/3 seconds. There are 18 partsin a minute and 1,080 parts in an hour. Most sources expresstime from calendar calculations in days, hours and parts,although some sources break the parts down into minutes.For example, the period between moladot could be writtenas 29 days, 12 hours, 44 minutes and 1 part (29d 12h 44m1p), because 793 parts is 44 minutes and 1 part (793 = 44times 18 parts plus 1 part) . This makes the resulting timeslook somewhat more familiar, but it increases the number ofcalculations, so we will stick with days, hours and parts.

The same shorthand can be used to express the time when amolad occurs. The time is normally expressed as a day of theweek, along with the hours and parts (or hours, minutes andparts). For example, the time of a molad might be expressedas 2d 12h 1005p (or 2d 12h 55m 15p), meaning that itoccurs on Monday (the second day) at the 12th hour and1005 parts.The "hours" used to calculate the molad are standard 1/24 ofa day hours. Note that this differs from the "hours" used forritual scheduling, which are 1/12 of the time from sunrise tosunset. For example, at Pesach (Passover), we are requiredto stop eating chametz at the end of the "fourth hour "of themorning on Nissan 14, that is, at the end of 1/3 of the timebetween sunrise and sunset. These "seasonal hours" varydepending on the time of the year; molad hours are constant.The time for the molad is Jerusalem Solar Time, which isnot necessarily the same as your local time. It is also notnecessarily the same as the time on the clock, even inJerusalem. This fact has no effect on your calculations, but isworth knowing.The Jewish "day" starts at sunset, rather than at midnight. Ifyou read the story of creation in Genesis Ch. 1, you willnotice that it says, "And there was evening, and there wasmorning, one day." From this, we infer that a day beginswith evening, that is, at sunset. Accordingly, most sourcesdiscussing the molad use 6PM of the preceding evening asthe "zero hour." In our example, 2d 12h 1005p, the 12hmeans the 12th hour after 6PM, that is, 6AM. If a moladoccurs at 2d 4h 0p, this means that it occurs at 10PM onSunday night, because the second day (Monday) begins at6PM of the preceding evening (Sunday). Some sources,however, use the more familiar Western conventions and usemidnight as the zero hour. Be very careful to check whichsystem is being used when you rely on times given by anysource! If the time is referred to as "Rambam time" orsomething similar, then you know it uses 6PM as the zerohour. On this page, I am using Rambam time, but some well-respected Orthodox sources in America use midnight as theirzero hour. As long as you are consistent, you will get thesame result under either system.

Calculating the CalendarHere is an overview of the steps involved in calculating thedate of Rosh Hashanah on the Jewish calendar:

1. Start with a known molad (and the correspondingsecular date, if you wish to convert your resulting dateto a secular date).

2. Determine the number of months between the knownmolad and Tishri of the year of the date you arecalculating.

3. Multiply the number of months by the length of themolad: 29d 12h 793p.

4. Add the result to the known starting molad.5. Apply the dechiyot (rules of postponement) to

determine the date of Rosh Hashanah for the year ofyour date.

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

If you want to calculate a date other than Rosh Hashanah,you will have to calculate either that year's Rosh Hashanah,the following year's Rosh Hashanah or both and use thisinformation to work out the date.We will now look at these steps in detail, illustrating thetechniques by calculating the dates of Rosh Hashanah andPesach (Passover) in the year 5775 (2014-2015) using 5732as our starting point. As I said above, if you are comfortablewith JavaScript, you may find that it is faster and easier tounderstand these concepts by viewing my code here. Thiscode is designed to illustrate calendar principles and is notthe most efficient code possible. If you choose to use it inyour own work despite this warning, would it kill you togive me credit and a link back?Step 1: Start with a Known MoladTo perform any calculations on the Jewish calendar, youneed a starting point, preferably the molad of Tishri for aspecific year, along with the corresponding secular date ifyou want to be able to convert the Hebrew date to secular. Itis not possible to work out a molad from first principles,because the first molad of creation (known as Molad Tohu)did not occur at 0d 0h 0p!I like to base my calculations on the molad of Tishri 5732,which occurred at 2d 7h 743p (using 6PM as the zero hour),and corresponded to the secular date September 20, 1971. Iuse this particular year because it is the first Molad Tishriafter 1/1/1970 (Javascript's 0 date) that is not subject todechiyot (postponements), which complicate secular dateconversions. If you will be calculating dates in the past andwould like to avoid the complications of subtracting dates,you may prefer to work with an earlier molad, such as Tishri5661 (9/24/1900), 2d 11h 9p, or even Molad Tishri 5558(9/21/1797), 5d 11h 607p. I'm sure our Christian friends areprimarily interested in knowing Molad Tishri 3762 (the year1), or some other year in that lifetime. Unfortunately, theprogram I use to calculate molads overflows after 3861 (theyear 100), so you'll have to work out the rest yourself:Molad Tishri 3869 (9/22/108, the earliest one I can work outthat is not subject to postponements) is 7d 8h 957p. A moreinteresting base from my perspective is Molad Tishri 4120(9/10/359), 5d 8h 29p, which is the first non-postponed year

after Rabbi Hillel II developed this calendar! Anycalculations before that calendar was developed do notnecessarily correspond to what people in those timesobserved. In addition, it is very complicated to convert aHebrew date to a secular date before the Gregorian calendarreforms, which took effect at different times in differentcountries (introduced in 1582 but not adopted in Americauntil 1752!).Step 2: Determine the Number of Months to Tishri ofYour YearThe next step is to determine how many months are betweenyour starting point and Tishri of the year of your end point.There are exactly 235 months in every 19-year cycle of leapyears (12 12-month years plus 7 13-month years), but if yournumber of years is not evenly divisible by 19, then you willhave to determine whether each remaining year is a regularyear (12 months) or a leap year (13 months).Fortunately, the leap year cycle is easily calculated. Leapyears occur in years 3, 6, 8, 11, 14, 17 and 19 of a 19-yearcycle, and the 19-year cycle begins in the year 1, so you cansimply divide the year number by 19 and examine theremainder. If the remainder is 3, 6, 8, 11, 14, 17 or 0 (the19th year of the cycle) then the year is a leap year.Otherwise, it is not.There are two cycles between 5732 and 5775, with aremainder of 5 years (5775 - 5732 / 19 = 2 remainder 5), thatis 470 months from 5732 to 5770. The remaining monthsbefore 5775 are:

Year Divided by 19 LeapYear? Months

5770 303 remainder13 No 12

5771 303 remainder14 Yes 13

5772 303 remainder15 No 12

5773 303 remainder16 No 12

5774 303 remainder17 Yes 13

Plus 2 cycles (2*235) 470Total 532

Step 3: Multiply the Number of Months by the Length ofthe Molad

Next, we multiply the number of months by the averagelength of the molad, which is 29d 12h 793p:

793p * 532 = 421,876p12h * 532 = 6,384h29d * 532 = 15,428d

Of course, we will then have to round up the smaller unitsinto the larger units, just as we would round 75 minutes into1 hour and 15 minutes. Here are the stages of this rounding:

421,876 parts / 1,080 parts per hour = 390 hoursremainder 676 parts(6,384 multiplied hours + 390 rounded hours) / 24hours per day = 282 days remainder 6 hours15,428 multiplied days + 282 rounded days = 15,710days

We now know the amount of time between our startingmolad and our ending molad: 15,710d 6h 676pStep 4: Add the Result to the Starting MoladNext, we add the elapsed time calculated above to thestarting date to get the ending date. Our starting molad is 2d12h 1005p (using 6PM as 0h). We will not add the days yet,for reasons that will soon become clear.

676 elapsed parts + 743 starting parts = 1,419 parts6 elapsed hours + 7 starting hours = 13 hours

Now we need to do some more rounding:1,419 calculated parts / 1,080 parts per hour = 1 hourremainder 339 parts13 calculated hours + 1 rounded hour / 24 hours perday = 0 days remainder 14 hours15,710 calculated days + 0 rounded days = 15,710days

At this point, we should note the number of days elapsedbetween our starting point and our ending point: 15,710days. We must note this at this point in the calculation, afterthe hours are rounded into the days but before the weekdayof the starting molad is added to the number of days. Thisnumber of days will be necessary to determine the Gregoriandate. Note that if the hours are more than 24 at this point,you will need to round those hours into the days to get theelapsed time. In this calculation, however, the number ofhours is only 14.Let's finish the calculation of the molad, adding the days anddetermining the day of the week:

(15,710 calculated days + 2 starting days) / 7 days per

week = 2244 weeks remainder 4 daysThe remainder of 4 days gives us the day of the week for ourmolad, so the resulting molad is: 4d 14h 339p, that is,Wednesday in the 14th hour (8 am) and 339 parts, with15,710 elapsed days. Note that if the remainder is 0 days, themolad is 7d (Shabbat), because 7 days / 7 days per week = 1week remainder 0 days.Step 5: Apply the DechiyotThere are four rules of postponement known as dechiyot,pronounced d'-khee-YOHT, where "kh" is a throat-clearingnoise (singular: dechiyah). These rules postpone the date ofRosh Hashanah, but do not affect the calculated time of themolad. One of the dechiyot is a general rule of roundingwhile the rest are designed to prevent oddities in the lengthof the year and the date of Rosh Hashanah.Dechiyah 1: Molad ZakeinThe first dechiyah is molad zakein, meaning an "old" molad.If the molad occurs at or after noon (that is, 18h where 6PMis 0h or 12h where midnight is 0h), the molad is consideredto be "old" and we round to the next day. This rule is quitecommonly applied, affecting a quarter of all years (half ifyou use midnight as the 0 hour).If noon seems a bit early to be considering the molad "old,"remember that the Jewish "day" starts at sunset. The rule ofmolad zakein simply means that a molad at or after noonrelates to the "day" that starts at the next sunset (4-10 hourslater) rather than the previous sunset (14-20 hours earlier).This rationale is clear from the Rambam notation, where6PM is 0h and a Molad Zakein is one that occurs at or after18h in a 24h day.Interestingly, Molad Zakein is the reason why you will getthe same result regardless of whether you use 6PM ormidnight as your zero hour. With midnight as your zerohour, Molad Zakein applies to molads after 12h, applying inhalf of all years. With 6PM as your zero hour, Molad Zakeinonly applies in one quarter of all years, but molads between6PM and midnight are already considered to be part of thenext day, so the result is the same!Our molad occurs at 14h in Rambam notation, so it is not amolad zakein and Rosh Hashanah stays on the calculateddate for now.Note that when dechiyot like this apply, a day must be addedto the elapsed time for purposes of calculating the Gregorianequivalent date, but the molad does not change. Theunchanged molad is used for purposes of calculatingsubsequent years and for certain religious purposes. For

example, in 5760, the calculated molad was 6d 21h 801p.Molad Zakein pushed Rosh Hashanah to the next day, but ifyou were to calculate a subsequent date using 5760 as yourbase, you would calculate from 6d (Friday), not fromSaturday. This is why it is best to start with a molad that isnot subject to postponements.Dechiyah 2: Lo A"DU RoshThe second dechiyah is known as Lo A"DU orLo A"DU Rosh. This rule states thatRosh Hashanah cannot occur on a Sunday(Day 1), a Wednesday (Day 4) or a Friday (Day 6). Theword Lo means "Not," and the word A"DU is a way ofpronouncing Alef-Dalet-Vav, letters with the numericalvalues 1, 4 and 6 (seeHebrew Alphabet - Numerical Values). If the calculatedmolad occurs on one of these days of the week, RoshHashanah is postponed by a day to prevent other problemswith the calendar. If Rosh Hashanah fell on a Wednesday orFriday, then Yom Kippur would fall on a Friday or Sunday,which is undesirable. If Rosh Hashanah fell on a Sunday, theHoshanah Rabbah would fall on a Saturday, making itimpossible to observe some of the day's customs.This dechiyah is also commonly applied, as you mightimagine. It applies to three out of seven days, so one wouldexpect it to occur almost half of the time.Note that the dechiyot of molad zakein and Lo A"DU Roshcan work in combination: a molad at 5d 19h 0m 0p(Thursday at 1PM) is rounded to Friday by the rule of MoladZakein, then postponed to Saturday by the rule of Lo A"DURosh, even though the original molad was on a valid day ofthe week. On the other hand, a molad at 4d 19h 0m 0p(Wednesday at 1PM) is rounded to Thursday by MoladZakein, and Lo A"DU Rosh does not apply: even though themolad occurred on Wednesday, Molad Zakein has alreadymoved it off of that date so Lo A"DU Rosh is not necessary.This is why the rule of Molad Zakein must be checkedbefore the rule of Lo A"DU Rosh.Our molad occurs on 4d, Wednesday, so it is postponed toThursday. You should add 1 to your elapsed time so yoursecular date conversion will be correct. There are now15,711 elapsed days.Dechiyah 3: GataradThe remaining two dechiyot are much less commonlyapplied.Dechiyah Gatarad holds that if Molad Tishri in a simple (12-month, non-leap) year occurs on a Tuesday at 9h 204p orlater, Rosh Hashanah is postponed to the next day (a

Wednesday, which by the effect of Lo A"DU Rosh wouldthen be postponed to Thursday).The name, Gatarad, is a mnemonic for therule. In Hebrew, Gatarad it is spelledGimel-Teit-Reish-Dalet. Using letters asnumerals, Gimel is 3, and represents Tuesday. Teit is 9 andrepresents the 9th hour (that is, 9h in Rambam notation, but3h in midnight-based notation). Reish is 200 and Dalet is 4,representing 204 parts.Why does such a complicated rule exist? This rule preventsthe possibility that a year might be 356 days, an invalidlength. Consider: a Molad Tishri at 3d 9h 204p would not bepostponed by Molad Zakein or Lo A"DU Rosh. Add 12lunar cycles (354d 8h 876p) to the next year's RoshHashanah and you get 7d 18h 0p with 354 days elapsed.Molad Zakein applies to the following year, postponingRosh Hashanah to the next day, a Sunday, with 355 dayselapsed. Lo A"DU Rosh is then triggered, postponing RoshHashanah and leaving 356 days elapsed and making thecurrent year an invalid length. Gatarad takes days away fromthe following year and adds them to the preceding year, soboth years are a valid length.Note that Gatarad invariably triggers Lo A"DU Rosh.Gatarad only applies when Rosh Hashanah is Tuesday andGatarad postpones Rosh Hashanah to Wednesday. Lo A"DURosh then postpones Rosh Hashanah to Thursday. Someprogrammers like to check Gatarad before checking LoA"DU Rosh; others check Gatarad after Lo A"DU Rosh butuse this rule to add two days (the Gatarad day plus theresulting Lo A"DU Rosh day). Either way, if Gataradapplies, Rosh Hashanah falls on Thursday.Note also that this rule is not combined with Molad Zakein.If Molad Zakein applies to the current year, Gatarad isunnecessary; thus Gatarad applies only to molads between9h 204p and 17h 1079p.As you might imagine, this rule is not commonly applied. Itapplies only in non-leap years (12 out of 19 years) when themolad occurs on Tuesday (1 out of 7 days) between the 9thhour and the 18th hour (9 out of 24 hours). It occurs aboutthree times a century. It last occurred in 5745 (1984-85) andwill not occur again until 5796 (2035-36).Dechiyah 4: BetutkafotLike Dechiyah Gatarad, this rule is not very commonlyapplied and is designed to prevent a year from having aninvalid length. Dechiyah Betutkafot prevents a leap-yearfrom having 382 days (too few days) by postponing RoshHashanah of the non-leap year following the leap year.

Also like Dechiyah Gatarad, thename of the rule tells you howit is calculated: if Molad Tishriin a year following a leap year occurs on Monday (Beit, 2)after the 15th hour (Teit-Vav, 15 in Rambam notation, but 9hin midnight-based notation) and 589 parts (Tav-Kaf-Pei-Teit,589), then it is postponed to the next day. The rule is appliedonly if the actual molad occurs on Monday, not if it ispostponed to Monday. Like Gatarad, the rule really onlyapplies to molads before noon (18h), because Molad Zakeinhandles the postponements for molads at or after noon.Unlike Gatarad, Betutkafot does not trigger Lo A"DU Rosh,because Betutkafot postpones Rosh Hashanah from aMonday to a Tuesday and Tuesday is an acceptable day forRosh Hashanah.The reasoning behind this rule is similar to the reasoningbehind Gatarad: the 13 lunar cycles of the preceding year are383d 21h 589p. If this year's Molad Tishri occurs after 2d15h 589p, then the preceding year's Molad Tishri must haveoccurred on or after 3d 18h 0p. This is 384 elapsed days, butthe preceding year's Molad Tishri was a Molad Zakeinpostponing Rosh Hashanah to Wednesday, which triggers LoA"DU, moving Rosh Hashanah to Thursday. The twopostponements shorten the preceding year to 382 days.Dechiyah Betutkafot postpones the current year's RoshHashanah by one day to increase the preceding year to apermissible 383 days.This is the rarest of the four dechiyot, applying only in theyear after a leap year (7 out of 19 years) when the moladoccurs on Monday (1 out of 7 days) between the 15th hourand the 18th hour (3 out of 24 hours). It applies once ortwice a century. The last time it applied was 5766 (2005-06).It will not apply again until 6013 (2252-2253)!Step 6: Add Elapsed Days to Gregorian Starting DateTo determine the Gregorian date for Rosh Hashanah, youmust take the elapsed days calculated in Step 4, add anyadditional days triggered by the dechiyot in Step 5, and addthis number of days to the date of Rosh Hashanah for yourknown molad. For those following the script: JavaScriptdoesn't do a great job of handling dates, but the ConvertGregfunction adds elapsed dates and returns a properly formatteddate (American m/d/yyyy format). If you're trying to do thiswithout writing a program, a spreadsheet such as MicrosoftExcel should be able to add a number of days to a date.

Rosh Hashanah Calculator

To calculate the date of Rosh Hashanah for any year after5732 using the principles above:

1. Calculate the number of months between 5732 andyour year.

2. Calculate the amount of time elapsed in those months.3. Add the elapsed time to the molad of 5732 to

determine the molad of your year, stopping to note theelapsed days before adding the day of week from the5732 molad.

4. Determine whether any dechiyot apply and if so, addthem to the elapsed days determined above.

5. Add the days elapsed to the date of Rosh Hashanah in5732.

These steps are performed by the function CalcRH in theJavaScript, used by the form below. Just type the Hebrewyear and the secular date will appear, using only thefunctions discussed above. Click the button below to try it!

Hebrew Year 5772 Get Date!Secular Date

Calculating Days Other Than Rosh Hashanah

The principles and JavaScript above are sufficient to allowyou to convert Rosh Hashanah to a Gregorian date for anyyear. However, if you want to calculate a date other thanRosh Hashanah, you will have to calculate either that year'sRosh Hashanah, the following year's Rosh Hashanah or bothand use this information to work out the date. Theinformation you need varies depending on the month of thedate you are calculatingTishri

Tishri is the month of Rosh Hashanah, so you simplyadd the date of the month to Rosh Hashanah andsubtract 1 (because Rosh Hashanah is Day 1).

CheshvanCheshvan is the second month of the calendar year,and the preceding month of Tishri is always 30 days,so you simply take the current Rosh Hashanah, add 29days (30 - 1 for Rosh Hashanah) and add the date ofthe month.

KislevKislev is the hardest month to calculate. You cannotsimply work forward from the current year's RoshHashanah, because the preceding month of Cheshvancan be 29 or 30 days, nor can you work backwardfrom the next year's Rosh Hashanah, because Kislevitself can also be 29 or 30 days. To calculate the lengthof Kislev, you need to know the date of RoshHashanah of both the current year and the next year,then calculate the difference between them todetermine the length of the current year. If the year is

353, 354, 383 or 384 days, then Cheshvan is 29 daysand you can determine a date in Kislev taking thecurrent Rosh Hashanah, adding 58 days, then addingthe date of the month. If the year is 355 or 385 days,then Cheshvan is 30 days and you can determine adate in Kislev by taking the current Rosh Hashanah,adding 59 days, then adding the date of the month.

Tevet, ShevatThe remaining months of the year are of unchanginglength, but the number of months varies depending onwhether the year is a leap year! Tevet and Shevat arebest calculated by working backwards from thefollowing year's Rosh Hashanah and subtracting anadditional 30 days in a leap year. Tevet's offset in anon-leap year is -266; Shevat's is -237.

Adar, Adar I and Adar IIAdar is always offset -207 from the following RoshHashanah; however, in regular years, Adar is the 12thmonth of the year (starting from Nissan), and in leapyears, is known as Adar II and is the 13th month ofthe year. Adar I, the extra month inserted as the 12thmonth in leap years, is always offset -237 days fromRosh Hashanah.

Nissan, Iyar, Sivan, Tammuz, Av, ElulThe remaining months of the year are all ofunchanging length and not affected by leap years.Simply subtract the appropriate number of days fromthe following year's Rosh Hashanah and add the dateof the month.

The form below uses the functions above to calculate thedates of major Jewish holidays for any Hebrew year.

Holiday Hebrew YearRosh Hashanah (1 Tishri) 5772

Get Date!

Encoding the YearCalendar scholars use a system of encoding to describe eachJewish year. This encoding consists of three Hebrew lettersthat serve as a shorthand for important features of thecalendar, and once you work out the code, you knoweverything you need to know about the calendar. You don'tneed to know the encoding system to be able to calculate thecalendar, but it may help you understand important featuresof the calendar.The first letter is either Pei or Mem. Pei stands for theHebrew word P'shuta (simple), and refers to a 12-monthregular year. Mem stands for Me'uberet, and refers to a 13-month leap year.

The second letter indicates which day of the week RoshHashanah occurs. Letters of the Hebrew alphabet also serveas numerals (see Hebrew Alphabet - Numerical Values), andthis letter indicates whether Rosh Hashanah occurs on aMonday (Beit, that is, 2), a Tuesday (Gimel, 3), a Thursday(Hei, 5) or a Saturday (Zayin, 7). Why not Alef (1), Dalet (4)or Vav (6)? Because Dechiyah Lo A"DU Rosh, discussedabove, prevents Rosh Hashanah from falling on Sunday,Wednesday or Friday.The third letter tells you the length of the year, which can be353, 354 or 355 days (in a leap year, 383, 384 or 385). Thisvariation comes in part from the length of the molad cycles(which add about 8 or 21 hours to the time of day each year,which sometimes rolls over to another day) and in part fromthe application of the dechiyot. A year's length can beencoded as Cheit for Chaseir (deficient or lacking, a 353 or383 day year), Kaf for K'Seder (in order, a 354 or 384 dayyear) or Shin for Shaleim (whole or complete, a 355 or 385day year). In a Chaseir year, both Cheshvan and Kislev have29 days. In a Shaleim year, both Cheshvan and Kislev have30 days. In a K'Seder year, Cheshvan has 29 days and Kislevhas 30 days.Under this system of encoding, the current year (5765) iscoded Mem-Hei-Cheit, because it is a leap year (Mem),Rosh Hashanah started on a Thursday (Hei), and the yearwill have 383 days (Cheit). Next year (5766) would beencoded as Pei-Gimel-Kaf because it will be a regular (non-leap) year, it will start on a Tuesday, and it will have 354days.Some people code the years differently: the day of RoshHashanah as the first letter (instead of the second), the lengthof the year as the second letter (instead of the third), and theday of the week that Pesach (Passover) starts as the third.This third letter can be Alef (1, Sunday) through Zayin (7,Saturday). The advantage of this system is that it tells youthe day of the week that both Pesach and Rosh Hashanahoccur, which has some effect on their observances, and oncethese are known, we can infer the days of the other majorfestivals (Sukkot and Shavu'ot). The disadvantage is thatnothing in this system tells you whether the year is a leapyear, although this can be inferred if you know the calendarwell enough.Although there are many theoretical permutations of thesethree-letter codes, only 14 of them are actually possiblegiven the constraints of calendar calculations. This meansthat there are only 14 different possible layouts for an annualJewish calendar. Keep in mind, though, that these 14different layouts don't necessarily correspond to the sameGregorian days, but they do correspond to the distribution ofweekly Torah readings. For example, in the year 5765, a

Mem-Hei-Cheit year, the Torah portion Emor was read on 5Iyar, which was May 14, 2005. The next Mem-Hei-Cheityear will be 5768, and Parshat Emor will be read on 5 Iyar inthat year too, but 5 Iyar will occur on May 10, 2008.Contrast this with 5766, a Pei-Gimel-Kaf year, when Emorwill be read on 15 Iyar (May 13, 2006).The following table shows which parshiyot are combined inwhich year encodings:

Year Encoding Parshiyot CombinedMem-Beit-Cheit Chukat-Balak

Matot-MaseiNitzavim-Vayeilech

Mem-Beit-Shin Matot-MaseiMem-Gimel-Kaf Matot-MaseiMem-Hei-Cheit noneMem-Hei-Shin Nitzavim-VayeilechMem-Zayin-Cheit Matot-Masei

Nitzavim-VayeilechMem-Zayin-Shin Chukat-Balak

Matot-MaseiNitzavim-Vayeilech

Pei-Beit-Cheit Vayakhel-PekudeiTazria-MetzoraAchrei Mot-KedoshimBehar-BechukotaiMatot-MaseiNitzavim-Vayeilech

Pei-Beit-Shin Vayakhel-PekudeiTazria-MetzoraAchrei Mot-KedoshimBehar-BechukotaiChukat-BalakMatot-MaseiNitzavim-Vayeilech

Pei-Gimel-Kaf Vayakhel-PekudeiTazria-MetzoraAchrei Mot-KedoshimBehar-BechukotaiChukat-BalakMatot-MaseiNitzavim-Vayeilech

Pei-Hei-Kaf Vayakhel-PekudeiTazria-MetzoraAchrei Mot-KedoshimBehar-BechukotaiMatot-Masei

Pei-Hei-Shin Tazria-MetzoraAchrei Mot-KedoshimBehar-BechukotaiMatot-Masei

Pei-Zayin-Cheit Vayakhel-PekudeiTazria-MetzoraAchrei Mot-KedoshimBehar-BechukotaiMatot-Masei

Pei-Zayin-Shin Vayakhel-PekudeiTazria-MetzoraAchrei Mot-KedoshimBehar-BechukotaiMatot-MaseiNitzavim-Vayeilech

Accuracy of the Jewish CalendarAt one time, the accuracy of the Jewish calendar wasproverbial. But how accurate is it really?The average lunar month on the Jewish calendar is 29d 12h793p. The average lunar month as calculated by modernastronomers is 29d 12h 44m 2.8s, that is, 29d 12h 792.84p.so the variation is less than two tenths of the smallest unit ofmeasurement recognized by the system, about half of asecond. That is quite remarkably accurate. Of course, thoselost half-seconds do add up: within in a century, you're offby 10 minutes.How well does the calendar correspond to the solar year?The rabbis recognized long ago that the calendar gains 1h485p in every 19-year cycle, adding up to a day every 300years or so. This was important to the rabbis in schedulingcertain rituals that are based on the solar year rather than thelunar year. We can see this effect when we examine the datesof Rosh Hashanah over time.Rabbi Hillel II developed the Jewish calendar in the Jewishyear 4119. Using his calendar methods as described above,and artificially assuming that the Gregorian calendar we usetoday was in effect at that time, the date of Rosh Hashanahranged from August 29 to September 28 between the years4100 and 4200 (the 42nd century). In the present Jewishcentury (the 58th), the dates of Rosh Hashanah range fromSeptember 5 to October 5, a gain of 6 or 7 days. This isconsiderably more accurate than the Julian calendar used byChristians in Rabbi Hillel's time (which had to be correctedby 11 days a few centuries ago), but you can see that it isgaining some time.The discrepancy in the Jewish calendar, however, is still lessthan a lunar month and is therefore as accurate as it is

possible to be in a lunisolar calendar. In fact, it takes about9300 years for this discrepancy to accumulate to a fullmonth of time. The rabbis were aware of the problem, butwere quite confident that a new Sanhedrin will beestablished long before this discrepancy becomesproblematic. We still have more than 3500 years to go.

Suggestions for Further ReadingThe book that most people recommend for learning aboutthe Jewish calendar is Rabbi Nathan Bushwick's Understanding the Jewish Calendar. I ordered this bookwhile I was writing this page; it took about a month toarrive, and I confess I was a bit disappointed by it. Abouthalf of the book was basic astronomy that I learned in fourthgrade, and most of the calendar calculations I had learnedbefore the book arrived. Nevertheless, the book did havesome interesting insights and thorough citation to Torah,Talmud and Rambam that you may find useful or interesting.

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