Units of Time:

Hour is divided into 1080 parts called halaqim (1 halaqim = 3 1/3 seconds)
Abbreviations used here: dy=day, hr=hour, hq=halaqim, mn=minutes, sc=seconds
 

Month:

Starts at new moon, i.e., when moon comes closest to being between earth and sun (molad)
Mean time between molads is 29dy, 12hr, 793hq (29dy 12hr, 44 mn, 3 1/3 sc)
 

Year:

19-year cycle used since 19 solar years is almost exactly 235 lunar months
A cycle consists of 12 common years (12 months) and 7 leap years (13 months)
Leap years occur at 3rd, 6th, 8th, 11th, 14th, 17th, and 19th year in cycle
 

Number of days in month:
 

1. Tishri 2. Heshvan 3. Kislev

30 29 30

4. Tevet 5. Shevat 6. Adar 6a. VeAdar

29 30 29 (*)   0 (*)

7. Nisan 8. Iyyar 9. Sivan

30 29 30

10. Tamuz 11. Ab 12. Elul

29 30 29

   (*) In leap years Adar has 30 days and VeAdar has 29 days
 

Number of days in year:

normal common year = 354 days
normal leap year = 384 days
(1 day deviation for complete or defective years -- see below)
 

Computing molad (new moon) of Tishri for any year:

Day starts at 6 PM (called 0 hr)
Molad of Tishri in year 1 occurred on Monday at 5hr, 204hq (5hr, 11mn, 20 sc)  i.e., evening before Monday daytime at 11 min and 20 sec after 11 PM
(See section below entitled "In the Beginning")

Knowing the time of the first molad Tishri, the mean time between molads,  and the number of molads (months) in a year, the molad Tishri for any given year can be computed.

The following table simplifies this computation:

       1 month =   29dy, 12hr, 793hq (  29dy, 12hr, 44mn,  3 1/3 sc)
      12 months =  354dy,  8hr, 876hq ( 354dy, 8hr, 48mn, 40   sc)
      13 months =  383dy, 21hr, 589hq ( 383dy, 21hr, 32mn, 43 1/3 sc)
     235 months = 6939dy, 16hr, 595hq (6939dy, 16hr, 33mn,  3 1/3 sc)
 

Defective, normal, and complete years:

From the table above, note that 12 months is longer than a common year (by 8hr, 876hq) and 13 months is less than a leap year (by 2hr, 491hq)

To compensate for the error that this would introduce, the number of days in Heshvan and Kislev are adjusted in each year so that start of following year falls on the molad Tishri.  Specifically, if Tishri 1 of the following year would come too late, one day is taken out of Kislev (defective year); if it would come too early, one day is added to Heshvan (complete year).  If neither of these corrections are necessary, the year is called a normal year.
 

number of days in Heshvan Kislev common year leap year

complete year 30 30 355 385

normal year 29 30 354 384

defective year 29 29 353 383

Computing the start of year (Rosh Hashanah):

Sometime a defective or complete year is used not to start the following year on the molad Tishri, but rather to start that following year one or two days beyond the molad Tishri for religious reasons.  For example, it would be undesirable to have certain holidays occur on or adjacent to the Sabbath.

The following rules are used for delaying the start of the year:

(1) If molad Tishri occurs on Sunday, Wednesday, or Friday, Tishri 1 must be delayed by one day for the following reasons:

Wednesday or Friday would cause Yom Kippor (Tishri 10) to fall on Friday or Sunday making it impossible to prepare food (because of Sabbath restrictions) on either the day before or the day after the Yom Kippor fast.

Sunday would cause the seventh day of Succoth (Hoshana Rabba) to fall on the Sabbath.

(3) If molad Tishri in a common year falls on Tuesday at 9 hr 204 hq (i.e., 3:11:20 AM) or later, then Tishri 1 is delayed by one day for the following reason:

Molad Tishri of following year would occur on Saturday at or after 18hr (noon)
Therefore following year must be delayed one day by (2) and then one more day by (1)
This makes the common year in question too long (356 days)

(4) If molad Tishri following a leap year falls on Monday at 15 hr 589 hq (9:32:43 1/3 AM) or later, Tishri 1 is delayed by one day for the following reason:

Molad Tishri of the leap year occurred on or after Tuesday at 18hr (noon)
Therefore Tishri 1 of that leap year was delayed one day by (2) and one more day by (1)
This would make that leap year too short (382 days)

Converting to Julian or Gregorian Dates

The above rules enable the number of days in each year to be calculated starting from creation

The rules for determining number of days in each Gregorian year are simple and well known (365 for common years, 366 for leap years, a leap year is any year that is divisible by 4 except that if it is divisible by 100 it must also be divisible by 400 to be a leap year).  Prior to the Gregorian calendar, the calendar in common use was the Julian calendar.  It differed from the Gregorian one in that it did not have the century rule.  And when the switch was made from the Julian to Gregorian calendar, several days were skipped over to correct the accumulated error to date.

To convert to Julian Calendar dates, the only additional piece of information needed is the Julian Calendar date corresponding to at least one Jewish Calendar date.  Specifically, the Julian Calendar date for Tishri 1, of the year 1 is October 7, 3761 B.C.E. (before the common era).  This will be shown on the Jewish Calendar Converter as -3761.

The Gregorian Calendar date (corrected for the days lost when switching from the Julian calendar) is September 7, 3761 B.C.E.
 

In the Beginning ...

Note that the origin for the above calculations was chosen so that the Molad of Tishri in the year 1 occurred on a Monday.  That seems strange if you believe that creation started on that date, because it would mean that the seventh day (the day of rest) was on a Sunday.

There are actually two incorrect assumptions in the preceding paragraph according to Jewish tradition.  First is that the creation started on Tishri 1.  Another interpretation is that Tishri 1 marks the completion of creation -- namely the first Sabbath.  In other words creation started on the 24th of Elul and ended six days later on the 29th of Elul, the day before Tishri 1.  But that would mean that Tishri 1 in the year 1 should have been a Saturday and not a Monday.

The second incorrect assumption is that the first year was year 1.  Actually year 1 was a fictitious year introduced to make the calculations correct.  Creation started on 24th of Elul in the year 1 (year 1 prior to 24 Elul did not exist) and the first Sabbath was on Tishri 1 in the year 2.  And, indeed, based on the origin chosen for Tishri 1 in the year 1, the calculations have Tishri 1 in the year 2 falling on a Saturday.

This leads to two different ways of counting years.  One is the counting method in common use, starting with the year that never existed.  That method of counting is called Aera Mundi.  The other is the counting method used in the Talmudic and Geonic* period and starts with the year that began upon the completion of creation.  That method of counting is called Aera Adama.

* The Geonic period is the period in Jewish history, approximately 700-1000 CE when the Geonim (the heads of the two Yeshivot, the Jewish centers of learning in Babylon ) were the transmitters of traditional Judaism.

There is yet one more wrinkle to this story.  Tishri 1 in the year 2 falls on a Saturday only if you believe that the year 2 was subject to the four delay rules described above.  In particular, since the molad Tishri of year 2 falls on a Friday, Tishri 1 of that year should have been delayed by rule 1 so that Yom Kippor wouldn't be on the day after the Sabbath.  However Adam and Eve would not yet have sinned as of the start of that year, so there was no predetermined need for them to fast on that first Yom Kippor, and the delay rule would not have been needed.  And if year 2 was not delayed, the Sunday to Friday of creation would not have been from 24-29 Elul but rather from 25 Elul to 1 Tishri.  In other words, Tishri 1 in the year 2 is not the first Sabbath, but rather it is the day that Adam and Eve were created.
 

End of Computations

The above contains all the information necessary for performing the calendar computations.  The following sections, although interesting, have no bearing on the computations but rather are consequences of it.
 

Notation for Designating Year Types

A year can be completely specified by three characters xyz where

   x = day of the week of Rosh Hashanah (1 = Sunday, ..., 7 = Saturday)

   y = denotes defective, normal, complete as follows:
         H (for Haser) = defective
         K (for Kesidra) = normal
         S (for Shalem) = complete

   z = day of the week of Passover

Note that only the following year types are possible:
 

Common Years 2H3 2S5 3K5 5K7 5S1 7H1 7S3

Leap Years 2S7 2H5 3K7 5H1 5S3 7H3 7S5

Date Encodings used on Tombstones

The dates of death found on Jewish tombstones are encoded using a Hebrew equivalent of Roman numerals.  In particular, the encoding is as follows:
 
 

1  Alef	6  Vav	20 Kaf	70 Ayin	300 Shin
2  Bays	7  Zayin	30 Lamed	80 Pay	400 Taf
3  Gimel	8  Khess	40 Mem	90 Tsadi
4  Dalet	9  Tess	50 Nun	100 Kuf
5  Hay	10 Yud	60 Samekh	200 Raish

The encodings from 500 to 900 are:

500: Taf Kuf (400+100)
600: Taf Raish (400+200)
700: Taf Shin (400+300)
800: Taf Taf (400+400)
900: Taf Taf Kaf (400+400+100)

An alternate encoding from 500 to 900 involve the final (sufit) letters as follows.  This is not commonly used on tombstones.

500: final Khaf
600: final Mem
700: final Nun
800: final Fay
900: final Tsadi

Beyond 1000, numbers are broken into two parts separated by an apostrophe.  To the right of the apostrophe is the number of thousands and to the left is the number of units, both using the encoding shown above.  Sometimes the thousands part is omitted completely.

Examples:

  5699 = (from right to left) Hay apostrophe Taf Raish Tsadi Tess
  5761 = (from right to left) Hay apostrophe Taf Shin Samekh Alef

Although dates are usually written in a decimal notation (that is, one character representing the units column, another the tens column, etc.), this rule is sometimes violated just as long as the sum of the characters represents the desired result.  For example, 15 would be written as Yud (10) Hay (5) in decimal notation.  But these two letters are in the name of G-d, so the equivalent Tess (9) Vav (6) is sometimes used.  Same goes for 16 which is sometimes written as Tess (9) Zayin (7).
 

Jewish Calendar Creep

The Jewish calendar is slowly creeping through years.  That is, the start of the Hebrew year is occurring later and later each year.  Relative to the Gregorian calendar, the Jewish calendar is creeping one-day every 237 years.  Since we are nearly in the  year 6000, the calendar has already crept about 25 days since the time of creation.  And in just 40,000 more years it will creep six months so that instead of having Rosh Hashanah in September or October, we will have it in March or April.  I can't wait to see that!  But fear not -- in 80,000 years it will have crept a full year so that by the time the Hebrew year 86000 rolls around, our children's children will once again celebrate Rosh Hashanah in September.

The reason for the creep is that the ratio of the Earth's revolution around the sun (one year) to its rotation on its axis (one day) is not an integer.  The creep is caused by different values for this ratio being used in the Jewish and Gregorian calendars.  Here's the calculation of each ratio:

Jewish Calendar

Based on a 19-year cycle consisting of 235 lunar months. Each lunar month is defined as being 29 days, 12 hours, 793 halaqim.

Therefore the average number of days in a year is:

   (235*(29 days + 12 hours + 793 halaqim)) / 19 =
   (6939 days + 16 hours + 595 halaqim) / 19 =
   6939.6896/19 days =
   365.2468 days

Gregorian Calendar

Based on a 400 year cycle with 365 days each year plus an extra day every 4 years minus no extra day in year 100, 200, and 300.

Therefore the average number of days in a year is:

  (400*365 + 100 - 3)/400 = 365.2425

Difference

Average Jewish year exceeds average Gregorian year by .0042 days.  Therefore Jewish calendar will creep one day every 1/.0042 years which calculates to 238 years.
 

Gregorian Calendar Creep

Even the Gregorian calendar is not correct and indeed the seasons are creeping each year but at a much slower rate than 1 day in 238 years.  The actual ratio, as determined by astronomical calculations, is 365.2422 days.  That means that the average Gregorian year exceeds an astronomical year by .0003 days.  Therefore the seasons in the Gregorian calendar will creep one day every 1/.0003 years which calculates to 3333 years.  So in 600,000 years the accumulated creep will be 180 days and we will have winter in July in the northern hemisphere.  What a chilling thought.
 

Calendar Creep [second thoughts]

The above calculated the Jewish calendar creep relative to the Gregorian calendar.  What makes more sense is to calculate the creep of each calendar independently, relative to the seasons.  Those calculations are as follows:

Jewish Calendar

Based on a 19-year cycle consisting of 235 lunar months. Each lunar month is defined as being 29 days, 12 hours, 793 halaqim.

Therefore the average number of days in a year is:

   (235*(29 days + 12 hours + 793 halaqim)) / 19 =
   (6939 days + 16 hours + 595 halaqim) / 19 =
   6939.6896/19 days =
   365.2468 days

The actual ratio, as determined by astronomical calculations, is 365.2422 days per year.  That means that the average Hebrew year exceeds an astronomical year by .0046 days.  Therefore the Hebrew calendar will creep through the seasons one day every 1/.0046 years which calculates to 217 years.

Secular Calendar

The average number of days in a Julian year is 365.25.  That exceeds an astronomical year by .0078 days, which means a creep of one day every 128 years.

The average number of days in a Gregorian year is (400*365 + 100 - 3)/400 = 365.2425.  That exceeds an astronomical year by .0003 days, which means a creep of one day every 3,333 years.

Lunar Creep

In addition to the yearly creep relative to the seasons, there is also a monthly creep relative to the phases of the moon.

The length of a month in the Jewish calendar is defined as 29 days, 12 hours 793 halaqim which comes to 29.530594135 days.
The average length of a month by astronomical calculations is 29.530588853 days.
The error is .000005282 days/month, or 1 day every 15,776 years.
Every 236,652 years, start of month will creep by 15 days.
So in the year 237,000, the months will start with the full moon instead of the new moon.
 

Jewish Calendar Month versus Muslim Calendar Month

Both the Jewish Calendar and the Muslim Calendar have the months tied to the moon.
So how do the lengths of a month in the two calendars compare?

The Jewish month is defined to be 29 days, 12 hours, 793 halaqim.
   where a haliq (singular of halaqim) is 1/1080 of an hour
Based on that, the length of each year is dynamically adjusted so that the start of each year
   does not drift from the computed start of the first month in the year (Tishri)

The length of a Muslim month is computed as follows:
A normal year is 6 months of 30 days and 6 of 29 days for a total of 354 days
A leap year has an added day, and there are 11 leap years in every 30 year cycle
So 30 years = 354*30+11 = 10631 days
Since every year has 12 months, 30 years = 30*12 = 360 months
Therefore 360 months = 10631 days
which means
   1 month = 10631/360 days = 29 + 191/360 days
                = 29 days + (191/360)*24 hours = 29 days + 191/15 hours
               = 29 days + 12  11/15 hours = 29 days + 12 hours + (11/15)*1080 halaqim
               = 29 days + 12 hours + 792 halaqim

Conclusion: The Jewish Month and the Muslim Month differ by 1 haliq, or 3 1/3 seconds