The Hebrew calendar is constructed from the simple multiple of a constant value for the mean lunation period. The mean lunation period is an averaging of the time taken by the moon to cycle from lunar conjunction to lunar conjunction. The value used for the constant is 29d 12h 793p.

Hebrew years can be either 12 or 13 lunation periods.

Years of 12 lunation periods are usually called common years. Years of 13 lunation periods are usually called leap years.

Hence, the lunar length of all common years is

        12 * (29d 12h 793p) = 354d  8h 876p

and    the lunar length of all leap   years is

        13 * (29d 12h 793p) = 383d 21h 589p.

Hebrew years are arranged in a cycle of 19 years consisting of 12 common and 7 leap years. This cycle is known as a mahzor katan, ie, a small cycle, and is exactly 235 months long.

The mean lunar length of any 19 consecutive Hebrew years is
235 * (29d 12h 793p) = 6939d 16h 595p.

Let L(i) represent the mean lunar length of Hebrew year i.

Then the value of L(i) can be either 354d 8h 876p, if year i is common,
or 383d 21h 589p if year i is leap.

Since the common and leap years are arranged in a 19 year cycle, L(i) = L(i + 19*k) for any value of i and k.

The fixed Hebrew calendar has one, and only one, order for the arrangement of its common and leap years. The apparent order in which the leap and common years occur in a given cycle of 19 consecutive years depends entirely on which element of the arrangement pattern is selected as first.

Allowing L to represent a leap year and c to represent a common year, then the arrangement of leap years and common years, extracted from some starting year in a Hebrew year time line can be seen to be

If the pattern had started with the 4th year of the above time line then the arrangement from the first year of the pattern would appear to be

Looking at the 19 year cyclical arrangement of the leap years beginning from the first year of each pattern, the 19 year cycles appear to be

the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of the 19 year cycle

for the first selection and

the 3rd, 5th, 8th, 11th, 14th, 16th and 19th years of the 19 year cycle

for the second selection.

For some Hebrew year i let C(i) be a cycle of 19 consecutive Hebrew year lengths beginning with the year length L(i).

Then, C(i) = {L(i), L(i+1), L(i+2), ... , L(i+18)} and C(i+19*k) = C(i) since L(i+19*k) = L(i).

For some Hebrew year j let i-j = d.

Then L(j) = L(i-d) and C(j) = {L(i-d), L(i-d+1), L(i-d+2), ... , L(i-d+18)} = C(i-d).

The sequence of elements started at L(j) is the same as the sequence of elements started at L(i-d). Unless d is a multiple of 19, C(j) will appear to be different than C(i).

Since L(i) = L(i + 19*k) for any value of i and k, 19 apparent distributions exist for the common and leap years in a mahzor katan.

It is common practice today to associate C(1) with a 19 year cycle in which the order of leap years is the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th year of the cycle. This cycle is known by the acronym GUChADZaT, which is formed from the Hebrew letters gimel-vov-het-alef-daled-zayin-tet. The Hebrew letters also symbolize the digits 3, 6, 8, 1, 4, 7, and 9.

The table below shows the 19 intercalation patterns which can be observed relative to C(1).

  1  4  7  9 12 15 18

  2  3  3  2  3  3  3

 3d  7h  695p

  3  6  8 11 14 17 19

  3  3  2  3  3  3  2

 2d  5h  204p

  2  5  7 10 13 16 18

  3  3  2  3  3  3  2

 6d 14h    0p

  1  4  6  9 12 15 17

  3  3  2  3  3  3  2

 3d 22h  876p

  3  5  8 11 14 16 19

  3  2  3  3  3  2  3

 2d 20h  385p

  2  4  7 10 13 15 18

  3  2  3  3  3  2  3

 0d  5h  181p

  1  3  6  9 12 14 17

  3  2  3  3  3  2  3

 4d 13h 1057p

  2  5  8 11 13 16 19

  2  3  3  3  2  3  3

 3d 11h  566p

  1  4  7 10 12 15 18

  2  3  3  3  2  3  3

 0d 20h  362p

  3  6  9 11 14 17 19

  3  3  3  2  3  3  2

 6d 17h  951p

  2  5  8 10 13 16 18

  3  3  3  2  3  3  2

 4d  2h  747p

  1  4  7  9 12 15 17

  3  3  3  2  3  3  2

 1d 11h  543p

  3  6  8 11 14 16 19

  3  3  2  3  3  2  3

 0d  9h   52p

  2  5  7 10 13 15 18

  3  3  2  3  3  2  3

 4d 17h  928p

  1  4  6  9 12 14 17

  3  3  2  3  3  2  3

 2d  2h  724p

  3  5  8 11 13 16 19

  3  2  3  3  2  3  3

 1d  0h  233p

  2  4  7 10 12 15 18

  3  2  3  3  2  3  3

 5d  9h   29p

  1  3  6  9 11 14 17

  3  2  3  3  2  3  3

 2d 17h  905p

  2  5  8 10 13 16 19

  2  3  3  2  3  3  3

 1d 15h  414p

  1  4  7  9 12 15 18

  2  3  3  2  3  3  3

 6d  0h  210p

Note that C(19) = C(0).

The relative order cyclical pattern is formed from the number of years that have elapsed since the previous leap year.

The relative order for C(4) is shown to be 3 2 3 3 3 2 3. In Hebrew this numreic pattern is represented by letters gimel-bet-tet-bet-gimel. Obviously, tet representing 9, reflects 3+3+3.

Since there are 7 leap years in a mahzor katan, only 7 relative order patterns are produced.Hence, specifying a cycle by its relative order does not uniquely identify the cycle. For example, it is to be noted that gimel-bet-tet-bet-gimel is also a relative order for C(5) and C(6).

The table also includes the Epochal Moladot which are associated with the first 20 years of Yetsirah, that is, Creation, or Genesis, or Aera Mundi.

1. Pages 91-95 of Studies In Hebrew Astronomy and Mathematics by Solomon Gandz, KTAV Publishing House, Inc. (1970), show a study of the calendar cycles as discussed in 3 medieval sources. These are the Seder Olam, the Yesod Olam, and the Pirke di Rabbi Eliezer (PdRE).

In this discussion, Gandz analyzes ancient rabbinic sources which debate the issues of whether the mahzor katan ought to be either C(1), C(2), C(3), C(4), or C(12). Gandz does not dicuss the fact that cycles expressed in the relative order notation can refer to more than one leap year intercalation cycle.

Gandz's text also indicates the existence of a third manner of noting the leap year cycles. It seems to be the relative order notation, but with the first entry indicating the number of years elapsed from the first year of the cycle.

2. On page 48, under the subheading Historicalof the article Calendar, in the Encyclopedia Judaica (1982), the cycles C(1), C(2), C(3) and C(4) are shown together with their Hebrew acronyms. The article identifies the epochal moladot associated with C(1), C(2), and C(3) to be
2d 5h 204p, 6d 14h, and 3d 22h 876p respectively.

The article indicates that 4d 20h 408p (named daled-hof-tof-het) is also an epochal molad. This is a puzzling detail, since 4d 20h 408p represents the molad of Heshvan 0H. Since the article shows C(4), the value 2d 20h 385p would have been expected, since it represents the value of the molad of Tishrei 4H.

3. Pages 64-66 of Al-Biruni's The Chronology of Ancient Nations (1000g) as translated into English by Dr. E.C. Sachau in 1879g, indicate that only 3 leap year cycles were in use at about 1000g (4761H). These cycles were C(2) and C(3) "used by the Jews of Syria", and C(4) "most extensively diffused among the Jews; ... because they attribute its invention to Babylonia."

Although the cycle C(1) appears in documents that predate Al-Biruni's work by several centuries, it is not mentioned at all in the Chronology.

4. Chapters 6-8 of Maimonides' Hilchot Kiddush HaChodesh (1175g) describe the calculation of the Hebrew calendar using C(1) and BaHaRaD. Maimonides' instructions still remain in use today.

Let M(i) represent the known value of the molad of Tishrei for some year i.

Then M(i+1) = M(i) + L(i) is the molad of Tishrei for the subsequent year i+1.

Hence, M(i+1) - M(i) = L(i)

Similarly,

     M(i+2) - M(i + 1) = L(i + 1)
     M(i+3) - M(i + 2) = L(i + 2)

and by induction,

     M(i+k) - M(i+k-1) = L(i + k - 1)

Summing the above formulas from i+1 to i+k,

     M(i+k) - M(i) = L(i) + L(i+1) + L(i+2) + ... + L(i+k-1)
===> M(i+k) = M(i) + L(i) + L(i+1) + L(i+2) + ... + L(i+k-1)

Defining LSUM(i,k) to be the sum of the lunar lengths of k consecutive years beginning with the length of year i,

        LSUM(i,k) = L(i) + L(i+1) + L(i+2) + ... + L(i+k-1)
===>       M(i+k) = M(i) + LSUM(i,k)    

Since L(i) = L(i+19*k) for any i and k, every sequence of 19 consecutive Hebrew years has the mean lunar length 6939d 16h 595p.

Therefore, LSUM(i,19*j) = (6939d 16h 595p)*j

Expressing k in the form 19*j + r

M(i+k) = M(i  + 19*j  + r) 
       = M(i  + 19*j) + LSUM(i + 19*j, r)
       = M(i) + LSUM(i, 19*j) + LSUM(i,r)
       = M(i) + (6939d 16h 595p)*j + LSUM(i,r)

It is therefore not required to calculate LSUM(i,r) for more than the first 18 terms.

The expression for LSUM(i,r) represents the sum of the first r terms of C(i). Consequently, the evaluation of LSUM(i,r) is dependent on the order of the values of the length elements in C(i).

Before showing the tables for LSUM(i,r) these arithmetical reductions need to be explained.

The moladot are normally expressed in units representing the day of the week, the hour, and the number of halakim.

In traditional calculations, the moladot are remaindered using 1080 as the divisor. The quotient is then added to the hours. The hours are then remaindered by 24 and the quotient added to the days. Finally, the days are remaindered by 7. If the remainder for the day is zero, it is sometimes expressed as 7 depending on the context. Traditional texts show no entries for the halakim or the hours if their corresponding value after remaindering is zero.

A 19 Hebrew year cycle (mahzor katan) consists of 235 months. Since one month has the lunation period of 29d 12h 793p, 235 months have

235 * (29d 12h 793p) = 6815d 2820h  186,355p 
                     = 6815d 2820h+172h 595p 
                     = 6815d 2992h      595p
                     = 6815d+124d   16h 595p  
                     = 6939d        16h 595p
                     ==>  2d        16h 595p [after remaindering by 7d]

Consequently, it is sometimes more convenient to compute

        M(i+k) = M(i) + (6939d 16h 595p)*j + LSUM(i,r)

in the reduced form

        M(i+k) = M(i) + (2d 16h 595p)*j + LSUM(i,r)

M(i+0)    = M(i) + LSUM(i,0) [Rel'n 3]
M(i)      = M(i) + LSUM(i,0)       
LSUM(i,0) = 0

LSUM(i+19*j,k) = L(i+19*j)+L(i+1+19*j)+L(i+2+19*j) ... +L(i+k-1+19*j)
               = L(i)+L(i+1)+L(i+2)+ ... +L(i+k-1) [Rel'n 1]
               = LSUM(i,k)                         [Definition]

   M(5800) =    M(5783)   +  LSUM(7 , 17) [Rel'n 3 & 5]
           =   (2d 3h 6p) + (6d 10h 210p)
           ==> 1d 13h 216p  [after the traditonal reduction by 7d].

     M(5758)   = M(1)         + LSUM(1, 5757)      [Rel'n 3]
               = M(1)         + LSUM(1, 303*19)    
             ==> M(1)         + (2d 16h 595p)*303  [Rel'n 6]
             ==> (2d 5h 204p) + (2d 22h 1005p)
               =  5d 4h 129p  [after traditional reductions]

M(i+19*j+r) = M(i+19*j) + LSUM(i+19*j,r)                 [Rel'n 3]
            = M(i)      + LSUM(i,19*j)       + LSUM(i,r) [Rel'n 3 & 5]
            = M(i)      + (6939d 16h 595p)*j + LSUM(i,r) [Rel'n 6]
          ==> M(i)      + (   2d 16h 595p)*j + LSUM(i,r) [Trad'l Red'n]

Since,           M(i+k) = M(k+i)
       M(i) + LSUM(i,k) = M(k) + LSUM(k,i) [Rel'n 3]

     M(5759) = M(1+5758)
             = M(1)         +  LSUM(1      ,  5758        )     [Rel'n 3]
             = M(1)         +  LSUM(1      ,  303*19 + 1  )
             = M(1)         +  LSUM(1,1)   + (2d 16h 595p )*303 [Rel'n 7]
             = (2d 5h 204p) + (4d 8h 876p) + (2d 22h 1005p)
             =  2d 12h 1005p  [after traditional reductions]

     M(5759) = M(5758+1) 
             = M(5758)      +  LSUM(5758,1)                     [Rel'n 3]
             = M(5758)      +  LSUM(1+303*19,1)
             = M(5758)      +  LSUM(1,1)                        [Rel'n 5]
             = (5d 4h 129p) + (4d 8h 876p)
             =  2d 12h 1005p  [after traditional reductions]

       M(i) = M( j + i-j)
            = M( j ) + LSUM( j , i-j ) [Rel'n 3]
Hence, M(j) = M( i ) - LSUM( j , i-j )

M(i+k) = M(i+j+k-j) 
       = M(j+k+i-j)
       = M(j)+LSUM(j,k+i-j)               [Rel'n 3]
       = M(i)-LSUM(j,i-j  )+LSUM(j,k+i-j) [Rel'n 9 substitution of M(j)]

Since 5775 = 5760 + 15, and 5760 = 19*303 + 3,

M(5760+15) = M(5760) - LSUM(0,5760-0) + LSUM(0,15+5760-0)     [Rel'n 10]
           = M(5760) - LSUM(0,19*303+3) + LSUM(0,15+19*303+3)
           = M(5760) - LSUM(0, 3) - 303*(2d 16h 595p) 
                     + LSUM(0,18) + 303*(2d 16h 595p)         [Rel'n 7]
           = M(5760) - LSUM(0, 3) + LSUM(0,18)
           = (6d 21h 801p) - (0d 15h 181p) + (5d 7h 799p)
           =  4d 14h 339p    [after the traditional reductions]

         M(i) - LSUM(j,19*m+i-j) + LSUM(j,k+19*m+i-j)
       = M(i) - LSUM(j,i-j  ) - m*(6939d 16h 595p)
              + LSUM(j,k+i-j) + m*(6939d 16h 595p)     [Rel'n 7]
       = M(i) - LSUM(j,i-j  ) + LSUM(j,k+i-j)
       = M(i+k)                                        [Rel'n 10]

Therefore,

M(i+k) = M(i) - LSUM(j,19*m+i-j) + LSUM(j,k+19*m+i-j)

Since 5775 = 5760 + 15, and 5767 = 19*303 + 10,

M(5775) = M(5760+15)
        = M(5760) - LSUM(5767,5760-5767) 
                  + LSUM(5767,15+5760-5767)                  [Rel'n 10]
        = M(5760) - LSUM(10,  -7)+LSUM(10, 15 -7)            [Rel'n  5]
        = M(5760) - LSUM(10,19-7)+LSUM(10,15 +19-7)          [Rel'n 11]
        = M(5760) - LSUM(10, 12 )+LSUM(10, 27 )           
        = M(5760) - LSUM(10, 12) +LSUM(10, 8) + (2d 16h 595p) [Rel'n 7]
        = (6d 21h 801p) - (2d 12h 724p) + (4d 12h 747p) + (2d 16h 595p)
        = 4d 14h 339p     [after the traditional reductions]

M(5775) = M(5758+17) 
        = M(5758)      + LSUM(19*303+1, 17)
        = M(5758)      + LSUM(1, 17)                       [Rel'n 5]
        = M(5758)      + INT((235*17+1)/19)*(29d 12h 793p)
        = (5d 4h 129p) + 210*(29d 12h 793p)
        = (5d 4h 129p) + (6090d 2520h 166530p)
        = 4d 14h 339p  [after the traditional reductions]

M(5775) = M(5770) - LSUM(1, 5770 - 1)   + LSUM(1 ,5 + 5770 - 1) [Rel'n 10]
        = M(5770) - LSUM(1,19*303+13-1) + LSUM(1,5+19*303+13-1)           
        = M(5770) - LSUM(1, 13 - 1)     + LSUM(1 , 5 + 13 - 1 ) [Rel'n 11]
        = M(5770) - (INT((235*12 +1)/19) - INT((235*17 + 1)/19)) * (29d 12h 793p)
        = (0d 16h 853p) - ( 148 - 210 ) * (29d 12h 793p)
        = (0d 16h 853p) +  62*(29d 12h 793p)
        = (0d 16h 853p) + (1798d 744h 49166p)
        =  4d 14h 339p  [after the traditional reductions]

     M(i+k) = M(i) + L(i) + L(i+1) + L(i+2) + ... + L(i+k-1)

1. some value for year i
2. some value M(i) for its corresponding molad of Tishrei
3. some intercalation pattern C(i) beginning at year i 

1. i = 1
2. M(1) = 2d 5h 204p 
3. the intercalation pattern C(1) known as GUChADZaT 

First  Paged 30 Jan 2000
Next Revised 23 Jan 2002