Properties of Hebrew Year Periods
Properties of Hebrew Year Periods - Part 2

by Remy Landau

21. The Number of Months in Hebrew Year Periods
 21a. Leap Year Counts Relative to GUChADZaT
 21b. Leap Year Counts Possible
 22. Calculating the Number of Months
 22a. Examples Using Months Formulas 22.1 and 22.2
 22b. Hebrew Year Spans Can Have Up To 10 Lengths In Days
 22b. The Maximum Number of Lengths For Any Hebrew Year Span
 22c. An Example of Property 22b
 23. The 34 Day Maximum Variance
 24. R(M', 7d) When L" - L' = 34d
 25. R(M", 7d) When L" - L' = 34d
 26. Limits to m' and m" When L" - L' = 34d
 26a. A 34 Day Variance Example
 247 Hebrew Year Periods
 27. Year Spans That Exclude L" = M" - 2d
 27a. Extra Month Period Starts
 27b. Extra Month Period Stops

247 Hebrew Year Periods
 10 Lengths Hebrew Year Periods
 34 Day Variance Hebrew Year Periods

Properties of Hebrew Year Periods - Part 1

21. The Number of Months in Hebrew Year Periods

As explained in Cycles and Moladot, all periods of Hebrew years that are exact multiples of 19 years have exactly one number of months, namely, the number of 19 year periods multiplied by 235.

All other periods of Hebrew years have two numbers of months differing from each other by exactly one month.

Let N be the number of leap years in a given span of Y Hebrew years. Then the alternate number of leap years for that span can be either N - 1 or N + 1, but not both. The actual choice of the secondary number of leap years will depend entirely on where the count of leap years begins relative to a particular mahzor katan (19 year cycle).

This idea is illustrated in the following table which shows the number of leap years that can be counted for 19 years when the count begins at a particular year i of the mahzor katan known as GUChADZaT.

21a. Leap Year Counts Relative to GUChADZaT

Leap Year Counts in Spans 1-19 Years
Year i of GUChADZaT
SPAN
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
  1
 0
 0
 1
 0
 0
 1
 0
 1
 0
 0
 1
 0
 0
 1
 0
 0
 1
 0
 1
  2
 0
 1
 1
 0
 1
 1
 1
 1
 0
 1
 1
 0
 1
 1
 0
 1
 1
 1
 1
  3
 1
 1
 1
 1
 1
 2
 1
 1
 1
 1
 1
 1
 1
 1
 1
 1
 2
 1
 1
  4
 1
 1
 2
 1
 2
 2
 1
 2
 1
 1
 2
 1
 1
 2
 1
 2
 2
 1
 2
  5
 1
 2
 2
 2
 2
 2
 2
 2
 1
 2
 2
 1
 2
 2
 2
 2
 2
 2
 2
  6
 2
 2
 3
 2
 2
 3
 2
 2
 2
 2
 2
 2
 2
 3
 2
 2
 3
 2
 2
  7
 2
 3
 3
 2
 3
 3
 2
 3
 2
 2
 3
 2
 3
 3
 2
 3
 3
 2
 3
  8
 3
 3
 3
 3
 3
 3
 3
 3
 2
 3
 3
 3
 3
 3
 3
 3
 3
 3
 3
  9
 3
 3
 4
 3
 3
 4
 3
 3
 3
 3
 4
 3
 3
 4
 3
 3
 4
 3
 4
 10
 3
 4
 4
 3
 4
 4
 3
 4
 3
 4
 4
 3
 4
 4
 3
 4
 4
 4
 4
 11
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 5
 4
 4
 12
 4
 4
 5
 4
 4
 5
 4
 5
 4
 4
 5
 4
 4
 5
 4
 5
 5
 4
 5
 13
 4
 5
 5
 4
 5
 5
 5
 5
 4
 5
 5
 4
 5
 5
 5
 5
 5
 5
 5
 14
 5
 5
 5
 5
 5
 6
 5
 5
 5
 5
 5
 5
 5
 6
 5
 5
 6
 5
 5
 15
 5
 5
 6
 5
 6
 6
 5
 6
 5
 5
 6
 5
 6
 6
 5
 6
 6
 5
 6
 16
 5
 6
 6
 6
 6
 6
 6
 6
 5
 6
 6
 6
 6
 6
 6
 6
 6
 6
 6
 17
 6
 6
 7
 6
 6
 7
 6
 6
 6
 6
 7
 6
 6
 7
 6
 6
 7
 6
 6
 18
 6
 7
 7
 6
 7
 7
 6
 7
 6
 7
 7
 6
 7
 7
 6
 7
 7
 6
 7
 19
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7

In the above table it can be seen that a count of 8 Hebrew years, started relative to year 9 of the cycle GUChADZaT, contains 2 leap years, instead of 3.

It may also be noted that, relative to GUChADZaT, the column for the 9th year, consistently contains the shorter of the of the two periods possible and that the column for the 17th year, consistently contains the longer of the of the two periods possible.

21b. Month Counts

For any given span of Y years, there are only two possible counts for the number N of months. Consequently, it is more convenient to compress the above table as follows:

Number of Leap Years in Spans 1-19 Years
SPANLEAPSLEAPS
  1
  0
  1
  2
  0
  1
  3
  1
  2
  4
  1
  2
  5
  1
  2
  6
  2
  3
  7
  2
  3
  8
  3
  2
  9
  3
  4
 10
  3
  4
 11
  4
  5
 12
  4
  5
 13
  4
  5
 14
  5
  6
 15
  5
  6
 16
  5
  6
 17
  6
  7
 18
  6
  7
 19
  7
  7

22. Calculating the Number of Months

Let Y represent a given number of complete Hebrew years beginning from 1 Tishrei, and

R(X, 19)= the remainder of X divided by 19.

Then,

22.1 - the smaller number of months is given by

MINMONTHS(Y) = [235 * Y / 19]

While

22.2 - the larger number of months is given by

MAXMONTHS = (235 * Y + R(12 * Y, 19)) / 19

22a. Examples Using Months Formulas 22.1 and 22.2

Example 1 - 19 Hebrew Years

19 years have [19/19] * 235 = 235 months.

Example 2 - 247 Hebrew Years

247 years represent 13 * 19 years = 13 * 235 months = 3,055 months.

Example 3 - 120 Hebrew Years 


MINMONTHS(120) = [235 * 120 / 19] = 1484 (by 22.1 above)
MAXMONTHS(120) = (235 * 120 + R(12 * 120, 19))/ 19 = (28200 + 15) / 19 = 1485

247 Hebrew Year Periods shows the lengths of the smaller and the larger timings of the molad periods for the first 247 spans of Y Hebrew years.

22b. Hebrew Year Spans Can Have Up To 10 Lengths In Days

Given the time between two known moladot of Tishrei = M + m, the only possible lengths L in days appear to be 

       L = M + [f+m]    + p"           - p'           (Equation 4.5)
         = M + {0d, 1d} + {0d, 1d, 2d} - {0d, 1d, 2d}
         = M + {-2d, -1d, 0d, 1d, 2d, 3d} (Applying Equation 5.3 above)
However, 15. L = {M-2d, ..., M+3d} Can NOT Exist for any M + m, shows that for any given M+m the length L = M-2d cannot exist whenever the length L = M+3d is found, and vice versa.

Consequently, for any given M+m, it is possible only to have 

either L = M + {-2d, -1d, 0d, 1d, 2d}
or     L = M + {-1d,  0d, 1d, 2d, 3d}
as the maximum number of lengths for that specific molad period.

Since any span of Y Hebrew years, such that R(Y, 19) > 0, has 2 different molad lengths,
M' and M", the maximum number of lengths possible for any span of Hebrew years is
10 lengths in days.

22c. An Example of Property 22b

Property 22b shows that all periods of Hebrew years are limited to a maximum number of
10 possible lengths in days.

The very first span which can be found having 10 lengths in days is 137 Hebrew years.

137 YEAR SPANS 
  1,694 months =  50,024d 19h  902p
 
  1,695 months =  50,054d  8h  615p
M'+/-DAYSR(D, 7d)OCCURSM"+/-DAYSR(D, 7d)OCCURS
-2
       0d
 0
       0
-2
  50,052d
 2
     105
-1
  50,023d
 1
   6,297
 -1
  50,053d
 3
  42,865
0
  50,024d
 2
 153,412
 0
  50,054d
 4
 106,535
1
  50,025d
 3
 133,504
 1
  50,055d
 5
 164,130
2
  50,026d
 4
  66,656
 2
  50,056d
 6
  12,957
3
  50,027d
 5
   3,011
 3
       0d
 0
       0
The maximum variance is 33 days

The left hand side of the table has M' = 50,024d.
Since the left hand side has the length M' + 3d = 50,027d then due to Property 15
it does not, and cannot, have a length M' - 2d = 50,022d.

The right hand side of the table has M" = 50,054d.
Since the right hand side has the length M" - 2d = 50,052d then due to Property 15
it does not, and cannot, have a length M" + 3d = 50,057d.

23. The 34 Day Maximum Variance

As explained in Cycles and Moladot, for any given period of Y Hebrew years, R(Y, 19) > 0, and having N months, the length of the molad period is determined by N * (29d 12h 793p).

As shown above, Y Hebrew years, for which R(Y, 19) > 0, have two possible numbers of months differing by a count of one month.

Consequently, Y Hebrew years for which R(Y, 19) > 0 have two possible molad period lengths differing by 29d 12h 793p.

As shown by 15. L = {M-2d, ..., M+3d} Can NOT Exist for any M + m, the minimum possible length in days for any M + m, as measured from 1 Tishrei to some other 1 Tishrei,
can be L = M - 2d, while its maximum possible length in days can be L = M + 3d.

Also indicated, is that these two extreme lengths in days cannot exist for the same M + m.

For a given period of Y Hebrew years, such that R(Y, 19) > 0,

Let M' + m', as measured from 1 Tishrei to some other 1 Tishrei, be its shorter molad period.

For M' + m', assume that the length L' = M' - 2d exists.

Let M" + m", as measured from 1 Tishrei to some other 1 Tishrei, be Y's longer molad period.

Then, as shown by 22.1 and 22.2 above, M" + m" = M' + m' + 29d 12h 793p.

For M" + m", assume that the length L" = M" + 3d exists.

As shown by 18. Periods for Which L = {M-2d, ..., M+2d}, if L' = M' - 2d then m' < 8h 876p

Hence, M' + 29d + 12h 793p <= M" + m" < M' + 29d 21h 589p
Since the smallest and largest possible integral value of M" is the single value M' + 29d, consequently, M" = M' + 29d 

Since L' = M' - 2d is the smallest possible length in days for M' + m'
and   L" = M" + 3d is the largest  possible length in days for M" + m"
      L" - L' is the maximum variance in days for Y Hebrew years

Hence L" - L'  = M" +       3d - (M' - 2d)
               = M' + 29d + 3d -  M' + 2d
               = 34d
Therefore, the lengths of any period of Y Hebrew years, as measured from some
1 Tishrei to some other 1 Tishrei, can vary by at most 34 days.

24. R(M', 7d) When L" - L' = 34d
R(M', 7d) = {0d, 2d, 4d, 6d} when L' = M' - 2d (Property 17)  
R(M", 7d) = {0d, 2d, 4d, 6d} when L" = M" - 2d (Property 17)  

Since M'        =  M" - 29d when L" - L' = 34d (Shown in 23 above)
      R(M', 7d) = R(M"               - 29d, 7d)
                = R({0d, 2d, 4d, 6d} -  1d, 7d)
                =  {6d, 1d, 3d, 5d}

Since 6d is the only value common to R(M', 7d) and R(M" - 29d, 7d)

Therefore, R(M', 7d) = 6d whenever L" - L' = 34d.

25. R(M", 7d) When L" - L' = 34d
R(M', 7d) = {0d, 2d, 4d, 6d} when L' = M' - 2d (Property 17)  
R(M", 7d) = {0d, 2d, 4d, 6d} when L" = M" - 2d (Property 17)  

Since M"        =  M' + 29d when L" - L' = 34d (Shown in 23 above)
      R(M", 7d) = R(M'               + 29d, 7d)
                = R({0d, 2d, 4d, 6d} +  1d, 7d)
                =  {1d, 3d, 5d, 0d}

Since 0d is the only value common to R(M", 7d) and R(M' + 29d, 7d)

Therefore, R(M", 7d) = 0d whenever L" - L' = 34d.

26. Limits to m' and m" When L" - L' = 34d
Since L' = M' - 2d when L" - L' = 34d (as shown in 23 above)
      m' < 8h 876p                    (Property 18)
Since m" = m' + 12h 793p              (by definition above)
      m" < 8h 876p + 12h 793p = 21h 589p
Since L" = M + 3d when L" - L' =34d   (as shown in 23 above)
      m" > 15h 204p                   (Property 20)

Therefore, 15h 204p < m" < 21h 589p
and         2h 491p < m' <  8h 876p   (since m' = m" - 12h 793p by def'n)
whenever    L" - L' = 34d

26a. A 34 Day Variance Example

The 1,993 year span is the 71st span to have the 34 day maximum variance.

1,993 YEAR SPANS 
 24,650 months =   727,929d  3h  530p
 
 24,651 months =   727,958d 16h  243p
M'+/-DAYSR(D, 7d)OCCURSM"+/-DAYSR(D, 7d)OCCURS
-2
   727,927d
 4
  12,730
 -2
         0d
 0
       0
-1
   727,928d
 5
 182,846
 -1
         0d
 0
       0
0
   727,929d
 6
  51,941
 0
   727,958d
 0
 100,300
1
   727,930d
 0
 260,515
 1
   727,959d
 1
  19,055
2
         0d
 0
       0
 2
   727,960d
 2
  60,966
3
         0d
 0
       0
 3
   727,961d
 3
   1,119
The maximum variance is 34 days

34 Day Variance Hebrew Year Periods displays the first 71 spans that have the 34 day variance.

27. Year Spans That Exclude L" = M" - 2d

17. R(M, 7d) When Either L = M - 2d Or L = M + 3d finds that
when L = M -2d then R(M, 7d) = {0d, 2d, 4d, 6d}.

18. Periods for Which L = {M-2d, ...}, concludes that for a given number of Hebrew years, Y, of lunar period M + m, the length M - 2d can exist only when 0h <= m < 8h 876p.

However, Properties 17 and 18, taken together, do not garantee that the molad period M" + m", which contains the extra month for some given span of Y Hebrew years, can have L" = M" - 2d as one of its lengths.

When R(Y, 19) = {1, 3, 6, 9, 11, 14, 17} then it is not possible to have L" = M" - 2d.

For example, even though the period of 237 Hebrew years has L' = M' - 2d as one of its lengths, it does not have L" = M" - 2d as another of its lengths because R(237, 19) = 9.

237 YEAR SPANS 
  2,931 months =  86,554d  4h  123p
 
  2,932 months =  86,583d 16h  916p
M'+/-DAYSR(D, 7d)OCCURSM"+/-DAYSR(D, 7d)OCCURS
-2
  86,552d
 4
  11,645
 -2
       0d
 0
       0
-1
  86,553d
 5
 167,294
 -1
       0d
 0
       0
0
  86,554d
 6
  47,376
 0
  86,583d
 0
 117,938
1
  86,555d
 0
 245,429
 1
  86,584d
 1
  22,866
2
       0d
 0
       0
 2
  86,585d
 2
  74,774
3
       0d
 0
       0
 3
  86,586d
 3
   2,150
The maximum variance is 34 days

As shown in Cycles and Moladot this cycle of remainders is exactly the same as the leap year distribution pattern for C(17).

This unusual condition is very clearly demonstrated by the table 27a. Extra Month Period Starts.

When R(Y, 19) = {1, 3, 6, 9, 11, 14, 17} the table shows that the extra month is possible for the period Y only if it begins with a leap year.

Under these circumstances it is not possible to generate a 2 day postponement at the start of the period defined by M" + m", since Tishrei moladot on Tuesday past 15h 203p trigger the 2 day postponements only for 12 month years.

Consequently, as indicated by the length formula

4.5 L = D" - D' = M + [f+m] + p" - p'

no combination of values for [f+m] + p" - p' can equal -2d since p' cannot be 2d.

Therefore, it is not possible to have L" = M" - 2d when R(Y, 19) = {1, 3, 6, 9, 11, 14, 17}.

27a. Extra Month Period Starts

This table is identical to the one shown as 21a. Leap Year Counts Relative to GUChADZaT.

Some of the entries have been highlighted to better show the relationship to the leap years, as numbered from GUChADZaT, of the extra month counts in spans 1, 3, 6, 9, 11, 14, and 17.

It is to be noted that for the spans R(Y, 19) = {1, 3, 6, 9, 11, 14, 17} the extra month count, applying to M" + m", occurs only when the span begins in a leap year.

Extra Month Period Starts in Spans 1-19 Years
Year i of GUChADZaT
SPAN
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
  1
 0
 0
 1
 0
 0
 1
 0
 1
 0
 0
 1
 0
 0
 1
 0
 0
 1
 0
 1
  2
 0
 1
 1
 0
 1
 1
 1
 1
 0
 1
 1
 0
 1
 1
 0
 1
 1
 1
 1
  3
 1
 1
 1
 1
 1
 2
 1
 1
 1
 1
 1
 1
 1
 1
 1
 1
 2
 1
 1
  4
 1
 1
 2
 1
 2
 2
 1
 2
 1
 1
 2
 1
 1
 2
 1
 2
 2
 1
 2
  5
 1
 2
 2
 2
 2
 2
 2
 2
 1
 2
 2
 1
 2
 2
 2
 2
 2
 2
 2
  6
 2
 2
 3
 2
 2
 3
 2
 2
 2
 2
 2
 2
 2
 3
 2
 2
 3
 2
 2
  7
 2
 3
 3
 2
 3
 3
 2
 3
 2
 2
 3
 2
 3
 3
 2
 3
 3
 2
 3
  8
 3
 3
 3
 3
 3
 3
 3
 3
 2
 3
 3
 3
 3
 3
 3
 3
 3
 3
 3
  9
 3
 3
 4
 3
 3
 4
 3
 3
 3
 3
 4
 3
 3
 4
 3
 3
 4
 3
 4
 10
 3
 4
 4
 3
 4
 4
 3
 4
 3
 4
 4
 3
 4
 4
 3
 4
 4
 4
 4
 11
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 5
 4
 4
 12
 4
 4
 5
 4
 4
 5
 4
 5
 4
 4
 5
 4
 4
 5
 4
 5
 5
 4
 5
 13
 4
 5
 5
 4
 5
 5
 5
 5
 4
 5
 5
 4
 5
 5
 5
 5
 5
 5
 5
 14
 5
 5
 5
 5
 5
 6
 5
 5
 5
 5
 5
 5
 5
 6
 5
 5
 6
 5
 5
 15
 5
 5
 6
 5
 6
 6
 5
 6
 5
 5
 6
 5
 6
 6
 5
 6
 6
 5
 6
 16
 5
 6
 6
 6
 6
 6
 6
 6
 5
 6
 6
 6
 6
 6
 6
 6
 6
 6
 6
 17
 6
 6
 7
 6
 6
 7
 6
 6
 6
 6
 7
 6
 6
 7
 6
 6
 7
 6
 6
 18
 6
 7
 7
 6
 7
 7
 6
 7
 6
 7
 7
 6
 7
 7
 6
 7
 7
 6
 7
 19
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7

27b. Extra Month Period Stops

Some of the entries have been highlighted to better show the relationship to the leap years, as numbered from GUChADZaT, of the extra month counts in spans 1, 3, 6, 9, 11, 14, and 17.

It is to be noted that for the spans R(Y, 19) = {1, 3, 6, 9, 11, 14, 17} the extra month count, applying to M" + m", occurs only when the span ends in a leap year.

Extra Month Period Stops in Spans 1-19 Years
Year i of GUChADZaT
SPAN
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
  1
 0
 0
 1
 0
 0
 1
 0
 1
 0
 0
 1
 0
 0
 1
 0
 0
 1
 0
 1
  2
 1
 0
 1
 1
 0
 1
 1
 1
 1
 0
 1
 1
 0
 1
 1
 0
 1
 1
 1
  3
 1
 1
 1
 1
 1
 1
 1
 2
 1
 1
 1
 1
 1
 1
 1
 1
 1
 1
 2
  4
 2
 1
 2
 1
 1
 2
 1
 2
 2
 1
 2
 1
 1
 2
 1
 1
 2
 1
 2
  5
 2
 2
 2
 2
 1
 2
 2
 2
 2
 2
 2
 2
 1
 2
 2
 1
 2
 2
 2
  6
 2
 2
 3
 2
 2
 2
 2
 3
 2
 2
 3
 2
 2
 2
 2
 2
 2
 2
 3
  7
 3
 2
 3
 3
 2
 3
 2
 3
 3
 2
 3
 3
 2
 3
 2
 2
 3
 2
 3
  8
 3
 3
 3
 3
 3
 3
 3
 3
 3
 3
 3
 3
 3
 3
 3
 2
 3
 3
 3
  9
 3
 3
 4
 3
 3
 4
 3
 4
 3
 3
 4
 3
 3
 4
 3
 3
 3
 3
 4
 10
 4
 3
 4
 4
 3
 4
 4
 4
 4
 3
 4
 4
 3
 4
 4
 3
 4
 3
 4
 11
 4
 4
 4
 4
 4
 4
 4
 5
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 4
 12
 4
 4
 5
 4
 4
 5
 4
 5
 5
 4
 5
 4
 4
 5
 4
 4
 5
 4
 5
 13
 5
 4
 5
 5
 4
 5
 5
 5
 5
 5
 5
 5
 4
 5
 5
 4
 5
 5
 5
 14
 5
 5
 5
 5
 5
 5
 5
 6
 5
 5
 6
 5
 5
 5
 5
 5
 5
 5
 6
 15
 6
 5
 6
 5
 5
 6
 5
 6
 6
 5
 6
 6
 5
 6
 5
 5
 6
 5
 6
 16
 6
 6
 6
 6
 5
 6
 6
 6
 6
 6
 6
 6
 6
 6
 6
 5
 6
 6
 6
 17
 6
 6
 7
 6
 6
 6
 6
 7
 6
 6
 7
 6
 6
 7
 6
 6
 6
 6
 7
 18
 7
 6
 7
 7
 6
 7
 6
 7
 7
 6
 7
 7
 6
 7
 7
 6
 7
 6
 7
 19
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7
 7

To return to the top click here.
For other Additional Notes click here.
Hebrew Calendar Science and Myths

I'd love to hear from you. Please send your thoughts to:

Remy Landau 

First  Paged 16 Apr 2001
Next Revised  3 Oct 2003