247 Hebrew Year Periods
10 Lengths Hebrew Year Periods
34 Day Variance 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.
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.
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:
SPAN  LEAPS  LEAPS 

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 
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
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.
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 = {M2d, ..., M+3d} Can NOT Exist for any M + m, shows that for any given M+m the length L = M2d 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.
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'+/  DAYS  R(D, 7d)  OCCURS  M"+/  DAYS  R(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.
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 = {M2d, ..., 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 = {M2d, ..., 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 = 34dTherefore, the lengths of any period of Y Hebrew years, as measured from some
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.
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.
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
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'+/  DAYS  R(D, 7d)  OCCURS  M"+/  DAYS  R(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.
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 = {M2d, ...}, 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'+/  DAYS  R(D, 7d)  OCCURS  M"+/  DAYS  R(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
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}.
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.
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 
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.
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 
First Paged 16 Apr 2001 Next Revised 3 Oct 2003