Psalm 118:22 probably offers one of the most insightful metaphor related to the construction of the Hebrew calendar.
The calculation of the Hebrew calendar involves numbers that are expressed in whole days and fractions of days.
The molad of Tishrei is a number that is expressed in days and a fraction of the 24 hour day.
That fraction cannot be used in a calendar only produced in whole numbers of days. And so, that fraction is rejected in the construction of the day to day calendar.
However, it is that particular fraction which determines the
actual day of the week on which
Rosh Hashannah, the head of the year, is to begin.
As a consequence, it is also that fraction which ultimately determines the final length in days of any given period of Hebrew years.
Consequently, that fraction, viewed as the stone rejected by the builders of the calendar, is truly the corner stone of that edifice.
Properties of Hebrew Year Periods is, as much as possible, an elementary algebraic analysis of some of the arithmetical phenomenas of the fixed Hebrew calendar.
The analysis is useful in predicting results which can be verified by computer simulation over the full Hebrew calendar cycle of 689,472 years.
Most of the results can be seen in one or more of the entries in 247 Hebrew Year Periods.
Periods of Hebrew years, as measured from one Rosh Hashannah to another, is the subject of the analysis to be found in Properties of Hebrew Year Periods.
The analysis finds that while a given Hebrew year period has only one length in years, it can have up to 2 lengths in terms of months, and up to 10 lengths in terms of days.
The period of 137 Hebrew years is the first span to show all of these conditions.
137 YEAR SPANS  

1,694 months = 50,024d 19h 902p 
1,695 months = 50,054d 8h 615p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 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 maximum variance is the difference between the longest and the shortest possible lengths in days for a given period of Hebrew years. In the case of 137 years, the maximum variance is 33 days.
However, as shown by 23. The 34 Day Maximum Variance, the maximum possible variance for any period of Hebrew years is 34 days.
10 Hebrew years is the smallest span to have the maximum variance of 34 days.
10 YEAR SPANS  

123 months = 3,632d 6h 339p 
124 months = 3,661d 19h 52p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 7d  OCCURS  
2  3,630d  4  3,236 
2  0d  0  0  
1  3,631d  5  76,946 
1  0d  0  0  
0  3,632d  6  22,866 
0  3,661d  0  252,669 

1  3,633d  0  114,680 
1  3,662d  1  47,195 

2  0d  0  0  2  3,663d  2  162,407 

3  0d  0  0  3  3,664d  3  9,473 

The maximum variance is 34 days 
The twosided table arrangement shown above is derived from
the empirical observation shown in
21. The Number of Months in Hebrew Year
Periods
.
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 
The above table shows that all spans of Hebrew years that are not multiples of 19 years have two lengths which differ by one month. This is a direct consequence of the arithmetic of the Hebrew calendar.
The sixlined table arrangement shown in the year span tables above is derived from the result of the analysis in 4. The Length of Hebrew Year Periods .
Expressing the molad period of a given Hebrew year span as M + m, then the length in days for that number of years can be either M  2d, M  1d, M, M + 1d, M + 2d, or M + 3d.
However, 15. L = {M2d, ..., M+3d} Can NOT Exist for any M + m proves that, at most, only 5 lengths in days are ever possible for any given M + m.
The 4 year span is the smallest period of Hebrew years containing
the 5 lengths ranging from
M  1d to M + 3d on the left hand side of the table.
4 YEAR SPANS  

49 months = 1,446d 23h 1057p 
50 months = 1,476d 12h 770p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 7d  OCCURS  
2  0d  0  0  2  0d  0  0  
1  1,445d  3  78  1  1,475d  5  52,504 

0  1,446d  4  74,676 
0  1,476d  6  18,230 

1  1,447d  5  249,668 
1  1,477d  0  255,858 

2  1,448d  6  32,762 
2  0d  0  0  
3  1,449d  0  5,696 
3  0d  0  0  
The maximum variance is 32 days 
The 15 year span is the smallest period of Hebrew years containing
the 5 lengths ranging from
M  1d to M + 3d on the right hand side of the table.
15 YEAR SPANS  

185 months = 5,463d 3h 905p 
186 months = 5,492d 16h 618p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 7d  OCCURS  
2  0d  0  0  2  0d  0  0  
1  5,462d  2  95,519 
1  5,491d  3  16,044 

0  5,463d  3  128,700 
0  5,492d  4  85,926 

1  5,464d  4  87,356 
1  5,493d  5  227,836 

2  5,465d  5  15,017 
2  5,494d  6  30,086 

3  0d  0  0  3  5,495d  0  2,988 

The maximum variance is 33 days 
The 5 year span is the smallest period of Hebrew years containing
the 5 lengths ranging from
M  2d to M + 2d on the left hand side of the table.
5 YEAR SPANS  

61 months = 1,801d 8h 853p 
62 months = 1,830d 21h 566p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 7d  OCCURS  
2  1,799d  0  14  2  0d  0  0  
1  1,800d  1  9,856 
1  1,829d  2  30,531 

0  1,801d  2  67,945 
0  1,830d  3  155,318 

1  1,802d  3  25,353 
1  1,831d  4  225,267 

2  1,803d  4  5,696 
2  1,832d  5  169,492 

3  0d  0  0  3  0d  0  0  
The maximum variance is 33 days 
The 12 year span is the smallest period of Hebrew years containing
the 5 lengths ranging from
M  2d to M + 2d on the right hand side of the table.
12 YEAR SPANS  

148 months = 4,370d 12h 724p 
149 months = 4,400d 1h 437p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 7d  OCCURS  
2  0d  0  0  2  4,398d  2  1,600 

1  4,369d  1  19,306 
1  4,399d  3  62,864 

0  4,370d  2  219,452 
0  4,400d  4  114,554 

1  4,371d  3  121,135 
1  4,401d  5  109,163 

2  4,372d  4  39,275 
2  4,402d  6  2,123 

3  0d  0  0  3  0d  0  0  
The maximum variance is 33 days 
Each table also includes a statistical distribution of the occurrences of
a particular length of days over the full Hebrew calendar cycle
of 689,472 years. As proved by
13. Dehiyyah Molad Zakein's
Transparency,
These distributions are not affected by either the
presence or the absence of the Molad Zakein postponement rule.
Consequently, the Molad Zakein rule is not used in the
analysis related to the lengths of Hebrew year periods.
The analytical approach to the problems of the Hebrew year lengths produces a number of interesting and provocative side observations.
10. The 357 Day Year Paradox shows that If the Hebrew month had a molad period that was just 288 halakim (parts), or 16 minutes more than the traditional 29d 12h 793p, then the elimination of the 356 day year would have created a 357 day year.
27a. Extra Month Period Starts and 27b. Extra Month Period Stops demonstrate that Hebrew year spans always begin with and end with a leap year whenever the spans contain the extra leap month, and their number of years, after division by 19, leaves a remainder of either 1, 3, 6, 9, 11, 14, or 17.
Ultimately, many problems have been left with undone analysis.
First Paged 7 Aug 2001 Next Revised 30 Jan 2012