The Qeviyyot
The Qeviyyot

by Remy Landau
=================================

A keviyyah (pl. keviyyot) is a pair of numbers which consists of the day of the week on which Rosh Hashannah starts AND the length of the year the holiday inaugurates.

Some traditions also include the day of the week that Pesach begins in that year. That day of the week is easily derived by adding to the first day of Rosh Hashannah the following values:

```
1 for a 353 day year
2 for a 354 day year
3 for a 355 day year

3 for a 383 day year
4 for a 384 day year
5 for a 385 day year
```

The Hebrew calendar arithmetic develops only 14 keviyyot. This number of keviyyot is ultimately achieved through the rules which eliminate years of 356 and 382 days.

In the following, the calendar rules are progressively brought into play so as to allow an understanding of the evolution of the keviyyot.

The rules which will not be varied are:-

1. The lunation period is 29d;12h;793p.

2. Every 19 year cycle, beginning with year 1H contains 12 years of 12 months and 7 years of 13 months.

3. The leap years are the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of the 19 year cycle. The year in the cycle is the remainder after dividing the Hebrew year value by 19.

4. The extra month is 30 days long and is inserted immediately after the month of Shevat during a leap year.

5. The months alternate between 30 and 29 days. In years of 353 and 383 days one day is taken away from the month of Kislev, while in years of 355 and 385 days, one day is added to the month of Heshvan.

Each of the postponement rules will be included one at a time to determine their particular impact to the distribution of the keviyyot.

The tables are developed over one full 689,472 year cycle of the Hebrew calendar. In this way it is possible to determine the absolute frequency of the occurences of each keviyyah by dividing any of the totals with 689,472.

In the following tables it should be noted that 36,288 is exactly 1/19 of 689,472 and 98,496 is exactly 1/7 of 689,472.

The Keviyyot Under No Postponement Rules

When the calendar is calculated without any of the dehiyyot, then the following distribution of the keviyyot is seen

Keviyyot - Under No Postponement Rules
YEAR LENGTH IN DAYS
DAY 354 355 383 384 TOTALS
Sun 39369 22839 3712 32576 98496
Mon 39369 22839 3712 32576 98496
Tue 39369 22839 3712 32576 98496
Wed 39369 22839 3712 32576 98496
Thu 39369 22839 3712 32576 98496
Fri 39369 22839 3712 32576 98496
Sat 39369 22839 3712 32576 98496
TOTALS 275583 159873 25984 228032 689472

Additionally, the following number of postponements may be calculated

```689472 times with no postponements
0 "356 Rule" postponements
0 "382 Rule" postponements

Postponed by 0 days = 689472
Postponed by 1 day  = 0
Postponed by 2 days = 0
```

It may be seen that without any of the postponement rules there exists only 4 year lengths and all of the keviyyot are uniformly distributed over each day of the week.

The Effect of LO ADU on the Keviyyot

This rule requires the postponement of Rosh Hashannah by one day should molad of Tishrei occur either on Sunday, Wednesday, or Friday.

The rule's impact on the keviyyot is shown below.

Keviyyot - Under LO ADU Postponement Rule
YEAR LENGTH IN DAYS
DAY 353 354 355 356 382 383 384 385 TOTALS
Mon 39369 0 85047 0 0 40000 0 32576 196992
Tue 0 39369 0 22839 0 0 36288 0 98496
Thu 0 101577 22839 0 3712 36288 0 32576 196992
Sat 39369 0 85047 0 0 40000 0 32576 196992
TOTALS 78738 140946 192933 22839 3712 116288 36288 97728 689472

Additionally, the following number of postponements may be calculated

```393984 times with no postponements
0 "356 Rule" postponements
0 "382 Rule" postponements

Postponed by 0 days = 393984
Postponed by 1 days = 295488
Postponed by 2 days = 0
```

LO ADU cause a doubling of the number of year lengths from 4 to 8.

Surprisingly, the rule actually decreases the number of keviyyot from 28 down to 16. This result is a bit unexpected because 8 year lengths paired to 4 days of the week could have led to 32 possible keviyyot.

The Effect of the "356 Rule" on the Keviyyot

The "356 Rule" prevents years of 356 days by forcing a postponement based on the time of the molad of Tishrei.

When the molad Tishrei of a 12 month year occurs on Tuesday on or after 15h;204p (3d;15h;204p) then adding 354d;8h;876p (12 lunation periods) results in the time of the next molad of Tishrei as Sunday at 0h;0p or later. This is 355 days later.

Since Lo ADU forces the postponement of Rosh Hashannah to the next day, which is Monday, the year now becomes 356 days long. Hence, by forcing a postponement on or after 3d;15h;204p the 356 day year is eliminated. The postponement effectively adds one or 2 days to the previous year, and takes away those added days from the current year.

Keviyyot - Under LO ADU and "356 Rule" Postponements
YEAR LENGTH IN DAYS
DAY 353 354 355 382 383 384 385 TOTALS
Mon 39369 0 85047 0 40000 0 32576 196992
Tue 0 39369 0 0 0 36288 0 75657
Thu 0 124416 22839 3712 22965 0 45899 219831
Sat 29853 0 94563 0 40000 0 32576 196992
TOTALS 69222 163785 202449 3712 102965 36288 111051 689472

Additionally, the following number of postponements may be calculated

```371145 times with no postponements
22839 "356 Rule" postponements
0 "382 Rule" postponements

Postponed by 0 days= 371145
Postponed by 1 day = 295488
Postponed by 2 days= 22839
```

The Effect of the "382 Rule" on the Keviyyot

The "382 Rule" prevents years of 382 days by forcing a postponement based on the time of the molad of Tishrei.

When the molad of Tishrei of a 13 month year occurs on Wednesday on or after 0h;0p (4d;0h;0p) then adding 383d;21h;589p (the time period of 13 lunations) results in the time of the next molad of Tishrei as Monday at 21h;589p or later. This is 383 days later.

The next Rosh Hashannah would be allowed to start on Monday. However, since Lo ADU forces the postponement of the current Rosh Hashannah from Wednesday to Thursday, the year now becomes 382 days long. Hence, by forcing a postponement of the next Rosh Hashannah, whose molad occurs on or later than 2d;21h;589p the 382 day year is eliminated.

Keviyyot - Under LO ADU, "356 Rule" and "382 Rule" Postponements
YEAR LENGTH IN DAYS
DAY 353 354 355 383 384 385 TOTALS
Mon 39369 0 81335 40000 0 32576 193280
Tue 0 43081 0 0 36288 0 79369
Thu 0 124416 22839 26677 0 45899 219831
Sat 29853 0 94563 40000 0 32576 196992
TOTALS 69222 167497 198737 106677 36288 111051 689472

Additionally, the following number of postponements may be calculated

```367433 times with no postponements
22839 "356 Rule" postponements
3712 "382 Rule" postponements

Postponed by 0 days= 367433
Postponed by 1 day = 299200
Postponed by 2 days= 22839
```

The Effect of the Molad Zakein Postponement Rule

The molad zakein rule ensures that there will be a molad no later than the very last part of the first day of any month. It essentially forces a postponement of Rosh Hashannah to the next allowable day whenever the molad of Tishrei occurs on or later than 18h on any of the allowable days for the start of the holiday.

The calendar arithmetic indicates that the molad zakein rule would not have been required had the leap month been placed prior to the month of Heshvan in leap years.

See The Overpost Problem in the additional notes for an explanation of the logic behind the molad zakein rule.

The molad zakein rule requires the lowering by 6 hours of the time limit of the "356 Rule" to 3d;9h;204p and that of the "382 Rule" to 2d;15h;589p.

This rule has no effect whatever on the keviyyot as can be determined from a comparison with the preceding table.

Keviyyot - Under ALL Postponement Rules
YEAR LENGTH IN DAYS
DAY 353 354 355 383 384 385 TOTALS
Mon 39369 0 81335 40000 0 32576 193280
Tue 0 43081 0 0 36288 0 79369
Thu 0 124416 22839 26677 0 45899 219831
Sat 29853 0 94563 40000 0 32576 196992
TOTALS 69222 167497 198737 106677 36288 111051 689472

Additionally, the following number of postponements may be calculated

```268937 times with no postponements
22839 "356 Rule" postponements
3712 "382 Rule" postponements

Postponed by 0 days= 268937
Postponed by 1 day = 323824
Postponed by 2 days= 96711
```

The number of postponements due to the Molad Zakein rule when its limiting time is 18h is seen to occur exactly 1 out every 7 possible Rosh Hashannot.

This is a remarkable coincidence as can be determined by adjusting this limit and observing the corresponding changes in the frequency of the rule's occurrence.

Since 7 and 18 were considered numerologically significant values this unusual coincidence in the calendar system may perhaps explain why the scholars opted for the 18h time limit rather than the more optimal 18h;657p.

The Keviyyot Under the Ben Meir Rules

Ben Meir in 920g suggested the relaxation of the molad zakein rule by 642p.

In order to maintain the existing keviyyot, those 642p must also be added to the "356 Rule" and the "382 Rule". Hence, their limiting times become 3d;9h;846p and 2d;16h;151p.

Keviyyot - Under the Ben Meir Rules
YEAR LENGTH IN DAYS
DAY 353 354 355 383 384 385 TOTALS
Mon 39369 0 81335 40000 0 32576 193280
Tue 0 43081 0 0 36288 0 79369
Thu 0 124416 22839 26677 0 45899 219831
Sat 29853 0 94563 40000 0 32576 196992
TOTALS 69222 167497 198737 106677 36288 111051 689472

Additionally, the following number of postponements may be calculated

```278697 times with no postponements
22839 "356 Rule" postponements
3712 "382 Rule" postponements

Postponed by 0 days= 278697
Postponed by 1 day = 321384
Postponed by 2 days= 89391
```

As can be seen there are 9760 fewer postponements under the Ben Meir rules than under the traditional rules.

The Year Type Successions

353 and 383 day years are known as HASER, that is, deficient years. In such years the month of Kislev loses one day to become 29 days long. These years are denoted by the Hebrew letter het.

354 and 384 day years are known as KESIDRAH, that is, regular years. If we ignore the extra month of Adar I then all of the months of such years are seen to alternate regularly from 30 to 29 days. These years are denoted by the Hebrew letter chof.

355 and 385 day years are known as SHELEMAH, that is, abundant years. In such years the month of Heshvan is given an additional day to become 30 days long. These years are denoted by the Hebrew letter shin.

The calendar arithmetic shows that

```     deficient years cannot follow deficient years
regular years cannot follow regular years
but abundant years can follow abundant years.
```

The following table shows, over the full calendar cycle, the number of times that a year type in the left margin is followed by any other year type.

Pairwise Succession of Year Types
NUMBER OF PAIRWISE SUCCESSIONS
TYPE 353 354 355 383 384 385TOTALS
353 0 13776 9516 0 22965 22965 69222
354 16404 0 54491 40000 0 56602 167497
355 16404 54468 16381 66677 13323 31484 198737
383 0 53354 53323 0 0 0 106677
384 0 0 36288 0 0 0 36288
385 36414 45899 28738 0 0 0 111051
TOTALS 69222 167497 198737 106677 36288 111051 689472

Of interest is the fact that a regular leap year is always followed by an abundant year. A 384 day year is always followed by a 355 day year.

Leap years cannot follow leap years due to their predefined distribution in the 19 year cycle.

What is not immediately apparent is why any deficient year beginning on a Monday cannot be followed by a deficient leap year. Theoretically, such a pair of years would end on a Tuesday, which day is allowed for Rosh Hashannah.

```First  Paged  1 Aug 1997