How often do Hebrew years of 383 days go missing from all of the 19 year cycles in the full Hebrew calendar cycle?

Besides the regular leap years of **384 days**, the deficient leap years
of **383 days** are the only other kind of year to skip some of the
19 year cycles.

As shown in the *Additional Notes** *under the topic of
** The Keviyyot** 383 day years occur

Consequently, it is hard to imagine that any 19 year cycle can be found
without any 383 day year in it. However, over the full Hebrew calendar
cycle exactly **9 of the 19 year cycles** do not have a 383 day year.

The first such 19 year cycle to exclude a 383 day year will not begin until

**Rosh Hashannah 114,780H** corresponding to
**Tuesday 16 January 111,021g**.

Correspondent **Ari Z. Zivotofsky** did a random sampling of a number of
Rosh Hashannah dates mapped onto the Gregorian calendar. He then
calculated the Gregorian equivalent of these Rosh Hashannah dates
offset by 247 years. He was surprised to discover that all of the new
Gregorian dates were exactly ONE day later than their earlier counterparts.

**Ari Z. Zivotofsky** showed these examples

Rosh Hashannah 5665H occurred on Sat 10 Sep 1904g 5912H Sat 11 Sep 2151g Rosh Hashannah 5669H occurred on Sat 26 Sep 1908g 5916H Sat 27 Sep 2155g Rosh Hashannah 5724H occurred on Thu 19 Sep 1963g 5971H Thu 20 Sep 2210g Rosh Hashannah 5726H occurred on Mon 27 Sep 1965g 5973H Mon 28 Sep 2212g

Are all

Rosh Hashannah's separated by 247 years exactly one day apart in their Gregorian calendar equivalents?

**NO! **While many examples of the one day date difference easily
can be found, a number of additional examples can be shown which do not
conform to that difference.

Correspondent **Ari Z. Zivotofsky** did a random sampling of a number of
Rosh Hashannah dates mapped onto the Gregorian calendar. He then
calculated the Gregorian equivalent of these Rosh Hashannah dates
offset by 247 years. He was surprised to discover that all of the new
Gregorian dates were exactly ONE day later than their earlier counterparts.

**Ari Z. Zivotofsky** showed these examples

Rosh Hashannah 5665H occurred on Sat 10 Sep 1904g 5912H Sat 11 Sep 2151g Rosh Hashannah 5669H occurred on Sat 26 Sep 1908g 5916H Sat 27 Sep 2155g Rosh Hashannah 5724H occurred on Thu 19 Sep 1963g 5971H Thu 20 Sep 2210g Rosh Hashannah 5726H occurred on Mon 27 Sep 1965g 5973H Mon 28 Sep 2212g

Ari Z. Zivotofskythen also found these examplesRosh Hashannah 5662H occurred on Sat 10 Sep 1901g 5909H Sat 10 Sep 2148g Rosh Hashannah 5663H occurred on Thu 2 Oct 1902g 5910H Thu 2 Oct 2149g Rosh Hashannah 5664H occurred on Tue 22 Sep 1903g 5911H Tue 22 Sep 2150g

The years 5725H (1964g) and 5850H (2089g) were also mentioned as showing a 2 day Gregorian date difference after 247 years.

Correspondent **Winfried Gerum** made the following excellent analysis

Thank you very muchThe conjecture is, that 247 Hebrew years differ from 247 Gregorian years by one day.The length of the Gregorian year is

146,097/ 400 = 365.2425 days. The length of the Hebrew year is251,827,457/689472 = 365.246822.. days.Summing 247 "average" years, the difference between 247 Hebrew and Gregorian years is about 1.067... days.

So it is clear, that the difference between two Gregorian dates 247 Hebrew years apart cannot always be 1 day!

Correspondent **Jerrold Landau** attracted my attention to the fact
that, beginning with

Rosh Hashannah 5760H (Sat 11 Sep 1999g),
**4 out of the next 5** consecutive Rosh Hashannah's will begin on
**Saturday**.

After

Rosh Hashannah5764H (Shabbat27 September 2003g) when again will 4 out of 5 consecutiveRosh Hashannah's begin onShabbat?

The next pattern of of 5 consecutive Hebrew years in which 4 out of 5 begin
on Shabbat will not occur until **Rosh Hashannah 5936H
(Saturday 16 Sep 2175g)**.

The 5 Rosh Hashannah's are

5936H corresponding to Sat 16 Sep 2175g 5937H Sat 5 Oct 2176g 5938H Tue 23 Sep 2177g 5939H Sat 12 Sep 2178g 5940H Sat 2 Oct 2179g

Correspondent **Winfried Gerum** sent the following correct answer:

The next period with 4 out of 5 years beginning on Shabbat are the years 5936 through 5940.Three consecutive years cannot all commence on Shabbat. Therefore, periods with 4 out of 5 years starting on Shabbat must have their 3rd year starting on some day other than Shabbat.

Thank you very much **Winfried Gerum** for sharing these observations
with us.

Correspondent **Larry Padwa** also sent the following correct answer:

The next occurence of this will be in about 175 years. RH 5936 (Saturday 16-September 2175G) begins the next cycle.It is interesting to note that this pattern occurs with other days of the week much sooner.

Thank you also **Larry Padwa** for not only sharing your observations
with us but actually suggesting the next **Weekly Question**.

After

Rosh Hashannah5764H (Shabbat27 September 2003g) when will 4 out of 5 consecutiveRosh Hashannah's begin on the same day other thanShabbat?

Correspondent **Winfried Gerum** sent the following correct answer:

The next period of five years with four of them beginning on the same day of the week with Rosh Hashannah not falling on a Sabbath are the years 5771H through 5775H (2010g thru 2014g).

The four Rosh haShannas will all begin on Thursday while Rosh haShanna 5773H (2012g) will begin on Monday. As with all such five year periods the third year starts on a different day of the week.

Thank you very much **Winfried Gerum** for sharing these observations
with us.

In suggesting this question Correspondent **Larry Padwa** had made
the following observation:

It is interesting to note that this pattern occurs with other days of the week much sooner. RH 5771H (Thurs 9 Sep 2010g) is the first of four out of five consecutive RH's beginning on Thursday.

Thank you also **Larry Padwa** for not only sharing your observations
with us but once again suggesting the next **Weekly Question**.

After

Rosh Hashannah5775H (Thu 25 Sep 2014g) when next will 4 out of 5Rosh Hashannah's begin on the same day?

In suggesting this question, correspondent **Larry Padwa** had made
the following observation:

RH 5771H (Thurs 9-Sept 2010G) is the first of four out of five consecutive RH's beginning on Thursday, and those five years are immediately followed by RH 5776H (Mon 14-Sept 2015G) which is the first of four out of five consecutive RH's beginning on Monday.

A further curiosity is that in the string of Thursday RH's the non-Thursday RH is Monday, and in the string of Monday RH's the non-Monday RH is Thursday.

Thus,

in the ten years beginning 5771H (2010G), all RH's begin on Monday or Thursday.The pattern is:

Th,Th,Mo,Th,Th,Mo,Mo,Th,Mo,Mo.

Thank you **Larry Padwa** for once again having shared your amazing
observations with us.

Correspondent **Winfried Gerum** also sent the following thought
provoking comments:

The next such period is from

5776H through 5780H.Of these "Thursday"-Periods 14 overlap during a full calendar cycle. I.e., there are 14 periods of eight years with six out of eight Rosh Hashanot falling on a Thursday.

The most recent such period was the years 3173H .... 3180H.

The next such period will be the years

35883H through 35890H.Except for these 14 periods there are no other 8 year periods with six Rosh Hashanot on the same day of the week.

Thank you very much **Winfried Gerum** for sharing these observations
with us.

Do the numbers

2 4 7 10 12 15 18constitute a valid leap year distribution for the calculation of the Hebrew calendar?

**YES!**

The traditional literature of the Hebrew calendar, such as the 8th century
** Seder Olam**, and the 11th century work

The Encyclopedia Judaica, in its article on the ** Calendar**
very correctly notes that

Apparent variations in theordo intercalationis... are but variants of the selfsame order.

The leap year order that is used will depend entirely on the year of the
19 year cycle that is selected to be the ** first** year of the
year counts, ie, the

Since the Hebrew calendar uses only one pattern in which to arrange the 7 leap years of a 19 year cycle, there can only be at most 19 leap year distributions resulting from the choice of a specific epochal year within the mahzor katan. Assuming that the epochal year 1H is the first year of the 19 year cycle, then these distributions are:-

YEAR Leap Year Distribution ==== ====================== 1 3 6 8 11 14 17 19 2 2 5 7 10 13 16 18 3 1 4 6 9 12 15 17 4 3 5 8 11 14 16 19 5 2 4 7 10 13 15 18 6 1 3 6 9 12 14 17 7 2 5 8 11 13 16 19 8 1 4 7 10 12 15 18 9 3 6 9 11 14 17 19 10 2 5 8 10 13 16 18 11 1 4 7 9 12 15 17 12 3 6 8 11 14 16 19 13 2 5 7 10 13 15 18 14 1 4 6 9 12 14 17 15 3 5 8 11 13 16 19 16 2 4 7 10 12 15 18 17 1 3 6 9 11 14 17 18 2 5 8 10 13 16 19 19 1 4 7 9 12 15 18

The numbers **2 4 7 10 12 15 18** can be seen against the 16th year
of the 19 year cycle whose first year is also year 1H. Consequently, any
Hebrew calendar system whose year count begins relative to year 16H will be
required to use the leap year distribution **2 4 7 10 12 15 18**
if it is to maintain synchronization with the fixed Hebrew calendar.

Correspondent **Winfried Gerum** gave the following correct answer

... if one does not start counting19-year cycles in the year 1 but instead, the year 16, then leap years fall into the stated sequence of numbers.

Good work **Winfried Gerum**!

The Encyclopedia Judaica, in its article on the

cites the following leap year distributions as having been usedCalendar2 5 7 10 13 16 18 1 4 6 9 12 15 17 3 5 8 11 14 16 19 3 6 8 11 14 17 19Additionally, the EJ suggests that the following values were used at one time as the epochal

moladot4d 20h 408p 2d 5h 204p 6d 14h 0p 3d 22h 876pIs the EJ entirely correct?

**NO!**

In the ** Encyclopedia Judaica**, the epochal molad given as

The traditional literature of the Hebrew calendar, such as the 8th century
** Seder Olam**, and the 11th century work

The Encyclopedia Judaica, in its article on the ** Calendar**
very correctly notes that

Apparent variations in theordo intercalationis... are but variants of the selfsame order.

The leap year order that is used will depend entirely on the year of the
19 year cycle that is selected to be the ** first** year of the
year counts, ie, the

Since the Hebrew calendar uses only one pattern in which to arrange the 7 leap years of a 19 year cycle, there can only be at most 19 leap year distributions resulting from the choice of a specific epochal year within the mahzor katan.

The EJ article ** Calendar** mentions only the first 4 possible
leap year variations assuming that the epochal year 1H is the first
year of the 19 year cycle. These distributions are:-

YEAR Leap Year Distribution ==== ====================== 1 3 6 8 11 14 17 19 2 2 5 7 10 13 16 18 3 1 4 6 9 12 15 17 4 3 5 8 11 14 16 19

Consequently, the epochal moladot should be

2d 5h 204p (for the 1st year) 6d 14h 0p (for the 2nd year) 3d 22h 876p (for the 3rd year) and2d 20h 385p(for the 4th year) instead of the value4d 20h 408p

Which Hebrew year could most reasonably be represented as an

epochalyear with the presence of themolad4d 20h 408p?

The Encyclopedia Judaica article ** Calendar** mentions only the
first 4 possible leap year variations assuming that the epochal year 1H is
the first year of the 19 year cycle. These distributions are:-

YEAR Leap Year Distribution ==== ====================== 1 3 6 8 11 14 17 19 2 2 5 7 10 13 16 18 3 1 4 6 9 12 15 17 4 3 5 8 11 14 16 19

Consequently, the epochal moladot should be

2d 5h 204p (for the 1st year) 6d 14h 0p (for the 2nd year) 3d 22h 876p (for the 3rd year) and2d 20h 385p(for the 4th year) instead of the value4d 20h 408p

The moladot of Tishrei which correspond to the value **4d 20h 408p**
are for the Hebrew years

117357H(Thu 15 Jan 113598g)308062H(Thu 27 Apr 304305g)616124H(Thu 3 Dec 612370g)

These Hebrew years are respectively, the **13th, 15th, and 11th** years of the
mahzor katan using the leap year distribution ** GUChADZaT**.

Clearly, the above possibilities are not *sensible* in terms of
epochal years. However, the value **4d 20h 408p** also comes as the
molad of **Heshvan** for the year

Consequently, if the Encyclopedia Judaica were to correct this particular
passage in their well written article on the ** Calendar** the
two corrections that could be recommended would be that:-

1. The epochal molad of **4d 20h 408p** be replaced by the value
**3d 7h 695p** which represents the molad of **Tishrei 0H**;

2. The leap year distribution **3 5 8 11 14 16 19** be replaced
by **1 4 7 9 12 15 18** which represents the leap year distribution
for an epochal year beginning at year **0H**.

Which Hebrew month least frequently has the

moladof0d 0h 0p?

In the full Hebrew calendar cycle of 689472 years each and every month
will experience the molad **0d 0h 0p at least 2 times**.

Of all the months, **the leap month Adar** will experience the molad
**0d 0h 0p** exactly two times in the full Hebrew calendar cycle.
All of the other months will see this molad either 3 or 4 times.

Therefore, the **leap month Adar** least frequently has the molad
**0d 0h 0p**.

Is the Hebrew 5760H a leap year because it begins on the Gregorian date of September 11?

**YES!**

The last Rosh Hashannah to coincide with the Gregorian date September 11 will begin on Tuesday 7142H (3381g).

Until then, all of the Rosh Hashannah's coinciding with the Gregorian date September 11 will mark the start of a Hebrew leap year.

Correspondent **Winfried Gerum** had another quite correct opinion.

As the two calendars are not in any way related there is no reason why a Rosh Hashanna falling on the 11th day of September is forced to be a leap year. However, during the years 5350H through 7142H there are 63 Rosh Hashannah's falling on September 11th and all of them happen to be leap years. Prior to 5350H, there are some Rosh Hashannah's falling on September 11, but none of them is a leap year.

Good work **Winfried Gerum!**

The first time that a Rosh Hashannah began a leap year on the Gregorian date September 11 was Monday 4511H (750g).

First Begun 21 Jun 1998 First Paged 5 Nov 2004 Next Revised 5 Nov 2004