Tu B'Shevat cannot occur on which day, or days, of the week?

Correspondent Larry Padwa suggested this question, which is very timely because
Tu B'Shevat 5760H (the 15th day of Shevat 5760H) occurs on Sat 22 Jan 2000g.

Tu B'Shevat is a minor celebration also known as The New Year of Trees.

Since Tu B'Shevat is the 15th day of Shevat, it must always fall on the same day of the week as the 1st day of Shevat, because it is 2 complete weeks away from the 1st of the month.

The simplest way of answering the question is to look at the topic of The First Day of the Month found in the Additional Notes. The first table in that item shows the The Start of Month Distribution By Day of Week.

That table indicates no occurrences of Shevat beginning on either Sunday or Friday.

Correspondent Benjamin W Dreyfus correctly stated as follows

Tu Bishvat cannot fall on Friday or Sunday.

(It can be on Monday, Tuesday, Thursday, or Saturday in a leap year
and falls on the same day as the following Rosh Hashana, and can be on
Monday, Wednesday, Thursday, or Saturday in a common year.)

A Tu B' Shevat never occurs on a Friday or on a Sunday:

In a deficient year (353 days) Tu B' Shevat is the 131st day.
As this type of year begins either on a Shabbat or on a Monday,
Tu B'Shevat can be either a Thursday or a Shabbat.

In a regular year (354 days) Tu B' Shevat is the 132nd day.
As this type of year begins either on a Thursday or on a Thuesday,
Tu B'Shevat can be either a Wednesday or a Monday.

In an abundant year (355 days) Tu B' Shevat is the 133rd day.
As this type of year begins either on a Shabbat, a Thursday or on a
Monday, Tu B'Shevat can be either a Shabbat, a Thursday or a Monday.

In a deficient leap year (383 days) Tu B' Shevat is the 131st day.
As this type of year begins either on a Shabbat or on a Monday,
Tu B'Shevat can be either a Thursday or a Shabbat.
As this type of year begins either on a Shabbat, a Thursday or on a
Monday, Tu B'Shevat can be either a Thursday, Wednesday or a Shabbat.

In a regular year (384 days) Tu B' Shevat is the 132nd day.
As this type of year begins on a Tuesday
Tu B'Shevat can be only a Monday.

As this type of year begins either on a Shabbat, a Thursday or on a
Monday, Tu B'Shevat can be either a Shabbat, a Thursday or a Monday.
In an abundant year (385 days) Tu B' Shevat is the 133rd day.

This is most directly approached by considering the Pesach following
the Shevat in question.

The first day of Pesach cannot fall on Monday, Wednesday or Friday.

Now in a common year, Tu B'Shevat is 59 days before Pesach
(30 day Shevat and 29 day Adar). Thus in a common year Tu B'Shevat
cannot occur on the days of the week that are 59 days
(or 3 days when multiples of 7 are eliminated) before {Mon,Wed,Fri},
namely Friday, Sunday, and Tuesday.

In a leap year, Tu B'Shevat is 89 days before Pesach
(30 day Shevat, 30 day Adar I, and 29 day Adar II). Thus in a leap year
Tu B'Shevat cannot occur on the days of the week that are 89 days
(or 5 days when multiples of 7 are eliminated) before {Mon,Wed,Fri},
namely Wednesday, Friday, and Sunday.

The only days that are on both of the above lists are Sunday and Friday.

NOTE
----
I stated at the outset that this is the most direct approach. This is
because there are only two possibilities to consider--common or leap
year.

Another approach is to use the allowable days from the RH preceding the
Shevat in question; however this is slightly less direct because there
are three possibilities to consider corresponding to the lengths of the
intervening Cheshvan and Kislev
(i.e., deficient, normal or abundant year).

What is the frequency of occurrence of each of the Hebrew calendar's different 19 year period lengths?

Correspondent Dwight Blevins asked about the frequency of each of the Hebrew calendar's different 19 year period lengths.

Since the question did not specify the method of measurement, two entirely different answers are possible.

The first answer is that all periods of 19 consecutive Hebrew years have the mean lunar length of 6939d 16h 595p.

A 19 Hebrew year cycle (mahzor katan) consists of 235 months. Since one month has the lunation period of 29d 12h 793p, 235 months have

235 * (29d 12h 793p) = 6815d 2820h  186,355p 
                     = 6815d 2820h+172h 595p 
                     = 6815d 2992h      595p
                     = 6815d+124d   16h 595p  
                     = 6939d        16h 595p

Within the entire 689,472 year cycle of the Hebrew calendar, the possible number of days for any period of 19 consecutive years, and their frequency of occurrence within the calendar cycle are as follows

             number of days= 6,938 number of times=  11,263 
                             6,939                  311,544 
                             6,940                  250,123 
                             6,941                  113,011 
                             6,942                    3,531 

Which year, or years, of the 19 year cycle most frequently begin(s) the 6,938 day span of a 19 year period?

Within the entire 689,472 year cycle of the Hebrew calendar, the possible number of days for any period of 19 consecutive years, and their frequency of occurrence within the calendar cycle are as follows

             number of days= 6,938 number of times=  11,263 
                             6,939                  311,544 
                             6,940                  250,123 
                             6,941                  113,011 
                             6,942                    3,531 

Each of the shortest periods is distributed equally among each of the leap years in the 19 year cycle. That is, the shortest period of 19 years can occur 1,609 times starting from any of the leap years in a mahzor katan.

What, today, is the value of the year of the Aera Alexandri?

The Hebrew calendar was subject to several different ways of counting the years.

Some year counts would start at the time a new king of ancient Israel ascended the throne. Other year counts would begin on the basis of biblical events, such as the first year in the life of Adam.

The current year count is based on an interpretation of the biblical text as regards the number of years since Bereshit, i.e., Creation, or Yetsirah. This count is now valued at 5760. However, different schools of biblical studies establish different counts.

It is not known at what point in the history of the Hebrew calendar the Yetsirah count was first adopted as the universal count.

In the medieval work The Chronology of Ancient Nations (4760H/1000g) by the Muslim author Al-Biruni, the Yetsirah count is not reported as being used by any of the Jewish communities then known to the author.

Some 175 years later, in the work Hilchot Kiddush HaHodesh, Maimonides uses the Yetsirah count in relation to details of the Hebrew calendar rules.

However, both of these medieval scholars do mention the Aera Alexandri, and do leave behind some definite details which help to correlate that particular year count to the Yetsirah count.

In Hilchot Kiddush HaHodesh, at 11:16, Maimonides indicates that year 4938H is the year 1489 li'shtorot (of contracts). That year count, according to rabbinic tradition, was the number of years that had elapsed since Alexander the Great ascended the throne, thus beginning the
Aera Alexandri year count. This particular year count is also referred as the epoch of the Seleucids.

Since the difference between the Yetsirah count and the Aera Alexandri count is 3449, the current year of the Aera Alexandri is 5760 - 3449 = 2311.

In the Hebrew calendar, what is the proportion of 29 day months to 30 day months?

The proportion of 29 day months to 30 day months is 12,167 / 13,753 = 0.8847

Correspondents Ben Dreyfus, Larry Padwa, and Winfried Gerum all sent the correct answers, showing two different ways of approaching the problem. All three had one solution in common.

Ben Dreyfus expressed his solution as follows

In 353-day years there are 7 29-day months and 5 30-day months.
Likewise, 354 = 6,6;  355 = 5,7;  383 = 7,6;  384 = 6,7;  385 = 5,8.
 
Multiply these by the number of years of each type in the full cycle to
find the frequency of each month in the full cycle:
 
29: 69222*7+167497*6+198737*5+106677*7+36288*6+111051*5 = 4002943
30: 69222*5+167497*6+198737*7+106677*6+36288*7+111051*8 = 4524737
 
Therefore, 29-day months are 46.9% of the total,
and        30-day months are 53.1%.

It is very difficult to quarrel with successful results. However, the fraction of day method presented by Larry Padwa and Winfried Gerum is so elegant that it deserves a more rigorous mathematical approach to prove that it really does work.

For the full Hebrew calendar cycle of 689472 years, is it possible to show mathematically the actual proportion of 29 day months to 30 day months?

Let Cycle mean the full Hebrew calendar cycle.

The period of the molad is given by the constant 29d 12h 793p.

Let f stand for the fraction of the day represented by (12h 793p)/(24*1080) = 13753/25920.
Then the period of the molad can be expressed as 29 + f days.

Let x and y = number of 29 and 30 day months in the Cycle.

Then x*29 + y*30 = total number of days in the Cycle.

Since x + y = total number of months in the Cycle,

(29+f)*(x+y) = total number of days in the Cycle.

Consequently,

(29+f)*(x+y) = x*29 + y*30

leading to

y = f * (x + y)

Adding x to both sides of the equation,

        (x+y) - f*(x+y) = x
        (1-f) *   (x+y) = x

The proportion of 29 day months to 30 day months is then

        x = (1-f)*(x+y) = (1-f)
        y       f*(x+y)     f   
        

x/y = (25920-13753)/13753 = 12167/13753 = 0.88468

Several observations can be immediately made.

1. The number of 30 day months is directly related to the fraction f.
2. Because f > 0.5 there are more 30 day months than 29 day months.
3. The proportion of 29 day months to 30 day months is independent 
   of the number of days or months in the Cycle.
4. When the two month lengths are 1 day apart, then no matter what the
   length of the months, the same proportion x/y will result.

Using statistics relevant only to the 19 year periods, is it possible to find the proportion of 29 day months to 30 day months?

Using statistics relevant only to the 19 year periods, is it possible to find the proportion of 29 day months to 30 day months?

Let x and y respectively represent the number of 29 and 30 day months in any 19 year cycle.

Let d = the total number of days in any 19 year cycle.

Then x*29 + y*30 = d.

Now, x + y = the total number of months in any 19 year cycle = 235 so that y = 235 - x

Hence, x*29 + (235-x)*30 = d

leading to 7050 - d = x

The 19 year cycles can have either 6939, 6940, 6941, or 6942 days with an occurrence of either 17099, 13648, 5246, 295 times in the full Hebrew calendar cycle.

The following table can then be derived.

Therefore, the proportion of 29 day months to 30 day months is 4,002,943 / 4,524,737 = 0.8847

This topic was first touched upon in Question 29. But it seems that Winfried Gerum has the definitive answers.

It is assumed that both the Hebrew and Gregorian calendars keep their rules fixed over the intended period of time.

On what day will both the Hebrew and Gregorian calendars first show the exact same year?

Correspondent Winfried Gerum made a study of the time period during which the Hebrew and Gregorian year counts are equal.

This topic was first touched upon in Question 29. But it seems that Winfried Gerum has the definitive answers.

It is assumed that both the Hebrew and Gregorian calendars keep their rules fixed over the intended period of time.

The accuracy of the Hebrew calendar is fixed by the value of the mean lunation period coupled to the 19 year cycle of 235 lunar months.

That leads to an average Hebrew year length of 365.2468 days.

The mean Gregorian year is 365.2425 days.

Hence, the average Hebrew year is slower than the mean Gregorian year by about one day in every 230 years.

Consequently, the Gregorian year count, while still distant from the Hebrew year count, will eventually catch up to and exceed the Hebrew year count.

Correspondent Winfried Gerum predicts that the catchup will first occur on 28 Elul 317,675,828 corresponding to 1 January 317,675,828.

Winfried Gerum made another interesting observation.

On what day will both the Hebrew and Gregorian calendars first end the exact same year?

Correspondent Winfried Gerum made a study of the time period during which the Hebrew and Gregorian year counts are equal.

This topic was first touched upon in Question 29. But it seems that Winfried Gerum has the definitive answers.

It is assumed that both the Hebrew and Gregorian calendars keep their rules fixed over the intended period of time.

The accuracy of the Hebrew calendar is fixed by the value of the mean lunation period coupled to the 19 year cycle of 235 lunar months.

That leads to an average Hebrew year length of 365.2468 days.

The mean Gregorian year is 365.2425 days.

Hence, the average Hebrew year is slower than the mean Gregorian year by about one day in every 230 years.

Consequently, the Gregorian year count, while still distant from the Hebrew year count, will eventually catch up to and exceed the Hebrew year count.

Correspondent Winfried Gerum predicts that the catchup will first occur on 28 Elul 317,675,828 corresponding to 1 January 317,675,828.

Winfried Gerum also made the interesting observation that the Hebrew and the Gregorian calendars, if they do not change rules, will both first end the same year on the same day of
29 Elul 317,760,226 corresponding to 31 December 317,760,226.

What do Shabbat, Monday, and Wednesday have in common?

Correspondent Avi Veisz noted that

Shabbat, Monday, and Wednesday are excluded from Purim day.

The Talmudic sages prevented Purim from from being observed either on Monday, Wednesday, or Saturday.

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