Rosh Hodesh Elul will be observed on
Tuesday and Wednesday the 22nd and 23rd days of August.
The molad of Elul will occur on Tuesday at 16h 44m 15hl.
Hebrew Calendar Related Links
Convert Hebrew dates to Gregorian dates (and vice versa) using
Alan D. Corre's Hebrew  Gregorian Calendar
Map any Hebrew date to multiple calendar dates (and vice versa) using the
Calendar Converter
Shabbat Candle Lighting Times in Major Cities
Dvir Gassner's FREE Hebrew calendar software for Outlook
Jewish Calendar for OUTLOOK
Modern Lunar Phase Time Tables
Fred Espenak's Lunar Tables
The following shows the arithmetic rules of the Hebrew calendar and demonstrates some of the more intriguing calculation results. In many instances, the arithmetic results appear to overthrow long held assumptions related to the Hebrew calendar, thereby relegating the assumptions to the category of myths.
The arithmetic of the Hebrew calendar does not require any major mathematical skills. In fact, Maimonides once pointed out that
...the method of the fixed calendar is one which an average school child can master in 3 or 4 days. (Hilkhot Qiddush HaHodesh 11:4)
However, the development of the calendar's arithmetic rules embedded and demonstrated the considerable mathematical and scientific genius of the many unnamed scholars who devoted their skills to this unique problem.
Due to the nature of the arithmetic performed, and the lack of corroborating reference to many of the results, I can only suggest that what is being shown herein is reasonably correct to the best of my knowledge.
Hebrew Calendar Science and Myths uses the following notational and calculation conventions.
The current Hebrew calendar rules are ASSUMED as fixed for all time periods both past and future.
All Hebrew years will be suffixed with a capital H.
The current Gregorian calendar rules are ASSUMED as fixed for all time periods both past and future.
All Gregorian years will be suffixed with the lower case g.
All Julian years will be suffixed with the lower case j.
Hence, 5757H is the year that spans both 1996g and 1997g.
For purposes of convenience, a Gregorian year 0g is assumed to have existed between 1g and 1g. 0g spans 3760H and 3761H. 1g spans 3759H and 3760H. 1g spans 3761H and 3762H. In this convention 0g is a Gregorian leap year.
All Hebrew lunar days begin at exactly 18:00 hours which corresponds to hour 0 of the Hebrew calendar's day. So learn to recognize that in all subsequent calculations hour 0 is actually 18:00 or 6 p.m..
The rabbis also divided the hours into 1080 parts, thus making each part 3 and 1/3 seconds and each minute 18 parts. All calculation results are shown in days, hours and parts.
The days units will be denoted by numbers suffixed by the letter d.
The hours units will be denoted by numbers suffixed by the letter h.
The parts units will be denoted by numbers suffixed by the letter p.
Thus, 29 days, 12 hours, and 793 parts will be denoted as 29d 12h 793p.
The week days are numbered as follows
0 = Shabbat  1 = Sunday  2 = Monday  3 = Tuesday 
4 = Wednesday  5 = Thursday  6 = Friday  7 = Shabbat 
Although it might seem unecessary to number Shabbat as both 0 and 7, the traditonal literature does in fact use the numerical value 7 for that week day, and does represent that day by the Hebrew letter zayyin.
Unless otherwise noted, the term Hebrew Calendar will mean the day to day calendar derived from the traditional astronomical calculations of the lunar periods. The astronomical results are usually provided in days and fraction of the day. The Hebrew Calendar is provided only in integral days.
The time of birth of the new moon, ie, the MOLAD, is determined by the period of the MOLAD.
This period was determined to be 29 days, 12 hours, and 793 parts. The exact same value, in hexasegimal notation, is reputed by scholars to have been found in Claudius Ptolemy's Almagest [4:2] published about 150g.
The traditional time of the molad period is slower by about 1 day in every 15,304 years than the presently accepted time of the mean lunar conjunction.
The scholars Richard A. Parker and Waldo H. Dubberstein, in their paper Babylonian Chronology 626 B.C.A.D. 45, (The University of Chicago Press 1942), indicated that the Babylonians had been using a calendar system which used a cycle of 235 lunar months in a period of 19 years. They also indicated that the extra month lunar years were distributed in a cycle that is familiar to us today. That calendar knowledge apparently preceded Meton's discoveries by a number of years.
The ancient Greek astronomer Meton (c. 5th cent. b.c.e.) observed that 235 lunation periods practically equalled 19 solar years. He therefore suggested a cyclical method of distributing 7 extra lunar months into every period of 19 lunar years. It is not known if he borrowed the idea from the ancient Babylonians or determined that independently. Also, it is not really known how Meton actually distributed the extra month lunar years within this synchronized period of time.
Our scholars used a calendar cycle of 19 years consisting of 12 years of 12 lunar months each and 7 years of 13 lunar months each for a total of 235 lunar months. The Hebrew name for this cycle is mahzor qatan.
At some point in the history of the calendar, the beginning of the very first period of 19 years was determined, and years 3, 6, 8, 11, 14, 17, and 19 of the 19 year cycle were declared to be leap years of 13 months each.
This distribution of the leap years ensured that all Hebrew years in the 19 year cycle would begin, arithmetically at least, less than one lunar month after the start of their corresponding solar year in that cycle.
The distribution is easily remembered by the mnemonic GUChADZaT which stands for the Hebrew letters gimelvovhet alephdaledzayyentet.
A given Hebrew year is a leap year whenever its value divided by 19 leaves a remainder that is either 0, 3, 6, 8, 11, 14, or 17. In prezero arithmetic the number 19 corresponded to the zero remainder.
For example, the year 5757H (1996g/1997g) was a Hebrew leap year because after division by 19 the remainder is 0. That, by the way, also made year 5757H (1996g/1997g) the last year of the 303rd 19 year cycle.
In a Hebrew leap year a 30 day month is added to the year. This month is today known as the month of Adar I, or Adar Alef, or Adar Rishon, and is inserted immediately after the Hebrew month of Shevat. In our times, the insertion tends to take place in the February/March period of the Gregorian calendar year.
Presently, Hebrew leap years can begin no earlier than September 5 and no later than September 16, while Hebrew common years can begin no earlier than September 16 and no later than October 5.
For more detail please refer to
September 16
and also to Deriving GUChADZaT
BaHaRaD is the acronym given to the time of the mythical Molad shel Tohu. The new moon of the chaos is formally calculated to have taken place place on the 1st day of Tishrei in Hebrew year 1H, corresponding to
Monday 7 September 3760g.
The letters of the acronym BaHaRaD correspond to the time of Monday (bet); 5 hours (hey); and 204 parts (reshdaled).
By our conventions, we also denote BaHaRaD as 2d 5h 204p.
Since the Hebrew lunar days begin at 0 hours = 6 pm, 5 hours on Monday is actually 11 pm on the civilian Sunday.
For any given Hebrew year HY, you first count the number of months that have elapsed since 1 Tishrei 1H.
To calculate the number of months, Wolfgang Alexander Shocken, on page 35 of his book The Calculated Confusion of Calendars..., suggested the following formula which I've slightly modified:
The same formula can also be found somewhere around pages 9092 of Calendric Calculations by Nachum Dershowitz and Edward M. Reingold (Cambridge University Press 1997).
In June 2004, correspondent Edgar Efraim M. Rechtschaffen noted that HY is a leap whenever
the remainder of (235 * HY  234) / 19 is more than 11.
Once you begin to understand a little bit of The Gauss Pesach Formula, you can devise other variations of this month count.
For example, one possibility which I derived, shows the number of Hebrew months elapsed, up to Hebrew year HY since 1H, as
An interesting feature of my formula is that if the remainder of (12 * HY + 5) / 19 is less than 7 then year HY is a leap year.
You then multiply the mean lunation time of 29d 12h 793p by the integer result for the total number of months.
Adding the value of BaHaRad to the product provides you with the time of the molad of Tishrei for the Hebrew year HY.
That total when reduced to days (max of 6); hours (max of 23); parts (max of 1079) will give you the time of the molad for target year HY.
In June 2004, correspondent Edgar Efraim M. Rechtschaffen independently confirmed the correctness of this formula.
Thank you very much Edgar Efraim M. Rechtschaffen for sharing your fascinating insights into these calculation methods.
For Hebrew year HY, multiply 29d 12h 793p by the integer of ((235 * HY + 13) / 19) and add 3d 7h 695p to the resulting product.
The total, when reduced to days (max of 6); hours (max of 23); parts (max of 1079), will give you the time of the Tishrei molad for any target year HY.
Using HY = 5764 (Sat 27 Sep 2003g), INT((235 * 5764 + 13)/ 19) = 71,292 Then 71,292 * (29d 12h 793p) = 2,105,295d 2h 876p 2,105,295d 2h 876p + 3d 7h 695p = 2,105,298d 10h 491p 2,105,298d 10h 491p remaindered by 7d = 6d 10h 491p
The nontraditional approaches are just some of the many variants possible as a direct consequence of an article published in 1802g by the German mathematician Carl Friedrich Gauss. The mathematical formulas found in that article are shown in
The Gauss Pesach Formula.
Other formulas are also shown in Some Tishrei Moladot Formulas
See also The Moladot and Cycles and Moladot.
The above calculations provide the astronomical calendar of the moladot. The astronomical calendar has remained invariant since its introduction.
Since it is required that the Hebrew calendar be constructed in complete days, an initial approach at determining the Hebrew calendar based on the astronomical calendar would be to have the Hebrew years start on the weekday of the molad of Tishrei.
Assuming such an approach, then the above rules for calculating the Tishrei moladot lead to Hebrew years which could have either
were it not for the 4 postponement rules known as the Dehiyyot.
The Dehiyyot are very simple arithmetic rules whose functions are either to reduce the number of Qeviyyot, that is, Hebrew year types, or to ensure that no molad is ever preceded by the first day of its corresponding month.
The qeviyyah, that is, type, of any Hebrew year is defined by its length in days, and the week day corresponding to its Tishrei 1.
The 4 basic year lengths, described above, all begin on each of the 7 weekdays. Consequently, in the absence of all of the dehiyyot, there would be a total of 28 qeviyyot.
The effect of the dehiyyot on both the total number of the qeviyyot and the timing of the moladot will now be explained.
The day for Tishrei 1 may be postponed by up to two days depending on the time calculated for its molad.
The 4 special rules, each of which is called a dehiyyah (or postponement), function either to reduce the number of qeviyyot or to ensure that no molad is ever preceded by the first day of its corresponding month in the Hebrew calendar.
Dehiyyah Lo ADU Rosh is used to reduce the number of qeviyyot from 28 to 16.
Dehiyyah Lo ADU Rosh requires that the day of Tishrei 1 be postponed to the following day whenever the Molad of Tishrei occurs either on a Sunday, Wednesday or Friday.
The name ADU is the acronym formed from the Hebrew letters alef (=1 for Sunday), daled (=4 for Wednesday), and vov (=6 for Friday).
Dehiyyah Lo ADU Rosh has the negative consequence of increasing the number of Hebrew year lengths from 4 to 8. These lengths are either 353, 354, 355, 356, 382, 383, 384, or 385 days. The same lengths result even if only one day is specified for the postponement.
However, careful analysis of the arithmetical impacts of Dehiyyah Lo ADU Rosh proves that the rule actually reduces the number of qeviyyot from 28 to 16. The reduction leads to a far more practical Hebrew calendar, and therefore, easily overrides the negative consequences of the additional 4 Hebrew year lengths.
The elimination of 12 qeviyyot from the Hebrew calendar can be accomplished by disallowing for Tishrei 1 any 3 weekdays separated from each other by at least one week day. There are exactly 7 ways of making that choice, of which the combination ADU is one.
The combination of ADU may have been chosen because it has the religious effect of preventing Yom Kippur from occurring on either side of Shabbat, and Hoshannah Rabbah from occurring on Shabbat.
Dehiyyah Molad Zaqen ensures that the calculated time of any molad does not exceed the first day of any month in the Hebrew calendar.
Dehiyyah Molad Zaqen requires that Tishrei 1 be postponed to its next allowable week day whenever, on the day of the Molad of Tishrei, the time of the molad is 18 hours (ie, noon) or later.
The name for this rule is often translated into English as the old moon, or obsolete moon, or cancelled moon rule.
The qeviyyot would appear differently ordered, but not change, in a Hebrew calendar that excluded, or changed the value of, Dehiyyah Molad Zaqen.
Traditionally, Dehiyyah Molad Zaqen was considered as necessary so as to ensure the visibility of the new moon on the first day of Rosh Hashannah. However, this assumption is very doubtful, not only because of the calendar's arithmetical analysis, but also because many Hebrew calendar historians now indicate that Dehiyyah Molad Zaqen may have been implemented sometimes between the years 4596H (835/6g) and 4681H (920/1g). See, for example, Sacha Sterns's Calendar and Community (Oxford University Press 2001).
See further Understanding Dehiyah Molad Zaqen.
Dehiyyah GaTaRaD, which is not found in the Talmud, eliminates all of the 356day Hebrew years that resulted from the introduction of Dehiyyah Lo ADU Rosh.
Dehiyyah GaTaRaD requires that Tishrei 1 be postponed to Thursday whenever the Molad of Tishrei for a 12month year is on Tuesday at 9 hours and 204 parts or later.
The name GaTaRaD is the acronym formed from the Hebrew letters gimel (=3 for Tuesday), tet (=9), resh (=200), and daled (=4).
In the absence of Dehiyyah Molad Zaqen, the limiting time needed to eliminate 356day years would be 15h 204p on a Tuesday beginning a 12month year.
If the period of the molad had been at least 16 minutes longer, then the traditional calendar logic used to eliminate 356day years would have produced 357day years. This phenomenon is explained in Properties of Hebrew Year Periods  Part 1.
Dehiyyah BeTU'TeKaPoT, which is not found in the Talmud, eliminates all of the 382day Hebrew years that resulted from the introduction of Dehiyyah Lo ADU Rosh.
Dehiyyah BeTU'TeKaPoT requires that Tishrei 1 be postponed to Tuesday whenever the Molad of Tishrei following a 13month year is on Monday at 15 hours and 589 parts or later.
The name BeTU'TeKaPoT is the acronym formed from the Hebrew letters bet (=2 for Monday), tet (=9), vov (=6), tof (=400), kuf (=100), peh (=80), and tet (=9).
In the absence of Dehiyyah Molad Zaqen, the limiting time needed to eliminate 382day years would be 21h 589p on a Monday following a 13month year.
The 382day years can be eliminated simply by postponing to Tuesday any year beginning on a Monday following a 13month year. The number of Hebrew year lengths possible would remain either 353, 354, 355, 383, 384, and 385 days. However, this postponement would have the undesirable effect of producing the additional qeviyyah of 385day years beginning on Tuesday.
Together, the 4 Dehiyyot (Postponement Rules) lead to 6 possible Hebrew single year lengths which can be either 353, 354, 355, 383, 384, or 385 days long.
Beginning with Tishrei, the Hebrew months basically alternate between 30 and 29 days as follows:
Tishrei 30  Heshvan 29  Kislev 30  Tevet 29 
Shevat 30  Adar 29  Nisan 30  Iyar 29 
Sivan 30  Tammuz 29  Av 30  Elul 29 
For leap years the 30day month of Adar is added immediately after Shevat.
This particular placement of the leap month forces the use of Dehiyyah Molad Zaqen. Calendar arithmetic shows that if the leap month is placed prior to the month of Heshvan, then Dehiyyah Molad Zaqen is not required.
It is now necessary to compute the length of the year. Normally this is done by finding the Rosh Hashannah date of the subsequent year and differencing.
When the difference is either 355 or 385 days, then Heshvan gets a day to become 30 days.
When the difference is either 353 or 383 days, then Kislev loses a day to become 29 days.
The time of the Molad of Tishrei 5758H is 2,102,728d 4h 129p.
Dividing the days by 7 leaves a remainder of 5, which means that the Molad of Tishrei 5758H occurs on Thursday.
None of the dehiyyot apply to the Molad of Tishrei 5758H, and so, Rosh Hashannah 5758H begins on Thursday.
The time of the Molad of Tishrei 5759H is 2,103,082d 12h 1005p.
Dividing the days by 7 leaves a remainder of 2, which means that the Molad of Tishrei 5759H occurs on Monday.
None of the dehiyyot apply to the Molad of Tishrei 5759H, and so, Rosh Hashannah 5759H begins on Monday.
The difference in days between the years 5759H and 5758H is 2,103,082d  2,102,728d = 354d .
Hence, the length of Hebrew year 5758H, beginning on Thursday, is 354 days. Since we now have the qeviyyah for Hebrew year 5758H, it is possible to layout, not only all of its calendar details, but also most of the religious requirements that are calendar dependent, such as the occurrences of the Holidays, the Torah portions for any given day, the set of psalms to be read each day, and so on.
The above calculations do not include the Tequfot of Reb Shmuel which are prescribed in accordance to a completely different set of astronomical parameters, and so require additional arithmetic in order to be mapped onto the Hebrew calendar. This arithmetic, among other things, governs the addition or omission of certain liturgical phrases in such prayers as the Amidah.
See also Hebrew to Gregorian Date Conversion.
The annual Hebrew calendar period which begins on the first day of the 29day month of Adar and ends with the 29th day of Heshvan forms a constant period of 265 days.
It is within that period that may be found all of the biblically ordained festivals such as Pesach, Shavuot, Rosh Hashannah, Yom Kippur, Sukkot, and Shemini Atzeret.
The period of time beginning with the first day of Pesach on Nisan 15th up to and including Shemini Atzeret which occurs on the 22nd day of Tishrei is exactly 185 days long.
The period of time from the traditional first day of the vernal equinox which is normally March 21st up to and including the traditional first day of the autumnal equinox, usually September 21st, is also exactly 185 days long.
It would be interesting to know whether or not these two periods of time are the same length merely by coincidence.
It is to be noted that the starting day of the constant annual calendar period is fixed by the the first day of Tishrei for the immediately following Hebrew year and not from the day of Rosh Hashannah for the current Hebrew year.
In Hebrew, the six single year lengths give rise to the terms haser, qesidrah, and shalem.
Haser is the term applied to Hebrew years that are is either 353 or 383 days long. Such year lengths are accommodated by removing the 30th day of the month of Kislev. In English, the term haser is very often translated either as deficient or imperfect. The Hebrew letter het is used to denote either one of these year lengths.
Qesidrah is the term applied to Hebrew years that are is either 354 or 384 days long. Such year lengths are accommodated by keeping all of their month's lengths intact. In English, the term qesidrah is very often translated either as regular or intermediate. The Hebrew letter chof is used to denote either one of these year lengths.
Shalem is the term applied to Hebrew years that are is either 355 or 385 days long. Such year lengths are accommodated by adding a 30th day to the month of Heshvan. In English, the term shalem is very often translated either as abundant or perfect. The Hebrew letter shiyyen is used to denote either one of these year lengths.
Hebrew calendar arithmetic results in exactly 14 qeviyyot. This means that there are only 14 different ways in which to lay out the years of the Hebrew calendar.
The Tishrei 1 week day and the length of the year define the calendar layout required for that Hebrew year. Thus, when Hebrew years begin on
Mondays then they can have only either 353, 355, 383, or 385 days Tuesdays then they can have only either 354, or 384 days Thursdays then they can have only either 354, 355, 383, or 385 days Saturdays then they can have only either 353, 355, 383, or 385 days.
Calendar arithmetic develops the following pairwise sequences of Hebrew single year lengths.
13month years cannot immediately follow any other leap year.
Regular years cannot follow regular years.
Deficient years cannot follow deficient years.
Abundant years can follow abundant years.
These facts were well known as early as the year 1000g, as evidenced by the Muslim scholar AlBiruni's writings on the Hebrew calendar. (See E. C. Sachau's 1879g English translation of AlBiruni's The Chronology of Ancient Nations, and also Remy Landau's article AlBiruni's Hebrew Calendar Enigmas in Mo'ed, Journal for Jewish Studies, Vol 14, August 2003g).
384day years are always followed by 355day years.
A logical error, most often found in the Hebrew calendar literature, suggests that if a particular year is long (ie, abundant), then the next year must be shortened. Calendar arithmetic does in fact produce pairs of sequential abundant years.
Over the full and complete Hebrew calendar cycle, there are only 61 ways in which to arrange the sequences of Hebrew year lengths in all of its 19year cycles. This thought provoking observation is noted in Rev. Sherrard Beaumont Burnaby's 1901g book Elements of the Jewish and Muhammadan Calendars, beginning in Chapter 6, at page 146.
See also The Qeviyyot.
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.
Hebrew year spans that are multiples of 19 years have only one length in years, one length in months, but up to 5 lengths in days.
For any span of Hebrew years, the 9th year of the mahzor qatan always begins the shorter length in months.
For any span of Hebrew years, the 17th year of the mahzor qatan always begins the longer length in months.
The period of 137 Hebrew years is the first span to show all of the maximum number of lengths that are possible.
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 in days between the longest and the shortest lengths possible for a given period of Hebrew years. In the case of 137 years, the maximum variance is 33 days.
However, as shown in The 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 above 2 tables are fully explained in Properties of Hebrew Year Periods.
Properties of Hebrew Year Periods was developed in what appears to be a complete absence of related reference material. Should these eventually be found, or noted by correspondents, such relevant references will be credited.
The 19 year cycle does not cause the Hebrew calendar to repeat itself every 19 Hebrew years. The 19 year cycle only refers to the positions of the 13month years in those cycles. These years are the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of the cycle. Any of those years can be either 383, 384, or 385 days long.
Moreover all periods of 19 Hebrew years can be either 6938, 6939, 6940, 6941, or 6942 days each. Since none of these values are exact multiples of 7 it follows that no two consecutive periods of 19 years can begin on the same day of the week. Hence, the Hebrew calendar clearly does not repeat itself after every 19 years.
Incidentally, calendar arithmetic indicates that 19 consecutive Hebrew years cannot have the length of 6938 days if these years begin with a 12month year.
At one time some authorities suggested that the calendar would repeat itself after every 13 cycles of 19 years, that is, once every 247 years. However, simple arithmetic shows that the 247 year cycle is short by 905 halaqim (about 50 minutes) and therefore cannot be a full repetition cycle.
Periods of 247 Hebrew years can be either 90214d, 90215d, or 90216d. The period of 90,216d is a complete number of weeks and occurs 98% of the time over the full Hebrew calendar repetition cycle.
See further 247 Hebrew Year Periods for the statistical information.
The true calendar repetition cycle is 689,472 Hebrew years long, requiring 36,288 cycles of 19 years. This number of years was also known to AlBiruni as ealy as the year 1000g.
See also Proving the 689,472 Year Cycle.
After year 1H (on Monday, September 7, 3760g) the Molad of Tishrei will next occur at Baharad for the following Hebrew years
HYEAR  Civ. Date 

117,358H  04 Jan 113,599g 
308,063H  16 Apr 304,306g 
498,768H  26 Jul 495,013g 
689,473H  04 Nov 685,720g 
Of these 4 times only the year 689,473H leaves a remainder of 1 when divided by 19. Therefore, it is the very first year of a mahzor qatan and as such marks the repetition of the calendar as at year 1H.
An intriguing observation, not found anywhere in earlier Hebrew calendar literature, is that the Tishrei moladot are repeated exactly either 3 or 4 times over the complete Hebrew calendar repetiton cycle.
Over the full Hebrew calendar cycle of 689472 years, the Tishrei moladot that are for the 11th, 13th, and 15th years of the calendar's 19 year cycles are repeated exactly 3 times, while all of the other Tishrei moladot are repeated exactly 4 times.
The Tishrei Moladot above show Rosh Hashannah in seasons that appear to be anything but autumn. Also, the differences between the Hebrew and the gregorian years shown are NOT 3761.
The accuracy of the Hebrew calendar is fixed by the value of its mean lunation period coupled to the 19 year cycle of 235 lunar months.
That leads to an average Hebrew year length of aproximately 365.2468 days.
Since the mean tropical solar year is about 365.2422 days, the average Hebrew year is slower than the average solar year by about one day in every 216 years.
Hence, on average today, we celebrate our holidays, about 8 days later than did our ancestors in 4119H (358/9g), at the time that the fixed calendar rules were said to have been published by Hillel II.
Should no Hebrew calendar reform take place to account for the known astronomical differences, then over the next few millenia, all of our holidays will have drifted out of their anticipated seasons and Pesach could theoretically be observed in winter.
The actual repeatable cycle of the Gregorian calendar is 400 gregorian years, of which 97 years are gregorian leap years, that is, 366day years. Hence, the average Gregorian year is (400 * 365 + 97) / 400 = 365.2425 days long.
The Gregorian calendar is therefore slower than the mean tropical solar year by about 3 days in every 10,000 years.
So, if left unchecked, then the Gregorian calendar could also have all of its dates travel the seasons over very large periods of time.
The Hebrew calendar fully repeats itself in a cycle of 689,472 Hebrew years.
The Gregorian calendar fully repeats itself in a cycle of 400 Gregorian years.
Due to the fact that both the Hebrew years and the Gregorian years are of different lengths, the correspondence between the Hebrew and Gregorian calendars repeats itself in a cycle of
14,389,970,112 Hebrew years
which is also 14,390,140,400 Gregorian years!
In days, 14,389,970,112 Hebrew years = 5,255,890,855,047 days = 14,390,140,400 Gregorian years!
The Hebrew calendar fully repeats itself in a cycle of 689,472 Hebrew years.
The Julian calendar fully repeats itself in a cycle of 28 Julian years.
Due to the fact that both the Hebrew years and the Julian years are of different lengths, the correspondence between the Hebrew and Julian calendars repeats itself in a cycle of
1,007,318,592 Hebrew years
which is also 1,007,309,828 Julian years!
In days, 1,007,318,592 Hebrew years = 367,919,914,677 days = 1,007,309,828 Julian years!
The above mean values indicate that the average Hebrew year is slower than the average Gregorian year by about 1 day in every 231 years.
Because of the differences in the average years of these two calendars, Rosh Hashannah today cannot occur any earlier than September 5. Nor can Rosh Hashannah occur today any later than October 5.
The earliest possible Rosh Hashannah last occurred in 1899g and will next begin on that Gregorian date in 2013g.
The latest possible Rosh Hashannah last occurred on October 5, 1967g and will next occur on that Gregorian date in 2043g.
Rosh Hashannah, after the year 2089g, will not be able to occur any earlier than September 6.
The earliest possible Rosh Hashannah always begins the 17th year of a mahzor qatan.The latest possible Rosh Hashannah always begins the 9th year of a mahzor qatan.
In terms of the Hebrew calendar drift, the first gregorian dates for Rosh Hashannah always occur on the 9th year of a mahzor qatan, while the last gregorian dates for Rosh Hashannah always occur on the 17th year of a mahzor qatan.
See also The Rosh Hashannah Drift.
The gradual change over time in the difference between the Hebrew year value and the Gregorian year value usually goes unnoticed, since since most who map Hebrew dates onto the equivalent Gregorian dates normally do not use spans that exceed 10,000 years.
We are accustomed to determining the Hebrew year at Rosh Hashannah by adding the constant 3761. For example, by adding 3761 to Gregorian year value 1996g we get the Hebrew year value 5757H.
If we assume, for purposes of formal calculations, that the rules of both the Gregorian and the Hebrew calendar remain fixed indefinitely, then the assumption of the constant 3761 can be seen to be formally incorrect over indefinitely long time. That value can actually be seen to be decreasing over unrealistically large periods of time at the rate of 1 year in approximately every 84,500 Gregorian years.
by the year  the value will begin to be 

22,203g  3760 
106,716g  3759 
191,305g  3758 
.  . 
.  . 
613,756g  3753 
.  . 
I'll agree that this information is of absolutely no practical value for mostly everybody, except the computer programmer who runs his little calendar algorithm past the 22,203g mark and is led to believe that his program has some kind of weird bug because the difference between the Hebrew and the Gregorian years is not 3761. This phenomenon will also be a severe nuisance to all who reiterate the Gauss Pesach formula much past a certain number of years.
However, this very unexpected phenomenon can easily be traced back to the relative speeds between the Hebrew and Gregorian calendars. The Gregorian calendar is faster than the Hebrew calendar and so completes its years just a little bit more quickly, thereby increasing its years faster than the Hebrew calendar.
It is interesting to note that if no changes take place to either the Hebrew or Gregorian calendars, then the Hebrew year value, at Rosh HaShannah, will first equal the Gregorian year value on Shabbat 1 Tishrei 317,760,208 corresponding to 2 January 317,760,208.
I have not yet found other reference to this nonintuitive fact, but believe that the above results are reasonably correct.
In 1802g, the famous German mathematician Carl Friedrich Gauss (1777g1855g) published a formula which gave the Julian date of Pesach for any Hebrew year A.
The Gauss Pesach formula appeared without proof of its derivation in “Berechnung des jüdischen Osterfestes”, Monatliche Correspondenz zur Beförderung der Erd und HimmelsKunde, 5 (1802), 435437 – reprinted in: Carl Friedrich Gauss Werke (Königlichen Gesellschaft der Wissenschaften, Göttingen, 1874), vol. 6, pp. 8081.
A QBasic program which demonstrates one form of the Gauss Pesach formula is found in the Additional Notes under The Gauss Pesach Formula.
Given that the full Hebrew calendar cycle is 689,472 years it is possible to determine accurately the frequency of postponements, the occurrences of Rosh Hashannah, and other calendar related statistics.
An old Jewish tradition suggests that Tuesday is a good day because it was twice blessed at Bereshit (Creation). (See Genesis 1:913). Hence, it should follow that Tuesday would be the most popular day on which to start Rosh Hashannah. Amazingly, Tuesday ranks a very poor 4th place among the 4 permissible start days of the week.
In the full calendar cycle Rosh Hashannah begins on
Weekday  No. of Times  Frequency 

Thursday  219,831  31.9% 
Saturday  196,992  28.6% 
Monday  193,280  28.0% 
Tuesday  79,369  11.5% 
Moreover, Yom Kippur can never occur on a Tuesday!
But all is not lost. Even though Jewish tradition suggests that Pesach took place on a Thursday, it is Tuesday that is the most popular start day for Pesach!
Over the full Hebrew calendar cycle of 689,472 years:
o 268,937 (39.0%) occurrences of Rosh Hashannah are NOT postponed. o 323,824 (47.0%) occurrences of Rosh Hashannah are postponed by 1 day. o 96,711 (14.0%) occurrences of Rosh Hashannah are postponed by 2 days. o 295,488 (3/7) occurrences of Rosh Hashannah are postponed due to the Lo Adu rule. o 98,496 (1/7) occurrences of Rosh Hashannah are postponed due to Dehiyyah Molad Zaqen. o 22,839 (3.31%) occurrences of Rosh Hashannah are postponed due to Dehiyyah GaTaRad. These have been advanced into the 354 day years beginning on Thursdays. o 3,712 (0.54%) occurrences of Rosh Hashannah are postponed due to Dehiyyah BeTU'TeKaPoT. These have been advanced into the 354 day years beginning on Tuesdays. o Only 14 qeviyyot (year layouts) are needed.
Over the full Hebrew calendar cycle of 689,472 years:
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 
Shabbat  29853  0  94563  40000  0  32576  196992 
TOTALS  69222  167497  198737  106677  36288  111051  689472 
o 353day years occur 69,222 (10.0%) times. o 354day years occur 167,497 (24.3%) times. o 355day years occur 198,737 (28.8%) times. o 383day years occur 106,677 (15.5%) times. o 384day years occur 36,288 (1/19) times. o 385day years occur 111,051 (16.1%) times.
It is to be noted that the 355day year is the most frequently occurring kind of year, while the 384day year is the least frequently occurring type of year.
Not only do 384day Hebrew years only begin on Tuesdays but they occur exactly 1/19 of the time.
It is also interesting to note that Rosh Hashannah begins on Shabbat in exactly 2 out of every 7 years.
Dehiyyah Molad Zaqen is the arithmetical device which ensures that the first day of any Hebrew month never completes prior to the calculated time of its corresponding molad.
o Suppose that the molad of Tishrei for a 384day year is on Tuesday at 21h 68p.
o Then, the molad of Shevat for that year is on Tuesday at 0h 0p.
o Since the year is 384 days long, there are 118 days from 1 Tishrei to 1 Shevat.
o Therefore the first day of Shevat is on Monday, which is the day before its molad.
An extra day added between Tishrei and Shevat eliminates the precedence of the first day of the month over its molad. And, based on this example, that could be done by postponing to Tuesday the start of the following year whose molad of Tishrei is on Monday at 18h 657p.
Although analysis shows that the postponement can be done as late as 18h 657p on the day of the molad of Tishrei, Dehiyyah Molad Zaqen has been constructed to limit the time to no later than 17h 1079p.
Dehiyyah Molad Zaqen creates considerable puzzlement and debate among scholars, some of whom have questioned whether or not it was actually rooted in R. Zera's dictum found in the Talmud tractate Rosh Hashannah 20b. Traditional references imply that Dehiyyah Molad Zaqen has something to do with the visibility of the new moon on Rosh Hashannah, possibly over Jerusalem. Calendar arithmetic, however, suggests a more compelling but entirely different functionality for this rule.
Using the presently known fixed Hebrew calendar arithmetic rules, it is possible to determine that the closest that any molad can occur to the 2nd day of any Hebrew month is 36 and 5/9 minutes.
Theoretically, this maximum time first occurs on 1 Shevat 128,459H (124,700g).
When Dehiyyah Molad Zaqen is removed from the calendar rules, the calculated time of the molad can be seen to exceed the first day of some months by as much as 5 hours, 23 and 4/9 minutes, which by no coincidence is exactly 6 hours later than its current maximum value.
Support for this function of Dehiyyah Molad Zaqen appeared on page 37 of Wolfgang Alexander Shocken's book The Calculated Confusion of Calendars... in which he noted under the topic of Dehiyyah Molad Zaqen that
The calendar makers wanted to make sure that no month would begin before the actual setting in of the New Moon.It is unfortunate that Shocken did not explain how he came to that particular correct conclusion. Perhaps he felt that the matter was obvious and therefore would be an easy task for his readers to solve.
Although not immediately apparent, it can be shown if the leap month is placed prior to the month of Heshvan then there is no need whatever for Dehiyyah Molad Zaqen.
Modern historians believe, from evidence found in the Cairo Genizah, that Dehiyyah Molad Zaqen may have been introduced sometimes between the years 4596H (835/6g) and 4681H (920/1g). See, for example, Sacha Sterns's Calendar and Community (Oxford University Press 2001).
See also The Overpost Problem.
In 4681H (920g), Aaron ben Meir of Palestine sought formally to increase the Dehiyyah Molad Zaqen postponement limit to 18h 642p. That limit could have been increased up to 18h 657p. Saadia ben Yosef of Babylon, for what appears to be nonarithmetical reasons, opposed the change, thereby triggering a mahloqet (controversy) that may have lasted as much as five years. About 100 years ago, the scholar Joshua Heschel recovered from the Cairo Genizah in Egypt some of the 10th century correspondence related to this controversy.
Scholars of the MeirSaadia Calendar Controversy (mahloqet) have never once before noted that Dehiyyah Molad Zaqen could have had its limit set higher by at most 657 halaqim. This knowledge would have lent, and does lend, considerable support to the idea that Ben Meir had wanted to eliminate as many unecessary postponements as possible from the settings of the Hebrew years. In his classic book Saadia Gaon: His Life and His Works, (1926), Henry Malter suggested on page 80
Another, more acceptable explanation is that Ben Meir's real purpose was to reduce the number of postponements provided for in the accepted calendar.
Calendar arithmetic does show such a reduction. Applying the Ben Meir rules we find that, over the full calendar cycle of 689,472 years, a total of 9,760 unnecessary postponements are eliminated. The number of postponements drops from 420,535 to 410,775, representing a 1.42 percent reduction. Thus, by adding 642 halaqim to the rule's limit, it can be seen that Ben Meir had sought to optimize the time that would be permitted for the molad of Tishrei before declaring the requisite postponement.
As stated previously, the qeviyyot would appear differently ordered, but not change, in a Hebrew calendar that excluded, or changed the value of the limit of, Dehiyyah Molad Zaqen. This additional time to the rules meant that the 2 day Rosh Hashannah postponements which would otherwise have been required for each of the years 4683H (922g) and 4684H (923g) would not be invoked. In turn it meant that the ben Meir rule changes would cause all of the major Jewish festivals in 922g and 923g to be celebrated two days earlier than otherwise.
Ben Meir's proposed calendar change was met with vigorous opposition from Saadia ben Yosef (later known as Saadia Gaon). And so, according to some historical records, during the years 922g and 923g, those parts of the Jewish world which accepted ben Meir's rulings actually did celebrate the holidays of Pesach, Shavuot, Rosh Hashannah, Yom Kippur, and Sukkot two days earlier than the rest of the Jewish communities world wide.
Within Ben Meir's lifetime, the next Rosh Hashannah that could have been affected by his changes would have been 4688H (927g). However, no documents appear to exist which would have indicated another 2 day split in the observances of Rosh Hashannah.
It is important to note that technically, Ben Meir made no change whatever to the astronomical calendar of the moladot. He only proposed a small correction to the religious component of the Hebrew calendar. As such, he made no changes whatever to the fundamental structures of the Hebrew calendar.
Calendar arithmetic points to an interesting possibility as to why ben Meir would have chosen to proclaim his calendar changes in the year 922g.
First, his proposals would have taken effect for the years 922g, 923g, and also 927g. The next time that his changes would be seen would be 181 years later in the year 1108g and it would have only been for that year. Immediately after that the new rules would have taken effect in the years 1330g, 1334g, and 1335g.
So it seems that ben Meir's choice of 922g seized a priceless historical opportunity to have the suggested changes very much impressed during his own lifetime.
By one of these intriguing coincidences, under the fixed calendar rules, 922g actually marked the first time in the fixed calendar's history that the latest possible date for Pesach and Rosh Hashannah had advanced once again by one day. Under the existing calendar rules, Rosh Hashannah 4683H (922g) occurred for the very time in history on October 1 .
In the Julian calendar, this was the last time that Rosh Hashannah would occur on September 26.
Perhaps, among other things, Saadia might have known and wanted to cherish that particular historical moment.
To date, traditional scholarship of the issues in the MeirSaadia dispute appear to have concentrated almost entirely on what appeared to have been the political issues of the debate. Apparently, none of the scholars had ever thoroughly analyzed this bit of the calendar's arithmetic, and so they all essentially missed the key contribution to the calendar that Ben Meir had proposed, which in effect was the optimization of Dehiyyah Molad Zaqen.
By concentrating on the political issues, rather than on the technical merits of this debate, historians failed to reconstruct the calendar science which led Ben Meir to choose a value that had come to within 15 halaqim (50 seconds) of the maximum allowable limit. And that is unfortunate since such a study could have given us a much better feel for Ben Meir's genius with our calendar's calculations.
Scholars today are split on the issue as to whether or not there was a winner in this calendar quarrel. After the year 4688H (927g) the next time that the Ben Meir rules would have caused a split in the traditional Rosh Hashannah timing would have been 4689H (1108g), namely, 181 years later. So from the calendar arithmetic perspective, it seems that the debate may have died off in the crush of 181 years of medieval Jewish history.
The debate also seems to have very significantly halted any further progress in the science of the Hebrew calendar. A mere 75 years after the controversy, the Persian born Muslim scholar AlBiruni, in his writeup of the Hebrew calendar, would omit all references to Dehiyyah Molad Zaqen. It was not until 1802g that the next most significant advance was made to the Hebrew calendar's computus in the form of a formula published by the German mathematician Carl Friedrich Gauss.
See also The Ben Meir Years and The Gauss Pesach Formula.
The MeirSaadia confrontation of 920g was not about the accuracy of the calendar. Rather it was about a change which would have optimized the limiting time of Dehiyyah Molad Zaqen. Since then, no change has been made to the Hebrew calendar. However, there have been a number of suggestions for its reform so as to make its calculations more astronomically accurate.
Arithmetically, it is quite possible to improve the difference between the average Hebrew year and the known mean solar year.
On page 208 of Rabbinical Mathematics and Astronomy W. M. Feldman suggested a Hebrew calendar of 334 years in which the first 17 cycles of 19 years would remain the same. There would be a left over period of 11 years whose leap year distribution would be the same as it is now. After those 11 years the next cycle of 334 years would begin.
Feldman believed that this would make the Hebrew calendar accurate to within one day in 12,500 years.
This change in the calendar cycle does not affect any of the known religious requirements of the calendar. But it is only an arithmetical change and so suffers from the drawback of not allowing for some method of also introducing the changes that are always taking place in the astronomical parameters that at one time were believed to be eternal.
However, the most serious fault in the Feldman model is that it allows the Hebrew years to begin more than one month after the start of their corresponding solar years in the cycle. Deriving GUChADZaT in the Additional Notes explains this relatively unknown idea.
It is unknown when the Hebrew calendar will one day be improved so as to more accurately maintain the holidays in their proper seasons.
Since Mosheh is said to have lived exactly 120 years, considerable importance has traditionally been attached to the length of such life spans. Calendar arithmetic reveals that any 120 Hebrew years as measured from Rosh Hashannah to Rosh Hashannah can have the following lengths in days
120 YEAR SPANS  

1,484 months = 43,823d 9h 692p 
1,485 months = 43,852d 22h 405p 

M'+/  DAYS  MOD 7d  OCCURS  M"+/  DAYS  MOD 7d  OCCURS  
2  0d  0  0  2  0d  0  0  
1  43,822d  2  114,296 
1  43,851d  3  1,404 

0  43,823d  3  188,409 
0  43,852d  4  21,423 

1  43,824d  4  167,629 
1  43,853d  5  96,784 

2  43,825d  5  73,986 
2  43,854d  6  19,332 

3  0d  0  0  3  43,855d  0  6,209 

The maximum variance is 33 days 
The above table is fully explained in Properties of Hebrew Year Periods.
The shortest possible period of 120 Hebrew years is 43,822 days long, while the longest period is 43,855 days long. Hence the shortest period varies from the longest possible 120 Hebrew year period by only 33 days!
In the full Hebrew calendar cycle of 689472 years, the longest period of 120 Hebrew years begins on only 6,209 Rosh Hashannah's, that is, on only 0.9% of all the new years possible.
Between the years 1800g and 2100g only the years 5676H (1915g) and 5774H (2013g) can be found to begin the periods of 43,855 days. Incidentally, 5774H (2013g) will begin on September 5, which in our times, is the earliest possible Gregorian date for the first day of Rosh Hashannah.
Because the longest period of 120 Hebrew years is an exact multiple of 7, that is, 7 times 6265, the 121st Rosh Hashannah will always begin on the same day of the week begun for that 120 year period.
By one of these unbelievable coincidences, not only do all of the Rosh Hashannah's which begin the longest 120 year periods start on Thursday, and inaugurate 385day Hebrew years, but so does the very last year of each of these spans. The 121st years following these spans also begin on Thursday, and are all 354 days long.
It is to be hoped that some mathematically inclined individual will one day try to figure out why that is so.
See also Properties of Hebrew Year Periods.
It is a highly beloved Jewish tradition to wish each other biz a hunderdt und zwanzig, which is the Yiddish idiom meaning may you live until 120 years of age.
Now, the shortest possible period of 120 Hebrew years is 43,822 days long, while the longest period is 43,855 days long. Hence the shortest period varies from the longest possible 120 Hebrew year period by 33 days!
Because the shortest 120 Hebrew year period is 33 days less than the longest such span, and in order not to deny anyone a single precious day of life, it is now absolutely necessary to wish each other...
Ver 1 Paged 20 Feb 1997 Ver 2 Paged 16 Nov 2003 Next Revised 6 Aug 2017