© J R Stockton, ≥ 2011-03-29

Links within this site :-

**Merlyn Home Page**- Site Index, E-Mail, Copying**This Page**:-- Hebrew Calendar, A.M. 5700-6000 - condensed
- Pascal Date Routines
- For more date/time pages, and links, see Date and Time Index and Links

In the Hebrew Calendar, the Day starts at sunset (Genesis, Ch.1 v.5); but for lunar-based calendar calculation, the zero hour used is at 1800h (6 p.m.), civil time or Jerusalem meridian (2h 21m E of Greenwich) time. A Day is 24 hours, each of 1080 parts (halakhim). Consequently, the Week, Month, and Year all start at that time of day too.

Ordinarily, the mapping of days between "sunset-start" and "midnight-start" calendars ignores the quarter-day difference. Therefore, one can say "The first day of Pesach 5767 was Tuesday 3rd April 2007", although that Pesach actually began at around sunset on Monday 2nd April 2007.

The Days of the Week are the same as in the Gregorian calendar, apart from starting a few hours earlier; they have their own distinct names in Hebrew.

The Week ends with Saturday = Shabbat (which begins on Friday at local sunset). The count is Sunday = 1 to Friday = 6, Shabbat = 7 or (in some calculations) 0. (There is a "Planetary Week" beginning with Saturday = 1). The Bible, of course, has the Sabbath as the Seventh Day (and the seven-day cycle extends properly from present times back to the first chapter of the Book of Genesis, when using Archbishop Ussher's dating).

The Month starts on or soon after the calculated day of the New Moon
(Molod (or Molad)). A lunar month is taken as 29d 12h 793p (29d 12:44:03.33),
which, with non-cumulative complications (postponements), determines the
start of each Year (the earliest Molod was
Monday BC 3761-10-07 + 5h 204p
["Molad beharad"; or "Molad efes", *lit.* "the zeroth Molad"]).

Whitaker (1989) gives 29d 12:44:02.9 for the lunar month; the difference of 0.43 s is 0.17 ppm, or about one day in 16000 years. For Moon phases, and eclipses, see Astronomy / Astronautics 2.

The Month is always 29 or 30 days long.

The basic sequence of Month lengths is 30, 29, repeating. Months 2 and 3 vary in length, with 2 ≤ 3; the lengths of Months 2 and 3 follow from the length of the Year. Most Months can start on any day of the Week.

The Months are Tishri, Heshvan or Marcheshvan, Kislev; Tebet, Shebat, Adar, Adar II (Adar Sheni); Nisan, Iyar, Sivan; Tammuz, Ab, Elul - the English spellings seem to vary.

Note : There's apparently a Civil and a Religious year; they start six months apart (careful!). The Civil Calendar Year starts with Tishri and the Religious Year with Nisan, as I understand it.

The Year contains either 12 or 13 Months, in a 19-year cycle. Seven years of each nineteen are Leap (3, 6, 8, 11, 14, 17, 19) with the extra 30-day month Adar 1, placed sixth. There are thus 12×12 + 7×13 = 235 Months in every 19 consecutive years - a Metonic Cycle, closely approximating 19 astronomical years.

Leap := (Year mod 19) in [0, 3, 6, 8, 11, 14, 17] ; Leap := ((Year*7 + 1) mod 19) < 7 ; Leap := ((Year*7 + 13) mod 19) >= 12 ; Leap := ((Year*7 + 13) mod 19) div 12 { 0/1 } ;

A Regular Year thus has 353/354/355 days, and a Leap Year has 30 days more; the lengths all have different frequencies.

The first day of the Hebrew Calendar Year is 1st Tishri (Rosh Hashana).

The Lunar cycle gives a day on which the year may start; but if the calculated New Moon is after noon, the start is postponed by one day. That postponement may be also needed for consistency with other requirements.

Two festivals must avoid specific days of the week; so 10th Tishri avoids Friday and Sunday, and 21st Tishri avoids Saturday. Therefore, the start of the year can only be on a Saturday, Monday, Tuesday, or Thursday, and so may be (perhaps again) postponed by a day. These start days have rather unequal frequencies.

There are six possible year lengths, and four days of the week on which the year may start. But there are only fourteen possible year types, since ten combinations of start-day and year-length do not occur.

Only a 385-day year has an exact number of weeks. No year contains one or two "extra" days over a multiple of 7; the exact opposite of Gregorian/Julian years.

It is practical to use a numerical form such as A.M. 5768-09-27 if it is understood that years start with Nisan; there will at most be doubt as to whether -12- stands for Adar or for Adar I, and -13- will necessarily be Adar II.

But the form is inconvenient when the years start with Tishri, since the name of the month corresponding to numbers 07-12 would depend on whether the year happened to be Leap, which is by no means obvious.

I believe that month numbers are not used in stating dates.

The calendar is denoted by *Anno Mundi*, or A.M., meaning Year
of the World. I have read that there have been other A.M. calendars, but
only the Hebrew one is well-known.

The routines for the CMJD that starts a given Hebrew Year and for the Hebrew Year that contains a given CMJD which are now in hebclndr.pas have passed all my tests. A condensed calendar for A.M. 5700-6000 is in hebr-cal.txt.

As far as I see from Whitaker, special Dates are constant in Month-Day form, except for a couple that avoid the Sabbath. Month-Day dates from 1 Adar (in Leap Years, the later Adar) to 29/30 Heshvan (9 months, ending with month 2) are each at a constant offset from the start of the included Tishri (the important festivals are included); those in the intervening 3/4 months are harder to number.

The Creation is considered to have started on Elul 24, near the end of Year A.M. 1, in B.C. 3760; Adam was created on Elul 29, the last day of A.M. 1. Those dates are alternatively given as being one day later.

The first Sabbath (Saturday) was A.M. 2 Tishri 1.

Year A.M. 1 Day 1 contained the beginning of Julian B.C. 3761-10-07, Gregorian B.C. 3761-09-07, CMJD -2052003; it started at about JD 347997.25.

Year A.D. *X* begins during A.M. *3760+X*;
Year A.M. *3761+X* begins during A.D. *X*;
they will do so for tens of thousands of years.

Years counted from the start of A.M. 2 are
*Aera Adami*, A.A., rarely seen.

As there are just 14 types of Hebrew year, only 14 truly distinct Hebrew-only Diaries are possible.

The average Hebrew year length is about 365.2468 days - exactly, it is (29d 12h 793p)×(12×12 + 7×13)/19 = 365 + 121555/492480 = 365 + 24311/98496 = 35975351/98496 = 365.246822 (205977 907732 293697) recurring days. Here, 492480 = 24×1080×19.

From the above, it takes 98496 Hebrew years for the average
length in days to be exactly right, and this is a day under a
multiple of 7; hence the full Calendar can repeat after 689472 years.
It in fact does so; and
689472 = 2^{6}×3^{4}×7×19 years
= 2^{6}×3^{4}×7×(5×47)
= 8527680 months.

I find that it is possible for six consecutive years to contain all possible year lengths; from now, 5801 to 5806 first does so. The longest interval not containing all year lengths is 43 years; 5861 to 5903 is the next such, containing no 384-day year. They occur respectively 13104 and 2149 times per 689472 years. (Written A.M. 5762; to be checked)

The difference of about (365.2468222-365.2425000) days in average length makes a one-year difference between the Hebrew and Gregorian Calendars in about 84503 years; the exact interval is not a multiple of either Year.

The average Hebrew year length is 35975351/98496 days, which means that to get an exact number of days one needs 98496 *average* Hebrew years. The *average* is important.

However, 98496 consecutive actual Hebrew years will, I'm told, have four different lengths in days; but any 689472 = 7×98496 consecutive actual years will have a fixed number of days; from above, it is 7×35975351 = 251827457 days = 35975351 weeks.

In the Gregorian Calendar, there are 365.2425 = 146097/400 days per average year, and 400 *average* years are needed to get an exact number of days. So far, the circumstances are similar.

But the Gregorian Calendar, disregarding Easter, repeats every 400 years; *any* 400 consecutive years contain a total of 146097 days, which is exactly 20871 weeks.

Now for a Cyclic Diary Sequence, *i.e.*a set of diaries
to be used sequentially in perpetuity :-

(1) Julian without Easter needs 28 years, of 14 types;

(2) Gregorian without Easter (secular) needs 400 years,
of 14 types;

(3) Gregorian with Easter (religious) needs 5700000 years
(a multiple of 400), of 70 types;

(4) Hebrew alone needs 689472 years, of 14 types (as above).

Next there's the question of the relative lengths of the Hebrew and Gregorian years.

For an exact repeat of the joint secular or religious calendar, the period must be an exact multiple of either 400 or 5700000 years Gregorian AND 689472 years Hebrew - and those types of years are of differing numbers of days :-

- Secular : LCM(400×365.2425, 251827457) = 5255890855047 days = 14390140400 years Gregorian = 14389970112 years Hebrew (difference in year-count = 170288).
- Religious : LCM(5700000×365.2425, 251827457) = LCM(570×3652425, 251827457) = 74896444684419750 days = 205059500700000 years Gregorian = 205057074096000 years Hebrew (difference in year-count = 2426604000).

So, for an exact pattern repeat of the combined Hebrew and full Gregorian Calendars, it seems that one must wait some 205 million million years.

*HCF = Highest Common Factor ; LCM = Lowest Common Multiple.
Calculations by longcalc.*

Pesach [Passover] begins on 15 Nisan, always 163 days (23 weeks, 2 days) before the start of the next year; so the First Day of Pesach can only be Sunday, Tuesday, Thursday, or Saturday.

I have read that Gauss, who also worked on the date of Easter, gave a formula for the date of Pesach on the Julian Calendar; but I've not located any Web copy of the original.

The Date of Easter Sunday is historically linked with that of the First Day of Pesach, but is calculated differently. A comparison of their Full Moons is below at Hebrew and Gregorian Moons.

Pesach dates from my `hebclndr.exe 10`; some dates checked
independently.

First Day of Pesach (begins at previous sunset) :- 2000-04-20 Thu 2001-04-08 Sun 2002-03-28 Thu 2003-04-17 Thu 2004-04-06 Tue 2005-04-24 Sun 2006-04-13 Thu 2007-04-03 Tue 2008-04-20 Sun 2009-04-09 Thu 2010-03-30 Tue 2011-04-19 Tue 2012-04-07 Sat 2013-03-26 Tue 2014-04-15 Tue 2015-04-04 Sat 2016-04-23 Sat 2017-04-11 Tue 2018-03-31 Sat 2019-04-20 Sat 2020-04-09 Thu 2021-03-28 Sun 2022-04-16 Sat 2023-04-06 Thu 2024-04-23 Tue 2025-04-13 Sun 2026-04-02 Thu 2027-04-22 Thu Easter Sunday :- 2000-04-23 Sun 2001-04-15 Sun 2002-03-31 Sun 2003-04-20 Sun 2004-04-11 Sun 2005-03-27 Sun 2006-04-16 Sun 2007-04-08 Sun 2008-03-23 Sun 2009-04-12 Sun 2010-04-04 Sun 2011-04-24 Sun 2012-04-08 Sun 2013-03-31 Sun 2014-04-20 Sun 2015-04-05 Sun 2016-03-27 Sun 2017-04-16 Sun 2018-04-01 Sun 2019-04-21 Sun 2020-04-12 Sun 2021-04-04 Sun 2022-04-17 Sun 2023-04-09 Sun 2024-03-31 Sun 2025-04-20 Sun 2026-04-05 Sun 2027-03-28 Sun

Hebrew dates slowly drift later, on average, in the Gregorian year. If my calculations and imported data are correct, then, for years within the range AD 1754-9613 for which the First Day of Pesach is a Sunday :-

- A Sunday First Day of Pesach always precedes the Julian Easter Sunday.
- The last year with Gregorian Easter on a Sunday First Day of Pesach is AD 7485.
- The last year with Gregorian Easter after a Sunday First Day of Pesach is AD 6991.

If my Passover calculations are
correct in program `hebclndr`
and using the most recent rules * (Gregorian Easter : AD1582; Julian
Easter : ca.AD325; Hebrew : ca.AD1175, and Pesach [Passover] begins on
15 Nisan, always 163 days (23 weeks, 2 days) before the start of
the next year)*, then (in the range AD 1-11111) :-

- The last year with Julian Easter on the First Day of Pesach is AD 783, AM 4543. The last year with Julian Easter before the First Day of Pesach is AD 495, AM 4255. If the First Day of Pesach is after Julian Easter, it is the following Tuesday.
- The last year with Gregorian Easter on the First Day of Pesach is AD 7485, AM 11245. The last year with Gregorian Easter after the First Day of Pesach is AD 7466, AM 11266, for which the First Day of Pesach is on the previous Thursday.

- Remy Landau : extensive
**Hebrew Calendar**site, with weekly quiz - At webexhibits,
*via*: Calendars - Kees Couprie : Calendar Math - date converters
- Stephen P. Morse : Jewish Calendar Description, FAQ, Jewish Calendar Demystified, (+ .doc)
- I've been told : Zerubavel, E. 1985: The Seven Day Circle. The History and Meaning of the Week. New York & London, ISBN 0-02-934680-0
- Hermetic Systems : Calendrical and Astronomical Links
- Whitaker's Almanac

I have a little Pascal, but no JavaScript,
for the Hebrew Calendar - see `hebclndr` *via*
index.

*This section was moved to here from
The Calculation of Easter Sunday
on 2007-05-28.*

Easter Day is the Sunday following the calculated date of the Paschal (Passover) Full Moon, as above. The time of the Full Moon does not appear explicitly, and the New Moon is not used. Leap Years added an apparent irregularity to the Julian Paschal Full Moon. The Gregorian Paschal Full Moon is even less regular, since its behaviour is adjusted stepwise for better long-term agreement with the real Moon, and is further adjusted to preserve the set of possible PFM dates.

The Hebrew Calendar, including the date of
Passover (Pesach), depends on a regularly-behaved calculated New Moon
(a Molad) for its months, though the actual beginning of a month may be
postponed. The interval between molads is 29 days, 12 hours, and 793
halakim. One hour is exactly 1080 halakim; one halakim is exactly
3^{1}/_{3} seconds. Hebrew days begin at
the "previous" 18:00. Pesach begins on 15 Nisan, therefore at about
Full Moon. RL wrote that the molad of Iyar 5767H would occur on Tuesday
(2007-04-17?) at 18h 45m 13hl.

It should be possible to examine the degree of agreement between the Hebrew and Gregorian Moons.

Using program MJD_DATE 2007-04-15 - OddTests Paschal MeanMoon : Gregorian : 5700000 years have 70499183 lunar months (a prime number) Average 12.368277719298 per year, length = 29.53058690056025 days Molad 29d 12h 793p = 29 + (12 + 793/1080)/24 = 29.53059413580247 days Exact on average : 235 of those months in 19 Hebrew years The Hebrew Moon is therefore about 0.25 ppm slower than the Gregorian; Whitaker's Almanac 2001 : True Month = 29d 12h 44m 02.9s 29 + 12/24 + 44/1440 + 02.9/86400 = 29.53058912037037 days 5700000*365.2425 / 29.53058912037037 = 70499177.7005866 lunar months

The following probably needs detailed adjustment; consider it merely as a demonstration of possibility.

For each year, that shows the dates of the Paschal Full Moon and
Easter Sunday; then the interval in Hebrew Lunar Months from the first
Molad of Gregorian 1970 UTC to the PFM (here taken as at noon UTC, to
remove bias) and the Gregorian Hebrew date of the Hebrew Full Moon at
the middle^{*} of that Hebrew Lunar Month; and the difference
between the Full Moons both in real time and in respective dates. Note
that the above uses the previous Molod, rather than that of any
particular Hebrew Month; the HFM is not necessarily that of Pesach.

^{*} : Possibly should use the 14th Day.

No account need be taken of Time Zone or Summer Time; UTC is used throughout.

There is a related calculation, for Julian and Gregorian, by Stephen
P. Morse at Calculating
Easter using the Jewish Calendar in One Step. I have a program
`hebclndr`, *via* index,
for some Hebrew Calendar calculations.