Editing Hebrew calendar (section)

==Calculations==
===Leap year calculations===
{{See also|Golden number (time)}}
To determine whether a Jewish year is a leap year, one must find its position in the 19-year Metonic cycle. This position is calculated by dividing the Jewish year number by 19 and finding the [[remainder]]. (Since there is no year 0, a remainder of 0 indicates that the year is year 19 of the cycle.) For example, the Jewish year {{#time:xjY}} divided by 19 results in a remainder of {{#expr:{{#time:xjY}}mod 19}}, indicating that it is year {{#ifexpr:{{#time:xjY}}mod 19|{{#expr:{{#time:xjY}}mod 19}}|19}} of the Metonic cycle. The Jewish year used is the ''anno mundi'' year, in which the year of creation according to the Rabbinical Chronology (3761 BCE) is taken as year 1. Years 3, 6, 8, 11, 14, 17, and 19 of the Metonic cycle are leap years. The Hebrew mnemonic GUCHADZaT {{lang|he|גוחאדז״ט}} refers to these years,{{efn|In which the letters refer to [[Hebrew numerals]] equivalent to 3, 6, 8, 1, 4, 7, 9.}} while another memory aid refers to musical notation.{{efn|Intervals of the [[major scale]] follow the same pattern as do Jewish leap years, with ''do'' corresponding to year 19 (or 0): a [[whole step]] in the scale corresponds to two common years between consecutive leap years, and a [[half step]] to one common year between two leap years. This connection with the major scale is more plain in the context of [[19 equal temperament]]: counting the tonic as 0, the notes of the major scale in 19 equal temperament are numbers 0 (or 19), 3, 6, 8, 11, 14, 17, the same numbers as the leap years in the Hebrew calendar.}}

Whether a year is a leap year can also be determined by a simple calculation (which also gives the fraction of a month by which the calendar is behind the seasons, useful for agricultural purposes). To determine whether year ''n'' of the calendar is a leap year, find the remainder on dividing [(7&nbsp;×&nbsp;''n'')&nbsp;+&nbsp;1] by 19. If the remainder is 6 or less it is a leap year; if it is 7 or more it is not. For example, the {{Hebrew calendar/c|{{#time:xjY}}}} The {{Hebrew calendar/c|{{#expr:{{#time:xjY}}+1}}}}<ref>{{cite book |last1=Dershowitz |first1=Nachum |last2=Reingold |first2=Edward M. |title=Calendrical Calculations |title-link= Calendrical Calculations |date=2007 |edition=3rd |publisher=Cambridge University Press |page=91}}</ref> This works because as there are seven leap years in nineteen years the difference between the solar and lunar years increases by {{frac|7|19}} month per year. When the difference goes above {{frac|18|19}} month this signifies a leap year, and the difference is reduced by one month.

The Hebrew calendar assumes that a month is uniformly of the length of an average [[synodic month]], taken as exactly {{frac|29|13753|25920}} days (about 29.530594 days, which is less than half a second from the modern scientific estimate); it also assumes that a [[tropical year]] is exactly {{frac|12|7|19}} times that, i.e., about 365.2468 days. Thus it overestimates the length of the [[tropical year]] (365.2422 days) by 0.0046 days (about 7 minutes) per year, or about one day in 216 years. This error is less than the [[Julian Calendar|Julian years]] (365.2500 days) make (0.0078 days/year, or one day in 128 years), but much more than what the [[Gregorian Calendar|Gregorian years]] (365.2425 days/year) make (0.0003 days/year, or one day in 3333 years).

===Rosh Hashanah postponement rules===
Besides the adding of leap months, the year length is sometimes adjusted by adding one day to the month of Marcheshvan, or removing one day from the month of Kislev. Because each calendar year begins with [[Rosh Hashanah]], adjusting the year length is equivalent to moving the day of the next Rosh Hashanah. Several rules are used to determine when this is performed.

To calculate the day on which Rosh Hashanah of a given year will fall, the expected [[molad]] (moment of [[lunar conjunction]] or [[new moon]]) of Tishrei in that year is calculated. The molad is calculated by multiplying the number of months that will have elapsed since some (preceding) molad (whose weekday is known) by the mean length of a (synodic) lunar month, which is 29 days, 12 hours, and 793 parts (there are 1080 "parts" in an hour, so that one part is equal to {{frac|3|1|3}} seconds). The very first molad, the [[#Anno Mundi|molad tohu]], fell on Sunday evening at 11:11:20&nbsp;pm in the local time of [[Jerusalem]],<ref name=Tondering>{{cite web |url=https://www.tondering.dk/claus/cal/hebrew.php#newmoon |title=Calendar FAQ: the Hebrew calendar: New moon |first1=Trine |last1=Tøndering |first2= Claus |last2=Tøndering}}</ref>{{efn|UTC+02:20:56.9}} 6 October 3761 BCE ([[Proleptic Julian calendar]]) 20:50:23.1 [[Coordinated Universal Time|UTC]], or in Jewish terms Day 2, 5 hours, and 204 parts. The exact time of a molad in terms of days after midnight between 29 and 30 December 1899 (the form used by many spreadsheets for date and time) is
:-2067022+(23+34/3/60)/24+(29.5+793/1080/24)*''N''
where ''N'' is the number of lunar months since the beginning. ({{nowrap|''N'' {{=}} 71440}} for the beginning of the 305th Machzor Katan on 1 October 2016.) Adding 0.25 to this converts it to the Jewish system in which the day begins at 6&nbsp;pm.

In calculating the number of months that will have passed since the known molad that one uses as the starting point, one must remember to include any leap months that falls within the elapsed interval, according to the cycle of leap years. A 19-year cycle of 235 synodic months has 991 weeks 2 days 16 hours 595 parts, a common year of 12 synodic months has 50 weeks 4 days 8 hours 876 parts, while a leap year of 13 synodic months has 54 weeks 5 days 21 hours 589 parts.

Four conditions are considered to determine whether the date of Rosh Hashanah must be postponed. These are called the Rosh Hashanah postponement rules, or {{lang|he-Latn|deḥiyyot}}.<ref name=Ibbur>{{cite book|title=Sefer ha-Ibbur |volume= 2|chapter= 9,10 |author=R. Avraham bar Chiya ha-nasi |oclc=729982627 |date=1851 |language=he |location=London}}</ref><ref name=Tur>{{cite book|title=Tur, Orach Chaim ''(section 428)''|url=https://he.wikisource.org/wiki/%D7%98%D7%95%D7%A8_%D7%90%D7%95%D7%A8%D7%97_%D7%97%D7%99%D7%99%D7%9D_%D7%AA%D7%9B%D7%97}}</ref><ref name=HKC>{{cite book|author=Rambam|title=Hilchos Kiddush ha-Chodesh (chapters 6, 7, 8)|url=https://he.wikisource.org/wiki/%D7%A8%D7%9E%D7%91%22%D7%9D_%D7%94%D7%9C%D7%9B%D7%95%D7%AA_%D7%A7%D7%99%D7%93%D7%95%D7%A9_%D7%94%D7%97%D7%95%D7%93%D7%A9_%D7%95}}</ref><ref name=Feldman>{{cite book|author=W. M. Feldman|title=Rabbinical Mathematics and Astronomy|edition=2nd|publisher=Hermon Press|date=1965|chapter=Chapter 17: The Fixed Calendar}}</ref><ref name=Mandelbaum>{{cite book|author=Hugo Mandelbaum|chapter=Introduction: Elements of the Calendar Calculations|editor=Arthur Spier|title=The Comprehensive Hebrew Calendar|edition=3rd|date=1986}}</ref> The two most important conditions are:

*If the molad occurs at or later than noon, Rosh Hashanah is postponed a day. This is called {{lang|he-Latn|deḥiyyat molad zaken}} ({{lang|he|דְחִיַּת מוֹלָד זָקֵן}}, literally, "old birth", i.e., late new moon). This rule is mentioned in the Talmud,<ref name="epstein"/> and is used nowadays to prevent the molad falling on the second day of the month.<ref>{{cite web|last=Landau|first=Remy|url=http://hebrewcalendar.tripod.com/#25|title=Hebrew Calendar Science and Myth: 'The Debatable Dehiyah Molad Zaquen'|access-date=7 February 2015}}</ref> This ensures that the long-term average month length is 29.530594 days (equal to the molad interval), rather than the 29.5 days implied by the standard alternation between 29- and 30-day months.
*If the molad occurs on a Sunday, Wednesday, or Friday, Rosh Hashanah is postponed a day. If the application of {{lang|he-Latn|deḥiyyah molad zaken}} would place Rosh Hashanah on one of these days, then it must be postponed a second day. This is called {{lang|he-Latn|deḥiyyat lo ADU}} ({{lang|he|דְחִיַּת לֹא אד״ו}}), an acronym that means "not [weekday] one, four, or six".
:This rule is applied for religious reasons, so that [[Yom Kippur]] does not fall on a Friday or Sunday, and [[Hoshana Rabbah]] does not fall on [[Shabbat]].{{efn|This is the reason given by most [[halachic]] authorities, based on the [[Talmud]], Rosh Hashanah 20b and Sukkah 43b. [[Maimonides]] ([[Mishneh Torah]], Kiddush Hachodesh 7:7), however, writes that the arrangement was made (possible days alternating with impossible ones) in order to average out the difference between the mean and true [[lunar conjunction]]s.}} Since Shabbat restrictions also apply to Yom Kippur, if either day falls immediately before the other, it would not be possible to make necessary preparations for the second day (such as [[Shabbat candles|candle lighting]]).{{efn|The Talmud (Rosh Hashanah 20b) puts it differently: over two consecutive days of full Shabbat restrictions, vegetables would wilt (since they can't be cooked), and unburied corpses would putrefy.}} Additionally, the laws of Shabbat override those of Hoshana Rabbah, so that if Hoshana Rabbah were to fall on Shabbat, the Hoshana Rabbah ''aravah'' ritual could not be performed.<ref>Yerushalmi, [https://www.sefaria.org/Jerusalem_Talmud_Sukkah.4.1.3 Sukkah 4:1] (18a, 54b)</ref>
:Thus Rosh Hashanah can only fall on Monday, Tuesday, Thursday, and Saturday. The ''kevi'ah'' uses the letters ה ,ג ,ב and ז (representing 2, 3, 5, and 7, for Monday, Tuesday, Thursday, and Saturday) to denote the starting day of Rosh Hashana and the year.

Another two rules are applied much less frequently and serve to prevent impermissible year lengths. Their names are Hebrew acronyms that refer to the ways they are calculated:
* If the molad in a common year falls on a Tuesday, on or after 9 hours and 204 parts, Rosh Hashanah is postponed to Thursday. This is {{lang|he-Latn|deḥiyyat GaTaRaD}} ({{lang|he|דְחִיַּת גטר״ד}}, where the acronym stands for "3 [Tuesday], 9, 204").
* If the molad following a leap year falls on a Monday, on or after 15 hours and 589 parts after the Hebrew day began (for calculation purposes, this is taken to be 6&nbsp;pm Sunday), Rosh Hashanah is postponed to Tuesday. This is {{lang|he-Latn|deḥiyyat BeTUTeKaPoT}} ({{lang|he|דְחִיַּת בט״ו תקפ״ט}}), where the acronym stands for "2 [Monday], 15, 589".

===Deficient, regular, and complete years===
The rules of postponement of Rosh HaShanah make it that a Jewish common year will have 353, 354, or 355 days while a leap year (with the addition of Adar I which always has 30 days) has 383, 384, or 385 days.<ref name=companion/>

*A {{transliteration|he|chaserah}} year (Hebrew for "deficient" or "incomplete") is 353 or 383 days long. Both Cheshvan and Kislev have 29 days.
*A {{transliteration|he|kesidrah}} year ("regular" or "in-order") is 354 or 384 days long. Cheshvan has 29 days while Kislev has 30 days.
*A {{transliteration|he|shlemah}} year ("complete" or "perfect", also "abundant") is 355 or 385 days long. Both Cheshvan and Kislev have 30 days.

Whether a year is deficient, regular, or complete is determined by the time between two adjacent Rosh Hashanah observances and the leap year.

A Metonic cycle equates to 235 lunar months in each 19-year cycle. This gives an average of 6,939 days, 16 hours, and 595 parts for each cycle.<ref name=weinberg/> But due to the Rosh Hashanah postponement rules (preceding section) a cycle of 19 Jewish years can be either 6,939, 6,940, 6,941, or 6,942 days in duration. For any given year in the Metonic cycle, the molad moves forward in the week by 2 days, 16 hours, and 595 parts every 19 years. The [[greatest common divisor]] of this and a week is 5 parts, so the Jewish calendar repeats exactly following a number of Metonic cycles equal to the number of parts in a week divided by 5, namely 7×24×216&nbsp;=&nbsp;36,288 Metonic cycles, or 689,472 Jewish years. There is a near-repetition every 247 years, except for an excess of 50 minutes {{frac|16|2|3}} seconds (905 parts).

Contrary to popular impression, one's Hebrew birthday does not necessarily fall on the same Gregorian date every 19 years, since the length of the Metonic cycle varies by several days (as does the length of a 19-year Gregorian period, depending whether it contains 4 or 5 leap years).<ref>[https://outorah.org/p/5696/ Tzarich Iyun: Your Hebrew Birthday]</ref>

===Keviah===
{| class="wikitable" align="right" style="float; margin: 9px;"
! style="text-align:right" | Days in year →
| 353 || 354 || 355 || 383 || 384 || 385
|-
! Day of Rosh HaShanah
! colspan="6" | English ''Kevi'ah'' symbol
|-
| Monday (2) || 2D3 || || 2C5 || 2D5 || || 2C7
|-
| Tuesday (3) || || 3R5 || || || 3R7 ||
|-
| Thursday (5) || || 5R7 || 5C1 || 5D1 || || 5C3
|-
| Saturday (7) || 7D1 || || 7C3 || 7D3 || || 7C5
|}
There are three qualities that distinguish one year from another: whether it is a leap year or a common year; on which of four permissible days of the week the year begins; and whether it is a deficient, regular, or complete year. Mathematically, there are 24 (2×4×3) possible [[combination]]s, but only 14 of them are valid.

Each of these patterns is known by a {{lang|he|kevi'ah}} ({{langx|he|קביעה}} for 'a setting' or 'an established thing'), which is a code consisting of two numbers and a letter. In English, the code consists of the following:
* The left number is the day of the week of {{nowrap|1 Tishrei}}, Rosh Hashanah {{nowrap|(2 3 5 7; Hebrew: ב ג ה ז)}}
* The letter indicates whether that year is deficient (D, "ח", from {{Langx|he|חסרה|Chasera}}), regular (R, "כ", from {{Langx|he|כסדרה|Kesidra}}), or complete (C, "ש", from {{Langx|he|שלמה|Shlema}})
* The right number is the day of the week of {{nowrap|15 Nisan}}, the first day of Passover or Pesach {{nowrap|(1 3 5 7; Hebrew: א ג ה ז)}}, within the same Hebrew year (next Julian/Gregorian year)
The {{transliteration|he|kevi'ah}} in Hebrew letters is written right-to-left, so their days of the week are reversed, the right number for {{nowrap|1 Tishrei}} and the left for {{nowrap|15 Nisan}}.

The ''kevi'ah'' also determines the [[Torah reading]] cycle (which ''parshiyot'' are read together or separately.<ref name="Judaism 101">{{cite web|title=The Jewish Calendar: A Closer Look|url=http://www.jewfaq.org/calendr2.htm|publisher=Judaism 101|access-date=25 March 2011}}</ref>

=== The four gates ===
The ''keviah'', and thus the annual calendar, of a numbered Hebrew year can be determined by consulting the table of Four Gates, whose inputs are the year's position in the 19-year cycle and its [[#Rosh Hashanah postponement rules|molad Tishrei]].<ref name=biruni>{{citation |author=al-Biruni |title=The Chronology of Ancient Nations |url=https://archive.org/details/chronologyofanci00biru/page/150 |translator-last=Sachau |translator-first=C. Edward |year=1879 |orig-year=1000}}</ref>{{rp|150–152}}<ref>{{cite book |last=Bushwick |first=Nathan |year=1989 |title=Understanding the Jewish Calendar |location=New York/Jerusalem |publisher=Moznaim |isbn=0-940118-17-3 |pages=95–97}}</ref><ref>{{cite encyclopedia |last=Poznanski |first=Samuel |year=1910 |title=Calendar (Jewish) |encyclopedia=Encyclopædia of Religion and Ethics |editor-last=Hastings |editor-first=James |editor-link=James Hastings |title-link=Encyclopædia of Religion and Ethics |publisher=T. & T. Clark |location=Edinburgh |volume=3 |page=121 |url=https://archive.org/stream/encyclopaediaofr003hast#page/120/mode/2up |quote=limits, Qebi'oth [kevi'ot]}}</ref><ref>{{cite journal |last=Resnikoff |first=Louis A. |title=Jewish Calendar Calculations |page=276 |journal=Scripta Mathematica |volume=9 |year=1943}}</ref>{{efn|In the Four Gates sources ({{transliteration|he|kevi'ot}} cited here are in Hebrew in sources except al-Biruni): al-Biruni specified 5R (5 Intermediate) instead of 5D in leap years. Bushwick forgot to include 5D for leap years. Poznanski forgot to include 5D for a limit in his table although he did include it in his text as 5D1; for leap years he incorrectly listed 5C7 instead of the correct 5C3. Resnikoff's table is correct.}}<ref>{{cite web |first=Robert |last=Schram |date=1908  |title={{lang|de|Kalendariographische und Chronologische Tafeln|nocat=yes}} |url=https://archive.org/stream/kalendariograph00schrgoog#page/n231/mode/2up |pages= xxiii–xxvi, 190–238|publisher=Leipzig, J. C. Hinrichs }} Schram gives the type of Hebrew year for all years 1–6149 AM (−3760 to 2388 Julian/Gregorian) in a main table (3946+) and its adjunct (1+, 1742+) on pages 191–234 in the form 2d, 2a, 3r, 5r, 5a, 7d, 7a for common years and 2D, 2A, 3R, 5D, 5A, 7D, 7A for leap years. The type of year 1&nbsp;AM, 2a, is on page 200 at the far right.</ref> In this table, the years of a 19-year cycle are organized into four groups (called "gates"): common years after a leap year but before a common year {{nowrap|(1 4 9 12 15)}}; common years between two leap years {{nowrap|(7 18)}}; common years after a common year but before a leap year {{nowrap|(2 5 10 13 16)}}; and leap years {{nowrap|(3 6 8 11 14 17 19)}}.<ref>[https://hakirah.org/vol20AjdlerAppendices.pdf A Short History of the Jewish Fixed Calendar : Appendices].</ref>

This table<ref name="biruni" />{{rp|150}}<ref name=ajdler/>{{rp|183}} numbers the days of the week and hours for the limits of molad Tishrei in the Hebrew manner for calendrical calculations, that is, both begin at {{nowrap|6 pm}}, thus {{nowrap|7d 18h 0p}} is noon Saturday, with the week starting on {{nowrap|1d 0h 0p}} (Saturday 6pm, i.e. the beginning of Sunday reckoned in the Hebrew manner). The oldest surviving table of Four Gates was written by [[Muhammad ibn Musa al-Khwarizmi]] in [[824]].<ref>{{cite web | url=http://www.jphogendijk.nl/khwarizmi.html#JewCal | title=Muhammad ibn Musa (Al-)Khwarizmi (Or Kharazmi) (Ca. 780–850 CE) }}</ref>

{| style="border-collapse: collapse;"
|+ '''Four gates''' or '''Table of Limits'''
|- style="background-color: #F4F4F4;"
! style="border: 1px solid #B0B0B0; text-align: center;" rowspan="2" | molad <br /> Tishrei ≥
! style="border: 1px solid #B0B0B0; text-align: center;" colspan="4" | Year of 19-year cycle
|- style="background-color: #F4F4F4;"
! style="border: 1px solid #B0B0B0; padding: 0px 10px;" | 1 4 9 12 15
! style="border: 1px solid #B0B0B0; padding: 0px 30px;" | 7 18
! style="border: 1px solid; border-color: #B0B0B0 #707070; padding: 0px 10px;" | 2 5 10 13 16
! style="border: 1px solid #B0B0B0; padding: 0px 10px;" | 3 6 8 11 14 17 19
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 7d 18h 0p
| style="border: 1px solid #707070; |
| style="border: 1px #707070; border-style: solid hidden; text-align: center;" | '''2D3''' &nbsp; {{resize|135%|בחג}}
| style="border: 1px solid #707070;" |
| style="border: 1px #B0B0B0; border-style: solid solid hidden; text-align: center;" | '''2D5''' &nbsp; {{resize|135%|בחה}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 1d 9h 204p
| style="border: 1px #707070; border-style: solid solid hidden;" colspan="3" |
| style="border: 1px solid; border-color: #707070 #B0B0B0;" | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 1d 20h 491p
| style="border: 1px solid #707070;" |
| style="border: 1px #707070; border-style: solid hidden; text-align: center;" | '''2C5''' &nbsp; {{resize|135%|בשה}} 
| style="border: 1px #707070; border-style: hidden solid;" |
| style="border: 1px #B0B0B0; border-style: solid solid hidden; text-align: center;" | '''2C7''' &nbsp; {{resize|135%|בשז}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 2d 15h 589p
| style="border: 1px #707070; border-style: solid solid hidden;" colspan="2" |
| style="border: 1px solid #707070;" |
| style="border: 1px solid; border-color: #707070 #B0B0B0;" | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 2d 18h 0p
| style="border: 1px solid #707070;" |
| style="border: 1px #707070; border-style: solid hidden; text-align: center;" | '''3R5''' &nbsp; {{resize|135%|גכה}}
| style="border: 1px solid #707070;" |
| style="border: 1px #B0B0B0; border-style: solid solid hidden; text-align: center;" | '''3R7''' &nbsp; {{resize|135%|גכז}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 3d 9h 204p
| style="border: 1px #707070; border-style: solid hidden solid solid;" rowspan="3" |
| style="border: 1px #707070; border-style: solid hidden; text-align: center;" rowspan="3" | '''5R7''' &nbsp; {{resize|135%|הכז}}
| style="border: 1px #707070; border-style: solid solid solid hidden;" rowspan="3" |
| style="border: 1px solid; border-color: #707070 #B0B0B0;" | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 3d 18h 0p
| style="border: 1px solid; border-color: #707070 #B0B0B0; text-align: center;" | '''5D1''' &nbsp; {{resize|135%|החא}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 4d 11h 695p
| style="border: 1px #B0B0B0; border-style: solid solid hidden;" | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 5d 9h 204p
| style="border: 1px solid #707070;" |
| style="border: 1px #707070; border-style: solid hidden; text-align: center;" | '''5C1''' &nbsp; {{resize|135%|השא}}
| style="border: 1px solid #707070;" |
| style="border: 1px solid; border-color: #707070 #B0B0B0; text-align: center;" | '''5C3''' &nbsp; {{resize|135%|השג}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 5d 18h 0p
| style="border: 1px solid #707070;" |
| style="border: 1px #707070; border-style: solid solid hidden hidden;" colspan="2" |
| style="border: 1px #B0B0B0; border-style: solid solid hidden; | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 6d 0h 408p
| style="border: 1px #707070; border-style: solid solid hidden;" |
| style="border: 1px #707070; border-style: hidden hidden solid solid; text-align: center;" | '''7D1''' &nbsp; {{resize|135%|זחא}}
| style="border: 1px solid #707070;" |
| style="border: 1px #B0B0B0; border-style: hidden solid; text-align: center;" | '''7D3''' &nbsp; {{resize|135%|זחג}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding-left: 10px;" | 6d 9h 204p
| style="border: 1px #B0B0B0; border-style: hidden hidden solid solid;" rowspan="2" |
| style="border: 1px #707070; border-style: solid solid hidden hidden;" colspan="2" |
| style="border: 1px solid; border-color: #707070 #B0B0B0;" | {{resize|135%|&nbsp;}}
|- style="background-color: #FAFAFA;"
| style="border: 1px solid #B0B0B0; padding: 0px 10px;" | 6d 20h 491p
| style="border: 1px #B0B0B0; border-style: hidden hidden solid; text-align: center;" | '''7C3''' &nbsp; {{resize|135%|זשג}}
| style="border: 1px; border-style: hidden solid solid hidden; border-color: #A0A0A0 #707070;" |
| style="border: 1px solid #B0B0B0; text-align: center;" | '''7C5''' &nbsp; {{resize|135%|זשה}}
|}
====Incidence====
Comparing the days of the week of molad Tishrei with those in the {{transliteration|he|kevi'ah}} shows that during 39% of years {{nowrap|1 Tishrei}} is not postponed beyond the day of the week of its molad Tishrei, 47% are postponed one day, and 14% are postponed two days. This table also identifies the seven types of common years and seven types of leap years. Most are represented in any 19-year cycle, except one or two may be in neighboring cycles. The most likely type of year is 5R7 in 18.1% of years, whereas the least likely is 5C1 in 3.3% of years. The day of the week of {{nowrap|15 Nisan}} is later than that of {{nowrap|1 Tishrei}} by one, two or three days for common years and three, four or five days for leap years in deficient, regular or complete years, respectively.

{| class="wikitable"
|+Incidence (percentage)
! colspan="2" |common years
! colspan="2" |leap years
|-
|'''5R7'''
|18.05
|'''5C3'''
|6.66
|-
|'''7C3'''
|13.72
|'''7D3'''
|5.8
|-
|'''2C5'''
|11.8
|'''2D5'''
|5.8
|-
|'''3R5'''
|6.25
|'''3R7'''
|5.26
|-
|'''2D3'''
|5.71
|'''2C7'''
|4.72
|-
|'''7D1'''
|4.33
|'''7C5'''
|4.72
|-
|'''5C1'''
|3.31
|'''5D1'''
|3.87
|}

===Worked example===
Given the length of the year, the length of each month is fixed as described above, so the real problem in determining the calendar for a year is determining the number of days in the year. In the modern calendar, this is determined in the following manner.{{efn|The following description is based on the article "Calendar" in Encyclopaedia Judaica (Jerusalem: Ketter, 1972). It is an explanatory description, not a procedural one, in particular explaining what is going on with the third and fourth ''deḥiyyot''}}

The day of Rosh Hashanah and the length of the year are determined by the time and the day of the week of the Tishrei ''molad'', that is, the moment of the average conjunction. Given the Tishrei ''molad'' of a certain year, the length of the year is determined as follows:

First, one must determine whether each year is an ordinary or leap year by its position in the 19-year Metonic cycle. Years 3, 6, 8, 11, 14, 17, and 19 are leap years.

Secondly, one must determine the number of days between the starting Tishrei ''molad'' (TM1) and the Tishrei ''molad'' of the next year (TM2). For calendar descriptions in general the day begins at 6&nbsp;pm, but for the purpose of determining Rosh Hashanah, a ''molad'' occurring on or after noon is treated as belonging to the next day (the first ''deḥiyyah'').{{efn|So for example if the Tishrei molad is calculated as occurring from noon on Wednesday (the 18th hour of the fourth day) up until noon on Thursday, Rosh Hashanah falls on a Thursday, which starts Wednesday at sunset wherever one happens to be.}} All months are calculated as 29d, 12h, 44m, {{fraction|3|1|3}}s long (MonLen). Therefore, in an ordinary year TM2 occurs 12 × MonLen days after TM1. This is usually 354 calendar days after TM1, but if TM1 is on or after 3:11:20&nbsp;am and before noon, it will be 355 days. Similarly, in a leap year, TM2 occurs 13 × MonLen days after TM1. This is usually 384 days after TM1, but if TM1 is on or after noon and before {{fraction|2:27:16|2|3}}&nbsp;pm, TM2 will be only 383 days after TM1. In the same way, from TM2 one calculates TM3. Thus the four natural year lengths are 354, 355, 383, and 384 days.

However, because of the holiday rules, Rosh Hashanah cannot fall on a Sunday, Wednesday, or Friday, so if TM2 is one of those days, Rosh Hashanah in year 2 is postponed by adding one day to year 1 (the second ''deḥiyyah''). To compensate, one day is subtracted from year 2. It is to allow for these adjustments that the system allows 385-day years (long leap) and 353-day years (short ordinary) besides the four natural year lengths.

But how can year 1 be lengthened if it is already a long ordinary year of 355 days or year 2 be shortened if it is a short leap year of 383 days? That is why the third and fourth ''deḥiyyah''s are needed.

If year 1 is already a long ordinary year of 355 days, there will be a problem if TM1 is on a Tuesday,{{efn|This will happen if TM1 is on or after 3:11:20 am and before noon on a Tuesday. If TM1 is Monday, Thursday or Saturday, Rosh Hashanah in year 2 does not need to be postponed. If TM1 is Sunday, Wednesday or Friday, Rosh Hashanah in year 1 is postponed, so year 1 is not the maximum length.}} as that means TM2 falls on a Sunday and will have to be postponed, creating a 356-day year. In this case, Rosh Hashanah in year 1 is postponed from Tuesday (the third ''deḥiyyah''). As it cannot be postponed to Wednesday, it is postponed to Thursday, and year 1 ends up with 354 days.

On the other hand, if year 2 is already a short year of 383 days, there will be a problem if TM2 is on a Wednesday.{{efn|TM2 will be between noon and {{fraction|2:27:16|2|3}}&nbsp;pm on Tuesday, and TM3 will be between {{fraction|9:32:43|1|3}} and noon on Monday.}} because Rosh Hashanah in year 2 will have to be postponed from Wednesday to Thursday and this will cause year 2 to be only 382 days long. In this case, year 2 is extended by one day by postponing Rosh Hashanah in year 3 from Monday to Tuesday (the fourth ''deḥiyyah''), and year 2 will have 383 days.

=== Holidays ===
For calculated dates of Jewish holidays, see [[Jewish and Israeli holidays 2000–2050]]