Days of week on Hebrew calendar

The modern Hebrew calendar has been designed to ensure that certain holy days and festivals do not fall on certain days of the week. As a result, there are only four possible patterns of days on which festivals can fall. (Note that Jewish days start at sunset of the preceding day indicated in this article.)

Reasons

The modern Hebrew calendar has been arranged so that Yom Kippur does not fall on a Friday or Sunday, and Hoshana Rabbah does not fall on Shabbat.^[1] These rules have been instituted because Shabbat restrictions also apply to Yom Kippur, and if Yom Kippur were to fall on Friday, it would not be possible to make necessary preparations for Shabbat, including candle lighting, because the preceding day is Yom Kippur. Similarly, if Yom Kippur fell on a Sunday, it would not be possible to make the necessary preparations for Yom Kippur, including candle lighting, because the preceding day is Shabbat.^[2] Also, the laws of Shabbat override those of Hoshana Rabbah, so that if Hoshana Rabbah were to fall on Shabbat certain rituals that are a part of Hoshana Rabbah services (such as carrying willows, which is work) could not be performed in that year.

As a consequence, in the case of Yom Kippur, which falls on 10 Tishrei and cannot fall on a Friday or Sunday, the days in Cheshvan and/or Kislev are adjusted so that Rosh Hashanah, which falls on 1 Tishrei, does not fall on a Wednesday or Friday. And, in the case of Hoshana Rabbah, which falls on 21 Tishrei and cannot fall on a Saturday, Rosh Hashanah cannot be on a Sunday. This leaves only four days on which Rosh Hashanah is allowed to fall: Monday, Tuesday, Thursday and Saturday, which are also referred as the "four gates".

The four gates

Since three arrangements cannot occur within the fixed calendar, most holidays can each occur on one of four possible days.

All the holy days and festivals fall in the months of Nisan through Tishrei, months one to seven. These months always have the same number of days, alternating 30 and 29. The next two months are Cheshvan and Kislev. Both or either of these months can have either 29 or 30 days, allowing for adjustments to be made and the schedule in the coming year to be manipulated. The months of Tevet and Shevat, months ten and eleven, have 29 and 30 days respectively. Finally, in a regular year the month of Adar has 29 days, while in a leap year Adar I of 30 days is added before the regular Adar, which becomes Adar II of 29 days. The result is that the period from 1 Tevet to 29 Cheshvan is fixed, except that in a leap year Adar I of 30 days is added; and all adjustments are made using 30 Cheshvan and/or 30 Kislev.

The period from 1 Adar (or Adar II, in leap years) to 29 Cheshvan contains all of the festivals specified in the Bible - Purim (14 Adar), Pesach (15 Nisan), Shavuot (6 Sivan), Rosh Hashanah (1 Tishrei), Yom Kippur (10 Tishrei), Sukkot (15 Tishrei), and Shemini Atzeret (22 Tishrei). This period is fixed, during which no adjustments are made.

The result is that all dates from 1 Nisan through 29 (or 30) Cheshvan can each fall on one of four days of the week. Dates during Kislev can fall on any of six days of the week; during Tevet and Shevat, five days; and dates during Adar (or Adar I and II, in leap years) can each fall on one of four days of the week.

Gate	Frequency	Purim	Passover (first day)	Shavuot (first day)	17 Tammuz/ Tisha B'Av	Rosh Hashanah/ Sukkot/ Shmini Atzeret/ (first day)	Yom Kippur	Chanukah (first day)	10 Tevet	Tu Bishvat	Purim Katan (only in leap years)	Gate that follows
1	31.9%	Sun	Tue	Wed	Tue	Thu	Sat	Wed or Thu	Wed, Thu, or Fri	Tue, Wed, or Thu	Wed or Fri	Gates 1, 3, or 4
2	28.6%	Tue	Thu	Fri	Thu	Sat	Mon	Fri or Sat	Fri or Sun	Thu or Sat	Fri or Sun	Gates 1, 2, or 4
3	28.0%	Thu	Sat	Sun	Sun*	Mon	Wed	Sun or Mon	Sun or Tue	Sat or Mon	Sun or Tue	Gates 1, 2, or 3
4	11.5%	Fri	Sun	Mon	Sun	Tue	Thu	Mon	Tue	Mon	Tue	Gates 2 or 3
			*Postponed from Shabbat

*Postponed to not fall on Shabbat

With each gate, some unusual effects occur.

Gate 1 (31.9%)

Since Purim falls on Sunday, the 13th of Adar, known as the Fast of Esther, falls on Saturday. Usually, fasts other than Yom Kippur are postponed to the following Sunday. But as this Sunday is Purim, the fast is pushed back to the prior Thursday (the 11th of Adar).
No Yom Tov during the year (starting with Nisan) falls on Sunday, therefore havdalah during the Yom Tov kiddush is not recited at all during the course of the year.
During Tishrei, three holidays start on Thursday. In the Diaspora, the eruv tavshilin is prepared three times. (In Israel, it is only made on Erev Rosh Hashanah.)
In Israel, this is the only occasion when there is a "three-day holiday" (the two days of Rosh Hashanah followed by Shabbat).
The Fast of Gedaliah falls on Saturday. Since fasts other than Yom Kippur are not observed on Shabbat, this is observed on the following Sunday. This leaves an interval of just five days between fasts, the shortest ever on the Jewish Calendar.
Yom Kippur falls on Shabbat. This is the only occurrence in which a fast is ever observed on Shabbat. Avinu Malkeinu is not recited by Ashkenazim except during Ne'ila.
The 10th of Tevet has the potential to fall on Friday, the only public fast that can possibly be observed on a Friday. The fast is not broken until about an hour after Shabbat begins.
Purim Katan, the holiday that only occurs in leap years, has the potential to fall on Friday.

Gate 2 (28.6%)

The eruv tavshilin is prepared on Erev Pesach (diaspora only) and on Erev Shavuot.
In the diaspora, the second day of Shavuot falls on Shabbat, the only time Shavuot ever falls on Shabbat. (In Israel, Shavuot never falls on Shabbat.)
Rosh Hashanah, Sukkot, and Shemini Atzeret all start on Shabbat. On Rosh Hashanah, since blowing the Shofar is not permitted on Shabbat, the shofar is blown only on the second day. Tashlich is also postponed to the second day. On Sukkot, the Four Species, which are not taken on Shabbat, are not used on the first day.
Chanukah will begin on either Friday or Shabbat. If on Friday, then the Shabbat of Miketz will not be during Chanukah; this is the only case in which this will occur (and where Miketz's proper haftarah will thus be read). If Chanukah begins on Shabbat, there will be two Shabbatot of Chanukah, since the holiday is eight days long.
The 10th of Tevet has the potential to fall on Friday, the only public fast that can possibly be observed on a Friday. The fast is not broken until about an hour after Shabbat begins.
Rosh Chodesh Shevat and Tu Bishvat have the potential to fall on Shabbat. This means that on Rosh Chodesh Shevat, two Torah scrolls are used for the Shabbat morning Torah reading: one for the weekly parshah, and one for Rosh Chodesh reading.
Purim Katan, the holiday that only occurs in leap years, has the potential to fall on Friday.

Gate 3 (28.0%)

Rosh Chodesh Nisan coincides with Shabbat. Three Torah scrolls are used for the Shabbat morning Torah reading: one for the weekly parshah, another for the Rosh Chodesh reading, and a third for Parshat Hachodesh.
The eruv tavshilin is prepared prior to the final day(s) of Passover.
In the diaspora, the final day of Passover is Shabbat. In Israel, this day is not considered a part of Passover. Nevertheless, Chametz cannot be consumed because it cannot be purchased on Shabbat or Yom Tov for this consumption.
The 17th of Tammuz and Tisha B'Av fall on Saturday. Since fasts other than Yom Kippur are not observed on Shabbat, these are both observed on the following Sunday.
If this is a leap year, in Israel there are no doubled-up parshiot during the year (Tishrei-Elul).
Chanukah has the potential to begin on Sunday. This is the only occurrence in which the eve of Chanukah begins on Shabbat.
If both Cheshvan and Kislev have 30 days, then the 30th of Kislev (which is also the first day of Rosh Chodesh Tevet) will fall on Shabbat. Since this is also during Chanukah, three Torah scrolls are used for the Shabbat morning Torah reading: one for the weekly parshah, another for the Rosh Chodesh reading, and a third for the Chanukah reading.
If both Cheshvan and Kislev have 29 days, then Rosh Chodesh Shevat and Tu Bishvat will fall on Shabbat. This means that on Rosh Chodesh Shevat, two Torah scrolls are used for the Shabbat morning Torah reading: one for the weekly parshah, and one for Rosh Chodesh reading.

Gate 4 (11.5%)

Rosh Chodesh Adar (or Adar II) falls on Shabbat. Three Torah scrolls are used for the Shabbat morning Torah reading: one for the weekly parshah, another for the Rosh Chodesh reading, and a third for Parshat Shekalim.
Purim falls on Friday, and the Purim seudah is held earlier in the day. In Jerusalem, where Purim always occurs a day later, the observances are spread out over Friday, Shabbat, and Sunday (a "three-day Purim").
The Fast of the Firstborn is held on the Thursday before Shabbat (the 12th of Nisan).
Bedikas Chametz occurs on the night of the 13th of Nisan (Thursday night). Chametz is burned on the following Friday morning, but may be consumed throughout this day and up until Saturday morning, at which time any remaining chametz is flushed.
During the Shabbat morning on Erev Pesach, there is a custom two meals are consumed early in the morning in order to fulfill the mitzvah of Seudah Shlishit before the time in which Chametz can no longer be consumed.
Tisha B'Av is observed on the actual ninth day of Av on a Sunday, and there is no "week in which Tisha B'Av occurs" as a level of mourning prior to the start of Tisha B'Av.
No eruv tavshilin is prepared at any time during this year (Nisan-Tishrei).
If this is a leap year, in both the diaspora and in Israel there are no doubled-up parshiot during the year (Tishrei-Elul).
The first day of Rosh Chodesh Tevet falls on Shabbat. Three Torah scrolls are used for the Shabbat morning Torah reading: one for the weekly parshah, another for the Rosh Chodesh reading, and a third for the Chanukah reading.

Next occurrence of each gate

Update when Purim comes around during the year with the next occurrence of either gate.

Gate 1 (Thursday Rosh Hashanah): 2024
Gate 2 (Saturday Rosh Hashanah): 2020
Gate 3 (Monday Rosh Hashanah): 2019
Gate 4 (Tuesday Rosh Hashanah): 2021

Sources

Shulchan Aruch, Orach Chaim 428:1
The Complete Artscroll Siddur

References

^ 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 conjunctions.
^ 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.

External links

Exact Hebrew Calendar Full Cycle Weekday Frequencies

[1] 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 conjunctions.

[2] 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.

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