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.

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. 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. 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 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, months eight and nine. 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 29/30 Cheshvan and/or 29/30 Kislev.
The period from 1 Adar to 29 Cheshvan contains all of the festivals specified in the Bible - Purim, Pesach, Shavuot, Rosh Hashanah, Yom Kippur, Sukkot, and Shemini Atzeret. This period is fixed, during which no adjustments are made.
The result is that all dates from 1 Nisan through 29 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 can each fall on one of four days of the week.
*Postponed to not fall on Shabbat
With each gate, some unusual effects occur.

Gate 1 (31.9%)

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