Exact Hebrew Calendar Full Cycle Weekday Frequencies

The Hebrew calendar mean year = (235 months × traditional molad interval 29+¹³⁷⁵³/₂₅₉₂₀ days) / 19 years = (365+²⁴³¹¹/₉₈₄₉₆ days) / 7 = 52+¹²²⁸⁰⁷/₆₈₉₄₇₂ weeks, which indicates that it takes 689472 years to accumulate a whole number of weeks, completing one full calendar cycle. To confirm, there are 25920 parts per day × 29 days + 13753 parts = 765432+1 parts per molad interval. Multiply by 235 months per 19-year cycle = 179876755 parts. Divide by (25920 parts per day × 7 days per week) = 181440 parts per week = 991+¹³⁹⁴³/₃₆₂₈₈ weeks, so a full repetition cycle contains 36288 cycles × 19 years = 689472 years, as above.

The fraction ¹³⁷⁵³/₂₅₉₂₀ implies an underlying molad cycle of 25920 months in which 13753 are full (30 days) and 12167 are deficient (29 days). There are 47 such cycles in 98496 years, each lasting ⁹⁸⁴⁹⁶/₄₇ = 2095+³¹/₄₇ years, and there are 7 × 47 = 329 molad cycles in the full 689472-year calendar cycle.

There are 14 unique types of Hebrew calendar year. The following table shows the "character" of each year type, indicated by 3 or 4 symbols, along with their exact frequencies in the full 689472-year repetition cycle:

• A digit 2, 3, 5, or 7 denotes the first weekday of Rosh Hashanah
• The letter N, D, or P denotes the year length:
  • N = "Normal" year: Cheshvan has 29 days but Kislev has 30 days (full cycle frequency ²⁰³⁷⁸⁵/₆₈₉₄₇₂ ≈ 29.6% of years)
  • D = "Deficient" year: both have 29 days (full cycle frequency ⁵⁸⁶³³/₂₂₉₈₂₄ ≈ 25.5% of years)
  • P = "Perfect" (also called "Abundant") year: both have 30 days (full cycle frequency ⁷⁷⁴⁴⁷/₁₇₂₃₆₈ ≈ 44.9% of years)
• A digit 1, 3, 5, or 7 denotes the first weekday of Passover
• An asterisk denotes a leap year, in which the 30-day month Adar Rishon is inserted before the 29-day Adar (renamed to Adar Sheini).

The 4-digit Hebrew year number to the left of each "=" sign is an example of the year type that is to the right of the "=" sign, and the small number (in parentheses) was the first year of that type after the Hebrew calendar epoch. Below that is the total count of that year type and its exact frequency and approximate percentage in the full calendar cycle. The omitted weekdays (1, 4, 6) are disallowed as the first day of Rosh Hashanah. Blank cells are length and weekday combinations that never occur. Hebrew calendar days begin at local sunset. After civil midnight the corresponding English weekday names are: 1 = Sunday, 2 = Monday, ... 7 = Saturday.

To calculate the exact frequency for each weekday for each date, determine the weekday of each of the 14 example years, add the full cycle count of each year type to the appropriate weekday numerator, place each numerator over the appropriate denominator (all years, common years, or leap years), then reduce the fraction.

All dates from "the month before the month before Nisan" [Shevat or Adar Rishon] through the 29th of Cheshvan are fixed relative to the Rosh Hashanah that they span, so they all have 4 possible weekdays whose frequencies are affected only by that Rosh Hashanah starting weekday and equal the figures shown in the right-most "Total by Weekday" column. The 30th of Cheshvan exists only in perfect years and has only 3 possible weekdays, with the consequence that all permanent dates in Kislev have 6 possible weekdays. The 30th of Kislev exists only in non-deficient years. All dates in Tevet and Shevat have 5 possible weekdays. The first 29 days of the following month pairs have identical possible weekdays and frequencies, because they are always separated by a whole number of weeks, but their respective calendar year types differ: Iyar = Tishrei; Av = "month before Nisan" [Adar or Adar Sheini]; Cheshvan = "month before the month before Nisan" [Shevat or Adar Rishon].

The following tables list for each day of each Hebrew calendar month the exact weekday frequencies, the approximate percentages, and the applicable year type codes. Blank cells are weekdays upon which that day of that month can never occur.