There exists a very real distinction between the first day of the month and Rosh Hodesh.

The festival of Rosh Hodesh may be either one or two days long. The observance will always begin on the 30th day of a Hebrew month. If the Hebrew month is 29 days long, then that 30th day is the first day of the subsequent month. If the month is 30 days long, then the next day is the first day of the subsequent month and Rosh Hodesh will once again be observed, thus making the festival two days long.

For purposes of discussion, all references to Rosh Hodesh will imply the first day of the festival.

Also, even though the first day of Tishrei is Rosh Hashannah and practically not viewed as being Rosh Hodesh, it is considered to also be Rosh Hodesh in the ensuing statistical tables.

The full Hebrew calendar cycle of 689,472 years consists of 8,527,680 months. This total can be easily derived by multiplying the number of months in every 19 year cycle with the number of these cycles in the full Hebrew calendar cycle. This gives 235 months * 36,288 = 8,527,680 months.

Each day of the week will coincide with Rosh Hodesh. However, due to the initial Rosh Hashannah rule of Lo ADU it is not possible for some of the Rashei Hadashim to begin on certain days of the week. For example, Tishrei cannot begin on either Sunday, Wednesday, or Friday. Similarly, Rosh Hodesh Tevet never falls on a Sunday, while Rosh Hodesh Shevat avoids Sunday and Friday.

In here, the 30 day month of Adar is called Leap Adar and the weekday distribution of its Roshei Hadashim is appended to the folowing table.

The following table demonstrates the weekday distribution of the Roshei Hadashim for each month over the full Hebrew calendar cycle.

A comparison of the distribution by week day of the Roshei Hadashim and the first days of the months, as shown below, displays an unexpected result.

Except for a one day shift, the total distribution of the Roshei Hadashim by week day is exactly the same as the total distribution by week day of the first days of the months. The first day value for Saturday is found in Sunday for the Roshei Hadashim, and similarly for Sunday into Monday and so on to Friday into Saturday.

None of the Roshei Hadashim have a weekday distribution that is identical to that of the month of Tishrei.

The month of Tevet can have Rosh Hodesh begin on every single day of the week except for Sunday. This will be true for both the non-leap and the leap year distibutions.

The Roshei Hadashim for the months of Heshvan, Kislev, Adar, v'Adar, Iyar, and Tammuz never begin on Saturday.

The Roshei Hadashim for the months of Tishrei, Shevat, Nisan, Sivan, and Av all occur on the first day of the month. Hence, their week day distribution remains the same as that of the first day distributions.

The week day distribution of the Roshei Hadashim for the months of Heshvan, Adar, v'Adar, Iyar, Tammuz, and Elul are shifted by one day downwards from the distribution of the corresponding first days of the months.

Rosh Hodesh most likely starts on Tuesday. From the above table it can be seen that the Roshei Hadashim coincide some 17.2% of the time with Tuesday.

Similarly, Rosh Hodesh will least likely start day on Wednesday. From the above table it can be seen that the Roshei Hadashim coincide some 11.6% of the time with Wednesday.

The weekday distribution of the Roshei Hadashim for Heshvan, Kislev, Nisan, Iyar, Sivan, Tammuz, Av, and Elul have the same values as those for the month of Tishrei, although each may be shifted differently across the weekdays. That is because the Roshei Hadashim for each of these months is a constant distance away from the first day of the month of Tishrei.

The month of Kislev begins after a month whose length may be either 29 or 30 days long. However, Rosh Hodesh Kislev always begins 29 days after the first day of Hehsvan. Consequently, Rosh Hodesh Kislev, as opposed to the first day of Kislev, is always a constant distance of 59 days from the start of Tishrei.

The month of Adar has 3 different characteristics. The first is the 29 day month which occurs in all non-leap years. The second is the 30 day month of Adar which is inserted after the month of Shevat during Hebrew leap years. And the third is the 29 day month of v'Adar which always follows the leap month of Adar.

It is to be noted that v'Adar has the same weekday distribution values as those for the month of Leap Adar. However, the values for v'Adar are shifted two days later than those for Leap Adar.

Rosh Hodesh Leap Adar can start on a Wednesday, but not Rosh Hodesh v'Adar.

Rosh Hodesh Adar cannot start on Thursday in a leap year, and can never begin either on Monday or Saturday.

The following tables show the week day distribution of the first day of the months segregated by year lengths.

It is interesting to note that for the following year lengths, these months have exactly the same Rosh Hodesh distribution as the month of Tishrei

          353 days - Av
          354 days - Adar
          355 days - Shevat Tammuz
          383 days - Tammuz
          384 days - Adar   Sivan
          385 days - Shevat 

It is also interesting to note that the most likely week day for Rosh Hodesh varies by year length as follows

          353 days - Tuesday
          354 days - Wednesday Thursday Friday (each occurs 334,994 times)
          355 days - Tuesday
          383 days - Sunday
          384 days - Tuesday Saturday (each occurs 108,864 times)
          385 days - Tuesday

Even though Wednesday is the least likely day for a Rosh Hodesh, Monday is the only weekday that does not catch a Rosh Hodesh most frequently for any specific year type.

Each year type occurs with a particular frequency over the full Hebrew calendar cycle of 689,472 years. The big surprise is that over this cycle some of the week days actually catch the Roshei Hadashim in exact multiples of the frequency of the year types.

69,222

167,497

198,737

106,677

36,288

111,051

The 355, 384, and 385 day years are the only years which have week days that catch the Roshei Hadashim with the same frequency of occurrence as their corresponding years.

Some of the week days also catch the first days of the month with exactly double the frequency of the year types.

2*69,222

2*167,497

2*198,737

2*106,677

2*36,288

2*111,051

The 355 day years are unique in that none of its week days catch either the first day of a month or a Rosh Hodesh with double the frequency of occurrence as the 355 day years.

The 384 day years are the only ones to have a week day, namely Tuesday, catch Rosh Hodesh at triple the frequency of occurence of the 384 day years.

First  Paged 10 Jul 1997
Next Revised 10 Jul 1997