by Remy Landau
=================================

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.

Roshei Hadashim Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 193280 79369 0 219831 0 196992 689472
Heshvan 196992 0 193280 79369 0 219831 0 689472
Kislev 219831 0 196992 0 193280 79369 0 689472
Tevet 0 151093 68738 69853 127139 79369 193280 689472
Shevat 0 193280 26677 124416 138591 0 206508 689472
Adar 206508 0 193280 26677 124416 138591 0 689472
v'Adar 85899 0 72576 0 68864 26677 0 254016
Nisan 79369 0 219831 0 196992 0 193280 689472
Iyar 193280 79369 0 219831 0 196992 0 689472
Sivan 196992 0 193280 79369 0 219831 0 689472
Tammuz 0 196992 0 193280 79369 0 219831 689472
Av 0 219831 0 196992 0 193280 79369 689472
Elul 79369 0 219831 0 196992 0 193280 689472
Totals 1258240 1033845 1463854 989787 1345474 1153940 1282540 8527680
Leap Adar 72576 0 68864 26677 0 85899 0 254016

Some Observations on the Roshei Hadashim

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.

Distribution by Week Day of Month Starts and Roshei Hadashim
Sun Mon Tue Wed Thu Fri Sat
Month Starts 1033845 1463854 989787 1345474 1153940 1282540 1258240
Roshei Hadashim 1258240 1033845 1463854 989787 1345474 1153940 1282540

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 Roshei Hadashim Distributions by Keviyyot

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
355 days - Shevat Tammuz
383 days - Tammuz
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.

Roshei Hadashim of Same Frequency as Year Types
Year LengthOccurencesWeek Day
353
`69,222`
NONE
354
`167,497`
NONE
355
`198,737`
Wednesday
383
`106,677`
NONE
384
`36,288`
Su & Mo & Th
385
`111,051`
Wednedsday

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.

Week Day Firsts of Double Frequency as Year Types
Year LengthOccurencesWeek Day
353
`2*69,222`
Saturday
354
`2*167,497`
We & Th & Fr
355
`2*198,737`
NONE
383
`2*106,677`
Th & Fr
384
`2*36,288`
We & Fr
385
`2*111,051`
Sunday

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.

For 353 Day Years
Start of Rosh Hodesh Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 39369 0 0 0 0 29853 69222
Heshvan 29853 0 39369 0 0 0 0 69222
Kislev 0 0 29853 0 39369 0 0 69222
Tevet 0 0 0 29853 0 39369 0 69222
Shevat 0 0 0 0 29853 0 39369 69222
Adar 39369 0 0 0 0 29853 0 69222
v'Adar 0 0 0 0 0 0 0 0
Nisan 29853 0 39369 0 0 0 0 69222
Iyar 0 29853 0 39369 0 0 0 69222
Sivan 0 0 0 29853 0 39369 0 69222
Tammuz 0 0 0 0 29853 0 39369 69222
Av 0 39369 0 0 0 0 29853 69222
Elul 29853 0 39369 0 0 0 0 69222
Totals 128928 108591 147960 99075 99075 108591 138444 830664

For 354 Day Years
Start of Rosh Hodesh Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 0 43081 0 124416 0 0 167497
Heshvan 0 0 0 43081 0 124416 0 167497
Kislev 124416 0 0 0 0 43081 0 167497
Tevet 0 124416 0 0 0 0 43081 167497
Shevat 0 43081 0 124416 0 0 0 167497
Adar 0 0 43081 0 124416 0 0 167497
v'Adar 0 0 0 0 0 0 0 0
Nisan 0 0 0 0 43081 0 124416 167497
Iyar 124416 0 0 0 0 43081 0 167497
Sivan 43081 0 124416 0 0 0 0 167497
Tammuz 0 43081 0 124416 0 0 0 167497
Av 0 0 0 43081 0 124416 0 167497
Elul 0 0 0 0 43081 0 124416 167497
Totals 291913 210578 210578 334994 334994 334994 291913 2009964

For 355 Day Years
Start of Rosh Hodesh Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 81335 0 0 22839 0 94563 198737
Heshvan 94563 0 81335 0 0 22839 0 198737
Kislev 22839 0 94563 0 81335 0 0 198737
Tevet 0 0 22839 0 94563 0 81335 198737
Shevat 0 81335 0 0 22839 0 94563 198737
Adar 94563 0 81335 0 0 22839 0 198737
v'Adar 0 0 0 0 0 0 0 0
Nisan 22839 0 94563 0 81335 0 0 198737
Iyar 0 22839 0 94563 0 81335 0 198737
Sivan 81335 0 0 22839 0 94563 0 198737
Tammuz 0 81335 0 0 22839 0 94563 198737
Av 0 94563 0 81335 0 0 22839 198737
Elul 22839 0 94563 0 81335 0 0 198737
Totals 338978 361407 469198 198737 407085 221576 387863 2384844

For 383 Day Years
Start of Rosh Hodesh Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 40000 0 0 26677 0 40000 106677
Heshvan 40000 0 40000 0 0 26677 0 106677
Kislev 26677 0 40000 0 40000 0 0 106677
Tevet 0 26677 0 40000 0 40000 0 106677
Shevat 0 0 26677 0 40000 0 40000 106677
Adar 40000 0 0 26677 0 40000 0 106677
v'Adar 40000 0 40000 0 0 26677 0 106677
Nisan 26677 0 40000 0 40000 0 0 106677
Iyar 0 26677 0 40000 0 40000 0 106677
Sivan 40000 0 0 26677 0 40000 0 106677
Tammuz 0 40000 0 0 26677 0 40000 106677
Av 0 40000 0 40000 0 0 26677 106677
Elul 26677 0 40000 0 40000 0 0 106677
Totals 240031 173354 226677 173354 213354 213354 146677 1386801

For 384 Day Years
Start of Rosh Hodesh Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 0 36288 0 0 0 0 36288
Heshvan 0 0 0 36288 0 0 0 36288
Kislev 0 0 0 0 0 36288 0 36288
Tevet 0 0 0 0 0 0 36288 36288
Shevat 0 36288 0 0 0 0 0 36288
Adar 0 0 36288 0 0 0 0 36288
v'Adar 0 0 0 0 36288 0 0 36288
Nisan 0 0 0 0 0 0 36288 36288
Iyar 36288 0 0 0 0 0 0 36288
Sivan 0 0 36288 0 0 0 0 36288
Tammuz 0 0 0 36288 0 0 0 36288
Av 0 0 0 0 0 36288 0 36288
Elul 0 0 0 0 0 0 36288 36288
Totals 36288 36288 108864 72576 36288 72576 108864 471744

For 385 Day Years
Start of Rosh Hodesh Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 32576 0 0 45899 0 32576 111051
Heshvan 32576 0 32576 0 0 45899 0 111051
Kislev 45899 0 32576 0 32576 0 0 111051
Tevet 0 0 45899 0 32576 0 32576 111051
Shevat 0 32576 0 0 45899 0 32576 111051
Adar 32576 0 32576 0 0 45899 0 111051
v'Adar 45899 0 32576 0 32576 0 0 111051
Nisan 0 0 45899 0 32576 0 32576 111051
Iyar 32576 0 0 45899 0 32576 0 111051
Sivan 32576 0 32576 0 0 45899 0 111051
Tammuz 0 32576 0 32576 0 0 45899 111051
Av 0 45899 0 32576 0 32576 0 111051
Elul 0 0 45899 0 32576 0 32576 111051
Totals 222102 143627 300577 111051 254678 202849 208779 1443663

Summary For NON-LEAP Years
Start of Rosh Hodesh Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 120704 43081 0 147255 0 124416 435456
Heshvan 124416 0 120704 43081 0 147255 0 435456
Kislev 147255 0 124416 0 120704 43081 0 435456
Tevet 0 124416 22839 29853 94563 39369 124416 435456
Shevat 0 124416 0 124416 52692 0 133932 435456
Adar 133932 0 124416 0 124416 52692 0 435456
v'Adar 0 0 0 0 0 0 0 0
Nisan 52692 0 133932 0 124416 0 124416 435456
Iyar 124416 52692 0 133932 0 124416 0 435456
Sivan 124416 0 124416 52692 0 133932 0 435456
Tammuz 0 124416 0 124416 52692 0 133932 435456
Av 0 133932 0 124416 0 124416 52692 435456
Elul 52692 0 133932 0 124416 0 124416 435456
Totals 759819 680576 827736 632806 841154 665161 818220 5225472

Summary For LEAP Years
Start of Rosh Hodesh Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 72576 36288 0 72576 0 72576 254016
Heshvan 72576 0 72576 36288 0 72576 0 254016
Kislev 72576 0 72576 0 72576 36288 0 254016
Tevet 0 26677 45899 40000 32576 40000 68864 254016
Shevat 0 68864 26677 0 85899 0 72576 254016
Adar 72576 0 68864 26677 0 85899 0 254016
v'Adar 85899 0 72576 0 68864 26677 0 254016
Nisan 26677 0 85899 0 72576 0 68864 254016
Iyar 68864 26677 0 85899 0 72576 0 254016
Sivan 72576 0 68864 26677 0 85899 0 254016
Tammuz 0 72576 0 68864 26677 0 85899 254016
Av 0 85899 0 72576 0 68864 26677 254016
Elul 26677 0 85899 0 72576 0 68864 254016
Totals 498421 353269 636118 356981 504320 488779 464320 3302208

```First  Paged 10 Jul 1997