[Avodah] Tisha-Asar Mi Yode'a

[L Reich]

　From: Elozor Reich

Tisha-Asar Mi Yode'a or

Anniversaries, Birthdays & Cycles 

Many are aware that Hebrew and Civil birthdays don't usually correspond in most years, but that they often 

do coincide or come near to each other on any 19th anniversary or on any multiple of 19 years. This note explains this phenomenon and more.

The Hebrew calendar attempts to reconcile the astronomical length of the lunar month of over 29.5 days 

with the solar year of under 365.25 days and its four seasons. In use it ensures that Rosh Chodesh is 

always near the time of the astronomical new moon and that Pesach occurs in Spring and Rosh 

Hashonoh in Autumn. This is done by arranging a 19 year cycle, known as Machzor Koton. This 19 

year cycle consists of 12 year of 12 months, and 7 years of 13 months, known as Shnois Ha'Ibur, 

during which we have an extra Adar. 

The longer or Ibbur years are those which have a remainder of 3, 6, 8, 11,14, 17 & 19, when dividing the 

year number by 19, so giving us its position in the Machzor. The 19 year cycle or Machzor lasts for 235 months since 19 X 12 + 7 = 235, or 12 X 12 + 7 X 13 = 235. Currently (Year 5769) we are in the 

12th year of Cycle 304. The arithmetic for this is straight forward, e.g. if we divide 5769 by 19 we get a dividend of 303 – showing that we have completed 303 cycles and are now in cycle 304 – and a remainder 

of 12 – which is not one of the Shnois Ha'Ibur

(Although all the classical works use a time system of hours being divided into 1080 "Parts" (Chalokim) and each 

"Part" (Chelek) being divided into 76 "Moments" (Rego'im), we shall here make things more familiar by using 

minutes and seconds. A minute has 18 Chalokim; one Chelek equals 3⅓ seconds and one Rega is equivalent to 

five parts of a one hundred and fourteenth of a second , 5/114 sec.)

The accepted astronomical length of a lunar month for fixed calendar purposes is an average; individual 

months can vary considerably. This average, known as Molad Ho'emtzo'i, is 29 days, 12 hours, 44 Minutes 

& 3⅓ seconds. Multiply this by the above mentioned 235 gives us the astronomical length of a Machzor Koton. 6939 days, 16 hrs, 33 minutes & 3⅓ seconds.

We can now divide the last figure by 19 and get a very near approximation of the astronomical solar year. It come to 365 days 5 hours 55 minutes and 25.4 seconds. This year length (about 7 minutes longer than the 

true solar year, which is known to astronomers as the tropical year) is the basis of the Jewish Calendar and 

is commonly called Tekufas Rav Ada. The solar year of 365 days and exactly 6 hours is called Tekufas Shmuel. Tekufas Shmuel is equivalent to the old civil Julian Calendar which was replaced by the Gregorian 

one, which is shorter by three days in a 400 year period.

　

>From an Halachic viewpoint Tekufas Shmuel has only two uses. It decides the date of the start of the Tefiloh for rain (Tal U'Motor in Chutz Lo'oretz and is also the basis for the 28 year Machzor Godol, whose 

completion we commemorated this year. (It is also used by some in Minhagim connected with the avoidance of drinking water at the change of seasons.) However, Tekufas Rav Ada is the one which matters in settling the 19 year cycle. It governs all calendar dates and this is the one which we will continue to explore.

Now a Machzor Koton starts on Rosh Hashonoh of year 1 of the cycle and ends on the last day of Ellul 19 years later. Astronomically this is about 6939.7 days later. Since we can't split days in the real calendar, one might think that the calendrical length of a Machzor would be either 6939 or 6940 days. In fact it can also 

be 6941 and even, very rarely, 6942 days. The cause of this wide range is the fact that Rosh Hashonoh is decided by the Molad Ho'emtzo'i of the 1st of Tishri. By the basic rules of the Jewish Calendar Rosh Hashonoh can land on the same day as the Molad but can also be postponed one or two days (Molad Zokon, Lo Adu Rosh etc). If one Machzor starts on the day of the Molad and the following one is postponed by a day or two, then the calendrical length of the Machzor is extended. 


Furthermore, 19 Civil Years can include either 4 or 5 leap years, i.e. 6939 or 6940 days. The 4 year 

Civil Leap Year cycle is not linked to the Machzor Koton pattern, hence another contributor to the 'discrepancies' in the 19th anniversaries. 

To summarise; take any Hebrew Calendar date, move on to its 19th Hebrew anniversary, and you have, 

in effect, completed a Machzor Koton , but it may be 6939, 6940, 6941 or 6942 days later. Take the 

Civil Date of the same starting point, move on to its 19th Civil anniversary and it may be 6939 or 6940 

days later. This means that 19th anniversaries can show a difference of two (& very rarely three) days 

in their Hebrew and Civil dates. 

Let us now look at the Machzor Koton more closely. We start the cycle with the Hebrew and Civil years 

being level. Since the Civil Year of 365+ days is approximately 11 days longer than the average Hebrew 

Shono Peshuta (a non-leap year) of 354+ days, by the next Rosh Hashonoh the Hebrew year lags by 

11 days. After two years it is 22 behind. After three years it would 33 behind had we not made it a Shono Me'uberes (adding an extra Adar), which "pays off" 30 of the 33 days, but still leaving us three days "in 

arrear". This slipping and correcting continues throughout the 19 year cycle. The addition of an extra Adar 

in years 3,6,8,11,14,17 and 19 makes up for all the slippage and gets us back to our starting point at the 

end of the final leap year in year 19.

If you do this addition and subtraction for the whole of the cycle, you will discover that although we only 

get back exactly to par at the end of the 19th year, we come quite near to it at the end of years 8 and 11. 

After 8 years into the cycle, the Civil Calendar will have counted the passing of 2922 days; the 

corresponding Hebrew count (5 years of 12 months and 3 of 13) comes to around 2923 (non-leap years 

can be 353, 354 or 355 days and leap years 383, 384 or 385), so we are not far out. A similar exercise for the 

first 11 years shows 4017 or 4018 days in the Civil Calendar and around 4016 (7 years of 12 months 

and 4 of 13) in the Hebrew one.

All this arithmetic shows us that 19th anniversaries and their multiples will be spot on or near, and that multiples of 19 with the addition of 8, e.g. 27, 46 & 65, or with addition of 11, e.g. 30, 49 & 68 will 

either be spot on or not be far out.

Although I have not attempted a rigorous mathematical analysis of the probability ratio, an inspection of a sizeable sample shows as follows. On 19th anniversaries around one half land on the same Hebrew and 

Civil dates and an additional third within one day, leaving a small fraction more than one day out. On both 

19th + 8 and on 19th + 11 anniversaries around one eighth coincide precisely and over one half of the

remainder land within one or two days.

Readers interested in arithmetic might wish to explore the following. In examining the Machzor Koton it 

has been pointed out that we nearly reach equilibrium after 8 years with their 3 leap years, and after 11 

years with their 4 leap years, and reach a true balance after 19 years with their 7 leap years. Consider the relationship of this to the three fractions 3/8, 4/11 and 7/19. The difference between 3/8 and 4/11 is only 

1/88. 7/19, which lands between the two is 1/152 less than 3/8 and 1/209 more than 4/11. 

　

Elozor Reich)

　

　

　
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