Wednesday, March 04, 2009

The Accuracy (or lack thereof) of the Jewish Calendar

Rabbi Gershon Tannenbaum, in his weekly My Machberes column in the Jewish Press talks about the Jewish calendar. As a part of this column, he makes the following observation:

In 4104 (344 CE), Hillel wrote his formulas predicting the Hebrew calendar until the Hebrew year 6000 (2240 CE). In 1949, the National Institute of Time and Technology established the Atomic Clock to standardize times throughout the world. Since its establishment, the Atomic Clock has been readjusted several times. Hillel's calendar has never been readjusted and remains supremely accurate. The precision of the Molad announced in every shul during Rosh Chodesh Bentching, indicating the exact time of the new moon in Jerusalem, continues to be breathtaking.

Now, as my mother told me when I was young, it's okay to toot your own horn, but don't do it by putting down others - and especially when don't toot your own horn and put others down when you're wrong.

The Hebrew calendar is pretty accurate, no question about it - but compared to the Atomic clock, it just falls flat.

First, a little background. The Jewish calendar, unlike most calendars, is a luni-solar calendar. While the months are reckoned according to the phases of the moon, it is occassionally reconciled to the solar calendar. Because the solar calendar is longer than the lunar calendar by 11 days (approximately), an extra month is inserted into the calendar 7 out of 19 years. As a result, there are 235 month (12 x 19 + 7 extra months) in the Jewish calendar every 19 years. These 235 lunar months come out to the same length as 228 solar months.

But not exactly.

As it turns out, 235 lunar months does not exactly equal 228 solar months. In fact, the 235 lunar months come out (on average) to about two hours longer than the 228 solar months. Now, two hours over a 19 year span may not sound like much, but over the course of the centuries, they add up. In about 216 years, you have a one-day difference. Since we're 1,665 years away from when the calendar was first set up by Hillel, the differences between the solar and lunar calendar have built up to about eight days. That means that we celebrate our holidays about eight days later in the year than Hillel did. Carried far enough (assuming Moshiach doesn't come by then), we could, in theory, begin celebrating Pesach in the summer! The Hebrew calendar may have been pretty accurate considering the technology of the time that it was established, but to say that it remains "supremely accurate" is just plain bunk.

As for the atomic clock, it's probably unfair to compare it to the Hebrew calendar. Part of the reason for that is that the atomic clock does not reckon time by the motions of heavenly bodies but by the occelations of a cesium atom. A second is no longer 1/31557600 of a year, but is now defined as follows:

The second is the duration of 9,192,631,770* periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

In other words, instead of reckoning time by the sun, they now use cesium atoms (hence the name "atomic clock"). As such, comparing it to the Hebrew calendar would be like comparing apples and oranges... they judge time by two completely different standards.

However, we can look at the Gregorian calendar, which has been in use for over four hundred years. The average Gregorian year differs from the astrnomical year by about 26 seconds per year. As a result, the Gregorian calendar gains a day every 3300 years. That being said, let's see which calendar is more accurate:

Gregorian: gains 1 day in 3300 years
Hebrew: gains about 15 days in 3300 years (1 day every 216 years or so).

Clear winner: Gregorian.

Of course, Pope Gregory had a twelve hundred year head start on Hillel, so it's understandable that his calendar might be more accurate. But to state that the Jewish calendar is "supremely accurate" while descibing the secular calendar as less accurate is just plain wrong.

While we're at it, we should also take a look at this statement of Rabbi Tannenbaum's:

The precision of the Molad announced in every shul during Rosh Chodesh Bentching, indicating the exact time of the new moon in Jerusalem, continues to be breathtaking.

It may be breathtaking, but it's also wrong.

The molad is based on an average period between new moons of 29 days, 12 hours, 44 minutes and 3 1/3 seconds. However, that period is actually wrong. The true period for the molad is about 3/5 of a second longer than that. Again, 3/5 of a second per month may not sound like a lot, but over the course of the centuries, they add up. Currently, the true molad is about two and a third hours later than the molad according to the Jewish calendar. Therefore, the molad that is announced in shul is no longer the exact time the new moon is visible over Jerusalem, but the time the new moon is visible over Afghanistan.

The Wolf

* I wonder if people in Chabad would find some significance in that number.


Anonymous said...

Nu, so why don't you write a letter to the Jewish Press?

micha berger said...

What is breathtaking about the molad is /when/ it was accurate.

Tidal forces not only pull on the earth, but there is an equal and opposite pull on the moon. This causes the moon to slow down; ie months are growing in length.

Our value for the molad dates back to at least Galus Bavel, as the Babylonians use the same number. (I would argue they got it from us; as we see from navi, they drafted our nevi'im to serve as their court sages. But in any case, we have a record that far back.)

However, the molad was too long back then. It was only accurate to the nearest cheileq centuries later. In fact, at around the time Hillel haSheini (or whomever) set up the current standardized calendar!

Also, the molad is the average synodic month, the time it takes for the moon
to be in the same relation to the earth and the son, which is slightly
further than once around the planet, the sideral month. The actual length
of a given synodic month is between 29.26 and 29.80 days (-6.5 hr to +7 hr from the molad).

I once figured out that given a normal distribution, it would take roughly 24,000 years to get an average that was correct to the nearest cheileq. It's not a normal distribution, there is skew, so averaging a smaller set isn't likely to get you the right number.

Hipparchus's number had to be wrong. He
was averaging over 345 years, which means that had he been accurate,
his result would have been around cheileq and a half shorter than one
lunation would take in his day. (The values were from moving avgs of around 3 chalqim to 0 chalaqim too long, an average of 1-1/2 chalaqim of error due to the earlier months in his sample.)

Instead, his result was 15 sec or so too long, or off by 16-1/2 chalaqim from his methodology if it were carried out perfectly. And in 500 years, with the month lengthening by 3 sec/100 yr, one gets to the days we standardize the calendar.

(Admittedly "to the nearest cheileq" includes the 3/5 sec error in the post. But no irrational number can be captured to infinite precision. There has to be /some/ cut-off.)


Kylopod said...

The Jewish calendar, unlike most calendars, is a luni-solar calendar.

I read in Reuven Firestone's book An Introduction to Islam for Jews that the Arabs once used a luni-solar calendar. Islamic scholars regard the change to pure lunar as having been a deliberate attempt by Muslims to wean Arabs off seasonal paganism.

micha berger said...

And the Chinese calendar is lunar-solar (although their shanos me'ubaros are at a different point in the 19 yr cycle).

Arabs don't do much agriculture compared to shepherding; seasons and the solar year don't impact lifestyle as much.

Anonymous said...

I'd love to see Rabbi Gershon Tannenbaum's explanation for the "breathtaking" accuracy of Chazal's calculation for birchas hachama. Just like with their lunar calendar, it's based on an incorrect understanding of the mechanics and
history of the solar system, but unlike the lunar calendar it's so wrong as to be useless.

If you're going to brag the "accuracy" of Chazal's astronomical knowledge you can't exclude the stuff they bumbled.

Anonymous said...

It's interesting to see how much of our tradition's divisions (havdalah) of the sacred (kadosh) are dependent on the arbitrary time span in which they were invented.

Time: Shabbos and seasonal holidays are totally dependent on the narrow window of time with 365 days in a year. Go backwards or forwards a few tens of millions of years and the whole thing falls apart as the number of days in a year drastically changes as days change in length.

Space: the kedusha of eretz yisrael has boundaries, but those boundaries are only meaningful in the narrow window of time in which the movement of the continents is in that configuration. Go backwards or forwards a few tens of millions of years and the whole thing falls apart as the continents move about. All the lives lost over "holy land" are in the end futile, as Earth's geology reclaims it and puts is underwater, under a mountain, or into a mountain.

The people that formed Judaism had no idea about tectonic plates, the vast spans of time the world has been around or the real nature of the planet's motion over that history and future. It helps put thing in perspective to realize that.

micha berger said...

It's not "wrong". Rather, it was an accepted approximation of an irrational number.

To get more precise, eg using Tequfas R' Adda's estimate for the year, would mean only acknowledging that Hashem created the sun every 2,068,417 years. Rounding to the nearest cheileq would be once every 907,201 years.

But the true number of days per year is irrational. Full precision would require an infinite number of digits. SOME estimate has to be used. Why not one that reminds us that the sun too is a beryah?


Anonymous said...

It's wrong because they thought the sun orbited the earth and was created a few piddling thousand years ago, thus having a "point in the sky where it "started." It's wrong because the actual origin and behavior of the sun, both historical and current, differs from it. It's wrong because it's just not true, and it's useless because the flaws mean it doesn't even do what it's supposed to do.

I'll be saying birchas hachama, but there's no need to be under any illusions about its accuracy. You're right about one thing: the ikkar is to acknowledge the sun's status as a beryah. But to do that you don't have to argue for the correctness of birchas hachama, so why fight a lost battle?

micha berger said...

I still say you're overly interested in the word "wrong". It's about estimates and reminders, not science.

I know it's less fun to buy into the system than to feel proud of being able to look down at it, but that's leitzanus (in the technical sense - scoffing).

No one argued for the accuracy of birkhas hachamah -- you (or another "Anonymous") were the one to bring it up.


PS: If you wrote under a real identity, I probably would not tear into you like that. However, since this isn't tochachah berabbim -- no one knows who I am talking to -- I feel freer to do so.

Anonymous said...

If birchas hachama isn't wrong then nothing is wrong.

Why would you assume that I'm not proud of it and/or look down on it? It's not leitzanus to speak factually: birchas hachama is wrong. Leitzanus would be to say that chazal were stupid, something I did not and never would do (or even think, c"v). You're just oversensitive, rushing to defend Chazal's honor even when it's not being impuned. You can say they were wrong without meaning any insult.

micha berger said...

Birkhas haChamah isn't wrong because it doesn't aspire to be scientifically right.

You approach the world with the wrong postulates, and thus reach the wrong answer. People, particularly on the blogosphere, are in too big of a hurry to dismiss rather than trying to find a compatible worldview.

It's leitzanus to bring up your own topic just to cry "wrong wrong wrong". Progress is made by lifting myself up, not pushing everyone else down.


BrooklynWolf said...

Progress is made by lifting myself up, not pushing everyone else down.

And, interestingly, that was one of the points of my post. :)

The Wolf

DAG said...

Wolfie, Hillel would have been using the Julian Calendar. At that point in time the Julian Calender itself would have required an adjustment to reflect the true date.

The point is that there was even greater a discrepancy between the Julian Dates Hillel used and the Gregorian dates we use.

BrooklynWolf said...

Wolfie, Hillel would have been using the Julian Calendar.

Yeah, I knew that.

R. Tannenbaum wanted to make the case that the old is better than the new (the calendar vs. the Atomic clock). Since I felt it was unfair to use the atomic clock (since it defines time differently), I went to the next best thing -- the Gregorian calendar.

The Wolf

micha berger said...

Anonymous has now drifted from leitzanus to kefirah. "The people that formed Judaism..." Feh!

The rest of your post would make sense if Yahadus were a science, not a discipline for human beings who have souls. But since you're not working with the same postulates as the mesorah, it's not worth arguing details.


Ahavah said...

Karaites sight the moon in Jerusalem every month, and the Rabbinic calendar is generally off two days from the real sightings of the new moon.

DAG said...

I went to and tried numerous spellings and got ZERO hits for Birchas Hachama!

I think they realize this is a problem for them

BrooklynWolf said...


In all fairness, I don't think that's the same anonymous (I know, it's hard to tell. I wish people would just choose a name -- even a fake one). It has a different tone and writing style.

The Wolf

Yitzchak said...

First of all, the Gregorian calendar gains 12 days every 3300 years. THat's why he stopped the clock for 11 at the beginning.
Second, Re the eight days: Hillel took care of that by making adar sheni a "variable" month. Some years malei some chaser. It is a clumsy way to take care of it, but the fact that such an awkward way can be made to work is amazing in itself.

BrooklynWolf said...


I don't believe you're correct regarding the Gregorian calendar. I'm fairly certain that it's far more accurate than 12 days/3300 years. Do you have a cite for that?

Secondly, what do you mean "he stopped the clock for 11 at the beginning." Are you referring to Pope Gregory skipping 11 days? He did that to bring the calendar back to it's "proper date." It was a one-time adjustment and had to do with the inaccuracy of the Julian calendar, not any inaccuracy of the Gregorian.

Secondly, Adar and Adar Sheini are never variable. When Adar is by itself, it always has 29 days. When an extra Adar is added (Adar I), it always has 30 days.

The Wolf

DAG said...


Gregory adjusted those days against the Julian Calender, not the Gregorian,

Yitzchak said...

My mistake. I looked it up in the interim. Also my bad on the adars too. I meant that the regular variable months are used to compensate by making them both 30 days during certain rare leap years. I think it would be better for me to keep my mouth shut until I get to look at my references again.

Zach Kessin said...

Putting on my astronomy hat for a second...

First of all the atomic clock was adjusted not because the clock was inaccurate, but because the Earth was (is). The solar year is not a fixed length, there is some variation in length due to tides and other effects. The Atomic clocks are adjusted from time to time to keep in sync with the planet (because we can't do it the other way).

It should also be noted that the Sun-Earth-Moon system is rather complex and has a number of different motions that happen on long timescales. The earth's axis wobbles at about 1/72 of a degree per year, the moon moves away from the earth by about 2cm/year, the day gets longer to preserve angular momentum from the moon moving away etc.

And then there are tides from the other planets. Its a complex system and as we can messure all of these things to amazing accuracy these days

Zach Kessin said...

If you have some time to listen Prof Richard Pogge at OSU has a set of podcasts of his lectures in Intro Astronomy on the web (Telling time) (The Calendar)

They are about 45min each and well worth listening too. (Actually the whole series is worth listening too)

Jeff Eyges said...

I wonder if people in Chabad would find some significance in that

Any excuse to stay up all night drinking vodka.

BTW, Wolf, did you see this a few paragraphs up?

Thankfully, alcoholism is not a serious problem within the observant community.

Right - neither is child molestation.

Thank goodness we have the gedoylim to tell us how superior we are to the gentiles.

Anonymous said...

(I would argue they got it from
us; as we see from navi, they drafted our nevi'im to serve as their
court sages."

Makes perfect sense! Everyone knows how a court sage has the power and authority to introduce new month names that are immediately and widely accepted. I'm sure as soon as those Babylonians heard our holy Jewish months names spoken from the mouths of our holy neviim, they said to themselves: boy our current and existing month names sure do suck. Let's switch forthwith.

micha berger said...

Actually, the shift from doubling Ellu (what we call Elul) to doubling Addu, and from doing so whenever the king thought it was proper to doing so according to formula was in 499 BC. So, shortly after we got there they suddenly had a formula...

As for the month names, the gemara says we got those from the Babylonians, and kept them in order to remember the return from that exile. Much the way we originally numbered our months from the end of the Egyptian exile.

IMHO, this was part of the same trend by which Hadasah picked up a second name after the goddess Ishtar (Canaanite: Asheirah), and Malachi or Ezra ended up called by a tribute to Marduk (Canaanite: Molekh). We assimilated; we know that from the intermarriage rate.

But that doesn't mean religion flowed from them to us, when the Golden Age of Babylonian thought, both science and montheism, didn't happen until after we got there.

But why let facts get in the way of some good sarcasm?


Anonymous said...

I found this Jewish Press Machberes 10-15-2004:
The Perfect Molad

If we paid attention during Rosh Chodesh Benching (Sanctification of the New Moon) this past Shabbos, Bereishis, 24th Tishre, 5765, October 9th, 2004, we heard the molad (the exact instant when the new moon monthly revolution cycle would be seen) announced, usually in Yiddish, in an exact hour with no additional minutes or fractions thereof. “Der molad vet zein Donnerstag in der free 3 azeiger pinklicht” (The molad will be Thursday morning at 3 o’clock exactly).

Such precise announcement is seldom heard. The perfect molad is once in cycles of 1,080 months. In fact, the last time a perfect molad was announced, 87 years ago, was on Shabbos Shelach Lecha, 26th Sivan, 5677, June 16th, 1917, when the molad for Rosh Chodesh Tammuz was announced. The next such perfect molad will again take place 87 years from now, for Rosh Chodesh Adar 1, 5852, to be announced on Shabbos Mishpatim, 24th Shevat, January 2nd, 2092. Since the first perfect molad when the world was created 5,765 years ago, this is the 66th such perfect molad.

The NIST Time and Frequency Division, an operating unit of the Physics Laboratory of the National Institute of Standards and Technology (NIST), an agency of the U.S. Department of Commerce, located in Boulder, Colorado at the NIST Boulder Laboratories, continually updates its precision measurement of time.

NIST is presently undertaking several projects including the atomic-clock mission scheduled to fly on the International Space Station (ISS) in 2008. The mission, funded by NASA, involves a laser-cooled cesium atomic clock, and a time-transfer system using Global Positioning System (GPS) satellites. PARCS (Primary Atomic Reference Clock in Space) will fly concurrently with SUMO (Superconducting Microwave Oscillator), a different sort of clock that will be compared against the PARCS clock to test certain theory. The objectives of the mission include the accuracy of timekeeping on earth.

With literally additional billions of dollars being spent, and billions of dollars already spent, in achieving perfect calibration of time, our sages accomplished greater perfection and accuracy in predicting time by celestial observation thousands of years ago without the aid of NASA, computers, or atomic clocks !!

Anonymous said...

But why let facts get in the way of some good sarcasm?

And why let facts get in the way of some good faith?

Anonymous said...

1) The lunar conjunction occurs simultaneously everywhere on Earth--it has to do with the relative position of the Earth, Moon and Sun, not whether the moon happens to be above the horizon where you are standing.

2) The idea of a fixed molad is itself an approximation. the numbers cited above are the means, but because the orbits involved are elliptical, they vary a little from month to month.

3) because of the dechiyot (not having Rosh hashana fall on ad"u) we deliberately move rosh chodesh from the calculated molad by upt to a little shy of 2 days. This is corrected long term by adjusting marcheshvan and Kislev.

Lion of Zion said...


"why don't you write a letter to the Jewish Press?"

because then he wouldn't be anonymous.
besides, a letter? how about a front page essay!


great post, although i'm not following everything in the post and the comments

i generally tune out whenever someone talks about the supremacy of judaism contra [fill in the blank]. often they don't really know what they are talking about, or worse yet, sometimes their assertion actually weakens the very point they were trying to prove).

Lion of Zion said...


"Birkhas haChamah isn't wrong because it doesn't aspire to be scientifically right."

it's not necessarily a matter of whether it (or anything else, because i'm not making the point specifically about ברכת החמה) is scientifically right, but rather what some people try to pass off as right. (that this is genrally due, i'm sure, to ignorance rather than deceit, is irrelevant.)

DAG said...

This is only a problem to those who claim Chazal had superior and infallible scientific knowledge and use "proofs" of said knowledge as evidence to the objective truth of Judaism

micha berger said...

Between our host the wolf and this last commenter the lion, I am getting nervous... :-)

Lion of Zion: You are correct that the article Wolf is commenting upon is making the same error -- it's trying to force a scientific statement into a non-scientific pronouncement by the Sanhedrin. Much of which is actually due to counter-reformation -- we weren't so maximalist in our beliefs until we had to battle Reform. As an example both Darwin and the dating of the Big Bang set off increases in the number of O Jews who insist that the timeline of Bereishis 1 is literal. (A position that is hard to find amongst rishonim.)

Exacerbated by the fact that it demonstrates the dangers on one's Yahadus of not having a solid "secular" education.

That said, I am less bothered by the ignorant pious making that error than by someone who uses the same error to feel comfortable in his impiety.

I'm reminded of the first two katim (classes of people) discussed by the Rambam in his introduction to Peirush haMishnayos on Sanhedrin, pereq Cheileq.

He discusses three attitudes toward aggadic stories in Chazal. The first group think that all these stories are meant literally, find them absurd and therefore ridicule the Torah. The second kat think that all these stories are meant literally, believe them, and thereby belittle the Torah. The third kat understand that the rabbis were speaking in hints and parable.

It seems we have the same three classes WRT halakhah: one group assumes they know what the halakhah is referring to but believe they know better and use it to ridicule. The next assume they know what the halakhah is talking about, don't know better, and make halakhah absurd. The third group realize that if their observance is working, halakhah should be taken as a given (on the weight of that experience) and ideas about what it's talking about should be derived from that.


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