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Can there be 5, Consecutive 30-day Months in a Lunar Calendar?

The subject comes up from time to time that the timespan of the Biblical Flood, being 150 days, identified because the flood began and ended on the 17th of the month, means that there were 5, consecutive 30-day months necessary for the timing to work out. It’s an issue because the Hebrew calendar (both ancient and modern) alternates between 29 and 30 day months because the moon’s average synodic period is 29.53 days, and thus it makes absolute sense that one month would be 29 days, while the next would be 30 and so on. It is common knowledge that there sometimes needs to be two, 30-day months in a row, because the moon's synodic period is not exactly 29.5 days, so the difference accumulates to an additional day and causes what would otherwise have been a 29-day month following a 30 day month, to be 30 days instead. Also, the slow, cyclical change in the moon's synodic period from a low of about 29.27 days to a high of 29.84 days can also result in consecutive 29-day months or 30-day months. Still, there is certainly not a situation where FIVE 30-day months in a row can or must happen – at least that is known, or ever has been observed!

One flaw in the analysis of the timing of the flood is that it is rather a complete assumption that at the time of the flood, the alternating 29-30 day calendar was in use. For all we know, Noah simply used a fixed, 30-day calendar – and did not rely on the moon! After all, it rained 40 days and 40 nights, and perhaps the entire 150 days the Arc was afloat it was cloudy and overcast, and Noah could not have watched for the month by the moon anyway! Still, as a result of the 150 day period of the flood, many insist that it must be possible to have five, consecutive 30-day months by the Hebrew calendar for the scripture to work. So how do we prove that five, 30-day lunar months are or are not possible?

Let’s discuss the data:

The synodic period of the moon (also called the “synodic month”) averages 29.53 days. (In reality, we know this average to many more decimal places, to wit: 29.530589 days.) The synodic period is defined as the number of days between the same measurable phase of the moon. (Many sources leave out the word “measurable”, yet it is very important.) So this could be, for example, “new to new”, “full to full”, or “1st quarter to 1st quarter.”

But 29.53 days is only an average. The maximum synodic period is 29.84089 days, and the minimum is 29.26574 (Espenak, F. and Meeus, J. (2008), Five Millennium Catalog of Solar Eclipses). When the moon has completed one orbit, and has returned to the same, measurable phase, it is called a “lunation”. The length of a lunation then, averages 29.53 days, with a low possible of 29.26 and a high of 29.84. [Note: You never see a lunation of say, 29.84 days, followed by a lunation of 29.26 days. The change from the longest lunation to the shortest and vice versa happens over 7 lunations. The change in the length of the lunation month-to-month is actually quite small, about 0.14 days.]

So when might we expect a “high” and when might we expect a “low” in the number of days in a lunation? And is it possible for there to be “five, high period lunations” in a row so that the total number of days in 5 lunations totals 150 days - that is, that there can be 5, 30-day months in a row?

Let’s find out

We find that when New Moon occurs near the perigee of its orbit, the length of the lunation is at a minimum. Similarly, when New Moon occurs near apogee, the length of the lunation reaches a maximum. (See NASA Eclipse website). [The length of a lunation is also slightly affected by the speed of the earth as it orbits the sun, but I am not addressing the earth's impact here. Computations of the monthly lunation take that into account.]

Let’s look at the effect of the moon’s orbit on the length of a lunation. We can plot the lunations on a graph, and see the effect of a lunation visually. In the analysis which follows, I will be taking New Moon to New Moon (conjunction) but I want to emphasize that this analysis could also be made from Full Moon to Full Moon, or 1st Quarter to 1st Quarter, or any other “same” phase to “same” phase – again, so long as that phase is measurable. In fact, this analysis could also be done from “1st visible crescent” to “1st visible crescent” (or even last crescent to last crescent) if it were not for the fact that the first or last crescent is never seen month to month from the same place at the same lunar age. (More on this later.)

So in Figure 1, we see a plot of New Moons for a consecutive period of New Moons spanning about 25 years. This plot happens to be from 1944 to 1968, though the specific 25-year period is not important to this analysis.

Referring to Figure 1, let’s understand what we are seeing. The X-axis is simply time, starting at the left from sometime in 1944, and growing uniformly month after month to sometime in 1968. (Again the calendar years are not important to the topic at hand. Just understand that the X-axis is “time” – left to right.)

The Y-axis then is the important axis. It is the plot of the number of days for each successive lunation. The span is readily visible as it is marked from 29.2 to 29.8, thus it is plotted within the range of the minimum and maximum lunation. A “blue diamond” marks the event of a New Moon. One can follow one New Moon to the Next by simply following the blue line that connects the diamonds. I have also placed a red line at the average lunation – 29.53 days – so it is readily apparent that month to month, the number of days in a lunation is not often “at 29.53”, rather it is most often longer or shorter!


Figure 1.

Continuing to examine Figure 1, there are many important features to grasp. First, notice the plot is sinusoidal, yet not exactly uniform. No sequence of lunations is exactly in the same spot cycle to cycle - (see comment on the Metonic cycle, below). This is because the number of days between lunations is continuously variable. Yet, we can see from the plot, that even though a lunation is continuously variable, it is bounded. Just look at Figure 1, and you can see that no lunation is fewer than about 29.28 days, and no lunation is greater than about 29.84 days. (Now refer back to the citation above from Espenak, F. and Meeus, at the numbers of the minimum and maximum number for the lunation, and see that our plot, indeed, is within those bounds).

Notice also the length of the space between any two blue diamonds. Notice that this length, this span, between any two diamonds is not uniform! Sometimes two consecutive diamonds are very close together and sometimes they are much farther apart – yet there seems to be a maximum that they can be apart!

Indeed there is a maximum! One finds that no two consecutive lunations will differ in length by more than 0.14 days. That is to say that for any two consecutive lunations, if the number of days in each lunation is compared, they will never be more than 0.14 days apart! Two consecutive lunations can be exactly the same, for example, when the New Moon is symmetric on either side of apogee or perigee, but it would be exceedingly rare to see two consecutive lunations of exactly the same period. We can pretty much state that “no two consecutive lunations are exactly the same, with rare exceptions”.

But let’s continue to examine what we can see from this data. Take special care and examine the distance between any two diamonds. You can see that when the lunation is just greater than the mean of 29.53 days, the distance between it and the next blue diamonds is greater than the distance between two blue diamonds for those below the 29.53 line! This is very useful information!

Go back and read the data above quoted from NASA: “We find that when New Moon occurs near perigee, the length of the lunation is at a minimum. Similarly, when New Moon occurs near apogee, the length of the lunation reaches a maximum”. What we are seeing by our observations just from the plot is that the distance between two consecutive diamonds is this: When the lunation is below the 29.53 average, we are seeing the effects of the New Moon around perigee. Conversely when the New Moons are happening greater than the 29.53 average, we are seeing the effects of the New Moon around its apogee. Further, if we could look at the real numbers rather than estimating from the plot, we’d see even without NASA that the number of days difference between any two New Moons is always greater when the moon is near apogee, and fewer when near perigee, and we will use this fact in our analysis of the possibility of five, consecutive 30-day months.

Actually demonstrating Newton’s Laws

Before we go on, let me point out that the data revealed in Figure 1, is entirely consistent with Newton’s Laws of Motion, in that if a planet is near its aphelion, it is moving slower, and when near perihelion, it is moving faster. This real motion of the moon is far more complex as there are many physical effects impacting the true motion of the moon due to many factors beyond the scope of this article, yet still, we see the essence of Newton’s law of motion in the plot!

So now that we know that the moon is moving slowest around apogee, then we know that the absolute greatest number of days between any 5, consecutive New Moons, must be when the moon is around apogee – that is – on the “greater than 29.53 side of the plot” of Figure 1.

Let’s reproduce Figure 1 and pick out a data set to look at. Referring to Figure 2, look at the plot again, and let’s pick out the data:

We’ve marked one of the “high” lunations – one at 29.8292 days. Notice the 3 New Moons before that high point are both less than 29.8292 days, yet greater than 29.53, the average of all lunations. Notice also that the 3 New Moons after the highest have the same characteristics. What I want to do is consider the “maximum” of this very case, or at least for the case with the greatest lunations of any group of 5 consecutive months and see what that total can possibly be.

Before looking at the chosen data set, notice one more thing: For 5, consecutive months, for the total of ALL to be as great as possible, we must choose one of them to be the peak in the cycle. From this small data set plotted, we can see that if ONE exceeds 29.8, then no more than two can exceed 29.8, and it looks like only 4 in a row can exceed 29.7.

Admittedly, we need a larger data set to confirm these observations, but the point I am trying to drive home is that we are looking for the situation when 5, consecutive lunations are as great as possible. To total the greatest number of days, all 5 lunations must be as near as possible to the maximum possible lunation which is 29.84089 days. We already can see, just from this small data set, that all 5 will never be at or over 29.80 days, which, if they were, would total in excess of 149 days, so we already know that the maximum number of days in 5, consecutive lunations will likely be fewer than 149 days. This is very important in our quest, because if the total of 5 consecutive lunations can ever exceed 149 days, only then it would be possible to have 5, consecutive 30-day months and the calendar still work!


Figure 2.

Looking at real data

So let’s look at the data suggested from the dataset of Figure 2, and find out how many days are in the total of 5 consecutive lunations, 2 before, the 29.8292 day lunation, and the 2 after.

Pulling the data for the NM marked “29.8292”, and for the 6 other NMs in the red oval – 3 before and 3 after the peak - I’m going to tabulate this data, and then discuss it:

The columns are:

“Synodic month” – this is the number of days in that lunar month, NM to NM.

“Cumulative Synodic month” – this is simply adding the total time (number of days), cumulatively from the correct point.

“# of Days” – this is our entry for the number of days in the Hebrew month. Can only be 29 or 30, but what we are looking for is the effect of consecutive 30-day months.

“Cumulative # of days” – this is simply the cumulative number of days in our Hebrew months.

Data:


Figure 3.

Starting in Figure 3 with the month which was 29.55070 days long which will be for us an “anchor month” , I’ve verified the Hebrew month that particular month was 29 days – so I put it in YELLOW highlight – we KNOW that that month was 29 days long. (Note: It turns out it does not matter whether an “anchor month” was 29 or 30. What matters is that you are counting the effect of continuous 30 day months after that 29 or 30 day month.) So the first month, in this case, 29 days, is not counted.

Now, I want to assume that every month thereafter is 30 days long. (That is what we are looking for – can there be 5, consecutive, 30-day months?) So looking at the data, we see everything is fine – as annotated by “OK” in the far-right column till we see that the 5th consecutive 30-day month has added up to 150 days, yet the “real” moon has added up only to 148.753 days (that is, the real months can be no more than 149 days when rounded to whole days for calendar purposes), so our calendar is 1 day ahead if we insist on a 30-day month that last lunation (the red one – 30 days). But what we see is that that last calendar month must instead be 29 days so our calendar total is 149 to match the “real” moon of 148.753 days (rounded to 149) in 5 months. This illustrates that it is possible only to have 4 consecutive 30-day months and the calendar still work. But this is only ONE data set.

Do I need to show additional data sets? Well, it would be convincing, but it is not necessary because what is happening is that extra time between any consecutive synodic months, mentioned earlier, reveals that no two consecutive synodic months can be greater than 0.14 days apart – and is more often less than 0.14 days apart. To make a long story short, looking statistically at all the data, here is the bottom line: For any consecutive 5 lunations, the total number of days cannot exceed 148.88. Since the total number of days in 5 consecutive months of 30-days is ALWAYS exactly 150 days, then 5, 30-day calendar months is ALWAYS ONE day greater than the “real” moon of 5 consecutive lunations – regardless of what phase you use to measure.

Here is the supporting data for that statement:

For data spanning over 300 years, I isolated the 66 lunations which exceeded 29.8 days. I then found the average number of days between lunations for the 2 lunations on either side of that max – the required total of 5 lunations. The LARGEST average between lunations was 0.0889 days, while the smallest was 0.0699 days. Now, a “smaller” number is preferred, because a smaller number means each length of each consecutive lunation was the greatest, and closest to the # of days in the preceding month. In other words, the closer to the smaller number 0.0699 days difference in the length of lunations, the greater the total number of days in a span of 5 consecutive lunations.

So, since the smallest average number in a data set of 300 years was 0.0699, this means that the greatest number of days in five consecutive lunations, using the maximum possible lunation of 29.84089 days is: 148.79 days.

This is found this way:

(Remember, the MAX is 29.84089, so those lunations on either side must be no greater than 0.0699 days shorter, using 0.0699 as the "low" average. This ensures we are calculating a maximum possible 5-month total. )

(I recognize carrying 5 decimal places is ridiculous, as it is less than a second of time, but I did not need to deal with error bars or standard deviations in this analysis.)

This totals 148.78505 days (rounds to 149 whole calendar days) demonstrating that taking the maximum possible lunation, and assuming the greatest lunations possible on either side of that maximum lunation, we find that the total days in 5, consecutive months is 148.79 days. But, choosing an even more conservative number, 0.054 days between lunations which is the real average of ALL lunations in the 300 year dataset, the maximum possible number of days in 5, consecutive 30-day months is 148.88.

Since 148.88 is remains still less than 149, which is obviously more than 1 day LESS than 150, we can state definitively that there can never be 5, consecutive 30-day calendar months, because if you require it, at the end of the 5th month, your calendar is 1 day off and it will impact the 6th month!

Therefore the maximum possible number of consecutive 30-day lunar months is 4.

Now there is an objection I need to cover. I said “It does not matter whether the starting month, that is, the 'anchor month', was 29 or 30 days”. This could lead one to argue that “if the anchor month was 30 days, and there can be 4 consecutive 30-day months which follow it, then the total would be 5 consecutive months!” I agree that one might think this, at least on the surface.

But remember the conclusion: That no, five, consecutive lunations can exceed 148.88 days. So if the month you stared with was itself a 30-day month, then the 5th lunation including that starting 30-day month, will end up no greater than 148.88 days long, thus, you’d find that only 3 months after the 30-day month in which you began, you’d be forced to follow it with a 29 day, 5th month, to keep the calendar in-sync with the real moon.

Other work

Curiously, I could find little scholarly work on this issue. On the other hand, there seems to be much work done on this question in the Islamic world, but they mandate a “sighted crescent” to begin a month. One source I found said it is possible to have 5 consecutive months of 30-days on the Muslim calendar, but only as observed from the southern hemisphere. But this is an observational problem, not a physical reality. They might say: “I did not see the crescent because the sky was too bright at sunset, so I have to add a 30th day to the old month.” Not very sound science. Indeed, determining the start of the month by sighting the crescent is a really poor way to start the month. Even the ancient people watching the moon continuously, would have concluded the "sighted crescent" is not a reliable way to determine the month! They would know this by simply watching the old moon, i.e., the last crescent of the old month over time, and the day of the full moon month after month, year after year, decade after decade and tabulating their data.

Gautschy argues there are several instances in ancient times of 5, consecutive, 30-day months (and of 4, consecutive, 29-day months). She says: "Since 2001 BC only five lunar months with 30 days or four lunar months with 29 days can follow each other. Between 2001 BC and 2000 AD five lunar months with 30 days in a row took place 12 times. The occurrence of this phenomenon is not equally spaced in time: between 588 BC and 887 AD there was none, in the 2nd millenium BC on the other hand 6 times (1986, 1950, 1800, 1764, 1578 and 1128 BC)."

But Gautschy's computations are based on calculated times of sighting the visible crescent of the new moon and do not take into account that the natural variation of the moon's synodic period must also be manifested in observations of the visible crescent. That is to say, even if relying on the sighted crescent to establish the month, the natural rhythm of the moon dictates a return to a 29-day month after a maximum period of only 4 months, as I have shown above. This natural convergence to 29.5 days in the synodic month is also found by observing the new crescent even in a span as short as 5 months. This is also found by Schaefer, Archaeoastronomy, no. 17 (JHA, xxiii 1992).

I took one set of Gautschy's data, that of 1986 BCE, and after validating her values for the synodic period of the moon in that year, I found nothing different from my findings presented in this article. By assuming the lunar months of the five months [August - December (Julian), 1986 BCE] were each 30-days long by observation of the crescent, the new crescent in November was late by a full day. In other words, it is more likely than not that the new moon in November would have already been visible at sunset when the lunar calendar said it was the 29th of the 4th lunation (the lunation that began in October). Therefore the 4th month (lunation that began in October) would have been ended at 29 days. Even if the the new moon in November was not visible till sunset the 30th of the 4th lunation, the 5th lunation would only be 29 days because the new crescent in December would most definitely have been observed at sunset the 29th ending the 5th lunation at 29 days.

Comment on the Metonic cycle

While my statement above that "No sequence of lunations is exactly in the same spot cycle to cycle", at first glance, seems to be negated by the Metonic cycle, a 19 year period which was found by the ancient astronomer, Meton, (but known to the ancient Babylonians as well), to contain almost exactly 235 synodic months, I can say that my statement is valid. What the Metonic cycle means is that in exactly 19 tropical (solar) years, the moon will have the exact same phase and be in the same part of the sky that it was 19 years earlier. (The moon won't actually be in the exact spot, but will be very, very close to the eye.

The Metonic cycle is useful because the phenomenon permits a mathematical way to "insert" an extra (13th) month to a purely lunar 12-month calendar to keep the lunar calendar and solar calendar in-sync. This insertion of the extra, "intercalary", month is not an artificial construct for if you pick a day in the solar year on which you are going to call "the first day of the year", then the fact that the natural cycle of the moon's "12 month year" is about 11 days short of the solar year means that you will need to "add" a 13th month every 3rd year or so to "naturally" keep "lunar year" and "solar years" in-sync. The Metonic cycle is simply a way to mathematically apply that 13th month on a "schedule" so at the end of 19 years the two calendars "line-up".

Unfortunately, while the Metonic cycle works great mathematically, it does not work so well in reality. You see, to get the needed 13th month in the correct number of years of the 19-year cycle, they simply declared the month would be added in the 3rd, 5th, 8th, 11th, 13th, 16th, and 19th year of the cycle. Over the whole 19 years this works out fine. But in the real earth-moon-sun system, rigidly applying the 13th month in these fixed years may not meet the Biblical criteria of keeping Passover at its proper time of the year. The modern Jewish calendar, for example, which today implements the intercalary month in accordance with the Metonic cycle, will introduce a 13th month at the end of the Hebrew year 2015-2016 and instead of Passover falling in March, 2016 (as it should according to the moon), Passover will be in late April - a full month late. This error is an artifact of a rule which, while good over the whole 19 year period, is not necessarily good in any particular year in the cycle.

But, the point is that the statement "No sequence of lunations is exactly in the same spot cycle to cycle" is not incorrect. The Metonic cycle only does a pretty good job of mathematically matching 19 solar years with a whole number of synodic periods. That may sound like "exactly in the same spot cycle to cycle", but it is only close.

Final words

Some final words about applying this to a sighted crescent in the northern hemisphere. Despite Gautschy's findings (which were calculated for Egypt), it would never be possible to have 5, consecutive 30-day months by watching for the new crescent alone from anywhere in the Middle East. The new crescent is never visible at the same lunar "age" month-to-month, i.e., the same number of hours after conjunction, so the sighting of the crescent is a notoriously poor way to determine the beginning of the month. During the fall months, when the moon emerges from conjunction but remains very low in the sky after sunset it is possible that the new crescent may not be "sighted" till it is actually 3 days old! This means it is likely the very next month will only be 29 days long as its new moon will be sighted sooner after the next conjunction. In reality, as already stated, over the long term (several months) lunation observation by the sighted moon does average pretty quickly to 29.5 days. Even though the sighted moon is a terrible determinant of the start of the month for any particular month, it could be used by a society which did not spend much time comprehending all the signs of the moon during the month which would help a skilled observer know exactly how to start the month on the proper day - even if the new crescent is not seen!

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