Lunar calendars and "Qualifications"

Over at Multiple Musings I left a comment in response to the post on Easter. The blogger had written

Because first century Jews used a lunar calendar, every month was twenty-eight days long, beginning with the new moon and having the full moon on the 14th of the month.

I wrote in response

Not quite. A month of the moon's phases is twenty-nine and a half days long, so a lunar calendar--such as the Hebrew Calendar--that tracks the moon's phases will have 30-day and 29-day months, not 28-day months. If the first day is defined by the visibility of the new waxing crescent, as the ancient Babylonian (and probably the 1st century Jewish) calendar defined it, then the full moon will, as you say, occur around the 14th day. The Gregorian lunar calendar, used to determine Easter, attempts to approximate this scheme at the present day, though it is based on averages, not on the actual visibility of the new crescent.

In a calendar that begins its lunar months on the day of the lunar conjunction, however, as the present-day Chinese lunar calendar does, and, with some qualifications, the present-day Hebrew calendar does, the full moon will tend to be closer to the 15th of the month. So, for example, today, Thursday, April 9th, 2009, was the 13th day of the moon by the Gregorian lunar calendar, but the 15th day of Nisan (the 1st day of Unleavened Bread, popularly called "Passover") by the modern Hebrew calendar. This situation often occurs, when the Gregorian lunar calendar is a day or two behind the Hebrew calendar.

In this post, I thought I might elaborate on the "qualifications" mentioned there, and the agreement between the Hebrew and Gregorian lunar calendars.

The Gregorian lunar calendar, that is, the cycle used to compute Easter, was indeed devised (or, at least, Christoph Clavius claimed it was so devised) in such a way that its new moons would fall no earlier than the day after the mean lunar conjunction. This was deliberately done (Clavius wrote) in order to keep the full moon close to the 14th day of the lunar month, which Christian tradition associated with the full moon.

The present-day Hebrew calendar, to a first approximation, sets the first day of its lunar months to the day of the molad, an event which recurs at intervals of 29 days, 12 hours, and 793 "divisions", where a division is the 1080th part of an hour, or three and one-third seconds. In other words, the molad recurs at intervals of a synodic lunar month, a month of the moon's phases. Hence the molad can be, and sometimes is, identified with the mean lunar conjunction at some reference meridian. In the first part of the computation of when the first day of the Jewish numbered year, Tishri 1, should fall, the day is tentatively assigned to the day of the molad.

But now we come to the "qualifications". The year's starting day is moved to the day after the molad if the molad occurs at noon or later. So the start of the day is not permitted to occur 18 hours or more prior to the molad. If the molad is interpreted as a mean conjunction at some longitude in central Asia, then the rule that the start of the year must not precede the molad by 18 hours or more will have the effect of sometimes moving the start of the year to the day after the molad.

Furthermore, the year is not permitted to start on Sunday, Wednesday, or Friday. (This is equivalent to a rule that Nisan 15, the first Day of Unleavened Bread, can never fall on Monday, Wednesday, or Friday.) If the molad falls on one of these days, the 1st of Tishri is pushed back by a day. If the molad fell at noon or later on a Saturday, Tuesday, or Thursday, then it was pushed back onto Sunday, Wednesday, or Friday because of the late molad. But since the year cannot begin on one of these days, the start of the year is pushed back a second day in these cases. This will create a further tendency for the Hebrew Calendar's lunar months to begin later than otherwise. Hence we should expect to see the Hebrew Calendar to agree with the Gregorian lunar calendar some of the time due to these postponements, even though the Hebrew calendar initially fixes the year's start to the day of the molad, which can be interpreted as a mean conjunction, while the Gregorian lunar calendar was devised so that the new moons would always follow "the mean new moon of the astronomers." That this in fact occurs is shown by the following table, which compares Hebrew calendar and Gregorian calendar new moons to the true lunar conjunction at zero degrees longitude. The conjunction times are referred to a day starting at midnight.

Conjunction                                 Tishri 1                  Gregorian new moon

1995 Sept 24 16:55   Sept 25         Sept 25
1996 Sept 12 23:07   Sept 14         Sept 14
1997 Oct   1 16:51   Oct   2         Oct   2
1998 Sept 20 17:01   Sept 21         Sept 22
1999 Sept  9 22:02   Sept 11         Sept 11
2000 Sept 27 19:53   Sept 30         Sept 29
2001 Sept 17 10:27   Sept 18         Sept 19
2002 Sept  7 03:10   Sept  7         Sept  8
2003 Sept 26 03:09   Sept 27         Sept 27
2004 Sept 14 14:29   Sept 16         Sept 16
2005 Oct   3 10:28   Oct   4         Oct   4
2006 Sept 22 11:45   Sept 23         Sept 24
2007 Sept 11 12:44   Sept 13         Sept 13
2008 Sept 29  8:12   Sept 30         Oct   1
2009 Sept 18 18:44   Sept 19         Sept 21
2010 Sept  8 10:30   Sept  9         Sept 10
2011 Sept 27 11:09   Sept 29         Sept 28
2012 Sept 16 02:11   Sept 17         Sept 18
2013 Sept  5 11:36   Sept  5         Sept  7

2014 Sept 24 06:14   Sept 25         Sept 25
2015 Sept 13 06:41   Sept 14         Sept 14
2016 Oct   1 00:11   Oct   3         Oct   2
2017 Sept 20 05:30   Sept 21         Sept 22
2018 Sept  9 18:01   Sept 10         Sept 11
2019 Sept 28 18:26   Sept 30         Sept 29
2020 Sept 17 11:00   Sept 19         Sept 19
2021 Sept  7 00:52   Sept  7         Sept  8
2022 Sept 25 21:54   Sept 26         Sept 27
2023 Sept 15 01:40   Sept 16         Sept 16
2024 Oct   2 18:49   Oct   3         Oct   4
2025 Sept 21 19:54   Sept 23         Sept 24
2026 Sept 11 03:27   Sept 12         Sept 13
2027 Sept 30 02:36   Oct   2         Oct   1
2028 Sept 18 18:24   Sept 21         Sept 21
2029 Sept  8 10:44   Sept 10         Sept 10
2030 Sept 27 09:54   Sept 28         Sept 28
2031 Sept 16 18:47   Sept 18         Sept 18
2032 Sept  4 20:56   Sept  6         Sept  7

It can be seen that for the Hebrew month of Tishri to begin earlier than the corresponding Gregorian lunar month is fairly common in this sample, occurring seventeen times in thirty-eight years. For Tishri to begin on the same day as the corresponding Gregorian month is almost as common, occuring sixteen times. In the remaining five of the thirty-eight years, the Gregorian lunar month begins a day earlier than Tishri. It is possible for the Gregorian month to begin two days prior to Tishri 1, though no cases occur in the years listed here.

This is not quite the whole story, though. Under the rules currently in force for the Gregorian lunar calendar, a lunar month beginning in September, on September 27th or earlier, has 29 days. The Hebrew month of Tishri always has 30 days. This means that at the beginning of the next lunar month, the Hebrew calendar will be one day behind the Gregorian lunar calendar in every year in which the month of Tishri begins on the same day as the Gregorian new moon, if this day is September 27th or earlier. If the table showed a comparison for the month of Heshvan instead of for the month of Tishri, it would show the Hebrew month starting on the same day as the Gregorian month in sixteen years out of thirty-eight, starting a day later than the Gregorian in eighteen years out of thirty-eight, and starting earlier than the Gregorian in four years of the thirty-eight. So occasions when a Hebrew month begins earlier than the corresponding Gregorian lunar month are sometimes balanced by occasions in other years when some other Hebrew month begins later than the corresponding Gregorian lunar month.

In both lunar calendars, the day starts at 18:00 on the day prior to the one listed in the table. Comparison of the new moon dates to the times of the true conjunctions listed in the first column shows that the Hebrew month begins before the conjunction in seven years out of the thirty-eight (2002, 2009, 2013, 2018, 2021, 2022, and 2024), while the Gregorian lunar month begins before the true conjunction only once, in 2019. In the month following that shown in the table, the Hebrew month begins before the true conjunction in two years out of the thirty-eight , while the Gregorian begins before the true conjunction in four years. This may indicate that the Hebrew month begins before the true conjunction at 0 degrees longitude slightly more often than the Gregorian month does. But if so, the effect is slight.

Clearly the "postponements" in the Hebrew calendar are working to push the new moon past the conjunction. Hence my comment over at Multiple Musings exaggerated somewhat the tendency of the Hebrew new moon to occur closer to the lunar conjunction than the Gregorian new moon. This tendency is there, but it is often overridden by the Hebrew calendar's "postponements", as scholars from Maimonides on have noted. In particular, the two-day difference in the current month--Saturday, April 11th, 2009 is the 15th day of the moon by the Gregorian lunar calendar but the 17th of Nisan by the Jewish calendar--is probably due more to the vicissitudes of these postponements, and the different scheduling of 30-day and 29-day months in the two calendars, than to any underlying bias of the Jewish calendar toward the day of the mean conjunction.