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The Metonic Anomaly of 1699/1700 and the 371 year cycle

If you become interested in lunisolar calendars you will undoubtedly come across the Metonic system which is defined rather simply.

The Metonic system is a way to reconcile the movements of the sun and the moon over 19 year period. 

12 lunar months of 29.5 days is 354 days
1 solar year is 365.24 days

So 12 lunar months is 11 days short of a solar year. This means that over time the months drift backwards through the year in relation to the seasons. 

By including an extra 13th month in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years you end up with a total of 235 months, or more accurately 6,939.688 days. 19 solar years is 6,939.602 days which is a difference of only 0.086 days. Therefore on any given date the sun and the moon will be in the same position in the sky 19 years later. 

And that is about as much of an explanation as you can find online. Some sources will go into more detail, but there are a number of questions that remain unanswered. One key question is when in the leap year does the extra month occur? What about the remaining discrepancy? These are questions I have had to find my own answers too.

Working out the metonic cycle

The key breakthrough regarding the months was my realisation of the relationship between blue moons and the Metonic cycle. In effect a blue moon is a 13th full moon in a year. It is a description for full moons rather than “months” that begin at the new moon, but the pattern is exactly the same, only the 13th new moons occur in different years to the blue moons. You can not have a blue moon and a 13th new moon in the same year, because it takes  385.5 days for 13 new and full moons to occur, 18.26 days longer than the solar year. This helped me to understand and calculate exactly what season the 13th month should be included in each year one is needed.

During my research, however, I came across an interesting fact. During the winter of 1961/62 there were only 2 full moons, resulting in a blue moon both in the autumn beforehand and the spring afterwards. But wait a moment, blue moons are said to occur roughly every 2.5 years the same as Metonic leap years? How can a blue moon occur only half a year after another? If this can happen with full moons, can happen with new moons as well? This doesn’t fit the Metonic sequence. What is going on?

The 1699/1700 anomaly

After writing code to calculate when a 13th new or full (blue) moon would occur (when there are 4 between a solstice and equinox), I realised that there were indeed only 2 full moons that winter, so I started to look for the same thing happening with new moons and sure enough I found it. The last time there were only two new moons in the winter was the winter of 1699/1700 and the next is the winter of 2071/2. What does this mean for the Metonic cycle?

The Metonic cycle that started in 1684 had leap years in the following years:

3rd
6th
8th
11th
14th
16th
17th
19th

Instead of the ordinary 7 leap years in a cycle, this cycle had 8 leap years in it, including the extra leap year in the 16th year. What is even more interesting is what happens during the next couple of 19 year Metonic cycles. During the cycle that starts in 1703 the leap years fall in:

3rd
6th
8th
11th
14th
16th
19th

Again, this is not the ordinary metonic sequence. The leap year is now in the 16th year rather than the 17th year. And the cycle after that starting in 1722 follows the same pattern again, but the cycle after that one starting in 1741 returns the cycle back to normal. Later this century with the Metonic cycle that will begin in 2035 we see the same pattern again. In this cycle there will be 8 additional months as well.

This anomaly in the Metonic cycle occurs every 371 years and highlights for us a larger cycle that the Metonic cycle is but a part of.  It also gives us more understanding of the Metonic cycle than is readily available online. For example we can now say that this anomaly occurs every 19.52 Metonic cycles (371 / 19 = 19.52) which again raises a question. How can this anomaly occur halfway through a metonic cycle (0.52)?

To remain true to the definition of the Metonic cycle having leap years in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years there will always be a gap of 3,3,2,3,3,3,2 years between each leap year. We can now verify this pattern across all the Metonic cycles that make up the wider 371 year cycle and we notice yet another anomaly. In each 371 year period there is is an additional half a Metonic cycle, and it occurs a couple of cycles before the cycle with 8 leap months. Here is a list of the 4th new moons in a season around the anomaly of 1699/1700.

First a standard sequence

1618 Summer 3 Mon, 20 Aug 1618 06:36:21 GMT
1621 Spring 3 Fri, 21 May 1621 08:42:26 GMT
1623 Autumn 2 Wed, 22 Nov 1623 07:59:56 GMT
1626 Summer 3 Fri, 21 Aug 1626 21:50:01 GMT
1629 Spring 3 Tue, 22 May 1629 19:39:40 GMT
1632 Spring 3 Wed, 19 May 1632 01:46:01 GMT
1634 Summer 2 Wed, 23 Aug 1634 16:06:17 GMT

A half cycle that breaks the sequence

1637 Summer 3 Thu, 20 Aug 1637 01:58:06 GMT
1640 Spring 3 Sun, 20 May 1640 20:33:36 GMT
1643 Winter 3 Wed, 18 Feb 1643 15:50:06 GMT
1645 Summer 2 Mon, 21 Aug 1645 11:42:53 GMT

Another two standard sequences

1648 Summer 3 Tue, 18 Aug 1648 18:45:32 GMT
1651 Spring 3 Fri, 19 May 1651 10:48:24 GMT
1653 Summer 2 Sat, 23 Aug 1653 01:03:27 GMT
1656 Summer 3 Sun, 20 Aug 1656 02:12:55 GMT
1659 Spring 3 Wed, 21 May 1659 03:46:25 GMT
1662 Winter 3 Sat, 18 Feb 1662 16:07:30 GMT
1664 Summer 2 Thu, 21 Aug 1664 09:15:07 GMT

1667 Spring 3 Sun, 22 May 1667 22:38:57 GMT
1670 Spring 3 Mon, 19 May 1670 03:27:00 GMT
1672 Summer 2 Mon, 22 Aug 1672 17:38:52 GMT
1675 Summer 3 Wed, 21 Aug 1675 01:56:09 GMT
1678 Spring 3 Fri, 20 May 1678 14:56:07 GMT
1681 Spring 3 Sat, 17 May 1681 19:25:29 GMT
1683 Summer 2 Sun, 22 Aug 1683 09:58:08 GMT

A cycle with 8 13th months (the anomaly)

1686 Spring 3 Wed, 22 May 1686 06:13:12 GMT
1689 Spring 3 Thu, 19 May 1689 02:32:34 GMT
1691 Summer 2 Thu, 23 Aug 1691 16:49:12 GMT
1694 Summer 3 Fri, 20 Aug 1694 19:58:17 GMT
1697 Spring 3 Mon, 20 May 1697 10:07:52 GMT
1699 Autumn 2 Sat, 21 Nov 1699 14:52:33 GMT
1700 Spring 1 Tue, 18 May 1700 17:48:25 GMT
1702 Summer 2 Wed, 23 Aug 1702 08:25:19 GMT

Leap year moves from 17th to 16th year for 2 cycles

1705 Spring 3 Fri, 22 May 1705 19:55:55 GMT
1708 Spring 3 Sun, 20 May 1708 03:03:38 GMT
1710 Summer 2 Sun, 24 Aug 1710 17:25:09 GMT
1713 Summer 3 Mon, 21 Aug 1713 06:04:31 GMT
1716 Spring 3 Thu, 21 May 1716 10:19:26 GMT
1718 Autumn 2 Tue, 22 Nov 1718 06:09:03 GMT
1721 Summer 3 Fri, 22 Aug 1721 23:51:50 GMT

1724 Spring 3 Mon, 22 May 1724 17:17:30 GMT
1727 Spring 3 Tue, 20 May 1727 23:36:24 GMT
1729 Summer 2 Wed, 24 Aug 1729 14:03:52 GMT
1732 Summer 3 Wed, 20 Aug 1732 13:22:04 GMT
1735 Spring 3 Sun, 22 May 1735 10:12:04 GMT
1737 Autumn 2 Fri, 22 Nov 1737 04:25:22 GMT
1740 Summer 3 Mon, 22 Aug 1740 08:09:52 GMT

Back to normal sequence for a couple of hundred years (until 2009)

1743 Summer 3 Mon, 19 Aug 1743 15:33:24 GMT
1746 Spring 3 Fri, 20 May 1746 12:14:07 GMT
1748 Autumn 2 Wed, 20 Nov 1748 19:55:32 GMT
1751 Summer 3 Sat, 21 Aug 1751 02:12:13 GMT
1754 Spring 3 Wed, 22 May 1754 04:26:57 GMT
1757 Winter 3 Fri, 18 Feb 1757 13:23:11 GMT
1759 Summer 2 Wed, 22 Aug 1759 16:30:42 GMT

The 2052/3 and 2071/2 Anomalies

We can see the same thing happening again in our modern age as we lead up to the anomaly of 2052/3. Here again are the details of the metonic cycles either side of this event.

Last standard Metonic cycle

1990 Summer 3 Mon, 20 Aug 1990 12:37:50 GMT
1993 Spring 3 Fri, 21 May 1993 14:05:48 GMT
1995 Autumn 2 Wed, 22 Nov 1995 15:42:00 GMT
1998 Summer 3 Sat, 22 Aug 1998 02:03:22 GMT
2001 Summer 3 Sun, 19 Aug 2001 02:55:13 GMT
2004 Spring 3 Wed, 19 May 2004 04:51:25 GMT
2006 Summer 2 Wed, 23 Aug 2006 19:10:24 GMT

Half Metonic cycle breaking the sequence

2009 Summer 3 Thu, 20 Aug 2009 10:02:11 GMT
2012 Spring 3 Sun, 20 May 2012 23:44:29 GMT
2015 Winter 3 Wed, 18 Feb 2015 23:45:08 GMT
2017 Summer 2 Mon, 21 Aug 2017 18:30:05 GMT

A Standard cycle

2020 Summer 3 Wed, 19 Aug 2020 02:38:50 GMT
2023 Spring 3 Fri, 19 May 2023 15:52:27 GMT
2025 Summer 2 Sat, 23 Aug 2025 06:04:16 GMT
2028 Summer 3 Sun, 20 Aug 2028 10:41:55 GMT
2031 Spring 3 Wed, 21 May 2031 07:15:38 GMT
2034 Winter 3 Sat, 18 Feb 2034 23:08:26 GMT
2036 Summer 2 Thu, 21 Aug 2036 17:35:02 GMT

2 cycles with 8 13th months

2039 Summer 3 Fri, 19 Aug 2039 20:46:52 GMT
2042 Spring 3 Mon, 19 May 2042 10:54:22 GMT
2044 Summer 2 Tue, 23 Aug 2044 01:04:39 GMT
2047 Summer 3 Wed, 21 Aug 2047 09:11:29 GMT
2050 Spring 3 Fri, 20 May 2050 20:49:25 GMT
2052 Autumn 2 Thu, 21 Nov 2052 08:58:29 GMT
2053 Spring 1 Sun, 18 May 2053 03:40:28 GMT
2055 Summer 2 Sun, 22 Aug 2055 18:10:39 GMT

2058 Spring 3 Wed, 22 May 2058 10:21:27 GMT
2061 Spring 3 Thu, 19 May 2061 11:00:08 GMT
2063 Summer 2 Fri, 24 Aug 2063 01:15:03 GMT
2066 Summer 3 Sat, 21 Aug 2066 00:44:24 GMT
2069 Spring 3 Mon, 20 May 2069 18:03:15 GMT
2071 Autumn 2 Sat, 21 Nov 2071 18:53:46 GMT
2072 Spring 1 Wed, 18 May 2072 00:17:57 GMT
2074 Summer 2 Wed, 22 Aug 2074 14:53:53 GMT

Leap year moves from 17th to 16th year for 2 cycles

2077 Spring 3 Sat, 22 May 2077 02:36:01 GMT
2080 Spring 3 Sun, 19 May 2080 10:52:45 GMT
2082 Summer 2 Mon, 24 Aug 2082 01:11:45 GMT
2085 Summer 3 Mon, 20 Aug 2085 09:05:15 GMT
2088 Spring 3 Thu, 20 May 2088 18:43:09 GMT
2090 Autumn 2 Tue, 21 Nov 2090 12:42:14 GMT
2093 Summer 3 Sat, 22 Aug 2093 03:47:52 GMT

2096 Summer 3 Sat, 18 Aug 2096 02:52:45 GMT
2099 Spring 3 Wed, 20 May 2099 05:13:31 GMT
2101 Autumn 2 Mon, 21 Nov 2101 04:41:37 GMT
2104 Summer 3 Wed, 20 Aug 2104 17:19:26 GMT
2107 Spring 3 Sun, 22 May 2107 17:24:46 GMT
2109 Autumn 2 Fri, 22 Nov 2109 12:13:51 GMT
2112 Summer 3 Mon, 22 Aug 2112 11:07:16 GMT

Back to standard sequence

2115 Summer 3 Mon, 19 Aug 2115 23:25:10 GMT
2118 Spring 3 Fri, 20 May 2118 15:32:04 GMT
2120 Autumn 2 Thu, 21 Nov 2120 02:06:35 GMT
2123 Summer 3 Sat, 21 Aug 2123 08:42:29 GMT
2126 Spring 3 Wed, 22 May 2126 09:09:59 GMT
2129 Winter 3 Fri, 18 Feb 2129 20:38:36 GMT
2131 Summer 2 Wed, 22 Aug 2131 21:12:20 GMT

As you may have noticed, there is a subtle difference between these two examples of the anomaly. In the first example there were 2 standard sequences followed by a sequence with 8 13th months. In the second example there was one standard sequence followed by 2 cycles with 8 13th months. If we look forward to the next time this happens we can see that the pattern is the same as it is this century, with 2 cycles with 8 13th months, the first of which starts in 2409.

Conclusions

What is clear from this data is that the standard description of the Metonic cycle, while true for the vast majority of cycles, it cannot be consistently applied. Further, to stay in sync with the description over long periods of time there has to be a “half cycle” at some point in the 371 year pattern, as well as 2 cycles where the leap year swaps from 17 to 16, and at least one cycle (often 2) where there are 8 leap years.  This new information could impact the current calculation of exactly which cycle we are in for this age, but as always I feel that updating the calendar to reflect this information about the actual movements of the sun and moon to increase accuracy is the correct thing to do. I will be building this information into the calendar application over the coming weeks.

It also has implications for the next phase of my studies. I intend to look next at the pattern of festival days and in particular the “odd ones out”. Days where specific moon phases line up with solar events, eclipses or other rare events such as this break in the Metonic cycle that could be considered an epoch in the calendar. Importantly these are not made up events but rather events that become apparent only when looking at the actual patterns of the moon and sun over many centuries, and are dictated by their actual movements.

I hope that this more detailed information regarding the Metonic cycle is useful to other people too. It is apparent while reading online that many people take the description at face value, and have not noticed some of the issues with the system that are not properly articulated. I hope that by analysing and studying the Metonic cycle in detail, this site can contribute to other peoples work on the metonic cycle.

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