

The
Metonic Cycle  Dozens and Thirteens

AN
EASY QUESTION FOR ASTRONOMERS?
Here's
a very interesting question which many astronomers would initially
imagine is easy to answer:
"If
you saw the Full Moon above Orion tonight, when could you expect
to see another Full Moon in EXACTLY the same position among the
stars again??"
If
that's got you thinking, don't be surprised. The answer will not
roll of your tongue, even if you are an astronomer! The first time
I was asked that question, I was at a loss. You see, the Moon's
movements through the sky are not straightforward, like those of
its companion, the Sun. With the Sun, we know it takes 365 and a
quarter days to make a full journey through the Zodiac and return
to the same position in the sky again. It follows the same imaginary
line (Ecliptic) every time it does this journey. It's regular and
easy to follow.
The
picture on right shows a Full Moon above Orion on a sample date,
December 18, 2021, at 23:57. When will the Full Moon return to exactly
that position between Taurus' horns? 

TRYING TO WORK IT OUT
The Moon makes its full journey through the sky in, roughly speaking,
27 days (the exact decimal figure is 27.322 days). That's called
the Siderial Lunar Month, or the Tropical Lunar Month. Simple enough,
one would think. So the Moon returns to the same background stars
every 27.3 days. There's the answer to the question. But wait 
the question is "If you saw the Full Moon
above Orion tonight, when could you expect to see another Full Moon
in EXACTLY the same position among the stars again??"
And
herein lies the problem. While the Moon takes 27.3 days (Tropical
Month) to return to the same background stars, it does not return
to the same phase until two days after. In other words, the time
it takes the Moon to return from one Full phase
to the next is, roughly speaking, 29 and a half days. (Actual period
expressed as a decimal is 29.5306 days). That period is called a
Synodic Lunar Month.
So,
we have a Full Moon above Orion's outstretched hand. We know the
Moon will come back to this position in 27.3 days (Tropical Month),
but it won't be full until 29.5 days (Synodic Month). So when can
we expect to see it Full again, and back in the same position between
the tips of Taurus' horns? We will need to count the number of same
phases (Synodic Months) and the number of returns Orion's hand (Tropical
Months) to find out.

THE
LUNAR YEAR
12 synodic months, or 12 returns of the Moon to the same phase,
forms the period of time known as a lunar year. If you start from
the Full Moon closest to the time of Winter Solstice (Dec. 21st),
and count how many times the Moon returns to this position and how
many times it returns to the same phase, you will find that in the
time it has returned to Full Moon 12 times, it will have passed
Orion's hand 13 times. This is one lunar year. 12 same shapes, 13
returns to the same stars. This lunar year is exactly 354.372 days
long, which is a whole 11 days shorter than a Solar tropical year. 
So,
could that be the answer? Is 12 returns to the same shape, one Lunar
Year, the time it takes the Moon to return to exactly the same background
stars?
Let's try it out. Using an astronomy program such as SkyMap
Pro, go forward 354 days from our sample date, Dec. 18 2021,
and we get December 7, 2022. This is the first time since our start
date that the Full Moon is visible in the horns of Taurus, so it's
pretty close. But it's not bang on. Remember, we're looking for
the Full Moon in exactly the same position.
If
we wait another lunar year, another 12 returns to the same phase,
13 returns to the same shape, we see the Full Moon in this region
of the sky again, but its position is under the Pleiades, a bit
west from the original position. So we have seen 24 Full Moons and
26 returns to this part of the sky. This is two pure lunar years.
To get a more accurate return of the Full Moon to the horns of Taurus,
wait another Synodic Month. This takes us to December 26, five days
after Winter Solstice, the year 2023. We have seen 25 Full Moons
and 27 returns to Orion's hand. 

In
order to keep the lunar periods attached to the solar year (remember
we are watching for the Full Moon closest to Winter Solstice),
we have added 1 "pure" lunar year containing 12 synodic
months, 13 tropical months, with a period we will call the lunar
"leap" year  13 synodic months, 14 tropical months.
This is a very valuable 'first lesson' in learning the Metonic
cycle  "Dozens and Thirteens". We can express these
periods in an easytoremember fashion as follows:
12,I 
12
synodic months ending 11 days before 1
tropical year 
II,25 
25
synodic months ending 8 days after 2
tropical years 

In
this notation, developed by Charlie Scribner, the 12 comes before
the I because the 12 synodic months ends 11 days BEFORE 1 year.
In the second period, the 25 follows the II because the 25 sm ends
8 days AFTER 2 years.
We
use the period counts of same Moon phases and returns to the same
stars, called the Synodic month and Tropical month, to warn us when
to pay close attention to what the Sun is doing and to better manage
time. If we continue our series, we add another "pure lunar
year". This time, we will see the Full Moon for the 37th time,
and we've seen it pass Orion's hand 40 times. It's now December
14, 2024 and the Full Moon is this time located just above the upper
horn of Taurus. This gives us the third Metonic interval:
37,III 
37
synodic months ending 3 days before 3
tropical years 

The
numbers of tropical years in our evolving series have an interesting
quality. They are equal to the numbers of tropical months (in the
latest instance 40) minus the numbers of synodic months (37). So
3ty = 40tm
 37sm.
Remember
the formula: TY
= TM  SM 
The
number of tropical years equals the number of tropical months
minus the number of synodic months. 
When the synodic month and tropical month come back into phase with
one another, when same shapes return to the same stars, the synodic
and tropical months also come back into phase with the sun and his
seasons. The numbers of tropical years are equal to the numbers
of tropical months plus the number of synodic months. The three
different periods form what we now call an harmonic. Adding another
pure lunar year takes us to:
49,IV,53 
49
synodic months ending 14 days before 4
tropical years (53 tropical lunar months) 

If
we add a second lunar leap year to the series, we arrive at another
Metonic interval:
V,62,67 
62
synodic months ending 5 days after 5
tropical years (67 tropical lunar months) 

The
first lesson, "Dozens and Thirteens", continues:
74,VI,80 
74
synodic months ending 6 days before 6
tropical years (80 tropical lunar months) 
VIII,87,94 
87
synodic months ending 13 days after 7
tropical years (94 tropical lunar months) 
VIII,99,107 
99
synodic months ending 2 days after 8
tropical years (107 tropical lunar months) 



The
Full Moon back in Orion's hand after Metonic Interval V,62
 five days after 5 tropical years, date: 23 Dec. 2026. 
The
Full Moon in Orion's hand again, this time at Metonic Interval
VIII,99  2 days after 8 tropical years, date: 20 Dec. 2029. 

This
latest Metonic interval, VIII,99, brings the Full Moon
in Orion's hand to within just two days of the date of the same
Moon we saw eight years back. The original observation was made
on December 18th (2021), with the current observation on December
20th (2029) and since our very first Full Moon eight years ago we
have seen 99 Full Moons, and a whopping 107 returns of the Moon
to Orion's hand. That's a lot of moon watching!
Here's
an interesting fact: This VIII,99 Metonic subunit which brings the
same phase of the Moon back to the same part of the sky two days
after eight solar years, is actually made up of two of the smaller
intervals. You can add them up yourself to see how it works:
37,III,40 
37
synodic months ending 3 days before 3
tropical years 
V,62,67 
62
synodic months ending 5 days after 5 tropical
years 
VIII,99,107 
99
synodic months ending 2 days after 8 tropical
years 

37,III,40
+ V,62,67 = VIII,99,107 

Adding
37,III, the 3 days before, to VIII,99,
the 2 days after, finds the even stronger tie:
136,XI,147 
136
synodic months ending about a day before 11
tropical years 
Add
VIII,99 to 136,XI and find the answer to the question!:
XIX,235,254 
235
synodic months ending at the same time as 19
tropical years or 254 tropical
lunar months. 
This
is the Metonic Cycle, and it brings the Full Moon back to where
we first observed it, between the horns of Taurus all those long
19 years ago.


The
Full Moon back above Orion at the 11year Metonic interval
of 136 synodic months. This interval brings the same phase
to within one day of its original date. 
Finally,
the Full Moon returns to the exact position where we saw it
19 years before. Compare this with the very first image at
the top of the page. 

It's
most incredible. If you see the Moon tonight, watch closely its
position and phase, because you won't see it returning to that exact
position and phase for another 19 years, or 235 synodic months,
254 tropical lunar months. You might not even be alive the next
time it happens. Try it with a computer program like SkyMap. Just
pick a date and look at the phase and position of the Moon and add
19 years. Here's how it works out in terms of actual days:
365.24
days (solar tropical year) x 19 =
6939.56 days 
29.5306
days (lunar synodic month) x 235 =
6939.691 days 
27.322
days (lunar tropical month) x 254=
6939.788 days 

But
remember, you DO NOT have to know the day counts in order to see
the Metonic Cycle in action. It's the whole period counts which
give us the intervals. We don't think of 12,I 11 as being 354 days.
We think of it as being 12 returns of the Moon to the same shape,
in this case, Full Moon ,
and that this is 11 days before the Winter Solstice sunrise. Try
it with another example, this time the Full Moon on Spring Equinox,
2000, March 20, the old pagan Easter, with the Moon under Denebola,
the tail of Leo the Lion, in the stars of Virgo.


Full
Moon on March 20, 2000, the Spring Equinox, under Denebola
in the stars of Virgo. 
Full
Moon 19 years later, March 20, 2019, in exactly the same location
under Leo's tail. 

You
can try some of the other intervals too, and watch how the Full
Moon returns to this part of the sky. But remember, think of whole
period counts instead of big numbers of days. Add 12 synodic months,
one solar year minus 11 days.
If
you can look out a window and see a moon among the stars right now,
you will see this Moon return to the same shape and passing the
same stars in 19 tropical years, 235 synodic months, 254 tropical
months. We have seen how this can be uncovered visually, without
the need for complex mathematics and astronomical instrumentation,
and also how we do not need to know the actual day counts because
we can record the cycle with period counts  synodic months, tropical
months and tropical years. We don't even need to know about fractions.
This "Metonic Cycle" is named after a Greek, called Meton,
who lived in the 5th Century BC, and who claimed he discovered the
cycle on his own. It seems that simple visual observations are all
that's needed to see the cycle . . . and there's plenty of evidence
it was known and recorded long before Meton ever existed.
If
you have SkyMap, you can download the starter maps which I have
used for the examples on this page and open them with SkyMap (after
you save the file, go to SkyMap, click on 'File' > 'Open' and
locate the file). Here are the files:

PAGES
OF INTEREST:
Calendarstone:
See how the Metonic interval V,62,67 was recorded on a 5,000yearold
stone at Knowth.
Lunar stone:
More lunar calculations at Knowth. 

