Periods
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Systematics is a new way of seeing. What is understood by such a seeing is a Whole system which can be divided into parts called terms. Terms are distinct whilst operating "on the same level". The limited number of terms are sufficient to describe an aspect of a given System for the purpose of understanding it, and interacting with it. First proposed by J. G. Bennett, Systematics is conservatively delineated at systematics.org. Bennett constrained himself to systems having one to twelve terms which, taken together, can describe many facets of system behaviour and these first twelve types of systematics relate most easily to common situations whilst being small enough to suit human minds.
I have found within Systematics diagrams a strange relationship to the notion of conservation (e.g. Kirchoff's laws, which apply to dynamic systems in general) when astronomical time periods are seen as relationships (connectives) between astronomical terms. This may mean that astronomical periods involving sun moon and planets, when seen from earth (i.e. synodically), form parts of a whole (viz.holistic) system, exactly as Bennett posited for multiterm systems. Astronomic time may be an opportunity to learn about the systematics of whole systems, whilst learning more about the time system surrounding the Earth. It is an interesting fact: that whole systems cannot be seen without a special effort. Generally conceived as purely qualitative, Systematics here appears quantitative with regard to astronomical time.
Tetrad of the 19 year Metonic Period
In February 2009 I found an excellent example involving the nineteen year Metonic Period.
PREREQUISITE 1
You may want to study what is meant by a 4fold tetradic system in systematics.
The four terms can be expressed as:
Figure 1 The four terms of the Metonic system
with which the text will find conservation of quantative time periods
in between, as the lines shown.
PREREQUISITE 2
It is also good to view the YouTube slide presentation which prefixes what is to follow with discussion
Figure 2 Quantifying relationships between terms around the "outside"  Figure 3 Seeing the upper triangle as 254 divided by19 = 13.368 orbits per solar year 
Figure 4 Seeing the lower triangle as 10,000 divided by 366 equal to 27.32166 days of the lunar orbit 
Figure 5 The emergence of the Solar Year as composite horizontal connective of 365.242 days, the sum of the values in the upper and lower triangles 
My notes of the time were:
 We notice four effective terms which combine to give 254/19 x 10,000 / 366 = 365.242
 These support the goal of achieving the 19 year recurrence of the same configuration of sun, moon and starts.
 The instrumentality used to achieve this is the moon's orbit
 The ground of this tetrad is the rotation of the earth relative to the sun
 Direction then appears as the (vague term) solar logos of the solar day and year
The ambiguity of the solar logos can be resolved by increasing the number of terms so the solar logos becomes the solar day and the solar year. A five fold systematics is called the Pentad.
PREREQUISITE 3
You may want to study the pentad at systematics.org
"An entity has meaning in its own right. This gives it an unique character and an inner and outer range of significance. Nothing less is sufficient for an independent structure. The inner range of significance includes the potentialities of the entity and partly stems from the nature or kind of entity and partly from its history. Any real thing is potentially more than it ever actually is. This is true of situations as well as of entities such as a man."
It is this "inner range of significance" (between higher and lower nature) which is opened up between the solar day and the solar year. The "entity in its own right" is the lunar orbit and the outer range is between the unit of earth rotation (Chronon) and the 19 year Metonic.
The two triangular cycles of figure 3 and 4 above have split so as to form 13.368 orbits per solar year and 27.321 solar days per orbit. The outer range on the right gives the numerocity required to achieve such a state of affairs by dividing by 19 and 254 and multiplying by 366 and 10,000. The rotation of the earth divides up the earth orbital period by 365.242 days. Those solar days then give 6939 days in a metonic period of nineteen years.
In some way the moon becomes the measure which "eats" the rotational energy of the earth through tidal interaction. Longer term cyclicities like the metonic (such as the synodic preiodicities of the outer planets in particular) provide an environment of what I call emergent cohesion in which many resonant intervals between synodic time periods appear to be governed by pure numerical litims such as 19 years. Whilst we see anniversaies called synods and might arrange them within a calendar, the realities of what we call discrete mathematics cause pure numbers to function as "attractors" .
In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system.^{[1]} System values that get close enough to the attractor values remain close even if slightly disturbed.
https://en.wikipedia.org/wiki/Attractor
In the pentad, the topmost term is the "god" which feeds on the "thing in itself", because the pentad is a good presentation of Gurdjieff's cosmic principle Trogoautoegocrat  the law of reciprocal maintenance. The relatively small integer number nineteen is to be found in 13.369 orbits being 13 plus 7/19th of an orbit. This guarantees that after 19 repetitions there must be a whole number of orbits. The "attraction" of low number repeat cycles is their stability and simplicity which tends towards conservation of energy, in this case the gravitational energy within dynamic systems. The elements within a dynamic system might naturally find such synodic anniversaries but one could not rule out that in organising the world of time,
“This system, which maintains everything arisen and existing, was actualized by our ENDLESS CREATOR in order that what is called the ‘exchange of substances’ or the ‘Reciprocalfeeding’ of everything that exists, might proceed in the Universe and thereby that the merciless ‘Heropass’ [time] might not have its maleficent effect on the Sun Absolute. Gurdjieff, G.I. Beelzebub's Tales. 7.1367 
The Moon presents itself as a highly variable cycle of behaviours yet it largely exhausts its main cycles under a recurring nineteen fold regime. The creation of diversity reminds us of the extreme diversity found on the earth of living beings that live "under the moon", the sublunary world of geocentric astronomy. One can see that the geocentric view is not essentially "wrong" but is an equally valid perspective that happens to carry simplifying numerical messages lost to a world of heliocentric physics.
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recovered from http://web.archive.org/
This document was prepared by Richard Heath as a letter for Nature magazine and submitted on 14th April 1994 but remained unpublished. For readers of the Matrix of Creation (2nd ed, Inner Traditions Press, 2004) it marks the discovery of a unit of time proposed and named the Chronon, as being 1/10000th of the Moon's orbit and also the difference between the sidereal and tropical day of the Earth. The paper also documents a discovery made, with Robin Heath, later to be documented in his books: that one can divide up the solar year by its excess over the eclipse year to reveal an 18.618:19.618 ratio between these years, and many other interesting numerical facts not mentioned in this place. The puzzle here is a connection between the rotation of the Earth, the solar year and the precession of the Moon's orbit which (a) may be explainable by science (b) appears to have puzzled Megalithic astronomers and (c) should puzzle us today.
by Richard Heath and Robin Heath
We find that the Earth’s rotational day divides the year according to the 18.62 year cycle of the Lunar Nodes. From this we conclude that the Earth’s orbit, the Moon’s orbital precession and the Earth’s rotational velocity are most probably interconnected. The tropical solar year in days is factorised almost exactly by 18.618 times 19.618 and the Moon travels one ten thousandth of its orbit in the time difference between sidereal and tropical days.
We have been considering a range of numerical coincidences present in arithmetical and geometrical analyses of astronomical cycles involving the SunMoonEarth system. There is an apparently lawful relationship concerning the Earth, Sun and Moon, one that is most unusual.
The Earth ‘s rate of rotation is directly proportional to the ratio of angular velocities of the Sun and the Moon ~ orbital nodes; as seen from Earth.
The reason why this fact has been hidden is that we use the day or the degree to manipulate the data concerning these phenomena and since the day is implicated in the above law it obscures the relationship and the degree changes the numbers to further obscure it.
We now refer to the angle traveled by the sun on the ecliptic in one day as a DAY. If we convert the Moon’s average daily motion of 13.176 degrees per day to DAYs per day we obtain 13.368 DAYS per day. This is the sidereal frequency per year because DAYS per day is also revolutions per year. This shows the virtue of using DAYS over degrees.
As there is not a great deal of familiarity with the terms used in describing Sun and Moon phenomena, we will recap some terms (see Figure 1).
 The Moon crosses the Sun‘s path or ecliptic at two places, the lunar nodes.
 Full or new moons occuring near a node produce solar or lunar eclipse respectively.
 Whilst the Sun moves East day by day by about one degree, it precesses the Lunar Nodes in the opposite sense, i.e. retrograde. The Sun moves about 18.618 times faster than the precessing nodes.
 Whilst the Sun returns to the same place on the ecliptic after one Solar Year, it will return to a given node after a shorter period an Eclipse Year. In Earth days, a Solar Year is 365.242 units long whilst an Eclipse year is 346.620 units long.
The ratio of 1:18.618 between the angular velocity of the Sun and the Nodes means that after 18.618 solar years, the Nodes will return to the same part of the sky, a period called the Draconic Period. This means that after 1 solar year, the nodes have traveled by 1 /18.618 of the ecliptic. During 18.618 solar years there are 18.618 + 1 eclipse years, the + 1 being due to the complete revolution of the nodes in that period. Because there are 19.618 eclipse years in a Draconic Period, then the nodes must move by 1 /19.618 of the ecliptic in an eclipse year, i.e. before the Sun again meets a given Node. When we draw the ecliptic geocentrically as a circle and place a SunNode conjunction at the "top", then the eclipse and solar year node movements can be shown as in Figure 2.
The question then arises: What is the proportion of the whole circle, shown δ, between the end of the eclipse year and the end of the solar year? It is,
1 /18.618  1 /19.618  which is  19.618 18.618 18.618 x 19.618 
or  1 /365.248  of a solar year! 
In other words, the Nodes move in the excess of the solar year over eclipse year by the equivalent of one Earth DAY on the ecliptic. There are 365.2421 tropical days in a solar year. Thus we can state,
The solar year, in days, has the two factors 19.618 and 18.618 and both these numbers are generated by the Moon 's nodal motion with respect to the Sun.
The Earth’s rotation is the lowest common denominator in the numerosity generated and thus calibrates the cycles involved.
Because of the 18.618 ratio between Sun and Nodal motion, there must be 18.618 days difference between the eclipse and solar years. What also has to be true is that the eclipse year is (1 8.618)2 days long or 346.63 days, which is close to the 346.62 days given [Astrophysical Quantities, C.W. Allen, 1973]. We have come to refer to 18.618 days as a NODE DAY, the time it takes the nodes to move by one DAY on the ecliptic and a fundamental constant in SunEarthMoon astronomy.
We therefore reassert that,
The Earth ‘s rate of rotation is directly proportional to the ratio of angular velocities of the Sun and the Moon ‘s orbital nodes; as seen from Earth.
because,
The Earth rotates once, with respect to the Sun, in the time it takes the Sun, seen from Earth, to move by the amount the Lunar Nodes move in the excess of the solar year over the ed ipse year.
We found another calibrating factor when we moved to DAYs instead of degrees. When we divided 13.176 degrees per day by 360 degrees, we obtained the fraction of a lunar orbit per tropical day. The value is 0.0366009. The same calculation for the sidereal day yields 0.0365009, and both happen to be the number of days in a year of the other type of day, in 1 /10000 ths of a complete lunar orbit. The sidereal day is 365 ths and the tropical day is 366 1 /10000ths of a lunar orbit: the difference between the two is a calibration unit of 1 /10000th.
Therefore, we found that,
The Moon moves 1/10000th of its orbit in the time between one sidereal day and one tropical day. There are 365 such periods in a sidereal day and 366 in a tropical day.
The present understanding of the EarthMoon system does not account for either of the above calibration effects.
© 1994 Richard D Heath, all rights reserved
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The Metonic Period is the 19 year anniversary of the Moon which dominates the repetition of cyclic astronomical aspects in the sky. We know that the megalithic identified this period and others of similar length (the Saros and Nodal periods), because interrelated units of measure, especially the megalithic yard and royal cubit, can be found within monuments which recorded these periods as counted lengths.
How the Metonic Works
There is only numerical way to arrive at a system in which there is an anniversary between the sun, the moon and the stars over 19 years (as in the Metonic period), involving having a lunar orbital period (around the earth) which divides into the same period, in this case exactly 254 times.

 254 lunar orbits of (on average) 27.32166 days equals 6939.7 days
 19 solar years of 365.2422 days equals 6939.6 days and
 235 lunar months of 29.53059 days equals 6939.7 days
If the moon returns to the position of the sun after 19 years then the phase of the moon will be the same so that there will be 254 minus 19 equal to 235 lunar months in the Metonic observatory and it is in the nature of orbits which become commensurate with each other (as discrete, wholenumber gravitational systems), to fall into such resonant relationships rather than chaotic ones.
Signature Ratios within the Metonic
Since the solar year, lunar orbit and month are synchronous, the excess of the solar over lunar year will form a near rational (i.e. integer) fraction of months or orbits, in that case an excess of 7/19 (0.368) lunar months.

 The lunar orbit is 27.32166 days
 The lunar month is 29.53059 days
Similarly, the excess of the lunar month over the lunar orbit become the rational and fractional ratio found between their frequencies within the 19 year Saros cycle, that is 254/235; which has the fractional part, reciprocated of 1/0.08085 = 12.368 which is then N for the N:N+1 triangle describing the orbit relative to the month, of the moon.
However, the lunar orbit is also found to be 10,000 time periods long, the unit of time being the excess of the solar over the sidereal day***, which in Matrix of Creation I called the chronon after the god of time.

 *** The sidereal day is relative to the stars and it is the time for the earth to rotate once whilst the solar day is that time plus the time taken for a location upon the earth to catch up with the sun's motion within the solar year, this due to the earth's orbit around the sun.
Once the earth's rotation and lunar orbit came to be set to this chronon ratio of one solar excess to ten thousand solar excesses per orbit, the orbit of the moon can be visualised as containing 10,000 chrononinches as a length (the circumference of the Le Menec western cromlech near Carnac town in Brittany) and this actual length of 10,000 inches can be divided by the number of orbits in the Metonic so as to create the 39.37 inch metre standard (its legal ratio in the U.S.A [Berriman 1953, p21]) which is 10,000/254 inches.

 We will later look at how three quarters of this length of 39.37 inches equals 29.528 inches, one part in 9742 of the lunar month's average duration of 29.53059 days, a relationship seen very clearly at Le Manio's Quadrilateral [HEATH & HEATH, 2010 & HEATH 2014] marking a count of three solar years and three lunar years, and their difference of one megalithic yard of 261/8 dayinches.
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BUT HERE IS THE ACCOMPANYING DIAGRAM TO PUZZLE OVER.
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