Introduction

An interface can be documented to exist between time domains of the lunar orbit (synodic) and the solar orbit (tropical).

The indicated interface is remarkable in the regard that the spin of the Earth and both orbital periods (Moon and Sun) can be recited to work together to divide the timesteam into a functional arrangement.

The lunisolar system that is subsequently presented is based upon the definition of the solar-day at the epic of the 20th century--when the day was defined to be 86400 seconds in length. This means that the modern interface would probably have been even more accurate (by a tiny degree) toward the time range of first recorded history.

Note that modern astronomers have determined that the spin of the Earth has slowed down at the rate of about 0.0017 seconds (throughout the 20th century). The indicated slow down points to the possibility that Earth's spin rate may have been even fractionally shorter (or faster) in millennia of the recent past.

A lunar-month model

The phenomenon of the spin and orbital returns points to a time interface wherein the spin of the Earth (the solar-day rate) is synchronized with both the synodic Moon and the tropical Sun. Furthermore, a system that is fully functional for tracking time can be recognized from out of this three-way interface.

Of special significance here is that the interpretation of an interrelated lunisolar system is possible within the context of a common unit of time (7 days).

A lunisolar system that is predicated upon tracking solar days in 7-day units (or in week units) can at first be illustrated from the completion rate of the lunar month.

Because the lunar period is inherently equal to the span of time occupied by 29.53 days (average) then each quarter division of the lunar month (the 4 phases) can conveniently be defined/delimited by tracking 7 solar days across each of the quarter phases.

The following diagram is presented to more clearly show that each synodic revolution of the Moon can be cross-referenced to a day-unit model wherein each of the quarters of the lunar month are represented by a lunar-week unit of 7 days:

This formal representation of the lunar month (as diagrammed) is predicated upon a distribution of weeks (4 weeks per month). However, because the lunar lunar period of 29.53059 days is 1.53059 days longer than 4 week units of 28 days then a specific rate of non-week days must also be taken into account.

For the purposes of more clearly demonstrating a plausible weeks model of the Earth-Moon-Sun system, the cited non-week-day rate will hereafter be referred to as the LR rate of days.

A weeks model of the lunar month (4 weeks, or 28 days per month) can then be understood within the context of a secondary LR rate of non-week days. The indicated non-week-day rate (LR) is required to closely be equal to 1.53059 days per lunar period (on the average). This non-week-day rate is also required to closely be equal to 18.93074 days per year (on the average).

In summary, a day model for representing the lunar month can be interpreted within the context of a weeks count (4 weeks per lunar month). However, correlating 4 week units with each synodic revolution does require the addition of a secondary LR rate of days (1.53059 days per lunar month). In essence, a 28-day model for the synodic period of the Moon (29.53059 days) mandates the additional inclusion of LR days.

As is further shown below, the cited LR rate (or 1.53059 days per lunar period) can be interpreted to have a very specific function or a definite purpose within an interrelated lunisolar system. In fact, the required secondary LR time rate can be understood as an effective meter of the return rate of the tropical year. This means that--in a weeks model of the spin and orbital periods--the peculiar LR rate has a recognizable double function of pertaining to an annual time clock.

A solar-month model

A weeks interpretation of the spin-orbital phenomenon also sees the turn of each tropical or solar year (365.24219 days) in the context of a time grid of 7 days. Of significance here is that each annual quarter (91.31055 days) and each one of 12 month divisions (30.43685 days per annual division) can likewise be defined/delimited (on the average) by simply counting week units.

The following diagram illustrates the feasibility of measuring and metering out the span of the tropical year by time tracking a formal arrangement of week units:

A grid of 48 weeks (a calendar of weeks as diagrammed) can pace the return rate of each passing tropical year as long as the count of an additional week (a secondary SR rate) is included every 110 days.

For the purposes of clearly presenting a calendar model out of the orbital returns, the cited secondary weeks rate will hereafter be referred to as the SR rate of weeks.

As is further shown below, the required SR rate is also equivalent to the following three rates:

1. 7 days every 7th set of 7 lunar weeks.
2. 7 days every 7th month.
3. 7 days every 7th season.

Note that the rate of 7 days in 110 days is equal to the rate of 23.24268 days per year while the combined rate of a week in pace with the 7th lunar week, the 7th month, and the 7th season is equal to the rate of 23.24232 days per year.

This all means the shown template of 48 weeks can be fitted right on top of the span of time occupied by each tropical year. In essence, the weeks grid (as diagrammed) can just about exactly be correlated to each annual return (on the average) in the context of additionally counting a secondary SR rate of weeks:

Then, to be completely specific about the feasibility of counting annual and SR weeks, the modern tropical year can be recognized to inherently revolve in pace with a time span equal to 365.24219 days while the shown calendar model of weeks renews (on the average) right in association with a parallel span of time (365.24232 days per year). Each passing tropical year is thus so exactly synchronized with the completion rate of cited weeks calendar (on the average) that the skip or insertion of an additional calendar day (or days) is not ever warranted.

The rate of the modern tropical year does however clock at a tiny difference away from the average return of the cited weeks calendar. It is here significant that a difference of less than 1 second per month can be recited. However, it is also significant that the spin rate of the Earth appears to be slowing down by a fractional amount in correspondence with each passing century. The slowing spin factor thus indicates that the return of the tropical year would recently have been EXACTLY synchronized with a count of 48 annual weeks and SR weeks (as diagrammed and documented).

The Earth inherently spins 365 times per year, and the spin rate throughout recent millennia has slowed down at a rate of between 0.001 and 0.003 seconds per century. A given conclusion from these respective rates then follows: 1. A loss in the annual definition of at least 0.365 spin-seconds has been experienced (a per-century rate); and 2. The rate of 1 second of modern difference per month (12 seconds per year) when divided by a gain of 0.365 spin-seconds per century, points to a time of no (zero) difference with a calendar of weeks at only 33 centuries ago.

Significance of SR weeks

In the context of a lunisolar system modelled to account for both lunar weeks and annual weeks, the inherent definition of an SR rate of weeks can be recited to have a considerable amount of significance.

This rate can be stated to be gear driven by both of the apparent orbits; Sun and Moon; as is further shown below.

In addition to representing an inherent interface between the Sun and Moon, the stated SR rate of weeks can be interpreted to have the functional purpose of defining/delimiting specific month cycles, pentecontad cycles, and seasonal cycles.

To be more specific, it was shown in the previously presented section that a rate of SR weeks is necessary to pace a template of weeks with each annual return. However, this rate additionally points to a time cycle (or cycles) that clock at a different rate (or rates) than does the cycle of the year. Thus, in a weeks model of the orbital returns, a rate of SR weeks can be interpreted as a meter that pertains to defining a time cycle (or cycles) other than the year cycle.

Of significance here is that the required SR week can be intercalated by cycle rates that well suit an interrelated time model of the Moon and Sun. For example, a satisfactory (composite) rate for the addition of the required SR week can be derived in the context of counting the following time cycles:

1. 7 days at every 7th month of 30 days.
2. 7 days at every 7th pentecondad (a pentecontad is 7 lunar quarters).
3. 7 days at every 7th annual quarter (the 7th season).

Note that a composite annual SR rate of 23.24232 days can be achieved by accounting for 7 days in the context of 7 months, 7 pentecontads, and 7 annual quarters--where 12.17473 days per year and 7.06758 days per year and 4.00000 days per year are equal to 23.24232 days per year (rounded from expanded precision). Note that because the cited template of 48 annual weeks (and 6 lunar days) does inherently occupy 342 days on a per annum basis, and because the tropical year revolves every 365.24219 days, the required additional rate of SR weeks must come close to a rate that is equivalent to the difference (23.24219 days per year).

The indicated required addition of 7 days at every 7th month, every 7th pentecontad, and every 7th season then does quite perfectly correspond with a rate that is necessary in defining an annual set of weeks and days.

Significance of LR days

As was shown in the introductory sections, a template of 7-day units can be used to formally define/delimit each quarter division (or quarter phase) of the Moon. To here be more specific, the time span occupied by each lunar quarter is inherently equal to 7.38265 days (as an average unit of time).

Note: There are 4 distinct quarter phases of the Moon: 1.New phase; 2. First-quarter phase; 3. Full phase; and 4. Third-quarter phase. The quarter phases are easy to recognize on the basis of observation. At the new phase the Moon is dark and appears to be completely invisible; at full phase, the Moon is fully-illuminated and is round-shaped; and at the first quarter and at the third quarter, the Moon is half illuminated and is distinctly divided into half-parts (half-light and half-dark, or the reverse).

It is here significant that each quarter divison of the synodic revolution of the Moon can be correlated to a template of 7-day units--and to a rate of LR days--as follows:

Note that the turn of each quarter phase of the Moon can effectively be counted within the context of a lunar-week unit as long as a specific rate of LR days is counted in addition.

Of additional significance then is that in order to keep a template of 4 weeks (as diagrammed) in pace with the phases of the Moon an LR day (a non-week day) must routinely be added.

It is obvious that the required LR day must be added amid the count of lunar weeks at a rate that is closely equivalent to 0.38265 days per lunar quarter. (The required LR rate is also equal to 1.53059 days per lunar month).

In a weeks model that paces each quarter phase of the Moon, the requirement to intercalate by a rate that closely equals 1.53059 days per month (the LR rate) can be satisfied by adding a non-week day in correspondence with the following epochs in time:

1. The Sun's 30th day (a running solar-month cycle).
2. The Moon's 30th day (an alternate lunar-month cycle).
3. The 7th annual quarter (the 7th season).

A count of weeks can thus be adjusted into pace with each synodic revolution of the Moon (on the average) by adding a non-week day at each 30th day (both Sun and Moon), and by adding an additional non-week day at each 7th annual quarter.

By intercalating each of the 30th days as non-week days, and by intercalating an additional non-week day at the 7th seasons, a composite rate of 1.53055 days per lunar period can be achieved for the required LR rate. [Note that 0.984353 days per lunar period and 0.500000 days per lunar period and 0.046201 days per lunar period is equal to 1.53055 days per lunar period.] This composite rate then comes very close to bringing the cited weeks template into exact synchronization with each of the lunar quarters. A difference of only 0.9 seconds per lunar week is the inherent result.

Other plausible interpretations that can even more perfectly account for the required rate of lunar weeks are possible. For more information about tracking the synodic revolution of the Moon, refer to the content of subsequently presented sections. Refer also to the online publication entitled: 'Time Portals or Annual Gates'.

A lunisolar system

The currently presented Earth-Moon-Sun model is of special interest in regard of the doubled definitions of weeks, months, and seasons. However, in a time model that accounts for both lunar and solar weeks, not only are months and seasons doubly defined but even the years, and long-cycles of years are twofold defined.

The indicated double definition of 30th days makes a time track of the orbital returns almost inherent or automatic. It is here signficant that a simple method of counting or scribing day cycles is all that is required to effectively meter cycles of the Sun and Moon (weeks, months, seasons, and years). A more refined time tracking method might account for the spin and orbital phenomenon at the resolution of the half-day unit (or the division between night and day).

Then, to be more specific about the currently presented weeks model, the time span defined by the revolution of each tropical year (365.24219 days) can perfectly be measured and metered out in association with a fixed template of 48 weeks. This respective calendar grid requires a rate of annual weeks and non-annual weeks--as follows:

The following list of attendant rules or guides are presented to more explicitly show the feasibility of defining dual templates through which the average limits of lunar and annual quarters can ultimately be determined:

Interface of 4 weeks per lunar month

1. The 30th day of the Sun (an endless time cycle) must be leaped (or not counted) as a lunar-week day.
2. The 30th day of the Moon (an endless time cycle) must be leaped (or not counted) as a lunar-week day. Note the current interpretation sees every alternate period of the Moon to contain a 30th day. (The time division between 30th days is inherently equivalent to 2 lunar months and will hereafter be referred to as a lunar portal).
3. At every 7th season, an additional day must be leaped (or not counted) as a lunar-week day.

Note that in the context of intercalating every 30th day of the Sun, every 30th day of the Moon, and one more day each 7th season, a time template of 4 weeks can be recognized to almost exactly keep pace with the return rate of each passing lunar period.

Interface of 48 weeks per solar year

1. A day corresponding to the cited lunar portals must be leaped (or not counted) from amid the annual count of days (6 instances per year).
2. At every 7th solar month of 30 days, an entire week (7 days) must be leaped (or not counted) from the routine count of annual weeks.
3. At every 7th pentecontad, a week (7 days) must be leaped (or not counted) from the count of annual weeks.
4. At every 7th season, a week (7 days) must be leaped (or not counted) from the count of annual weeks.

Note that in the context of intercalating a day at lunar portals (6 times per year); and in the context of additionally intercalating 7 days every 7 months, 7 days every 7th set of 7 lunar weeks, and 7 days every 7 seasons; a count of 48 weeks (336 days) can be recognized to almost perfectly pace the return rate of each passing tropical year.

The interpretation of a lunisolar system then seems completely satisfactory within the context of a weeks interface (both with the cycle of the Moon and the circle of the Sun).

A running count of weeks

A plausible alternate interpretation to that shown above likewise sees a fixed template of always 48 weeks astride the tropical year. However, the 7-day unit according to this calendar model is understood as a running (unbroken) cycle). The feasibility of a continuous run of 7 days in a solar calendar is illustrated in the diagram shown below:

Note that this respective model (as diagrammed) does not require the addition of 6 lunar days per year. Instead, 7 days are required to routinely be added at each 7th solar portal (where a solar portal is defined as the span of time straddled by 2 zodiac divisions).

The precision of a lunisolar system that accounts for 7-day cycles in the additional context of solar-portal divisions is identically the same (in average time) as was shown for the previously presented model. (Note that the sum of 336.00000 days and 12.17474 days and 7.06758 days and 6.00000 days and 4.00000 days is equal to 365.24232 days per year).

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           A SEASONAL TRACK OF 7TH YEARS

          -------------------------------

           1. QW at 7 seasons
           2. QW at 7 seasons (3.5 years)
           3. QW at 7 seasons
           4. QW at 7 seasons (7.0 years)

          -------------------------------

             Total Time = 28 seasons

              THE INTERFACE OF         Number of
             49 SYNODIC MONTHS *        Earth's 
                                       Rotations
         __________________________    _________
 
          1   2   3   4   5   6   7      206.71
          8   9  10  11  12  13  14      413.43
         15  16  17  18  19  20  21      620.14
         22  23  24  25  26  27  28      826.86
         29  30  31  32  33  34  35     1033.57
         36  37  38  39  40  41  42     1240.28
         43  44  45  46  47  48  49     1447.00
         __________________________    _________

        * - Earth's rotation aligns
            with 49 lunar months.

          ---------------------------------------

                    A LUNISOLAR SYSTEM *
             (Template for each Tropical Year)

               ---------  --------------------
                 Annual          Annual
               Divisions         Weeks
               ---------  --------------------

                   1       1  2  3  4  5  6  7
                   2       8  9 10 11 12 13 14
                   3      15 16 17 18 19 20 21
                   4      22 23 24 25 26 27 28
                   5      29 30 31 32 33 34 35
                   6      36 37 38 39 40 41 42
                   7      43 44 45 46 47 48 49

               ---------  --------------------

                                 343 days

          ---------------------------------------

        * - Template requires an additional week:

             1. At 7th months (of 30 days).
             2. At 7th sets of 7 lunar weeks (LW).
             3. At 28th zodiac division.