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'.