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HORUS VOL III. Issue 1

Ring Counters and Calendrical Cycles
William J. Douglas

The Greeks, Romans, and Chinese. used an abacus in the form of sliding beads on wires which were based on or compatible with the decimal or duodecimal (base 12) system. The earliest evidence we have for the abacus is from the 3rd or 4th Century B.C. It may have originated as a means of keeping a record of calculations done by finger reckoning. The decimal and biquinary (two fives) are directly related, and by using the thumb as a pointer, one can count to 12 on the joints of the fingers of one hand. Another type of the abacus in its various forms was especially well suited for business transactions.

What kind of calculator did early man use in tracking the movements of the Sun, Moon, and the planets? When tracking the heavenly bodies, which return to a given starting point after a number of years, months, and days, a different kind of calculator was needed.

Ring counters long have been used to keep track of repeating cycles, such as those that occur in astronomy and calendrics. A prime example is the Stonehenge, where marker stones in the Aubrey, X, and Y holes could be advanced each day to keep track of the days of the year and the month. [See "Setting and Using the Stonehenge Nineteen Year Sun-Moon Calendar" by Alban Wall in HORUS II:3.]

Much more ancient is the Mallia disk which was found in the Minoan remains on Crete. It has been dated as belonging to the Middle Minoan period, which corresponds to the Egyptian Middle Kingdom.

[*!* Image: Diagram of Mallia Disk]

[*!* Image: The Mallia Disk. [From a photo by J. Carlson]]

[*!* Image: Table 1. The Ruling Planets of the Hours of the Days. The planets are listed in the order of their distance from the Earth, Saturn being the furthest, and the Moon being the closest. The hours of the day ruled by each planet change every day for seven days until the cycle is complete. LABELS: The Hours of the Days. The Planets. SATURN. JUPITER. MARS. SUN. VENUS. MERCURY. MOON.]

Made of limestone and set in a stone pavement, the disk had 34 cups around its perimeter, and a large bowl in its center. One of the cups was larger than the others. Charles Herberger [Archaeoastronomy VI, p. 116] suggests that the smaller cups were used to coordinate the lunar and solar calendars. By moving a marker from one cup to the next each New Moon, skipping the large cup, the 99 lunations in eight solar years were divided into three equal parts. Each time the large cup was passed over, an extra month was inserted in the lunar calendar to catch up with the solar calendar.

Ring counters have also been used in modem technology in the first digital computers using vacuum tubes, in order to simplify design and reduce power consumption. when a number is the product of two smaller numbers without a common factor a double ring counter was especially efficient. For example, the number 380 = 19 X 20 could be counted with a double ring counter, since 19 can not be divided by four or five.

Calendars often contain two or more cycles of different length which are counted independently and result in a larger cycle when they return to their starting position. For example, if the year consisted of twelve 30-day months (360 days) Friday would fall on the 13th day of the month every 210 = 30 X 7 = 210 days. The interaction of these multiple cycles is demonstrated below by ring counters, rather than the usual "cogwheels". Three examples are given: one from our own calendar, one from China and one from Mesoamerica.

The planetary week

A less obvious example from our calendar is the 24-hour day and the seven-day week. The seven 11 planets", which included the Sun and the Moon are listed in order according to their supposed distance in the first column of Table 1, along with the hours ruled by each planet.

Looking down the first column of hours, we see that on the first day Saturn is the ruler (astrologically speaking) of the first hour, Jupiter the second, Mars the third, etc., the Moon being the ruler of the seventh hour. In the second column, Saturn is the ruler of the eight hour, Jupiter the ninth, etc., the Moon being the ruler of the 14th hour. Continuing in the third column, Saturn is the ruler of the 15th hour, ending with the Moon for the 21st hour. In the fourth column, Saturn rules the 22nd hour, Jupiter the 23rd, and Mars the 24th hour, completing the first day.

The second day proceeds in a similar fashion, continuing in the fourth column with the Sun (the next planet) as the ruler of the first hour, the 24th hour (in the seventh column) being ruled by Mercury.

Continuing as above, the first hour of the third day is ruled by the Moon, the first of the fourth by Mars, the first of the fifth by Mercury, of the sixth by Jupiter, the first of the seventh being ruled by Venus. The table is completed with the Moon as the ruler of the 24th hour of the seventh day.

Note that the order of the planets by distance (Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon) was replaced with a new order obtained by skipping two each time: Saturn, Sun, Moon, Mars, Mercury, Jupiter, Venus, which is the order of our days of the week, Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday, Friday.

It is apparent by now that the days of the week are named after the planetary gods. The English (Anglo-Saxon or Scandanavian) names for Tuesday through Friday were Tiw's day, Woden's day, Thor's day, and Frig's day, which were respectively Mars, Mercury, Jupiter, and Venus. Our days of the week are named according to the planetary rulers; it is known as the planetary week.

[*!* Image: Figure 1. The Planetary Week]

The planetary week can be used to illustrate the principle of a double ring counter (Fig. 1). Seven dots representing the planets are arranged in a circle in order according to their distances in the same sequence as in Table 1. It is surrounded by a circle of 24 dots for the hours of the day, with hour number 1 aligned with Saturn. Markers are placed at the starting position on hour-1 and Saturn. Both markers are advanced clockwise in step. When the planet marker gets back to Saturn, the hour marker will be on hour-8, indicating that Saturn is the ruler of that hour of the day as was seen in the table above. This procedure is repeated until the markers return to their starting position.

The first time the hour marker returns to hour-1, the planet marker is at the Sun, the second time at the Moon, the third at Mars, then Mercury, Jupiter, Venus, and Saturn. Note that all the 7 X 24 possible combinations of hour numbers and planet names listed in the table are recapitulated in the ring counter in the same sequence; they are mathematically equivalent. If we connect these planet positions in sequence, as shown in Figure 2, we have a diagram which illustrates the derivation of the order of the days of the week from the order of the planets. This diagram is a heptagram (hepta = seven), a symbol which was known to the

Babylonians, where it is said to represent the 'seven regions' or heptamychos of the philosopher Pherecydes of Syros.

[*!* Image: Figure 2. The Heptagram]

The Chia-Tzu

The Chinese calendar also has a double cycle, which consists of the Ten Celestial Stems, and the Twelve Terrestrial Branches (Fig. 3). Starting with the Celestial marker at Tzu, and the Terrestrial marker at Chia, both markers are advanced for each count. The first pair is Chia Tzu, the second is 1 Chou, etc. Dividing by the common factor of two, we see that the number of unique pairs is 10 X 12 / 2 = 60, the last pair being Kuei Hai. Then the cycle repeats itself. The factor of 10 is convenient, since the decimal system seems so natural. The factor of 12 is also convenient, since it is much easier to divide a circle into 12 equal parts than into 10.

This cycle is the Kan-Chih, also called the Chia-Tzu, from the first pair of terms. (Our alphabet is named in the same fashion, from the Greek letter names, alpha, beta.) It was used as early as the late Shang dynasty (about 1500 BC) to count days. Later it was used for a year count, which was especially convenient in connection with measuring long periods of time by the conjunctions of Jupiter and Saturn, which occur every 20 years (in round numbers) and make a complete circuit of the heavens every 60 years. And Saturn itself completes two revolutions about the Sun in 60 years.

[*!* Image: Figure 3. The Chia-Tzu Ring Counter]

[*!* Image: Figure 4. The Tzolkin Ring Counter]

The 260-day cycle

At the time of the Spanish conquest the 260-day cycle was found throughout Mesoamerica, a large region extending from central Mexico southward into parts of El Salvador and Honduras, and to the northeast throughout the Yucatan peninsula. It is still in use today in parts of highland Guatemala.

The 260-day cycle consists of thirteen 20day periods. It is illustrated by the ring counter in Figure 4, in which both markers are advanced for each count, as in the planetary week counter.

Starting with markers at #1 and the Imix glyph, the first time the inner marker returns to #1, the glyph marker is at Ix, the second time it is at Manik, the third at Ahau, etc. A new cycle is started when the markers return to the starting position at 1 Imix. The number of unique pairs is 13 X 20 = 260.

[*!* Image: Table 2. The 260-Day Cycle]
Day Names The Tonalpohualli and Tzolkim
IMIX 1 8 2 9 3 10 4 11 5 12 6 13 7
IK 2 9 3 10 4 11 5 12 6 13 7 1 8
AKBAL 3 10 4 11 5 12 6 13 7 1 8 2 9
KAN 4 11 5 12 6 13 7 1 8 2 9 3 10
CHICCHAN   5 12 6 13 7 1 8 2 9 3 10 4 11
KIMI 6 13 7 1 8 2 9 3 10 4 11 5 12
MANIK 7 1 8 2 9 3 10 4 11 5 12 6 13
LAMAT 8 2 9 3 10 4 11 5 12 6 13 7 1
MULUC 9 3 10 4 11 5 12 6 13 7 1 8 2
OC 10 4 11 5 12 6 13 7 1 8 2 9 3
CHUEN 11 5 12 6 13 7 1 8 2 9 3 10 4
EB 12 6 13 7 1 8 2 9 3 10 4 11 5
BEN 13 7 1 8 2 9 3 10 4 11 5 12 6
IX or HIX 1 8 2 9 3 10 4 11 5 12 6 13 7
MEN 2 9 3 10 4 11 5 12 6 13 7 1 8
KIB 3 10 4 11 5 12 6 13 7 1 8 2 9
CABAN 4 11 5 12 6 13 7 1 8 2 9 3 10
EZNAB 5 12 6 13 7 1 8 2 9 3 10 4 11
CAUAC 6 13 7 1 8 2 9 3 10 4 11 5 12
AHAU 7 1 8 2 9 3 10 4 11 5 12 6 13

The results are summarized in Table 2 which lists all combinations of the 13 numbers and the 20 day-names, reading vertically in columns, from 1 Imix to 13 Ahau. Starting with 1 Imix, successive days are 2 Ik, 3 Akbal, 4 Kan, etc. After 13 Ben, the numerical sequence repeats, the next day being 1 lx. The last date is 13 Ahau - the completion of a 260-day cycle.

The glyphs in Figure 4 are the same throughout Mesoamerica, regardless of language or dialect, much as Chinese characters are in China.The names for the 20 days depend on the language. Scholars of Mayan calendrics use the Yucatec day-names and often refer to the 260-day cycle as the Tzolkin. The Nahuatal-speaking Aztecs called it the Tonalpohualli.

Why the 260-day cycle? The factors four and five in the count of 20, along with the count of 13 are fundamental throughout planetary astronomy and calendrics. In round numbers, for eight revolutions of the Earth around the Sun, there are 13 revolutions of Venus, which overtakes the Earth five times. Also, there are 13 revolutions of the Moon about the Earth (sidereal months) in a year, and conjunctions of Jupiter and Saturn occur at 20 year intervals. These are just a few typical examples.

Going along with the Pythagoreans, who were concerned with form (rather than matter) which was expressed in numbers, it would seem natural to construct a model which embodied the essential numbers in a form easy to memorize and use for computation. The Mesoamerican 260-day cycle could be the result of such an effort.

The astronomical nature of the 260-day cycle is revealed when we note that a double cycle of 520 days is very close to three eclipse half-years (3 X 171.31 = 519.93 days), the actual appearance interval of Venus is 263 days on the average, and Mars' synodic period is 3 X 260days. The idea that the numbers 20 and 13 were chosen because the Mayans went barefoot and counted on their fingers and toes has no merit.

The Calendar Round

The Mayan calendar system also included a 365-day calendar which consisted of eighteen 20-day periods (or "months"), and one five-day period. The 365-day count and the 260-day cycle are combined in another double cycle, which is illustrated as a table rather than a ring counter (Table 3). The names of the months are shown across the top of Table 3, and the days of the months, which are numbered from 0 to 19, are given in the column on the right. (Since the days of each month are numbered sequentially as in our calendar, a ring counter is not shown.) Since any day was not counted until it was completed, the first day is zero, which indicates completion of a previous cycle. The body of Table 3 gives the numbers of the 260-day cycle. It is read vertically in columns from left to right in contrast to our calendar, which is read across in rows. The first day is 1 Ik, 0 Pop, and the last in the first column is 7 Imfix, 19 Pop. Going to the top of the next column, the next day is 8 1k, 0 Uo.

Names of the Months

Names of the Days Pop Uo Zip Zotz Tzec Xul Yaxkin Mol Chen Yax Zac Keh Mac Kankin Muan Pax Kayab Cumhu Uayeb Day of the Month
IK 1 8 2 9 3 10 4 11 5 12 6 13 7 1 8 2 9 3 10 0
AKBAL 2 9 3 10 4 11 5 12 6 13 7 1 8 2 9 3 10 4 11 1
KAN 3 10 4 11 5 12 6 13 7 1 8 2 9 3 10 4 11 5 12 2
CHICCHAN 4 11 5 12 6 13 7 1 8 2 9 3 10 4 11 5 12 6 13 3
KIMI 5 12 6 13 7 1 8 2 9 3 10 4 11 5 12 6 13 7 1 4
MANIK 6 13 7 1 8 2 9 3 10 4 11 5 12 6 13 7 1 8
5
LAMAT 7 1 8 2 9 3 10 4 11 5 12 6 13 7 1 8 2 9
6
MULUC 8 2 9 3 10 4 11 5 12 6 13 7 1 8 2 9 3 10
7
OC 9 3 10 4 11 5 12 6 13 7 1 8 2 9 3 10 4 11
8
CHUEN 10 4 11 5 12 6 13 7 1 8 2 9 3 10 4 11 5 12
9
EB 11 5 12 6 13 7 1 8 2 9 3 10 4 11 5 12 6 13
10
BEN 12 6 13 7 1 8 2 9 3 10 4 11 5 12 6 13 7 1
11
IX or HIX 13 7 1 8 2 9 3 10 4 11 5 12 6 13 7 1 8 2
12
MEN 1 8 2 9 3 10 4 11 5 12 6 13 7 1 8 2 9 3
13
KIB 2 9 3 10 4 11 5 12 6 13 7 1 8 2 9 3 10 4
14
CABAN 3 10 4 11 5 12 6 13 7 1 8 2 9 3 10 4 11 5
15
EZNAB 4 11 5 12 6 13 7 1 8 2 9 3 10 4 11 5 12 6
16
CAUAC 5 12 6 13 7 1 8 2 9 3 10 4 11 5 12 6 13 7
17
AHAU 6 13 7 1 8 2 9 3 10 4 11 5 12 6 13 7 1 8
18
IMIX 7 1 8 2 9 3 10 4 11 5 12 6 13 7 1 8 2 9
19

Table 3.
First Year of the Calendar Round (the year 1 Ik).

The first day of the year is Ik 0 Pop, and the last day is 1 Kimi 4 Uyaeb. The last day of the month Cumhu is 9 Imix Cumhu.

The last day of the year is 1 Kimi, 4 Uayab. The first day of the next year is 6 Manik, 5 Pop, and the following years start on 11 Eb and 3 Caban, respectively, after which the cycle is repeated. These dates are known as the Year Bearers. With three more tables like Table 3 for each of the other Year Bearers one would have a perpetual calendar.

In this case we get the number of possible combinations by dividing 260 by the common factor of 5, which is multiplied by 365:

(260 / 5) X 365 = 52 X 365

Leaving the answer in this form, we can see that the count repeats itself every 52 years of 365 days. This 52-year cycle, which occurred throughout Mesoamerica, is referred to as the Calendar Round by scholars. There is solid evidence for widespread use of the 260-day and 365-day cycles as early as 500 B.C., and indications that this calendar system may have originated with the Olmecs much earlier.

The day count was continued without interruption, even though the Calendar Round came up 12.6 days short of 52 solar years. (Astronomers today use a strict day count called the Julian Period, not to be confused with the Julian year.) The error continued to accumulate amounting to a full solar year every 29 calendar rounds. The 365-day count was also adhered to in Egypt, even after the Canopus Decree ordered an additional day every four years to prevent the festivals from drifting through the seasons. This correction was introduced into the Roman Empire by Julius Caesar, and is thus called the Julian year.

Why was the 365-day year, often called the vague year, clung to so tenaciously, even though the Calendar Round or New Year drifted through the seasons? The 365-day year seems to be made to order for Venus, since the number of days in eight vague years falls between those in five synodic and 13 sidereal revolutions of Venus:

5 X 583.92 = 2919.6
8 X 365 = 2920
13 X 224.7 = 2921.1

For the Julian year we get:

8 X 36515 = 2922

With one day of error in 8 X 365 = 2920 days, Venus makes 13 revolutions about the Sun, and passes Earth five times. Note that eight vague years fall two days short of eight Julian years. Velikovsky has shown that the Greek version of the Canopus Decree speaks of Venus and its relation to Sothis (Sirius). [see Peoples of the Sea, p. 205 ff.1 By observing the helical risings of Venus relative to Sirius, the return of the heliacal rising of Venus to its starting point in the seasonal year could be determined.

Showing historically that the "Sothic" cycle pertained to Venus, exposed a major flaw in the current scheme of Egyptian chronology. It also showed clearly that scholars of both Egyptian and Mesoamerican cultures remain ignorant of the fact that both had a 365-day year related to Venus.

Especially curious is that in a recent study, Mesoamerican scholar Vincent Malmstrom came to the conclusion that the 260 and 365-day cycles were linked to initiate the Calendar Round about 235 B.C., without noting also that the Canopus Decree was issued in 238 B.C.

Conclusions

Efficient computation requires number systems which fit the problem. The decimal system, which has become second nature to us, is often not the best, and at times ill-suited. Planetary and lunar cycles were the kind that ring counters were used for in vacuum tube digital computers. This is only one example of the If rediscovery" of long forgotten calculation techniques brought about by new tools.

The 20-day month, which was at the heart of the Mesoamerican calendar, calls for further investigation. We used quotes when we introduced it above, since "month" actually refers to a synodic month of 29.53 days, which is the time required for the Moon to go through a complete set of phases. However, some of the Maya are known to have also referred to the 20-day period as a month, and the glyph for "twenty" is the Moon sign [Velikovsky suggested evidence that the 20-day month reflects a system much older than the 360-day system].

We showed above that the Calendar Round was a 52-year cycle which was specifically related to Venus. The termination of a Calendar Round was marked by the New Fire Ceremony, another subject worthy of further exploration. We shall return to these and related subjects in future issues of HORUS.

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