On previous pages, we have encountered some of the elements of the calendar system of the Maya, elements of which were shared with other civilizations in Mesoamerica.
One element of the Maya calendar was the Tzolkin, a series of 260 days which resulted from the interplay between two kinds of week, a week of 13 days and a week of 20 days. Perpetual calendars for these two weeks were given on this page.
The days of the week of 13 days were simply numbered from 1 through 13.
The days of the week of 20 days were named:
1) Imix 11) Chuen 2) Ix 12) Eb 3) Akbal 13) Ben 4) Kan 14) Ix 5) Chicchan 15) Men 6) Cimi 16) Cib 7) Manik 17) Caban 8) Lamat 18) Etznab 9) Muluc 19) Cauac 10) Oc 20) Ahau
Another element was the Haab. This was a year of 365 days without any leap years; thus, it was discussed, along with the ancient Egyptian vague year, in the page concerning Julian Day numbers.
This year consisted of eighteen months, each 20 days long, plus one epagomenal period of five days which went by the name Uayeb.
Its months were named:
1) Pop 10) Yax
2) Uo 11) Sac
3) Zip 12) Ceh
4) Zotz 13) Mac
5) Tzec 14) Kankin
6) Xul 15) Muan
7) Yaxkin 16) Pax
8) Mol 17) Kayab
9) Chen 18) Cumhu
Uayeb
The Tzolkin, and possibly also the Haab, date back at least to the Olmecs, a civilization which preceded the Maya, and may also have been used by the Zapotec.
There is a third component to the Mayan calendar, the Long Count.
In this system, we have:
Julian Julian Day Long Count Tzolkin Haab September 5, 3114 B.C. 584283 19.19.19.17.19 3 Cauac 7 Cumhu September 6, 3114 B.C. 584284 0. 0. 0. 0. 0 4 Ahau 8 Cumhu
according to the generally-accepted correlation between the Mayan calendar and Western dates. As the Long Count is a strict count of days, its first three numbers give the number of a 360-day year. However, 8,000 360-day years do not add up to an even number of 365-day years or 260-day years, and the Long Count is, at least according to one source I have encountered, to always begin on 4 Ahau 8 Cumhu.
Note that the date September 6, 3114 B.C. began at midnight of the previous day, J.D. 584283.5; at noon on that date, it was J.D. 584284, and at midnight, J.D. 584284.5. Thus, in my discussion of the Julian Day, I have treated that day as J.D. 584284; other sources will represent a calendar day by truncating the Julian Day number at the start of the day, referring to it as J.D. 584283. Thus, while I represent the day 0.0.0.0.0 as Julian Day 584284, I am correlating it to the day September 6, 3114 B.C. in the Julian calendar, and therefore I am using what is sometimes called the 584283 correlation.
It may also be noted that the year 2012 is not 8,000 years, even short years of 360 days, from 3113 B.C.; thus, the Mayan belief that the world is destroyed and created anew with the starting of each new calendar round would apply to Julian Day 3,464,649, or October 14, 4773 AD, Gregorian, if the Goodman-Martinez-Thompson correlation is correct.
I have chosen the form of the GMT correlation that begins the Long Count on September 6, 3114 B.C., as originally proposed by Goodman; Martinez proposed it to start one day later, and Thompson two days later. The original Goodman correlation has the advantage that it leads to the Tzolkin, as used by the Classical Maya, remaining synchronized to the Tzolkin as used by Maya communities today.
The Spinden correlation assigns the origin of the Mayan calendar to a date five 52-year calendar rounds earlier than September 7, 3114 B.C., and the Vaillant correlation assigns it to a date five 52-year calendar rounds later than September 6, 3114 B.C., so the Vaillant correlation and a Spinden - 1 correlation would agree with the Goodman correlation in both Haab and Tzolkin. Also, the Bowditch correlation matches the Tzolkin but not the Haab, and the Kreichgauer correlation is only a few days away from the Tzolkin as currently in use as well.
However, I had overlooked one thing, and it turns out the people claiming that the end of the world is coming in the year 2012 aren't depending on the world coming to an end every 52 years after all. The world, after all, didn't come to an end in 1960; it didn't even come to an end in, say, 1964.
Since the long count always begins on 4 Ahau 8 Cumhu, it follows that it needs to be a multiple of 52 years. 8,000 years isn't a multiple of 52, but 13 times 400 would be a multiple of 52. 5,200 short years of 360 days is 1,872,000 days; but it works out to 5,128 vague years of 365 days with 280 days left over. So having the long count end on 12.19.19.17.19 and begin over at 0.0.0.0.0 doesn't quite work out either.
Of course, if the five unlucky epagomenal days of Uayeb were not counted in the long count we could then wait 5,200 vague years of 365 days, and there would be no problem. However, a large number of monuments left by the Maya which include dates in the Long Count form accompanied by the corresponding Haab and Tzolkin eliminate that possibility quite conclusively.
On top of everything else, there are 20 baktun in a piktun, not 13. However, that the date 13.0.0.0.0 was believed to be the end of an age by the Maya apparently is, to my surprise, attested to some extent by reputable sources, such as the book Skywatchers, by Anthony Aveni. Apparently it is unclear if the baktun advances from 19 to 0 or from 13 to 1 from extant inscriptions. But that a start date of 4 Ahau 8 Cumhu is required for a new age is also affirmed by the sources I have recently been consulting. (I was hoping to debunk the 2012 panic thoroughly by showing it to be wholly a New Age invention without any genuine Mayan roots.) The Aztecs, on the other hand, did believe that every time the 52-year Calendar Round came to an end, there was a risk of the world coming to an end, which had to be averted by ceremony and ritual.
But if one waits long enough, one can find a date on which the world can come to an end:
Julian Julian Day Long Count Tzolkin Haab
September 5, 3114 B.C. 584,283 0. 1.12.19.19.19.19.17.19 3 Cauac 7 Cumhu
September 6, 3114 B.C. 584,284 0. 1.13. 0. 0. 0. 0. 0. 0 4 Ahau 8 Cumhu
July 19, 690 A.D. 1,973,280 0. 1.13. 0. 9.12.18. 5.16 2 Cib 14 Mol
July 21, 690 A.D. 1,973,282 0. 1.13. 0. 9.12.18. 5.18 4 Etznab 16 Mol
Gregorian
March 24, 1980 A.D. 2,444,323 0. 1.13. 0.12.18. 6.13.19 3 Cauac 7 Cumhu
March 25, 1980 A.D. 2,444,324 0. 1.13. 0.12.18. 6.14. 0 4 Ahau 8 Cumhu
December 21, 2012 A.D. 2,456,283 0. 1.13. 0.12.19.19.17.19 3 Cauac 2 Yax
December 22, 2012 A.D. 2,456,284 0. 1.13. 0.13. 0. 0. 0. 0 4 Ahau 3 Yax
March 11, 2032 A.D. 2,463,303 0. 1.13. 0.13. 0.19. 8.19 3 Cauac 7 Cumhu
March 12, 2032 A.D. 2,463,304 0. 1.13. 0.13. 0.19. 9. 0 4 Ahau 8 Cumhu
February 27, 2084 A.D. 2,482,283 0. 1.13. 0.13. 3.12. 3.19 3 Cauac 7 Cumhu
February 28, 2084 A.D. 2,482,284 0. 1.13. 0.13. 3.12. 4. 0 4 Ahau 8 Cumhu
October 13, 4772 A.D. 3,464,283 0. 1.13. 0.19.19.19.17.19 9 Cauac 12 Yaxkin
October 14, 4772 A.D. 3,464,284 0. 1.13. 1. 0. 0. 0. 0. 0 10 Ahau 13 Yaxkin
March 8, 15,595 A.D. 7,417,083 0. 1.13. 2. 7. 8.19.17.19 3 Cauac 7 Cumhu
March 9, 15,595 A.D. 7,417,084 0. 1.13. 2. 7. 9. 0. 0. 0 4 Ahau 8 Cumhu
January 12, 371,039 A.D. 137,240,283 0. 1.15. 7. 8.19.19.17.19 3 Cauac 7 Cumhu
January 13, 371,039 A.D. 137,240,284 0. 1.15. 7. 9. 0. 0. 0. 0 4 Ahau 8 Cumhu
January 5, 7,479,916 A.D. 3,733,704,283 0. 4. 2. 8 19.19.19.17.19 3 Cauac 7 Cumhu
January 6, 7,479,916 A.D. 3,733,704,284 0. 4. 2. 9. 0. 0. 0. 0. 0 4 Ahau 8 Cumhu
| | | | | | | | |
| | | | | | | | --- kin (day)
| | | | | | | ------ uinal (20-day month)
| | | | | | --------- tun (360-day short year)
| | | | | ------------ katun (20 years)
| | | | --------------- baktun (400 years)
| | | ------------------ pictun (8,000 years)
| | --------------------- calabtun (160,000 years)
| ------------------------ kinchiltun (3,200,000 years)
--------------------------- alautun (64,000,000 years)
The fact that the Mayans used a 20-day month is surprising, as it makes it appear that they ignored the phases of the moon in their calendar. This is not the case, however.
July 19, 690 A.D. is noted above as it is the date of a conjunction involving Mars, Jupiter, and Saturn that was recorded by the Maya.
The Tzolkin and the Haab calendars are the most well-known parts of the Mayan calendrical system, as they were also adopted by the Aztecs. The Long Count is the next most well-known part. Long Count dates were written using a system of positional notation for numbers which included a symbol for zero, in which a dot stood for one and a bar stood for five within the symbols for digits from 1 through 19. But Mayan inscriptions of dates often consisted of a Long Count date, a Tzolkin date, and then a group of glyphs called the Lunar Series, and finally the Haab date.
The decipherment of the glyphs in the Lunar Series took place only recently, and I present an attempt to describe them briefly.
As not all inscriptions including the Lunar Series included all the glyphs, the designations they were given by archeologists, in order, are G, F, Z, Y, E, D, C, X, B, and A.
Glyph G indicates one of nine lords of the underworld; the series of nine is synchronized with the last two digits of the Long Count. Glyph F is simply a marker for glyph G.
The glyphs Z and Y together (in a somewhat complicated manner) the position of the day in a regular succession of seven days, just like the days of our own week. This seven-day cycle, in combination with the nine-day cycle indicated by glyph G, and the 13-day cycle in the Tzolkin calendar, produced an 819-day cycle.
Glyph E is the first vigesimal digit, and glyph D the second vigesimal digit, of the Moon's age. Although some have suggested that the Maya reckoned the Moon's age from the full Moon, the GMT correlation is consistent with it being reckoned from the new Moon.
Glyph C indicates the place of the current lunation in a recurring cycle of 18 lunations.
Glyph X indicates the place of the current lunation in a cycle of either 6 lunations or 5 lunations. The rule for this sequence was changed at different times in Mayan history; originally, the cycle was always 6 lunations long, then it changed to one that seems to have been either 6 lunations or 5 lunations in length with equal frequency. A mixture of 6 lunation cycles with only a few 5 lunation cycles would have been useful in predicting eclipses, and so it appears that this was not the purpose of glyphs C and X. But the value of glyph X usually correlates with whehter the current lunation is 29 or 30 days long (as glyph A indicates).
Glyph B appears to be a marker for glyphs C and X which takes different forms due to grammatical rules.
Glyph A indicates whether the current lunation is 29 days long or 30 days long.
The age of the Moon was not calculated numerically in a fashion that did not alter over the life of the Mayan calendar. But it does not appear to have been derived from observation alone, either. Instead, 29-day months and 30-day months usually alternated. One study has led to the conclusion that during the classical period, the Maya used a cycle in which 81 lunar months included 38 months of 29 days and 43 months of 30 days. In Copan, the capital of one Mayan kingdon, it appears that a cycle of 149 lunar months with 70 months of 29 days and 79 months of 30 days was used instead.
While the Aztecs are said to have mainly used only the Haab and Tzolkin portions of the calendar, they did give names to the Nine Lords of the Night, and only their names, rather than those used by the Maya, are today known with confidence:
1) Xiuhtecuhtli 2) Iztli 3) Pilcintecuhtli 4) Cinteotl 5) Mictlantecuhtli 6) Chalchiuhtlicue 7) Tlazolteotl 8) Tepeyollotl 0,9) Tlaloc
and a perpetual calendar for this part of the cycle can be constructed:
1) Tll Xiu Izt Pil Cin Mic Cha Tlz Tep
2) Xiu Izt Pil Cin Mic Cha Tlz Tep Tll
3) Izt Pil Cin Mic Cha Tlz Tep Tll Xiu
4) Pil Cin Mic Cha Tlz Tep Tll Xiu Izt
5) Cin Mic Cha Tlz Tep Tll Xiu Izt Pil
6) Mic Cha Tlz Tep Tll Xiu Izt Pil Cin
7) Cha Tlz Tep Tll Xiu Izt Pil Cin Mic
8) Tlz Tep Tll Xiu Izt Pil Cin Mic Cha
9) Tep Tll Xiu Izt Pil Cin Mic Cha Tlz
MAR 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
28 29 30 31
APR 1 2 3 4 5
6 7 8 9 10 11 12 13 14
15 16 17 18 19 20 21 22 23
24 25 26 27 28 29 30
MAY 1 2
3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29
30 31
JUN 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24 25
26 27 28 29 30
JUL 1 2 3 4
5 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22
23 24 25 26 27 28 29 30 31
AUG 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
28 29 30 31
SEP 1 2 3 4 5
6 7 8 9 10 11 12 13 14
15 16 17 18 19 20 21 22 23
24 25 26 27 28 29 30
OCT 1 2
3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29
30 31
NOV 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24 25
26 27 28 29 30
DEC 1 2 3 4
5 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22
23 24 25 26 27 28 29 30 31
JAN 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
28 29 30 31
FEB 1 2 3 4 5
6 7 8 9 10 11 12 13 14
15 16 17 18 19 20 21 22 23
24 25 26 27 28 29
with the table indicating which row at the top to use for each year of the century having this form:
Gregorian
6 2 7 3 8 4 9 5 1 1600 2000 2400 2800 3200 3600
8 4 9 5 1 6 2 7 3 1700 2100 2500 2900 3300 3700
1 6 2 7 3 8 4 9 5 1800 2200 2600 3000 3400 3900
3 8 4 9 5 1 6 2 7 1500 1900 2300 2700 3100 3500 4000
Julian
1 6 2 7 3 8 4 9 5 200 500 800 1100 1400 1700
4 9 5 1 6 2 7 3 8 0 300 600 900 1200 1500
7 3 8 4 9 5 1 6 2 100 400 700 1000 1300 1600
00 01 02 03 04 05 06
07 08 09 10 11
12 13 14 15 16 17 18
19 20 21 22 23
24 25 26 27 28 29 30
31 32 33 34 35
36 37 38 39 40 41 42
43 44 45 46 47
48 49 50 51 52 53 54
55 56 57 58 59
60 61 62 63 64 65 66
67 68 69 70 71
72 73 74 75 76 77 78
79 80 81 82 83
84 85 86 87 88 89 90
91 92 93 94 95
96 97 98 99
For years B.C. in the Julian calendar, the following chart is the one to use:
2 7 3 8 4 9 5 1 6 3399 3099 2799 2499 2199 1899 1599 1299 999 699 399 99
5 1 6 2 7 3 8 4 9 3299 2999 2699 2399 2099 1799 1499 1199 899 599 299
8 4 9 5 1 6 2 7 3 3199 2899 2599 2299 1999 1699 1399 1099 799 499 199
99 98 97 96 95 94
93 92 91 90 89 88
87 86 85 84 83 82
81 80 79 78 77 76
75 74 73 72 71 70
69 68 67 66 65 64
63 62 61 60 59 58
57 56 55 54 53 52
51 50 49 48 47 46
45 44 43 42 41 40
39 38 37 36 35 34
33 32 31 30 29 28
27 26 25 24 23 22
21 20 19 18 17 16
15 14 13 12 11 10
09 08 07 06 05 04
03 02 01 00
As for the seven lords of the Earth, again based on the correlation of 0.0.0.0.0 to September 6, 3114 B.C., just as, since that date is 4 Ahau 8 Cumhu, the 13-day cycle started three days earlier, on September 3, 3114 B.C., that date being 1 Caban 5 Cumhu, the 7-day and 9-day cycles also start on that date. September 3, 3114 B.C. was a Friday, so that is the day ruled by the first lord of the Earth.
Because the Maya are known to have made incredible achievements in their tracking of the motions of the heavenly bodies, some extravagant claims have been made concerning their achievements. Thus, some have claimed that the 260-day Tzolkin cycle reflects an inner harmony of the entire Solar System, and might attribute it to a supernatural or extraterrestrial source.
Although, as recorded in the Dresden Codex, the Maya did work out some remarkable relationships between the phases of the Moon and the synodic period of Venus with the Haab and the Tzolkin, the general consensus is that the 260-day Tzolkin was not designed as a way to produce these relationships, but was simply a pre-existing period used for purposes of divination and established before the flowering of Mayan astronomical achievement.
It may also be noted that the orbit of Venus around the Sun takes 224.701 days, and its synodic period (the average time between recurrences of a given relationship between the Earth and Venus around the Sun, such as maximum elongations) is 583.92 days. Any simple connection between either of those periods and the 260-day Tzolkin cycle is not immediately obvious.
However, 584 days is a multiple of 4 days, and so it is not relatively prime to the 20-day cycle. Thus, 65 Venus synodic periods of 584 days will correspond to 146 Tzolkin cycles of 260 days. This, in itself, is not remarkable; any two cycles will have a least common multiple.
The Haab and Tzolkin calendars synchronize every 52 Haab years of 365 days, and every 73 Tzolkin cycles of 260 days, this being the Calendar Round. As 73 is half of 146, two calendar rounds do lead to a recurrence of the 584-day synodic period of Venus, and this is, at the least, a coincidence.
But perhaps not as big a coincidence as it might seem. 104 is a multiple of eight, 52 being four times thirteen as we know from playing cards, and the positions of the planet Venus in the sky very nearly repeat themselves every eight years.
In about 2,922 days, the Earth orbits the Sun eight times, and in about 2,921 days, Venus orbits the Sun thirteen times, thus the Earth overtakes Venus five times in this span, but the positions of Venus change slightly from one eight-year period to the next. Five times 584 days gives 2,920 days, eight times the vague year of 365 days, so the Haab, rather than the Tzolkin, might be considered as being tied to the planet Venus, even though, as a year, it is obviously also tied to the Sun.
The Maya were aware of the synodic cycle of Venus, and did use 584 days as an approximation to its duration. The Dresden Codex, one of the few surviving Mayan books, includes tables of the position of Venus. What is unclear, however, is that this cycle formed the basis of their Tzolkin calendar cycle. The Lunar Series glyphs found in many inscriptions include the day of the lunar month and the position of the day in both a 7-day week and a 9-day week, but they do not give an indication of the position of the day within the 584 day cycle either.
One element of the well-known parts of the Mayan calendar that might be affected by Venus, however, is the Haab portion. Eight years are also approximately five synodic periods of Venus. Five times 584 days is 2,920 days, which is also eight times 365 days. Thus, the omission of the leap year could be credited to Venus; the only contribution made by the 260-day cycle is that 260 is divisible by 4, which allows two calendar rounds to be a multiple of eight vague years.
There is some evidence from monuments, but not of a conclusive nature, that suggests the Maya might have 29 calendar rounds of 52 Haab each amounted not only to 1,508 Haab, but to 1,507 tropical years, leading to an approximation to the tropical year of 365.242 days, closer than that used in our present Gregorian calendar.
Also, the synodic period of Mars is 779.94 days. 780 days is 260 days times three. Thus, if anything, the 365-day period is both a Sun year and a Venus year, and the 260-day period is a Mars year rather than a Venus year as some have called it. Because of the limited amount of surviving records, we cannot be certain that the Maya kept track of Mars, or of Jupiter, Saturn, and Mercury.
That there would be cycles available to them for this, however, is true:
Tropical year 365.2422 1,507 * 365.242 = 1,508 * 365 = 2,117 * 260
Synodic month 29.53059 405 * 29.53086 = 46 * 260
Synodic periods of:
Mercury 115.88 9 * 115.56 = 4 * 260
7 * 117 = 819
22 * 116.14 = 7 * 365
Venus 583.92 5 * 584 = 8 * 365
61 * 583.93 = 137 * 260
Mars 779.94 780 = 3 * 260
Jupiter 398.88 39 * 399 = 19 * 819
140 * 398.89 = 153 * 365
88 * 398.86 = 135 * 260
Saturn 378.09 13 * 378 = 6 * 819
We will see below that the 819-day cycle was formed by taking the 13-day week of the Tzolkin, and combining it with a 9-day week indicating the lords of the night and a 7-day week indicating the lords of the earth; this cycle was used for divination of the affairs of kings even as the 260-day cycle was used for divination by the ordinary people, and it could be interpreted as relating to the motions of Jupiter and Saturn.
In addition to a relationship between the synodic period of Mercury and the Tzolkin, a less accurate one to the Haab is also shown that illustrates the curious fact that the year is about pi times the synodic period of Mercury.
Possible relationships between the synodic periods of Venus and Jupiter to the 260-day Tzolkin are shown that have been used as part of the basis for a claim that this period had relationships to all the Solar System cycles. However, continued fraction methods could obtain equally good relationships for almost any other number of days.
After three Saros cycles, also known as an exeligmos, which amounts to 19,756 days, eclipses will tend to recur in the same geographical location. This is only four days short of 19,760 days, which is 76 Tzolkin cycles of 260 days. Thus, it is possible to claim that the Tzolkin cycle somewhat simplifies the calculation of eclipses as well, an odd four days each cycle being relatively easy to calculate. However, the Saros cycle does not appear to have been used by the Maya.
The Dresden Codex also shows that the Maya used an approximation of the lunar month in which 81 lunar months were 2,392 days long. Repeating this approximation five times obtains 405 lunar months, and 11,960 days, with an accumulated error of about 1/9th of a day by which the 405 lunar months are shorter than 11,960 days, and 46 Tzolkin cycles of 260 days.
The lunar tables in the Dresden Codex divided the lunations into groups of 6 lunations which are usually 177 days long, but occasionally 178 days long, and the occasional group of 5 lunations which is 148 days long. This was done to predict days on which an eclipse was possible.
If one slides this sequence of 11,960 days against itself at a displacement of 6,585 days, a Saros cycle being 6,585 and a third days, one will find the two sequences usually match; this is a natural consequence of the fact that it is intended to keep track of eclipses. But the sequence was not restarted after 6,585 days or 19,756 days, but at its end, 11,960 days, and so the Saros cycle was not what was used by the Maya; they were less accurate in predicting eclipses than the ancient Greeks as far as we know.
The 405 lunation cycle, in addition to being five times a cycle of 81 lunar months in 2,392 days, is also three times a cycle of 135 lunations called the Tritos. This is the next closest approximation to a coincidence between the synodic month and half of the draconic month after the Saros; beyond the Saros, the Inex, of 358 lunations, is an even closer approximation to the Saros to the relationship between those two cycles.
Ideally, then, a table of 405 lunations should repeat the exact same pattern of alternating 29 and 30 day lunations five times, and the same pattern of 6-lunation and 5-lunation eclipse semesters six times. The one in the Dresden Codex did not indicate the length of individual lunations, and the pattern of eclipse semesters was nearly, but not exactly, repeated.
135 lunations would include twenty semesters of six lunations and three semesters of five lunations. In the Dresden codex, all the semesters of five lunations are 148 days long. Of the twenty semesters of six lunations, eighteen are 177 days long, and two are 178 days long.
2,392 days would include 43 lunations of 30 days, and 38 lunations of 29 days.
How closely could one work these two cycles together?
Because the five lunation semester is fully consistent with alternation of 29 and 30 day lunations, its location isn't constrained by the distribution of pairs of 30 day lunations, except to avoid one within such a short semester, so an even distribution should be possible (the one in the Dresden Codex was not as even as possible).
81 lunations, consisting of 38 pairs of one 30-day lunation and one 29-day lunation, plus 5 extra 30-day lunations to distribute evenly, could follow this pattern:
8*(30 29) 30 7*(30 29) 30 8*(30 29) 30 7*(30 29) 30 8*(30 29) 30
If we impose three repetitions of a maximally-regular repetition of an alternation between five and six lunation eclipse semesters on five repetitions of that sequence, we may have to make some slight adjustments to fit within the assortment of semester types shown above. Of course, other sequences of three 30-day months and two 29-day months could produce a 148-day eclipse semester.
A regular pattern of semesters might be like this:
7*(6) 5 6*(6) 5 7*(6) 5
Let us assume we are dealing with twelve types of eclipse semesters:
29 30 29 30 29 30 177 A 29 30 29 30 30 29 177a B 29 30 30 29 30 29 177b C 30 29 30 29 30 29 177c D 29 30 29 30 30 148a 1 29 30 30 29 30 148b 2 30 29 30 29 30 148 3 30 29 30 30 29 148c 4 30 30 29 30 29 148d 5 30 29 30 29 30 30 178 x 30 29 30 30 29 30 178a y 30 30 29 30 29 30 178b z
With these types of eclipse semesters, we would be able to build stretches of 29 and 30 day lunations, with occasional flips in the parity of the sequence involving two 30-day lunations in a row, so we have flexibility with which to at least approximate the regular sequences above.
Since five lunations in 148 days lead to three 30-day lunations and only two 29-day lunations, it's easy to associate a pair of consecutive 30-day lunations with a short semester. This can also be avoided, but then the six-lunation semester before must end, and the six-lunation semester after must begin, with a 29-day lunation. Meeting this constraint while including only two 178-day periods of six lunations in a 135-lunation Tritos will be the challenge.
Taking the regular pattern of five and six lunation semesters and placing it against the regular pattern of 29 and 30 day lunations, the following image shows one possible sequence:
This sequence includes some 147-day five lunation semesters, unlike the sequence given in the Dresden Codex, and, as noted, that sequence was not as exactly regular as this one.
But much of the sequence fits into the plan of possible semesters shown above. As well, while the succession of 29 and 30 day lunations follows the actual lengths of lunations closely, and thus shouldn't be altered, because the purpose of the semesters is to indicate the days that have a higher chance of an eclipse, they could be altered. Since a sequence of six-lunation and five-lunation semesters only approximates the relation between the synodic month and the draconic month, eclipses would not be likely at the parts of that sequence which were furthest out of step with a semester of uniform size, just a bit less than six lunations.
If the cycle is started so that the midpoint of the sequence of semesters coincides with a full moon (if it is lunar eclipses that are being checked for) that is exactly on the ecliptic, then the start and end days of a five-lunation semester will be those where the full moon is furthest from the ecliptic, and thus the short semester could be moved. In the diagram above, in the three cases where we would wish to do so, we would also be able to eliminate an adjacent 178-day semester by doing so.
The diagram above shows a blue triangle at one such midpoint; red triangles indicate successive Tritos cycles after that point, a green triangle indicates one Saros cycle after that point, and a purple triangle indicates an Inex cycle; because the blue triangle does not begin the chart, it is placed where it would be on another repetition of the diagram. As the Saros and Inex cycles are more accurate than the Tritos, they show places where a new repetition might be shifted to as the sequence loses synchronization.
If we make the adjustements needed to avoid 147-day semesters, the resulting sequence becomes:
..!....*..!...*....!..*....!..*....!.oO....oO!......oO......*..!....*
where * and o stand for 148-day semesters, . and O stand for 177-day semesters, and ! stands for a 177-day semester. The symbols o and O are used where a five-lunation semester was moved to avoid it being 147 days in length.
The actual sequence in the Dresden Codex is:
...*...!.....*!...*.......*..!......*!...*.......*..!.....*.!...*....
The second 148-day semester out of every three is considerably delayed, and the 178-day semesters are more evenly distributed.
Can the even distribution of 29-day and 30-day lunations above be made to correspond with this sequence?
For a first attempt, one might try and slide the sequence of semesters along so as to fit as closely as possible:
Dresden: ...*...!.....*!...*.......*..!......*!...*.......*..!.....*.!...*....
Regular: ...*..!...*....!..*....!..*....!.oO....oO!......oO......*..!....*..!.
!..*....!.oO....oO!......oO......*..!....*..!....*..!...*....!..*....
..oO......*..!....*..!....*..!...*....!..*....!..*....!.oO....oO!....
To break the Dresden Codex sequence into five aligned pieces, so as to determine possible locations for pairs of 30-day lunations, one would have to expand the semesters into individual lunations. So, * becomes *****, ! becomes !!!!!!, each showing a stretch with an excess 30-day lunation, and . would alternate between ...... and ======, since a pair of 30-day lunations could also be placed at the boundary between two normal semesters.
That gives us this sequence:
......======......*****......======......!!!!!!......======......======......**** *!!!!!!......======......*****......======......======......======......*****.... ..======!!!!!!......======......======......======*****!!!!!!......======......** ***......======......======......======......*****......======!!!!!!......======. .....======......*****......!!!!!!......======......*****......======......======
One thing we can do to solve for the location of 30 and 29 day locations is to note the cases where a 178-day semester overlaps a 177-day semester with an offset of only one lunation. In each such case, a 30-day lunation must be given up, and a 29-day lunation gained, when going from the 178-day semester to the 177-day semester. This should allow us to quickly determine if a consistent sequence of 81 lunations always repeating is even possible. There are no cases of an exact overlap of a 178-day semester with a 177-day semester, making such a solution obviously impossible.
......======......*****......======......!!!!!!......======......======......**** *!!!!!!......======......*****......======......======......======......*****.... ..======!!!!!!......======......======......======*****!!!!!!......======......** ***......======......======......======......*****......======!!!!!!......======. .....======......*****......!!!!!!......======......*****......======......====== SL LSL LS SL LS SL LS SL LSL LS
Our immediate deductions give us, because of the way in which the 178-day semesters are scattered over the 81-lunation cycle, six places in that cycle where there must be a pair of 30-day lunations; since we only have five such pairs to distribute, we can conclude that the series of semesters in the Dresden Codex is not consistent with any alignment of a maximally-regular distribution of 29 and 30 day lunations.
......======......*****......======......!!!!!!......======......======......****
*!!!!!!......======......*****......======......======......======......*****....
..======!!!!!!......======......======......======*****!!!!!!......======......**
***......======......======......======......*****......======!!!!!!......======.
.....======......*****......!!!!!!......======......*****......======......======
SL LSL LS SL LS SL LS SL LSL LS
!!!! !!!! !!!! !!!! !!!! !!!!
......======......*****......======......!!!!!!......======......======......****
*!!!!!!......======......*****......======......======......======......*****....
..!!!!!!======......======......======......======*****!!!!!!......======......**
***......======......======......======......*****......!!!!!!======......======.
.....======......*****......!!!!!!......======......*****......======......======
S L L S SL LS SL LS S L L S
!!!! !!!!
......======......*****......======......!!!!!!......======......======......****
*!!!!!!......======......*****......======......======......======......*****....
..======!!!!!!......======......======......======*****!!!!!!......======......**
***......======......======......======......*****!!!!!!......======......======.
.....======......*****......!!!!!!......======......*****......======......======
SL LSL LS SL LS SL LS SL LS
!!!! !!!! !!!! !!!! !!!!
An attempt to solve this by moving two 178-day semesters by one place leads to places where, because two 29-day lunations in a row are impossible, pairs of 30-day lunations are now fixed in locations that are too close together.
Moving one 178-day semester by two places allows it to be placed where there are no overlaps of the type used to make the initial deductions. An overlap with a 148-day semester implies a 30-day lunation that is already indicated for other reasons.
This would seem to allow a recurrent sequence, even if not one that is maximally regular. However, there is also one case where two 148-day semesters overlap with an offset of one semester. As this means what is given up must be of the same type as what is gained, this again forces a sixth location for two consecutive 30-day lunations.
Above, we encountered the Tritos, a period which matches a recurrence of half the draconic month with the synodic month, that the Maya made use of in the Dresden Codex, and it was noted that two other longer periods which match even closer recurrences, the Saros and the Inex, could be used to take partial sequences of eclipse semesters from a repeated Tritos that could be repeated with better accuracy.
After the Inex, the next very close approximation to the ratio between (half) the draconic month and the synodic month is reached at 4,161 lunations, and the next one after that is only a bit larger at 4,519 lunations.
With the Tritos cycle at 135 lunations, the Saros at 223, and the Inex at 358, the cycle of 4,519 lunations would consist of twelve Inex cycles and one Saros, which is equivalent to twelve Tritos cycles and thirteen Saros cycles, and the cycle of 4,161 lunations of eleven Inex cycles and one Saros. The longer cycle is about twenty times more accurate than the shorter, so the next good eclipse cycle will also be quite long: 94541, 99060, and 193,601 lunations are the next few attractive cycles.
The Saros, however, is also close to a multiple of the anomalistic month, and so while other eclipse cycles will relate eclipses of some sort, a Saros will tend to relate total eclipses to total eclipses and annular eclipses to annular eclipses. As well, the elliptical orbit of the Moon also affects when it reaches a particular phase, and when it reaches its nodes. Thus, an eclipse is almost certain to be followed by another eclipse one Saros cycle later, and this eclipse is likely to be of the same kind. A series of days separated by an Inex that have some eclipses on some of those days can also have intervening days without an eclipse, but most of the time there will be eclipses as well, although the character of the eclipses will vary. Thus, the Saros cycle is much more striking, although, since the Inex more accurately approximates the relation between the two cycles it reflects, the series of eclipses related by the Inex will continue on for a longer period.
The number of lunations in a Saros is 223, and that in an Inex is 358; these two numbers are relatively prime. Any lunation can be reached by a combination of those two numbers, so it's not surprising that any lunation associated with an eclipse can be reached that way. But because these are valid eclipse cycles, far-out combinations that reach numbers that aren't close to multiples of the Saros and Inex individually are not required. A graphic chart showing the sequence of solar eclipse types separated by the interval of a Saros vertically, and an Inex horizontally, is presented on this page, this type of graphic chart was justly called the Saros-Inex Panorama by George van der Bergh.
The synodic, draconic, and anomalistic months are not constant, but change in length over time, and the diagram on that page covers such a length of time that it reflects this, with the sequence of eclipses forming a curved arc.
Just as it was possible to repeat the Saros three times in order to obtain the Exeligmos, which is close to a multiple of the day, so as to find repeated eclipses in the same location, it might be possible to do the same thing with the other eclipse cycles to bring them into correspondence with the anomalistic month, in order to obtain an eclipse cycle that is genuinely beyond the Saros.
Given that the synodic month is 29.530588853 days long, and half the draconic month is 13.6061104085 days long, the ratio between these two periods is 2.17039168185. Thus, some successive approximations to coincidences between these cycles are as shown below:
Synodic Half Draconic Days Anomalistic
months months months
1 2.17039168 29.5306 1.072 Month
6 13.02235009 177.1835 6.430 Eclipse semester
135 293.00288771 3986.6295 144.681 Tritos
223 483.99734505 6585.3213 238.992 Saros
358 777.00022210 10571.9508 383.674 Inex
4161 9030.99978820 122876.7802 4459.401
4519 9808.00001030 133448.7310 4843.074 Square Year
94541 205190.99999419 2791851.4008 101320.886
99060 214999.00000449 2925300.1318 106163.960
193601 420189.99999868 5717151.5325 207484.846
The 99060-lunation cycle comes close to being a multiple of the anomalistic month, but not as close as the Saros. Five repetitions of the 4161-lunation cycle, however, will provide a closer recurrence of the anomalistic month, as well as a recurrence of the half draconic month that is still more than twice as good as a single Saros cycle. This cycle, 20,805 lunations, is known as the Selenid I cycle. It was named by George van den Bergh (who also coined the name of the Inex), who, through study of the famous Canon of Eclipses by Oppolzer, wrote of how the interplay between the Saros and the Inex is reflected in the pattern of eclipses.