Extracto
1.2 The Elements of the Maya Calendars and the Number Systems Used within
1.2.2 Numbers Used with the Maya Calendar Systems
1.2.5 The Distinction Between k'atun- and ajaw-Periods
1.2.6 An Intermittent Resume and Retrospect
1.3.1 Introduction with Regard to the Calendar Structures
1.3.3 Pre-Study about the Structure of the Yucatecan Calendar
1.3.4 The Association Table for the Calendar Round, the Long and the ajaw-Periods Count
1.3.5 The Improvement of the Average Solar Year by the Leap Day Corrections
1.3.6 More Data from Monuments of the Rio Bec, the Chenes and the Puuc Region
1.3.7 The Chronicle of Dates from the Yucatecan Calendar
1.3.8 A Western Point of View of the Mechanics and the Counting of Intercalated Days With the Maya Calendars
1.3.9 The Patrons of the Double Calendar Rounds and The Preliminary Structure of the Yucatecan Calendar
1.3.10 Examples from Northern Yucatán of the Maya Classic Calendar Defined by the Year Bearer Group “B”
1.3.11 The Last Step Towards the Eternal Solar Calendar: The Transformation of the ajaw-Period from 20 tun to 24 haab and the Effect of the Discontinuities at the End of Every Double Calendar Round on the ajaw-Period Count
1.4 The Sudden Death of the Lived Yucatecan Calendar
Events on the Part of the Yucatecos
Events on the Part of the Spaniards
Correlating the Events
Table for the Comparison of the Events, and the Conclusions Drawn
1.5 The Uniqueness of Dates Within the Maya Calendar Systems
1.6 End Note to the Maya Calendar Systems
Survey
Major Steps of the Calendar Reforms
Anticipation
Bibliography
Preface
This treatise about the Maya calendar, emphasizing the Yucatecan calendar, is an extract and a summary of the studies on the number as well as on the calendar systems of the Maya. The studies have been continuously enhanced and expanded over the last years.
A first summary of the studies has been published in 2012 under the title:
„Die Kalenderreform der Maya und die Korrelation zur Umrechnung der Datierungen in den Gregorianischen Kalender
oder:
Weshalb der für den 21. Dezember 2012 auf Basis des Klassischen Kalenders prognostizierte Weltuntergang nicht stattfindet.“
That summary resulted in a subdivision of the Maya Calendar Systems into three distinct calendars and a revision of the GMT-correlation for the Maya and the Christian calendars.
For the “world ending date” to take place supposedly in 2012, the revised correlation revealed that the Great Cycle, 13.0.0.0.0, should have come to an end already in 1934.
This treatise concerns
in Volume 1
Part 1
The Maya Calendar Systems consisting of the three calenders:
the Pre-Classic
the Classic and
the Yucatecan Calendar
in Volume 2 (in preparation)
Part 2
the revised correlation for the Maya and the Gregorian Calendar.
Part 3
Supplementary data from the monuments, the codices and the Chilam Balam-Books
a chronicle of all data referred to in these treatises.
Acknowledgments
I am very obliged
to the Philosophische Fakultät der Universität Bonn, that it issues thesisses via internet, in particular:
Daniel Graña-Behrens (2002):
Die Maya-Inschriften aus Nordwestyukatan, Mexiko,
and Antje Gunsenheimer (2002):
Geschichtstradierung in den yukatekischen Chilam Balam-Büchern:
Eine Analyse der Herkunft und Entwicklung ausgewählter historischer
Berichte
and as well to the librarians of the Verbund der deutschen Büchereien, since without their services I would not have been able to take hold of most of the literature required for the study and this treatise.
I am also obliged for the courtesy to use, scan, reprint, or copy material such as figures and sketches in particular:
MIT Press to let me reprint a short excerpt including the thereby attached sketch from
Karl Menninger (1970) : “Number Words and Number Symbols: A Cultural History of Numbers” - (on pages 76/77),
University of Oklahoma Press, Norman, to reuse material by
Fray Diego Durán (1579/1971) : “Book of the Gods and Rites and the Ancient Calendar” (on pages XXIII, 386, 465,469 and Plate 55),
Philipp Reclam jun. Verlag GmbH
Diego de Landa: “Bericht aus Yucatán”
“Kreislinie: Der Krieg der Katunes (circular diagram)” (on page 133), and “die Buchstaben / Glyphen der 20 heiligen Tage (Uinal Hunekeh) sowie der 18 Monate“ (on pages 82 and 83 respectively on pages 95 to 131)
MAIRDUMONT Business Solutions Print & Digital
John L. Stephens (1841/1843 – 1980) : “In den Städten der Maya“ Figure 59 (on page 154), (F. Catherwood's originally Fig. 13)
© 2015 Mit freundlicher Unterstützung MairDumont, D-73760 Ostfildern
Finally I like to take the opportunity again to thank Ute in particular, since she had been so wonderfully patient with me all these years of study.
Illustrations
Fig. 1.2.3.7 Standardized Head Variant and Geometric Period Glyphs
Fig. 1.2.3.9 Plaque of Leiden, Rear Side (drawing by Ernst Förstemann 1903:553)
Fig. 1.2.3.13 (see previous page)
Part 1 The Maya Calendar Systems, whilst emphasizing the Yucatecan Calendar the World's Very First Eternal Solar Calendar
1.1 Introduction
„It has been known for very many years that the positions in the months held by each day were one less in Yucatecan at the time of the Spanish conquest than they had been in the cities of the Central area during the Classic or Initial Series Period. That is to say, combinations such as 12 Kan 1 Pop, 6 Imix 3 Yax, or 13 Ahau 17 Mol were used at that time in Yucatan, whereas in the Central area during the Classic Period those dates would normally have been written 12 Kan 2 Pop, 6 Imix 4 Yax, and 13 Ahau 18 Mol.
It has been generally supposed that the backward shift of one place in month positions was a late innovation, not long antedating the Spanish conquest, and perhaps connected in some way with the Mexican domination of much of the Maya area. We believe that this shift occurred at a quite early date in Campeche and, perhaps, elsewhere, and that it is characteristic of the Puuc area.”
Tatiana Proskouriakoff and J. Eric S. Thompson (1947:143)
T. Proskouriakoff and J. E. S. Thompson's statement is in line with the general conviction of most Maya scholars, that there was only one calendar utilized by the Maya, the Classic. Its structure was build up on only four day signs as opening days of the years; the year bearers: ik’, manik’, eb and kaban.
However and in addition to the shifts described above there are data with up to four (cyclic?) shifts in position as documented by Daniel Graña-Behrens (2002) from monuments of Northwestern Yucatán, to be exact of the region of the Rio Bec, Chenes, and Puuc styles of architecture; and such data concerning the “Yucatecan Calendar” were also discovered in the Post-Classic codices, the Dresdensis and the Peresianus, as well as in the narratives of the Post-Columbian books of Chilam Balam.
Up till now most scholars relate the shifts in the month position to modifications due to local adjustments to the calendar. But are these shifts in the month position really only locally? Or are these, accordingly Arlen F. Chase (1986:108), “shifts in the sets of year bearers” and is it, hence, “logical to assume, in accord with the nature of the Maya calendar, that such shifts were cyclical”? and may it additionally “be suggested that the year bearer sets changed in a regular cycle every baktun – perhaps at the end of the twelfth katun of each baktun”?
Or operated the “calendrical mechanics” under even a third approach, and in a totally different way?
The basic study, this treatise reflects the main ideas of, was initiated by considering the shifts in the month position as systematic discontinuities in the Maya Calendars. Latter had reminded the author of the discontinuities with the Christian Calendars.
The upcoming curiosity on the subject was enhanced by the statement of Diego de Landa (1566/1966:133):
“They have their year as perfect as ours, consisting of 365 days and 6 hours. ... From these six hours one day was made every four years, and so they had every four years the year of 366 days”
Today most scholars deny the correctness of de Landa's statement due to the inequality with the structure of the Classic Calendar.
Hence, the quest with this study was not only to search for the base of the shifts, but also for whether the backward shifts in month positions, the discontinuities, can rather be related to corrections by leap days, and thus rule, whether de Landa's statement is either right or wrong.
1.2 The Elements of the Maya Calendars and the Number Systems Used within
The structure of the calendar systems the Maya utilized consist all of the so-called Calendar Round in permutation with a longtime count.
1.2.1 The Calendar Round
The Basic Elements of the Calendar Round
The Calendar Round is construed by the spiritual calendar, the tzolk'in, and the solar one, the haab.
The tzolk'in consists of the concurrently running thirteen sacred numbers and the twenty sacred day signs, forming a cycle of 260 days.
The haab consists of eighteen month of twenty days each and a short month of five days, thus in total of 365 days.
The permutation of tzolk'in and haab constitute a cycle of 52 years, the least common multiple of 260 and 365 days: the Calendar Round.
The names of the twenty sacred day signs are noted as:
imix, ik’, ak’bal, k’an, chikchan, kimi, manik’, lamat, muluk, ok, chuen, eb, ben, hix, men, kib, kaban, etz’nab, kawak and ajaw.
All Yucatecan words are written throughout this treatise accordingly the orthography from 1989 (Nikolai Grube 2000:466), except where used in quotations.
The names of the eighteen month of the haab are:
pop, wo, zip, zotz’, zek, xul, yaxk’in, mol, ch’en, yax, zak, keh, mak, k’ank’in, muwan, pax, k’ayab, kumk’u;
the short month of 5 days is wayeb.
The Numbers of the Solar and the Sacred Calendars
The number systems used in Mesoamerica were of Base 20.
Until now most of the scholars have considered and counted all common calendar dates as elapsed periods, i. e. the day (k'in), the month (winal / uinal), the short year (tun) of 360 days, or the solar years (haab) of 365 days.
Victoria R. Bricker and Helga-Maria Miram explained (2002:39): „During the Classic period of Maya history (ca. 300-950), the twenty days in each normal month were numbered from 0 to 19, and those in Uayeb were numbered from 0 to 4. Therefore, the first day of the haab was 0 Pop (expressed metaphorically as the “seating” of Pop), the second day was 1 Pop, ... , and the last day (of the haab) was 4 Uayeb. In other words, the days of the month were counted in terms of elapsed time, not current time, the day receiving a number only after it had been completed.“
Harvey M. Bricker and V. R. Bricker remarked to this kind of counting at another time (2011:67): “The months of the Maya haab differed from the months of the European year not only in their number and length, but also in their use of elapsed time, instead of current time, for counting days.”
But the day name, the date of the tzolk'in was considered differently. Since it is composed of one of the sacred numbers and of one of the sacred day signs and since both are not elements of a counting sequence, the sacred number was treated as a name, thus, always as an element of current time.
Oliver La Farge wrote (1934:115): “It must be remembered ... , that the elapsed time system of counting applied, and applies, to measures of time, such as the uinal and the haab, but not to the day-lords of the tzolkin with their ceremonial numbers. Of necessity in their very nature, these must be counted currently.”
However, as we will see in the chapter “Notation Accordingly the Yucatecan Method”, the count of current days of the month as well as current longtime periods were introduced at a later time.
The Year Bearers and Other Characteristics of the Calendar Round
Juan Pio Pérez (1843/1990:293f) evaluated the characteristics, the properties of the Calender Round and concluded:
1) The 365 days of the solar year divided by 13, the quantity of sacred numbers, results in 28 and a remainder of 1. Thus, the last day of the year will carry the same sacred number as the first day and in order to stay with the steadily progressing counting, the next year will start with one number higher than the previous one.
And since the Calendar Round starts always with the sacred number 1, all the following years will begin with a number ascending from 1 until 13 is reached. Thereafter the count will start again with 1. When the end of the Calendar Round has been reached the cycle 1 to 13 has been completed four times.
2) Since the division of the 365 days by 20, the number of sacred day signs, equals 18 with a remainder of 5 and since, in addition (see F. G. Lounsbury 1978:765) these 20 day signs themselves are divisible by 5 without a remainder, the day signs fall in groups or sets of four out of the twenty. Each four of a group are called the bearer of the year, ah cuch haabob.
The groups of year bearers are shown in the table below:
3) The combination of characteristics 1) and 2) results in: the first section of thirteen years of the Calendar Round starts with the first year bearer of a group. And since the 13th year of the first section begins with the first year bearer of the group, then the second section of the Calendar Round starts with the second year bearer, the third section accordingly with the third year bearer and the fourth section with the fourth year bearer.
4) Since the sacred day signs correspond in number with the days in the month, it follows, that the day sign of the first day of the year being known, the day signs of the first day of all the successive months and, thus, all day signs of the year are equally known.
In order to identify the year bearer group for a Calendar Round date, the table below has been devised. This table is based on the characteristic 4) of the Calendar Round, that each month of the year starts with the year bearer.
In setting up the base for this table the impediment mentioned before turned up again: most of the dates started the solar year with 0 pop, as with the Classic Calendar, but some, originating in the region of the Rio Bec, the Chenes, and the Puuc styles of architecture, begin it with 1st pop. This dual arrangement, as will be explained later (see page 80), is based on numbering a day either as an elapsed day, thus, by a cardinal number, e. g. “0”, or by a current day, thus, by an ordinal number, i. e. “1st” for the same day.
Looking up the year bearer group for a Calendar Round date in the table above one follows the line of the coefficient of the month along to the intersection with the group patron of the sacred day sign, and from there up to the top, to the year bearer group of the Calendar Round.
Hence, only two parameters define the year bearer group of the Calendar Round: the coefficient of the month and the group patron of the sacred day sign.
As far as the application of the year bearers is concerned, David Stuart remarked (2005:3): “In Postclassic Yucatan the four year bearer days were the so-called “K'an set” (K'an, Muluk, Ix and Kawak) (see Tozzer 1941:135) ... . In the codices the year bearer days are thought to be shifted back one position to the “Ak'bal set” (Ak'bal, Lamat, Ben and Etz'nab), ... .”
And referring to a passage from Stela 18 at Naranjo (2005:1), which “includes a fascinating record of the data ”1 Ik' Seating of Pop,” corresponding to the Long Count 9.14.14.7.2” D. Stuart noted (2005:3): ”To posit that the Naranjo passage is an actual year bearer record necessitates proposing yet another shift to a new set of days (the “Ik' set”) that fell on the seating of Pop. This, I admit, would be a rash conclusion to draw from the Naranjo evidence alone.
Nevertheless, it is significant that among the Kiche, Mam, Ixil and Pokomchi Maya the true year bearers correspond (with obvious local variations in the names) to Ik', Manik, Eb, and Kaban (Tedlock 1982:92) – the very same system I propose may be in use at Naranjo and other lowland sites in Classic times.”
And he added (2005:4):“Turning again to the Classic period; we find more indications that Ik', Manik, Eb and Kaban were the year bearer days of that era.”
Returning to the page (2005:3) D. Stuart writes: ”In the four New Year pages (25-28) of the Dresden Codex ... we find that the “Ik' set” of days is repeated thirteen times at the upper left margin of each page, before the striding possum (... ) who carries a patron deity of the New Year to one of the four world directions. The event in each case seems to be tal-iiy, “he arrived” (Bricker 1986:110). Although the shifted “Ak'bal set” of traditional Yucatecan year-bearer days are given in the lower left margin of these same pages (again in a line of thirteen signs) that does not demonstrate that the Dresden conforms to the Postclassic system; if anything, the upper registers may offer evidence that the arrival of the new year fell on the seating of the first month. However, in the Paris Codex – clearly a later document than the Dresden – we find the Yucatecan system well represented in its own New Year pages.
Turning again to the Classic period, we find more indications that Ik', Manik, Eb and Kaban were the year-bearer days of that era”
Returning to the statement of T. Proskouriakoff and J. E. S. Thompson (see page 13) on the shifts in the position of the month a test with one pair of data reveals, that
“12 Kan 1 Pop” respectively “12 Kan 2 Pop”
indicate by the tables above, that the dates appertain to different year bearer groups:
2 Pop to the year bearer group “B”,
1 Pop to the year bearer group “C” and
1st Pop to the year bearer group “D”.
From this test, it seems, that instead of a shift in the position of the month rather a shift in the year bearer groups will meet the association requirements with respect to the shifts in the Maya Calendars.
The year bearer group is determined from the given data of the Calendar Round as described above;
the year bearer, however, who starts the first year of the Calendar Round, i. e. the primus inter pares of a particular group and, thus, the patron of the Calendar Round, is, in general, not known, except for the period, handed down in the literature of the time the Spaniards conquered Yucatán.
In order to overcome this locked-in situation the groups of year bearers shall become, for the time being, the group patrons of a Calendar Round, and thus, the base of the preliminary structure of the Yucatecan Calendar. By developing the structure further and by attaching to it the only known patron of the Calendar Rounds and stepping backwards from it, the structure of the calendar shall be established at the end.
For that period the Books of Chilam, e. g. of Tizimin (Antje Gunsenheimer 2002:75) as well as the Relación de las cosas de Yucatan written by Fray Diego de Landa (1566/1966:136), point out that the primus inter pares is k'an.
The above presented relationship of the sets of year bearers as Patrons of the Calendar Round require to recognize and establish an other characteristic of the Calendar Round:
5) Each Calendar Round is reigned by a distinct patron deity.
1.2.2 Numbers Used with the Maya Calendar Systems
Number Words and Number Signs of the Maya
The names of the Maya vigesimal number words are given for “1” to “10” in the first three lines below and for “11” to “19” in the fourth and fifth. The figures in brackets express the way the number words are composed (Karl Menninger 1970:60):
The vigesimal Maya number words include a numbering system to the Base 10, therefore numbers “1” to “11” are unique while “13” to “19” are composed of first the numbers “3” to “9” with the “10” directly attached.
For the number signs the Maya used two systems, the so-called head variant types and the so-called bar-and-dot types.
The signs of the head variant type for “1” to “12” are singular faces with the “10” outstanding as a skull, the head of the death god. The signs for “13” to “19” are composed with the upper parts of the heads for “3” to “9” and with the yaw bone, as the truncated sign for “10”, for the lower part.
The death god and the expression “is dead” in connection with numbers has been employed by people all over the world to indicate the end of a numerical order. Richard C. E. Long explains (1924: 2: 355): „It would appear that the ... death sign. ... (has) a meaning of “finished” ... An exact parallel can be found in several of the Melanesian languages of New Guinea for this use of “dead” in a numerical sense, e. g., in the Dobu language, which has a vigesimal numeral system. Here the expression for “five” is “hand is dead”, meaning that the count of the fingers is finished, and the expression for “twenty” is “man is dead”, meaning that the count of fingers and toes is finished.
In many other languages the idea is expressed by “man is finished”.
The other sign system of numbers the Maya used consists of bar-and-dots. Hereby the dot represents the value of “1” and the bar the value of “5”. All other numbers are composed by stacking the horizontal bars and lining up dots on top of the bars.
Thus, the highest number of the vigesimal base, the “19”, consists of three stacked bars and of four lined-up dots on top:
Number Words of Other Mesoamerican Nations
The bar-and-dot number signs correspond with the number words for most of the other nations in Mesoamerica. Therefor, it seems, they all had a common originator.
The correspondence is most evident for the number words of the Mixe, given hereunder for “1” to “10” in the first three lines and for “11” to “19” in lines four to seven (Alvin Schoenhals 2009: Tontontepec Mixe). Here again the figures in brackets express the way the number words are composed:
The number words for “1” through “10” are unique while all others are composites based on first the “10”, but for the numbers greater than “15” first the “10” and then the “5”, and finally the numbers smaller than “5”.
It seems that the bar-and-dot numeral system was developed from the oldest way of communicating numbers: the finger counting. The print of the finger tip in wet sand or clay marked the value “1” and the print of the fist marked the value “5”. Thus, this numeral system seems to be very old.
And here it is appropriate to add a reflection of Karl Menninger (1970:54):
“One would naturally be inclined to suppose, in all innocence, that the human mind, when it took the trouble to record its ideas and concepts, would have devised similar systems of writing words and numbers, “seven” and “7.” But this did not happen, neither in our western culture nor anywhere else in the world. The early system of writing numerals is everywhere the older of the two sisters.”
The Mixe are a nation distributed all over the area of the Isthmus of Tehuantepec. The perished nation of the Olmec lived once in this area. Whether the Mixe are descendants of the Olmec or whether they immigrated to the area and adopted the number systems from the Olmec has not been cleared yet.
Ordinal Number Words of the Maya
The Maya expressed ordinal numbers by prefixing the cardinal numbers with tu.
Floyd G. Lounsbury stated (1978:762): “Ordinal numbers are formed by preposing the third-person possessive pronoun to the cardinal number. The preposition contracts with the pronoun; tu ... is such a contraction ... of the preposition ti and the ordinal-forming pronoun u.”
And in another treatise he supplemented (1990:292): “u before a consonant, but u y-, or simply a prefixed y-, before vowels”.
The Different Ways of Counting Number Words greater than Twenty in Meso-america
As the base vigesimal number words are composed very differently by the Maya in comparison to the other nations in Mesoamerica, so are the number words greater than twenty.
Karl Menninger (1970:76) explains the difference with respect to the number words of the Maya: “... a very remarkable manner of counting, which once prevailed in two areas of the world, the Germanic north of Europe and the [region of the Maya] in ... Mexico; ... This method of counting expresses 24, for example, not in the usual manner as either “4 and 20” or “4 and 2 tens,” but as “4 from the 3rd ten”; “
He adds: “The common method of counting places the units (in this case 4) upon the next lower rank level (in this case on the 2nd ten): this may be called counting from the lower level, or undercounting. But the other method places the units in the interval of 20 ... 30, between the two rank levels, and thus within the third interval of tens: this is counting from the upper rank level, or overcounting, ...”
And he continues: “If we think of the numbers [969] shown along the line [of the figure above] as proceeding in order from 0 running left to right, we can readily see how the method of undercounting really does build up the number from below in descending order: 9 H 6 T 9 U, whereas overcounting begins with the unit 9 and then suspends these in ascending order from the next level above: 9 U in the 7th T of the 10th H. In so doing it reverses the succession of ranks. Undercounting sees the number 969 by itself, resting upon the number line; overcounting, on the other hand, regards 969 as a point within an expanse of numbers arranged one inside the other, the tens within the hundred interval. It indicates these intervals verbally by the order in which the number word is arranged.”
Overcounting, as executed in the old Germanic north of Europe, has remained with us in counting the centuries, hence, the 20th century started on 1st of January 1900 and ended on 31st of December 1999.
How number words in access of twenty were composed by the Maya in overcount is explained by F. G. Lounsbury (1978:762): “In Yucatec as it was spoken at the time of the Spanish conquest, ... , the predominant method was to name the intervening quantity and to place it in ordinal-numbered score or other power of twenty. Thus, for example:
forty-one was “one in the third score” (hun tu yox kal), and
379 was “nineteen in the nineteenth score”
(bolonhalun tu bolonlahun kal).”
Please note again:
All other nations in Mesoamerica constitute their number words as undercount.
Number Signs Equal to or Greater than Twenty
Number signs of the head variant type greater than “20” were not handed down by the Maya.
The numbers signs in access of “19” of the bar-and-dot system were written in positional notation. The positional notation requires a sign for ranks, which have no value. For it, the Maya used in their Codices an oval sign, however, with “a more pointed outline on its two ends” (D. Stuart 2012:9); see example below. This sign can be formed while attaching the finger tips of the thumb and the pointer; in some countries, e. g. Germany, that sign is still in use as “finger counting” to indicate “Nothing”.
The following example shows how each higher rank is stacked on top of the lower one:
The Principles of and Examples for “Systems of Measurements”
Our way of counting is based on the gradation by steps of 10 or as in the case throughout ancient Mesoamerica by steps of 20. But there is another base: the grouping of higher ranks for systems of measurements (K. Menninger 1970:41) in either ascending or descending order, depending on the way of counting, as exemplified hereunder:
the time of the clock is composed by undercounting of elapsed periods by the three bundles
The written timely announcement is given normally abbreviated, without the measurements, just as cardinal numbers separated by colon.
The date of the calendar, however, is composed by overcounting of current periods by the
as given in the olden times, or abbreviated by ordinal numbers, as written in most European countries
21.1. (in the year of the Lord) 1678
Hereby the dot is the indication for the ordinal number; it is not used to separate the numbers (please compare with the vigesimal numbers of elapsed time of the longtime count – see e. g. page 29).
1.2.3 The Calendars and Their Longtime Counts Exemplified in Their Notations for the Day Count, the Long Count and the Short Count
„Eine der charakteristischen Gemeinsamkeiten der mesoamerikanischen Kulturen – über alle sprachlichen und kulturellen Grenzen hinweg – ist das kalendarische System.“
Hanns J. Prem und Berthold Riese (1986:382)
The above citation reminds us, that although this treatise concerns mainly a study of the Maya Calendar Systems, we nevertheless should always keep in mind the close relationship with the other Mesoamerican calendars.
All calendar systems used by the Maya differ at first glance - by looking at the Calendar Round and at the longtime count - only in the notation of the longtime counts.
The longtime count of the Preclassic calendar was a linearly progressing day count in undercount and positional notation. The one of the Classic consisted of a count of cyclic measurements grouped in descending ranks, while the Yucatecan Calendar was developed as a count of a new cyclic measurement, the ajaw-Period, grouped in ascending ranks but with, in addition, a “dynamic” Calendar Round. Latter will be derived and demonstrated starting by the chapter “The Structure of the Calendar Round of the Maya Calendar and its Expansion Leading to the Yucatecan Calendar” (page 70).
The lunar calendar, on the other side, the very first calendar of the Maya, will be addressed as of page 38.
1.2.3.1 The Day Count of the Preclassic Calendar
The Oldest Calendar Dates Known Today
At the time the oldest calendar dates were carved by Maya onto monuments in the highlands of Guatemala, dates of a similar structure were marked on stone by non-Maya people at the Golf-Coast of Mexico and the adjacent Isthmus of Tehuantepec.
The three examples above were handed down from
Fig. 1.2.3.1 - at the left
Tak'alik Ab'aj, Guatemala, Stela 2 – Cut-out from the Fragment
the truncated day-count composes of the so-called Initial Series Introductory Glyph (ISIG) and the numbers 7. 6?.?.?.? or 7.11?.?.?.? or 7.16?. ?. ?. ?
(redrawn by the author after Miguel Orrego Corzo et al. 2001:790, 803)
Fig. 1.2.3.2 - in the middle
Tres Zapotes, Veracruz, Mexico, Stela C, Cut-out from the back-side
the ISIG is followed by the date 7.16.6.16.18 6 etz’nab
(the day sign name is given here as the one in Yucatec)
(redrawn by the author after J. Marcus 1976:52)
Fig. 1.2.3.3 - at the right
Chiapa de Corzo, Chiapas, Mexico, Stela 2 - Fragment
it has been suggested that the truncated date could read (7.16.)3.2.13 6 Acatl
(redrawn by the author after J. Marcus 1976: 51)
Please note:
The above bar-and-dot signs of the longtime count have been transcribed into Indian-Arabic numbers – as is the convention - by separating the vigesimal digits by a dot.
The reading direction depicts the notation of the ranks from the highest to the lowest as of the undercount numbers. The longtime counts of these calendars are considered as elapsed days.
The bar-and-dot signs of the tzolk'in day name are not written horizontally as with the Day Count but vertically prefixed to the day signs.
The longtime count dates: the first one is of Maya, the next two are of non-Maya origin. Converted accordingly to the Goodman-Martinez-Thompson (for short GMT)-correlation the dates reveal that the carvings originate from the first century B.C.
Joyce Marcus remarked upon the age of the calendars (1992:95): “The fact that many widely separated groups from northern Mexico to Honduras, all speaking different languages, had similar calendrical structures suggests that the calendars were of great antiquity. Both calendars [the secular of 365 days and the sacred of 260 days] go back to at least 400 B.C. in the Valley of Oaxaca, and both may already have been ancient at that time.”
David Stuart wrote (2011:173): “ We tend to view the [Day /] Long Count calendar as a hallmark of Maya civilization, but there's good evidence that the Maya borrowed the system from their early Mesoamerican neighbors. While ... the origins of the system remain obscure, the first examples of it come from the Isthmus of Tehuantepec, to the west of the Maya region, more or less where the Olmec thrived centuries before the Maya. Did the earlier Olmec invent it? We cannot say, but the geographic distribution of the earliest [Day /] Long Count date is suggestive.”
J. Marcus' and D. Stuart's statements as well as the notation by undercount numbers in positional notation point at an adoption of the bar-and-dot number sign system as well as the calendar system by the Maya.
Another date scholars tried to decipher for a long time, is from
El Baúl, District Escuintla, Guatemala, on Stela 1
It diverts from the other three notations since the ISIG is missing and the date begins with the tzolk'in name day 12 eb.
Walter Lehmann, at his time the stela was called – stone relief “Piedra Herrera”, was the first who reconstructed the date. He wrote with astonishment (1926:175), that the date of the „echten Mayainschrift ... [in] ungewöhnlicher Weise mit [dem] Tageszeichen-Datum 12 Eb beginnt“. And he continued: „die Form des Tageszeichens, obwohl in calculiformer, mayaartiger Umrahmung, erinnert ... eher an ein mexikanisches als an ein Maya-Zeichen, auf das vier kleinere Hieroglyphen folgen, anschließend 7.19.7.8.12“.
Michael D. Coe describes the engraving (1957:600): „The sculptured side shows a rather stiffly posed figure in left profile, ... To the left of the figure are two vertical columns which once contained glyphs. The column to the extreme left has a series of bar-and-dot numerals below a number 12, which itself is in association with a glyph in the form of a fleshless jaw. ... As on Stela C of Tres Zapotes, the numerical values for the cycles are without accompanying period glyphs.”
(The „period glyphs“ will be explained in chapter 1.2.3.3, page 43f)
Joyce Marcus wrote about the weathered inscription (1976:53): „The text ... opens with an apparent day sign with a superfix of 12. The two dots and two bars appear above a fleshless jawbone, which is recognizable as the day Eb in the Maya calendar; but the presentation of the day sign before the Initial Series or Long Count date is most unusual. Four small hieroglyphs immediately follow the day 12 Eb, and they appear to be paired. Below these tiny hieroglyphs, we have a Long Count date without the Initial Series Introducing Glyph.“
With reference to the Day Count she concluded, that the decipherment of M. D. Coe (7.19.15.7.12 12 Eb) and Tatiana Proskouriakoff (7.18.14.8.12 12 Eb) suit better than the one of W. Lehmann. She regretted, that, since the date of the solar calendar is not given, the correct Day Count can not be checked.
The structure of the notation seems to be derived from the build-up of the overcount number words of the Maya, which starts with rank (0) and attaches higher ranks, here recognized as the longtime count.
Michael D. Coe assessed in his revue “Cycle 7 Monuments in Middle America“ (1957:597-611) the level of knowledge about the early longtime counts of the above mentioned stelae and commented: “ ...certain essential features of the Maya time system, especially the Long Count, and the carving of dated stelae, were also found among neighboring peoples in Middle America, and the belief was early expressed, mainly by Mexican archeologists, that the Long Count was not a Maya invention. On the other side, most Mayanists, especially Morley (1946), maintained Maya priority in all matters relating to the Long Count. Among the latter, Thompson has consistently opposed acceptance of monuments which seem to bear Long Count dates prior to any recorded by the Classic Maya; ...“
M. D. Coe added (1957:598): “It has been recognized for some time, however, that outside the lowland Maya area proper there are several stone monuments with inscribed dates, which, if read in the Maya Long Count system, would considerably antedate the Leyden Plate itself [see page 46f]. These monuments are scattered through a region that extends from Veracruz to the highlands of Guatemala.“
And he listed the reasons of J. Eric S. Thompson, published in the years 1941 and 1943, for rejecting an early association of dates from outside the central Maya region, the Peten;
“(1) none have accompanying glyphs for the cycles, but are recorded only in bar-and-dot numerals, unlike Classic May inscriptions;
(2) all have been found outside the Classic area;
(3) the associated art style and iconography is late and often “Mexican”;
(4) even if we accept the dates as contemporary there is no assurance that the starting point for the Maya Long Count, 4 Ahau 8 Cumhu, was used;
(5) these monuments may have been erected by peoples who used a 400-day year.”
M. D. Coe continued: “This paper will present evidence that a rejection of these Dates is now untenable.”
And he stated further about the oldest calendar dates – the ones presented on the monuments above:
Tres Zapotes, Stela C
“The jaguar-monster mask is indubitably Olmec-La Venta [style], but as recognized by [Philip] Drucker (1952:205-9), there are some minor divergences from the florescence of Olmec art known from La Venta; ... Stylistically, ... , Stela C would be a product of the La Venta - Middle Tres Zapotes period of Olmec civilization. ... [hence,] then, Stela C is not a late “Mexican” product, but a late Formative monument ...”
El Baul, Stela 1
“Examining the stylistic associations of the monument, [T.] Proskouriakoff (1950:174-5) finds that they suggest “a fairly early date.” ... [and] ... closest stylistic resemblances are to the monuments of Izapa and San Isidro Piedra Parada (Colomba) [which is one of the haciendas now part of the archaeological site of Tak'alik Ab'aj]. ... There are at least 15 carved stelae at Izapa (...), all in a style which to Proskouriakoff (1950:177) is of particular interest because it “seems to form a link between the Maya style, the style of La Venta, and that of Monte Alban.”
The Baul stela, then, has styllistic associations which appear to be early.“
Tak'alik Ab'aj, Stele 2
“ ... below a voluted design like that on Stela 1, El Baul, two figures with elaborate headdresses face each other, separated by a vertical inscription. ... Stylistically, the stela is very early; the reasons given by Proskouriakoff (1950:176-7) for the early placement of Stela 2, ..., can be summarized as follows:
(1) The arrangement of two figures flanking a vertical column of glyphs recalls some of Kaminaljuyu sculptures and the altar at Polol.
(2) The best preserved figure shows a costume related to the earliest costumes of the Maya. The chain-like element hanging from the belt is characteristic of Cycle 8 (Thompson 1943:103 has pointed out this chain on the Leyden Plate).
(3) The ornamental detail of the costume is scroll-like, of early Maya type and related to the Baul monument.
(4) The relief is simple, with little gradation in modeling and no textural effects.”
M. D. Coe (1957:606) concluded: „We have examined the archeological and stylistic associations of four monuments which bear what may be Cycle 7 dates, and find that of these, three can best be assigned to the 7th Cycle of elapsed time since 4 Ahau 8 Cumhu in the Long Count: Stela C at Tres Zapotes, Stela 2 at Colomba, and Stela 1, El Baul. One of the four, Stela 2 at Piedra Labrada, almost certainly bears not a date, but the name of a Teotihuacan goddess, and therefore must be dismissed from consideration.“
He added (1957:607): “The supposition that a 400-day year, instead of the 360-day tun, was used by the carvers of these inscriptions rests on very slender evidence. Of all the peoples of Middle America, only the Quiche-Cakchiquel of the Guatemalan highlands are definitely known to have used a commemorative year of 400 days.“
At the end M. D. Coe set the focus at the stela cult which was initiated by M. W. Stirling (1943:73): „In the light of present evidence, the general fact appears to stand out that there was an early spread of stela cult extending from the southeastern Mexican coast across the Isthmus of Tehuantepec to the Pacific coast region of southern Mexico, and possibly into Guatemala.”
And he continued: „While Stirling (ibid) holds that “it is not necessary to postulate that a single people or linguistic group was involved,” it must be admitted that the direction of the flow through time and space seems to have been out from the Olmec-La Venta culture of the Gulf Coast, then across to the Pacific and south. Quite possibly the Olmec-La Venta people, who seem to have originated the Long Count and the stela “cult,” set up what was originally a migration route but later became the path for trait diffusion lasting as late as Early Classic times.“
In Tak'alik Ab'aj another monument was found with a similar appearance and style as Stela 2 , that is Stela 5 (Miguel Orrego Corzo y Christa Schieber de Lavarreda 2001:790, 803). It carries two Day Count dates 8.2.2.10.5 and 8.4.5.17.11, each with a ISIG but no date of the ritual nor the solar calendar.
There are three dates, later than the above ones, available from the state of Veracruz, Mexico, from a stela, which has two inscriptions, and another from a statuette:
La Mojarra, Stela 1
was saved from the Acula river in 1986 at a place half way between Cerro de las Mesas and Tres Zapotes.
The first date is inscribed in Glyphs A1-A9
The Introductory Glyphe (A1) is followed by the glyph of the patron of the seventeenth month (A2) with the coefficient 3 for the elapsed days attached at the right side (A3), the day count (A4) to (A8), and the day name (A9 +9a), that is “13 snake”; thus the date is:
day 3 of the seventeenth months 8.5.3.3.5 (in the Day Count) on the day 13 snake (with the Maya the 17. month was k'ayab, and the equivalent sacred sign for snake is chikchan.)
(Terrence Kaufman and John Justeson 2001: 2.34 and 2.71)
The structure of this date from the Isthmus reminds of the Maya date from El Baul, but for the missing ISIG.
The second date from La Mojarra, Stela 1 reads in Glyphs M8-M16:
It was the day 15 of the first month; the long count was 8.5.16.9.7 , and the day was 5 Deer.
(Terrence Kaufman and John Justeson 2001: 2.40 and 2.71)
Please note, that the day [5] is effaced, the 1. month is equivalent to pop in Maya, while Deer is the Maya sign of manik'.
The Calendar Round, the Year Bearer Group “B” and the Short Year
Both dates from Stela 1 of La Mojarra note, for the first time to this date, the elapsed days of the month and the day sign. These two elements are the first complete indications for the Calendar Round. And both elements allow to determine the Year Bearer Group as “B” in both cases.
The Tuxtla Statuette was found in 1902 on a field in the vicinity of Catemaco, District San Andres Tuxtla, Veracruz about 25 km away from the archaeological site of Tres Zapotes.
The date reads:
ISIG (the Day Count) 8.6.2.4.17 the day sign is not decipherable
Joyce Marcus commented on and reconstructed the date (1976:56): “In most respects, the Initial Series on the Tuxtla Statuette is quite similar to that recorded on Stela C at Tres Zapotes. The Introducing Glyph carries a trinary superfix, in this case three scrolls. The Introducing Glyph is followed by a column of horizontally placed bar and dot numerals. If we again assume position-value notation, we would reconstruct this date as follows:
If we assume that the above number of days elapsed is counted from the base 4 Ahau 8 Cumku, the Calendar Round to be reached is 8 Caban (0 Kankin). Looking at the last number in the column, we notice that there is indeed a vertical 8 as a coefficient to the day sign. Although the day sign does not look like Caban as written by the Maya, it should logically be Caban because that is the 17th day of the possible 20 day names; the month position and month are not given. The Initial Series date 8.6.2.4.17 8 Caban (0 Kankin) would correspond to A.D. 162 in our calendar [using the GMT- correlation].”
J. Marcus determined the Calendar Round data on two assumptions known to be true for the Classic Calendar (see page 43f), but have not yet been approved for the Preclassic Calendar:
the count of the numbers on rank (1) is from “0” to “17”, thus, the year becomes a short year of 360 days and not from “0” to “19”, what would give a year of 400 days,
the Calendar Round starts on 4 ajaw 8 kumk'u as does the Classic Calendar.
The first assumption will be checked hereunder with all the data from above, on the condition, that all obey to the same calendrical structure with respect to the redundancy of the Calendar Round as well as the Day Count, as it is the case but for the different base of counting, of the Classic Calendar.
Within the table the delta of the remaining integer is calculated by the difference in total days of the day counts divided by 13.
The outcome of the comparison, using the factors “400” or “360” for the days of rank (2), of the differences of the values from the number signs with a reference date (e. g. Tres Zapotes, Stela C) and of the differences of the total days from the Day Count results in the data of the above table.
The conclusion from these data is: The Day Count of the Preclassic Calendar written as position-value notation applies to the numbers “0” to “17” for the second rank. Thus, the first assumption is correct.
In addition, since the data from Stela 1 of La Mojarra result in the Year Bearer Group “B”, it is esteemed, that all data from the Gulf Coast and the Isthmus of Tehuantepec are of the Group “B”.
But the second assumption, that the Calendar Round starts on 4 ajaw 8 kumk'u, can not be verified alone by the conformity with regard to the Year Bearer Group “B”. The assumption can only be checked for continuity of the calendars with data of the Classic Calendar, for which the initial date, the Maya world creation day, had been handed down. Hence, the checking shall be carried out at the end of chapter 1.2.3.3 (page 56).
The Latest Inscription in the Day Count
The Pestác Stela
The latest inscription of a Pre-Classic Maya date, handed down up till now, is the one from a Stela found at Pestác.
Frans Blom reported (1935:190): “... in 1928, a visit was made to the ruins of Toniná in Chiapas, Mexico. ...
At a distance of 2 km due north of the ruins of Toniná, close by a ranch-house named, Pestác lay a monument, broken in two parts and somewhat weathered. It contains inscriptions on both sides and is unique in that its initial series date is given in bar and dot numerals, by position and not accompanied by period-value glyphs.
The Initial series is on the front side and reads: 9-11-12-9-0, probably followed by the month and day (1 Ahau 8 Cumku). The lower part of the monument is broken off and we did not locate this.
On the reverse are a series of glyphs containing secondary series and a supplementary calculation.”
J. E. S. Thompson stated with respect to the stela (1943:103), that the Initial Series Introductory Glyph “shows the variable element denoting the patron deity of the month.”
The Initial Series Introducing Glyph
As we have seen, already the oldest calendar dates known today begin the longtime count with an Initial Series Introductory Glyph (ISIG).
F. G. Lounsbury stated correspondingly (1978:810): “Two pieces (Stela C of Tres Zapotes and the Tuxtla Statuette) exhibit early forms of the “initial-series introducing glyph,” even neither of them has the chronological series that is so introduced in a position which is initial to the inscription. (It might thus better be called a “day-number introducing glyph.”) Both introducing glyphs have a tripartite superfix similar (one very much so, and one somewhat less so) to that which is a standard part of the glyph in Classic Maya inscriptions. They also have instances of the “variable element” in them, one definitely and the other possibly corresponding to the appropriate calendrical twenty-day “month.” Thus the introducing glyph, in at least two of its standard components, is pre-Maya. The inclusion of a variable element, especially the identifiable one in Stela C of Tres Zapotes, implies that the vigesimally based subdivision of the calendar year were already instituted, even though they are not named in separate year-day specifications as they are in Maya inscriptions.
The 260-day almanac and the 365-day year, as well as bar-and-dot numerals, have a still older history. Inscriptions from the Valley of Oaxaca give evidence of these as far back as about the middle of the first millennium B.C. ... These inscriptions give evidence of the 365-day year by naming years for their “year bearers,” the almanac days on which they begin. These, restricted to four days of the veintena, but combining with all positions in the trecena, name the 52 years of the calendar round and fix dates within this cycle. This is a practice for which there is no evidence in Classic Maya inscriptions, but for which there is good evidence in all three Maya codices (Dresden, Paris, and Madrid), as well as in Central Mexican sources, and which was very much in evidence in Yucatan and elsewhere at the time of conquest and until recently. It is an element of Mesoamerican calendrical practice which has had a history of approximately two and a half millennia.”
F. G. Lounsbury bridges a vast period, hurries on and anticipates the script. But chapter 1.2.3.3 will catch up with him.
Conclusions
The citations illustrate the long and controversial debate about the uncertainties as far as the age of the monuments and the origins of style are concerned. Today the antiquity of the artifacts is undoubted. The calendars shall thus be called the Preclassic Calendar System with a day count as the longtime measure.
And this calendar consisted finally of most of the elements we will find with the Classic Calendar.
The comparison of the notations from Tak'alik Ab'aj, Tres Zapotes and Chiapa de Corzo with the ones from El Baúl and La Mojarra demonstrates the reforms with respect to the number and the sequence of the calendar elements. And it indicates that the transfer of reform-ideas were not one-directional from the Gulf to the Pacific Coast but also vice versa.
The longtime count by cardinal numerals in positional notation in undercount is set up for all these calendars of Mesoamerica as a Day Count of elapsed time. But this Day Count contrasts with the longtime count of the Classic Calendar as we will see in chapter 1.2.3.3.
1.2.3.2 Notations of the Lunar Calendar and of the Secondary or Supple- mentary Series
The Lunar Notation During the Transition Period from the Preclassic to the Classic Calendar
The name for “month” in Yucatec is somehow puzzling. Juan Pío Pérez by studying ancient texts concluded (1846/2001:213):
“In the Yucatec language, the month was called u, which also meant “moon.” This corroborates the presumption that the Indians started out with the computation of lunations as a scale to fix the solar course, designating the months as moons; ...”
And Herbert J. Spinden commented (1924:9):
The Maya called the twenty-day unit or period by a name which is cognate with moon, thereby indicating that they regarded this twenty-day unit as a reformed month.”
Before the reform of the Classic Calendar took shape the Maya calendar scribes disengaged from the Preclassic Calendar notation as indicated by the following example:
Without Provenience, (Metropolitan Museum of Art, New York),
“Hauberg” Stela, (see next page)
vertical left side of monument
ISIG 12 k'ank'in (Glyph A2) Secondary Series (Glyphs A3/A5) 3 k'an (Glyph A6)
Linda Schele (1985:135 and 1990:1) expressed her view on the notation by: “The first six glyphs record the date, but the components are ordered in an unorthodox fashion, as follows: ISIG, haab, G5 (the Lord of the Night), unknown, 17C (seemed to record that seventeen lunations had ended), tzolkin”.
The transformation into the Long Count date had been derived by L. Schele with the support of Floyd Lounsbury, Peter Mathews, John Justeson, and David Stuart in two elaborated papers of 1985 and 1990. Based on the Calendar Round and on the secondary series data L. Schele, P. Mathews, and F. Lounsbury concluded as for the date (1990:4): “8.7.17.14.4 3 K'an 12 K'ank'in is the correct reading”.
In the earlier paper of the two L. Schele (1985:135) had exposed that the miniature stela was classified by Easby and Scott in 1970 as “Protoclassic, probably A. D. 100 – 200” and “tied iconographically to both Izapan art and Early Classic monuments at Tikal”, and by Greene, Rands, and Graham in 1972 as: “this unique miniature stela appears to date from a time before the characteristic Peten formalism [of the notation of the Classic Calendar] had crystallized”.
L. Schele added: “new findings in and understanding of the archaeology and iconography of the late Preclassic period have made it possible and important to place the Hauberg within the large context of the evolution of Maya civilization as an example of an intermediate stage of development, ...”
L. Schele referred also to a personal communication with D. Stuart, who in 1985 “has pointed out the existence of several Classic monuments, Quirigua Stela U, El Peru Stela 13, and Copan Stela 16, that have the same haab-tzolkin reversal of C[alendar]R[ound] dates. The use of the ISIG with a CR but without a L[ong]C[ount] notation is also known from a number of Classic monuments, including Tikal Stela 4 and the Copan Stela 16. The unorthodox order and the abbreviated format of the Hauberg date, therefore, has precedents on monuments of known provenience from the Late Preclassic and the Early Classic periods.”
The Lunar Notation in Later Calendars
While the calendar priests of the Maya emancipated from the longtime count of the Mesoamerican Calendar and emerged with the reform resulting in the Long Count (see page 43f) they included with the notation of the Calendar Round the so called Secondary, Supplementary or Lunar Series. These data are placed normally following the date of the tzolk'in and before the date of the haab.
The name Secondary or Supplementary Series – given by Charles P. Bowditch (1910:109) - goes back to a time when there was no perception about these data. The name Lunar Series came into use after John Edgar Teeple had made a first attempt to decipher the hieroglyphs in 1930.
Linda Schele, Nikolai Grube, and Federico Fahsen explained (1992:2): “Epigraphers have been studying the supplementary series for almost a century. Morley (1916) completed the first extensive study of the lunar series. He collected eighty examples, arranged them in a chart with each text placed in a horizontal row and all similar parts compared in vertical columns. Since Morley saw more regularity in the columns to the right of his charts than in those to the left, he assigned alphabetic designations (A-G) going from right to left on the charts. Because of his strange assumption, the alphabetic order of the supplementary series reverses the reading order of the glyphs. Glyphs G and F are read first, Glyph A last. He used X as the designation of a variable glyph that occurred between his Glyph C and D.
E. Wyllys Andrews IV (1938) added Glyph Z and Y to the series as designations for a pair of glyphs Morley had included in the original charts, but not labeled. Andrews retained the reversed alphabetic order so that Z is read before Y. ...
Because there are nine variables in the series Thompson postulated in 1929 that Glyphs F and G referred to the Nine Lords of the Night known from Aztec sources. Schele associated in 1991 Glyph F with the headband glyph and the name of the Jester God of royal headbands.”
L. Schele et al. remarked (1992:2): “It now seems fairly clear that Glyph F and G record a series of nine headdresses worn by the patrons of the particular day.”
Morley had suggested that Glyph E and D record information about the moon. Teeple confirmed Morley's suggestion by deducing from data of Palenque that Glyphs E and D record the age of the moon.
With respect to glyph C Linda Schele et al. wrote (1992:4): “ Glyph C consists of a number in either ordinal or cardinal form combined with a flat hand glyph that records the verbal action. One of several heads may appear above the hand along with a moon sign. Teeple (1930) first recognized that the number with C must refer to a series of six lunations, although he and other early researchers could find only the number 2 through 6. Schele (1978) identified T4, the sign that usually precedes the form that should be 1C, as phonetic na and “first.” Glyph C1 is then be written u na, “first,” in the series.”
At the end of the paragraph Schele et al. picked up again the counting of numbers in either ordinal or cardinal form by stating: “We also found evidence that some scribes used glyph C to record elapsed lunations, while others preferred to record the current lunations.”
Recording either cardinal or ordinal numbers, concerning the coefficients of the month in the solar calendar, will be addressed again in chapter 1.2.3.4 .
J. E. S. Thompson commented about the notation of the lunar age (1937:183): “It was a very common practice in Maya writing to combine the two lowest units of a Secondary Series by eliminating the Kin glyph and attaching its coefficient to the left of the Uinal glyph. .... Since Secondary Series are read in ascending order, in contrast to the descending order of the Initial Series, the coefficient to the left of the lowest digit represents the numerical value of the lowest unit.” (see also and compare with the Distance Numbers – page 52)
This numerology indicates – as we will see again with the distance numbers of the Classic calendar the fashion of how the Maya grouped their numbers of higher ranks in overcounting. Thus, these dates make evident, that the lunar series is a Maya invention.
The latest lunar date is handed down with the inscription from:
Chichen Itza, Yucatan, Mexico
Structure 5C4, Lintel n. n., Glyphs A1-B8
[ISIG sak] 10 pih 2 ? 9 tun 1 winik 9 k'in 9 muluk tu 7 sak (Secondary Series)
(Daniel Graña-Behrens 2002:404 – Analyse [204])
Year Bearer Group “B”
The Secondary Series expresses in Glyph B7 with the glyph “E” the actual number of moon days, in the overcounting fashion as (“5” “20”).
With the notation of the Yucatecan Calendar (see chapter 1.2.3.4 page 57f) the last date of the lunar series is handed down from:
Xcalumkin, Campeche, Mexico
Structure 5D30, Room C, Panel 2, Glyphs A1-A15
ISIG ? 9 bak'tun 15 k'atun 12 tun 6 winal 9 k'in (Secondary Series)
7 muluk 1 k'ank'in tu 13 tun 2 ajaw
(D. Graña-Behrens 2002:419 – Analyse [229])
Year Bearer Group “C”
The Secondary Series expresses in glyph “D” the age of the moon: it is two days.
From the last recording it seems that the priests of the Yucatecan calendar stopped to work with the lunar calendar while the priests of the Classic Calendar continued with it for at least about 140 years as is noted by the latest handed down engravings, that is the one from Chichen Itza.
The First Calendar of the Maya Was a Lunar One
The notation of the lunar series with the Classic Calendar astonishes. The astonishment seems to have been shared by by J. E. Teeple who remarked (1928/2001:242): “Apparently at this stage the Maya were undergoing the effort which almost all people have gone through: They once had a lunar calendar, later discovered a better approximate solar calendar, and were endeavoring by interpolation of whole moon months from time to time to keep the lunar in some sort of relation to the solar calendar. This is a familiar picture in history”.
And in deed, in ancient Egypt the solar calendar replaced the lunar one; in ancient Rome the Egyptian solar calendar was adopted and it replaced their lunar calendar; with many nations in Northern Europe the Christian calendar, based on the Roman solar calender, replaced the moon calendar during the time of Christianization. Today the lunar calendar prevails still as Hebrew and as Muslim calendar. And in the Far East the Asian peoples still celebrate their New Year accordingly the lunar calendar.
Following J. E. Teeple's remark it is assumed that the lunar calendar antedated the solar calendar. The insertion of a newly developed lunar calendar into the Maya Classic solar calendar would have been an anachronism.
1.2.3.3 The Long Count of the Classic Calendar
The Development of the Longtime Notations of the Classic Calendar
The development of the notations for this calendar concerns the longtime count in two fashions:
the Long Count in undercounting and the Distance Numbers in overcounting.
The Grouping and Bundling of Descending Ranks Created the Long Count
The reform which resulted in the Classic Calendar modified the longtime count of the Preclassic Calender, i. e. the linearly progressing day count assigned as positional undercount numbers, in such a way as to change it into a system of measurement, groupings and bundles of descending rank, in order to overcome the problem of reading the day count number signs in undercounting with the Maya overcounting number words on one side and to suit the Maya concept of time as a cyclic process on the other.
That the Maya regarded these measures as bundles will be demonstrated later in this chapter.
The notation for the new longtime count is composed first, as with the Preclassic Calendar, of the so-called Initial Series Introductory Glyph (ISIG), then of five so-called Period Glyphs with coefficients. The ISIG and the individual periods were named by scholars the Long Count. The system of measurement, the grouping of descending rank, consists of the measurements:
There are two different kinds of “standardized” Period Glyphs, the geometric and the head variant ones. The following ones are taken from Charles P. Bowditch (1910, Plates and Periods as indicated):
Fig. 1.2.3.7 Standardized Head Variant and Geometric Period Glyphs
Examples of the Period Glyphs of the “head-variant” type are shown below with Stela 29 of Tikal and with the Plaque of Leiden, the geometric form with the Lintel 11 of Oxkintok (see page 49) and with the tablet from the Temple of the Cross, Palenque, Chiapas, Mexico (see page 50), here, by the five glyphs following the ISIG in the upper left corner on the right hand side of the double row.
All periods are counted as elapsed time from “0” to “19” except for the winal which is counted from “0” to “17”.
Please note:
In order to transcribe the dates of Base 20 into Indian/Arabic numbers of Base “10” the elapsed Long Count is abbreviated and consists only of the coefficients, separated by a point, a period mark. The point of this notation shall NOT be interpreted as an indicator for an ordinal number but as a partition sign as, by contrast, the colon with the elapsed time count of the clock (see page 26).
An example for such a transcribed and abbreviated date is given below with the up till now oldest date in the Long Count, found by archeological diggings in the lowlands (Nikolai Grube 2000:160).
Tikal, Petén, Guatemala, North-Acropolis, Stela 29, Rear Side
ISIG 8 bak'tun 12 k'atun 14 tun 8 uinal 15 k'in as well as one dot of the numerical coefficient of the month
abbreviated as ISIG 8.12.14. 8.15
As to the notation J. Marcus remarked (1976:59): „For the first time, we have the bar and dot system of numeration set up vertically in order to serve as prefixes to the period glyphs. And she added: “ ... [the] specified period glyphs are of the “head-variant” type.“
L. Satterthwaite (1960:37) noted that the period glyphs for the bak'tun, k'atun, and tun are glyphs of bird heads; the uinal glyph is the head of a frog; the period glyph for the kin is largely missing, but by analogy with later monuments it should be the head of the “sun god”.
The number signs are used in the same way, as we have seen already for the day name of the tzolk'in with the Preclassic Calendar, however, they contrast with the positional notation of the horizontal numbers of the Day Count.
The earliest complete date is handed down on the Plaque of Leiden:
Plaque of Leiden, Rear Side - found at Puerto Barrios, Izabal, Guatemala
ISIG (yaxk'in) 8 bak'tun 14 pih (“bundle” for k'atun) 3 tun 1 winal 12 k'in
1 eb 0 yaxk'in
(Ernst Förstemann 1903:553; Analysis by L. Schele et al. - see below)
abbreviated as ISIG 8.14. 3. 1.12 1 eb 0 yaxk'in
Fig. 1.2.3.9 Plaque of Leiden, Rear Side (drawing by Ernst Förstemann 1903:553)
The Long Count starts in line (1) with the Initial Series Introductory Glyph which carries in a variable field the month sign indicator or the sign of the patron of the month, here yaxk'in.
With the line (2) the count of the period glyphs with their coefficients begins, whereby the k'atun glyph in line (3) is substituted by the glyph which reads phonetically as pih, the “bundle” (for details see analysis below).
The tzolk'in date follows in line (7).
The left side of line (8) records the Secondary Series, here “5 Lord of Night”.
The right side of line (8) and the left side of line (9) show – for the first time up till now (André Cauty and Jean-Michel Hoppan 2006, Vol. 2:25) – the phrase: “cumlahi yaxk'in - seated yaxk'in” - transposed – as it is commonly done in the literature - prosaically to “0” yaxk'in.
Please note that cumlahi translates in the first place as the phrase: to enthrone a Lord.
And this phrase was still in use according to the Codex Pérez at the time of the Spanish conquest.
There it reads with respect to the date of the day Ah Pula Tutul Xiu died (page 60):
“canil kaan cumlahi pop” - four k'an seated pop .
Cyclic Processes
With the Plaque of Leiden the scribes not only demonstrated the cyclic process of the individual periods of the longtime count but also with the notation of the date itself. Since at the beginning the variable sign of the ISIG carries the sign of the patron of the month yaxk'in and the notation ends with that same date of the solar month, the cycle of the month is closed.
The patronages of the cycles of the eighteen months, standing out of the vigesimal system, seem to have been of greater concern to the calendar priests. As seen above they described the first day of the month by the idiomatic expression “cum” - the seating of the patron - but they also defined the last day of the month by the collocation of the tun sign – (J. E. S Thompson 1960:120) “used as a prefix or as a prefatory
glyph” - with the “winal” sign as it is known from the southern lowland. J. E. S Thompson presented such combinations (1960: Fig. 19 – Ends of Month - 21. to 27.) from Palenque, Chiapas, Mexico - Piedras Negras, Peten, Guatemala – Yaxchilan, Chiapas, Mexico and Naranjo, Peten, Guatemala.
J. E. S Thompson pointed out (1960:121): “There is linguistic confirmation for reading the tun sign in these cases as “end of” or “last of”.” And he refers to R. L. Roys, who had called his attention “to a very similar use of the word tun in Yucatec” by supplying quotations from the dictionaries. Roys concluded, “that the most frequent meaning of tun in the texts seems to be “then, after that,” but it can mean “finally”. The word seems to be used frequently with expressions of finality to give added emphasis.”
Thus, the concern seems to have been that the patronage of the month not only had a defined beginning but also a defined ending, and hence, the relay from one month to the next took place in an organized way.
The phrase ti/k'a tun is transposed nowadays in the literature prosaically to the day 19 or – as we will see on page 80 – to the 20th of the month.
The “Bundle”
As far as the bak'tun glyphs are concerned Linda Schele and Nikolai Grube pointed out (1993:1): “The geometric form (...) consists of a double “kawak” sign often appearing with a suffix (Justeson and Fox 1984) read as hi. The head variant is a bird head with a hand substituting for its lower jaw.
David Stuart (1987:11-13) deciphered both of these glyphs as phonetic pi. With the hi sign attached, the “bak'tun” glyphs read phonetically as pih.”
L. Schele and N. Grube explained further: “We have known for many years that the glyphs in the “bak'tun” context do not record a word for “four-hundred tuns.” The mini-conference team (i. e. Schele, Mathews, Lounsbury, and Kelley) became aware of this when they realized that the “bak'tun” context do not record a word for “four-hundred tuns” ... when they realized that the “bak'tun” and “k'atun” heads are interchanged on the Leiden Plaque (...), and that the double “kawak” pi combination occurs on Tikal Stela 31 (...) in the position of a cycle that must refer to a k'atun, rather than a bak'tun.”
By analyzing various pictures which included the pi sign the mini-conference group, and thereafter L. Schele and N. Grube came to the conclusion: “that “bundle” fits each of these contexts, and that cycles above twenty tuns were known by the generic term of “bundle.”
As for the double cauac sign (J. E. S Thompson 1960:274) consisting of the “quarter-circle (as in early examples) surrounded with circlets” one example is handed down as Glyph A2 from
Oxkintok, Yucatán, Mexico
Structure 3C3, Lintel 11, Glyphs A1-B4
ISIG [month patron] pax? 9 pih 2 [ k’atun] 11 tun 16 winal 17 k’in 11 kaban
reconstructed: 9.2.11.16.17 11kaban 15 pax
(D. Graña-Behrens 2002:411- Analyse Nr. [216])
Year Bearer Group “B”
Somewhat astounding with this notation is that the coefficients of the period glyphs are given not as prefixes but as superfixes.
Above examples corroborate that the protagonists of the Classical Calendar reform introduced the grouping of measurements and the expansion by bundling the tunob, for the cycles greater than 20, in order to replace the ranks of the Day Count for a better readability of their overcount number words.
Fig. 1.2.3.13 (see previous page)
Palenque, Chiapas, Mexico
drawing by F. Catherwoods – Fig 13 Tablet off the Back Wall of Altar, Casa No. 2, today called Temple of the Cross,
here only the left half of the Left Panel is shown
in: John L. Stephens (1841/1843/1980: 154 – drawing number 59)
(copy of drawing courtesy of MAIRDUMONT GmbH & Co. KG)
The engraving, sketched by F. Catherwoods in 1840, from the Temple of the Cross at the site of Palenque and similar drawings from various sites, all printed in John L. Stephens' Volume One of his “Expedition to Central America, Chiapas, and Yucatan”, were of such an excellent quality that they were utilized to decipher the head variant type numbers, as the ones on the upper left side in column 1 and lines 3 to 6 as well as 8 and 9, about fifty years after they had been published.
Although the period glyphs had been identified quite early the expression, the term, used for bak'tun, the combination of the Yucatec number word bak for “four-hundred” with the word for “(short-)year”, tun, is a more modern invention (J. Eric S. Thompson 1960:147) – (compare with the explanation on page 48).
The Initial Series Introductory Glyph
The five pairs of glyphs with a numeral in the first place, which stand mostly at the begin of a text, like the ones from the Panel of Palenque, were named by Alfred P. Maudslay in 1897 the “Initial Series”, abbreviated as “IS”. The glyph proceeding these five pairs, standing at the head of the inscription, was then named the “Initial Series Introductory Glyph”.
J. E. S. Thompson described this glyph (1960:153): “The constant elements of the IS introductory glyph are the tun sign, the upper prefix, and the pair of lateral elements of the lower prefix. These last are formed of the so-called comb symbol, which in a few inscriptions is replaced by pairs of fishes. It has been suggested (Thompson, 1944) that since the fish and comb signs represent the word “count”, the whole glyph, less the variable element and the first prefix, has the meaning of “the count of tuns” (u xocan [or xocol] tunob], ...”
and furthermore: “The variable element at the center of the prefix is the glyph of the deity who rules or is closely associated with the month in which the IS falls.”
The purpose of the variable element was recognized by Hermann Beyer in 1931. He, thus, called it the “month-sign indicator”.
In 1943 J. E. S. Thompson published a paper about excavations at the coffee farms of Santa Margarita and San Isidro Piedra Parada – now part of the archeological site of Tak'alik Ab'aj.
Among the findings was Stela 2 (briefly described in chapter 1.2.3.1 see page 28).
J. E. S. Thompson stated about the inscribed date (1943:102): “The inscription opens with a glyph , ... , which is pretty clearly the Initial Series Introductory Glyph of Maya inscriptions save that it lacks the variable element denoting the patron of the month of the Initial Series, and its flanking comb-like elements.”
J. E. S. Thompson's remarks leads to the conclusion, that even the early Initial Series Introductory Glyph is an indicator for the following upon longtime count.
And as far as the Maya number systems are concerned, the ISIG of the Preclassic Calendar reveals in addition, that the following date is undercount, thus starting with the highest rank and stepping down to the lowest.
That same indication is also required for the Long Count of the Classical Calendar, especially in cases where the glyphs for pih – bundle – are used for the two highest ranks, which thus, require clamping and defining the ranks:
The Grouping and Bundling of Ascending Ranks Created the Distance Numbers
In long inscriptions on monuments, as on the Tablet from the Temple of the Cross, Palenque, additional longtime dates were written as so-called Distance Numbers.
These are build up – in contrast to the Long Count dates - in overcounting. Hence, the number starts with the rank (0) and continues up to the rank (4).
As shown with the example from Palenque, Chiapas, Mexico, Temple 18 Jambs, the Distance Number Introductory Glyph, abridged as DNIG, is followed by a composite glyph with the coefficient for the k'in and the winal date but without the k'in sign.
J. Eric S. Thompson addressed this abbreviation (1960:159): “In the majority of distance numbers the kin glyph is suppressed, and its coefficient is attached to the uinal glyph. With few exceptions the kin coefficient is to the left of the uinal sign; that of the uinal itself is placed above the glyph. ... In those cases in which the kin sign is not suppressed, special glyphs were used, and still other forms were utilized to express distance numbers of less than 20 days.“
Special “Number” Signs Used with the Long Count
Two Different Number Signs of the Head Variant Type for “13”
The assumption the Maya adopted the Mesoamerican Preclassic Calender exposes for the ritual calender, it seems, the reason why the Maya had two different number signs of the head variant type for “13”.
One sign, the composite one corresponds in its structure with the Maya number word of oxlahun (thirteen), thus, consists of the upper part of the face sign for ox and of the lower part of the sign for lahun (ten), the skull, that is the lower yaw bone (see also page 22).
The manor of formation of the number word and of the composite number sign is correspondent, and it seems, that this number sign is very old.
The other sign is a singular one. It seems, that it was created in order to be added to the Mayan original sequence of the twelve singular number signs of the head variant type and, thus, comply with the thirteen sacred numbers.
Evidence for the adoption of the ritual calender by the Maya envisaged Nicholas P. Dunning by stating (1992:177): “The day names are of ancient origin, and a few have little obvious semantic content in Classical Yucatecan Maya language.”
The No-Value Signs
The oval no-value sign of the bar-and-dot number system was in continuous use with calendrical calculations in the Codices up to about the time of the Spanish conquest. But with the inscriptions at monuments a variety of no-value signs was designed by the protagonists of the Classic Calendar reform.
The three-pedaled flower was described by Charles P. Bowditch (1910:96) as a flaring sign; it resembles very much the darling flower of the Maya – the Plumeria.
D. Stuart advised (2012/06:7): “All of these (signs) can be phonetically read as the syllable mi (...) or perhaps as the logogram MIH. As word signs these would correspond to the root mih and its cognates, widespread in Mayan Languages with the meaning of “nothing” (...).”
With regard to the expression “Nothing” please recall that the oval sign with the pointed outline on its two ends might have been taken from the “finger counting” and with that particular meaning originating from the inventors of the positional notation (see page 25).
Anna Blume pointed out (2011:58), that the first three-pedaled flowers were used as zero place holders with the Long Count of the Stelae 18 and 19 of Uaxactun, Peten, Guatemala, dating back to 8.16. 0. 0. 0 .
Other examples of the “nothing” sign, shown on previous figures:
are given with the Tablet from the Temple of the Cross, Palenque, in column 1 and line 7, represented by the “shell-hand”, and
with the distance number from the Temple 18 Jambs, Palenque, as the Glyph D8 left side, representing the three-pedaled flower.
The Maya World Creation Day, the Base for the Calendar Round and the Long Count
With reference to the base of the Long Count David Stuart wrote (2011:216): “If we were to choose one principal “Creation” date from Classic Maya mythology, it would be the Long Count 13.0.0.0.0 4 Ahaw 8 Kumk'u, corresponding to August 11, 3114 BC [accordingly the GMT-correlation]. Now, it may not be completely accurate to say that this was a true Creation date, because ancient religious texts tell of events and episodes that took place long before this date – sometimes millions of years before, in fact. But it is fair to say that the ancient Maya must have considered 4 Ahaw 8 Kumk'u to be the beginning date of our current era, when, according to some sources, the gods of the cosmos “were set in order.” With that, all else was possible.
This 4 Ahaw 8 Kumk'u date is explained in considerable detail in the inscriptions of a tall monument erected at the ruins of Quiriguá, Guatemala, known to archaeologists as Stela C. ... “
Finally it shall be noted:
that the Calendar Round date 4 ajaw 8 kumk'u is part of the Year Bearer Group “B”, and that all dates within the Classic Calendar are given in that same group, the year bearer set of ik’, manik’, eb, kaban.
The Classic Calendar's Redundancy by the Calendar Round and the Long Count
There is one conformity concerning both the Calender Round and the Long Count; from their initial day on the Calendar Round and the Long Count progress linearly in steps of one elapsed day and thus are redundant. Harvey M. Bricker and Victoria R. Bricker (2011:72) point at the practical side: “The redundancy inherent in this notation has been exploited by epigraphers for interpreting inscriptions in which part of the long count or the calendar round has been damaged or effaced. If part of the long count cannot be read, the calendar round (if it is intact) can be used for restoring the missing value and vice versa.”
The Continuity of the Day Count of the Preclassic and the Long Count of the Classic Calendar
For the transition of the Preclassic to the Classic Calendar there are no data available which would bridge directly that period. The data from the Gulf Coast and the Isthmus of Tehuantepec indicated (see page 35), that they belong to calendars of the same structure, i. e. the base for the Calendar Round is the Year Bearer Group “B” and the base for the longtime counts is the day count. Since these characteristics apply also for the Classic Calendar a check for continuity and compatibility will be performed by comparing a set of data from the Preclassic with one of the Classic Calendar, assuming the second rank counts only from “0” up to “17”.
The differences of the remaining integer from the longtime counts – calculated from the total number of days divided by 13 as the integer rest - and the difference from the number signs – calculated as (1 minus 6 plus 13) are both equally 8, thus verify, that both calendar systems have the same base point to start from and count the second rank only from “0” to “17”.
Hence, the conclusion is: Both calendar systems, the Mesoamerican one of the Gulf Coast and of the Isthmus of Tehuantepec as well as the Maya Preclassic and the Classic one are continuous and compatible.
1.2.3.4 The Notation Accordingly the Yucatecan Method or: The Short Count
An Other Longtime Count in Progress based on the ajaw-Period
The reform for the Yucatecan Calendar was conducted, as will be demonstrated by the applicable data from D. Graña-Behrens' collection of verified data, only in the northwestern part of the Yucatán peninsula, to be exact, in the region of the Rio Bec, Chenes and Puuc styles of architecture. It resulted in a “dynamic” Calendar Round, i. e. of which the dates appertain not only to the set of the Year Bearer Group “B” but also to “others”. This “dynamic” Calendar Round will be discussed in more detail in chapter 1.3 (page 70f). Here, at first, the longtime count of the Yucatecan calendar reform will be looked at.
The notation of the oldest dates selected from D. Graña-Behrens' collection indicate that they consist of the so-called Long Count and the Calendar Round as with the Classic Calendar. This notation was developed little by little till it was finally composed of first the day name of the tzolk'in followed, as an apposition, by the day and the month of the haab as well as the Short Count, a longtime count consisting of the bundle of tun enclosed in the bundle of a new longtime cycle of 20 tun, the ajaw-Period.
But the new count seems to have been developed by starting with the day name, from singular ajaw-glyphs and the appropriate coefficient attached.
The oldest singular ajaw-glyph date from the Lowland is engraved at
Loltun, Yucatán, Mexico, “Hunacab” mouth of the Cave
3 ahaw tsuts? – „end?“
(D. Graña-Behrens 2002:308 Analyse Nr.[ 19] )
To associate such a day name with a Long Count date is difficult. Thus, it is not astounding that T. Proskouriakoff (1950:154f,190) estimated a time around 8.14.0.0.0 based on a stylistic analysis of photos which did not allow to read the date properly while Nikolai Grube and Linda Schele (1994:2) considered that the date is appropriate as early as 8.3.0.0.0 and even as 7.10.0.0.0.
Later in the development the notation of the singular ajaw-glyph was replaced by a ajaw-sign and a specification to indicate the half term of a period with a coefficient in overcounting. While the singular ajaw-glyph with the coefficient has to be considered as a current day, the ajaw-sign together with the half-term specification emerges as a current ahaw-period.
Etzna, Campeche, Mexico,
„Pequeña Acrópolis“, Stela 1 (Fragment), Glyphs A1-C6
[ISIG mak] 9 bak'tun 14 [ k’atun] 10 tun 0 winal 0 k’in [secondary series]
5 ahaw? 2 mak tu tan lam (at “the half of”) 4 ahaw
(D. Graña-Behrens, 2002:421 Analyse Nr. [231])
This date and the date below demonstrate the cautious, initial steps during the development, which included both longtime notations as well as the secondary series:
Xcalumkin, Campeche, Mexico, Structure 5D30, Room C, Wall Panel 2, Glyphs A1-A15
[ISIG] ? 9 pih ? k’atun 12 tun 6 winal 9 k’in 7 muluk [secondary series]
1 k’ank’in tu 13 tun 2 ahaw
with the reconstructed Long Count: 9.15.12.6.9 7 muluk 1 k’ank’in
(D. Graña-Behrens 2002:419 - Analyse Nr. [229] )
Year Bearer Group „C“
Simultaneously another notation, based also on the ajaw-Period, was developed.
The first day of an ajaw-Period in overcounting, and thus each tun within, is the day ajaw (see chapter 1.2.5).
In order to address the first day of a tun within an ajaw-Period the new notation specifies only the tun and the ajaw-Period, each with the coefficient as a cardinal number and not yet – as we will see later (page 94) by a prefixed coefficient with tu, as to make it an ordinal number.
Hobomo, Campeche, Mexico
Fragment 3, Glyph pA1-pA2
13 tun 12 ahaw
two conversions are given 9.10.13.0.0 or 10.3.13.0.0
(D. Graña-Behrens 2002: 341, Analyse Nr. [83])
Corrected accordingly to overcounting: 9.10.12.0.0 or 10.3.12.0.0
In order to complete this reform the notation was set up in such a way as to specify the day name of the tzolk'in, followed by an apposition consisting of the opening phrase k'in and the date of the current day of the haab, the current tun within the current ajaw-Period.
The dates of the Preclassic Calendar were written as a single row from top to bottom, the one of the Classic Calendar and continued as longer inscriptions as double rows from left to right, and the ones of the Yucatecan Calendar – as we will see hereunder – also as single lines from left to right.
The earliest example of this type of notation this far comes from:
Sisila, Campeche, Mexico, Structure 35, Inner Room, Caption of Front Gate
Glyphs:
two conversions are indicated 9.16.4.10.18 or 10.9.8.3.18
(D. Graña-Behrens 2002:365 - Analyse Nr. [132])
Year Bearer Group C
The numbers of “10” (Glyph 14) and of the weathered “5” (Glyph 19) are given as head-variant type numbers.
This date shows for the first time the solar calendar date as apposition and the coefficient of the month as an ordinal number.
It does not surprise that the structure of the apposition, Glyphs 12 to 15 and 19 to 22, corresponds with the way the Maya number words greater than twenty were structure in overcounting, i. e. by starting with the lowest rank, followed by the others in ascending order.
This new longtime count was named by J. E. S. Thompson (1937) “the notation accordingly the Yucatecan method”. It represents the so-called Short Count.
This notation was still in use in Northwestern Yucatán at the time of conquest by the Spaniards, as it is handed down in the Codex Pérez as well as in the Chilam Balam of Tizimin and Chumayel with the date Ah Pula Tutul Xiu died.
The Codex Pérez reads
(translation to German by Antje Gunsenheimer (2002:68) and respectively to English by John L. Stephens (in J. P. Pérez 1843/1990:341)):
oxlahun ahau cimci Ah Pula
The Codex Pérez has the year in the Christian Calendar added:
The Length of the Period of the Short Count
The new longtime notation fixes dates only within a span of 13 by 20 or 260 tun. Therefore, other specifications, i. e. archaeological have to be consulted, or as with the narratives in the books of Chilam Balam structured tables have to be taken into account in order to sort the chronology, or - as we will find out in chapter 1.3.12 – that the “dynamic” Calendar Round will fix the chronology.
1.2.4 Continuity and Compatibility of the Maya Calendar Systems
The two reforms by the Maya of the Mesoamerican respectively of the Preclassic Calendar System resulted in two new longtime counts, hence, two transitions. However, modifications to the Calendar Round have not been considered so far.
For the first transition from the Preclassic to the Classic Calendar the check with one date of each calendar, the date from Tres Zapotes, Stela C and the date from the Plaque of Leiden, has proven that both calenders are continuous and compatible.
For the second transition from the Classic Calendar to the Yucatecan Calendar the date from Xcalumkin of the Wall Panel 2, Room C (see the following chapter) proves by itself as being continuous and compatible since it is given in both notations, the one of the Classic and the one of the Yucatec Calendar.
1.2.5 The Distinction Between k'atun- and ajaw-Periods
The reform by which the Yucatecan Calendar System was engineered provided a smooth transition from the Long Count to the short Count as indicated by the date:
Xcalumkin, Structure 5D30, Room C, Wall Panel 2
[ISIG] ? 9 pih ? 15 k’atun 12 tun 6 winal 9 k’in 7 muluk [secondary series]
1 k’ank’in tu 13 tun 2 ahaw
The date begins with the notation of the longtime count of elapsed periods of the Classic Calender and continues with the notation of the current periods of the Yucatecan Calendar. Hereby the tzolk'in date is shared by both counts. Both notations describe this day imbedded in the longtime counts whereby the count of the Classic Calendar is carried out in undercounting, evidenced by the descending order and the term “12 tun” of elapsed time, while the one of the Yucatecan is in overcounting as is evidenced by the ascending order and the term of “the 13th tun” of current time.
The following example from the Books of Chilam Balam concerning the above distinction of elapsed and current time irritates due to the interpretation of the period count:
Ralph L. Roys' translated from the third chronicle of the Chilam Balam of Chumayel (1933:142): “11 Ahau. On the first day the stone was taken at Colox-peten.“
And he added the foot-note: “This taking of the stone evidently refers to the Maya custom of setting up a monument every 7200 days to commemorate the katun that has just passed.“
Obviously Roys considered the “katun” as a period ending count, a count of elapsed periods, as with the Long Count. In that case the first day of that katun is 12 imix.
But since the predominant day of the calender was the world creation day 4 ajaw 8 kumk'u, the day to commemorate the world creation day should fall on a day ajaw. This becomes true by considering the ajaw-Period as a current period in overcounting, because then, the celebration for the first day of the 11th ajaw-Period falls on the day of the 13th ahaw.
The irritation vanishes while examining the chronology on how the knowledge about the k'atun count was accumulated.
The literature records the expression k'atun, for the first time with Diego de Landa's Relación de las cosas de Yucatán, translated by Alfred M. Tozzer (1566/1966:166): “Not only do the Indians keep track of the year and the month, as has been said and pointed out above, but they had a certain way of counting the periods of time and their affairs by ages, which they did by periods of twenty years (877)*, counting thirteen twenties by means of one of the twenty letters of the months called Ahau, not in regular order but inverted, as ... [can] be seen in the ... circular diagram [below] (878)*. They call these katuns in their language, and by them they kept the account of their ages marvelously well.”
)* for Tozzer's footnotes see the following pages, after two other citations.
de Landa's wheel shows a peculiar sequence of counting the ajaw-Periods: it is not linearly progressing but starts with the coefficient 13 and then steps down by 2 to 11, 9, 7, 5, 3, 1 and continues with 12, 10, 8, 6, 4, 2.
Juan Pio Pérez (1843/1990:299f) was next to report about periods starting from the day ahaw: “Besides the cycle of 52 years, ... , there was another great cycle peculiar to the Yucatecos, who referred to its periods for dating their principal epochs and the most notable events of their history. It contained 13 periods of 24 years each, making together 312 years. ... It is incontrovertible that those periods, epochs, or ages, took the name of Ajau Katun, because they began to be counted from the day Ajau, ... ; but as these days and numbers were taken from years which had run their course, the periods of 24 years could never have an arithmetical order, but succeeded each other to the numbers 13, 11, 9, 7, 5, 3, 1, 12, 10, 8, 6, 4, 2.”
Philipp J. J. Valentini (1880:19) pointed at the different timely length of the Ajau Katun with de Landa and Pérez: “ ... , the Mayas counted a great epoch of 260 years, the so called Ahau Katun, subdivided into 13 smaller periods each of 20 years, with the simple name Ahau.”
In a foot-note Valentini remarked about the sources of his knowledge: “The first (Cogolludo) gives Katun the meaning of a period of twenty years.” And about the second (Landa) he refers as: “ ... his intention was to state that each of the images of the thirteen Ahaues, depicted on the surface of the wheel, represented twenty years, this being a period which they also called Katunes.”
Having discussed Juan Pio Pérez statements Valentini concluded: “ ... Señor Perez intended to establish the fact that the ancient Maya cycles were composed of 24 and 312 years respectively. He does so in manifest contradiction to the prevalent opinion that they consisted of 20 and 260 years.”
And he pointed at another – in his mind – contradiction with regard to the above mentioned authorities, Cogolludo and de Landa (1880:23): “ ... Señor Perez arrives at the division into great epochs of 52 years used in Mexico as well as in Yucatan. This statement appears hazardous in the highest degree when compared with the statements made by the before-mentioned authorities. They claim for Yucatan an epoch of 20 and 260 years respectively; and Landa, who wrote with the first impression of the conquest still fresh in his mind, and whose information came directly from the natives themselves, agrees with them. ...”
After further discussions, however, he summed up (1880:26): “Nevertheless, the data which we possess of the ancient Maya Calendar are not so complete as to disprove emphatically that the cycle of 24 and 312 years respectively was never used by the Maya chronologers.”
Neither Pérez nor Valentini seems to have had a closer look at the order of the numbers given by de Landa since the order can be achieved by either periods of 20 tun (but not years) or 24 haab.
A. M. Tozzer had added footnotes to his translation (1941/1966:166,167):
“(877) We have already seen this period, known as the katun. It is not, as previously noted, “twenty years” but 20 tun (20x360 days) or 7200 days, and is designated by the day Ahau, one of the twenty days, not “letters” of the month, together with its numerical coefficient, which came on the last day of the period. Each katun thus ended with the day Ahau with one of thirteen numbers. A katun ending with the same number did not recur until approximately 256 years (7200x13=93,600 days = 256x365+160 days).
In the writing of Pio Perez and some other authors of his time and based upon explicit statements in the Chilam Balam of Mani and other eighteenth Century documents, the k'atun was mistakenly considered to have been twenty-four years in length. For a discussion of this see Seler (Bedeutung des Maya-Kalenders für die historische Chronologie (1895), Eng. ed., Trans., Bul. 28, 329-30), who states that marginal notations in the old text of the Mani gave the k'atun twenty-four years. Roys denies this, stating that this length of the katun is included in the actual text.”
“(878) ... During the later periods of Maya history the Katun Count was represented by a wheel as shown here by Landa. There are 13 Ahaus (faces of a man) each with a number. ... The wheel forms a repeating series. After you go around the wheel with its thirteen spokes, you return to the same point of departure and start again. Bowditch [Charles P., 1910:] (325-34) gives illustrations of these wheels taken from several Chilam Balam Books. ... In the wheel given here the words in the center are translated, “They call this count in their language, Uazlazon Katun, which means 'the war of the Katuns'.” In a copy of this wheel by Berendt (1868, given by Bowditch, fig. 61) Uazlazon Katun is translated by Berendt, “the revolution of the ages” or “the revolution or wheel of the katun.” Martinez H[ernández Juan, Diccionario de Motul] (1929:886), adding in brackets under the Motul definition of uaçaklom, “the turn or return of the katuns,” seems to agree with Berendt.
The Maya also had another type of wheel which represented the passage of the 52-year period, each of the four dominical days or year bearers bearing the numbers from 1 to 13 (4x13=52). Bowditch, 327-31. It is worthy of note, writes Roys, that “no known pre-Spanish representation of a katun wheel or any other circular chronological diagram has as yet been found.”
However, the k'atun-Period became relevant again when the dates on the monuments of northwestern Yucatán were explored.
The discussion about the length of the “katun” will be dismissed here but will be taken on again in chapter 1.5 (page 135).
The importance of the 52 year epoch, the Calendar Round, will be discussed hereafter.
J. Eric S. Thompson studied the dates of glyphs from Chichen Itza which were written, as he called it, accordingly the Yucatecan method and came to the following conclusion (1937:179):
“The Initial Series lintel at Chichen Itza records the date 10. 2. 9. 1. 9 9 Muluc 7 Zac. The front of the lintel opens in A1-B1 with glyphs which have been read as 10 Tuns, 1 Ahau. ... These two glyphs can be read as recording that the Initial Series fell in a Tun 10 of a Katun which ended on 1 Ahau. In other words the day Ahau with its coefficient may ... refer ... to the day on which the Katun, in which this Tun occurs, comes to an end.”
He continued: “A Calendar Round date followed by the information as to the number of the current Tun and the day on which the current Katun ends is fixed without any doubt in the Long Count, ...”
Here, Thompson distinguished between the count of a current tun and a current k'atun but in the following he came to a different result.
J. E. S. Thompson concluded (1960:182f): “ It is now taken for granted that Maya periods, be they tun, katun, or baktun, are not counted until they are completed, and that they are named for the day on which they end. Goodman held the opposite view, namely that the katun was named for its beginning day. ... the evidence for a reckoning by the ending day is very strong, for throughout the books of Chilam Balam the completion of the katun receives constant attention.”
And he continued: “Roys has called my attention to a passage in Tizimin (R. L. Roys 1933: 13), which gives the prophecy for the last tun of Katun 5 Ahau. One sentence reads: ' ... 4 Cauac would be the turn of the fold of the katun, the time when he gives up his mat, his throne. There comes another mat, another throne, another reign. The burden of 5 Ahau falls. He looks back, when he took what was granted to him. Gone is his cup, gone is his mat, gone is the bearer of his command.'”
And Thompson added: “As 4 Cauac is the day before 5 Ahau and is placed in the last tun of the katun, there is good evidence in this passage that Katun 5 Ahau ended on the day 5 Ahau.”
Considering this passage, however, as a period accordingly the method of overcounting then a new view emerges: the 5. ajaw-Period ended with the day 4 cauac and had started with the day 7 ajaw.
Also, by treating the ajaw-Period as an elapsed time count with the day ajaw being the last one, it follows, that there exist an ambivalence with the world creation day 4 ajaw 8 kumk'u. Considering on one side the very first day of the Maya era as the base for the Long Count while the ajaw-Period on the other side would have started one day later. And that does not agree with the conclusion that the compatibility and continuity of the Classic and the Yucatecan Calendars is given as demonstrated by the date from Xcalumkin, as well as, indicated by the example from the Books of Chilam Balam about the day “4 Cauac” above. Thus, it has to be concluded, that the very first day of the Maya era, 4 ajaw, is the base not only for the Long Count but also for the ajaw-Period count.
The bundle of twenty tun within the Long Count has been detected as being counted as elapsed short years, which are named k'atun and counted by themselves in twenties.
The bundle of 20 tun within the Short Count, however, contains current short years and are assembled up to thirteen.
In order to distinguish the two counts of bundles, the bundles of the Short Count are named ajaw-Periods throughout this treatise.
1.2.6 An Intermittent Resume and Retrospect
The Number Systems
Calendar systems dependent on the numerology and the counting system applied. In Mesoamerica all nations used number systems to the Base “20”.
The vigesimal number system of the Maya included also a number system to the Base “10”. Thus, their number words and their number signs of the head variant type were singular up to “12” and from there on composites, i. e. as of “13” to “19” included. The latter were composed first of the numbers “3” to “9” followed by the “10”. Number words greater than “20” were composites accordingly the method of overcounting.
Number signs of the head variant type greater than “20” were not handed down, they are not known.
The other known ancient number system of Mesoamerica had bar-and-dot number signs, whereby the dot represented the value “1” and the bar the value “5”. The bar signs were stacked and the dot signs were lined up on top of the bars until the highest basic number “19” was reached. The numbers for “20” and greater were written in vertical positional notation starting with the highest rank at the top.
In order to fill ranks without value this notation required an adequate sign; from the Maya Codices we learn that this sign has been of oval form.
The “Static” Calendar Round and the Longtime Counts of the Pre-Classic and Classic Calendars
The Preclassic Calendar system of Mesoamerica constituted a Calendar Round and a Day Count.
The Calendar Round consisted of the ritual calendar and of the civil, the solar calendar. The ritual calendar was constituted of the cycle of the thirteen sacred numbers and the cycle of the twenty sacred signs, whereas the civil calendar of the cycle of eighteen “month” of twenty days each as well as a “short month” of five days. All three cycles run in permutation up to a period of 52 years.
The Calendar Rounds of the Preclassic and the Classic Calendars appertained solely to the Year Bearer Group “B”, hence, it is considered as the “Static” Calendar Round.
The Day Count of the Preclassic Calendar was not totally vigesimal, since the second rank was counted only from “0” to “17” while the other ranks were counted from “0” to “19”, each of elapsed days, as the handed down positional notations in undercounting reveal on monuments. This way the comparable period of time, the solar year, was not counted as 365 but as 360 days, the short year - tun.
There is any indication to believe that the Maya got acquainted with this Preclassic Calendar system at the Isthmus of Tehuantepec and / or the highlands of what is now Guatemala and adopted this calendar system as well as the bar-and-dot number sign system with its positional notation. Evidence for the adoption results from the different number systems: the longtime count of the calendar had the bar-and-dot sign numbers in undercounting while the number word and the head variant type, “old” numbers of the Maya were overcounting. Thus, it does not wonder the Maya modified the longtime count as described in this treatise.
But what kind of calendar had the Maya before? Anywhere over the world the lunar calender was replaced by the solar calendar. This seems to be also the case with the Maya since after the transition period, when the Preclassic Calender came out of use, the Classic Calendar was installed with restored(?) elements of a lunar calendar.
The Maya reformed the longtime count of the Preclassic Calendar system by introducing a system of measurement in order to replace the positional notation of the Day Count. The measurements started analogous to the method of undercounting of the Day Count with the highest rank, thus, bak'tun, followed by k'atun, tun (the short year of 360 days), winal and ended with k'in, the day.
And finally they designed another sign for the number “13”, additionally to the composite one, which corresponds with the Maya number word, they invented a singular head variant type in order to complete the singular sign series of the thirteen sacred numbers of the tzolk'in.
The reform of the calendar system and the integration of the elements of the Lunar Calendar, the bar-and-dot number system in undercounting as well as the singular head variant type number “13” are considered strong evidences for the adoption of the Mesoamerican Preclassic Calendar by the Maya.
The next reform was entertained only in the region of the Rio Bec, Chenes and Puuc styles of architecture. Hereby a conceptually total new longtime count was developed. It is based on the ajaw-Period, the time in between the celebrations for the world creation day every twenty tun. The notation of the date is carried out accordingly the Yucatecan method, i. e. at first the day name of the ritual calendar, tzolk'in, was written, followed by an apposition consisting of the date of the solar calendar, haab, the tun and the ajaw-Period. Hereby the latter two periods were counted in short years of current time. The counting of the measurements within the apposition reminds of the way the Maya number words greater than twenty were composed, and thus, the notation was in accordance with the fashion the Maya arranged the ranks of quantities: beginning with the lowest and stepping up to the highest level.
The transition from the Preclassic to the Classic Calendar as well as the transition from the Classic to the Yucatecan Calender are compatible and continuous, hence, all the data from the three calendars as well as the data from the Isthmus of Tehuantepec are continuous and compatible.
Outlook
Juan Pio Pérez after having discussed the Calendar Round, he addressed (1843/1990:299) “another great cycle peculiar to the Yucatecos, who referred to its periods for dating their principle epochs and the most notable events in their history. It contained 13 ... “ ajaw-Periods. Later he stated (1843:302): „The origin and use of this species of age, epoch, or cycle, and (the time) when it commenced, are not known. Neither the Mexican nor the Toltecan authors, nor those who corrected the chronological system for the computation of time, ever used it, nor had their writers any knowledge of its existence. The few and incomplete manuscripts which exist in this peninsula make no mention of it; ...”
His note indicated and foresaw, why it was so cumbersome to tackle the puzzle of this Calendar.
The dates from the monuments of the Rio Bec, Chenes and Puuc region indicate another divergence concerning the Calendar Round of the Yucatec Calendar. The Year Bearers are not any more solely of the Group “B” but as has been shown above by the dates from Xcalumkin, Etzna, Sisila of the Group “C”. And as indicated with the date of the death of Ah Pula from the Chilam Balam of Tizimin respectively of Chumayel the Year Bearers are of the Group “D”.
In addition three other subjects need to be elucidated from here on:
is there a common structure for the calendars?
was there a switch from cardinal to ordinal numbers in the coefficients of the month?
and was there a change in the length of the ajaw-Period from 20 tun to 24 haab?
These matters will be treated in the chapters to follow.
1.3 The Structures of the Calendar Round of the Maya Calendars and its Expansion Leading to the Yucatecan Calendar
1.3.1 Introduction with Regard to the Calendar Structures
Characteristic for the Classic Maya Calendar is the Long Count and that all data sets of the Calendar Round appertain to the Year Bearer set ik’, manik’, eb, kaban.
In Yucatán, however, we find sets of data, noted like the Classic Calendar as well as accordingly the Yucatecan method but of other Year Bearer sets.
In a first approach it seems appropriate to treat the data sets of the Year Bearer Group “B” as belonging to the Classic Calendar, while the ones of the varying groups as belonging to the Yucatecan Calendar.
From D. Graña-Behrens (2002) collection of verified data we learn that the conversion of the Year Bearer Group “B” data from the notation accordingly the Yucatecan method into the Long Count results in only one date, while the conversion of most data sets from the Rio Bec, the Chenes and the Puuc region result in multiple solutions.
The question arises: Can, there, a common structure be established for the calenders with regard to the different long-time counts as well as to the varying Year Bearer Groups within the Calendar Rounds? And will it be, thus, possible to reduce the degrees of freedom and, hence, the number of solutions?
In order to relate these different data sets as well as to associate the Year Bearer Groups of the Calendar Rounds with the Long Count and the ajaw-Period Count, a common numeric base is required. For that purpose the Calendar Round will be digitized and the resultant tables of the two longtime counts will be superimposed.
1.3.2 The Numerical Scaling of the Calendar Round;the Resultant Three Longtime Tables Lead to the Association Table
The Long and the Short Count originate as of their base point 4 ajaw 8 kumk'u. From this day on a scale, stepped up on k'atun- respectively ajaw-Periods of twenty tun each, will be devised.
For the Calendar Round, however, since it intersects only with the base point, there exists, accordingly the Pérez characteristic (3) – (see page 18), four distinctly different timely distances to the begin of the very first and complete Calendar Round, each due to one of the four year bearers of the group “B”. The distance differ of each other by thirteen years.
Hence, to set up the digitized structure for the Calendar Round, the primus inter pares, the patron of the Calendar Round, is urgently requested.
The methods of calendar calculus, the Maya used, are not handed down (F. G. Lounsbury 1978:769, André Cauty 2006/2:20).
Hence, the Calendar Round data will be digitized in the following way:
the days and the years of the Calendar Rounds as well as the Calendar Round itself shall be converted as elapsed time periods of the Long Count, thus, the days will be numbered from 0 to 364, the years from 0 to 51 and the Calendar Rounds from 0 up to the number as required.
In order to start the scaling kaban has been chosen as the primus.
To determine the number of the day within the solar year, the date 8 kumk'u provides the information of 17 elapsed month plus 8 elapsed days resulting in 348 elapsed days; the day name of the first year of the Calendar Round results from the sacred days: the current days, that is (348 + 1) divided a) by the 13 sacred numbers is equal to 26 and a remainder of 11, and b) by the 20 sacred signs to equal 17 and a remainder of 9, thus, starting with kaban being the sign (1) (see Table of Year Bearer Groups – page 18), then the sign (9) is chikchan; and in order to determine the number of the years within the Calendar Round the table below is utilized:
The first preliminary basic calendar structure table is build up for the two longtime counts as well as the digitized Calender Round based on the patron B-kaban by steps ascending with k'atun- and ajaw-Periods of 20 tun each.
The second preliminary basic calendar structure table is build up with respect to the Long and the digitized Calender Round Count, latter based on the patron B-kaban, on steps ascending each by 1 Calendar Round.
For the four possible patrons of the Calendar Round, all deviating by 13 years, the following table has been accumulated for the digitized World Creation Day starting dates.
Which one of the four patrons is the primus inter pares can only be resolved by the selected data and the Association Table (see chapter 1.3.4, page 89).
1.3.3 Pre-Study about the Structure of the Yucatecan Calendar
The Data Base Available to Establish the Preliminary Structure of the Yucatecan Calendar
To check on the “discontinuities”, that is on “other” Year Bearer Groups than “B”, dates selected from D. Graña-Behrens collection, were first expanded by the information on the Calendar Round Group, and then placed into the Association Table to be checked against the structure obtained by superimposing the Basic Structure Tables (1) and (2) (see pages 72f) as well as the starting dates for all four patrons of the “B”-Group.
Data which Exhibit the Introduction of the ajaw-Period:
Data “at the Half” of an ajaw-Period
Besides the development of the Calendar Round to a system of cyclic changes in patronages an other development emerged: the introduction of the ajaw-Period as a longtime count with the measure as current time.
D. Graña-Behrens' collection of data contains three dates with the specification ti, ta u, respectively tu tan lam, “at the Half” of an ajaw-Period (see also page 58).
Etzna, Campeche, Mexico
(employed already as an example on page 58)
„Pequeña Acrópolis“, Stela 1 (Fragment), Glyphs A1-C6
[ISIG mak] 9 bak'tun 14 [ k’atun] 10 tun 0 winal 0 k’in [secondary series]
5 ahaw? 2 mak tu tan lam (at “the half of”) 4 ahaw
(D. Graña-Behrens, 2002:421 Analyse Nr. [231])
Since the 4th ajaw-Period begins on the day 9.14.0.0.0 and ends on the day 9.14.19.17.19 at “the half of “ is 9.14.10.0.0 .
Accordingly this example the following will be converted as indicated:
Dzibilchaltun, Yucatán, Mexico
south east of Structure 33, Stela 9
ti tan lam-wa 5 ahaw
at “the half of” 5 ahaw
converted to 9.6.10.0.0 and 10.0.10.0.0 , whereby the first one, the preferred is
(D. Graña-Behrens, 2002:353 Analyse Nr. [109])
Since the 5th ajaw-Period begins on the day 9.7.0.0.0 the conversion results in 9.7.10.0.0.
Etzna, Campeche, Mexico
“Pequeña Acrópolis“, Stela 21, Glyphs A1-C1
11 ahaw 17 ch'en ta u tan lam 10 ahaw
11 ahaw 17 ch'en at “the half of” 10 ahaw
Long Count: 9.11.10.0.0 11 ahaw 17 ch'en
(D. Graña-Behrens 2002:369 - Analyse Nr. [140] )
the Year Bearer Group is “C”
- Citar trabajo
- Armin A. Brandes (Autor), 2016, The Maya Calendar Systems Vol. 1, Múnich, GRIN Verlag, https://www.grin.com/document/334164
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