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Topic Study Group n° 20_André Cauty
‘How to transform an Aztec xihuitl (an 18 period year) into a solar calendar?’
Points to be presented and discussed
1.- A calendar is a social convention/system constructed to be able : 1) to divide/discretize the continuous time
into periods measured in days and 2) to distinguish and to define the days by assigning them a unique and
unambiguous expression called a date.
2.- The Mesoamericans used a 260 day week dated by 260 expressions of the form αX where α belongs to a set
of 13 integers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13) and X to a set of 20 sings/words of day that rely to scripts
and languages of each people or culture. Note that every number an each sign of day refers to entities which
inhabit the 7 seas, the 9 hells or the 13 heavens. Therefore each expression αX refers to religious believes and
myths of origin which is why it had a tremendous value for divination. In the manner of horoscopes, the diviner
will tell the fate of a person born on a day dated αX, or the future of an action started on a day dated αX. It was
customary to name αX: a person, a place, a year… Tonalpohualli is the nahuatl name of the divinatory week of
260 days also called the divinatory year; its yucatecan name is tzolkin; both could be translated “the story of
destinies” or “the account of days”.
3.- Like everyone else, the Mesoamericans undergo diurnal and annual variations of solar radiation; they
distinguished the days, months, seasons, years, centuries, double centuries and so on. Note that a month has 20
days, a year has 18 months,and a century has 52 years.
4.- The Mesoamericans used a year we refer to as ‘festive’. It has 19 periods, i.e. 18 months of 20 days each and
1 residue of n days. Theoretically, each period has a name, and all 19 are immutably ordered. So, we can refer to
the festive year by the ordered list I, II, III, etc., XIX.
5.- Here is the Duran’s festive year :
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
XIII
XIV
XV
XVI
XVII
XVIII
Where are the errors?
Note. Duran’s codex does not have a page for the residue
6.- As you can see, if used as a calendar, the Duran’s year presents a serious problem. We know that the days of
the Duran’s year are dated by means of αX expressions, so we have only 260 different labels to stick on 360
(365 or 366) bottles. Whatever the way to go around it, 100 (105 or 106) days will received a αX date already
given.
The Duran’s year is not a calendar.
7.- Like others, Duran was a monk-ethnologist who wanted both to study and understand the indigenous beliefs
and eradicate them (may be for that more effectively). I postulate that he wanted to transform the festive
indigenous year into a calendar as close as possible of the Julian calendar: a calendar of 18 months of 20 days
each and a residue of 5 days.
8.- Perhaps without realizing it, Duran opened a way to transform the xihuitl (festive year) into a calendar. He
distributed the 360 days of his year in 18 numbered months and he recorded their dates on 18 pages. Each page
contains the 20 αX dates of the days of the month written on the page. The result is a kind of array, of 18
columns, 20 lines and 360 boxes. Let T1 that array:
Y→
↓β
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
I
II
III
IV
V
VI
VII VIII IX
X
XI
XII XIII XIV XV XVI XVII XVIII ↓X
1
2
3
4
5
6
7
9
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
*
12
1
2
3
4
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
XIII
XIV
XV
XVI
XVII
XVIII
XIX
XX
Table T1 is a festive year of 360 days. The columns are the months (graded from I = Atlcahualo to XVIII =
Izcalli). The boxes are the days. Each is filled by the αX date of tonalpohualli (the constituent X is encoded by a
roman numeral listed at the right side of the line from I = Cipactli to XX = Xochitl). We follow the arrow of time
down the columns one after another, from left to right. A frame is used. Green for the 260 “well dated by Duran”
days, blue when a number is wrong, and white for months contaminated by an error that propagates. The αX
clearly outliers are marked in blue, red or a *.
9.- Reconstruction of a xihuitl (calendar) from Duran’s data. The festive year of
duran has 360 days dated by 260 αX expressions. So it is necessary to add a
residue of 5 days and to disambiguate. The transformed xihuitl would be
isomorphic to the Maya ha’ab, the solar calendar of 365 βY dates in use during
the Classic period. The array disposition allows the invention of an avatar of the
365 βY a Aztec/Mayan/Spanish could have used in the sixteenth century. Just
number the rows and columns. Each of the 360 days of the table could then be
identified/dated bye its coordinates (α
αX, β Y). Coordinates are unique and unambiguous. So T1 can attribute to
each day two dates : a date in the tonalpohualli, for instance 13 Tochtli, and a date in the calendar of 360
coordinates (α
αX, βY) that can be translated as 7 Ochpaniztli in nahuatl (13 Lamat 7 Zac, en yucatan).
This shows that Duran opened the door, but failed to build a calendar year in part because he did not add the
residue of five days and also because it retains the 260 αX expressions instead of using the 365 βY coordinates
easily translated in dates Let’s finish Duran’s job.
10.- A xihuitl that sounds like a ha’ab
Let’s look at the use of ha’ab in dating.
The Mayan texts contain a lot of equalities that link dates and durations. The equations are of the form [4 Ahau 8
Cumku] + Σ(ci Pi ) = (αX, βY) + Σ(ci Pi ) = (αX, βY).
Examples: Stela C of Quiriguá (Izabal, Guatemala)
DATE MAYA DE TYPE CR OU (α
αX, βY)
ASSOCIEE A UN COMPTE LONG
Transcriptions of the CL:
[0.0.0.;0.0. 4 Ahau 8 Cumku]
+ (translation)
9 x 144 000 + 1 x 7 200 + 0 x 360 years ; 0 x 20 + 0 days
= 1 303 200 jours
= 6 Ahau 8 Yaxkin
(1 303 200 = 2 modulo 13 = 150 modulo 365)
On compte
les katun pour
Yaxkin :
9-baktun 1-katun
0-tun
;
0-uinal
0-kin
and 1 303 200 = 26/08/455
(proleptic Gregorian and correlation constant = 584 283)
6 Ahau
8 Yaxkin
and panels 29, 30 et 31 of Yaxchilan (Chiapas, Mexico)
[Origine]
B1A4
EF1
EF2
H3G4
H4G5
J2I3
J3I4
KL3
KL4
Direction et pas de translation (nombre de distance)
[13.0.0.;0.0.]
9-baktun 13-katun 17-tun ; 12-uinal 10-kin
- 1-tun ; 1-uinal 17-kin
[9. 13. 16. 10. 13.]
+ 2-katun 3-tun ; 5-uinal 10-[kin]
[9. 16. 1. 0. 0.]
+ 12-tun ; 0-uinal 0-[kin]
[9. 16. 13. ; 0. 0.]
+ 7-tun ; 0-uinal 0-[kin]
[9. 17. 0. ; 0. 0.]
Date image
[4 Ahau 8 Cumku]
8 Oc 13 Yax
Corrélation (584283
3114 av. J.-C
25/08/709
1 Ben 1 Ch’en
24/07/708
11 Ahau 8 Tzec
01/05/752
2 Ahau 8 Uo
28/02/764
13 Ahau 18 Cumku 22/01/771
11.- How would Aztec date an event at the time of Durán ?
This is a date carved on the Monolith of the dedication (Mexico). The date has
two components: 7 Acatl and 8 Acatl. First is the αX date in the tonalpohualli
of an event; this event is the day of the dedication. The second αX is the date of
a day. A day decided to give its name to a year. This αX is the eponymous of a
year; in that case the year of the dedication.
Returning to the case of the Mayas, we observe that they associated various
types of dates: the 260 dates of the divinatory week, the 365 dates of the solar
year, and the number of days since the origin 4 Ahau 8 Cumku of their
chronology.
I invite you to do a thought experiment (Gedankenexperiment). With or without a computer, you simply write
the sequence of days from the origin. You write the date of the next day, then the next one, and so on, without
omissions nor repetitions : 0.0.0.,0.0. 4 Ahau 8 Cumku; 0.0.0.,0.1. 5 Imix 9 Cumku; 0.0.0.,0.2. 6 Ik 10 Cumku;
...; 0.2.12.,13.0. 3 Cauac 7 Cumku; 0.2.12.,13.1. 4 Ahau 8 Cumku; etc.
After 18 980 iterations: a) you come back over the same dates and b) you find that some couples have never been
outside, for instance 4 Ahau *8 Cumku; 4 Ahau *10
Cumku; 4 Ahau *11 Cumku; 4 Ahau *12 Cumku. And
they are never been documented. Now do the same with the
four star couples. Each couple generates 18 980 other star
couples. Finally, we get five calendar cycles, the first is
known as the Calendar Round, CR, and it is the real
calendar used by the Mayas during the Classic. The other
four, with star dates, were appartently stillborn calendars.
To my knowledge, the Mayan doucments (monuments,
furniture, codex) of the classical period do not contain a
single star date. These observations led Americanists to present the CR as a gear mechanism and discover a
theorem that solves the soothsayer’s problem: say on what day of the divinatory week falls on a given day of the
year.
Take for example the New Year, the first day of first month of the year, and suppose that it’s falling this year on
the 13 Ik . Next year, the New Year will fall on 1 Manik. Then successively on: 2 Manik, 3 Eb, 4 Caban, 5 Ik,
etc.
And it never falls on a star date. For the soothsayer it’s very convenient, because knowing on what day of the
divinatory week falls the New Year is to know the color of the year, is to know its numerologist and symbolic
properties. It’s easy to prove that the New Year fall only on four signs, so there are 52 different New Year dates.
That is called the Aztec century, SA. Now, if we decide to make the first day of the first month of a year the
eponymous of that year, we should have at our disposal 52 labels to distinguish and define the 52 years of the
SA. The set of these labels is totally ordered. And its four signs X are called the Year’s Bearers.
Observations
Notion de date étoilée. Soient αX et βY des dates tzolkin et ha’ab quelconques. J’appelle date étoilée tout
couple (α
αX,*βY) qui n’appartient pas au corpus des 18 980 dates du CR maya en usage à l’époque classique.
Une date non étoilée est bien écrite ou orthodoxe. La date αXβ Y = 9 Ahau 8 Cumku de l’origine de la
Chronologie maya en CL est une date non étoilée car elle appartient au CR ; elle est donc bien écrite ou
orthodoxe. Une date est non étoilée (/bien écrite) ssi elle respecte la règle dite Règle d’Orthodoxie de la
Chronologie maya, ROCm. C’est une contrainte de cooccurrence portant sur les seuls constituants X et β des
dates CR. Sans la ROCm, le CR totaliserait 5 x 18 980 dates, au lieu de 2.12.;13.0.. Le tableau suivant
(Thompson;1960: 123) donne toutes les liaisons X-β qui définissent les 18 980 dates αXβY bien écrites1 :
P0
P1
P2
P3
P4
Ik
Akbal
Kan
Chicchan
Cimi
Manik
Lamat
Muluc
Oc
Chuen
Eb
Ben
Hix
Men
Cib
Caban
Edznab
Cauac
Ahau
Imix
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Censé expliquer le mécanisme du Calendrier Rituel maya et le cours des dates qu’il délivre, l’engrenage de la
figure intitulée Combinación de los dos ciclos calendáricos en la rueda maya délivre (seulement) des dates
étoilées.
Prenons pour le comprendre les dates que la couleur a mises en avant : 9 Ahau *19 Cumku et 13 Kan *3
Uayeb. La ROCm dit : a) quand une date contient le signe X = Ahau, alors les α Ahau βY sont bien écrites ssi
le rang β est égal à 3, 8, 13 ou 18 ; et b) : quand une date contient le rang β = 19, alors les dates αX 19 Y sont
bien écrites ssi le nom de jour X appartient au quarteron {Cimi, Chuen, Cib, Imix}. En résumé :
a) 9 Ahau *19 Cumku est une date étoilée car :
X = Ahau ε β χ {3, 8, 13, 18} qui ne contient pas *19
β = 19ε X χ {Cimi, Chuen, Cib, Imix} sans Ahau *
b) 13 Kan *3 Uayeb n’est pas bien écrite car :
X = Kan ε β χ {2, 7, 12, 17} qui ne contient pas *3
β = 3 ε X χ {Chicchan, Oc, Men, Ahau} sans Kan *
Re-synchroniser les cycles. Le résumé a) ou b) définit
toutes les corrections pouvant conduire à régulariser une date
étoilée et lui faire ainsi respecter la ROCm.
Dans la métaphore de l’engrenage, faire une correction revient à remettre la pendule à l’heure. Cela se fait en
deux temps : 1) découpler les roues αX et β Y et 2) les raccoupler sur une position respectant la ROCm.
Soit par ex. la date étoilée 1 Imix *1 Pop (Sabloff;1994). Une fois découplé,
on peut choisir de remonter l’engrenage à partir de α = Imix ou de β = 1. Dans le
1er cas (respectivement le second), on met Imix (respect. 1) en regard avec un β
(respect. un X) du quadruplet Q = {4, 9, 14, 19} (respect. du quarteron P1 =
{Akbal, Lamat, Ben, Edznab}). D’où par ex. les types suivants de solutions :
dent (1) Imix et creux 4[5] (Pop)
dent (12) Edznab et creux 1[5] (Pop)
les parenthèses peuvent être renseignées par n’importe quelle valeur de 1 à 13 (modulo 5)
1
Environ 98% de 4 milliers de dates CR relevées (monuments, codex) sont non étoilées et 86% du 2 % des dates étoilées le
sont car β est trop grand ou trop petit d’une unité 1.
12.- Received from Mayas or created independently, the Aztec have had at their disposal that kind of century
which distinguished and defined 52 years by 52 αX labels (52 dates of the day eponymous of these 52 years).
The system of eponymous of years was common among the Aztecs. A proof is done by the first figure in the
Durán’s codex. It is a sort of wheel (or spiral) which proves that the Aztecs knew very well the structural
properties of the cycle of the 52 eponymous.
The wheel has 52 boxes filled with 52 αX labels. The layout
divides them into four classes of thirteen (5+8) elements,
each characterized by three features : a) a sign X with his
gloss in Spanish, b) a color and c) a cardinal point marked
by the position on the sheet and a commentary in Spanish. In
total, four classes of thirteen years:
Acatl/green/east, Tecpatl/red/North,
Calli/yellow/west, Tochtli/blue/south.
The wheel does not contain instructions or methods to
construct it. It does not say for example how and in what
order we have to read the 52 dates. Through trial and error,
we can convince ourselves that a spiral reading gives
meaning to the whole wheel. From the center, on the green
branch (1 Acatl) and rotating in the forward direction
(counterclockwise) we obtain the following: 1 Acatl, 2
Tecpatl, 3 Calli, 4 Tochtli, etc. This order of reading gives
the 52 consecutive years of a Aztec century, SA, the same as
those we find in the Mendoza codex written between 1541 and 1542.
I say in conclusion that the Aztecs distinguished, defined and date the 52 years of their century. They dated by
means of the date of the day agreed upon as eponymous.
Among the Mesoamericans, the way how the eponymous was agreed differs with people, cultures and times. So
differently that the specialists seem to have lost their Latin.
Apparently, it is never one day of the residue. Often this is the first day of the first month of the year, but it is
also, according to the authors consulted, the twentieth and last day of the fourth, fifth or 18th month.
Mais distinguer les années et les baptiser αXP n’est pas équivalent à associer en un tout calendaire les cycles 260
et 365.
ANNEXE : 1) LES VINGTAINES ET DIVINITES ASSOCIEES DE L’ANNEE FESTIVE (SAHAGÚN)
ANNEXE 2 : LES “SIGNES” AZTEQUES DES 18 MOIS DE VINGT JOURS ET LEURS NOMS EN NAHUATL CLASSIQUE
ANNEXE 3 : INCOHERENCES DANS LE MOIS XII TEOTLECO