Revisit Aztec and Maya calendars

Invitation to revisit the Mesoamerican Calendars

July 16-20, 2012 Daejeon CC, Korea

1000 km

The one called the Real Calendar

DG5:

안드레CAUTY보르도프랑스

문화와 수학done durin

g HPM

2012’ c

ongress

Daejeon, S

outh K

orea, J

uly th

e 21th

July 16-20, 2012 Daejeon, Korea

Εὕρηκα

Inter-Inter-dépaysedépaysementment

Εὕρηκα

Εὕρηκα

bi-uprooting

« uprooting » sounds like a ‘quiproquo’ or a misunderstanding that …

…calls to co-built inter-cultural bridges

a Mayan case where the commutation aB/Ba produces a subtle switch; that of the operation (protraction/ addition) and of the type (additive/protractive) of numeration.

« uprooting » calls to…

bolon tu Ø-kal Ø-kal catac bolon

…co-built inter-cultural bridges

Εὕρηκα!

Εὕρηκα!

Using theoretical concepts[1] by Ferdinand de Saussure, Bernard Pottier and André Martinet we have solved four intertwined identification’s problems:


Ø- = ca ‘2nd’ and Ø- = hun ‘1’

1. That of the indetermination of the SIGNIFIER, Sa, (‘signifiant’) 2. That of the syntactic ambiguity of the FORMAL CONSTITUENT, Sy, OF THE SIGNIFIED (‘forme du signifié’)3. That of the semantic vagueness of the MEANINGFUL CONSTITUENT, Sé, OF THE SIGNIFIED (‘signifié’)4. That of the analysis in units of the DOUBLE ARTICULATION[1] Signifier (Sa) Sa

linguistic sign: Si = ——————————————————————— = ———— Formal constituent (Sy) / Meaningful constituent (Sé) Sy / Sé

1st articulation: Phrases are formed of Morphemes2nd articulation: Morphemes are formed of Phonemes (or Graphemes)

Εὕρηκα!

Εὕρηκα!

our bridge was the discovery of a subtle Mayan switch between two different numerical processes shown and done by an interpretation in the metalanguage provided by the Yucatan language of the Maya scribe.


Ø- = ca ‘2nd’ and Ø- = hun ‘1’

is it strange? not so strange! the same subtle switch occurs in spoken French where “vingt-quatre” (twenty-four) refers to an additive composition 20 + 4 while “quatre vingt(s)” (eighty) refers to a multiplicative determination 4 x 20.

cognitive bridges build themselves in a meta-linguistic space of translation (an inter-disciplinary and inter-cultural translation)

^

our bridge is founded on a real transcription of the source-texts into Yucatec (or other Maya scribe’s natural language) this Mayan language is the real and final interpreter of all Maya’s cognitive productions

(the reason: natural languages are the only symbolic systems including their own meta-language)

the

space for

translation and meta-

linguistic processes is on the floor of

the Yucatec language, the ultimate interpreter

uprooting leads to a METALANGUAGE of TRANSLATION, built from inter-cultural/disciplinary places where the I/YOU-speakers of all communication’s act can built cognitive bridges, can develop new ideas and propose alternative interpretations…

Welcome in the Mayan nightsWelcome in the Mayan

nights

supernatural scribes… doing with math's texts

k1196numbers ? calculations? Yes! but for what?

Mayan numbers are bricks to construct the calendars which give to the Mesoamerican soothsayer his most used tool to “see” the invisible world and to predict the fate (good, evil or indifferent) of everybody, everything, every cycle of time…the divinatory week is among the Mesoamericans the most used, as well as the oldest, calendar

British Museum: linteau 47 de Yaxchilán

Par l’autosacrifice de son sang, la reine obtient une vision de son ancêtre qui nous est montré sortant de la gueule d’un serpent qui interprète les volutes de fumée des offrandes brûlées.

as all Mesoamericans did, Mayas and Aztecs used the same divinatory week of 13 x 20 = 260 expressions X (13 integers and 20 day signs)

is an integer, X a day sign, and the sequence of the X sounds like: 2 Sunday, 3 Monday… and follows the rule s(X) = s()s(X)

Mayan and Aztec examples of the 13 integers

The 20 Mayan and Aztec day signs X

Mayan Aztec

X dates were used to name persons, things or time cycles like days/years…

… and to predict his/her/its fate: “the natives of this sign will be rich”

Mayas and Aztecs used the same divinatory week in order to qualify the future or to celebrate the rulers or the cities history

13

Edznab

Ahau

Eb

Kan

? Cib

28 2 24 13

13 Ahau + 28 = 2 [Lamat] + 24 = 13 [Eb]

Here a very common Mayan divinatory practice done by small translations in the divinatory week:

X + d = ’X’ (d in additive numeration)

another common Maya use: XY + d = ’X’’Y’(duration d is done in dispositional numeration):

E2 F2 H3 G4 H4 G5 (1 Ben 1 Ch’en) + [10-[kin] 5-uinal 3-tun 2-katun] = (11 Ahau 8 Tzec)

the equations of the form XY + (ciPi) = ’X’’Y’

as we can imagine every people develops its own arithmetic and calendar skills and habits:

1.- Occidental dates:Mardi 14 juillet 1789, Saint Bonaventure

Sunday, July 8, 2012

2.- Aztec quantities (additive numeration) : cenxiquipilli xochocótzotl una talega de ocozote o goma de color 8 000 packages of resin of copalm (a tribute due to the Triple Alliance)

3.- Mayan duration (positional numeration): 1.5.14.;4.0. [-tun] or 1.5.14.;4.0. [-kin]

(1 x 400 + 5 x 20 + 14 x 1)-tun ; (4 x 20 + 0 x 1)-kin 1 x 144000 + 5 x 7200 + 14 x 360 + 4 x 20 + 0 x 1 = 185 120

July

a famous French date:

mardi 14 juillet 1789St Bonaventure, évêque,un jour après le dernier quartier de Lune

Tuesday, July 14th, 1789St Bonaventure, bishop,A day after the last district of the Moon

Almanach royal, année commune M.DCC.LXXXIX etM.DCC.LXXXX., BN de France, Lc25-28

the Mixtec and us did not have

the same dating system or the same calendar

the above Mixtec date: crocodile 13 house 7 had shape: (XXP

Mayas and Aztecs did not have

the same dating system or the same calendar

IDP

Wha

t a c

ogni

tive

diffe

renc

e!

Mayan CR dates follow the (X, Y)-model: 6 Ahau, 13 Yaxkin

Aztec SA dates follow the (X, XP)-model: 7 Acatl, 8 Acatl

Mixtec dates follow the (X, XP)-model: ‘crocodile’ 13, ‘house’ 7

so, Aztec and Mayan chronicles are very different:

the firsts look like comic strips

the second like texts full of equations

Azt

ec c

hro

nic

lelo

oks

like

a c

om

ic s

trip

XP dates

Mayan chroniclelooks like a text full of equations

Sous Yax, on compte les katun : 9-baktun 13-katun 17-tun 12-uinal 10-kin,le 8 Oc 13 Yax […] ; 1-tun 1-uinal 17-jours avant, le 1 Ben 1 Ch’en […] ;

2-katun 3-tun 5-uinal 10-jours plus tard, le 11 Ahau 8 Tzec […] ; 12-tun 0-uinal 0-jours plus tard, le 2 Ahau 8 Uo […] ;

7-tun 0-uinal 0-jours plus tard, le 13 Ahau 18 Cumku Fin du Katun 13

equations of the formX Y + (ci Pi) = ’X’ ’Y ’

today, Mayan arithmetic and Mesoamerican calendars are just SECS, strange extinct cognitive species, to be discovered, reinvent and rebuilt

Quoting Proust: “partons à la recherche des nombres perdus” let’s go to the search of the lost numbers

Tulum by Frederick Catherwood (1799 - 1854)

the search for the ‘lost numbers’ began at the time of the discovery of America (1492), it continues today and will continue tomorrow…this task was (and still is) specially hard not only because of the cultural gap between Mesoamerican and Spanish but also because the new rulers of America tried to remove the divinatory week (a very diabolic work, they said) and to transform the festive year used by the Natives into a calendar which sounds like the Julian (then Gregorian) one; first steps of the process leading to the imposition of the Spanish calendar…

DiscoveryConquests

SpoliationColonization

TrainingEvangelizationIncomprehension Education, Translation…

Situat

ions

and

rela

tions

of ser

vitu

de

but now: who and how rebuilt the disappeared Mesoamerican ideas?

“Tienen su año perfecto como el nuestro de 365 días y seis horas […] De estas seis horas se hacía cada cuatro años un día, y así tenían de cuatro en cuatro años el año de 366 días”

Diego de Landa(1524-1579) Bishop of the Yucatan

They had a year perfect as ours of 365 days and six hours […]. Of these six hours they made every four years one day, and so they had from four to four years one year of 366 days.

Colonial opinion

“El calendario mesoamericano era el

resultado de la combinación entre un

ciclo de 365 días, llamado en nahuatl

xiuhpohualli o ‘cuenta del año’ (ha'ab en maya),

y otro ciclo de 260 días, llamado tonalpohualli o

‘cuenta de los días’

(tzolkin en maya)”

Rafael Tena, ‘El calendario mesoamericano’, Arqueología mexicana, Vol. 7, nº 41, México, 2000, p. 5-6

The Mesoamerican calendar was the result of the combination of a cycle of 365 days called xiuhpohu-alli in Nahuatl […] (ha'ab in Maya), and an another cycle of 260 called tonalpohualli […] (tzolkin in Maya)

Modern opinion

Modern opinion: 260 x 365 = 52 / 18 980

“Se requería el transcurso de 18 980 días nominales, equivalentes a un ‘siglo’ de 52 años, para que se agotaran todas las posiciones posibles de un día cualquiera del tonalpohualli dentro del xiuhpohualli, y viceversa […]. Cada uno de los 52 años

tenia su nombre propio…” 18 980 days, equivalent to a 'century' of 52 years, they had to wait for each date Tonalpohualli goes through all possible positions of the xiuhpohualli, and vice versa […] Each of 52 years had its own name…

They said that all Mesoamericans share the same calendar: a combination of the divinatory week of 13 x 20 days with the solar year supposed, without proofs, to be a calendar counting 365 dated days. A combination often described as a mechanism borrowed from the industrial world!

They said that the Aztecs had intertwined these two cycles and obtained a calendar of 18 980 days similar to the Calendar Round used by the Mayas or a century of 52 years that it is called the "Siècle Aztèque", SA (Aztec century)

dangerous simplifying ideas…

…and amazing hybrids

caution!

Waldeck! What did you see?

The weight of time and habits makes difficult for Waldeck to see, admit and understand the Mayan creations which were for his time so strange and so radically different.

Whether called a “monstrous hybrid”, “Waldeck’s elephant”, or even the “self indulgence of the ethno-X”, the underlying concept reflects not only the ineluctable fact of projecting one’s own frames of reference and one’s own forms of knowledge upon the foreign work that we are trying to understand, but rather the human failure to submit all readings and interpretations to systematic and collective criticism, a kind of criticism that has to be interdisciplinary and interethnic or intercultural.

hybridize is not totally inevitable

All Mesoamerican calendars were founded on 2/3 main cycles(strongly intertwined among the Mayas):

1.- the divinatory week which was giving the color of the times and days 2.- a solar and/or festive year imposed by Nature and/or Rulers & Priests

Gods are time's bearers, they bear the packages of time

1. The divinatory week/almanac

2a. The undergone seasonal year

2b. The Mayan discretized festive year

If the residual period is well fixed and defined,

the Mesoamerican Festive year, FY, counts 365 days bundled in 18 months of 20 days and

1 residue of 5 days. It is the Mesoamerican way

to divide ‘discrétiser’ the continuous of the

time (tropic year) and to register it in the space

of the 6 cardinal points (Zenith, Nadir, North,

West, South and East).

Mesoamerican Festive Year

East

SpringSummer

Autumn

Winter

260 x 365.242…

they say that “all Mesoamericans combined these two essential cycles: the divinatory week of 260 days and the year”, that is OK! but they speak about a year which is not well defined by the historical sources…

Call to revisit the main thesis

Important historical dataAlmost everywhere and always in Mesoamerica, the days of the FY were not dated by means of an annual calendar, but they all were distinguished and defined by the highly symbolic ‘color’ conferred to each of them by its expression X in the divinatory week (it is the case for instance in the Duran’s descriptions of the Aztecan year).

dated at all; so, we have to observe that the fact of using a festive year FY of 365 days does not imply the obligation to name them all; documents said that people dated only 360 days, the 360 days which belong to the 18 months. If a calendar, or as a calendar, the Festive Year would have only 360 dates, in some cases.

Some colonial documents claimed that the 5 (or 6) aciagos days were not

More important historical dataIn 4th century, the Mayas had developed and were using a system of writing to uniquely define the days of the festive year. Making that, their festive year became a calendar with 365 dates, all written in the innovative shape Y

At the same time, the Mayas had formed the product of two important cycles they used in such a particular way that the coupling 260 x 365 gives a product of 18 980 elements. In this manner, they created a new calendar which dates are the 18 980 pairs (X, Y). The Mayan name of that calendar is yet unknown. Scientists say : ‘Calendrier Rituel’ or Calendar Round, CR.

Finally, the Mayas put in narrow correspondence the Calendar Round CR and the Long Count CL as shown by thousands of initial series. In conclusion: the Maya disposed of several intertwined calendars.

More important historical data…

Between the 4th and the 10th century, the Mayas used not only the 260 ancient X dates, but also the 365 innovative Y dates.

Their festive year had became the following calendarwhich gave the X and the Y correlated dates:

Le tableau montre une année maya de type [(28 x 13) + 1] = [(18 x 20) + 5] et de 1er jour (2 Eb, 0 Pop). Les 365 jours sont organisés en treizaines (distinguées par une couleur) et en mois (de 20 jours, matérialisés par les colonnes). Chaque case contient le rang (de 1 à 13) de la date tzolkin, X, du jour dont le signe X se trouve (en bleu) en dernière colonne. La date ha’ab, Y, est donnée par les coordonnées (en rouge). Le calendrier se lit par colonne (haut/bas, gauche/droite) : 2 Eb 0 Pop, 3 Ik 1 Pop, etc., jusqu’au 2 Cib 4 Uayeb.

More important historical dataduring the colonial period, the Mesoamericans (in particular Aztecs and Mayas) did not use the Y or the (X, Y) calendar dates. Spanish and Natives crossed their calendars and the Aztec festive year, FY, of 18 months:

became the following crossed calendars:

Y ↓ X

I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI XVII XVIII

I 1 8 2 9 3 10 4 11 5 12 6 12 5 12 6 13 7 1

II 2 9 3 10 4 11 5 12 6 13 7 13 6 13 7 1 8 2

III 3 10 4 11 5 12 6 13 7 1 8 1 7 1 8 2 9 3

IV 4 11 5 12 6 13 7 1 8 2 9 2 8 2 9 3 10 4

V 5 12 6 13 7 1 8 2 9 3 10 3 9 3 10 4 11 5

VI 6 13 7 1 8 2 9 3 10 4 11 4 10 4 11 5 12 6

VII 7 1 8 2 9 3 10 4 11 5 12 5 11 5 12 6 13 7

VIII 8 2 9 3 10 4 11 5 12 6 13 6 12 6 13 7 1 8

IX 9 3 10 4 11 5 12 6 13 7 13 7 13 7 1 8 2 9

X 10 4 11 5 12 6 13 7 1 8 1 8 1 8 2 9 3 10

XI 11 5 12 6 13 7 1 8 2 9 2 9 2 9 3 10 4 11

XII 12 6 13 7 1 8 2 9 3 10 3 10 3 10 4 11 5 12

XIII 13 7 1 8 2 9 3 10 4 11 4 11 4 11 5 12 6 13

XIV 1 8 2 9 3 10 4 11 5 12 5 12 5 12 6 13 7 1

XV 2 9 3 10 4 11 5 12 6 13 6 * 6 13 7 1 8 2

XVI 3 10 4 11 5 12 6 13 7 1 7 12 7 1 8 2 9 3

XVII 4 11 5 12 6 13 7 1 8 2 8 1 8 2 9 3 10 4

XVIII 5 12 6 13 7 1 8 2 9 3 9 2 9 3 10 4 11 5

XIX 6 13 7 1 8 2 9 3 10 4 10 3 10 4 11 5 12 6

XX 7 1 8 2 9 3 10 4 11 5 11 4 11 5 12 6 13 7

according to Durán (who did mistakes: months XI and XII)

ci-dessus les données calendaires européennes et indigènes insérées par Landa dans une année vague solaire maya (en désuétude à cette époque coloniale). Les couleurs servent à identifier les 12 mois de l’année julienne, dont le Nouvel an (lundi 1er Janvier) se trouve en case (9, IX) laissée en blanc ; en calendrier maya la date CR de ce jour aurait été un 12 Ben 11 Ch’en. Le tableau commence en juillet (jaune), le 16 Juillet.

…and according to Landa:

Mayas used intertwined calendars,the four more used were:

1. Long Count, CL, open and isomorphic to the set 2. tzolkin of 13 x 20 days, which provides 260 dates X 3. ha'ab ‘year’ which provides 365 dates Y4. Calendar Round, CR, which provides 18 980 dates of the

form (X, Y)

4 in

tert

win

ed c

alen

dar

s CL ‘long count’, tzolkin ‘divinatory calendar’ ha’ab ‘annual calendar’, CR ‘calendar round’

0.0

.0.;

0.0

. (4

Ah

au

, 8

Cu

mk

u)

The monuments prove that the Mayas located an event by placing its day at least with regard to four calendars:

Date CL 9-baktun 16-katun 10-tun; 0-uinal 0-kin Date tzolkin X = 1 Ahau Date ha’ab Y = 3 Zip Date CR (1 Ahau, 3 Zip)

1 Ahau3 Zip

9.16.10.; 0.0.

Stèle F de Quiriguá(Izabal, Guatemala)

15/03/761

tzolkin

ha’ab

CL4 intertwined calendars

75 C

R

74 C

R

│ │ │

Graphic: wheel tzolkin makes turn wheel ha’ab and gearing CR makes move ax CL

Theorem Whatever the integer P, the almanac date of the Pth day of the festive year is of the form αXP, where α [1, 13] and where XP belongs to a class (modulo 5) of four X almanac day signs.

Corollary. Every day of the vague year is associated with a set of 13 x 4 = 52 dates almanac; the tzolkin date of this day would be done, year after year, by the rule: s(αXP ) = s(α) s(XP ) = [(α + 1), (XP + 1) / (X + 5)].

Example The value P = 0 defines the 1st day of the 1st month of the Mayan festive year. Applied to this day, the theorem states, first, that the Mayan New Year is associated with four tzolkin XP=0 day signs. Second, its

corollary says that each New Year date αXP=0 distinguishes and defines a

ha’ab year in the group of 52 years that make up the CR.

Consequence The system of dates αXP supplied a practical means to

label the years of a CR: by making αXP the eponym for the year.

Definition The 4 XP are said the Year’s bearers. For instance the set (Ik, Manik, Eb, Caban) in duty during the Classic.

Theorem of the Mayan soothsayer

Theorem Whatever the integer P, the almanac date of the Pth day of the festive year is of the form αXP, where α [1, 13] and where XP belongs to a class (modulo 5) of four X almanac day signs.


Corollary. Every day of the vague year is associated with a set of 13 x 4 = 52 dates almanac; the tzolkin date of this day would be done, year after year, by the rule: s(αXP ) = s(α) s(XP ) = [(α + 1), (XP + 1) / (X + 5)].

Example The value P = 0 defines the 1st day of the 1st month of the Mayan festive year. Applied to this day, the theorem states, first, that the Mayan New Year is associated with four tzolkin XP=0 day signs. Second, its

corollary says that each New Year date αXP=0 distinguishes and defines a

ha’ab year in the group of 52 years that make up the CR.


Consequence The system of dates αXP supplied a practical means to

label the years of a CR: by making αXP the eponym for the year.

Definition The entities corresponding to the XP=0 are the Year’s bearers. For instance, the set P0 = {Ik, Manik, Eb, Caban} used during the Classic.


Note. The celebration of the Year’s bearers is attested among the Mayas from the first century (Late-Preclassic murals of San Bartolo) to the colonial period (codex of Madrid)

10 Cauac

1 Cauac

5 Cauac

9 Cauac

13 Cauac

4 Cauac

8 Cauac

12 Cauac

3 Cauac

7 Cauac

11 Cauac

2 Cauac

6 Cauac

11 Kan

2 Kan

6 Kan

10 Kan

1 Kan

5 Kan

9 Kan

13 Kan

4 Kan

8 Kan

12 Kan

3 Kan 7 Kan/*11 Kan

12 Muluc

3 Muluc

*7 Muluc

*11 Muluc

*2 Muluc

*6 Muluc

*10 Muluc

*1 Muluc

*5 Muluc

*9 Muluc

*13 Muluc

*4 Muluc

*8 Muluc

13 Hix

4 Hix

8 Hix

12 Hix

3 Hix

7 Hix

11 Hix

2 Hix

6 Hix

10 Hix

1 Hix

5 Hix

9 Hix p. 34 p. 35 p. 36 p. 37

Madrid’s codex shows the cycle of the 52 Year’s Bearers of a colonial CR with

P2 = { Cauac, Kan, Muluc, Hix }

Decisive historical factTo my knowledge and except really rare exceptions, all the Mayan dates of classic period verify the Mayan Soothsayer’s theorem, the corollary of which may be advantageously expressed in terms of a rule that I called Rule of Orthodoxy of the Mayan Chronology, and that was translated by Thompson (1960) into the following easy to use array :

It is a constraint of co-occurrence concerning only the constituents X and of the dates CR. Without the ROCm, the CR would add up 5 x 18 980 dates, instead of 2.12.; 13.0. = 18 980.

How to use the table to decide if a CR date X Y is correct or not? Let the CR date 9 Ahau 19 Cumku proposed by R. Tena. Is it correct? In its table form, the ROCm shows the following: a) when a date contains the sign X = Ahau, then the date Ahau Y are correct if and only if, iff, the rank is 3, 8, 13 or 18; and b): when a date contains the rang = 19, then the dates X 19Y are correct iff the day name X belongs to the set {Cimi, Chuen, Cib, Imix}. We conclude that the given date is not correct, it’s a stared-date: 9 Ahau *19 Cumku. And the same is true for all the 18 980 dates engendered by the device. The gear mechanism proposed by Tena produces a clone of the *Calendar Round.

ROCm

Aztec Century or SA (siècle)a consequence of the soothsayer’s theorem?

Unlike the Mayas, the Aztecs did not used the Long Count and did not have a numeration to express numbers higher than 160,000.

Aztecs did not develop the Y expressions to date the days of their Festive Year (18 x 20 + n); for that they used only the X expressions [1]

Aztecs did not used intertwined calendars, so the Soothsayer’s theorem does not apply in their case.

[1] they don’t used any equivalent of annual calendar dates like “July 13th” but only equivalent expressions of the day of the week like “Sunday or Saint John’s day”

The 20 days of an Aztec month described by Duran (1537-1588)

the soothsayer’s theorem did not apply in the Aztec case, nevertheless…

the Aztecs used its corolary and named their years by the X dates of a fixed day[1] in a way which gave them a cycle of 4 x 13 = 52 years XP.

Historical data do not allow to assert that these 52 years had all the same number of days or that their 360 + n days were all counted and dated

[1] called eponymous or pth day of the year)

The 52 years of an Aztec century described by Duran (1537-1588)

1 Acatl 2 Tecpatl 3 Calli 4 Tochtli5 Acatl 6 Tecpatl7 Calli 8 Tochtli

reading’s instructions:

The count that is called

the Real Calendar

the year of 365.242… days

Look at the sunrises supplied to the Maya a kind of calendar (sometimes

called ‘calendar of horizon’)

SpringSummer Winter

Autumn

subterranean observatories

supplied other Mesoamerican calendars

Su

mm

er s

ols

tice

(Ju

ne)

Su

mm

er s

ols

tice

(Ju

ne)Eq

uin

oxe

s

(Mar

ch, S

epte

mb

er)

Winter s

olstice

(December)

Zenith(May, July)

Année vague n° 1 Année vague n° 2 Année vague n° 3

105 j 260 j 105 j 260 j 105 j 260 j

Disposant d’un tel héliographe, les rois et les prêtres n’avaient nul besoin d’un calendrier de l’année vague, ni même de marquer les jours qui passent.

* d(13 Août, 30 Avril/1Mai) = 260/261 selon que Février compte 28/29 j.

Année vague n° 4 Année vague n° 5 Année vague n° 6

105 j 261 j 105 j 260 j 105 j 260 j

The enlightened period includes: summer solstice, 2 passages at the Zenith; it lasts 105 days (5 months and 5 days). The dark one lasts 260 days in normal year and 261 days in leap year:

It’s a kind of live solar calendar which gives directly and continuously the progress of days and periods of the tropic year

It is certain that Mesoamericans

13/08 30/04

01/05 21/06 12/08

Solsticedark during 260/261 days and enlightened during 105 days

had heliographs which cut the year in 2 parts:

and it is likely that…

some Mesoamerican cities were able to have a « real » calendar, counting 366 days every 4 years

… they have used this practical tool

With the exception of the intertwined calendars used by Mayas during the Classic period, Mesoamerican rulers had at their disposal different guard-times which were likely independent:

The cultural week of 260 almanac dates

The natural heliograph (‘verdadero calendario’)

The cultural vague year of 18/19 periods

The cultural cycle of 52 eponyms of years

If it was the case…

conclusions

classique mayaen calendrier maya classique

on écrit aX Y mais* on n’a pas d’éponymes XP

soit 18 980 expressions pour dater les 18 980 jours du CR

et ces expressions sont fonctionnellement liées

aux durées/dates compte longX Y ci

postclassique en calendrier aztèque on écrit X X mais* on

n’écrit pas de dates Y soit 260 x 52 = 13 520

expressions pour dater les 18 980/18 993 jours du siècle aztèque

ni compte long ni contrôle des dates ou constituants

* éponyme identique au porteur (X de la date tzolkin du jour 0 Pop = 1er jour de la 1ère période)

* les spécialistes diffèrent sur la définition de l’éponyme, celle du jeu de porteurs et sur la valeur de la durée séparant ces deux jours

Th

ank

you

for attention!