maya calendar and aztec calendar: a new presentation and interpretation of their very differences; and a possible understanding of the colonial expression "real calendar"

1. Invitation to revisit the
Mesoamerican Calendars
The one called the Real Calendar
July 16-20, 2012 Daejeon CC, Korea
1000 km

2. s s 1h
t
re 2
ng the
c o ly
2’ J u
01 a,
2 e
H P M K or
g h
in o u t
ur S
d ,
ne e o n DG5:
d o ej
Da 문화와 수학
안드레
C A UTY
보르도
프랑스

3. bi-uprooting
July 16-20, 2012 Daejeon, Korea
κα
ρη
Εὕ
κα
ρη
Inter-dépays ement Εὕ

4. « uprooting » sounds like a ‘quiproquo’ or a
misunderstanding that …
…calls to co-built inter-cultural bridges

5. « uprooting » calls to…
bolon tu Ø-kal Ø-kal catac bolon
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.
…co-built inter-cultural bridges

6. α! bolon tu Ø-kal Ø-kal catac bolon
ηκ
Εὕρ Ø- = ca ‘2nd’ and Ø- = hun ‘1’
Using theoretical concepts[1] by Ferdinand de Saussure,
Bernard Pottier and André Martinet we have solved four
intertwined identification’s problems:
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 Morphemes
2nd articulation: Morphemes are formed of Phonemes (or Graphemes)

7. α!
ηκ bolon tu Ø-kal Ø-kal catac bolon
Εὕρ Ø- = ca ‘2nd’ and Ø- = hun ‘1’
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.
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.

8. 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

9. 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…

10. nights
Welcome in the Mayan
nights

11. supernatural scribes…
doing with math's texts

12. numbers ? calculations? Yes! but for what?
k1196

13. 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

14. 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.
Br ish M int u 7 a án
it useum: l ea 4 de Y xchil

15. as all Mesoamericans did, Mayas and Aztecs
used the same divinatory week of 13 x 20 = 260
expressions αX (13 integers and 20 day signs)
α is an integer, X a day sign, and the sequence of the αX
sounds like: 2 Sunday, 3 Monday… and follows the rule s(αX) =
s(α)s(X)

16. Mayan and Aztec examples of the 13 integers α

17. The 20 Mayan and Aztec day signs X
Mayan Aztec

18. αX 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”

19. Mayas and Aztecs used the same divinatory week
in order to qualify the future or to celebrate the
rulers or the cities history
Here a very common Mayan divinatory practice done
by small translations in the divinatory week:
αX + d = α’X’ (d in additive numeration)
13
Edznab
Ahau 28 2 24 13
Eb
Kan
? Cib
13 Ahau + 28 = 2 [Lamat] + 24 = 13 [Eb]

20. another common Maya use: αXβY + d = α’X’β’Y’
(duration d is done in dispositional numeration):
the equations of the form αXβ Y + Σ(ciPi) = α’X’β ’Y’
E2 F2 H3 G4 H4 G5
(1 Ben 1 Ch’en) + [10-[kin] 5-uinal 3-tun 2-katun] = (11 Ahau 8 Tzec)

21. 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

22. July
a famous French date:
mardi 14 juillet 1789
St Bonaventure, évêque,
un jour après le dernier
quartier de Lune
J
Tuesday, July 14th, 1789
St Bonaventure, bishop,
A day after the last district of
the Moon
Almanach royal, année commune M.DCC.LXXXIX et
M.DCC.LXXXX., BN de France, Lc25-28

23. 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: (Xα, XP α)

24. Mayas and Aztecs did not
have
the same dating system or the same
What a cognitive difference!
calendar
IDP

25. Mayan CR dates follow the (αX, β Y)-model: 6 Ahau, 13 Yaxkin
Aztec SA dates follow the (αX, αXP)-model: 7 Acatl, 8 Acatl
Mixtec dates follow the (Xα, XPα)-model: ‘crocodile’ 13, ‘house’
7

26. so, Aztec and Mayan chronicles are
very different:
the firsts look like comic strips
the second like texts full of equations

27. Aztec chronicle
αXP dates
looks like a comic strip

28. Mayan chronicle
looks 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

29. equations of the form
αX β Y + Σ (ci Pi) = α’X’ β ’Y ’

30. 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

31. Tulum by Frederick Catherwood (1799 - 1854)

32. 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…

33. d de
an itu
ns erv
tio f s
ua s o
it n
S o Discovery
a ti Conquests
rel Spoliation
Colonization
but now: who and how
rebuilt the disappeared
Mesoamerican ideas?
Training
Evangelization
Incomprehension
Education, Translation…

34. Colonial opinion
“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”
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.
Diego de Landa
(1524-1579)
Bishop of the Yucatan

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

36. 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…

37. n !
u tio dangerous simplifying ideas…
ca
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)
…and amazing hybrids

38. 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.

39. hybridize is not totally inevitable
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.

40. 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

41. 1. The divinatory week/almanac

42. 2a. The undergone seasonal year

43. 2b. The Mayan discretized festive year

44. Mesoamerican 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 East
E Spring
(Zenith, Nadir, North, Summer S
Winter
West, South and East). Autumn

45. 26
0
x
36
5.
Call to revisit the main thesis 24
2…
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…

46. Important historical data
Almost 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 αX in the divinatory week (it is the case for
instance in the Duran’s descriptions of the Aztecan year).
Some colonial documents claimed that the 5 (or 6) aciagos days were not
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.

47. More important historical data
In 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 β 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 (αX, βY). 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.

48. More important historical data…
Between the 4th and the 10th century, the Mayas used
not only the 260 ancient αX dates, but also the 365
innovative β Y dates.
Their festive year had became the following calendar
which gave the αX and the β Y correlated dates:

49. 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 α (de 1 à 13) de la date tzolkin, αX, du jour dont le signe X se trouve (en bleu)
en dernière colonne. La date ha’ab, β 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.

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

51. according to Durán (who did mis takes : months XI
and XII)
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

52. …and ac cording to Landa:
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.

53. 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 αX
3. ha'ab ‘year’ which provides 365 dates β Y
4. Calendar Round, CR, which provides 18 980 dates of the
form (αX, β Y)

54. 4 intertwined calendars
CL ‘long count’, tzolkin ‘divinatory calendar’
ha’ab ‘annual calendar’, CR ‘calendar round’
0.0.0.;0.0. (4 Ahau, 8 Cumku)

55. 4 intertwined calendars
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 αX = 1 Ahau
Date ha’ab β Y = 3 Zip
Date CR (1 Ahau, 3 Zip)
ha’ab
tzolkin
1 Ahau
3 Zip
CL
CR
CR
│
74
75
│ 9.16.10.; 0.0. │
‌
Stèle F de Quiriguá
(Izabal, Guatemala) Graphic: wheel tzolkin makes turn wheel ha’ab and gearing CR makes move ax CL
15/03/761

57. Theorem of the
Mayan soothsayer the P
Theorem Whatever the integer P, the almanac date of day of the
th
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.

58. 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.

59. Theorem of the
Mayan soothsayer
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.

60. Theorem of the
Mayan soothsayer
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)

61. Madrid’s codex shows the cycle of the
52 Year’s Bearers of a colonial CR with
P2 = { Cauac, Kan, Muluc, Hix }
10 Cauac 11 Kan 12 Muluc 13 Hix
1 Cauac 2 Kan 3 Muluc 4 Hix
5 Cauac 6 Kan *7 Muluc 8 Hix
9 Cauac 10 Kan *11 Muluc 12 Hix
13 Cauac 1 Kan *2 Muluc 3 Hix
4 Cauac 5 Kan *6 Muluc 7 Hix
8 Cauac 9 Kan *10 Muluc 11 Hix
12 Cauac 13 Kan *1 Muluc 2 Hix
3 Cauac 4 Kan *5 Muluc 6 Hix
7 Cauac 8 Kan *9 Muluc 10 Hix
11 Cauac 12 Kan *13 Muluc 1 Hix
2 Cauac 3 Kan *4 Muluc 5 Hix
6 Cauac 7 Kan/*11 Kan *8 Muluc 9 Hix
p. 34 p. 35 p. 36 p. 37

62. Decisive historical fact
To 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
β 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.

63. ROCm
How to use the table to decide if a CR date αX β 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 α
Ahau β Y are correct if and only if, iff, the rank β is
3, 8, 13 or 18; and b): when a date contains the
rang β = 19, then the dates αX 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.

64. 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 β Y
expressions to date the days of
their Festive Year (18 x 20 + n);
for that they used only the αX
expressions [1]
Aztecs did not used intertwined
calendars, so the Soothsayer’s
theorem does not apply in their
case.
they don’t used any equivalent of annual
[1] The 20 days of an Aztec month
calendar dates like “July 13th” but only
equivalent expressions of the day of the described by Duran (1537-1588)
week like “Sunday or Saint John’s day”

65. the soothsayer’s theorem did not apply
in the Aztec case, nevertheless…
the Aztecs used its corolary
and named their years by
the αX dates of a fixed
day[1] in a way which gave
them a cycle of 4 x 13 = 52
years αXP.
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
The 52 years of an Aztec century
described by Duran (1537-1588)
[1] called eponymous or pth day of the year)

66. reading’s
instructions:
1 Acatl 2 Tecpatl
3 Calli 4 Tochtli
5 Acatl 6 Tecpatl
7 Calli 8 Tochtli

67. The count that is called
the Real Calendar
the year of 365.242… days

69. Look at the sunrises supplied to the
Maya a kind of calendar (sometimes
called ‘calendar of horizon’)
Spring
Summer Winter
Autumn

71. subterranean observatories
supplied other Mesoamerican calendars

72. tice
Summer sols
(June)

73. te
W (D
in ece
s
r s mb
tic
e
ol er)
(Ma Equi
rch nox
, Se es
p te
mb
er)
e
Zenith
tic
Summer sols
(May, July)
(June)

74. 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:
Anné e vague n° 1 Anné e vague n° 2 Anné e vague n° 3
105 j 260 j 105 j 260 j 105 j 260 j
Anné e vague n° 4 Anné e vague n° 5 Anné e vague n° 6
105 j 261 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.

75. It is certain that Mesoamericans
had heliographs which cut the year in 2 parts:
01/05 21/06 12/08
13/08
Solstice
30/04
dark during 260/261 days and enlightened during 105 days
It’s a kind of live solar calendar which gives
directly and continuously the progress of
days and periods of the tropic year
and it is likely that…

76. … they have used this practical tool
some Mesoamerican cities were able to
have a « real » calendar, counting 366 days
every 4 years

77. If it was the case…
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

78. conclusions
classique maya postclassique
en calendrier maya classique en calendrier aztèque
on écrit aX β Y mais* on écrit αX αXγ mais* on
on n’a pas d’éponymes αXP n’écrit pas de dates β Y
soit 18 980 expressions pour soit 260 x 52 = 13 520
dater les 18 980 jours du CR expressions pour dater les
et ces expressions sont 18 980/18 993 jours du
fonctionnellement liées siècle aztèque
aux durées/dates compte long ni compte long ni contrôle
αX β Y Σ ci des dates ou constituants
* éponyme identique au porteur (X de * les spécialistes diffèrent sur la
la date tzolkin du jour 0 Pop = 1er jour définition de l’éponyme, celle du jeu de
porteurs et sur la valeur de la durée
de la 1ère période)
séparant ces deux jours

79. Thank you
for attention!