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Revisit Aztec and Maya calendars
 

Revisit Aztec and Maya calendars

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

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

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  • Culture et Mathématiques (munhwa wa suhag)
  • HPM experiences: Change of scenery, uprooting. Disorientation, cognitive dissonance, commitive conflict causing a reorientation. On the left: the Aztec’s point of view; on the right: the Epistemologist’s point of viewD
  • Kerr 1196
  • Good, evil or indifferent. Relever les défis d’une divination qui trouve ses figures dans le mouvement des astres et dans les visions aussi difficilement arrachées aux mondes invisibles que la main n’attrape un poisson.
  • Let’s see what was common. ID P/A indicateur de date postérieur / antérieur
  • les scribes aztèques dressaient des listes d’années (sur la figure, les années successives : 6 Calli , 7 Tochtli , 8 Acatl ) diversement mises en page parfois en tableau. Sur ce canevas, il inscrivaient les événements non par un texte mais par des dessins conventionnels et sans les relier, comme les Mayas, par des équations jouant sur la dualité date/durée.
  • Le rapport de servitue (codex Kingsborough)
  • They have his perfect year as ours of 365 days and six hours […] Of these six hours it was done every four years one day, and this way they had of four in four years the year of 366 days.
  • The Mesomerican calendar was the result of the combination between a cycle of 365 days, called in nahuatl xiuhpohualli or ' account of the years ' (ha'ab in Maya), and another cycle of 260 days, called tonalpohualli or ' account of the days ' (tzolkin in Maya) .
  • La unidad cultural de los pueblos mesoamericanos se refleja en varios rasgos que Paul Kirchhoff definió en 1943 como el complejo mesoamericano. Let’s see how did Aztecs and Mayas date an event.
  • Or bundled in 24 trecenas and a 1 day
  • Il y a aussi de bonnes raisons de penser que les savants connaissaient et utilisaient aussi une autre année (zodiacale) divisée en 28 treizaines (avec ou sans un résidu), soit une année de 364 jours.
  • Il y a aussi de bonnes raisons de penser que les savants connaissaient et utilisaient aussi une autre année (zodiacale) divisée en 28 treizaines (avec ou sans un résidu), soit une année de 364 jours.
  • Il y a aussi de bonnes raisons de penser que les savants connaissaient et utilisaient aussi une autre année (zodiacale) divisée en 28 treizaines (avec ou sans un résidu), soit une année de 364 jours.
  • Il y a aussi de bonnes raisons de penser que les savants connaissaient et utilisaient aussi une autre année (zodiacale) divisée en 28 treizaines (avec ou sans un résidu), soit une année de 364 jours.
  • Il y a aussi de bonnes raisons de penser que les savants connaissaient et utilisaient aussi une autre année (zodiacale) divisée en 28 treizaines (avec ou sans un résidu), soit une année de 364 jours.
  • Comments:   Scribal workshop  Iconographic Elements:  Ah K'hun, orator, Bundle, tribute, cache, Codex - book, Conch shell, Inscription - other text, Mirror, Ruler, king, cacique, governor, potentate, Scribes, artists, carvers, Ink or paint container, Sabak Kuch, Throne
  • Plaque de Leyde : première attestation du zéro mayas (il s’agit du zéro ordinal des dates, dans la date ha’ab 0 Yaxkin). Le CR possède 4 clones, chacun de 18 980 éléments)
  • A chaque tour du Calendrier Rituel, l’engrenage aux 2 roues ( tzolkin x ha’ab ) ‘gradue’ la droite CL du Compte Long : 0 1 CR 2 CR 3 CR 4 CR 5 CR etc. Dans cette graduation, le contact des 3 dates (CL 9.16.10. ; 0.0. = 1 414 800 , date tzolkin 1 Ahau et date ha’ab 3 Zip ) se trouve entre les graduations 74 CR (1 404 520) et 75 CR (1 423 500). Le mécanisme implique l’invariance des nombres de jours de chaque cycle ou unité (le tzolkin a toujours 260 j, le ha’ab toujours 365, le tun toujours 360, et le CR toujours 18 980 jours, et il implique aussi que les mouvements se fassent sans glissement et sans changement les conditions initiales
  • Comments:   Scribal workshop  Iconographic Elements:  Ah K'hun, orator, Bundle, tribute, cache, Codex - book, Conch shell, Inscription - other text, Mirror, Ruler, king, cacique, governor, potentate, Scribes, artists, carvers, Ink or paint container, Sabak Kuch, Throne
  • Each was associated with a direction (north, west, sud, east) and a color (white, black, yellow, red)
  • Each was associated with a direction (north, west, sud, east) and a color (white, black, yellow, red)
  • Each was associated with a direction (north, west, sud, east) and a color (white, black, yellow, red)
  • Each was associated with a direction (north, west, sud, east) and a color (white, black, yellow, red)
  • Voir par exemple ANotational Explanationfor Maya Calendar RoundDates Suchas 11 Eb*16 Mac by VICTORIA R. BRICKER (sur wayeb)

Revisit Aztec and Maya calendars Revisit Aztec and Maya calendars Presentation Transcript

  • Invitation to revisit the Mesoamerican Calendars The one called the Real Calendar July 16-20, 2012 Daejeon CC, Korea1000 km
  • 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
  • bi-uprootingJuly 16-20, 2012 Daejeon, Korea κα ρη Εὕ κα ρηInter-dépays ement Εὕ
  • « uprooting » sounds like a ‘quiproquo’ or amisunderstanding that … …calls to co-built inter-cultural bridges
  • « uprooting » calls to… bolon tu Ø-kal Ø-kal catac bolona Mayan case where the commutation aB/Ba produces asubtle switch; that of the operation (protraction/ addition)and of the type (additive/protractive) of numeration. …co-built inter-cultural bridges
  • α! bolon tu Ø-kal Ø-kal catac bolon ηκΕὕρ Ø- = ca ‘2nd’ and Ø- = hun ‘1’Using theoretical concepts[1] by Ferdinand de Saussure,Bernard Pottier and André Martinet we have solved fourintertwined identification’s problems:1. That of the indetermination of the SIGNIFIER, Sa, (‘signifiant’)2. That of the syntactic ambiguity of the FORMAL CONSTITUENT, Sy, OFTHE 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) Salinguistic sign: Si = ——————————————————————— = ———— Formal constituent (Sy) / Meaningful constituent (Sé) Sy / Sé1st articulation: Phrases are formed of Morphemes2nd articulation: Morphemes are formed of Phonemes (or Graphemes)
  • α! ηκ bolon tu Ø-kal Ø-kal catac bolonΕὕρ Ø- = ca ‘2nd’ and Ø- = hun ‘1’our bridge was the discovery of a subtle Mayan switchbetween two different numerical processes shown and doneby an interpretation in the metalanguage provided by theYucatan language of the Maya scribe.is it strange?not so strange! the same subtle switch occurs inspoken French where “vingt-quatre” (twenty-four)refers to an additive composition 20 + 4 while“quatre vingt(s)” (eighty) refers to a multiplicativedetermination 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 cognitivebridges, can develop new ideas and propose alternativeinterpretations…
  • nights Welcome in the Mayan nights
  • supernatural scribes… doing with maths texts
  • numbers ? calculations? Yes! but for what? k1196
  • Mayan numbers are bricks to construct thecalendars which give to the Mesoamericansoothsayer his most used tool to “see” theinvisible world and to predict the fate (good, evilor indifferent) of everybody, everything, everycycle of time…the divinatory week is among the Mesoamericansthe most used, as well as the oldest, calendar
  • 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
  • as all Mesoamericans did, Mayas and Aztecsused the same divinatory week of 13 x 20 = 260expressions αX (13 integers and 20 day signs)α is an integer, X a day sign, and the sequence of the αXsounds like: 2 Sunday, 3 Monday… and follows the rule s(αX) =s(α)s(X)
  • Mayan and Aztec examples of the 13 integers α
  • The 20 Mayan and Aztec day signs XMayan Aztec
  • αX dates were used to name persons, thingsor 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 weekin order to qualify the future or to celebrate therulers or the cities historyHere a very common Mayan divinatory practice doneby 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]
  • 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)
  • as we can imagine every people develops itsown arithmetic and calendar skills and habits:1.- Occidental dates: Mardi 14 juillet 1789, Saint Bonaventure Sunday, July 8, 20122.- 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
  • 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
  • the Mixtec and us did not havethe same dating system or the same calendarthe above Mixtec date:crocodile 13 house 7 had shape: (Xα, XP α)
  • Mayas and Aztecs did not have the same dating system or the sameWhat a cognitive difference! calendar IDP
  • Mayan CR dates follow the (αX, β Y)-model: 6 Ahau, 13 Yaxkin Aztec SA dates follow the (αX, αXP)-model: 7 Acatl, 8 AcatlMixtec dates follow the (Xα, XPα)-model: ‘crocodile’ 13, ‘house’ 7
  • so, Aztec and Mayan chronicles arevery different: the firsts look like comic strips the second like texts full of equations
  • Aztec chronicleαXP dates looks like a comic strip
  • Mayan chroniclelooks like a text full of equationsSous 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 formαX β Y + Σ (ci Pi) = α’X’ β ’Y ’
  • today, Mayan arithmeticand Mesoamericancalendars are just SECS,strange extinct cognitivespecies, to be discovered,reinvent and rebuiltQuoting Proust: “partons à larecherche des nombres perdus”let’s go to the search of the lostnumbers
  • Tulum by Frederick Catherwood (1799 - 1854)
  • the search for the ‘lost numbers’ began at the timeof the discovery of America (1492), it continuestoday and will continue tomorrow…this task was (and still is) specially hard not onlybecause of the cultural gap between Mesoamericanand Spanish but also because the new rulers ofAmerica tried to remove the divinatory week (avery diabolic work, they said) and to transformthe festive year used by the Natives into acalendar which sounds like the Julian (thenGregorian) one; first steps of the process leading tothe imposition of the Spanish calendar…
  • d de an itu ns erv tio f s ua s o it n S o Discovery a ti Conquestsrel Spoliation Colonization but now: who and how rebuilt the disappeared Mesoamerican ideas? Training Evangelization Incomprehension Education, Translation…
  • Colonial opinion“Tienen su año perfectocomo el nuestro de 365 díasy seis horas […] De estasseis horas se hacía cadacuatro años un día, y asítenían de cuatro en cuatroaños el año de 366 días”They had a year perfect as ours of 365 days andsix hours […]. Of these six hours they madeevery four years one day, and so they had fromfour to four years one year of 366 days. Diego de Landa (1524-1579) Bishop of the Yucatan
  • Modern opinion “El calendario mesoamericano era el resultado de la combinación entre un ciclo de 365 días, llamado en nahuatl xiuhpohualli o ‘cuentadel año’ (haab en maya), y otro ciclo de 260 días, llamado tonalpohualli o ‘cuenta de los días’ (tzolkin en maya)”The Mesoamerican calendar wasthe result of the combination of acycle of 365 days called xiuhpohu-alli in Nahuatl […] (haab in Maya),and an another cycle of 260 calledtonalpohualli […] (tzolkin in Maya) Rafael Tena, ‘El calendario mesoamericano’, Arqueología mexicana, Vol. 7, nº 41, México, 2000, p. 5-6
  • Modern opinion: 260 x 365 = 52 / 18 980“Se requería el transcurso de 18 980 díasnominales, equivalentes a un ‘siglo’ de 52años, para que se agotaran todas lasposiciones posibles de un día cualquieradel tonalpohualli dentro del xiuhpohualli, yviceversa […]. Cada uno de los 52 añostenia su nombre propio…”18 980 days, equivalent to a century of 52 years, they had to wait for eachdate Tonalpohualli goes through all possible positions of the xiuhpohualli,and vice versa […] Each of 52 years had its own name…
  • n ! u tio dangerous simplifying ideas… caThey said that all Mesoamericans share the same calendar: a combinationof the divinatory week of 13 x 20 days with the solar year supposed,without proofs, to be a calendar counting 365 dated days. A combinationoften described as a mechanism borrowed from the industrial world!They said that the Aztecs had intertwined these two cycles and obtaineda calendar of 18 980 days similar to the Calendar Round used by theMayas or a century of 52 years that it is called the "Siècle Aztèque", SA(Aztec century) …and amazing hybrids
  • Waldeck! What did you see?The weight of time and habits makes difficult for Waldeck to see, admit and understandthe Mayan creations which were for his time so strange and so radically different.
  • hybridize is not totally inevitableWhether called a “monstrous hybrid”, “Waldeck’s elephant”,or even the “self indulgence of the ethno-X”, the underlyingconcept reflects not only the ineluctable fact of projectingone’s own frames of reference and one’s own forms ofknowledge upon the foreign work that we are trying tounderstand, but rather the human failure to submit allreadings and interpretations to systematic and collectivecriticism, a kind of criticism that has to be interdisciplinaryand interethnic or intercultural.
  • 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 days2.- a solar and/or festive year imposed by Nature and/or Rulers & Priests Gods are times bearers, they bear the packages of time
  • 1. The divinatory week/almanac
  • 2a. The undergone seasonal year
  • 2b. The Mayan discretized festive year
  • Mesoamerican Festive Year If the residual period is well fixed and defined, the MesoamericanFestive year, FY, counts 365 days bundled in 18 months of 20 days and1 residue of 5 days. It is the Mesoamerican way to divide ‘discrétiser’ the continuous of thetime (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
  • 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…
  • Important historical dataAlmost everywhere and always in Mesoamerica, the days of the FYwere not dated by means of an annual calendar, but they all weredistinguished and defined by the highly symbolic ‘color’ conferred to eachof them by its expression αX in the divinatory week (it is the case forinstance in the Duran’s descriptions of the Aztecan year).Some colonial documents claimed that the 5 (or 6) aciagos days were notdated at all; so, we haveto observe that the fact ofusing a festive year FY of365 days does not implythe obligation to namethem all; documents saidthat people dated only360 days, the 360 dayswhich belong to the 18months. If a calendar, oras a calendar, the FestiveYear would have only 360dates, in some cases.
  • More important historical dataIn 4th century, the Mayas had developed and were using asystem of writing to uniquely define the days of the festiveyear. Making that, their festive year became a calendar with365 dates, all written in the innovative shape β YAt the same time, the Mayas had formed the product of twoimportant cycles they used in such a particular way thatthe coupling 260 x 365 gives a product of 18 980elements. In this manner, they created a new calendarwhich dates are the 18 980 pairs (αX, βY). The Mayan nameof that calendar is yet unknown. Scientists say : ‘CalendrierRituel’ or Calendar Round, CR.Finally, the Mayas put in narrow correspondence theCalendar Round CR and the Long Count CL as shown bythousands of initial series. In conclusion: the Maya disposedof several intertwined calendars.
  • More important historical data…Between the 4th and the 10th century, the Mayas usednot only the 260 ancient αX dates, but also the 365innovative β Y dates.Their festive year had became the following calendarwhich gave the αX and the β Y correlated dates:
  • Le tableau montre une année maya de type [(28 x 13) + 1] = [(18 x 20) + 5] et de 1erjour (2 Eb, 0 Pop). Les 365 jours sont organisés en treizaines (distinguées par unecouleur) et en mois (de 20 jours, matérialisés par les colonnes). Chaque case contientle 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). Lecalendrier 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 particularAztecs and Mayas) did not use the β Y or the (αX, β Y)calendar dates. Spanish and Natives crossed their calendarsand the Aztec festive year, FY, of 18 months: became the following crossed calendars:
  • according to Durán (who did mis takes : months XIand 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 7VIII 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 12XIII 13 7 1 8 2 9 3 10 4 11 4 11 4 11 5 12 6 13XIV 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 2XVI 3 10 4 11 5 12 6 13 7 1 7 12 7 1 8 2 9 3XVII 4 11 5 12 6 13 7 1 8 2 8 1 8 2 9 3 10 4XVIII 5 12 6 13 7 1 8 2 9 3 9 2 9 3 10 4 11 5XIX 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
  • …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.
  • 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. haab ‘year’ which provides 365 dates β Y 4. Calendar Round, CR, which provides 18 980 dates of the form (αX, β Y)
  • 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)
  • 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
  • Theorem of the Mayan soothsayer the PTheorem Whatever the integer P, the almanac date of day of the thfestive year is of the form αXP, where α ∈ [1, 13] and where XP belongs toa 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 afteryear, 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 Mayanfestive year. Applied to this day, the theorem states, first, that the MayanNew Year is associated with four tzolkin XP=0 day signs. Second, itscorollary says that each New Year date αXP=0 distinguishes and defines aha’ab year in the group of 52 years that make up the CR.Consequence The system of dates αXP supplied a practical means tolabel 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 soothsayerTheorem Whatever the integer P, the almanac date of the Pth day of thefestive year is of the form αXP, where α ∈ [1, 13] and where XP belongs toa class (modulo 5) of four X almanac day signs.
  • Theorem of the Mayan soothsayerCorollary. 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 afteryear, 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 Mayanfestive year. Applied to this day, the theorem states, first, that the MayanNew Year is associated with four tzolkin XP=0 day signs. Second, itscorollary says that each New Year date αXP=0 distinguishes and defines aha’ab year in the group of 52 years that make up the CR.
  • Theorem of the Mayan soothsayerConsequence The system of dates αXP supplied a practical means tolabel 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 fromthe first century (Late-Preclassic murals of San Bartolo) to the colonialperiod (codex of Madrid)
  • Madrid’s codex shows the cycle of the52 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 Hix13 Cauac 1 Kan *2 Muluc 3 Hix 4 Cauac 5 Kan *6 Muluc 7 Hix 8 Cauac 9 Kan *10 Muluc 11 Hix12 Cauac 13 Kan *1 Muluc 2 Hix 3 Cauac 4 Kan *5 Muluc 6 Hix 7 Cauac 8 Kan *9 Muluc 10 Hix11 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
  • Decisive historical factTo my knowledge and except really rare exceptions, all theMayan dates of classic period verify the Mayan Soothsayer’stheorem, the corollary of which may be advantageouslyexpressed in terms of a rule that I called Rule of Orthodoxy ofthe 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 andβ of the dates CR. Without the ROCm, the CR would add up 5 x 18 980dates, instead of 2.12.; 13.0. = 18 980.
  • ROCmHow to use the table to decide if a CR date αX β Yis correct or not? Let the CR date 9 Ahau 19Cumku proposed by R. Tena. Is it correct? In itstable form, the ROCm shows the following: a) whena date contains the sign X = Ahau, then the date αAhau β Y are correct if and only if, iff, the rank β is3, 8, 13 or 18; and b): when a date contains therang β = 19, then the dates αX 19Y are correct iffthe day name X belongs to the set {Cimi, Chuen,Cib, Imix}. We conclude that the given date is notcorrect, it’s a stared-date: 9 Ahau *19 Cumku. Andthe same is true for all the 18 980 datesengendered by the device. The gear mechanismproposed by Tena produces a clone of the*Calendar Round.
  • 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”
  • 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)
  • reading’sinstructions:1 Acatl 2 Tecpatl3 Calli 4 Tochtli5 Acatl 6 Tecpatl7 Calli 8 Tochtli
  • The count that is calledthe Real Calendar the year of 365.242… days
  • Look at the sunrises supplied to theMaya a kind of calendar (sometimes called ‘calendar of horizon’) Spring Summer Winter Autumn
  • subterranean observatoriessupplied other Mesoamerican calendars
  • ticeSummer sols (June)
  • 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 ticSummer sols (May, July) (June)
  • The enlightened period includes: summer solstice, 2passages at the Zenith; it lasts 105 days (5 monthsand 5 days). The dark one lasts 260 days in normalyear 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.
  • It is certain that Mesoamericans had heliographs which cut the year in 2 parts: 01/05 21/06 12/0813/08 Solstice30/04 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…
  • … they have used this practical toolsome Mesoamerican cities were able tohave a « real » calendar, counting 366 daysevery 4 years
  • If it was the case…With the exception of the intertwinedcalendars used by Mayas during theClassic period, Mesoamerican rulershad 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
  • 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 aztèqueaux 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
  • Thank you for attention!