Some interesting remarks on the Incan calendar system from this site on Incan civilization.

..The Inca had a calendar based on the observation of both the Sun and the Moon, and their relationship to the stars. Names of 12 lunar months are recorded, as well as their association with festivities of the agricultural cycle.

There is no suggestion of the widespread use of a numerical system for counting time, although a quinary decimal system, with names of numbers at least up to 10,000, was used for other purposes. The organization of work on the basis of six weeks of nine days suggests the further possibility of a count by triads that could result in a formal month of 30 days.

A count of this sort was described by Alexander von Humboldt for a Chibcha tribe living outside of the Inca Empire, in the mountainous region of Colombia. The description is based on an earlier manuscript by a village priest... The smallest unit of this calendar was a numerical count of three days, which, interacting with a similar count of 10 days, formed a standard 30-day month.

Every third year was made up of 13 moons, the others having 12. This formed a cycle of 37 moons, and 20 of these cycles made up a period of 60 years, which was subdivided into four parts and could be multiplied by 100. A period of 20 months is also mentioned. Although the account of the Chibcha system cannot be accepted at face value, ...it is suggestive of devices that may have been used also by the Inca.

In one account, it is said that the Inca Veracocha established a year of 12 months, each beginning with the New Moon, and that his successor, Pachacuti, finding confusion in regard to the year, built the sun towers in order to keep a check on the calendar. Since Pachacuti reigned less than a century before the conquest, it may be that the contradictions and the meagerness of information on the Inca calendar are due to the fact that the system was still in the process of being revised when the Spaniards first arrived.

Despite the uncertainties, further research has made it clear that at least at Cuzco, the capital city of the Inca, there was an official calendar of the sidereal-lunar type, based on the sidereal month of 27 1/3 days. It consisted of 328 nights (12X271/3) and began on June 8/9, coinciding with the helical rising (the rising just after sunset) of the Pleiades; it ended on the first Full Moon after the June solstice (the winter solstice for the Southern Hemisphere). This sidereal-lunar calendar fell short of the solar year by 37 days, which consequently were intercalated. This intercalation, and thus the place of the sidereal-lunar within the solar year, was fixed by following the cycle of the Sun as it strengthened to summer (December) solstice and weakened afterward, and by noting a similar cycle in the visibility of the Pleiades.


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Regards,


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