9.16.0.2.0

Long count glyphs: Inscriptions almost always begin with a long count called the "initial series" date.  This date is introduced by a distinctive "initial series introductory glyph" (ISIG), followed by the period glyphs and numbers. Period glyphs are usually "head" types, as at the left. Glyphs are usually written in two columns,  and read from left to right in each row, but in some inscriptions, the date is written in a single row.  The "head" style glyphs are rather variable in detail. More abstract "symbol" type period glyphs are also encountered. But since the periods are always listed in order from baktun to k'in, they can be read even if they are not individually recognizable.

Here  is another example of a long count, using the symbol forms of the period glyphs. It counts 1394815 days, and correlates to 29 June 706 AD: 

In the Dresden Codex, the only Maya glyph book that includes long counts, the introductory and period glyphs are omitted. The number glyphs are in a single column, and the order in which they are written is enough to determine their meaning. The codex style date at the right reads 9.16.0.2.0. Note the distinctive codex form of the zero glyph.  The long count is actually a modified base 20 number system: All periods except the tun are 20 times the previous period. The Maya used place holding arithmetic and the concept of zero before they were invented in the Old World.

The base date was almost certainly set in late summer  for astronomical reasons.  In Maya Cosmos, Linda Schele noted that in mid-August, the Milky Way rises high in the sky.  This likely represents the "World Tree" or "Raised Up Sky" (Wakah chan) in Classical Creation accounts. See The Maya Creation Myth and the Milky Way  at this web site.  Vincent Malmström suggests that August 13 was chosen as the base date because the sun is at the zenith on this date at Izapa, an important pre-Classical site in Chiapis, where he argues the Maya calendar originated.  See Malmström's theory of calendrical origins on-line.

Whether there is also astronomical significance in the choice of 3114 BC as the base date is more debatable.  Victoria Bricker has pointed out that if the base date is 11 Aug 3114 BC,  the current "great cycle" of the long count will end on the winter solstice ( LC 13.0.0.0.0 =  21 Dec 2012).  She suggests that the base date was chosen so the LC  would end on the solstice.  John Major Jenkins has elaborated this theory.  Due to precession of the equinoxes, in the early 21st century  the position of the sun at the solstice moves into the centre of the Milky Way.  He suggests that this ties completion of the long count cycle to the creation symbolism associated with the beginning of the count.  However, whether the Maya were aware of precession  is still regarded by many archaeoastronomers as an open question.  See Jenkins,  The How and Why of the Mayan End Date in 2012 AD on-line.

A rival theory of the origin of the long count was proposed by John Teeple in 1930, and recently revived by Malmström.  This theory assumes that the inventors of the Long count would have chosen a base date that was both a full number of katuns and a full number of haabs, approximate solar years, in the past.  The LC base date  was on the haab date 8 Kumk'u.  Thus the long count would have been introduced on  a zero katun count that fell on 8 Kumk'u.  Because katuns end on a particular haab date only once every 73 katuns, the only likely candidate is LC  7.6.0.0.0  = 14 September 236 BC.  Earlier 8 Kumk'u katun ends occurred  before the rise of Mesoamerican civilization, and later examples are long after the LC appeared in inscriptions.  If this explanation of the base date is correct, 3114 BC has no astronomical meaning.  See Malmström,  The Astronomical Insignificance of 13.0.0.0.0. on-line
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Long count names:  The last initial series was recorded at Tonina, on  10.4.0.0.0 (20 Jan 909 AD).  Since it was no longer in use when the Spanish arrived, we are uncertain about some LC terminology.  Only the period names katun, winal, and kin are known with assurance. Tun is a Yucatec word for "year,"   but both LC and calendar round "years" seem to have sometimes been called haab. Baktun is only a plausible Yucatec name.  Linda Schele noted that  the phonetic value of the symbol form of the baktun glyph is pi.  The ISIG glyph has recently been read tzik haab’, "the count of years."

cc ISIG

The Yucatec names of the months are: Pop, Wo, Sip, Sotz', Sek, Xul, Yaxk'in, Mol, Ch'en, Yax, Sak, Seh, K'ank'in, Muan, Pax, K'ayeb, Kumk'u. The name glyphs, ending with the Wayeb glyph, are illustrated in order at the right. 

The post-Classical Maya numbered the days of the month from 1 to 20, but in Classical inscriptions, the last day of the month was written as the day on which the new month was "seated" (chum).  Thus the last day of Yaxk'in was usually written as chum Mol, rather than as 20 Yaxk'in. This amounts to taking chum Mol as the first day of Mol, and numbering the days of the month from 0 (chum) to 19. 
 

The calendar round  combines the tzolk'in and haab dates. The lowest common multiple of  the 260 days in the tzolk'in and 365 days in the haab is 18980. This is 52 haab, just short of 52 years in our calendar. Thus a combined tzolk'in and haab date repeats only after this lapse of time.

The beginning of the long count occurred on the calendar round date 4 Ahaw 8 Kumk'u. The calendar round date of any long count can be calculated by adding cycles of 260 and 365 days from 4 Ahaw  and 8 Kumk'u to reach the long count position. Once again, the arithmetic is simple in principle, but a calendar calculator program will make the task much easier. 

In dates on monuments, the calendar round usually follows immediately after the long count (as in the example above). However, in some inscriptions, the tzolk'in date and haab date are separated by 5-8 glyphs called the "lunar series". The LS is the date in yet another cycle, a lunar calendar. 

Right: The day 3 Xochitl in the Aztec tonalpohauli, equivalent to 3 Ahaw in the Maya tzolk'in
 

Why the 260 day cycle assumed such importance is unknown.  Suggestions include the length of agricultural seasons and the human gestation period.  The most convincing are astronomical.  Early in the 20th C., Zelia Nuthall noted that a missionary report, the Manuscript of Serna, asserts that the tonalpohauli approximates the time Venus is visible to the naked eye as morning star.   The 365 day "year" is clearly a solar calendar.  Anthony Aveni suggests  that the Calendar Round  "synthesizes the primary solar and Venus intervals."

Another popular theory in the early years of Mesoamerican studies equated the 260 day cycle to the time between summer and spring zenith passages of the sun at the latitude of Copan.  This theory lost support when it was discovered that the calendar round antedates the foundation of Copan by nearly a millennium.  It has recently been revived by Vincent Malmström, however.  He notes that 260 days is also the time between zenith passages at Izapa, which flourished from the "Olmec" era to the time when Maya civilization emerged.  More about Malmström's theory of calendrical origins on-line.

An interesting  recently published theory notes that multiples of the tzolk'in closely approximate  several astronomical cycles that interested the Maya,  including the solar year, lunar month, and periods of Venus nd Mars.  Robert D. Penden, "The Maya Calendar --- Why 260 Days?",  an on-line article,  argues that 260 days is an optimal choice of a ritual period fitting astronomical cycles. 

As Anthony Aveni has suggested, given the Mesoamerican love of commensuration and the sacred significance attributed to the meshing of cycles of time, more than one explanation of the length of the tzolk'in may have been known to the scribes. 

More about the calendar: The long count and calendar round were the basic elements of the Maya calendar, used to date  inscriptions and time rituals. But the outline above is not a complete account of Maya calendrics.  Some "advanced" topics in Maya calendrics include:

27th day of the Lunar Cycle

2. Lunar cycle -- The "Lunar Series" glyphs sometimes appended to long counts recorded the days elapsed since new moon, and kept track of lunar months of alternating 29 and 30 day length.  (See Lunar Astronomy in the Inscriptions at this web site and Robert Kihm's Lunar Glyphs at Astra's Stargate web page)

3. U kahlay katunob --- "The count of katuns" or "short count" replaced the long count for recording historical events in the post-Classical era. It is 13 katuns long. Tzolk'in dates of katun endings are always a day Ahaw, one of 1 Ahaw to 13 Ahaw. Each of these tzolk'in dates names a katun in the u kahlay katunob.  (See  U kahlay katunob: The short count and katun prophecy at this web site)

4. 819 day count --- Some inscriptions count back from the Initial Series date to one of the "stations" of the 819 day count. These dates are multiples of 819 days apart, and each is associated with a colour and a direction.  (See Gregory Reddick's 819 Day Cycle, and an example of an 819 day count inscription from a Palenque inscription at the Mesoweb site).

5. Tzolk'in almanacs --- The codices consist mostly of tzolk'in almanacs, used for divination and to time rituals.  Each almanac counts through a series of tzolk'in dates with ritual or augural significance.  See an example from the Dresden Codex here, and further discussion at the Madrid Codex Project website.  See also Tzolk'in Augury at this web site. 

6.  Other astronomical cycles ---  The Maya keep track of astronomical cycles in addition to the lunar month, including the periods of Venus and Mars, and eclipse cycles.  These cycles are recorded in tables in the Dresden Codex glyph book.

7. Maya Mathematics --- Although the Maya number system was functionally a modified base-20 system,  the Maya scribes may have conceived of it in a more complicated fashion, as a  composite of three distinct counts.  See Michael Closs, "the nature of the Maya chronological count."  But  however it may have been conceived and developed,  as  W. French Anderson, "Arithmetic in Maya numerals" shows,  quite sophisticated calculations are possible using the Maya number system. 
 

Dresden Codex Almanac

The Maya were deeply concerned to locate all events . . . within a cosmological framework designed to insure the regeneration of life. . . . Time past and time future were fixed in a discernible pattern that could be read and predicted. In addition to this nearly five-thousand year cycle [of the calendar] were a series of other cycles, which the Maya marked, celebrated, and sometimes feared  in detail. (David Carrasco, Religions of Mesoamerica )