[Aztlan] Calendrics, architecture

Mon May 1 15:04:54 CDT 2006

Harold,

I agree the Pakhomov's article does not "help us to understand the Mayan calendar." But your characterization of the assemblage of cycles as not constituting an ancient Mayan calendar (actually Olmec in origin) is not entirely accurate. Certainly the various cycles you mention enjoy a certain independent nature, but there is also the question of a RELATIVE alignment between the cycles - as an orientation between the various period origins of the cycles. Clearly the Calendar Round of haab and tzolkin repeat every 52 haab years, and therefore the synchronization constitutes an enlarged 18,980 day calendric base period - and therefore also expressly a 52 haab year calendar in its own right. This extends all the way out to the 5200 tuns or 260 katuns that form the (13.0.0.0.0) cycle, because this is expressly a count of 360-day tun years, and not simply a linear count of days. If the Mayans intended to simply count a series of days, there would have been no need to truncate the 2nd viget of the vigesimal positional notation system to base 18. They could simply have used a pure vigesimal notation to express a count of days, but chose instead to truncate the 2nd viget of the numerical counting system to yield a 360-day tun counter. 

The relative orientations between these cycles have meaning, and further the tropical year itself has a period origin of its own - say an Aug 13th vertical sun transit or winter solstice station - and this too must align in some fashion to the other period origins of the other cycles incorporated. It is simply not the case that all these cycles were simply thrown together in an adhoc manner, but were expressly aligned to each other, and how this was done is still in contention.

However, you are right, "there is no persuasive evidence of which I am aware that the pre-contact Mayans
attempted to adjust their day counts to the solar year by means of intercalation of days or any other means". I certainly did not convey such a possibility in what I wrote. Mesoamericans expressly shunned such practices with the exception of the more exotic example of the self-adjusting Venus calendar. Nevertheless, this is not the same thing as saying that they were ignorant of the leap year phenomenon - but simply that they expressly chose not to use such an adjustment methodology. Indeed they preferred to align the various cycle origins in a one-time arrangement forever fixed, never to be changed by adjustments of any kind EVER. The dates themselves were NEVER the target of an adjustment.   

Nevertheless, orbital calibration spans are a different matter all together - where an orbital circulation interval is delineated by means of two dates (within the confines of any one of the cycles that constitute the calendar), and in a sense there are two parallel spans bound together in this construction. First the interval as a day count in whatever calendric base period you choose (i.e. in years and a remainder of days), and then the integral orbital circulation span itself that the calendric span represents. For example, 1,507 tropical years is an integral tropical circulation span, while the 1,508 haab years that corresponds to this interval is an equivalent  representative calendric calibration span, and binding these two spans together constitutes an orbital calibration span. Or a lunar circulation span of 37 lunations is bound to its equivalent representative calendric span of 1092 days or approximately 4-days short of 3 tropical years. Here the notion of adjusting a "multi-year" calendric base period interval is used to match the actual orbital phenomenon being tracked. A Venus Round of 104 haab years is 5-days longer than 65 Venus orbital circulations (or Venus years), and therefore the equivalent calendric span interval of 365*8*13 days must be foreshortened by 5-days to represent the integral Venus orbital circulation interval. It is not the calendar that is being adjusted to accommodate this Venus orbital migration, but rather the calibration span expressed within the calendar by means of two dates to mark its extremities on the time-line. The orbital calibration span is subjected to adjustment, that is then expressed within the calendar using dates to designate its extremities - the calendar itself is not the target of the adjustment. 

My statement paraphrased as: "simply describe the passage of time by decorating the time-line with a continuous
series of dates", was an attempt to differentiate mere dates that appear along a time-line without periodic adjustment, as opposed to a series of dates that are expressly modified by adjustments as in the Julian calendar case. In other words the former is a CONTINUOUS series of dates, while the Julian calendar is a DISCONTINUOUS series of dates along the time-line, and therefore my statement expressly means the Mayan dating system is DEVOID of periodic adjustments based on orbital origin migration or the leap year phenomenon. I hope that clarifies the meaning of my statement. While devoid of periodic adjustments, however, nevertheless the leap year phenomenon played a role in the original inauguration of the great cycle. This is perhaps where the confusion lies - the calendar is fixed without recourse to periodic solar alignments, but the original setup of the cycle used the phenomenon to establish this fixed alignment initially, which is the element that is in contention. 

So how were the various cycles of the Mayan calendar initially oriented relative to each other? The thrust of my claim is that an accurate estimate of the tropical leap year phenomenon was used in initially establishing the great cycle's absolute alignment, which was then forever thereafter fixed without further adjustment. A complete haab seasonal migration round in 1,507y or 29 calendar rounds, and a complete seasonal migration round within the tun in 68y, formed the underpinnings for that relative alignment between the haab, tun and tropical year, thus establishing the haab basis of the Long Count. A 161-day tzolkin period residual over a 1,461-day leap year interval in conjunction with an Aug 13th vertical sun transit & Full Moon conjunction event at the start of the cycle, formed the foundations for a tzolkin basis of the Long Count, and this also included the 49.5 lunation span of 1,462-days (tropical and lunar stations were separated by 1-day).  

Cheers Cliff

  ----- Original Message ----- 
  From: Harold H. Green 
  To: vgray (gotsky) 
  Sent: Sunday, April 30, 2006 4:12 PM
  Subject: Re: [Aztlan] Calendrics, architecture

  I suggest that Pakhomov's article does not "help us to understand the Mayan calendar." There is
  actually not an ancient "Mayan calendar" as such, but rather a series of day counts (260-, 360-, 365-day counts,
  the Long Count [being a linear count of days from completion of the previous era on 13.0.0.0.0
  4 Ajaw 8 Kumku], the lunar count embodied in the so-called supplementary series, the 819-day
  count, the 9-day so-called "Lords of the Night" count, and perhaps others). While the ancient Maya
  did have the means to determine accurately the length of the solar year (as with the solar tracker
  in the Palace Tower at Palenque), there is no persuasive evidence of which I am aware that the pre-contact Maya
  attempted to adjust their day counts to the solar year by means of intercalation of days or any other means.

  As I think it can be demonstrated that the day counts are cosmically derived, it also may not assist understanding
  to say that these counts "simply describe the passage of time by decorating the time-line with a continuous
  series of dates.

  I find that a good source on Maya calendrics is Floyd Lounsbury's "Maya Numberation, Computation, and Calendrical
  Astronomy" in Vol. 15, Supp. 1 of the Dictonary of Scientific Biography (1978). Another source that I like is Gerardo Aldana's 
  “Preface” to Oracular Science: Uncertainty in the History of Maya Astronomy," Ph.D. Dissertation, Harvard University (2001).

  Hal Green
  Vashon, WA

  On Apr 30, 2006, at 3:02 PM, vgray ((gotsky)) wrote:

    I have read part of Vladimir Pakhomov's work, and this largely describes calendars that are structured relative to the heavens. I must point out there is a difference between structured calendars and dating systems. Dating systems simply describe the passage of time by decorating the time-line with a continuous series of dates, while structured calendars attempt to orient the dates themselves to the celestial sphere. The Julian calendar is expressly contrived to convert the "haab" into a fixed tropical calendar, while the Mayan calendar is merely a dating system for the purpose of decorating the passage of time with a series of continuous dates, without modifying the system of dating to fit the celestial sphere. Or at least the Mayan calendar does not periodically adjust the dating system in some manner as to fix it's divergence relative to the celestial sphere - the true tropical calendar is distinct from the dating system itself. This difference is not obvious, and scholars often confuse a "series of dates" with a "structured tropical calendar", and extrapolate that lack of a date adjustment methodology implies a tropical calendar solution does not exist. The underlying heuristics of the Mayan calendar cannot be understood in the terms usually associated with a structured calendar solution, as Vladimir Pakhomov suggests, and the leap year phenomenon must be addressed in different terms.

    Actually the Mayan calendar does not contain the leap year phenomenon at all, in terms of the three calendric periods used in its construction (I.e. 365, 360, 260 days), but rather uses an entirely different and unique leap year solution, while keeping the calendric base periods fixed. The Julian calendar periodically adjusts the calendric period to fix alignment of the calendar's dates to the tropical stations (approximately), but the Mayan calendar expressly shuns period adjustments (to simplify calendric arithmetic), and rather fixes two different elements: (a) the time-line's orientation to specific tropical events, and (b) uses fixed orbital calibration spans to describe tropical/lunar orbital behavior outside the dating system itself. In other words the Julian calendar uses periodic adjustments, while the Mayan calendar uses a one-time adjustment to fix haab, tun, and tzolkin basis, and to align the 13-baktun-cycle to the celestial sphere is a unique way. The Julian calendar attempts to create a fixed tropical calendar by periodic adjustments, while the Mayan calendar fixes the alignment of a fixed cycle interval to the heavens at its initial inauguration, changes nothing thereafter, and describes the tropical calendar by means of a pair of dates used to delineate a tropical calibration span. The two systems are almost opposite in character, fixing different elements of the temporal equation, and it would be a mistake to address the Mayan solution in Eurocentric terms.

    This difference in approach may be viewed as follows. The Julian calendar fixes the calendar's orientation to the heavens, fixing the relationship between solar calendar dates and tropical stations, so winter solstice always appears on Dec 21st every year (diverging only very slowly 1-day every 128y). But the Mayan calendar simply uses the dating system as a fixed unchanging means of decorating the time-line with dates, with no attempt made at orienting the dates themselves to the celestial sphere, and therefore in this sense there is no leap year solution incorporated into the dating system. In other words the Julian calendar converts the dating system to a tropical calendar, while the Mayan calendar expressly maintains the tropical calendar outside the purview of the dating system itself in a sense. This approach of differentiating the dating system from the tropical calendar is a uniquely Mesoamerican calendric technique, that other civilizations around the world did not adopt, and indeed has primarily misled scholars into believing the Mayan calendar does not incorporate a leap year solution of any kind. It is becoming increasingly clear Mesoamericans understood the tropical calendar to a greater depth than their contemporaries around the world.

    The Mayan tropical calendar exists in the form of a series of well designed tropical calibration spans, expressed using two dates to delineate the span's extremities - so the underlying tropical orbital heuristics are embedded within the span's makeup and not within the dating system itself. This is necessitated by the fact Mayans did not possess a numerical fraction notation, partitioning the interval between 0 and 1, but rather expressed calendric factions using calendric period residuals as a means of partitioning much larger numbers - for example a haab period residual of 73-days is a 1/5th fraction of 365-days. In this way a high degree of precision is obtainable where orbital circulation divergences are accumulated over much larger expanses of time. The precision is really quite remarkable, as in an example given earlier, where a 206 year tropical span is equated with a 209 tun calendric interval, where it is plain that the 52 Julian leap days that should have occurred over 208y did not materialize. But rather only 50 leap days occurs in 206y expressing the fact that 1½ leap days are missing, but also this interval of 209-tuns is also 3 complete rounds of seasonal migration with the tun constituting 1,080 "tun leap days" - equating 50 haab leap days with 1,080 tun leap days. Associated with this tropical calibration is also the span of 68y equated to a 69-tun calendric interval, which is approximately one complete seasonal migration round within the tun, over an interval that is also expressive of 17 haab leap days. This period is an important universal period, not just because it is one seasonal round within a tun calendric base period, but also because it defines a way for describing the divergence between the haab and tun calendric period origins. Elsewhere I have made the claim a Mayan haab basis is derived using a one-time reorientation of an idealized 13-baktun-cycle arising at 0-Pohp, where a great cycle is arranged to start at a TUN PERIOD ORIGIN 17-days before a HAAB PERIOD ORIGIN, and therefore the cycle begins at 0-Poph - 17 = 8-Kumku. This offers a one time orientation between the haab, tun and tropical calendar that is rooted in the leap year phenomenon, as a relative period origin alignment.

    Another example is a 42y tropical span that equates to 59 tzolkin, and this is 1-day short of 519½ lunations, and also 43.5y equating to 538 lunations. The latter tropical lunar calibration span measures the interval between a summer solstice & Full Moon conjunction event and a winter solstice & Full Moon conjunction event, and such calibration spans offer greater flexibility than the structured calendar approaches used in the Julian/Gregorian calendars. Without the need for mutilating the dating system with periodic adjustments that complicate calendric arithmetic. But more than this, identifies a technique where tropical/lunar conjunctions are used, as a means of harnessing the precision of astronomy. It is not just a question of a 1,461 day leap year interval, but rather this interval in conjunction with the 1,462-day interval that describes a lunar calibration span of 49½ lunations, with a ¼-day lunar period residual. Far more significant than the tropical leap year phenomenon in isolation, is a tropical/lunar conjunction resultant - over just 4*1462 days (~16y) it is clear the interval must be corrected by subtracting 1-day to describe 198 lunations, thus establishing a 3-day divergence between the tropical and lunar stations (and not 4-days). Seventeen such corrected spans must elapse over 272y before a further +1-day correction is needed, to give an integral lunar span of 3,366 lunations, and this predicts 17x3 + 1 = 52 days of tropical period residual must persist, but in reality two missing leap days increases the tropical period residual to 54-days. In other words 68 Julian leap days yields to 66 actual leap days, and a computed 52-day tropical station divergence then also yields to a 54-day resultant. Such tropical/lunar calibration spans form the foundations of the Mayan calendar, and also offers a deeper appreciation of both tropical and lunar phenomena than the leap year phenomenon alone, and further explains why a haab adjustment methodology, or a structured lunar calendar aligned to the tropical calendar, was never instituted by the Mayans. It is in these terms that the issue of orbital station migration is understood, and the precision achieved cognizable as a result of harnessing tropical/lunar conjunctions, and it seems quite likely that Mesoamericans shunned a haab adjustment methodology, long before Julius Caesar adopted the Julian calendar.

    The Mayans adopted in my mind a superior system for studying the phenomenon of orbital station migration generally. I respectively submit that Vladimir Pakhomov's work is expressly primarily a structured calendar heuristic, and is not a sufficient basis for describing Mesoamerican calendric science, which is quite unique in several important respects. In particular the Mesoamerican technique of using tropical and lunar conjunctions to harness the precision of astronomy over the longer term, and as a means of compensating for the lack of a numerical fraction notation system, preempting the design of both tropical and lunar calendars. What the Mayans might have understood in terms of mathematics is unknown, because their books were burnt, and also the question of exotic numerical structures is largely unknown ill understood. Vladimir Pakhomov's work discusses exotic numerical structures that do not appear to be particularly useful in studying the Mayan calendar. Although I find his work interesting for different reasons to do with structure theory generally.

    Cheers Cliff

    ----- Original Message ----- From: "Vladimir Pakhomov" <calendar.message at gmail.com>
    To: "'vgray (gotsky)'" <vgray at gotsky.com>; "'Jorge P?rez de Lara'" <jorgepl at estudioelias.com>; "'Aztlan'" <aztlan at lists.famsi.org>
    Sent: Saturday, April 29, 2006 1:34 PM
    Subject: RE: [Aztlan] Calendrics, architecture and the "obvious"

    -----Original Message-----
    From: aztlan-bounces at lists.famsi.org [mailto:aztlan-bounces at lists.famsi.org]
    On Behalf Of vgray (gotsky)
    Sent: Monday, March 27, 2006 5:03 AM
    ...
    Counting leap days is not difficult, and given that tropical station
    divergence can be tracked over 208y, as for instance by noticing that the
    year-bearer system itself leaps ahead one day every 4y, as depicted in a
    208y table within the Madrid Codex...
    -------------------------

    If the Mayan calendar has the leap days then my article
    http://www.pakhomov.com/calendar.html
    http://www.pakhomov.com/seals.html
    will help us to understand the Mayan calendar.

    Vladimir Pakhomov

    _______________________________________________
    Aztlan mailing list
    Aztlan at lists.famsi.org
    http://www.famsi.org/mailman/listinfo/aztlan