[Aztlan] Perhaps Calendrics, architecture
vgray (gotsky)
vgray at gotsky.com
Sun May 14 23:35:35 CDT 2006
Michael,
I would like to add some minor points to your argument that Mesoamericans
could have arrived at a reasonable estimate for the tropical year. The
detection of the leap year phenomenon using vertical sun transits, must have
occurred as early as San Lorenzo and La Venta 900 - 1200 BC, because
aligning structures to tropical stations must yield this discovery quickly.
This is 1000y before the oldest Long Count date. I find it rather incredible
that anybody could dispute the early Olmecs would have discovered the
effect, given sun zenithal passages and alignment of structures, and quite
honestly I discount such criticism as scholarly blindness. Once the 1-day
every 4y effect is known trivial arithmetic takes over some of the necessary
work.
Given a Julian estimate of 1-day every 4y, it is trivial to estimate the
lower bound interval for a complete haab seasonal round, since 365 leap days
yields 365 * 4 = 1460 tropical years or 1,461 haab years. This is beyond
reproach - Mesoamericans therefore knew that the interval could not be less
than 1,460 tropical years - and this is deduced without extensive
observations beyond the mere realization that the tropical station leaps
ahead 1-day every 4y. Mere arithmetic accomplishes quite a lot. Of course
the Julian estimate also amounts to 13 leap days every calendar round (CR),
and over 28 CR this would yield 28*13 = 364 leap days. Astronomical
observations would only be needed over 128y to improve upon the above simple
deductions, as you mention in your argument. It is also possible that over
104y or two calendar rounds that a missing 1-day appears imminent. This
implies only 12.5 leap days every calendar round, and 28 calendar rounds
yields 12.5 x 28 = 350 leap days, which means 15 additional leap days must
be accommodated over 60y, so that a haab seasonal round is now 1460 + 60 =
1520y. An average between the two estimates above is 1492y assuming the
result must be divisible by 4 (i.e. 8 missing leap days). Note Mesoamericans
did not possess a numerical fraction notation capability, which partitions
the interval between zero and unity, but period residuals within a calendric
base period allows the partitioning of larger numbers (like 365), and this
is an adequate fraction notation capability for the purposes of calendrics.
For example a span of several haab years with a 73-day remainder, expresses
a 1/5 fraction of the haab interval, without a need for a numerical fraction
notation to express the factional component of the span. This feature of a
long tropical calibration span acting in lieu of a numerical fraction
notation is perhaps lost on some scholars.
Detecting that there is 1 leap day missing every 128y (not 104y) gives the
necessary precision, which also yields 32 leap days within 132y, or 8 leap
days occur in 33y rather than 32y. This immediately offers a new boundary
estimate with 11 missing leap days in 132 * 11 = 1,452y, which is short of a
complete seasonal migration round in the haab, since only 32 * 11 = 352 leap
days persist over this interval. The 13 leap days required to complete the
haab seasonal round amounts to 52y, and therefore the new lower bound
estimate is 1452 + 52 = 1504y. It is unlikely that precision better than
this could be accomplished, but there are certain conventional assumptions
that could be adopted. For example, the choice of 128y for the interval
representing one missing leap day over say 124y or 132y might not be
discernible, without observations over yet longer periods of time, but
conventionally the choice of interval could be limited to possibilities that
are exactly divisible by 4, like 120, 124, 128, 132, etc.
However, there is one interval that could be adopted conventionally, which
is a period of 29 calendar rounds or 1508 haab years, and therefore a 1,507y
haab seasonal round. As shown 28 calendar rounds as a lower bound estimate
is arrived at easily - but too small based on the Julian estimate of the
leap year effect. But the same Julian estimate also suggests that 29
calendar rounds would yield 377 Julian leap days - 12 leap days too many.
Once the missing leap day every 128y becomes evident, and the lower bound
estimate is raised to 1,504y, then it is plausible to believe that exactly
29 calendar rounds could be adopted conventionally for the haab seasonal
round interval. After all the error is only a conventional 1/4 leap day. Any
estimate of 112 to 144y for the interval for a missing leap day, could be
assumed to be representative of 29 calendar rounds, and back calculation
would then yield a 128y average for the missing leap day conventional
interval.
It is certainly within the capability of Mesoamericans to play with numbers,
and look at the boundary possibilities, and make choices that appease the
arithmetic. Ease of calendric arithmetic must be a primary goal. If 29 CR is
adopted for the haab seasonal round interval, then approximately 1508/4 =
~377 Julian leap days persists, and 1508/12 = 128 rounded up to the nearest
multiple of 4.
The final step of simply adopting the 29 calendar round interval as the
final haab seasonal migration round seems likely. Even the 25 leap days
every 2 CR estimate yields 12.5 * 29 = 362.5 leap days (yes I know pesky
fractions again), so the haab seasonal round estimate is in error by no more
than 12y (i.e. ~1520y). Then over time the model may be tested against
observations over long expanses of time. The passage of 1000y from the early
Olmecs to the first Long Count date is long enough to test a model.
To suggest that the early Olmecs were incapable of developing a 29 CR
arithmetic model for the haab seasonal round, which was then subsequently
tested over a 1000 years before the first Long Count date was struck, cannot
be taken seriously. The model persisted, and tests over a protracted
observation interval was then patiently conducted, of that I have no doubt.
That is the real significance of a penchant for aligning large public
structures to the heavens. The lunar orbital interval is even more fun,
because 1461 days is a leap year interval but 1462 days is 49.5 lunations,
and over just 16y a -1-day adjustment becomes apparent. So that 1462*4 - 1 =
5847 days = 198 lunations = 1461*4 + 3 days, with a calibrated 3-day
divergence between the lunar and tropical stations relative to an integral
lunar span. If 198 lunation spans are concatenated, it is not until 272y has
elapsed will a +1-day adjustment be needed to maintain an integral lunar
span. In other words, 17 * (1462*4 - 1) + 1 = 3,366 lunations = 17 *
(1461*4 + 3) + 1, and the expected 17*3 = 51 day divergence between the
lunar and tropical stations is seen to be 54 days after adjustment - hence 2
missing leap days are evident. The concatenated result may be deduced very
easily knowing there is a missing leap day every 128y. Playing with
arithmetic and observations over time yields the fact that 29 CRs is an
integral 18,639 lunations. This too is well within Mesoamerican purview. By
using lunar & tropical station conjunctions both tropical and lunar
estimates may be derived together, and this increases the level of precision
obtainable.
A Full Moon & Aug 13th conjunction calibrates the above 198 lunation orbital
calibration span over just 16y, to provide a calibrated 3-day divergence
between the Full Moon and tropical station, so that 17 concatenations gives
a predicted 51-day divergence. This proves that 2 missing leap days persist
in 272y or 1 missing leap day every 136y when observations are carried out
to 272y. The concatenated span is just 1-day short of being an integral
lunar span which is quite remarkable.
Cheers Cliff
----- Original Message -----
From: "Michael Finley" <mjfinley at shaw.ca>
To: "Sid Hollander" <sid.hollander at gmail.com>; <aztlan at lists.famsi.org>
Sent: Saturday, May 13, 2006 11:56 PM
Subject: Re: [Aztlan] Perhaps Calendrics, architecture
> Hmm -- let me play devil's advocate for a moment here, Sid.
>
>
> 1. What is a calendar? Is it even useful to define it? The Classical
> Maya seem to have been interested in all shorts of cycles of time, some
> astronomical, some (probably) not. We think of the modified solar cycle
> called the Gregorian Calendar as a privileged cycle, the one we use to
> keep track of time --- while other cycles, like the period of Venus, we
> think of as things we measure with the calendar. I'm not sure the Maya
> privileged any cycle, thought of any one of them as THE cycle for time
> keeping the way we do. As I suggested, I think what is most distinctive
> about Mesoamerican "calendars" is the effort to make all cycles of
> interest commensurate by finding common multiples of them. Either the
> entire system of interlocking cycles is the calendar --- or better -- the
> concept of "calendar" just doesn't apply.
>
> Maybe I should add: Sure, we have the initial series "dates" on monuments
> that combine LC, CR and sometimes lunar & 819 day counts, and this looks
> privileged. Maybe. But really, if you consider that what used to be called
> "the stelae cult" was primarily a way for Classical rulers to link
> themselves with the mythic age and Creation, perhaps all we're really
> looking at in the LC is a cycle designed for that limited and specific
> purpose. In the Post-Classic, when political power came to be welded by
> aristocratic lineages, this particular cycle lost most of its purpose, and
> fell into disuse, at least on public monuments.
>
>
> 2. "Is 20 is the basic unit in Mesoamerican counting" or "Is 20 the the
> basic unit to count Mesoamerican days"? Again, is an answer needed?
> Michael Closs has suggested that the Maya number system was really a
> combination of several different counting methods. The Classical
> numerologists likely had no notion of a "base 20 number system," even
> though their method of counting amounted to one. The question itself
> imposes modern concepts on a numerological tradition that probably
> conceived arithmetic quite differently than we do. Anyway, "20" was a
> significant number --- for the tzolk'in, for the long count, for the
> notation system -- whether we regard it as, in a strict sense, the basic
> unit of counting or not.
>
>
> 3. When I wrote "that neither astronomy nor archeology stand in the way
> of the hypothesis that the long count was contrived to end (or complete
> a Great Cycle of 13 baktuns) on a solstice," I wasn't requiring
> anything more (or less) of the sage fellow who invented the long count
> than that, having decided that 13 baktuns was an appropriate length for
> the Great Cycle, he back-dated its beginning so that it would end on
> winter solstice. Whether --- and why --- he would have wanted to do that,
> I do not comment on (I just note that Victoria Bricker thinks he did). The
> choice of 13 doesn't seem to me to necessarily need much explanation ---
> Assuming the Tzolk'in and Short Count had already been devised, 13
> seems like a significant number, so I can easily imagine a
> number-conscious Mayan finding good ritual reasons for pressing it
> into another role.
>
> BUT -- this was my main point --- to back-date with accuracy, he'd have
> to have known the length of the solar year with enough precision to avoid
> "rounding error" over a span of many baktuns. (The tropical or solar year
> is defined as the time between spring equinoxes or winter solstices, and
> this is the "year" that is most accessible to obervation). And he was
> accurate (assuming he was aiming at the solstice) because the completion
> of 13 baktuns will in fact occur within a day or two of winter solstice
> in the infamous year 2012 AD. My main problem with the notion that he was
> really aiming at a solstice is doubt that such an accurate estimate of the
> length of the year could have been known at the time the long count was
> invented. This would have taken a long period of observation and record
> keeping, the latter requiring some method of recording passage of time
> on par with the long count, but in place before it. Maybe the short count
> was in place, and good enough for the purpose, maybe not. (For an
> argument that it may have been good enough after all --- see
> http://members.shaw.ca/mjfinley/solaryear.html ).
>
>
> The same considerations apply to any other hypothesis about development of
> the "calendar" that assumes early access to precise astronomical
> knowledge.
>
>
> Michael Finley
>
>
> Sid Hollander wrote:
>
>> What Michael Finley says is plausible to me, and I feel there are other
>> approaches but I'd like to explore his approach a bit further for the
>> time
>> (no pun intended) being.
>>
>> I think it would be helpful if we were to adopt a definition of what we
>> mean
>> by calendar and in particular, Maya Calendar System. To that I offer the
>> following definitons
>>
>>
>> A calendar is a system for naming periods of time, typically days.
>> These names are known as calendar dates. The dates may be based on the
>> perceived motion of astronomical objects. Other definitions point out
>> that a
>> 'calendar' is also a physical device (often paper) (Sid adds: Sometimes
>> Amate) that illustrates the system (for example, a desktop calendar) —
>> this
>> is the most common usage of the word.
>>
>> In terms of the Maya, I would include all of the following as components
>> of
>> the Maya 'calendrical' system (in no particular order): LC, Tzolkin,
>> Haab,
>> 819, Lords/Gods of the Night, Patron of Meses and the Lunar components of
>> CDE).
>>
>> Feel free to offer amendments .I would hesitate to include the 'physical
>> devices' mentioned above as part of the system although they may have
>> served as development tools for the defining of the calendar but this
>> too, I
>> is up for modification.
>>
>> I'd like to begin with some comments/questions evolving from Michaels
>> post
>> of May 10.
>>
>>
>> "For example, it seems to me that neither astronomy nor archeology stand
>> in
>> the way of the hypothesis that the long count was contrived to end (or
>> complete a Great Cycle of 13 baktuns) on a solstice ………"
>>
>>
>>
>> The contention is that the Maya designed their LC so that the base 20
>> positions (with modification) would fit in a convenient multiple of
>> irregular occurring events (i.e. some 182's and some 183's. ). I need
>> help
>> here! I am not an astronomer! I think that there are 2 solstice per
>> year. Taking
>> a year to be 365 or 365.2422 and dividing each by 2 I get 182.5 and
>> 182.6211 respectively. It beats the heck out of why one would want to
>> 'grow' either of these numbers into a bigger number, x, so that this x
>> could
>> be fit into some positional notation system i.e. the LC as we know it.
>> 1,872,000
>> (whereever it comes from) isn't even remotely close to being divisible by
>> 182.5 or 182.6211 (giving answers of 10257.5342 and
>> 10250.7323respectively). Is it that time between soltices have changed
>> since LC
>> invention? Where am I going astray? Where does the 1872000 number come
>> from (since it was used to define the LC positions). I might add that
>> it
>> is indeed fortunate that the number 187200 was selected/elected and not
>> either of the nearby neighbors 1871983, 1872001 or 1872007 which could
>> not
>> be factored for they are prime.
>>
>> "Given that 20 is a basic unit in Mesoamerican counting, why not also
>> record
>> 20 year periods?"
>>
>>
>> Is "20 is the basic unit in Mesoamerican counting" or "Is 20 the the
>> basic
>> unit to cont Mesoamerican days"?
>>
>> We would not want to say that 12 was the Western worlds' base of counting
>> after inspecting the number "1G. 3D.2E" and determining that its
>> equivalent
>> is 182 (Hint: ones column is called eggs)
>>
>> Is there evidence to substantiate that 20 was the basic unit in
>> Mesoamerican counting apart that it was used in the calendar? I ask this
>> because of the apparent 'evidence' reflected in the face glyphs that
>> indicate a base 10 awareness and because of my limited knowledge.
>>
>> While I am in the questioning frame of mind, am I to assume by your
>> inclusion of the Haab prior to the ibvention of zero that your Haab month
>> days run from 1 to 19 and then Seating? Just thought I'd stir the pot a
>> bit!
>>
>>
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