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Calendar FAQ, v. 2.9 (modified 4 April 2008) Part 3/3

( Part1 - Part2 - Part3 )
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Archive-name: calendars/faq/part3
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Last-modified: 2008/04/04
Version: 2.9
URL: http://www.tondering.dk/claus/calendar.html

See reader questions & answers on this topic! - Help others by sharing your knowledge
                   FREQUENTLY ASKED QUESTIONS ABOUT
                              CALENDARS
                             Part 3 of 3

Version 2.9 - 4 April 2008

Copyright and disclaimer
------------------------
        This document is Copyright (C) 2008 by Claus Tondering.
        E-mail: claus@tondering.dk. (Please include the word
        "calendar" in the subject line.)
        The document may be freely distributed, provided this
        copyright notice is included and no money is charged for
        the document.

        This document is provided "as is". No warranties are made as
        to its correctness.

Introduction
------------
        This is the calendar FAQ. Its purpose is to give an overview
        of the Christian, Hebrew, Persian, and Islamic calendars in
        common use. It will provide a historical background for the
        Christian calendar, plus an overview of the French
        Revolutionary calendar, the Maya calendar, and the Chinese
        calendar.

        Comments are very welcome. My e-mail address is given above.

Contents
--------

In part 1 of this document:

        1. What Astronomical Events Form the Basis of Calendars?
           1.1. What are equinoxes and solstices?
        2. The Christian Calendar
           2.1. What is the Julian calendar?
                2.1.1. What years are leap years?
                2.1.2. What consequences did the use of the Julian
                       calendar have?
           2.2. What is the Gregorian calendar?
                2.2.1. What years are leap years?
                2.2.2. Isn't there a 4000-year rule?
                2.2.3. Don't the Greeks do it differently?
                2.2.4. When did country X change from the Julian to
                       the Gregorian calendar?
           2.3. What day is the leap day?
           2.4. What is the Solar Cycle?
           2.5. What is the Dominical Letter?
           2.6. What day of the week was 2 August 1953?
           2.7. When can I reuse my 1992 calendar?
           2.8. What is the Roman calendar?
                2.7.1. How did the Romans number days?
           2.9. What is the proleptic calendar?
           2.10. Has the year always started on 1 January?
           2.11. Then what about leap years?
           2.12. What is the origin of the names of the months?

In part 2 of this document:

           2.13. What is Easter?
                2.13.1. When is Easter? (Short answer)
                2.13.2. When is Easter? (Long answer)
                2.13.3. What is the Golden Number?
                2.13.4. How does one calculate Easter then?
                2.13.5. What is the Epact?
                2.13.6. How does one calculate Gregorian Easter then?
                2.13.7. Isn't there a simpler way to calculate Easter?
                2.13.8. Isn't there an even simpler way to calculate
                        Easter?
                2.13.9. Is there a simple relationship between two
                        consecutive Easters?
                2.13.10. How frequently are the dates for Easter repeated?
                2.13.11. What about Greek Orthodox Easter?
                2.13.12. Did the Easter dates change in 2001?
           2.14. How does one count years?
                 2.14.1. How did Dionysius date Christ's birth?
                 2.14.2. Was Jesus born in the year 0?
                 2.14.3. When does the 3rd millennium start?
                 2.14.4. What do AD, BC, CE, and BCE stand for?
           2.15. What is the Indiction?
           2.16. What is the Julian period?
                 2.16.1. Is there a formula for calculating the Julian
                         day number?
                 2.16.2. What is the modified Julian day number?
                 2.16.3. What is the Lilian day number?
           2.17. What is the correct way to write dates?
        3. ISO 8601
           3.1. What date format does the Standard mandate?
           3.2. What time format does the Standard mandate?
           3.3. What if I want to specify both a date and a time?
           3.4. What format does the Standard mandate for a time
                interval?
           3.5. Can I write BC dates and dates after the year 9999
                using ISO 8601?
           3.6. Can I write dates in the Julian calendar using ISO 8601?
           3.7. Does the Standard define the Gregorian calendar?
           3.8. What does the Standard say about the week?
           3.9. Why are ISO 8601 dates not used in this Calendar FAQ?
           3.10. Where can I get the Standard?
        4. The Hebrew Calendar
           4.1. What does a Hebrew year look like?
           4.2. What years are leap years?
           4.3. What years are deficient, regular, and complete?
           4.4. When is New Year's day?
           4.5. When does a Hebrew day begin?
           4.6. When does a Hebrew year begin?
           4.7. When is the new moon?
           4.8. How does one count years?
           4.9. When was Passover in AD 30?
        5. The Islamic Calendar
           5.1. What does an Islamic year look like?
           5.2. So you can't print an Islamic calendar in advance?
           5.3. How does one count years?
           5.4. When will the Islamic calendar overtake the Gregorian
                calendar?
           5.5. Doesn't Saudi Arabia have special rules?

In part 3 of this document:

        6. The Persian Calendar
           6.1. What does a Persian year look like?
           6.2. When does the Persian year begin?
           6.3. How does one count years?
           6.4. What years are leap years?
        7. The Week
           7.1. What is the origin of the 7-day week?
           7.2. What do the names of the days of the week mean?
           7.3. What is the system behind the planetary day names?
           7.4. Has the 7-day week cycle ever been interrupted?
           7.5. Which day is the day of rest?
           7.6. What is the first day of the week?
           7.7. What is the week number?
           7.8. How can I calculate the week number?
           7.9. Do weeks of different lengths exist?
        8. The French Revolutionary Calendar
           8.1. What does a Republican year look like?
           8.2. How does one count years?
           8.3. What years are leap years?
           8.4. How does one convert a Republican date to a Gregorian one?
        9. The Maya Calendar
           9.1. What is the Long Count?
                9.1.1. When did the Long Count start?
           9.2. What is the Tzolkin?
                9.2.1. When did the Tzolkin start?
           9.3. What is the Haab?
                9.3.1. When did the Haab start?
           9.4. Did the Mayas think a year was 365 days?
        10. The Chinese Calendar
           10.1. What does the Chinese year look like?
           10.2. What years are leap years?
           10.3. How does one count years?
           10.4. What is the current year in the Chinese calendar?
        11. Frequently Asked Questions about this FAQ
           11.1. Why doesn't the FAQ describe calendar X?
           11.2. Why doesn't the FAQ contain information X?
           11.3. Why don't you reply to my e-mail?
           11.4. How do I know that I can trust your information?
           11.5. There is an error in one of your formulas!
           11.6. Can you recommend any good books about calendars?
           11.7. Do you know a web site where I can find information
                 about X?
        12. Date


6. The Persian Calendar
-----------------------

The Persian calendar is a solar calendar with a starting point that
matches that of the Islamic calendar. Apart from that, the two
calendars are not related. The origin of the Persian calendar can be
traced back to the 11th century when a group of astronomers (including
the well-known poet Omar Khayyam) created what is known as the Jalaali
calendar. However, a number of changes have been made to the calendar
since then.

The current calendar has been used in Iran since 1925 and in
Afghanistan since 1957. However, Afghanistan used the Islamic calendar
in the years 1999-2002.


6.1. What does a Persian year look like?
----------------------------------------

The names and lengths of the 12 months that comprise the Persian year
are:

1. Farvardin   (31 days)      7. Mehr    (30 days)  
2. Ordibehesht (31 days)      8. Aban    (30 days)  
3. Khordad     (31 days)      9. Azar    (30 days)  
4. Tir         (31 days)     10. Day     (30 days)  
5. Mordad      (31 days)     11. Bahman  (30 days)  
6. Shahrivar   (31 days)     12. Esfand  (29/30 days)  

(Due to different transliterations of the Persian alphabet, other
spellings of the months are possible.) In Afghanistan the months are
named differently.

The month of Esfand has 29 days in an ordinary year, 30 days in a leap
year.


6.2. When does the Persian year begin?
--------------------------------------

The Persian year starts at vernal equinox. If the astronomical vernal
equinox falls before noon (Tehran true time) on a particular day, then
that day is the first day of the year. If the astronomical vernal
equinox falls after noon, the following day is the first day of the
year.


6.3. How does one count years?
------------------------------

As in the Islamic calendar (section 5.3), years are counted since
Mohammed's emigration to Medina in AD 622. At vernal equinox of that
year, AP 1 started (AP = Anno Persico/Anno Persarum = Persian year).

Note that contrary to the Islamic calendar, the Persian calendar
counts solar years. In the year AD 2008 we have therefore witnessed
the start of Persian year 1387, but the start of Islamic year 1429.


6.4. What years are leap years?
-------------------------------

Since the Persian year is defined by the astronomical vernal equinox,
the answer is simply: Leap years are years in which there are 366 days
between two Persian new year's days.

However, basing the Persian calendar purely on an astronomical
observation of the vernal equinox is rejected by many, and a few
mathematical rules for determining the length of the year have been
suggested.

The most popular (and complex) of these is probably the following:

The calendar is divided into periods of 2820 years. These periods are
then divided into 88 cycles whose lengths follow this pattern:

29, 33, 33, 33, 29, 33, 33, 33, 29, 33, 33, 33, ...

This gives 2816 years. The total of 2820 years is achieved by
extending the last cycle by 4 years (for a total of 37 years).

If you number the years within each cycle starting with 0, then leap
years are the years that are divisible by 4, except that the year 0 is
not a leap year.

So within, say, a 29 year cycle, this is the leap year pattern:

Year            Year              Year              Year
 0  Ordinary      8  Leap          16  Leap          24  Leap
 1  Ordinary      9  Ordinary      17  Ordinary      25  Ordinary
 2  Ordinary     10  Ordinary      18  Ordinary      26  Ordinary
 3  Ordinary     11  Ordinary      19  Ordinary      27  Ordinary
 4  Leap         12  Leap          20  Leap          28  Leap 
 5  Ordinary     13  Ordinary      21  Ordinary
 6  Ordinary     14  Ordinary      22  Ordinary
 7  Ordinary     15  Ordinary      23  Ordinary

This gives a total of 683 leap years every 2820 years, which
corresponds to an average year length of 365 683/2820 = 365.24220
days. This is a better approximation to the tropical year than the
365.2425 days of the Gregorian calendar.

The current 2820 year period started in the year AP 475 (AD 1096).

This "mathematical" calendar currently coincides closely with the
purely astronomical calendar. In the years between AP 1244 and 1531
(AD 1865 and 2152) a discrepancy of one day is seen twice, namely in
AP 1404 and 1437 (starting at vernal equinox of AD 2025 and 2058).
However, outside this period, discrepancies are more frequent.


7. The Week
-----------

The Christian, the Hebrew, the Islamic, and the Persian calendars all
have a 7-day week.


7.1. What is the origin of the 7-day week?
------------------------------------------

Digging into the history of the 7-day week is a very complicated
matter. Authorities have very different opinions about the history of
the week, and they frequently present their speculations as if they
were indisputable facts. In short, nothing can be said with certainty
about the origin of the 7-day week.

The first pages of the Bible explain how God created the world in six
days and rested on the seventh. This seventh day became the Jewish
day of rest, the Sabbath, Saturday.

Extra-biblical locations sometimes mentioned as the birthplace of the
7-day week include: Babylon, Persia, and several others. The week was
known in Rome before the advent of Christianity.


7.2. What do the names of the days of the week mean?
----------------------------------------------------

An answer to this question is necessarily closely linked to the
language in question. Whereas most languages use the same names for
the months (with a few Slavonic languages as notable exceptions),
there is great variety in names that various languages use for the
days of the week. A few examples will be given here.

Except for the Sabbath, Jews simply number their week days.

A related method is partially used in Portuguese and Russian:

   English    Portuguese      Russian        Meaning of Russian name
   -------    ----------      -------        -----------------------
   Monday     segunda-feira   ponedelnik     After "do-nothing"
   Tuesday    terca-feira     vtornik        Second
   Wednesday  quarta-feira    sreda          Middle
   Thursday   quinta-feira    chetverg       Fourth
   Friday     sexta-feira     pyatnitsa      Fifth
   Saturday   sabado          subbota        Sabbath
   Sunday     domingo         voskresenye    Resurrection

Most Latin-based languages connect each day of the week with one of
the seven "planets" of the ancient times: Sun, Moon, Mercury, Venus,
Mars, Jupiter, and Saturn. French, for example, uses:

   English     French         "Planet"
   -------     ------         --------
   Monday      lundi          Moon
   Tuesday     mardi          Mars
   Wednesday   mercredi       Mercury
   Thursday    jeudi          Jupiter
   Friday      vendredi       Venus
   Saturday    samedi         Saturn
   Sunday      dimanche       (Sun)

The link with the sun has been broken in French, but Sunday was
called "dies solis" (day of the sun) in Latin.

It is interesting to note that also some Asiatic languages (for
example, Hindi, Japanese, and Korean) have a similar relationship
between the week days and the planets.

English has retained the original planets in the names for Saturday,
Sunday, and Monday. For the four other days, however, the names of
Anglo-Saxon or Nordic gods have replaced the Roman gods that gave
name to the planets. Thus, Tuesday is named after Tiw, Wednesday is
named after Woden, Thursday is named after Thor, and Friday is named
after Frigg.


7.3. What is the system behind the planetary day names?
-------------------------------------------------------

As we saw in the previous section, the planets have given the week
days their names following this order:

   Moon, Mars, Mercury, Jupiter, Venus, Saturn, Sun

Why this particular order?

One theory goes as follows: If you order the "planets" according to
either their presumed distance from Earth (assuming the Earth to be
the centre of the universe) or their period of revolution around the
Earth, you arrive at this order:

   Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn

Now, assign (in reverse order) these planets to the hours of the day:

   1=Saturn, 2=Jupiter, 3=Mars, 4=Sun, 5=Venus, 6=Mercury, 7=Moon,
   8=Saturn, 9=Jupiter, etc., 23=Jupiter, 24=Mars

The next day will then continue where the old day left off:

  1=Sun, 2=Venus, etc., 23=Venus, 24=Mercury

And the next day will go

  1=Moon, 2=Saturn, etc.

If you look at the planet assigned to the first hour of each day, you
will note that the planets come in this order:

   Saturn, Sun, Moon, Mars, Mercury, Jupiter, Venus

This is exactly the order of the associated week days.

Coincidence? Maybe.


7.4. Has the 7-day week cycle ever been interrupted?
----------------------------------------------------

There is no record of the 7-day week cycle ever having been broken.
Calendar changes and reform have never interrupted the 7-day cycles.
It is very likely that the week cycles have run uninterrupted at least
since the days of Moses (c. 1400 BC), possibly even longer.

Some sources claim that the ancient Jews used a calendar in which an
extra Sabbath was occasionally introduced. But this is probably not
true.


7.5. Which day is the day of rest?
----------------------------------

For the Jews, the Sabbath (Saturday) is the day of rest and
worship. On this day God rested after creating the world.

Most Christians have made Sunday their day of rest and worship,
because Jesus rose from the dead on a Sunday.

Muslims use Friday as their day of rest and worship. The Qur'an
calls Friday a holy day, the "king of days".


7.6. What is the first day of the week?
---------------------------------------

The Bible clearly makes the Sabbath (correspondiong to our Saturday)
the last day of the week. Therefore it is common Jewish and Christian
practice to regard Sunday as the first day of the week (as is also
evident from the Portuguese names for the week days mentioned in
section 7.2). However, the fact that, for example, Russian uses the
name "second" for Tuesday, indicates that some nations regard Monday
as the first day.

In international standard ISO 8601 (see chapter 3) the International
Organization for Standardization has decreed that Monday shall be the
first day of the week.


7.7. What is the week number?
-----------------------------

International standard ISO 8601 (see chapter 3) assigns a number to
each week of the year. A week that lies partly in one year and partly
in another is assigned a number in the year in which most of its days
lie. This means that

        Week 1 of any year is the week that contains 4 January,

or equivalently

        Week 1 of any year is the week that contains the first
        Thursday in January.

Most years have 52 weeks, but years that start on a Thursday and leap
years that start on a Wednesday have 53 weeks.

Note: This week numbering system is not commonly used in the United
States.


7.8. How can I calculate the week number?
-----------------------------------------

If you know the date, you can calculate the corresponding week number
(as defined in ISO 8601) as descibed below. (The divisions are integer
divisions in which the remainder is discarded.)

For dates in January and February, calculate:

    a = year-1
    b = a/4 - a/100 + a/400
    c = (a-1)/4 - (a-1)/100 + (a-1)/400
    s = b-c
    e = 0
    f = day - 1 + 31*(month-1)

For dates in March through December, calculate:

    a = year
    b = a/4 - a/100 + a/400
    c = (a-1)/4 - (a-1)/100 + (a-1)/400
    s = b-c
    e = s+1
    f = day + (153*(month-3)+2)/5 + 58 + s

Then, for any month continue thus:

    g = (a + b) mod 7
    d = (f + g - e) mod 7
    n = f + 3 - d

We now have three situations:

    If n<0, the day lies in week 53-(g-s)/5 of the previous year.
    If n>364+s, the day lies in week 1 of the coming year.
    Otherwise, the day lies in week n/7 + 1 of the current year.

This algorithm gives you a couple of additional useful values:

    d indicates the day of the week (0=Monday, 1=Tuesday, etc.)
    f+1 is the ordinal number of the date within the current year.

 
7.9. Do weeks of different lengths exist?
-----------------------------------------

If you define a "week" as a 7-day period, obviously the answer is
no. But if you define a "week" as a named interval that is greater
than a day and smaller than a month, the answer is yes.

The ancient Egyptians used a 10-day "week", as did the French
Revolutionary calendar (see section 8.1).

The Maya calendar uses a 13 and a 20-day "week" (see section 9.2).

The Soviet Union has used both a 5-day and a 6-day week. In 1929-30
the USSR gradually introduced a 5-day week. Every worker had one day
off every week, but there was no fixed day of rest. On 1 September
1931 this was replaced by a 6-day week with a fixed day of rest,
falling on the 6th, 12th, 18th, 24th, and 30th day of each month (1
March was used instead of the 30th day of February, and the last day
of months with 31 days was considered an extra working day outside
the normal 6-day week cycle). A return to the normal 7-day week was
decreed on 26 June 1940.


8. The French Revolutionary Calendar
------------------------------------

The French Revolutionary Calendar (or Republican Calendar) was
introduced in France on 24 November 1793 and abolished on 1 January
1806. It was used again briefly during the Paris Commune in 1871.


8.1. What does a Republican year look like?
-------------------------------------------

A year consists of 365 or 366 days, divided into 12 months of 30 days
each, followed by 5 or 6 additional days. The months were:

1. Vendemiaire           7. Germinal
2. Brumaire              8. Floreal
3. Frimaire              9. Prairial
4. Nivose               10. Messidor
5. Pluviose             11. Thermidor
6. Ventose              12. Fructidor

(The second e in Vendemiaire and the e in Floreal carry an acute
accent. The o's in Nivose, Pluviose, and Ventose carry a circumflex
accent.)

The year was not divided into weeks, instead each month was divided
into three "decades" of 10 days, of which the final day was a day of
rest. This was an attempt to de-Christianize the calendar, but it was
an unpopular move, because now there were 9 work days between each day
of rest, whereas the Gregorian Calendar had only 6 work days between
each Sunday.

The ten days of each decade were called, respectively, Primidi, Duodi,
Tridi, Quartidi, Quintidi, Sextidi, Septidi, Octidi, Nonidi, Decadi.

The 5 or 6 additional days followed the last day of Fructidor and were
called:
      1. Jour de la vertu (Day of virtue)
      2. Jour du genie (Day of genius)
      3. Jour du travail (Day of labour)
      4. Jour de l'opinion (Day of opinion)
      5. Jour des recompenses (Day of rewards)
      6. Jour de la revolution (Day of the revolution)  (the leap day)

(Different sources refer to these days as either "jour" ("day") or
"fete" ("celebration" or "feast").)

Each year was supposed to start on autumnal equinox (around 22
September), but this created problems as will be seen in section 8.3.


8.2. How does one count years?
------------------------------

Years are counted since the establishment of the first French Republic
on 22 September 1792. That day became 1 Vendemiaire of the year 1 of
the Republic. (However, the Revolutionary Calendar was not introduced
until 24 November 1793.)


8.3. What years are leap years?
-------------------------------

Leap years were introduced to keep New Year's Day on autumnal
equinox. But this turned out to be difficult to handle, because
equinox is not completely simple to predict.

In fact, the first decree implementing the calendar (5 Oct 1793)
contained two contradictory rules, as it stated that:
  - the first day of each year would be that of the autumnal equinox
  - every 4th year would be a leap year

In practice, the first calendars were based on the equinoxial
condition.

To remove the confusion, a rule similar to the one used in the
Gregorian Calendar (including a 4000 year rule as described in section
2.2.2) was proposed by the calendar's author, Gilbert Romme, but his
proposal ran into political problems.

In short, during the time when the French Revolutionary Calendar was
in use, the following years were leap years: 3, 7, and 11.


8.4. How does one convert a Republican date to a Gregorian one?
---------------------------------------------------------------

The following table lists the Gregorian date on which each year of the
Republic started:

Year  1: 22 Sep 1792            Year  8: 23 Sep 1799
Year  2: 22 Sep 1793            Year  9: 23 Sep 1800
Year  3: 22 Sep 1794            Year 10: 23 Sep 1801
Year  4: 23 Sep 1795            Year 11: 23 Sep 1802
Year  5: 22 Sep 1796            Year 12: 24 Sep 1803
Year  6: 22 Sep 1797            Year 13: 23 Sep 1804
Year  7: 22 Sep 1798            Year 14: 23 Sep 1805


9. The Maya Calendar
--------------------

(I am very grateful to Chris Carrier for providing most of the
information about the Maya calendar.)

Among their other accomplishments, the ancient Mayas invented a
calendar of remarkable accuracy and complexity. The Maya calendar was
adopted by the other Mesoamerican nations, such as the Aztecs and the
Toltec, which adopted the mechanics of the calendar unaltered but
changed the names of the days of the week and the months.

The Maya calendar uses three different dating systems in parallel, the
"Long Count", the "Tzolkin" (divine calendar), and the "Haab" (civil
calendar).  Of these, only the Haab has a direct relationship to the
length of the year.

A typical Mayan date looks like this: 12.18.16.2.6, 3 Cimi 4 Zotz.

12.18.16.2.6 is the Long Count date.
3 Cimi is the Tzolkin date.
4 Zotz is the Haab date.


9.1. What is the Long Count?
----------------------------

The Long Count is really a mixed base-20/base-18 representation of a
number, representing the number of days since the start of the Mayan
era. It is thus akin to the Julian Day Number (see section 2.16).

The basic unit is the "kin" (day), which is the last component of the
Long Count. Going from right to left the remaining components are:

   uinal  (1 uinal = 20 kin = 20 days)
   tun    (1 tun = 18 uinal = 360 days = approx. 1 year)
   katun  (1 katun = 20 tun = 7,200 days = approx. 20 years)
   baktun (1 baktun = 20 katun = 144,000 days = approx. 394 years)

The kin, tun, and katun are numbered from 0 to 19.
The uinal are numbered from 0 to 17.
The baktun are numbered from 1 to 13.

Although they are not part of the Long Count, the Mayas had names for
larger time spans. The following names are sometimes quoted, although
they are not ancient Maya terms:

   1 pictun = 20 baktun = 2,880,000 days = approx. 7885 years
   1 calabtun = 20 pictun = 57,600,000 days = approx. 158,000 years
   1 kinchiltun = 20 calabtun = 1,152,000,000 days = approx. 3 million years
   1 alautun = 20 kinchiltun = 23,040,000,000 days = approx. 63 million years


9.1.1. When did the Long Count start?
-------------------------------------

Logically, the first date in the Long Count should be 0.0.0.0.0, but
as the baktun (the first component) are numbered from 1 to 13 rather
than 0 to 12, this first date is actually written 13.0.0.0.0.

The authorities disagree on what 13.0.0.0.0 corresponds to in our
calendar. I have come across three possible equivalences:

13.0.0.0.0 =  8 Sep 3114 BC (Julian) = 13 Aug 3114 BC (Gregorian)
13.0.0.0.0 =  6 Sep 3114 BC (Julian) = 11 Aug 3114 BC (Gregorian)
13.0.0.0.0 = 11 Nov 3374 BC (Julian) = 15 Oct 3374 BC (Gregorian)

Assuming one of the first two equivalences, the Long Count will again
reach 13.0.0.0.0 on 21 or 23 December AD 2012 - a not too distant future.

The date 13.0.0.0.0 may have been the Mayas' idea of the date of the
creation of the world.


9.2. What is the Tzolkin?
-------------------------

The Tzolkin date is a combination of two "week" lengths.

While our calendar uses a single week of seven days, the Mayan
calendar used two different lengths of week:
   - a numbered week of 13 days, in which the days were numbered from
     1 to 13
   - a named week of 20 days, in which the names of the days were:

        0. Ahau         5. Chicchan     10. Oc          15. Men
        1. Imix         6. Cimi         11. Chuen       16. Cib
        2. Ik           7. Manik        12. Eb          17. Caban
        3. Akbal        8. Lamat        13. Ben         18. Etznab
        4. Kan          9. Muluc        14. Ix          19. Caunac

As the named week is 20 days and the smallest Long Count digit is 20
days, there is synchrony between the two; if, for example, the last
digit of today's Long Count is 0, today must be Ahau; if it is 6, it
must be Cimi. Since the numbered and the named week were both "weeks",
each of their name/number change daily; therefore, the day after 3
Cimi is not 4 Cimi, but 4 Manik, and the day after that, 5 Lamat. The
next time Cimi rolls around, 20 days later, it will be 10 Cimi instead
of 3 Cimi. The next 3 Cimi will not occur until 260 (or 13*20) days
have passed. This 260-day cycle also had good-luck or bad-luck
associations connected with each day, and for this reason, it became
known as the "divinatory year."

The "years" of the Tzolkin calendar are not counted.


9.2.1. When did the Tzolkin start?
----------------------------------

Long Count 13.0.0.0.0 corresponds to 4 Ahau. The authorities agree on
this.


9.3. What is the Haab?
----------------------

The Haab was the civil calendar of the Mayas. It consisted of 18
"months" of 20 days each, followed by 5 extra days, known as
"Uayeb". This gives a year length of 365 days.

The names of the month were:
        1. Pop           7. Yaxkin      13. Mac
        2. Uo            8. Mol         14. Kankin
        3. Zip           9. Chen        15. Muan
        4. Zotz         10. Yax         16. Pax
        5. Tzec         11. Zac         17. Kayab
        6. Xul          12. Ceh         18. Cumku
        
In contrast to the Tzolkin dates, the Haab month names changed every
20 days instead of daily; so the day after 4 Zotz would be 5 Zotz,
followed by 6 Zotz ... up to 19 Zotz, which is followed by 0 Tzec.

The days of the month were numbered from 0 to 19. This use of a 0th
day of the month in a civil calendar is unique to the Maya system; it
is believed that the Mayas discovered the number zero, and the uses to
which it could be put, centuries before it was discovered in Europe or
Asia.

The Uayeb days acquired a very derogatory reputation for bad luck;
known as "days without names" or "days without souls," and were
observed as days of prayer and mourning. Fires were extinguished and
the population refrained from eating hot food. Anyone born on those
days was "doomed to a miserable life."

The years of the Haab calendar are not counted.

The length of the Tzolkin year was 260 days and the length of the Haab
year was 365 days. The smallest number that can be divided evenly by
260 and 365 is 18,980, or 365*52; this was known as the Calendar
Round. If a day is, for example, "4 Ahau 8 Cumku," the next day
falling on "4 Ahau 8 Cumku" would be 18,980 days or about 52 years
later.  Among the Aztec, the end of a Calendar Round was a time of
public panic as it was thought the world might be coming to an
end. When the Pleiades crossed the horizon on 4 Ahau 8 Cumku, they
knew the world had been granted another 52-year extension.


9.3.1. When did the Haab start?
-------------------------------

Long Count 13.0.0.0.0 corresponds to 8 Cumku. The authorities agree on
this.


9.4. Did the Mayas think a year was 365 days?
---------------------------------------------

Although there were only 365 days in the Haab year, the Mayas were
aware that a year is slightly longer than 365 days, and in fact, many
of the month-names are associated with the seasons; Yaxkin, for
example, means "new or strong sun" and, at the beginning of the Long
Count, 1 Yaxkin was the day after the winter solstice, when the sun
starts to shine for a longer period of time and higher in the
sky. When the Long Count was put into motion, it was started at
7.13.0.0.0, and 0 Yaxkin corresponded with Midwinter Day, as it did at
13.0.0.0.0 back in 3114 B.C. The available evidence indicates that the
Mayas estimated that a 365-day year precessed through all the seasons
twice in 7.13.0.0.0 or 1,101,600 days.

We can therefore derive a value for the Mayan estimate of the year by
dividing 1,101,600 by 365, subtracting 2, and taking that number and
dividing 1,101,600 by the result, which gives us an answer of
365.242036 days, which is slightly more accurate than the 365.2425
days of the Gregorian calendar.

(This apparent accuracy could, however, be a simple coincidence. The
Mayas estimated that a 365-day year precessed through all the seasons
*twice* in 7.13.0.0.0 days. These numbers are only accurate to 2-3
digits. Suppose the 7.13.0.0.0 days had corresponded to 2.001 cycles
rather than 2 cycles of the 365-day year, would the Mayas have noticed?)


10. The Chinese Calendar
------------------------

Although the People's Republic of China uses the Gregorian calendar
for civil purposes, a special Chinese calendar is used for determining
festivals. Various Chinese communities around the world also use this
calendar.

The beginnings of the Chinese calendar can be traced back to the 14th
century BC. Legend has it that the Emperor Huangdi invented the
calendar in 2637 BC.

The Chinese calendar is based on exact astronomical observations of
the longitude of the sun and the phases of the moon. This means that
principles of modern science have had an impact on the Chinese
calendar.

I can recommend visiting Helmer Aslaksen's web site at
http://www.chinesecalendar.net for more information about the Chinese
calendar.


10.1. What does the Chinese year look like?
-------------------------------------------

The Chinese calendar - like the Hebrew - is a combined solar/lunar
calendar in that it strives to have its years coincide with the
tropical year and its months coincide with the synodic months. It is
not surprising that a few similarities exist between the Chinese and
the Hebrew calendar:

   * An ordinary year has 12 months, a leap year has 13 months.
   * An ordinary year has 353, 354, or 355 days, a leap year has 383,
     384, or 385 days.

When determining what a Chinese year looks like, one must make a
number of astronomical calculations:

First, determine the dates for the new moons. Here, a new moon is the
completely "black" moon (that is, when the moon is in conjunction with
the sun), not the first visible crescent used in the Islamic and
Hebrew calendars. The date of a new moon is the first day of a new
month.

Secondly, determine the dates when the sun's longitude is a multiple
of 30 degrees. (The sun's longitude is 0 at Vernal Equinox, 90 at
Summer Solstice, 180 at Autumnal Equinox, and 270 at Winter Solstice.)
These dates are called the "Principal Terms" and are used to determine
the number of each month:

Principal Term 1 occurs when the sun's longitude is 330 degrees.
Principal Term 2 occurs when the sun's longitude is 0 degrees.
Principal Term 3 occurs when the sun's longitude is 30 degrees.
etc.
Principal Term 11 occurs when the sun's longitude is 270 degrees.
Principal Term 12 occurs when the sun's longitude is 300 degrees.

Each month carries the number of the Principal Term that occurs in
that month.

In rare cases, a month may contain two Principal Terms; in this case
the months numbers may have to be shifted. Principal Term 11 (Winter
Solstice) must always fall in the 11th month.

All the astronomical calculations are carried out for the meridian 120
degrees east of Greenwich. This roughly corresponds to the east coast
of China.

Some variations in these rules are seen in various Chinese
communities.


10.2. What years are leap years?
--------------------------------

Leap years have 13 months. To determine if a year is a leap year,
calculate the number of new moons between the 11th month in one year
(i.e., the month containing the Winter Solstice) and the 11th month in
the following year. If there are 13 new moons from the start of the
11th month in the first year to the start of the 11th month in the
second year, a leap month must be inserted.

In leap years, at least one month does not contain a Principal Term.
The first such month is the leap month. It carries the same number as
the previous month, with the additional note that it is the leap
month.


10.3. How does one count years?
-------------------------------

Unlike most other calendars, the Chinese calendar does not count years
in an infinite sequence. Instead years have names that are repeated
every 60 years.

(Historically, years used to be counted since the accession of an
emperor, but this was abolished after the 1911 revolution.)

Within each 60-year cycle, each year is assigned a name consisting of
two components:

The first component is a "Celestial Stem":

        1. jia          6. ji
        2. yi           7. geng
        3. bing         8. xin
        4. ding         9. ren
        5. wu          10. gui

These words have no English equivalent.

The second component is a "Terrestrial Branch":

        1. zi (rat)             7. wu (horse)
        2. chou (ox)            8. wei (sheep)
        3. yin (tiger)          9. shen (monkey)
        4. mao (hare, rabbit)  10. you (rooster)
        5. chen (dragon)       11. xu (dog)
        6. si (snake)          12. hai (pig)

The names of the corresponding animals in the zodiac cycle of 12
animals are given in parentheses.

Each of the two components is used sequentially. Thus, the 1st year of
the 60-year cycle becomes jia-zi, the 2nd year is yi-chou, the 3rd
year is bing-yin, etc. When we reach the end of a component, we start
from the beginning: The 10th year is gui-you, the 11th year is jia-xu
(restarting the Celestial Stem), the 12th year is yi-hai, and the 13th
year is bing-zi (restarting the Terrestrial Branch). Finally, the 60th
year becomes gui-hai.

This way of naming years within a 60-year cycle goes back
approximately 2000 years. A similar naming of days and months has
fallen into disuse, but the date name is still listed in calendars.

It is customary to number the 60-year cycles since 2637 BC, when the
calendar was supposedly invented. In that year the first 60-year cycle
started.


10.4. What is the current year in the Chinese calendar?
-------------------------------------------------------

The current 60-year cycle started on 2 Feb 1984. That date bears the name
bing-yin in the 60-day cycle, and the first month of that first year
bears the name gui-chou in the 60-month cycle.

This means that the year wu-zi, the 25th year in the 78th cycle,
started on 7 Feb 2008.



11. Frequently Asked Questions about this FAQ
---------------------------------------------

This chapter does not answer questions about calendars. Instead it
answers questions that I am often asked about this document.


11.1. Why doesn't the FAQ describe calendar X?
----------------------------------------------

I am frequently asked to add a chapter describing the Japanese
calendar, the Ethiopian calendar, the Hindu calendar, etc.

But I have to stop somewhere. I have discovered that the more calendars
I include in the FAQ, the more difficult it becomes to ensure that the
information given is correct. I want to work on the quality rather
than the quantity of information in this document. It is therefore not
likely that other calendars will be added in the near future.


11.2. Why doesn't the FAQ contain information X?
------------------------------------------------

Obviously, I cannot include everything. So I have to prioritize. The
things that are most likely to be omitted from the FAQ are:

- Information that is relevant to a single country only.
- Views that are controversial and not supported by recognized
  authorities.


11.3. Why don't you reply to my e-mail?
---------------------------------------

I try to reply to all the e-mail I receive. But occasionally the
amount of mail I receive is so large that I have to ignore some
letters. If this has caused your letter to be lost, I apologize.

But please don't let this stop you from writing to me. I enjoy
receiving letters, even if I can't answer them all.


11.4. How do I know that I can trust your information?
------------------------------------------------------

I have tried to be accurate in everything I have described. If you are
unsure about something that I write, I suggest that you try to verify
the information yourself. If you come across a recognized authority
that contradicts something that I've written, please let me know.


11.5. There is an error in one of your formulas!
------------------------------------------------

Is that a question?

Anyway, I very much doubt that that the formulas given in this
document are wrong. Most of the formulas have been tested for all
possible dates between AD 1 and AD 10,000.

When people think they have found an error in a formula, the most
frequent reason is that they have used ordinary division instead of
integer division.


11.6. Can you recommend any good books about calendars?
-------------------------------------------------------

This is a big question because there are so many excellent books. At
this point I shall only recommend three books:

Edward M. Reingold & Nachum Dershowitz: "Calendrical Calculations.
The Millennium Edition". Cambridge University Press 2001.
ISBN 0-521-77752-6.
http://emr.cs.iit.edu/home/reingold/calendar-book/second-edition/index.html

  This book contains a lot of information about a huge number of
  calendars. As the title indicates, it has a strong emphasis on
  algorithms for calendrical calculations, so if you want to use your
  computer to compute calendars, this is a great book.

Bonnie Blackburn & Leofranc Holford-Strevens: "The Oxford Companion to
the Year". Oxford University Press 1999. ISBN 0-19-214231-3.

  A very thorough (900+ pages) book about the history of calendars.
  The book includes a large collections about customs related to each
  day of the year.

Finally, I have myself written a book:
Claus Tondering: "Julius og Gregor". 2007. ISBN 978-87-92032-58-4. 

  Unfortunately, this book is in Danish, but if you can read the
  Scandinavian languages, this is a useful handbook. One of its main
  strengths is a huge collection of tables of various calendars.


11.7. Do you know a web site where I can find information about X?
------------------------------------------------------------------

Probably not.

Good places to start your calendar search include:

   http://www.calendarzone.com
   http://personal.ecu.edu/mccartyr/calendar-reform.html


12. Date
--------

This version 2.9 of this document was finished on

        Friday before the second Sunday after Easter, the 4th day of
        April anno ab Incarnatione Domini MMVIII, indict. I, epacta
        XXII, luna XXVII, anno post Margaretam Reginam Daniae natam
        LXVIII, on the feast of Saint Isidore of Seville.
             
        The 28th day of Adar II, Anno Mundi 5768.

        The 27th day of Rabi' al-awwal, Anno Hegirae 1429.

        The 16th day of Farvardin, Anno Persico 1387.

        The 28th day of the 2nd month of the year wu-zi of the 78th
        cycle.

        Julian Day 2,454,561.

--- End of part 3 ---
 year wu-zi of the 78th
        cycle.

        Julian Day 2,454,561.

--- End of part 3 ---

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