The Calendar

Table of contents

• Table of contents
• General introduction
  • What is a calendar?
  • The Julian date
  • The day, the month, the year
• The Gregorian calendar
  • The Gregorian solar calendar
  • The Gregorian ISO 8601 calendar
  • The Gregorian computus
    • Dominical letter
    • Roman indiction
    • Golden number
    • Julian epact
    • Epact
    • Easter
  • The Gregorian lunar calendar
    • General description
    • Which years are embolismic
    • Initial data
    • Days
    • Length of the year
    • Per cycle statistics
    • Correspondance with the Gregorian solar calendar
    • Gregorian epact tables and the “tabular” Gregorian lunar calendar
    • On which month does the lunar year start?
  • A computer script
• The Mayan calendar
  • The Haab
  • The Tzolkin
  • The Short Count
  • The Long Count
  • The Lords of the Night
  • The Mayan correlation
  • When does the Long Count end?

General introduction

What is a calendar?

I won't try to give a formal definition of what a calendar is, but the important thing is that a calendar counts days (that is, solar—or synodic—days). It is not, strictly speaking, a time scale, or, if it is, it is referred to Universal Time (or some translation of it to account for the longitude of the locus where it is applied), not, say, Ephemeris Time. But essentially a calendar does not subdivide the day, and does not take into account hours, minutes and seconds, and days are measured in the most “naïve” way, counting the number of sunrises or sunsets. Of course, the exact moment at which the calendar changes day may be ill-defined, may depend on the place, and may vary from calendar to calendar (the Jewish calendar, for example, counts the start of a day at sunset): this may introduce a shift of one day in either direction when correlating two calendars, but we will consider this unessential and insist on the fiction that a day is a day.

The Julian date

The simplest kind of calendar is the Julian date. This counts the number of days since January 1 of the year ~4713 (aka 4713 before the Common Era, or −4712) of the proleptic Julian calendar, at 12 Noon (Universal Time, or of whatever time scale is being used: we have just explained that this is not the problem at hand), or November 24 of the year ~4714 of the proleptic Gregorian calendar. This is an incommodious definition because (a) nobody who was alive in the year 4713 before the Common Era lives to tell us about that point, (b) the length of the day has changed slightly since that time, making an exact count of days very impractical, (c) people get confused about the year ~4713 which is −4712, and (d) neither the Gregorian, nor the Julian calendar, were used at that time, hence the word “proleptic”. It is much better to say that at the epoch J2000 (which is January 1 of the year 2000 in the Gregorian calendar, at 12 Noon, and I deliberately disregard the less-than-a-minute difference between Coordinated Universal Time and Terrestrial Time, here) the Julian date was 2451545, and forget about point zero. The Julian date is then a simple day count and has no subtleties like years, months, leap counts or whatever. One unpleasant feature about the Julian date, however, is that it counts dates from noon (so if we wish to designate a day by the integral part of the Julian date, it will change at noon), whereas nearly all calendars either change day at sunrise, or sunset, or midnight, but certainly not at noon. The Julian date is pratical for astronomy, but this bit about changing at noon is an annoyance when it comes to comparing calendars. Some people prefer to use the modified Julian date, which is defined as the Julian date minus 2400000.5, so that it changes at midnight rather than noon.

Trivia: The “Julian” in “Julian date” refers to Julius Scaliger, who invented the scale in 1583, and is therefore not the same “Julian” as in “Julian calendar”, which refers to Caius Julius Cæsar, who introduced the calendar in question (on Julian date 1704986.5).

The day, the month, the year

Here we briefly discuss the physical values of the day, month and year (not, that is, those of some particular calendar).

The year is the period of the Earth's revolution around the Sun. Now there are several kinds of years according as one counts from fixed star to fixed star, or from equinox to equinox, or from perihelion to perihelon, and so on. But the most important kind for the calendar, since men are usually more interested about seasons than about the positions of the stars or the perihelion of the Earth's orbit, is the tropical year, counted from equinox to equinox. There are various (smallish) periodic perturbations in the Earth's movement, but the mean tropical year is a well-defined quantity, whose value can be measured accurately (or predicted from other measured values used to build a planetary theory): namely, to the first order,

tropical year = 31556925.25s − 0.532s·t

where t is Terrestrial Time (actually, Barycentric Dynamical Time) expressed in Julian centuries (that is, 3155760000 seconds) measured from J2000. In other words, at J2000, the length of the tropical year was 31556925.25 seconds, and it was a little more than half a second longer one century before this.

The month is the period of the Moon's revolution around the Earth. Again, there are several kinds of months according as one counts from fixed star to fixed star, equinox to equinox, perigee to perigee, and so on, but the most important kind for the calendar, because the obvious observable phenomenon is the phase of the Moon, counts the position of the Moon from that of the Sun, and is called the synodic month. Here the perturbations are much more important than for the year. We have, for the secular part, and to the first order in t:

synodic month = 2551442.877s + 0.0187s·t

Now we move to the solar day. Unfortunately, here there is no good value for the duration of the mean solar day: this is because irregularities are so large and so unpredictable that it doesn't make much sense to attempt a Poisson series expansion for the Earth's rotation over a long time scale, as is done for the Earth's revolution or the Moon's revolution. The following relation (due to Spencer Jones), therefore, is more or less conventional:

solar day = 86400.00198s + 0.00164s·t

Note that the mean solar day was equal to 86400 seconds roughly a little over a century ago. This is because the definition of the SI second was chosen to agree with the second of ephemeris time, which was itself based on the theory of ephemerides published by Newcomb at the end of the XIXth century, mostly based on data of the previous 50 years or so on the assumption that the length of the day was always 86400 seconds.

Now we can compute:

tropical year / solar day = 365.2421820 − 0.0000131·t

Here a good rational approximation is 1461/4, or 1461 solar days in 4 tropical years: this is the basis of the Julian calendar. A further Euclid's approximant to the ratio is 12053 days in 33 years (that is, use 8 leap years of 366 days and 25 regular years of 365 days in a cycle of 33 years). Beyond that, the varations of the lengths over historical times are too significant.

For the length of the month in days:

synodic month / solar day = 29.53058818 − 0.00000034·t

A good, and very simple, rational approximation is simply 59/2, in other words 59 days in two synodic months, or alternate a “full” lunar month of 30 days and a “hollow” one of 29 days. A very good approximation is that there are 1447 days in 49 lunar months (that is, 26 full and 23 hollow).

The Gregorian calendar

The Gregorian solar calendar

Years in the Gregorian (solar) calendar are either 365 or 366 days long. Those with 365 days in them are called “regular” and those with 366 days in them are called “leap”. Years receive a number, given consecutively so that the year 2000 starts on Julian date 2451544.5 (plus or minus 0.5) according to the time zone; conventionally, years with negative numbers are written with a tilde and omitting the year 0 (so that the year 0 is written ~1, the year 1 is ~2, the year 2 is ~3 and so on); alternatively, positive year numbers are called “of the common era” and the tilde can be replad by the indication “before the common era”.

A Gregorian year is leap if and only if its number is divisible by four except if it is divisible by 100 but not 400. Thus, the year 2000 is leap, and so is 2004, but 2100 is not leap.

Each Gregorian year contains twelve months, numbered 1 through 12: these are unrelated to the Moon's phases. In a regular year, they contain respectively 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30 and 31 days; in a leap year, the second month has 29 days rather than 28. These months have conventional names: January, February, March, April, May, June, July, August, September, October, November and December. In each month, the days are numbered from 1 through the number of days of the month. A date in the Gregorian calendar is typically written as follows: the four digits of the year, dash, the two digits of the month, dash, and the two digits of the day. For example, the first day of the Gregorian year 2000, starting on Julian date 2451544.5, is written 2000-01-01.

Besides the months and years, the Gregorian calendar is also typically used with a 7-day period called the week. This period is applied systematically and without variation. The seven days, which are often associated with the numbers 0 through 6, bear conventional names: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday (sometimes Sunday is placed last, and is then numbered 7 instead of 0; this is of course irrelevant). The day 2000-01-01 is a Saturday.

Note that the Gregorian calendar has a period of 400 years: this period contains 303 regular years and 97 leap years, to a total of 146097 days, or precisely 20871 weeks. So after a 400 year cycle, the days return as they were.

Note: In the original Gregorian calendar, the extra day in leap years was inserted after February 23, causing two days to be labeled February 24 (the second one called February 24bis). Using this practice would cause a shift of one day of the last five days of February in leap years, but of course no long-term shift.

The Gregorian ISO 8601 calendar

The Gregorian ISO 8601 calendar is tightly related to the usual Gregorian (solar) calendar, but it uses weeks instead of months. The week days are exactly the same as those of the usual Gregorian calendar. The weeks start on Monday (day 1) and end on Sunday (day 7). Each year contains an integral number of weeks: either 52 (“short year”) or 53 (“long year”). Week 1 of the ISO 8601 year Y is the one which starts on the Monday on or immediately before January 4 of the year Y of the Gregorian calendar, and this determines the start of the ISO 8601 year, whose weeks are then numbered consecutively from 1 through 52 or 53, until the previous day of the first day of the ISO 8601 year Y+1. For example, since 2000-01-01 of the Gregorian calendar was a Saturday, 2000-01-03 was a Monday, so ISO 8601 year 2000 starts on 2000-01-03; as for 2000-01-01 of the Gregorian calendar, it is day 6 of week 52 of 1999 in the ISO 8601 calendar.

The ISO 8601 calendar can be defined without reference to the Gregorian solar calendar: it just needs to be said that the year 2000 (day 1 of week 1 of 2000) starts on Julian date 2451546.5, and that the following table is used to determine whether year Y is short (52 weeks, written ‘S’ below) or long (53 weeks, written ‘L’ below) in function of the residue mod 400 of its number Y:

The table also gives the Gregorian date of start of the ISO 8601 year (day 1 of week 1, that is); naturally, if the date given is in December, it must be understood as part of the previous year. As can be seen, there are 71 long years and 329 short years in the 400-year cycle of the calendar.

The Gregorian computus

Dominical letter

The first day (January 1) of the Gregorian calendar year is labeled ‘A‘, the second is labeled ‘B’, the third is ‘C’ and so on until January 7 which is ‘G’ and then back to ‘A’: label every day in this manner except February 29 which receives no label:

The dominical letter is the letter label of the Sundays in the year; leap years receive two dominical letters (one for January and February and one for March through December). For example, the dominical letters of 2000 are ‘BA’; that of 2001 is ‘G’, that of 2002 is ‘F’, that of 2003 is ‘E’, those of 2004 are ‘DC’, and so on.

The dominical letter of any year of the Gregorian cycle is given by the following table:

Of course, given the dominical letter of the year and the letter label of the day, it is obvious to find the day of the week of any given Gregorian date:

Roman indiction

Roman indiction is a 15-year cycle (introduced by emperor Constantine for tax reasons) which is part of the computus but of no interest whatsoever except to decorate calendars. Numbers go from 1 through 15 perpetually, and year 2000 has Roman indiction 8.

Golden number

The golden number is a 19-year cycle related to the Meton cycle. It is used in computing the epact. Numbers go from 1 through 19 perpetually, and year 2000 has golden number 6. Years having golden number 19 are called “hollow” (for the purposes of the Gregorian lunar calendar).

Julian epact

The Julian epact, which in the Gregorian calendar is not the epact, is computed from the golden number by means of the following table:

Epact

The epact is the fundamental basis of all lunar computations in the Gregorian calendar, whether the full Gregorian lunar calendar or simply the date of Easter. It is something related to the age of the Moon on the first of March, but we shall simply consider it as a device for computation.

The (Gregorian) epact is obtained by adding a Gregorian correction to the Julian epact, everything being computed modulo 30. The Gregorian correction depends only on the quotient of the year number by 100: it is obtained by summing two terms, a “solar correction” which subtracts one every centennial year not divisible by 400 (that is, every leap year that is omitted from the Gregorian calendar with respect to the simple Julian rule), and a “lunar correction” which adds one to the epact eight times every twenty-five hundred years. This is summarized as follows:

To extend this array, repeat the “solar correction” line with periodicity four centuries, the “lunar correction” line with periodicity twenty-five centuries (note that it is not periodic with period three centuries, contrary to what may seem: this common mistake causes a problem in the year 4200 and following), add them to produce the “total correction” line which therefore has a periodicity of one hundred centuries. The “running correction” line is the sum modulo 30 of the total correction line up to that point plus an initial value of 29 for the year 2000 (and its periodicity is of three thousand centuries).

The epact is the sum modulo 30 of the Julian epact and the Gregorian correction. There is, however, one exception: if the sum in question is 25 and the golden number is between 12 and 19, or, equivalently, if the sum is 25 but the sum 24 also appears in the same 19-year cycle, then the epact is not 25 but “25*”. When the sum is 25 for a year with a golden number between 1 and 11, it is a regular 25. We can draw the following table of epacts for a few centuries:

The full periodicity of the cycle of epacts is 5700000 years (300000×19).

Easter

Easter is a mobile day within the Gregorian calendar. It is determined from the calendar of the year using the epact. Specifically, Easter is the Sunday strictly following the first ecclesiastical full moon (called the Paschal moon) that occurs on or after March 21. The date of the Paschal moon is given according to the epact of the year and the date of Easter according to the latter and the dominical letter (the second letter when the year is a leap year) by the following table:

Take a few examples: the year 2000 has golden number 6, hence epact 24, and dominical letters BA, hence Easter of 2000 fell on 2000-04-23; the year 2001 has golden number 7 hence epact 5, and dominical letter G, so Easter of 2001 fell on 2001-04-15; the year 2002 has golden number 8 hence epact 16, and dominical letter F, so Easter of 2002 fell on 2002-03-31; the year 2003 has golden number 9 hence epact 27, and dominical letter E, so Easter of 2003 fell on 2003-04-20; the year 2004 has golden number 10 hence epact 8, and dominical letter DC, so Easter of 2004 falls on 2004-04-11.

What follows is a precomputed table of the date of Easter for years 1800 through 2299.

The Gregorian lunar calendar

Important note: There are various minor variations on the Gregorian lunar calendar which introduce differences in the lengths of the first three months and the thirteenth in the lunar year. Here we present the variant (the “regular” Gregorian lunar calendar) which makes every lunar month either 29 or 30 days long: this seems to be the only truly sane variant, the one that it would make reasonable sense to use in practice; it is also slightly easier to describe. Other variants, which make the start of the months follow blindly the indication of epacts (the “tabular” lunar calendar), introduce occasional months of 31 or even 28 days, and do not constitute a real (or at any rate, sensible) calendar. See below for more about this.

Terminology: By “lunar calendar” what is really meant is “luni-solar calendar”, since the months of the Gregorian lunar calendar are aligned with the phases of the moon (to a tolerable precision, that is, within a few centuries' range) and the years coincide more or less (in the sense that they don't deviate in the long term) with those of the Gregorian solar calendar, which are themselves essentially solar (tropical) years. No matter what, luni-solar calendars are always complicated, and for historical reasons the Gregorian luni-solar calendar is even more devilish than some.

General description

The Gregorian lunar calendar uses years of twelve or thirteen months, each month having 29 or 30 days. Years with thirteen months in them are called “embolismic”; precise rules will be given further on to determine when a year is embolismic, but for the moment let us say roughly that, within a century, seven out of nineteen years are embolismic (in accordance with the Meton cycle).

The first month in the year normally has 30 days; the exception to this is when the year has golden number 1 and the previous year (which has golden number 19) was not embolismic. This is because years with golden number 19 are supposed to be “hollow”, i.e., they remove one day (following the Meton-Callippus cycle): the day in question is normally removed from the embolismic month (which then has 29 days instead of 30), but when there is no embolismic month to remove a day from, the day has to be deducted from the first month of the following year, making it 29 days long. This will not happen until the year 3116, however (because, until 3115, all years with golden number 19 are embolismic).

The second month in the year normally has 29 days. However, certain years are “leap” and the second month then has 30 days. The leap years are determined using a rule very similar to that for the Gregorian solar calendar, except that the centennial correction is not quite the same (it is rarer): a year is (lunar) leap when its number is divisible by four except when it is divisible by 100 and its quotient by one hundred is congruent to 2, 5, 8, 11, 14, 18, 21 or 24 modulo 25. Of course, one will recognize, here, the solar and lunar corrections of the epact: centennial years with the solar correction but no lunar correction are leap in the lunar but not the solar calendar, and those with the lunar correction but no solar correction are leap in the solar but not the lunar calendar; centennial years with both corrections are not leap at all, and those with neither correction are leap in both calendars. So, for example, 2000 is lunar leap (as well as solar leap), 2100 is not lunar leap (nor is it solar leap), 2200 and 2300 are lunar leap (but not solar leap), 2400 is not lunar leap (but it is solar leap), and so on; for non-centennial years, the rule is simply to check for divisibility by four, and is the same for lunar and solar (so 2104, 2108, 2112 and so on, are all leap years).

The third month in the lunar year always has 30 days, the fourth always has 29 days, the fifth always has 30, the sixth always has 29, the seventh always has 30, the eighth always has 29, the ninth always has 30, the tenth always has 29, the eleventh always has 30 and the twelfth always has 29.

The thirteenth month, or embolismic month, when it exists, normally has 30 days. However, it has 29 days in those years having golden number 19 (the “hollow years”).

Trivia: The months of the Gregorian lunar calendar do not bear any names as far as I know. It would be a worthy and interesting challenge to invent some. More as a joke than anything else (but then, the very idea of resurrecting the Gregorian lunar calendar is itself more of a joke than anything else), and for various stupid reasons (some of which of the very “private” joke kind) I propose: Terminus, Lipidus, Venuch, Amber, Pook, Jupe, Tibery, Claudy, Septil, Octil, Novil, Decil and sometimes Mercuary.

Which years are embolismic

To determine whether a year is embolismic (i.e., has thirteen months rather than twelve), various rules can be given. Here's one that isn't too awfully complicated: years having an epact of 16, 17, 18, 19, 20, 21, 22, 23, 24 or 25* (but not 25!) are always embolismic. Years having an epact of 0 through 11, or 25, 26, 27, 28 or 29 (but not 25*!), are never embolismic. When the epact is 12, 13, 14 or 15, the epact of the following year needs to be checked: years with epact 12 through 15 are embolismic exactly when the following one is not embolismic (actually, this sounds more complicated then it really is: the epact of the following year can only be 22, 23, 24, 25, 25*, 26, 27 or 28, so it will never be necessary to apply the rule recursively).

It turns out that years with epacts 12 and 13 are rarely embolimisc, those with epact 15 are very frequently embolismic. More precisely, the only years with epact 15 that are not embolismic are those just before a centennial year applying the solar correction but not the lunar correction (i.e., a centennial year that is lunar leap but not solar leap), which have a golden number between 11 and 18 (the next year then has epact 25*). Embolismic years with epact 13 are those with golden number 19 (except those just before a centennial year with only the solar correction), as well as those just before a centennial year with only the lunar correction, that have a golden number between 1 and 10. Embolismic years with epact 12 are exceedingly rare: the next such case is the lunar year 37999 (which has epact 12 and starts on December 20 of 37998 in the Gregorian solar calendar whereas the next year, 38000, has epact 25 and starts on January 7 of 38000, so evidently there are thirteen months in the lunar year 37999; of course this is more or less a joke since by that time the calendar will have long since entirely ceased to agree with the Sun and the Moon anyway); it happens exactly when the year has golden number 19 and is just before a centennial year with only the lunar correction. The case which is truly split is that of epact 14: barring complications just before a centennial year, years with epact 14 are embolismic when their golden number is between 1 and 10, or 19.

We summarize all this as follows:

Epacts 0 through 11

Never embolismic.

Epact 12

Embolismic only when the next year has epact 25, which is exceedingly rare: this happens when the year (that with epact 12) has golden number 19 and occurs just before a centennial year that applies the lunar correction but not the solar correction.

Epact 13

Embolismic only when the next year has epact 25 or 26. This happens in two cases: either when the year (that with epact 13) has golden number 19 and the following one is not a centennial year applying only the solar correction, or when the year (with epact 13) has golden number between 1 and 10 or 19, and the next year is a centennial year applying only the lunar correction.

Epact 14

The most complicated case. The simplest thing is just to check the epact of the following year: if it is 24 or 25* then the year with epact 14 is not embolismic, whereas if it is 25, 26 or 27, the year with epact 14 is embolismic. For years with epact 14 that are not just before a centennial year, the year (with epact 14) is embolismic when its golden number is between 1 and 10 or equal to 19. The same holds just before a centennial year with neither correction, or with both the solar and lunar corrections. Years with epact 14 just before a centennial year with only the solar correction are embolismic only when their golden number is 19. Years with epact 14 just before a centennial year with only the lunar correction are (always) embolismic.

Epact 15

Embolismic except when the following year has epact 25*: this happens exactly when the year (with epact 15) has golden number between 11 and 18 and the following year is a centennial year with only the solar correction.

Epacts 16 through 24, and 25* (but not 25)

Always embolismic.

Epacts 25 through 29 (but not 25*)

Never embolismic.

Here's yet another summary:

(Naturally, “pre-Solar” means just before a centennial year with only the solar correction, and “pre-Lunar” means just before a centennial year with only the lunar correction. Furthermore note that “Except if 11≤g.n.≤18” is entirely synonymous with “If 1≤g.n.≤10 or g.n.=19”.)

This may seem impossibly complicated. Actually, within a given century there is always a simple and fixed pattern of embolismic years in a cycle of 19 years (that is, according to the golden numbers), that repeats itself systematically, for all years between 00 and 98 in the century (year 99 requires more care as it is at the shift of the century); even beyond century borders, it is common for the same pattern to continue, either because there is no Gregorian correction (or because lunar and solar corrections cancel out) or because the shift in epacts is not sufficient to change the embolismic nature of the years. For example, for all years from 1898 through 2609, the seven embolismic years in the ninteen-year cycle are those having golden numbers 3, 6, 8, 11, 14, 17 and 19. Here is a simple table for determining, within a few centuries, which years are embolismic (note that the intervals are deliberately chosen to overlap):

Finally, we give yet another way to determine whether a year is embolismic. Define the “depact” of a year in function of its epact by the following simple table:

Then: a year is embolismic exactly when the depact of the following year is smaller than it. Obviously the epact could have been done away with and the depact could have been used everywhere instead of the epact, mutatis mutandis, and the embolismic rule would then have been very simple.

Initial data

To formally complete the description of the Gregorian lunar calendar, we need one correspondance point. The year 2000 of the Gregorian lunar calendar (golden number 6, epact 24; this year is embolismic) starts on 1999-12-08 of the Gregorian solar calendar, which is the day which starts on Julian date 2451520.5. After this, simply work through the number of days in each month: since 2000 is leap (both in the solar and in the lunar calendars) and not hollow, the number of days in each month are: 30, 30, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30 (total 385), so the lunar year 2001 starts on Julian date 2451905.5 or 2000-12-27 of the Gregorian solar calendar; the year 2001 (which is neither leap nor hollow nor embolismic) then has days of 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30 and 29 days (total 354).

Days

Within each month, days are numbered from 1 through 29 or 30.

Remark: Perhaps it would make more sense to number the days from 2 through 30 in the first month of years where this first month has only 29 days.

An ecclesiastical full moon occurs on the fourteenth day of each month.

Length of the year

The Gregorian lunar year can be of the following lengths:

353 days

As 29+29+30+29+30+29+30+29+30+29+30+29 for non-embolismic regular years following a non-embolismic hollow year (5719, 5738, 5757, 5795, …).

354 days

As 30+29+30+29+30+29+30+29+30+29+30+29 for non-embolismic regular years (2001, 2003, 2006, 2007, …) or
as 29+30+30+29+30+29+30+29+30+29+30+29 for non-embolismic leap years following a non-embolismic hollow year (5700, 5776, 5852, 5928, …).

355 days

As 30+30+30+29+30+29+30+29+30+29+30+29 for non-embolismic leap years (2004, 2012, 2020, 2028, …).

383 days

As 30+29+30+29+30+29+30+29+30+29+30+29+29 for embolismic hollow years (2013, 2051, 2070, 2089, …) or
as 29+29+30+29+30+29+30+29+30+29+30+29+30 for embolismic regular years following a non-embolismic hollow year (3135, 3154, 3173, 3211, …).

384 days

As 30+29+30+29+30+29+30+29+30+29+30+29+30 for embolismic regular years (2002, 2005, 2011, 2019, …) or
as 30+30+30+29+30+29+30+29+30+29+30+29+29 for embolismic hollow leap years (2032, 2108, 2184, 2260, …) or
as 29+30+30+29+30+29+30+29+30+29+30+29+30 for embolismic leap years following a non-embolismic hollow year (3116, 3192, 3268, 3344, …).

385 days

As 30+30+30+29+30+29+30+29+30+29+30+29+30 for embolismic leap years (2000, 2008, 2016, 2024, …).

Again, the “hollow” character of a year only decreases its length when that year is embolismic; otherwise, it decreases the length of the (first month of the) next year.

Per cycle statistics

The Gregorian lunar calendar has a cycle length of 5700000 years. Out of these 5700000 years, 2099183 are embolismic and 3600817 are not, 1406760 are leap and 4293240 are not, 300000 are hollow and 5400000 are not. This cycle of 5700000 years contains 70499183 months, of which 37405943 are of 30 days and 33093240 of 29 days, giving a total of 2081882250 days, which is (fortunately!) the same as the number of days in the same 5700000 years for the Gregorian solar calendar (1382250 and 4317750 regular years).

Correspondance with the Gregorian solar calendar

The following table gives the Gregorian solar date of the start of the first five months of the Gregorian lunar year, of the possible thirteenth month and of the following year.

Here, “g.n.=1” means years having golden number one, “Solar” means centennial year which apply the solar correction but no lunar correction, “S =1” mean those which additionally have golden number one, “Lunar” means centennial year which apply the lunar correction but no solar correction, “L =1” mean those which additionally have golden number one; “sLeap” means a year which is leap in the solar calendar; “Embol” means an embolismic year; “E&H” means a year which is both embolismic and hollow (golden number is 19).

The Paschal moon is the fourteenth day of the fourth month of the Gregorian lunar calendar except for years with epacts 24 and 25*, in which case it is the fourteenth day of the fifth month.

The following table gives the correspondance between the Gregorian solar and lunar calendars (as the first day of every lunar month) for a few years:

Just add 13 days to every date in this table to get the list of ecclesiastical full moons in the corresponding date range.

Gregorian epact tables and the “tabular” Gregorian lunar calendar

Sometimes the Gregorian lunar calendar is described as follows: simply take the following table

(note that the label “25” has been omitted every other time between “26” and “24”) and introduce a start of lunar month on every day whose label equals the epact of the year (for years with epact 25 use the labels 24 when 25 is skipped; for years with epact 25* use the label 25 when it exists or 26 when there is no 25).

Does the calendar defined by this table coincide with that which we have been describing at length? The answer is yes for start of months ranging from March 1 through December 24 inclusive: these are exactly given by this table. In particular, there is no problem with using this table to compute the date of Easter. There are occasional differences, however, around December 31 and around February 29 as well as all through the first two months when a lunar or solar correction is applied; to explain this in more detail, let us call “tabular Gregorian lunar calendar” the calendar which blindly follows the above table, and “regular Gregorian lunar calendar” the one which we have described so far (and which has, in particular, only months of 30 and 29 days). Here are a few things to note about the tabular Gregorian lunar calendar:

• Months of 31 days happen in many leap years (the regular Gregorian lunar calendar, on the other hand, has only months of 30 and 29 days, in accordance with the Meton-Callippus cycle). Months of 28 days even occur (for example the month which starts on 15199-12-31).
• Sometimes the table “skips” a Moon, because the Epact of one year is 18 or 19 and that of the next is 1 or 2. The skip from 19 to 1 occurs when the former year has golden number 19 and epact 19 (e.g. 1614) and the next year has golden number 1 and epact 1 (e.g. 1615): then according to the table there would be a lunar month starting on 12-02 of the former year, and one starting on 01-30 of the latter year, but nothing in between; this is remedied by artificially giving 12-31 the label “19” for such a year. This skip from 19 to 1 also occurs when the latter year has the lunar correction and no solar correction as well has having epact 1 (this happens from 16399 to 16400 for example; the difference is that the regular lunar calendar, this time, puts the new moon on 01-01). The skip from epact 18 to epact 1 is even rarer (doesn't happen until 106400) and so is the skip from 19 to 2 (not until 273600). It is not clear what should be done in those cases.
• Conversely, sometimes the table double-counts a Moon because the epact goes from 20 to 0, the latter year having solar correction. This happens in 4200, for example. It is also unclear what should be done in this case.

Hopefully this should convince that the tabular Gregorian lunar calendar is not a satisfactory calendar. Despite its apparent simplicity (merely one table!) it is really much more complicated than the regular Gregorian calendar which we have described, and it is far more difficult to give rules to compute the lengths of the months.

The reason for this should be clear. Various corrections are applied to the lunar calendar: an occasional leap year, an occasional hollow years, and some solar and lunar corrections. Now the regular Gregorian lunar calendar makes these corrections at sensible lunar dates; the tabular Gregorian lunar calendar, on the other hand, insists on applying these corrections at certain specific dates of the Gregorian solar calendar, which means they could occur at nasty places in the lunar year. Take the case of leap years, for example: the regular Gregorian lunar calendar adds the leap day at the end of the second lunar month (making it 30 days instead of 29); the tabular Gregorian lunar calendar, on the other hand, adds the leap day in the same place as in the Gregorian solar calendar (i.e. at the end of February, or between the 23th and the 24th of February, according to the variant used for the solar calendar): this can fall in the second or third lunar month, so sometimes the third lunar month becomes 31 days long. The case is similar for hollow years: the regular Gregorian lunar calendar removes a day either from the thirteenth month of the lunar year or from the first month of the following year (when there is no thirteenth month to remove from); the tabular Gregorian lunar calendar, on the other hand, systematically removes a day from the lunar month containing December 31, and is confused when this happens just at the month shift. Solar and lunar corrections are even more clear-cut: the tabular Gregorian lunar calendar makes them on December 31 also, whereas the regular Gregorian lunar calendar simply sees them as part of the general leap year rule (and the leap day is always added or removed at the end of the second lunar month).

Fortunately, all these differences remain within a bounded part of the year, and there is no difficulty from March 1 until the end of the lunar year.

To the author's opinion, the tabular Gregorian lunar calendar makes no sense as a self-standing calendar, merely as a device for placing new moons on the Gregorian solar calendar. The regular Gregorian lunar calendar (a) is simpler (even though its relation to the Gregorian solar calendar is ever so slightly more complex), and (b) can be described on its own, without any reference to the Gregorian solar calendar (merely use the epact to determine which years are embolismic, remember the leap year rule and the golden number to describe which years are hollow, and the length of the months follows). Do not trust the apparent simplicity of the above epact table.

Still, the table can be useful with the regular Gregorian lunar calendar: merely use it for dates between March 1 and December 24 inclusive, and then count back or forth as many days as required by the calendar rules to fill in the missing months.

Alternatively, we can also draw a table of epacts for use with the regular Gregorian lunar calendar—it just has to be a little more complicated:

This takes care of almost everything. Just beware with the solar and lunar corrections: they should be applied only on from February 5 (proceed with special care around epact 24, 25, 25* or 26, however: for these truly nasty cases it is best to count back from a later month than try to make sense of the table). Also note that for the start of a month to be indicated by the “December” at the last line of the table does not mean that month belongs to the end of that lunar year: it can also indicate the start of the next lunar year; see the next section for more about this.

On which month does the lunar year start?

We have given detailed rules on how to determine whether a given lunar year is embolismic. Sometimes disagreeing rules can be found on this subject, for example “a year is embolismic when its epact of the following year is smaller” (this is not a simpler formulation of the same criterion: it gives an altogether different set of embolismic years, although the total number of embolismic years in the 5700000-year cycle is obviously the same). The reason for this is that one can hesitate on which lunar month starts the lunar year. However, the only sane choice is that which always makes the third month 30 days in length, the fourth 29, and so on; if we agree with this choice, our pattern of embolismic years is the only one possible. This makes the first month of a given lunar year start between December 7 of the previous year and January 6 of the corresponding solar year, except for years with the solar correction (which might start as early as December 6) and years with the lunar correction (which might start as late as January 7). In case of doubt, remember this: epacts 22, 23, 24, 25* make the lunar year start very early whereas epacts 25, 26, 27, 28 make the lunar year start very late. The precise tables have already been given.

A computer script

A computer script to compute the Gregorian solar and lunar calendars of any given year, as well as the computus, has been written in Python by the author of this page. It's source code can be downloaded and is placed in the Public Domain.

The Mayan calendar

The Mayan calendar is actually the conjunction of three or four independent (but related) calendars: the Haab and the Tzolkin, which together form the Short Count, the Long Count, and possibly the Lords of the Night, which we now describe in turn. The correspondance between these systems is absolutely certain and confirmed by every existing Mayan inscription; on the other hand, the correspondance between the Mayan calendar(s) and our calendar is not completely certain (but nearly so).

The Mayan calendars are of the simplest kind there is, because they count days very blindly, without using any sort of intercalation (e.g., “leap years”) whatsoever, and without taking into account the fact that they drift with respect to every astronomical phenomenon (such as seasons). (This does not mean that the Mayans did not care about astronomical phenomena: quite the contrary, and we find precise indications of the phases of the Moon and the cycles of Venus. But the calendar proper does not regard these.)

Note: There is strong evidence that the so-called “Mayan” calendar is in fact not Mayan but Olmec in origin. The Tzolkin seems to be the most ancient system, the Haab having appeared later on, and the Long Count even later. The names we give for the months, however, are Mayan and not Olmec (for the original Olmec names are unknown).

The Haab

The Haab is the Mayan civil calendar. It names days within a year of 365 days (always, and with no exception). The Haab consists of 18 months of 20 days each: Pop, Uo, Zip, Zotz, Tzec, Xul, Yaxkin, Mol, Chen, Yax, Zac, Ceh, Mac, Kankin, Muan, Pax, Kayab, and Cumku, plus 5 “soulless” days called Uayeb. Within each month, days are numbered from 0 through 19: so we have 0 Pop, then 1 Pop, then 2 Pop, and so on until 19 Pop, which is followed by 0 Uo, 1 Uo and so on, and after 19 Cumku we have the five Uayeb (which may be named 0 Uayeb through 4 Uayeb), followed by 0 Pop again.

The years of the Haab are not counted: there is nothing to distinguish one 0 Pop from another, except that the Tzolkin would be different and so would the Long Count.

The Tzolkin

The Tzolkin is the Mayan religious calendar. It is formed by the conjunction of a 20-day cycle (“uinal”) and a 13-day cycle (sometimes called the Mayan “week”), pursued independently, so that together they form (by the Chinese remainder theorem) a 260-day cycle. The days of the 20-day cycle bear the names of gods: Ahau, Imix, Ik, Akbal, Kan, Chicchan, Cimi, Manik, Lamat, Muluc, Oc, Chuen, Eb, Ben, Ix, Men, Cib, Caban, Etznab and Caunac; those of the 13-day cycle bear numbers, from 1 through 13. So after 4 Ahau (say), we have 5 Imix then 6 Ik then 7 Akbal and so on until 13 Muluc, then 1 Oc, 2 Chuen, 3 Eb, 4 Ben up to 10 Caunac, 11 Ahau and so on.

As with the Haab, the cycles of the Tzolkin are not counted.

The Short Count

The Short Count is the conjunction of the Haab and the Tzolkin. The duration of its cycle is (by the Chinese theorem again) the least common multiple of 365 and 260, or 18980 days: every 73 rounds of the Tzolkin, or 52 of the Haab (that's a little below 52 years) there occurs a day that has the same label both in the Haab and the Tzolkin; and, equivalently, the two labels determine the day precisely within a 52-year period. This period has also been called the “Mayan century”. The Aztecs (which used the Mayan Haab and Tzolkin but had lost the Long Count) believed that every time the Haab was 8 Cumku and the Tzolkin 4 Ahau (the dates for the Mayan epoch, see below), the world might come to an end: but if the Sun rose on that day, the world was granted a 52-year extension.

The Long Count

The Long Count consists of a certain number of nested cycles: 20 days (kin) form an uinal, 18 uinal (360 days) form a tun, 20 tun (around 20 years) form a katun, and 20 katun (around 394 years) form a baktun. The Long Count is the count of a number of days since the so-called Mayan epoch: first the number of full baktun elapsed since the epoch, then the number of full katun after that, then the number of tun, then of uinal, and then of kin; so essentially it amounts to counting the number of days from the epoch in base 20 except that the antepenultimate number is worth only 18 of the penultimate rather than 20. We write 0.0.0.0.0 for the epoch, which would be followed by 0.0.0.0.1, then 0.0.0.0.2 and so on until 0.0.0.0.19, then 0.0.0.1.0, and so on until 0.0.0.17.19, then 0.0.1.0.0 and so on.

The Long Count is correlated with the Haab and the Tzolkin as follows: the Mayan epoch (0.0.0.0.0) is labeled 8 Cumku in the Haab and 4 Ahau in the Tzolkin. In particular, this means that the last number in the Long Count label simply determines the god name of the Tzolkin (0 means “Ahau”, 1 means “Imix” and so on): this 20-day period is called the “uinal” both in the Tzolkin and the Long Count.

The Lords of the Night

The nine Lords of the Night are a series of nine glyphs found on Mayan carvings, which repeat themselves systematically. They have been called “Lords of the Night” by association with other legends found in Mexico, but the precise correspondance is unknown, so we just call them the first, second, third, and so on up ninth, Lord of the Night, or G1 through G9 (because these data are found in the so-called glyph “G” of Mayan stelae). The Lord of the Night for the Mayan epoch (0.0.0.0.0 in the Long Count) would then be the ninth.

The Mayan correlation

Even when the precise relation between all these cycles has been understood, it remains to determine what their relation is to some fixed time scale. This is the so-called “correlation problem”.

One plausible solution to the problem places the Mayan epoch (0.0.0.0.0 in the Long Count, 8 Cumku in the Haab, 4 Ahau in the Tzolkin, rule of the ninth Lord of the Night) on September 6 of the year ~3114 (3114 before the common era, or −3113) in the proleptic Julian calendar, or August 11 (also of ~3114) in the proleptic Gregorian calendar, the day which starts on Julian date 584282.5 (or slightly later if we take into account the fact that the Mayan calendar was used in America). This makes January 1, 2000, of the Gregorian calendar the day 12.19.6.15.2, Haab 10 Kankin, Tzolkin 11 Ik, rule of the fifth Lord of the Night. This correlation was suggested in 1905 by John Goodman, and at the time met very little success; it was resurrected in 1926 by Juan Martínez Hernández who displaced it forward by one day (placing the Epoch on August 12 of ~3114 in the proleptic Gregorian calendar), and then one further day in 1927 by Sir John Eric Sydney Thompson, placing the Epoch on August 13 of ~3114 in the proleptic Gregorian calendar (Julian date 584285 at noon). Later (around 1935), Thompson disavowed his correction and proclaimed that Goodman's initial computation was correct. But others have claimed that Thompson's initial correction was correct. The author of this page does not know which of the two correlations is adopted by modern Mayan specialists (Goodman's original 584283, which Thompson finally favored, or Thompson's later withdrawn proposal of 584285), but there is hardly any doubt that the correct value is one of these or, at any rate, one very close to them. Other correlations have been proposed of 489384 (Morley?), 482699 (Smiley?), or even 774078 (Weitzel?): they are almost certainly wrong. For what it is worth, the calendar package in the Emacs program uses the Goodman-Thompson correlation (584283): not that it is necessarily better informed, but that should let Emacs users correct the data if they wish to use another correlation.

Of course, one way to be positively certain of the correlation would be to find the record of an astronomical event such as a solar eclipse. Apparently no such record has ever been found.

When does the Long Count end?

Much is often made of the fact that December 21, 2012, or perhaps December 23 if we accept Thompson's withdrawn correction to Goodman's correlation, marks the date 13.0.0.0.0 of the Mayan Long Count. Obviously I disregard astrological interpretations of this date. But it is sometimes called the “end” of the Mayan calendar. Why is this?

There is a priori no reason why 13.0.0.0.0 should be special: the Long Count, as we have seen, works in twenties (with the exception of the 18 uinals in a tun, which are evidently there to make the tun not too far from a solar year); so the special date, if there is to be one, should be 20—and not 13—baktun from the Mayan epoch (this gets us sometime in 4772), and even then there is no reason why we can't invent larger units than the baktun: 20 baktun in a piktun, 20 piktun in a kalabtun, 20 kalabtun in a kinchiltun, 20 kinchiltun in an alautun, 20 alautun in a hablatun, and so on. This is great fun, but not very serious (already a piktun is over 7885 years, not to mention the others). Interestingly, though, I am told that one Mayan carving was found (at Yaxchilan) which records the Long Count date using units larger than the baktun: it says 13.13.13.13.13.13.13.13.9.15.13.6.9; this seems to imply that the Mayans (or, at any rate, one Mayan) thought that the units beyond the baktun in the Long Count were all equal to 13 throughout the historical period. So maybe there is something special with 13 after all. I have read claims that the 13 stood for zero (and that the Mayan epoch should be written 13.0.0.0.0 instead of 0.0.0.0.0, just as that date in 2012): but this doesn't seem to make sense since the Mayans clearly knew of the number zero, and weren't afraid to use it. Maybe something special should be associated in the Mayan mythology with date 13.13.13.13.13 (all cycles being 13), which would be on October 20 (or 22) of 2282. But this is unconvincing.

One thing is certain, however, about 13.0.0.0.0: the Tzolkin will have cycled an integer number of times (7200 times, to be precise) since the Mayan epoch, so it will be back to 4 Ahau. The Haab, however, will not be 8 Cumku as on 0.0.0.0.0 but 3 Kankin, which means the Mayans probably wouldn't have thought of it as the end of anything.

It is indeed probable that the Mayans attached no special significance with 13.0.0.0.0 or any possible “end” of the Long Count.

As to the starting date of the Long Count, i.e., the Mayan epoch, it seems to have been chosen as a day of passing of the Sun directly overhead the sacred center of Izapa, and as to the year perhaps that was chosen an integer number of baktun in the past so that the Haab had cycled (nearly) twice through the seasons between the epoch and the time the Long Count was put in effect; if this hypothesis is correct, it would put the beginning of the use of the Long Count at 7.13.0.0.0, or, more precisely, on 7.12.19.16.10 (which would have been the Mayans' estimation of 3016 solar years past the Epoch, and also 8 Cumku; in other words, the hypothetical reasoning is this: the Sun goes overhead Izapa on 8 Cumku on August 11 or 13 or whereabouts of ~98 of the proleptic Gregorian calendar, then someone estimates when the same event, Sun overhead on 8 Cumku, last happened, and finds 550785 and 1101570 days in the past, and since the latter is just 30 days short of a very round number in the Olmec numeration, places the Epoch that far back in the past; this is pure speculation, of course). But it is by no means certain that anyone saw that as the creation of the world or some such thing.