Going from right to left the remaining components are:. The kin, tun, and katun are numbered from 0 to The uinal are numbered from 0 to The baktun are numbered from 1 to 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:. Logically, the first date in the Long Count should be 0. The authorities disagree on what I have come across three possible equivalences:.
Assuming one of the first two equivalences, the Long Count will again reach While our calendar uses a single week of seven days, the Mayan calendar used two different lengths of week:. Ahau 1. Imix 2. Akbal 4. Kan 5. Chicchan 6. Cimi 7. Manik 8. Lamat 9. Muluc Oc Chuen Eb Ben Ix Men When necessary, years are represented in positional notation, where each overt viget represents a coefficient and the opaque viget the positional multiplier or year glyph.
The format varies somewhat according to the style of the calendrist, with the overt viget placed on the top or on the left as in the figure, which is literally translated 14 Pictun about centuries. Historians number the sacred year in month numbers one through 13 and day glyphs corresponding to numbers one through In this format there are no month glyphs.
The day numbers, glyphs and names are:. Each succeeding day causes both the day number and glyph number to advance by one, e. Since 13 and 20 have no common divisors, this system uniquely represents all days of the sacred year.
Historians number the vague year in day numbers zero through 19 and month glyphs corresponding to numbers one through 19, where the last month represents the five uncounted days of the haab. The month numbers, glyphs and names are:. Each succeeding day causes the day number to advance by one. When the day number wraps from 19 to zero or in the month of uayeb from 4 to zero , the month glyph advances by one. For instance, a Maya day written 0 Pop is followed by 1 Pop, A calendar round date consists of the sacred day concatenated with the vague day, e.
A unique combination of sacred day and vague day recurs only once in the calendar round, where the beginning of each round is established at the correspondence 2 Ik 0 Pop.
Historians write a Maya long count calendar round date in the form, e. The Gregorian epact is obtained by adding a Gregorian correction to the Julian epact, everything being computed modulo This is summarized as follows:. The epact is the sum modulo 30 of the Julian epact and the Gregorian correction. When the sum is 25 for a year with a golden number between 1 and 11, it is a regular We can draw the following table of epacts for a few centuries:.
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 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 has golden number 6, hence epact 24, and dominical letters BA, hence Easter of fell on ; the year has golden number 7 hence epact 5, and dominical letter G, so Easter of fell on ; the year has golden number 8 hence epact 16, and dominical letter F, so Easter of fell on ; the year has golden number 9 hence epact 27, and dominical letter E, so Easter of fell on ; the year has golden number 10 hence epact 8, and dominical letter DC, so Easter of falls on What follows is a precomputed table of the date of Easter for years through 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.
See below for more about this. No matter what, luni-solar calendars are always complicated, and for historical reasons the Gregorian luni-solar calendar is even more devilish than some. The Gregorian lunar calendar uses years of twelve or thirteen months, each month having 29 or 30 days. 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 will not happen until the year , however because, until , all years with golden number 19 are embolismic. The second month in the year normally has 29 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 and its quotient by one hundred is congruent to 2, 5, 8, 11, 14, 18, 21 or 24 modulo 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, is lunar leap as well as solar leap , is not lunar leap nor is it solar leap , and are lunar leap but not solar leap , 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 , , 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 The thirteenth month, or embolismic month, when it exists, normally has 30 days.
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. To determine whether a year is embolismic i. 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.
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 Embolismic years with epact 12 are exceedingly rare: the next such case is the lunar year which has epact 12 and starts on December 20 of in the Gregorian solar calendar whereas the next year, , has epact 25 and starts on January 7 of , so evidently there are thirteen months in the lunar year ; 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 barring complications just before a centennial year, years with epact 14 are embolismic when their golden number is between 1 and 10, or 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 through , the seven embolismic years in the ninteen-year cycle are those having golden numbers 3, 6, 8, 11, 14, 17 and 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. 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. To formally complete the description of the Gregorian lunar calendar, we need one correspondance point.
The year of the Gregorian lunar calendar golden number 6, epact 24; this year is embolismic starts on of the Gregorian solar calendar , which is the day which starts on Julian date After this, simply work through the number of days in each month: since 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 , so the lunar year starts on Julian date 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.
The Gregorian lunar calendar has a cycle length of years. Out of these years, are embolismic and are not, are leap and are not, are hollow and are not.
This cycle of years contains months, of which are of 30 days and of 29 days, giving a total of days, which is fortunately! For example: a double star goes into eclipse every When is the next? Well, you could get out your calendar and count days, but it's far easier to convert all the quantities in question to Julian day numbers and simply add or subtract. Julian days simply enumerate the days and fraction which have elapsed since the start of the Julian era , which is defined as beginning at noon on Monday, 1st January of year B.
This date is defined in terms of a cycle of years, but has the additional advantage that all known historical astronomical observations bear positive Julian day numbers, and periods can be determined and events extrapolated by simple addition and subtraction.
Julian dates are a tad eccentric in starting at noon, but then so are astronomers and systems programmers! But even the Julian day convention bears witness to the eurocentrism of 19th century astronomy—noon at Greenwich is midnight on the other side of the world.
But the Julian day notation is so deeply embedded in astronomy that it is unlikely to be displaced at any time in the foreseeable future. It is an ideal system for storing dates in computer programs, free of cultural bias and discontinuities at various dates, and can be readily transformed into other calendar systems, as the source code for this page illustrates. While any event in recorded human history can be written as a positive Julian day number, when working with contemporary events all those digits can be cumbersome.
Modified Julian Days are widely used to specify the epoch in tables of orbital elements of artificial Earth satellites. Since no such objects existed prior to October 4, , all satellite-related MJDs are positive. The Julian calendar differs from the Gregorian only in the determination of leap years, lacking the correction for years divisible by and in the Gregorian calendar.
In the Julian calendar, any positive year is a leap year if divisible by 4. Negative years are leap years if the absolute value divided by 4 yields a remainder of 1. Days are considered to begin at midnight. In the Julian calendar the average year has a length of The calendar thus accumulates one day of error with respect to the solar year every years. Being a purely solar calendar, no attempt is made to synchronise the start of months to the phases of the Moon.
In addition, there are constraints on which days of the week on which a year can begin and to shift otherwise required extra days to prior years to keep the length of the year within the prescribed bounds. This isn't easy, and the computations required are correspondingly intricate.
Years are classified as common normal or embolismic leap years which occur in a 19 year cycle in years 3, 6, 8, 11, 14, 17, and Further, years may be deficient , regular , or complete , having respectively , , or days in a common year and , , or days in embolismic years.
Days are defined as beginning at sunset, and the calendar begins at sunset the night before Monday, October 7, B. Days are numbered with Sunday as day 1, through Saturday: day 7. The average length of a month is Such is the accuracy that more than 13, years elapse before a single day discrepancy between the calendar's average reckoning of the start of months and the mean time of the new Moon.
Alignment with the solar year is better than the Julian calendar, but inferior to the Gregorian. The average length of a year is The Islamic calendar is purely lunar and consists of twelve alternating months of 30 and 29 days, with the final 29 day month extended to 30 days during leap years.
Leap years follow a 30 year cycle and occur in years 1, 5, 7, 10, 13, 16, 18, 21, 24, 26, and Days are considered to begin at sunset. The calendar begins on Friday, July 16th, C. The names for the days are just their numbers: Sunday is the first day and Saturday the seventh; the week is considered to begin on Saturday. Since the calendar is fixed to the Moon, not the solar year, the months shift with respect to the seasons, with each month beginning about 11 days earlier in each successive solar year.
The calendar presented here is the most commonly used civil calendar in the Islamic world; for religious purposes months are defined to start with the first observation of the crescent of the new Moon. The modern Persian calendar was adopted in , supplanting while retaining the month names of a traditional calendar dating from the eleventh century.
The calendar consists of 12 months, the first six of which are 31 days, the next five 30 days, and the final month 29 days in a normal year and 30 days in a leap year. Days begin at midnight in the standard time zone. There is no leap year rule; day years do not recur in a regular pattern but instead occur whenever that number of days elapse between equinoxes at the reference meridian.
The calendar therefore stays perfectly aligned with the seasons. No attempt is made to synchronise months with the phases of the Moon. There is some controversy about the reference meridian at which the equinox is determined in this calendar.
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