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A perpetual calendar is a calendar valid for many years, usually designed to look up the day of the week for a given date in the past or future.
For the Gregorian and Julian calendars, a perpetual calendar typically consists of one of three general variations:
Such a perpetual calendar fails to indicate the dates of moveable feasts such as Easter, which are calculated based on a combination of events in the tropical year and lunar cycles. These issues are dealt with in great detail in computus .
An early example of a perpetual calendar for practical use is found in the Nürnberger Handschrift GNM 3227a . The calendar covers the period of 1390–1495 (on which grounds the manuscript is dated to c. 1389). For each year of this period, it lists the number of weeks between Christmas and Quinquagesima. This is the first known instance of a tabular form of perpetual calendar allowing the calculation of the moveable feasts that became popular during the 15th century. [5]
The chapel Cappella dei Mercanti, Turin contains a perpetual calendar machine made by Giovanni Plana using rotating drums.
Offices and retail establishments often display devices containing a set of elements to form all possible numbers from 1 through 31, as well as the names/abbreviations for the months and the days of the week, to show the current date for convenience of people who might be signing and dating documents such as checks. Establishments that serve alcoholic beverages may use a variant that shows the current month and day but subtracting the legal age of alcohol consumption in years, indicating the latest legal birth date for alcohol purchases. A common device consists of two cubes in a holder. One cube carries the digits zero to five. The other bears the digits 0, 1, 2, 6 (or 9 if inverted), 7, and 8. This is sufficient because only one and two may appear twice in date and they are on both cubes, while the 0 is on both cubes so that all single-digit dates can be shown in double-digit format. In addition to the two cubes, three blocks, each as wide as the two cubes combined, and a third as tall and as deep, have the names of the months printed on their long faces. The current month is turned forward on the front block, with the other two month blocks behind it.
Certain calendar reforms have been labeled perpetual calendars because their dates are fixed on the same weekdays every year. Examples are The World Calendar, the International Fixed Calendar and the Pax Calendar. Technically, these are not perpetual calendars but perennial calendars. Their purpose, in part, is to eliminate the need for perpetual calendar tables, algorithms, and computation devices.
In watchmaking, "perpetual calendar" describes a calendar mechanism that correctly displays the date on the watch "perpetually", taking into account the different lengths of the months as well as leap years. The internal mechanism will move the dial to the next day. [6]
Perpetual calendars use algorithms to compute the day of the week for any given year, month, and day of the month. Even though the individual operations in the formulas can be very efficiently implemented in software, they are too complicated for most people to perform all of the arithmetic mentally. [7] Perpetual calendar designers hide the complexity in tables to simplify their use.
A perpetual calendar employs a table for finding which of fourteen yearly calendars to use. A table for the Gregorian calendar expresses its 400-year grand cycle: 303 common years and 97 leap years total to 146,097 days, or exactly 20,871 weeks. This cycle breaks down into one 100-year period with 25 leap years, making 36,525 days, or one day less than 5,218 full weeks; and three 100-year periods with 24 leap years each, making 36,524 days, or two days less than 5,218 full weeks.
Within each 100-year block, the cyclic nature of the Gregorian calendar proceeds in the same fashion as its Julian predecessor: A common year begins and ends on the same day of the week, so the following year will begin on the next successive day of the week. A leap year has one more day, so the year following a leap year begins on the second day of the week after the leap year began. Every four years, the starting weekday advances five days, so over a 28-year period, it advances 35, returning to the same place in both the leap year progression and the starting weekday. This cycle completes three times in 84 years, leaving 16 years in the fourth, incomplete cycle of the century.
A major complicating factor in constructing a perpetual calendar algorithm is the peculiar and variable length of February, which was at one time the last month of the year, leaving the first 11 months March through January with a five-month repeating pattern: 31, 30, 31, 30, 31, ..., so that the offset from March of the starting day of the week for any month could be easily determined. Zeller's congruence, a well-known algorithm for finding the day of the week for any date, explicitly defines January and February as the "13th" and "14th" months of the previous year to take advantage of this regularity, but the month-dependent calculation is still very complicated for mental arithmetic:
Instead, a table-based perpetual calendar provides a simple lookup mechanism to find offset for the day of the week for the first day of each month. To simplify the table, in a leap year January and February must either be treated as a separate year or have extra entries in the month table:
Month | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Add | 0 | 3 | 3 | 6 | 1 | 4 | 6 | 2 | 5 | 0 | 3 | 5 |
For leap years | 6 | 2 |
The following calendar works for any date from 15 October 1582 onwards, but only for Gregorian calendar dates.
Years of the century | Example 1 Gregorian 31 March 2006: Greg century 20(c) and year 06(y) meet at A in the table of Latin square. The A in row Mar(m) meets 31(d) at Fri in the table of Weekdays. The day is Friday. Example 2 BC 1 January 45: BC 45 = -44 = -100 + 56 (a leap year). -1 and 56 meet at B and Jan_B meets 1 at Fri(day). Example 3 Julian 1 January 1900: Julian 19 meets 00 at A and Jan_A meets 1 at Sat(urday). Example 4 Gregorian 1 January 1900: Greg 19 meets 00 at G and Jan_G meets 1 at Mon(day). | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
00 | 01 | 02 | 03 | 04 | 05 | |||||||||||||||
06 | 07 | 08 | 09 | 10 | 11 | |||||||||||||||
12 | 13 | 14 | 15 | 16 | ||||||||||||||||
17 | 18 | 19 | 20 | 21 | 22 | |||||||||||||||
23 | 24 | 25 | 26 | 27 | ||||||||||||||||
28 | 29 | 30 | 31 | 32 | 33 | |||||||||||||||
34 | 35 | 36 | 37 | 38 | 39 | |||||||||||||||
40 | 41 | 42 | 43 | 44 | ||||||||||||||||
45 | 46 | 47 | 48 | 49 | 50 | |||||||||||||||
51 | 52 | 53 | 54 | 55 | ||||||||||||||||
56 | 57 | 58 | 59 | 60 | 61 | |||||||||||||||
62 | 63 | 64 | 65 | 66 | 67 | |||||||||||||||
68 | 69 | 70 | 71 | 72 | ||||||||||||||||
73 | 74 | 75 | 76 | 77 | 78 | |||||||||||||||
79 | 80 | 81 | 82 | 83 | ||||||||||||||||
84 | 85 | 86 | 87 | 88 | 89 | |||||||||||||||
90 | 91 | 92 | 93 | 94 | 95 | |||||||||||||||
96 | 97 | 98 | 99 | |||||||||||||||||
Centuries | Latin square | Months | ||||||||||||||||||
Julian | Greg. | |||||||||||||||||||
-4 | 3 | 10 | 17 | — | — | F | E | D | C | B | A | G | Jan | Apr | Jul | |||||
-3 | 4 | 11 | 18 | 15 | 19 | G | F | E | D | C | B | A | Jan | Oct | ||||||
-2 | 5 | 12 | 19 | 16 | 20 | A | G | F | E | D | C | B | May | |||||||
-1 | 6 | 13 | 20 | — | — | B | A | G | F | E | D | C | Feb | Aug | ||||||
0 | 7 | 14 | 21 | 17 | 21 | C | B | A | G | F | E | D | Feb | Mar | Nov | |||||
1 | 8 | 15 | 22 | — | — | D | C | B | A | G | F | E | Jun | |||||||
2 | 9 | 16 | 23 | 18 | 22 | E | D | C | B | A | G | F | Sep | Dec | ||||||
Days | Weekdays | |||||||||||||||||||
1 | 8 | 15 | 22 | 29 | Mon | Tue | Wed | Thu | Fri | Sat | Sun | |||||||||
2 | 9 | 16 | 23 | 30 | Tue | Wed | Thu | Fri | Sat | Sun | Mon | |||||||||
3 | 10 | 17 | 24 | 31 | Wed | Thu | Fri | Sat | Sun | Mon | Tue | |||||||||
4 | 11 | 18 | 25 | Thu | Fri | Sat | Sun | Mon | Tue | Wed | ||||||||||
5 | 12 | 19 | 26 | Fri | Sat | Sun | Mon | Tue | Wed | Thu | ||||||||||
6 | 13 | 20 | 27 | Sat | Sun | Mon | Tue | Wed | Thu | Fri | ||||||||||
7 | 14 | 21 | 28 | Sun | Mon | Tue | Wed | Thu | Fri | Sat |
Julian centuries | Gregorian centuries | Days of the week | Months | Days | |||||||||||||||||
04 11 18 | 19 23 27 | Sun | Mon | Tue | Wed | Thu | Fri | Sat | Jan | Apri | Jul | 01 | 08 | 15 | 22 | 29 | |||||
03 10 17 | Mon | Tue | Wed | Thu | Fri | Sat | Sun | Sep | Dec | 02 | 09 | 16 | 23 | 30 | |||||||
02 09 16 | 18 22 26 | Tue | Wed | Thu | Fri | Sat | Sun | Mon | Jun | 03 | 10 | 17 | 24 | 31 | |||||||
01 08 15 | Wed | Thu | Fri | Sat | Sun | Mon | Tue | Feb | Mar | Nov | 04 | 11 | 18 | 25 | |||||||
00 07 14 | 17 21 25 | Thu | Fri | Sat | Sun | Mon | Tue | Wed | Feb | Aug | 05 | 12 | 19 | 26 | |||||||
–1 06 13 | Fri | Sat | Sun | Mon | Tue | Wed | Thu | May | 06 | 13 | 20 | 27 | |||||||||
–2 05 12 | 16 20 24 | Sat | Sun | Mon | Tue | Wed | Thu | Fri | Jan | Oct | 07 | 14 | 21 | 28 | |||||||
Years | 00 | 01 | 02 | 03 | 04 | 05 | |||||||||||||||
06 | 07 | 08 | 09 | 10 | 11 | ||||||||||||||||
12 | 13 | 14 | 15 | 16 | |||||||||||||||||
17 | 18 | 19 | 20 | 21 | 22 | ||||||||||||||||
23 | 24 | 25 | 26 | 27 | |||||||||||||||||
28 | 29 | 30 | 31 | 32 | 33 | ||||||||||||||||
34 | 35 | 36 | 37 | 38 | 39 | ||||||||||||||||
40 | 41 | 42 | 43 | 44 | |||||||||||||||||
45 | 46 | 47 | 48 | 49 | 50 | ||||||||||||||||
51 | 52 | 53 | 54 | 55 | |||||||||||||||||
56 | 57 | 58 | 59 | 60 | 61 | ||||||||||||||||
62 | 63 | 64 | 65 | 66 | 67 | ||||||||||||||||
68 | 69 | 70 | 71 | 72 | |||||||||||||||||
73 | 74 | 75 | 76 | 77 | 78 | ||||||||||||||||
79 | 80 | 81 | 82 | 83 | |||||||||||||||||
84 | 85 | 86 | 87 | 88 | 89 | ||||||||||||||||
90 | 91 | 92 | 93 | 94 | 95 | ||||||||||||||||
96 | 97 | 98 | 99 |
# | Julian centuries (mod 7) | Gregorian centuries (mod 4) | Dates | 01 08 15 22 29 | 02 09 16 23 30 | 03 10 17 24 31 | 04 11 18 25 | 05 12 19 26 | 06 13 20 27 | 07 14 21 28 | Years of the century (mod 28) | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6 | 05 12 19 | 16 20 24 | Apr | Jul | Jan | Sun | Mon | Tue | Wed | Thu | Fri | Sat | 01 | 07 | 12 | 18 | 29 | 35 | 40 | 46 | 57 | 63 | 68 | 74 | 85 | 91 | 96 | |||
5 | 06 13 20 | Sep | Dec | Sat | Sun | Mon | Tue | Wed | Thu | Fri | 02 | 13 | 19 | 24 | 30 | 41 | 47 | 52 | 58 | 69 | 75 | 80 | 86 | 97 | ||||||
4 | 07 14 21 | 17 21 25 | Jun | Fri | Sat | Sun | Mon | Tue | Wed | Thu | 03 | 08 | 14 | 25 | 31 | 36 | 42 | 53 | 59 | 64 | 70 | 81 | 87 | 92 | 98 | |||||
3 | 08 15 22 | Feb | Mar | Nov | Thu | Fri | Sat | Sun | Mon | Tue | Wed | 09 | 15 | 20 | 26 | 37 | 43 | 48 | 54 | 65 | 71 | 76 | 82 | 93 | 99 | |||||
2 | 09 16 23 | 18 22 26 | Aug | Feb | Wed | Thu | Fri | Sat | Sun | Mon | Tue | 04 | 10 | 21 | 27 | 32 | 38 | 49 | 55 | 60 | 66 | 77 | 83 | 88 | 94 | |||||
1 | 10 17 24 | May | Tue | Wed | Thu | Fri | Sat | Sun | Mon | 05 | 11 | 16 | 22 | 33 | 39 | 44 | 50 | 61 | 67 | 72 | 78 | 89 | 95 | |||||||
0 | 11 18 25 | 19 23 27 | Jan | Oct | Mon | Tue | Wed | Thu | Fri | Sat | Sun | 06 | 17 | 23 | 28 | 34 | 45 | 51 | 56 | 62 | 73 | 79 | 84 | 90 | 00 |
A calendar date is a reference to a particular day represented within a calendar system. The calendar date allows the specific day to be identified. The number of days between two dates may be calculated. For example, "25 January 2025" is ten days after "15 January 2025". The date of a particular event depends on the observed time zone. For example, the air attack on Pearl Harbor that began at 7:48 a.m. Hawaiian time on 7 December 1941 took place at 3:18 a.m. Japan Standard Time, 8 December in Japan.
A leap year is a calendar year that contains an additional day compared to a common year. The 366th day is added to keep the calendar year synchronised with the astronomical year or seasonal year. Since astronomical events and seasons do not repeat in a whole number of days, calendars having a constant number of days each year will unavoidably drift over time with respect to the event that the year is supposed to track, such as seasons. By inserting ("intercalating") an additional day—a leap day—or month—a leap month—into some years, the drift between a civilization's dating system and the physical properties of the Solar System can be corrected.
The International Fixed Calendar was a proposed reform of the Gregorian calendar designed by Moses B. Cotsworth, first presented in 1902. The International Fixed Calendar divides the year into 13 months of 28 days each. A type of perennial calendar, every date is fixed to the same weekday every year. Though it was never officially adopted at the country level, the entrepreneur George Eastman instituted its use at the Eastman Kodak Company in 1928, where it was used until 1989. While it is sometimes described as the 13-month calendar or the equal-month calendar, various alternative calendar designs share these features.
The World Calendar is a proposed reform of the Gregorian calendar created by Elisabeth Achelis of Brooklyn, New York in 1930.
The Julian day is a continuous count of days from the beginning of the Julian period; it is used primarily by astronomers, and in software for easily calculating elapsed days between two events.
As a moveable feast, the date of Easter is determined in each year through a calculation known as computus paschalis – often simply Computus – or as paschalion particularly in the Eastern Orthodox Church. Easter is celebrated on the first Sunday after the Paschal full moon. Determining this date in advance requires a correlation between the lunar months and the solar year, while also accounting for the month, date, and weekday of the Julian or Gregorian calendar. The complexity of the algorithm arises because of the desire to associate the date of Easter with the date of the Jewish feast of Passover which, Christians believe, is when Jesus was crucified.
The epact used to be described by medieval computists as the age of a phase of the Moon in days on 22 March; in the newer Gregorian calendar, however, the epact is reckoned as the age of the ecclesiastical moon on 1 January. Its principal use is in determining the date of Easter by computistical methods. It varies from year to year, because of the difference between the solar year of 365–366 days and the lunar year of 354–355 days.
Dominical letters or Sunday letters are a method used to determine the day of the week for particular dates. When using this method, each year is assigned a letter depending on which day of the week the year starts with. The Dominical letter for the current year 2025 is E.
Calendar reform or calendrical reform is any significant revision of a calendar system. The term sometimes is used instead for a proposal to switch to a different calendar design.
The determination of the day of the week for any date may be performed with a variety of algorithms. In addition, perpetual calendars require no calculation by the user, and are essentially lookup tables. A typical application is to calculate the day of the week on which someone was born or a specific event occurred.
The Tabular Islamic calendar is a rule-based variation of the Islamic calendar. It has the same numbering of years and months, but the months are determined by arithmetical rules rather than by observation or astronomical calculations. It was developed by early Muslim astronomers of the second hijra century to provide a predictable time base for calculating the positions of the moon, sun, and planets. It is now used by historians to convert an Islamic date into a Western calendar when no other information is available. Its calendar era is the Hijri year. An example is the Fatimid or Misri calendar.
The Doomsday rule, Doomsday algorithm or Doomsday method is an algorithm of determination of the day of the week for a given date. It provides a perpetual calendar because the Gregorian calendar moves in cycles of 400 years. The algorithm for mental calculation was devised by John Conway in 1973, drawing inspiration from Lewis Carroll's perpetual calendar algorithm. It takes advantage of each year having a certain day of the week upon which certain easy-to-remember dates, called the doomsdays, fall; for example, the last day of February, April 4 (4/4), June 6 (6/6), August 8 (8/8), October 10 (10/10), and December 12 (12/12) all occur on the same day of the week in the year.
The Pax calendar was invented by James A. Colligan, SJ in 1930, as a perennializing reform of the annualized Gregorian calendar.
The ISO week date system is effectively a leap week calendar system that is part of the ISO 8601 date and time standard issued by the International Organization for Standardization (ISO) since 1988 and, before that, it was defined in ISO (R) 2015 since 1971. It is used (mainly) in government and business for fiscal years, as well as in timekeeping. This was previously known as "Industrial date coding". The system specifies a week year atop the Gregorian calendar by defining a notation for ordinal weeks of the year.
An annual calendar is a representation of the year that expires with the year represented, or that must be altered annually to remain current. The term takes different but related meanings across two contexts. One is for static (synchronic) calendars, such as wall calendars or calendar systems. The other is for dynamic (diachronic) calendars, such as digital calendars or timepieces. Static representations of the Gregorian calendar year are annual, because the weekdays of Gregorian dates vary from year to year. The calendar representing one year will not serve for the next year. With perennial calendars, the same representation of the year serves for every year. Perpetual calendars, in this context, are computation devices for determining the weekdays of dates in any given year, or for representing a wide range of annual calendars.
The Gregorian calendar is the calendar used in most parts of the world. It went into effect in October 1582 following the papal bull Inter gravissimas issued by Pope Gregory XIII, which introduced it as a modification of, and replacement for, the Julian calendar. The principal change was to space leap years differently so as to make the average calendar year 365.2425 days long, more closely approximating the 365.2422-day "tropical" or "solar" year that is determined by the Earth's revolution around the Sun.
The Hanke–Henry Permanent Calendar (HHPC) is a proposal for calendar reform. It is one of many examples of leap week calendars, calendars that maintain synchronization with the solar year by intercalating entire weeks rather than single days. It is a modification of a previous proposal, Common-Civil-Calendar-and-Time (CCC&T). With the Hanke–Henry Permanent Calendar, every calendar date always falls on the same day of the week. A major feature of the calendar system is the abolition of time zones.
A perennial calendar is a calendar that applies to any year, keeping the same dates, weekdays and other features.
The Solar Hijri calendar is the official calendar of Iran. It is a solar calendar and is the one Iranian calendar that is the most similar to the Gregorian calendar, being based on the Earth's orbit around the Sun. It begins on the March equinox as determined by the astronomical calculation for the Iran Standard Time meridian and has years of 365 or 366 days. It is sometimes also called the Shamsi calendar, Khorshidi calendar, or Persian calendar. It is abbreviated as SH, HS, AP, or, sometimes as AHSh, while the lunar Hijri calendar is usually abbreviated as AH.
A computus clock is a clock equipped with a mechanism that automatically calculates and displays, or helps determine, the date of Easter. A computus watch carries out the same function.