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The ISO week date system is a leap week calendar system that is part of the ISO 8601 date and time standard. The system is used (mainly) in government and business for fiscal years, as well as in timekeeping.

The system uses the same cycle of 7 weekdays as the Gregorian calendar. Weeks start with Monday. ISO years have a year numbering which is approximately the same as the Gregorian years, but not exactly (see below). An ISO year has 52 or 53 full weeks (364 or 371 days). The extra week is called a leap week, a year with such a week a leap year.

A date is specified by the ISO year in the format YYYY, a week number in the format ww prefixed by the letter W, and the weekday number, a digit d from 1 through 7, beginning with Monday and ending with Sunday. For example, 2006-W52-7 (or in its most compact form 06W527) is the Sunday of the 52nd week of 2006. In the Gregorian system this day is called December 31, 2006.

The system has a 400-year cycle of 146,097 days (20,871 weeks), with an average year length of exactly 365.2425 days, just like the Gregorian calendar. Since non-leap years have 52 weeks, in every 400 years there are 71 leap years.

Relation with the Gregorian calendarEdit

The ISO year number deviates from the number of the Gregorian year on, if applicable, a Friday, Saturday, and Sunday, or a Saturday and Sunday, or just a Sunday, at the start of the Gregorian year (which are at the end of the previous ISO year) and a Monday, Tuesday and Wednesday, or a Monday and Tuesday, or just a Monday, at the end of the Gregorian year (which are in week 01 of the next ISO year). In the period 4 January–28 December and on all Thursdays the ISO year number is always equal to the Gregorian year number.

Mutually equivalent definitions for week 01 are:

  • the week with the year's first Thursday in it
  • the week with the year's first working day in it (if Saturdays, Sundays, and 1 January are no working days)
  • the week with January 4 in it
  • the first week with the majority (four or more) of its days in the starting year
  • the week starting with the Monday in the period 29 December - 4 January
  • the week with the Thursday in the period 1 - 7 January
  • If 1 January is on a Monday, Tuesday, Wednesday or Thursday, it is in week 01. If 1 January is on a Friday, Saturday or Sunday, it is in week 52 or 53 of the previous year.

Note that while most definitions are symmetric with respect to time reversal, one definition in terms of working days happens to be equivalent.

The last week of the ISO year is the week before week 01; in accordance with the symmetry of the definition, equivalent definitions are:

  • the week with the year's last Thursday in it
  • the week with December 28 in it
  • the last week with the majority (four or more) of its days in the ending year
  • the week starting with the Monday in the period 22 - 28 December
  • the week with the Thursday in the period 25 - 31 December
  • the week ending with the Sunday in the period 28 December - 3 January
  • If 31 December is on a Monday, Tuesday, or Wednesday, it is in week 01, otherwise in week 52 or 53.

The following years have 53 weeks:

  • years starting with Thursday
  • leap years starting with Wednesday

ExamplesEdit

  • 2005-01-01 is 2004-W53-6
  • 2005-01-02 is 2004-W53-7
  • 2005-12-31 is 2005-W52-6
  • 2007-01-01 is 2007-W01-1 (both years 2007 start with the same day)
  • 2007-12-30 is 2007-W52-7
  • 2007-12-31 is 2008-W01-1
  • 2008-01-01 is 2008-W01-2 (Gregorian year 2008 is a leap year, ISO year 2008 is 2 days shorter: 1 day longer at the start, 3 days shorter at the end)
  • 2008-12-29 is 2009-W01-1
  • 2008-12-31 is 2009-W01-3
  • 2009-01-01 is 2009-W01-4
  • 2009-12-31 is 2009-W53-4 (ISO year 2009 is a leap year, extending the Gregorian year 2009, which starts and ends with Thursday, at both ends with three days)
  • 2010-01-03 is 2009-W53-7

Examples where the ISO year is three days into the next Gregorian year Edit

  • "{{ISOWEEKDATE|2009|12|31}}" gives "2009-W53-4" [1]
  • "{{ISOWEEKDATE|2010|1|1}}" gives "2009-W53-5" [2]
  • "{{ISOWEEKDATE|2010|1|2}}" gives "2009-W53-6" [3]
  • "{{ISOWEEKDATE|2010|1|3}}" gives "2009-W53-7" [4]
  • "{{ISOWEEKDATE|2010|1|4}}" gives "2010-W01-1" [5]

Examples where the ISO year is three days into the previous Gregorian year Edit

  • "{{ISOWEEKDATE|2008|12|28}}" gives "2008-W52-7" [6]
  • "{{ISOWEEKDATE|2008|12|29}}" gives "2009-W01-1" [7]
  • "{{ISOWEEKDATE|2008|12|30}}" gives "2009-W01-2" [8]
  • "{{ISOWEEKDATE|2008|12|31}}" gives "2009-W01-3" [9]
  • "{{ISOWEEKDATE|2009|1|1}}" gives "2009-W01-4" [10]

The system does not need the concept of month and is not well connected with the Gregorian system of months: some months January and December are divided over two ISO years.

Week numberEdit

Anchor dates with a fixed week number in any year other than a leap year starting on Thursday (DC)
W01W02W03W04W05W06W07W08W09W10W11W12W13W14W15W16W17W18W19W20W21W22W23W24W25W26W27W28W29W30W31W32W33W34W35W36W37W38W39W40W41W42W43W44W45W46W47W48W49W50W51W52W53
Jan04Jan11Jan18Jan25 Feb01Feb08Feb15Feb22 Mar01Mar08Mar15Mar22Mar29 Apr05Apr12Apr19Apr26 May03May10May17May24May31 Jun07Jun14Jun21Jun28 Jul05Jul12Jul19Jul26 Aug02Aug09Aug16Aug23Aug30 Sep06Sep13Sep20Sep27 Oct04Oct11Oct18Oct25 Nov01Nov08Nov15Nov22Nov29 Dec06Dec13Dec20Dec27

The day of the week for these days are related to Doomsday because for any year, the Doomsday is the day of the week that the last day of February falls on. These dates are one day after the Doomsdays, except that in January and February of leap years the dates themselves are Doomsdays. In leap years the week number is the rank number of its Doomsday.

All other month dates can fall in one of two weeks, except for 29 December through 2 January which can be in W52, W53 or W01, i.e. either in the first week of the new year or the last week of the old year, which can have two different designations.

Mapping of month dates to weeks in common years
Type Jan01Jan02Jan03Jan04Jan05Jan06Jan07Jan08Jan09Jan10Jan11Jan12Jan13Jan14Jan15Jan16Jan17Jan18Jan19Jan20Jan21Jan22Jan23Jan24Jan25Jan26Jan27Jan28Jan29Jan30Jan31 Feb01Feb02Feb03Feb04Feb05Feb06Feb07Feb08Feb09Feb10Feb11Feb12Feb13Feb14Feb15Feb16Feb17Feb18Feb19Feb20Feb21Feb22Feb23Feb24Feb25Feb26Feb27Feb28 Mar01Mar02Mar03Mar04Mar05Mar06Mar07Mar08Mar09Mar10Mar11Mar12Mar13Mar14Mar15Mar16Mar17Mar18Mar19Mar20Mar21Mar22Mar23Mar24Mar25Mar26Mar27Mar28Mar29Mar30Mar31 Apr01Apr02Apr03Apr04Apr05Apr06Apr07Apr08Apr09Apr10Apr11Apr12Apr13Apr14Apr15Apr16Apr17Apr18Apr19Apr20Apr21Apr22Apr23Apr24Apr25Apr26Apr27Apr28Apr29Apr30 May01May02May03May04May05May06May07May08May09May10May11May12May13May14May15May16May17May18May19May20May21May22May23May24May25May26May27May28May29May30May31 Jun01Jun02Jun03Jun04Jun05Jun06Jun07Jun08Jun09Jun10Jun11Jun12Jun13Jun14Jun15Jun16Jun17Jun18Jun19Jun20Jun21Jun22Jun23Jun24Jun25Jun26Jun27Jun28Jun29Jun30 Jul01Jul02Jul03Jul04Jul05Jul06Jul07Jul08Jul09Jul10Jul11Jul12Jul13Jul14Jul15Jul16Jul17Jul18Jul19Jul20Jul21Jul22Jul23Jul24Jul25Jul26Jul27Jul28Jul29Jul30Jul31 Aug01Aug02Aug03Aug04Aug05Aug06Aug07Aug08Aug09Aug10Aug11Aug12Aug13Aug14Aug15Aug16Aug17Aug18Aug19Aug20Aug21Aug22Aug23Aug24Aug25Aug26Aug27Aug28Aug29Aug30Aug31 Sep01Sep02Sep03Sep04Sep05Sep06Sep07Sep08Sep09Sep10Sep11Sep12Sep13Sep14Sep15Sep16Sep17Sep18Sep19Sep20Sep21Sep22Sep23Sep24Sep25Sep26Sep27Sep28Sep29Sep30 Oct01Oct02Oct03Oct04Oct05Oct06Oct07Oct08Oct09Oct10Oct11Oct12Oct13Oct14Oct15Oct16Oct17Oct18Oct19Oct20Oct21Oct22Oct23Oct24Oct25Oct26Oct27Oct28Oct29Oct30Oct31 Nov01Nov02Nov03Nov04Nov05Nov06Nov07Nov08Nov09Nov10Nov11Nov12Nov13Nov14Nov15Nov16Nov17Nov18Nov19Nov20Nov21Nov22Nov23Nov24Nov25Nov26Nov27Nov28Nov29Nov30 Dec01Dec02Dec03Dec04Dec05Dec06Dec07Dec08Dec09Dec10Dec11Dec12Dec13Dec14Dec15Dec16Dec17Dec18Dec19Dec20Dec21Dec22Dec23Dec24Dec25Dec26Dec27Dec28Dec29Dec30Dec31
A W52 W−01W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01
B/B* W52/3 W−01W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01
C W53 W−01W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01
D W01 W−53W02 W−52W03 W−51W04 W−50W05 W−49W06 W−48W07 W−47W08 W−46W09 W−45W10 W−44W11 W−43W12 W−42W13 W−41W14 W−40W15 W−39W16 W−38W17 W−37W18 W−36W19 W−35W20 W−34W21 W−33W22 W−32W23 W−31W24 W−30W25 W−29W26 W−28W27 W−27W28 W−26W29 W−25W30 W−24W31 W−23W32 W−22W33 W−21W34 W−20W35 W−19W36 W−18W37 W−17W38 W−16W39 W−15W40 W−14W41 W−13W42 W−12W43 W−11W44 W−10W45 W−09W46 W−08W47 W−07W48 W−06W49 W−05W50 W−04W51 W−03W52 W−02W53 W−01
E W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01W01 W−52
F W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01W01 W−52
G W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01W01 W−52
Mapping of month dates to weeks in leap years
Type Jan01Jan02Jan03Jan04Jan05Jan06Jan07Jan08Jan09Jan10Jan11Jan12Jan13Jan14Jan15Jan16Jan17Jan18Jan19Jan20Jan21Jan22Jan23Jan24Jan25Jan26Jan27Jan28Jan29Jan30Jan31 Feb01Feb02Feb03Feb04Feb05Feb06Feb07Feb08Feb09Feb10Feb11Feb12Feb13Feb14Feb15Feb16Feb17Feb18Feb19Feb20Feb21Feb22Feb23Feb24Feb25Feb26Feb27Feb28Feb29 Mar01Mar02Mar03Mar04Mar05Mar06Mar07Mar08Mar09Mar10Mar11Mar12Mar13Mar14Mar15Mar16Mar17Mar18Mar19Mar20Mar21Mar22Mar23Mar24Mar25Mar26Mar27Mar28Mar29Mar30Mar31 Apr01Apr02Apr03Apr04Apr05Apr06Apr07Apr08Apr09Apr10Apr11Apr12Apr13Apr14Apr15Apr16Apr17Apr18Apr19Apr20Apr21Apr22Apr23Apr24Apr25Apr26Apr27Apr28Apr29Apr30 May01May02May03May04May05May06May07May08May09May10May11May12May13May14May15May16May17May18May19May20May21May22May23May24May25May26May27May28May29May30May31 Jun01Jun02Jun03Jun04Jun05Jun06Jun07Jun08Jun09Jun10Jun11Jun12Jun13Jun14Jun15Jun16Jun17Jun18Jun19Jun20Jun21Jun22Jun23Jun24Jun25Jun26Jun27Jun28Jun29Jun30 Jul01Jul02Jul03Jul04Jul05Jul06Jul07Jul08Jul09Jul10Jul11Jul12Jul13Jul14Jul15Jul16Jul17Jul18Jul19Jul20Jul21Jul22Jul23Jul24Jul25Jul26Jul27Jul28Jul29Jul30Jul31 Aug01Aug02Aug03Aug04Aug05Aug06Aug07Aug08Aug09Aug10Aug11Aug12Aug13Aug14Aug15Aug16Aug17Aug18Aug19Aug20Aug21Aug22Aug23Aug24Aug25Aug26Aug27Aug28Aug29Aug30Aug31 Sep01Sep02Sep03Sep04Sep05Sep06Sep07Sep08Sep09Sep10Sep11Sep12Sep13Sep14Sep15Sep16Sep17Sep18Sep19Sep20Sep21Sep22Sep23Sep24Sep25Sep26Sep27Sep28Sep29Sep30 Oct01Oct02Oct03Oct04Oct05Oct06Oct07Oct08Oct09Oct10Oct11Oct12Oct13Oct14Oct15Oct16Oct17Oct18Oct19Oct20Oct21Oct22Oct23Oct24Oct25Oct26Oct27Oct28Oct29Oct30Oct31 Nov01Nov02Nov03Nov04Nov05Nov06Nov07Nov08Nov09Nov10Nov11Nov12Nov13Nov14Nov15Nov16Nov17Nov18Nov19Nov20Nov21Nov22Nov23Nov24Nov25Nov26Nov27Nov28Nov29Nov30 Dec01Dec02Dec03Dec04Dec05Dec06Dec07Dec08Dec09Dec10Dec11Dec12Dec13Dec14Dec15Dec16Dec17Dec18Dec19Dec20Dec21Dec22Dec23Dec24Dec25Dec26Dec27Dec28Dec29Dec30Dec31
AG W52 W−01W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01W01 W−52
BA W52 W−01W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01
CB W53 W−01W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01
DC W01 W−53W02 W−52W03 W−51W04 W−50W05 W−49W06 W−48W07 W−47W08 W−46W09 W−45W10 W−44W11 W−43W12 W−42W13 W−41W14 W−40W15 W−39W16 W−38W17 W−37W18 W−36W19 W−35W20 W−34W21 W−33W22 W−32W23 W−31W24 W−30W25 W−29W26 W−28W27 W−27W28 W−26W29 W−25W30 W−24W31 W−23W32 W−22W33 W−21W34 W−20W35 W−19W36 W−18W37 W−17W38 W−16W39 W−15W40 W−14W41 W−13W42 W−12W43 W−11W44 W−10W45 W−09W46 W−08W47 W−07W48 W−06W49 W−05W50 W−04W51 W−03W52 W−02W53 W−01
ED W01 W−53W02 W−52W03 W−51W04 W−50W05 W−49W06 W−48W07 W−47W08 W−46W09 W−45W10 W−44W11 W−43W12 W−42W13 W−41W14 W−40W15 W−39W16 W−38W17 W−37W18 W−36W19 W−35W20 W−34W21 W−33W22 W−32W23 W−31W24 W−30W25 W−29W26 W−28W27 W−27W28 W−26W29 W−25W30 W−24W31 W−23W32 W−22W33 W−21W34 W−20W35 W−19W36 W−18W37 W−17W38 W−16W39 W−15W40 W−14W41 W−13W42 W−12W43 W−11W44 W−10W45 W−09W46 W−08W47 W−07W48 W−06W49 W−05W50 W−04W51 W−03W52 W−02W53 W−01
FE W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01W01 W−52
GF W01 W−52W02 W−51W03 W−50W04 W−49W05 W−48W06 W−47W07 W−46W08 W−45W09 W−44W10 W−43W11 W−42W12 W−41W13 W−40W14 W−39W15 W−38W16 W−37W17 W−36W18 W−35W19 W−34W20 W−33W21 W−32W22 W−31W23 W−30W24 W−29W25 W−28W26 W−27W27 W−26W28 W−25W29 W−24W30 W−23W31 W−22W32 W−21W33 W−20W34 W−19W35 W−18W36 W−17W37 W−16W38 W−15W39 W−14W40 W−13W41 W−12W42 W−11W43 W−10W44 W−09W45 W−08W46 W−07W47 W−06W48 W−05W49 W−04W50 W−03W51 W−02W52 W−01W01 W−52

Advantages Edit

  • The date directly tells the weekday.
  • All years start with a Monday and end with a Sunday.
  • When used by itself without using the concept of month, all years are the same except that leap years have a leap week at the end.
  • The weeks are the same as in the Gregorian calendar.

Disadvantages Edit

Each equinox and solstice varies over a range of at least seven days. This is because each equinox and solstice may occur any day of the week and hence on at least seven different ISO week dates. For example, there are summer solstices on 2004-W12-7 and 2010-W11-7.

It cannot replace the Gregorian calendar, because it relies on it to define the new year day (Week 1 Day 1).

Leap year cycle Edit

The three types of long or leap-week year are D (Thursday–Thursday), DC (Thursday–Friday) and ED (Wednesday–Thursday).

Dominical letters and Doomsdays
DL Doomsday 31 December
A, BA Tuesday Sunday
B, CB Monday Saturday
C, DC Sunday Friday
D, ED Saturday Thursday
E, FE Friday Wednesday
F, GF Thursday Tuesday
G, AG Wednesday Monday
400-year cycle of years by dominical letter, grouped by Olympiad, long years highlighted
Years 16xy 17xy 18xy 19xy
20xy 21xy 22xy 23xy
00 BA C E G
01295785 G B D F
02305886 F A C E
03315987 E G B D
04326088DC FE AG CB
05336189 B D F A
06346290 A C E G
07356391 G B D F
08366492 FE AG CB ED
09376593D F A C
10386694 C E G B
11396795 B D F A
12406896 AG CB ED GF
13416997 F A C E
14427098 E G B D
15437199D F A C
167244 CB ED GF BA
734517 A C E G
744618 G B D F
754719 F A C E
764820ED GF BA DC
774921 C E G B
785022 B D F A
795123 A C E G
805224 GF BA DC FE
815325 E G B D
825426D F A C
835527 C E G B
845628 BA DC FE AG
Long years per 400-year cycle
(28-year subcycles arranged horizontally)
DC D D ED D
+004 +009 +015 +020 +026
+032 +037 +043 +048 +054
+060 +065 +071 +076 +082
+088 +093 +099
+105 +111 +116 +122
+128 +133 +139 +144 +150
+156 +161 +167 +172 +178
+184 +189 +195
+201 +207 +212 +218
+224 +229 +235 +240 +246
+252 +257 +263 +268 +274
+280 +285 +291 +296
+303 +308 +314
+320 +325 +331 +336 +342
+348 +353 +359 +364 +370
+376 +381 +387 +392 +398

There are 13 Julian 28-year subcycles with 5 leap years each, and 6 remaining leap years in the remaining 36 years (the absence of leap days in the Gregorian calendar in 2100, 2200, and 2300 interrupts the subcycles). The leap years are 5 years apart 27 times, 6 years 43 times and 7 years once. (A slightly more even distribution would be possible: 5 years apart 26 times and 6 years 45 times.)

The Gregorian years corresponding to the 71 long, ISO leap-week years can be subdivided as follows:

Thus 27 ISO years are 5 days longer than the corresponding Gregorian year, and 44 are 6 days longer. Of the other 329 Gregorian years (neither starting nor ending with Thursday), 70 are Gregorian leap years, and 259 are non-leap years, so 70 week years are 2 days shorter, and 259 are 1 day shorter than their corresponding month years.

Alternative leap year rules Edit

Karl Palmen Edit

The best leap week calendars to convert to Gregorian would be those based on the ISO week or a similar week numbering scheme. Karl Palmen thought of another that interlocks with the 28-year subcycle that such calendars have, which is interrupted by a dropped leap day in three out of four century years.

+002   +008   +014   +020   +026
+030   +036   +042   +048   +054
+058   +064   +070   +076   +082
+086   +092   +098   +104   +110   +116   +122
+126   +132   +138   +144   +150
+154   +160   +166   +172   +178
+182   +188   +194   +200   +206   +212   +218
+222   +228   +234   +240   +246
+250   +256   +262   +268   +274
+278   +284   +290   +296   +302   +308   +314
+318   +324   +330   +336   +342
+346   +352   +358   +364   +370
+374   +380   +386   +392   +398

In each row, the leap years are six years apart and the first of each row is four years after the last of the previous row. Each row covers 28 years unless it contains a dropped Gregorian leap day, in which case it covers 40 years. The rows are synchronized to the 28-year cycles that occur in any week-number calendar like ISO week date. In particular, the 2nd year of each row is a leap year starting on Tuesday (GF). A simpler leap week rule would have rows alternating between 28 and 34, but this would be harder to convert to and from the Gregorian calendar.

The following variation, offset by 12 years, matches all even-numbered years with 53 ISO weeks.[1] The italic years are a year later than ISO and the underlined years are a year early. Gregorian leap years are shown in bold.

+014   +020   +026   +032   +038
+042   +048   +054   +060   +066
+070   +076   +082   +088   +094   +100   +106
+110   +116   +122   +128   +134
+138   +144   +150   +156   +162
+166   +172   +178   +184   +190
+194   +200   +206   +212   +218   +224   +230
+234   +240   +246   +252   +258
+262   +268   +274   +280   +286
+290   +296   +302   +308   +314   +320   +326
+330   +336   +342   +348   +354
+358   +364   +370   +376   +382
+386   +392   +398   +004   +010

As with Karl Palmen's original proposal, every row has leap-week years six years apart. The first year of each row is four years after the previous. Each row has five, unless it spans a dropped leap year, then it has seven. Unlike the original proposal, it is not symmetrical.

To get a leap week rule where every leap week occurs in a Gregorian leap year and furthermore every Gregorian leap year that also has 53 ISO weeks is included among them, take the previously mentioned proposal, make the leap weeks in the 1st and 3rd columns occur two years later, the leap week in the 5th and 7th column occur two years earlier. Then move +100 to +096 and +200 to +204.

+016   +020   ---   +028   +032   +036   ---
+044   +048   ---   +056   +060   +064   ---
+072   +076   ---   +084   +088   +092   +096   <<<   +104   ---
+112   +116   ---   +124   +128   +132   ---
+138   +144   ---   +152   +156   +160   ---
+166   +172   ---   +180   +184   +188   ---
+196    >>>  +204   +208   +212   +216   ---   +224   +228   ---
+236   +240   ---   +248   +252   +256   ---
+264   +268   ---   +276   +280   +284   ---
+292   +296   ---   +304   +308   +312   ---   +320   +324   ---
+332   +336   ---   +344   +348   +352   ---
+360   +364   ---   +372   +376   +380   ---
+388   +392   ---   +000   +004   +008   ---

The Gregorian leap years with 53 ISO weeks remain in bold. Other changes can be made to the years not shown in bold. For example, the 4th row could be changed to

+108   ---   +116   +120   ---   +128   +132   ---

Christoph Päper Edit

Constraints:

  • Long leap year: Every year that has a leap week also has a leap day. 
  • Olympiad: The number of each leap year is divisible by four (as established over two millennia ago).

This way, years with dominical letter D never occur, just DC and ED. Since we need 97 leap days and just 71 leap weeks, there are 26 leap days per cycle in short years (i.e. ones without a 53rd week).

There are two distributions that are as even as possible under these constraints:

  • Round down
+000   +004   +008     –    +016   +020     –   
+028   +032   +036     –    +044   +048     –   
+056   +060   +064     *    +072   +076     –   
+084   +088   +092     –    +100   +104     –   
+112   +116   +120     –    +128   +132     –    +140   +144     –    
+152   +156   +160     –    +168   +172     –   
+180   +184   +188     –    +196   +200     *    
+208   +212   +216     –    +224   +228     –    
+236   +240   +244     –    +252   +256     –    +264   +268     –    
+276   +280   +284     –    +292   +296     –    
+304   +308   +312     –    +320   +324     *    
+332   +336   +340     –    +348   +352     –    
+360   +364   +368     –    +376   +380     –    +388   +392     –
  • Round up
  –    +020   +024     –    +032   +036   +040   
  –    +048   +052     –    +060   +064   +068
  *    +076   +080     –    +088   +092   +096   
  –    +104   +108     –    +116   +120   +124     –    +132   +136 
  –    +144   +148     –    +156   +160   +164 
  –    +172   +176     –    +184   +188   +192 
  *    +200   +204     –    +212   +216   +220 
  –    +228   +232     –    +240   +244   +248     –    +256   +260   
  –    +268   +272     –    +280   +284   +288   
  –    +296   +300     –    +308   +312   +316   
  –    +324   +328     *    +336   +340   +344   
  –    +352   +356     –    +364   +368   +372   
  –    +380   +384     –    +392   +396   +000     –    +008   +012   

Rounding down matches 27, rounding up just 20 of the 71 long years according to ISO 8601.

In a final step, one would need to select three of the empty places, which do not contain a “leap+leap year” but are a Julian “leap-day year”, to have no leap day (like 100, 200 and 300 by Gregorian rules), e.g. – as marked above with asterisks – 068, 204 and 328 for round-down and 072, 196 and 332 for round-up.

Other week numbering systems Edit

For an overview of week numbering systems see week number. The US system has weeks from Sunday through Saturday, and partial weeks at the beginning and the end of the year. An advantage is that no separate year numbering like the ISO year is needed, while correspondence of lexicographical order and chronological order is preserved.

External links Edit

  1. List of ISO leap years Dick Henry – leap weeks are called Newton

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