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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 calendar[]

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, considering 1 January:

  • years starting with Thursday (D, DC)
  • leap years starting with Wednesday (ED)

Alternatively, considering 31 December:

  • years ending with Thursday (D, ED)
  • leap years ending with Friday (DC)

Collectively, considering both:

  • years starting (DC) or ending (ED) on a Thursday, or both (D)

Examples[]

  • 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[]

  • "{{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[]

  • "{{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 number[]

Dates with a fixed week number
1st row: in every common year and every leap year other than a leap year starting on Thursday (DC)
2nd row: in every leap year
W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 W11 W12 W13 W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26 W27 W28 W29 W30 W31 W32 W33 W34 W35 W36 W37 W38 W39 W40 W41 W42 W43 W44 W45 W46 W47 W48 W49 W50 W51 W52 W53
Common Jan04 Jan11 Jan18 Jan25 Feb01 Feb08 Feb15 Feb22 Mar01 Mar08 Mar15 Mar22 Mar29 Apr05 Apr12 Apr19 Apr26 May03 May10 May17 May24 May31 Jun07 Jun14 Jun21 Jun28 Jul05 Jul12 Jul19 Jul26 Aug02 Aug09 Aug16 Aug23 Aug30 Sep06 Sep13 Sep20 Sep27 Oct04 Oct11 Oct18 Oct25 Nov01 Nov08 Nov15 Nov22 Nov29 Dec06 Dec13 Dec20 Dec27
Leap Jan04 Jan11 Jan18 Jan25 Feb01 Feb08 Feb15 Feb22 Feb29 Mar07 Mar14 Mar21 Mar28 Apr04 Apr11 Apr18 Apr25 May02 May09 May16 May23 May30 Jun06 Jun13 Jun20 Jun27 Jul04 Jul11 Jul18 Jul25 Aug01 Aug08 Aug15 Aug22 Aug29 Sep05 Sep12 Sep19 Sep26 Oct03 Oct10 Oct17 Oct24 Oct31 Nov07 Nov14 Nov21 Nov28 Dec05 Dec12 Dec19 Dec26

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.

Dates of ISO weeks in common years
Type Dec. January February March April May June July August September October November December Jan. DL
29 30 31 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 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 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 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 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 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 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 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 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 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 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 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 01 02 03
A W52 W−01 W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 A
B/B* W52/3 W−01 W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 B/B*
C W53 W−01 W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 C
D W01 W−53 W02 W−52 W03 W−51 W04 W−50 W05 W−49 W06 W−48 W07 W−47 W08 W−46 W09 W−45 W10 W−44 W11 W−43 W12 W−42 W13 W−41 W14 W−40 W15 W−39 W16 W−38 W17 W−37 W18 W−36 W19 W−35 W20 W−34 W21 W−33 W22 W−32 W23 W−31 W24 W−30 W25 W−29 W26 W−28 W27 W−27 W28 W−26 W29 W−25 W30 W−24 W31 W−23 W32 W−22 W33 W−21 W34 W−20 W35 W−19 W36 W−18 W37 W−17 W38 W−16 W39 W−15 W40 W−14 W41 W−13 W42 W−12 W43 W−11 W44 W−10 W45 W−09 W46 W−08 W47 W−07 W48 W−06 W49 W−05 W50 W−04 W51 W−03 W52 W−02 W53 W−01 D
E W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 W01 W−53 E
F W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 W01 W−52/3 F
G W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 W01 W−52 G
Dates of ISO weeks in leap years
Type Dec. January February March April May June July August September October November December Jan. DL
29 30 31 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 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 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 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 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 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 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 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 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 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 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 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 01 02 03
AG W52 W−01 W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 W01 W−52 AG
BA W52 W−01 W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 BA
CB W53 W−01 W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 CB
DC W01 W−53 W02 W−52 W03 W−51 W04 W−50 W05 W−49 W06 W−48 W07 W−47 W08 W−46 W09 W−45 W10 W−44 W11 W−43 W12 W−42 W13 W−41 W14 W−40 W15 W−39 W16 W−38 W17 W−37 W18 W−36 W19 W−35 W20 W−34 W21 W−33 W22 W−32 W23 W−31 W24 W−30 W25 W−29 W26 W−28 W27 W−27 W28 W−26 W29 W−25 W30 W−24 W31 W−23 W32 W−22 W33 W−21 W34 W−20 W35 W−19 W36 W−18 W37 W−17 W38 W−16 W39 W−15 W40 W−14 W41 W−13 W42 W−12 W43 W−11 W44 W−10 W45 W−09 W46 W−08 W47 W−07 W48 W−06 W49 W−05 W50 W−04 W51 W−03 W52 W−02 W53 W−01 DC
ED W01 W−53 W02 W−52 W03 W−51 W04 W−50 W05 W−49 W06 W−48 W07 W−47 W08 W−46 W09 W−45 W10 W−44 W11 W−43 W12 W−42 W13 W−41 W14 W−40 W15 W−39 W16 W−38 W17 W−37 W18 W−36 W19 W−35 W20 W−34 W21 W−33 W22 W−32 W23 W−31 W24 W−30 W25 W−29 W26 W−28 W27 W−27 W28 W−26 W29 W−25 W30 W−24 W31 W−23 W32 W−22 W33 W−21 W34 W−20 W35 W−19 W36 W−18 W37 W−17 W38 W−16 W39 W−15 W40 W−14 W41 W−13 W42 W−12 W43 W−11 W44 W−10 W45 W−09 W46 W−08 W47 W−07 W48 W−06 W49 W−05 W50 W−04 W51 W−03 W52 W−02 W53 W−01 ED
FE W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 W01 W−53 FE
GF W01 W−52 W02 W−51 W03 W−50 W04 W−49 W05 W−48 W06 W−47 W07 W−46 W08 W−45 W09 W−44 W10 W−43 W11 W−42 W12 W−41 W13 W−40 W14 W−39 W15 W−38 W16 W−37 W17 W−36 W18 W−35 W19 W−34 W20 W−33 W21 W−32 W22 W−31 W23 W−30 W24 W−29 W25 W−28 W26 W−27 W27 W−26 W28 W−25 W29 W−24 W30 W−23 W31 W−22 W32 W−21 W33 W−20 W34 W−19 W35 W−18 W36 W−17 W37 W−16 W38 W−15 W39 W−14 W40 W−13 W41 W−12 W42 W−11 W43 W−10 W44 W−09 W45 W−08 W46 W−07 W47 W−06 W48 W−05 W49 W−04 W50 W−03 W51 W−02 W52 W−01 W01 W−52 GF

Months[]

The assignment of weeks to months does not form part of the ISO standard. It only assigns weeks to years and thereby implicitly to the months of January and December. If week months were defined to begin on the nearest Monday to the first day of the Gregorian month, or, in other words, weeks belonged to the calendar month the majority of their days (and thus their Thursday) are in, then seemingly irregular patterns would be the result.

Weeks per month, year and quarter depending on the weekday of 1 January
1 Jan: Mon Tue Wed Thu Fri Sat Sun usual
01 4 5 5 5 4 4 4 57%
02 4+ 4 4 4 4 4 4 96¾%
03 5– 4 4 4 4+ 5 5 57%
04 4 4 4+ 5 5– 4 4 71½%
05 5 5 5– 4 4 4 4+ 57%
06 4 4 4 4 4+ 5 5– 67¾%
07 4 4+ 5 5 5– 4 4 57¼%
08 5 5– 4 4 4 4+ 5 57¼%
09 4 4 4 4+ 5 5– 4 71½%
10 4+ 5 5 5– 4 4 4 57%
11 5– 4 4 4 4 4+ 5 71½%
12 4 4 4+ 5 5 5– 4 57%
Year 52 52 52+ 53 52 52 52 82¼%
Q1 13 13 13 13 12+ 13 13 89¼%
Q2 13 13 13 13 13 13 13 100%
Q3 13 13 13 13+ 14– 13 13 86%
Q4 13 13 13+ 14– 13 13 13 85½%


Many ordinal weeks would always be in the same month, but some would fluctuate between two months.

Weeks that change their month depending on the weekday of 1 January or Dominical Letter
Jan01 Friday Saturday Sunday Monday Tuesday Wednesday Thursday
DL C CB B BA A AG G GF F FE E ED D DC
W05 February January
W09 March February
W13 April March
W18 May April
W22 June May
W26 July June
W31 August July
W35 September August
W40 October Sept.
W44 November October
W48 December November
W53 December

Months could also be assigned regularly to the weeks, but the months of such a mapping will deviate from the Gregorian months by more days. See Week & Month Calendar for an example.

Weeks of the months and their ordinal number
Month DLs 1st week 2nd week 3rd week 4th week Last week 4 weeks 5 weeks
January AG–G W01 W02 W03 W04 GF–C W04 GF–C DC–F
DC–F W05
February GF–C W05 W06 W07 W08 G–C W08 G–F GF
DC–F W06 W07 W08 W09 DC–GF W09
March G–C W09 W10 W11 W12 C W12 C–GF G–CB
DC–GF W10 W11 W12 W13 DC–CB W13
April C W13 W14 W15 W16 C–E W17 E–CB C–ED
DC–CB W14 W15 W16 W17 DC–ED W18
May C–E W18 W19 W20 W21 A–C W21 A–ED E–AG
DC–ED W19 W20 W21 W22 DC–AG W22
June A–C W22 W23 W24 W25 C W25 C–G AG–CB
DC–AG W23 W24 W25 W26 DC–CB W26
July C W26 W27 W28 W29 F–C W30 F–CB C–FE
DC–CB W27 W28 W29 W30 DC–FE W31
August F–C W31 W32 W33 W34 B–C W34 B–FE F–BA
DC–FE W32 W33 W34 W35 DC–BA W35
September B–C W35 W36 W37 W38 D–C W39 D–BA B–DC
DC–BA W36 W37 W38 W39 DC W40
October D–C W40 W41 W42 W43 G–C W43 G–DC D–GF
DC W41 W42 W43 W44 DC–GF W44
November G–C W44 W45 W46 W47 B–C W47 B–GF G–BA
DC–GF W45 W46 W47 W48 DC–BA W48
December B–C W48 W49 W50 W51 E–C W52 E–BA B–ED
DC–BA W49 W50 W51 W52 DC–ED W53

Advantages[]

  • 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[]

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[]

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
01 29 57 85 G B D F
02 30 58 86 F A C E
03 31 59 87 E G B D
04 32 60 88 DC FE AG CB
05 33 61 89 B D F A
06 34 62 90 A C E G
07 35 63 91 G B D F
08 36 64 92 FE AG CB ED
09 37 65 93 D F A C
10 38 66 94 C E G B
11 39 67 95 B D F A
12 40 68 96 AG CB ED GF
13 41 69 97 F A C E
14 42 70 98 E G B D
15 43 71 99 D F A C
16 72 44 CB ED GF BA
73 45 17 A C E G
74 46 18 G B D F
75 47 19 F A C E
76 48 20 ED GF BA DC
77 49 21 C E G B
78 50 22 B D F A
79 51 23 A C E G
80 52 24 GF BA DC FE
81 53 25 E G B D
82 54 26 D F A C
83 55 27 C E G B
84 56 28 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[]

Karl Palmen[]

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[]

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[]

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[]

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