Leap Week Calendars

 

by Karl Palmen


Introduction

There are many alternative calendars in which the year has a whole number of weeks so that each year has an identical calendar, except for intercalation. One of these, The World Calendar, nearly became a successful calendar reform.

The World Calendar and several others are leapday calendars in which the year has either 365 or 366 days. One or two days are added to the weeks of each year to ensure that there are a whole number of weeks in a year. This causes an interruption in the 7 day week, which has been uninterrupted for thousands of years and will continue be observed as such by some religious groups. If several of these alternative calendars were to be adopted, there would be times in which they would disagree over the day of the week.

Some alternative calendars have a whole number of weeks in a year and preserve the 7 day week without interruption. Such a calendar cannot be a leapday calendar. Instead it may be a calendar whose year has either 364 or 371 days (52 or 53 weeks) and so may be called a leap week calendar.

Because a leap week calendar uses the existing 7 day week, one does not need legislation to be able to use one. Many organisations such as universities already use some kind of leap week calendar to schedule their activities.


Examples of Leap Week Calendars

Here are many of the various leapweek calendars that have been proposed:


Leap Week Rules

A leap week calendar cannot use the same rules for the leap week as leap day calendar uses for the leap day. In choosing your leap week rule, there is a trade-off between simplicity and low variation against the seasons. Although, the new year of a leap week calendar must vary one week against the seasons, the simpler leap week rules vary the new year two to three weeks.

So there are two good reasons for sticking with a leapday calendar rather than changing to a leapweek calendar, (1) a very small variation against the seasons AND (2) a simple leapday rule. These would have to be sacrificed, if one were to have a calendar, that is fixed relative to a strict 7-day week.

In the following leap week rule descriptions I talk about how much the new year varies against the Gregorian calendar or the seasons. This variation, I define as the difference between the latest and earliest new year in the calendar cycle.

In the Gregorian calendar, the 400 year cycle has a whole number of weeks, providing 71 weeks in excess of 52 per year. This means a leap week calendar can have 71 leap weeks in 400 years.

The ISO week provides a leap week rule that is simple, only if you use it alongside the Gregorian calendar. A year has a leap week if and only if the corresponding Gregorian year begins and/or ends with a Thursday and so has 53 Thursdays. By definition, its new year varies just 6 days against the Gregorian Calendar (3 days early to 3 days late).

The Pragmatic Civil Calendar either uses the ISO week, or another rule, where the nth leap week year is 400*n/71 rounded to the nearest whole number. This rule is not simple, but provides a variation of about just one week against the seasons. This is actually slightly better than the ISO week, because the Gregorian calendar itself varies about 2 days against the seasons.

Colligans Pax Calendar has a comparatively simple leap week rule. A year has a leap week if and only if the last two digits are divisible by six or are '99' or with the exception of years divisible by 400 are '00'. The Pax year of 1901 would begin on Sunday 6 Jan 1901 and the Pax year of 2096 would begin on Sunday 18 December 2095, so the new year varies 19 days against the Gregorian calendar.

A simpler cycle, which I discovered and call the 5:40:400 leap week rule has a leap week in those years that are divisible by 400 or are divisible 5, but not divisible by 40. It is used by Bob McClenon's Reformed Weekly Calendar. If the new year for 2001 matches the Gregorian Calendar, then 1965 would begin of Monday 21 December 1964 and 2036 would begin on Monday 7 January 2036, so the new year varies 17 days against the Gregorian calendar. I've produced a simple pair of tables to calculate the 5:40:400 new year.

The Ziobro Calendar had a similar rule. In this rule, the leap week years are the those divisible by 5 but not divisible by 50 or with remainder of 175 when divided by 400. This also varies 17 days against the Gregorian calendar. Chris Carrier (who invented the Bonavian Calendar) proposed an alternative leap week rule, which is effectively the same as Ziobro's rule here, but 175 years earlier. Because of the different relationship with the Gregorian leap days, it varies 18 days against the Gregorian calendar. The Ziobro calendar has been modified since 2000-12-31 to use the 5:40:400 rule with synchronous new year 2001.

The Bonavian Civil Calendar does not use the Gregorian cycle. Instead, it uses an 896 year cycle, equivalent to a Julian calendar modified so that years divisible by 128 are not leap years. A year has a leap week if and only if it is divisible by 28 but not 896 or has a remainder of 5,11,16 or 22 from dividing by 28. Its new year varies about 2 weeks against the seasons.

The Hermetic Leap Week Calendar has 71 leap weeks in a 400 year cycle as in the Gregorian cycle with the leap years are spread as evenly as possible. The 400-year cycle is divided into 71-hexades such that the third year of each hexade is the only year in the hexade with a leap week. More about this leap year rule is found here.

Brij Bhushan Vij has proposed an 834-year cycle, where a year has a leap week if and only if it is divisible by six or is one of nine additional years per 834-year cycle. See his MS Word document for a table of these additional leap years. Each of these additional leap years occurs either 90 or 96 years after the previous such year and all have odd numbers divisible by 3.


Leap Week Cycles

There are many cycles that a leap week calendar can use. Here I give a complete (within certain limits) listing of leap week cycles, which may be used by a leap week calendar.

Starting from the Julian cycle, used with exception by the Bonavian Calendar, progressively more accurate cycles are

Years  Leap weeks   Mean Year
 28         5         365.25
 17         3         365.2353
 45         8         365.2444
 62        11         365.2419
The 62 year cycle is the shortest one in the range of 365.2416 and 365.2427 in which the mean year drifts both ways against the seasons. It is also divisible by 5 days. Six 62 year cycles (372 years) has a whole number of 210 day cycles.

The listing below has all leap week cycles that are

  1. are no more than 1000 years
  2. have a mean year length between that of the 62 year cycle and the Gregorian 400 year cycle
  3. are not a multiple of a shorter cycle
They are all formed by mixing the 62 year cycle with the 45 year cycle. The ratio is of 45 year cycles against 62 year cycles. The Gregorian Cycle is the only one where the number of years is divisible by 10.
Ratio  Years  Leap Weeks  Mean Year    Comments
0/1      62      11       365.241936  Pure 62 year cycle
1/15    975     173       365.242051
1/14    913     162       365.242059
1/13    851     151       365.242068
1/12    789     140       365.242079
1/11    727     129       365.242091
1/10    665     118       365.242105
1/9     603     107       365.242123
1/8     541      96       365.242144
1/7     479      85       365.242171
2/13    896     159       365.2421875  Bonavian Cycle (7*128 yrs)
1/6     417      74       365.242206   Half a Vij 834-year Cycle
2/11    772     137       365.242228
1/5     355      63       365.242254
2/9     648     115       365.242284   Good for base 6 or 12 counting
3/13    941     167       365.242295
1/4     293      52       365.242321   Six Jubilees of 364 day years
3/11    817     145       365.242350
2/7     524      93       365.242366
3/10    755     134       365.242384
4/13    986     175       365.242394   17yr cycle corrected by extra leapweek
1/3     231      41       365.242424   Seven 33-year cycles
4/11    862     153       365.242459
3/8     631     112       365.242472
2/5     400      71       365.2425     Gregorian Cycle
Some other interesting cycles
2/55   3500     621       365.242      A modified Gregorian cycle
8/45   3150     559       365.242222   Half New Orthodox cycle
14/45  3420     607       365.242398   Half MPSLC Cycle
1/2     169      30       365.242604

Copyright 2005-2014 Karl Palmen

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