## Leap Week Calendars

## by Karl Palmen

## Introduction

There are manyalternative calendarsin 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 either outside of or 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 to 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:

- Bonavian Civil Calendar (12 Months of 4 or 5 weeks)
- Colligan's Pax Calendar (13 months of 4 weeks + leap week )
- Pragmatic Civil Calendar (12 part-week months 30 or 31 days, leap week spread over 7 months)
- Hanke-Henry Permanent Calendar (12 part-week months of 30 or 31 days + leap week at end of year).
- Symmetry 454 Calendar (12 months of 4 or 5 weeks arranged in normally symmetrical quarters).
- Hermetic Leap Week Calendar (12 months of 4 or 5 weeks and years grouped into hexades of six or five years to determine leap week years)
All the above examples have months with day of month numbers. The ISO Week Date calendar has no months and instead numbers the weeks from 01 to either 52 or 53.

After years of consideration, I made my own proposal, the Week and Month Calendar, which has months and weeks, but

no day of month numbers. The year has 12 months of 4 or 5 weeks with the same names as the Gregorian months that occur at approximately the same time and with weeks named after letters of the Greek alphabet, which along with the day of week replaces the hard to remember day of month number. The years coincide with those of the ISO week date.

## 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 weekagainst the seasons, the simpler leap week rules vary the new yeartwo 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 weekprovides 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 ifthe corresponding Gregorian year begins and/or ends with a Thursdayand so has 53 Thursdays. By definition, its new year varies justagainst the Gregorian Calendar (3 days early to 3 days late). The6 daysHanke-Henry Permament Calendarand myMonth and Week Calendaruse the same leap week rule.The

Pragmatic Civil Calendareither uses the ISO week, or another rule, where thenth leap week year is400*n/71 rounded to the nearest whole number. This rule is not simple, but provides a variation of about justagainst the seasons. This is actually slightly better than the ISO week, because the Gregorian calendar itself varies about 2 days against the seasons.one week

Colligans Pax Calendarhas a comparatively simple leap week rule. A year has a leap week if and only ifthe 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 variesagainst the Gregorian calendar.19 daysA simpler cycle, which I discovered and call the

5:40:400 leap week rulehas a leap week in those years that aredivisible by 400or aredivisible 5, but not divisible by 40. It is used by Bob McClenon'sReformed 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 variesagainst the Gregorian calendar. I've produced a simple pair of tables to calculate the 5:40:400 new year.17 daysThe

Bonavian Civil Calendardoes 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 ifit is divisible by 28 but not 896 or has a remainder of 5,11,16 or 22from dividing by 28. Its new year varies aboutagainst the seasons.2 weeksThe

Hermetic Leap Week Calendarhas 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 Vijhas proposed an 834-year cycle, where a year has a leap week if and only ifit is divisible by six or is one of nine additional yearsper 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 11 365.2419The 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

allleap week cycles that areThey are all formed by mixing the 62 year cycle with the 45 year cycle. The

- are no more than 1000 years
- have a mean year length between that of the 62 year cycle and the Gregorian 400 year cycle
- are not a multiple of a shorter cycle
`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 CycleSome other interesting cycles2/55 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 30 365.242604Irv Bromberg suggests a 389-year cycle with 69 leap weeks and a mean year of 365.241645... days, which is equal to seven 62-year cycles with on 45-year cycle removed, to be accurate a tropical year beginning at the June solstice.

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