The Hermetic Leap Week Calendar

A proposal for calendar reformby Peter Meyer First published 2007-01-08 CE = 2007-03-01 LPW = 2007-01-15 LPM

## Introduction

Aleap week calendaris a calendar which divides the sequence of days (nychthemerons) into years, weeks and days such that

- every week has a fixed number of days
- every year consists of an integral number of weeks, though that number may differ slightly from year to year and
- the calendar year more or less stays in sync with the seasons.
Although "weeks" which do not have seven days are possible (see, e.g., here), the term

leap week calendarusually denotes a calendar in which the weeks have seven days. Since the length of the seasonal year is approximately 365.242 days (the exact value depends on where in the seasonal year one locates the beginning of the year) this implies that a year usually has 52 weeks, since 52 x 7 = 364 days. 52 7‑day weeks are 1.242 days short of a seasonal year, so some years must have more than 52 weeks. Usually a leap week calendar will have years consisting of either 52 or 53 weeks. Years of 51 or 54 weeks are possible, but this is undesirable because it lessens the degree to which the calendar year stays in sync with the seasonal year.Karl Palmen has briefly surveyed a number of leap week calendars in his article Leap Week Calendars, where he considers their various merits.

## Definition of the Hermetic Leap Week Calendar

Alocal nychthemeron, a.k.a. aday, is a the period from midnight to midnight at the place where this calendar is (to be) used. Ayearin this calendar is a sequence of 364 or 371 days. Years are divided into 7‑dayweeks, so each year consists either of 52 or 53 weeks. A year which has 52 weeks is said to be anormalyear, and a year which has 53 weeks is said to be aleapyear. Years are numbered according to astronomical notation: ..., -1, 0, 1, 2, ...The Hermetic Leap Week Calendar is related to Earth time by the selection of a particular day as the first day of year 1 of the calendar. This day is chosen to be the first Monday after the northern winter solstice of the year 0 CE. (This solstice occurred on Julian day number 1,721,414 = 0-12-20 CE, a Wednesday, so day 1 of year 1 is JDN 1,721,419 = 0-12-25 CE, i.e., December 25th in the year 1 BCE.)

A

hexadeis defined as a sequence of five or six consecutive years. Ashorthexade consists of five years, and alonghexade consists of six years. Clearly the sequence of years consists of a sequence of consecutive hexades. The actual sequence of hexades (in this calendar) is defined as follows:

(Thus the first hexade is long, because 1*71 mod 100 = 71, which is not < 26; the second hexade is long, because 7*71 mod 100 = 97, which also is not < 26; the third hexade is short, because 13*71 mod 100 = 23 < 26; and so on.)

- The first hexade begins with year 1.
- A hexade is short if and only if the number of its first year, Y, is such that (Y*71) mod 100 < 26.
The distribution of leap years is defined as follows:

- Every hexade has one and only one leap year.
- The third year in a hexade is a leap year.
Dates in this calendar are designated by the year number, the week number (1‑53) and the day number (1‑7), written thus: Y‑W‑D LPW, where the week number consists of two digits, e.g., "1433‑03‑2 LPW".

The seven days of the week are named according to the local custom for naming days of the week ("Monday", etc.). Day numbers correspond to names as follows: 1 = Monday, 2 = Tuesday, ..., 7 = Sunday.

This completes the definition of the calendar.

Historical note: (1) This definition originally used the rule that a hexade is short if and only if the number of its first year, Y, is such that (Y*71) mod 200 < 71. Karl Palmen subsequently showed that this can be replaced by the simpler form of the rule used above. (2) The original definition identified day 1 of year 0 with -1-12-27 CE and specified the fifth year of a hexade as the leap year. The leap year was changed to the third for a slightly better fit with the northern winter solstice.

Actually the rule for when a hexade is short can, in practice, be simplified:

Hexades have 5 years if and only if, when you take the last two digits of the number of the year which begins the hexade, and multiply this by 71, you get a number whose last two digits denote a number less than 26.

For example, if a hexade begins with, say, year 1934 then you don't need to multiply 1934 by 71, you just need to multiply 34 by 71, to get 2414. The last two digits denote the number 14. Since this is less than 26, the hexade has only five years, not six.

## Properties of the Calendar

The names of the days in the Hermetic Leap Week Calendar are the same as the usual names of the days, since the first day of the first week of year 1 LPW is a Monday.The numbering of the years is the same as the numbering of years in the Common Era Calendar (that is, the proleptic Gregorian Calendar with astronomical year numbering), since the first day of year 1, 1‑01‑1 LPW, is 0‑12‑25 CE, so almost all of year 1 LPW coincides with year 1 CE.

Consider now the question of the average length of the calendar year. We can ascertain this if we find a a pattern of short and long hexades, which is repeated. Such a pattern exists if we can find a year which is the first year in a hexade and whose number equals 1 mod 100. The easiest way to look for such a year is to work through the hexades. The initial hexades are as follows:

Hexade Y Y*71 Y*71 mod 100 Type 1 1 71 71 long 2 7 497 97 long 3 13 923 23 short 4 18 1278 78 long 5 24 1704 4 short 6 29 2059 59 long 7 35 2485 85 long 8 41 2911 11 short 101 = 1 mod 100, but when we come to year 101 we find that it is the fifth year of a hexade, so is not the year we're looking for. 201 = 1 mod 100, but when we come to year 201 we find that it is the fifth year of a hexade, so that won't do. 301 = 1 mod 100, but when we come to year 301 we find that it is the second year of a hexade, so we're out of luck again. 401 = 1 mod 100, and —

voila!— year 401isthe first year of a hexade. So the pattern of hexades and years repeats every 400 years.Counting the long and short hexades in the first 400 years reveals that there are exactly 26 short hexades and 45 long hexades. There are 4*364 + 371 = 1827 days in a short hexade, and 5*364 + 371 = 2191 days in a long hexade, so in 26 short hexades and 45 long hexades there are exactly 146,097 days. Thus the average year length in this calendar is 146,097 = 365.2425 days. (This is the same as the average length of the year in the Gregorian Calendar.)

The complete hexade pattern of this calendar is given here.

Consider now the question of how closely this calendar stays in sync with the seasons. Computation reveals that over the range of years 1200-4000 the average difference between the Julian day number of new years day and the Julian day number of the (preceding) northern winter solstice is 3.99 days and that this difference ranges from 0 to 8. In other words, during this period the northern winter solstice always occurs either on new years day or on one of the last eight days of the calendar year. During this period the northern winter solstice will occur during the last week of the year on average in about 99 years out of 100. When it does not it occurs either on new years day or exactly 8 days before.

New years day always occurs in December of the preceding Gregorian year. Computation reveals that during the period 1600-4000 new years day occurs on December dates in the following proportions:

21 22 23 24 25 26 27 28 29 30 0.25% 8.00% 14.00% 14.49% 14.00% 14.49% 14.03% 14.24% 6.25% 0.25% New years day 2007 in this calendar occurred on 2006‑12‑25 CE, and this article was published on 2007‑01‑08 CE, so the date of its publication in the Hermetic Leap Week Calendar was 2007‑03‑01 LPW.

## Ease of Use

This calendar is very easy to use provided one keeps track of years which occur at the start of a hexade. When a hexade begins, its first year has a number, Y. By definition the hexade is long if and only if (Y*71) mod 100 < 26. Even for very large numbers it is easy to calculate the mod 100 value — it is simply the number minus the largest multiple of 100 which is less than or equal to it, which is simply the number signified by the last two digits. Comparing the result with 26, one knows whether the hexade is short or long. In any case the next leap year is the third year in the hexade (so it occurs two years after the year marking the start of the hexade). After five or six years the next hexade begins, and one simply repeats this calculation. For example, the current hexade begins with this year, 2007 (so the leap year of this hexade is the 3rd, namely, 2009). 2007*71 = 142,297, so the mod 100 value is 97. Since this is not less than 26, this hexade is long, so the next hexade begins 6 years after the year 2007, thus with the year 2013.The Gregorian dates of new years day for the six years in the current hexade, and whether the year is a normal year (52 weeks) or a leap year (53 weeks), are as follows:

2007-01-1 LPW 2006-12-25 CE normal 2008-01-1 LPW 2007-12-24 CE normal 2009-01-1 LPW 2008-12-22 CE leap 2010-01-1 LPW 2009-12-28 CE normal 2011-01-1 LPW 2010-12-27 CE normal 2012-01-1 LPW 2011-12-26 CE normal A listing of all dates in this calendar for the year 2007, and their corresponding Gregorian dates, is given here.

## Alternative Definition of this Calendar

After this article was first published on this website I informed several people, including Karl Palmen, who pointed out that the calendar can be defined in terms of a rule as to when a year Y is a leap year (rather than when a hexade is a long hexade). This way of defining the calendar has the advantage that one can know where the leap years occur without knowing where the hexades begin.Taking (as above) the first day of year 1 to be the day with Julian day number 1,721,419 (a.k.a. 0‑12‑25 CE), and using the same concepts of

short hexadeandlong hexade, define the calendar as follows:

- The third year in a hexade is a leap year.
- A year Y is a leap year if and only if (71*Y + 203) mod 400 < 71.
A proof that this definition implies the one given above will be given by Karl Palmen in a separate article. Since both definitions completely determine the distribution of leap years, they are equivalent.

## A Calendar of Months

People are accustomed to calendars which divide the year into "months" which somewhat resemble the period of a lunation (approximately 29½ days). So what follows is a reformulation of the Hermetic Leap Week Calendar in terms of months.The calendar is defined as above. The 52 or 53 weeks of the year are divided into months as follows:

Month

number1 2 3 4 5 6 7 8 9 10 11 12 Number of weeks

in the month5 4 4 5 4 4 5 4 4 5 4 4 or 5 The 12th month has 4 weeks in a normal year and 5 weeks in a leap year.

Thus the weeks are partitioned into quarters as follows:

1st quarter 5 4 4 2nd quarter 5 4 4 3rd quarter 5 4 4 4th quarter 5 4 4 or 5 This is the same week structure as possessed by the Bonavian Civil Calendar, another leap week calendar, but one which has a different leap year rule (thus the leap years are distributed differently) and in which weeks begin on Sunday, not Monday.

This division of the 52 or 53 weeks in a year allows the following division of the year into semesters, terms and quarters:

Semesters (6 months each) 26 weeks 26 or 27 weeks Terms (4 months each) 18 weeks 17 weeks 17 or 18 weeks Quarters (3 months each) 13 weeks 13 weeks 13 weeks 13 or 14 weeks Months customarily have names. In the Hermetic Leap Week Calendar the following names (of stars) are used:

A mnemonic rhyme for the lengths of the months in the Gregorian Calendar is needed ("Thirty days hath September ...") because they are so irregular (31, 28, 31, 30, ...). For the Hermetic Leap Week Calendar no rhyme is needed, because the month lengths (in weeks) are regular: four sets of 5, 4 and 4 weeks (with the 12th month having 5 weeks in leap years).

Month number Month name Month type 1 Arcturus long 2 Bellatrix short 3 Canopus short 4 Deneb long 5 Elnath short 6 Fomalhaut short 7 Girtab long 8 Hadar short 9 Izar short 10 Jabbah long 11 Kochab short 12 Lesath short or long So this calendar was first published on 2007-01-15 LPM, the 15th day of Arcturus, 2007 LPM, which is the same day as denoted by the date 2007‑03‑01 LPW. The calendar designations "LPM" and "LPW" show whether the date is expressed respectively as year‑month‑day or as year‑week‑day.

A listing of all dates in this calendar for any selected year in month-and-day form, and their corresponding Gregorian dates, is given here.

## Advantages of Changing to this Calendar

Any proposal for a new calendar should not only be easy to understand, or at least easier to understand than the currently used calendar, but there also has to be some advantage to switching to the new calendar. So I list here some advantages (I use 'GC' to denote the Gregorian Calendar and 'LPM' to denote the Hermetic Leap Week Calandar with dates expressed as year-month-day):

When a new calendar is proposed, in the context of calendar reform, it has to be shown, not only that it offers advantages over the calendar currently in use, but also that a changeover can be carried out without causing calendrical confusion.

- The months of the GC are highly irregular: 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31 days

whereas the months of the LPM are regular: 5, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4 weeks.

- When a month has an additional day (GC) or an additional week (LPM) the rule for this in GC is complicated:

"An extra day is added at the end of the second month every four years, except in years whose number is divisible by 100 except in years whose number is divisible by 400."

Whereas the rule for this in LPM is utterly simple: "The third year of a hexade is a leap year."

This rule presupposes an understanding of hexades, but that is not difficult: A hexade is a period of five or six years. A hexade has five years if and only if the number made from the last two digits of its year, when multiplied by 71, gives a number whose last two digits are less than "26". This year, 2007 in LPM (and GC), is the start of a hexade. From which it is simple to infer that the next leap year is 2009. Since 07*71 = 497, and 97 is not less than 26, the hexade beginning in 2007 has six years, so the next hexade begins in 2013, so the next leap year after the leap year of 2009 is 2015.

- The conventional 7‑day‑week cycle does not fit exactly into a Gregorian year (there are always one or two days left over), whereas it does fit exactly into a LPM year (a year always has exactly 52 or 53 weeks). This means that it is difficult to know which day of the week a GC date falls on, but it is easy for date in LPM. For example, what day of the week does 2007‑10‑10 CE fall on? No easy way to answer except to consult a printed calendar. But for 2007‑10‑10 LPM we note that, since 10 = 7 + 3, the 10th day of the 10th month occurs on the 3rd day of the 2nd week, thus on a Wednesday.

- The irregularity of the structure of the GC makes it very difficult to formulate schedules of events occurring on certain days of the week which can be re‑used from year to year. E.g., if a course begins on the first Monday of the third month then this can be anything from March 1st to March 7th in GC. But in the LPM the first Monday of the third month is always Canopus 1st.

- The irregularity of the structure of the GC makes it very difficult to design schedules which can be used in any quarter (of three months), term (of four months) or semester (of six months). This is much easier in LPM because the 52 or 53 weeks in a year can be partitioned into quarters as 13+13+13+13 (+14 in a leap year), into terms as 18+17+17 (+18 in a leap year), and into semesters as 26+26 (+27 in a leap year).

- GC dates are currently expressed in a variety of formats:
month-day-year(U.S.),day-month-year(Europe and most of the rest of the world) andyear-month-day(ISO 8601 date format). This creates major confusion for Europeans reading dates in U.S. publications and for Americans reading dates in European publications. In LPM dates are always expressed asyear-month-day, so no confusion can arise.

- The adoption of this calendar would completely eliminate Friday the 13ths.

In the case of the Hermetic Leap Week Calendar, the facts that (a) years in this calendar almost entirely overlap years in the Gregorian Calendar and (b) the weeks of this calendar coincide with conventional weeks currently used with the Gregorian Calendar, make this transition feasible. No major social changes would be required. Weekends remain as usual. During a transition period of, say, six years, dates would need to be shown in both calendars, e.g., 2011‑12‑14 LPM = 2011‑12‑11 CE.

## Date Conversion Software

Click on this link to see a full-year display of dates in the Hermetic Leap Week Calendar for the current year.Conversion of dates in the Hermetic Leap Week Calendar to and from Julian day numbers, ordinal dates and dates in the Gregorian, Julian and ISO 8601 Week calendars can be performed using the Easy Date Converter software.

Custom software for particular business applications (either a PC application or a web page) is available.

Properties of the Hermetic Leap Week Calendar The Hermetic Lunar Week Calendar Calendar Studies Hermetic Systems Home Page