# Properties of the Hermetic Leap Week Calendar

## Introduction

This document first briefly defines the Hermetic Leap Week Calendar, which was invented by Peter Meyer in January 2007, and then shows some interesting properties of this calendar.

## Concise Definition

### Leap Week Rule

The years of the Hermetic Leap Week Calendar are grouped into periods of five or six years, which are called hexades.

A short hexade consists of five years, and a long hexade consists of six years. The third year of a hexade is its only year with a leap week. One hexade begins with year 1. Each hexade has an indicator that is formed by multiplying the number of its first year or just its last two digits by 71 modulo 100. A hexade is short if and only if its indicator is less than 26.

The hexade that begins with year 1 is a long hexade because its indicator is 71 (1*71 mod 100 = 71) which is not less than 26.

Note that the third year of a hexade, which has the leap week, is two years after the start year of the hexade.

Year 1 began on Monday 25 December 1 BC in the Proleptic Gregorian Calendar.

### Dates

Dates in this calendar are designated by the year number, the week number (1‑52) 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".

Alternatively the year may divided into months of 4 or 5 weeks, in which case, the date is of form Y‑M‑D LPM. This article will not concern itself with the internal structure of the years.

Although according to the definition there are no dates in this calendar with a year number less than 1, this article will show how such dates can be defined.

## Ease of Use

The Hermetic Leap Week Calendar is easy to use provided you know when the current hexade began. Then you know which year is the 3rd year of the current hexade and so its only leap week year. Also you can then use the hexade indicator to work out how many years there are in the hexade and hence when the next hexade begins.

If you don't know when the hexade began, you need to go back to year 1 or to the most recent year that you know was the first year of hexade and then work out each of the subsequent hexades one by one to the current hexade. However there is another way of dealing with this situation that will be mentioned below.

Working out the hexade indicator for a hexade of known start year is very easy. It is done by multiplying the last two digits of the start year by 71 and taking the last two digits of the result. The last digit of the hexade indicator is the same as the last digit of the start year. So one need only calculate the first digit of the hexade indicator. If the last two digits of the start year are AB, then the first digit of the hexade indicator is (A + 7*B mod 10).

For example, given that 1996 is the start year of a hexade, A=9 and B=6, so the first digit of the hexade indicator is 9 + 7*6 mod 10 = 1 and hence the hexade indicator is 16, so indicating a short hexade and the next hexade starting at year 2001.

This section deals more with the hexade indicator of various hexades, eventually leading to a way to ascertain leap week years without using hexades at all.

### The Values of the Hexade Indicator

If the current hexade is short, it has 5 years, so the indicator of the next hexade is either 55 (=71*5 mod 100) greater or 45 less than the indicator of the current hexade. Because it's a short hexade its indicator is less than 26, so it must always be 55 greater (and never 45 less).

If the current hexade is long, it has 6 years, so the indicator of the next hexade is either 26 (=71*6 mod 100) greater or 74 less than the indicator of the current hexade. If it is 74 less then the indicator of the next hexade is less than 26 and so the next hexade is short, else it is 26 more. This leads to:

1. No two consecutive hexades are short.
These rules uniquely determine the lengths and hexade indicators of all subsequent hexades. This will lead to the hexades following a cycle in which both the hexade indicators and lengths repeat. Also the indicator of a long hexade belonging to a cycle is at least 55.

Any hexade that belongs to a cycle must have a short hexade occurring sometime before it (or the same hexade of a later cycle). After any short hexade the hexade indicator is at least 55 and continues to go up for each subsequent long hexade before the next short hexade.

No hexade belonging to a cycle has an indicator of value 26 to 54 inclusive. All other hexades do belong to a cycle.

One can use the hexade indicator to reckon hexades in reverse. The hexade indicator of a hexade after a short hexade is in the range 55 to 80 inclusive, because 55 is added after a short hexade. The hexade after a long hexade has an indicator of 0 to 26 if short, else its indicator is in the range 71 to 99, because 26 is added to an indicator of at least 55. So the rule for working backwards is:

A hexade is the hexade after a short hexade if and only if its hexade indicator is in the range 55 to 80 inclusive.

For example, the hexade beginning with year 2001 has an indicator of 71, which is in the range of 55 to 80 inclusive, hence the previous hexade is short and so begins with 1996 and makes 1998 into a year with a leap week.

This rule allows you to reckon hexades in reverse and even hexades before year 1, provided Y mod 100 is taken to be non-negative for all Y.

For example -4 mod 100 is 96. This means you must add to the year number a multiple of 100 sufficiently large to make it non-negative before taking the last two digits.

## The Hexade Type From Last Two Digits of First Year

One can bypass the hexade indicator and simply look up the hexade type from a list of the last two digits of the start year.

Each short hexade has a start year ending in:

```00 03 10 13 17 20 24 27 31 34
41 44 48 51 55 58 62 65 72 75
79 82 86 89 93 96
```

Each long hexade has a start year ending in:

```01 04 05 07 08 11 14 15 18 21
22 25 28 29 32 35 36 38 39 42
45 46 49 52 53 56 59 60 63 66
67 69 70 73 76 77 80 83 84 87
90 91 94 97 98
```

These lists can be created by noting that 31*71 mod 100 = 1, so the last two digits Y of the start year of a hexade with indicator J is given by Y = (31*J) mod 100. Also we make use of the fact that the indicator of a long hexade is at least 55 (which occurs for year 5).

## A Direct Leap Week Year Rule

It is not necessary to think in terms of hexades to determine whether a particular year has a leap week. Year Y has a leap week if and only if
```(71*Y + 203) mod 400 < 71
```

I call the value of `(71*Y+203) mod 400` the accumulator of year Y. This rule can then be stated as:

A year has a leap week if and only if its accumulator is less than 71.

By means of this rule one can easily discover whether a particular year has a leap week and it also allows one to construct an efficient calendar conversion algorithm.

### Proof of the Direct Leap Week Rule

Suppose a calendar has its leap week years defined by this rule. Then by examining the accumulator of successive years it can be seen that each year with a leap week has the next one occurring five or six years after it. Hence the years can be grouped into hexades of 5 or 6 years so that the 3rd year of each hexade is its only year with a leap week. Each hexade begins two years before a year with a leap week.

Year 3 has an accumulator of (213+203 mod 400) = 16, which is less than 71, so it has a leap week, hence it is the 3rd year of a hexade and so there is a hexade beginning with year 1 as required by the hexade rule.

Now let's consider the year that occurs five years after the leap year of the hexade beginning with year Y. This year occurs 7 years after year Y, hence its accumulator is (71*Y+203+497) mod 400 which is equal to (71*Y+300) mod 400. This accumulator modulo 100 is then equal to the hexade indicator of the hexade beginning with year Y.

Hence if a hexade begins at year Y then its indicator is equal to the accumulator of year (Y+7) modulo 100. Because this year (Y+7) is five years after a year with a leap week, its accumulator must be in the range 0 to 70 shifted by adding 355 mod 400, which is 0 to 25, or 355 to 399, inclusive.

If the hexade is short then the year (Y+7) must also have a leap week and so its accumulator is less than 26. Therefore the short hexade has an indicator less than 26. If the hexade is long then the year (Y+7) must be a year before a year with a leap week, hence its accumulator must be at least 400-71 = 329 and so is in the range 355 to 399 inclusive. Therefore the long hexade has an indicator greater than 26.

Therefore, the hexades obey the rules of the Hermetic Leap Week Calendar. Since only one sequence of hexades can obey the rules of the Hermetic Leap Week Calendar, it must be the sequence in which a year has a leap week if and only if its accumulator is less than 71.

## List of Leap Week Years

The leap week years of each 400-year cycle are:
```003 009 015  020 026  031 037 043  048 054 060
065 071 077  082 088  093 099 105
110 116 122  127 133  138 144 150  155 161 167
172 178 184  189 195  200 206 212  217 223 229
234 240 246  251 257  262 268 274
279 285 291  296 302  307 313 319  324 330 336
341 347 353  358 364  369 375 381  386 392 398
```
The respective accumulators are:
```16 42 68  23 49  04 30 56  11 37 63
18 44 70  25 51  06 32 58
13 39 65  20 46  01 27 53  08 24 60
15 41 67  22 48  03 29 55  10 26 62
17 43 69  24 50  05 31 57
12 38 64  19 45  00 26 52  07 33 59
14 40 66  21 47  02 28 54  09 35 61
```

## Hexade Families, Clans and Tribes

The leap week years in the list above are grouped to reflect the structure of the 400-year cycle and how the 71 hexades themselves can be grouped. In each row I have grouped the leap week years into groups where the leap week years are six years apart. Between the groups they are five years apart. Each row is arranged so that the second group in the row is the only one with just two leap week years in it.

These groups of three or two leap week years can be considered to be groups of hexades, each ending with its only short hexade. I call such a group a hexade family. Each hexade family corresponds to a row of Peter's Hexade Pattern table.

A hexade family has either three hexades of 17 years or two hexades of 11 years. In each 400-year cycle there are 26 hexade families 19 of 17 years and seven short hexade families of 11 years. Here is a list of the hexade families:

```001-017, 018-028, 029-045, 046-062,
063-079, 080-090, 091-107
108-124, 125-135, 136-152, 153-169,

170-186, 187-197, 198-214, 215-231,
232-248, 249-259, 260-276,
277-293, 294-304, 305-321, 322-338,
339-355, 356-366, 367-383, 384-400.
```
I define a hexade clan as a period of four or three hexade families such that the second hexade family is the only short hexade family. Each of the seven rows of the above table forms a hexade clan.

There are five hexade clans of 62 years and two of 45 years. The hexade clans are the cycles of 62 and 45 years that all accurate leap-week cycles are a mixture of, as shown in my list of leap week cycles. The hexade clans themselves form two groups of 169 and 231 years (shown as blocks), such that the second hexade clan of is the only one of 45 years. I call each of these two groups a hexade tribe.

The hexade indicator of the first year of a hexade tells you whether that year is the first year of a hexade family, clan or tribe and how many years that group has.

```Hexade Indicator    Group
of First Year
1 to 26             Short Hexade (5 years)
55 to 73            Long Hexade Family (17 years)
67 to 71            Long Hexade Clan (62 years)
70                  Hexade Tribe of 231 years
71                  Hexade Tribe of 169 years
72 or 73            Short Hexade Clan (45 years)
81 to 99            Long Hexade precedeed by another Long Hexade (6 years)
```

A long and a short hexade family together form a 28-year Julian Calendar cycle of 1461 weeks. Each such period begins with a year whose hexade indicator is between 67 and 80 inclusive. Two consecutive 28-year periods occur if the first has a hexade indicator of 79 or 80 for its first year, which is year 249 or 80 respectively of the 400-year cycle. The two 28-year periods in either pair are not identical.

I've stated that year Y has a leap week if and only if
```(71*Y + 203) mod 400 < 71
```
One could change 203 to any non-negative integer `K` less than 400.
```(71*Y + K) mod 400 < 71
```
But not all these values of `K` lead to a hexade calendar where a hexade beginning with year `Y` is short if and only if
```71*Y mod 100 < 26.
```
All such cases are determined by which year `L` of the hexade has a leap week and which year `F` is the first year counting from year 0 that starts a hexade. It turns out that `F` must be one of 0,1,3 or 4. It can't be greater than 5 because no hexade has more than 6 years. It can't be 5 because its hexade indicator is 55 so is in the range 55 to 80 (why?), indicating that the previous hexade is short and so starts with year 1. It can't be 2 because no hexade can start with a year ending in 02 (why?).

Here I list the values of `K` that result in hexades for `B=71` and the corresonding values of `F` and `L`. A leap week occurring in the last year of a hexade is indicated by `L=0`.

 ``` K F L K F L 003 4 3 203 1 3 Hermetic Leap Week Calendar 016 1 0 216 4 0 032 3 4 232 0 4 045 0 1 245 3 1 061 1 5 261 4 5 074 4 2 274 1 2 103 3 3 303 0 3 116 0 0 316 3 0 132 1 4 332 4 4 145 4 1 345 1 1 161 0 5 361 3 5 174 3 2 374 0 2 ```

The value of K determines the period of about 7 days that an equinox or solstice would occur for a given epoch. Peter requires the northern winter solstice to occur in or very close to the last week of the year. This is achieved with K near to 200.

## Historical Note

Peter Meyer originally defined the Hermetic Leap Week Calendar so that the first hexade began with year 0 and the fifth year of a hexade was its leap week year, and he used the rule that a hexade beginning with year `Y` is short if and only if
```71*Y mod 200 < 71
```
I pointed out that this was equivalent to
```71*Y mod 100 < 26
```
which is much simpler. It is the case F=0, L=5, hence K=161. Peter then discovered that a better accord of the last week of the calendar to the northern winter solstice was achieved by making the first hexade begin with year 1 and making the 3rd year of the hexade the leap week year. This led to the current rules.