The Archetypes Calendar An accurate lunisolar calendarwith connections to the Chinese Calendar by Peter Meyer

### Definition of the Archetypes Calendar

The Archetypes Calendar denotes days by means of three numbers, year-month-day, where the day number ranges from 1 through 30, the month number ranges from 1 through 13, and the year number is an integer (-2, -1, 0, 1, 2, 3, ...).

Each year has 12 or 13 months, and each month has 29 or 30 days. Most years have 12 months. A year with 13 months is called a long year.

All odd-numbered months have 30 days and all even-numbered months have 29 days, except that in some years the 10th month has 30 days; such a year is called a leap year (by analogy with leap years in the Common Era Calendar). A long year may also be a leap year and vice-versa.

ARC years are grouped into consecutive periods of 1,803 years called ARC periods. The first year in an ARC period has position 1, the second has position 2, and so on up to position 1803. The linear numbering of years is related to the cycles of ARC periods, and to the Julian day number system (and thus to empirical time), as follows:

(a) The ARC years 443 through 2245 constitute an ARC period.

(b) The Julian day number of the first day of this ARC period is 897,474.

More generally, an ARC year numbered y occupies position ((y + 1360) mod Y) + 1 in some ARC period, where Y = 1803. For example, ((4300 + 1360) mod 1803) + 1 = 252, so the year 4300 ARC has position 252 in some ARC period.

The rules for when a year is a long year and when a year is a leap year are as follows, where L1 = 664 and L2 = 350:

(i)  A year with position p is a long year if and only if ((p*L1) + (Y-1)/2) mod Y < L1.

(ii)  A year with position p is a leap year if and only if ((p*L2) + (Y-1)/2) mod Y < L2.

More clearly, these rules are:

(i)  A year with position p is a long year if and only if (664*p + 901) mod 1803 < 664.

(ii)  A year with position p is a leap year if and only if (350*p + 901) mod 1803 < 350.

For example, the the year 4300 ARC has position 252 (as shown above), so taking p = 252 we find that 664*252 + 901 = 168229, and 168229 mod 1803 = 550, which is less than 664, so ARC year 4300 is a long year. Also 350*252 + 901 = 89101, and 89101 mod 1803 = 754, which is not less than 350, so ARC year 4300 is not a leap year. Thus 4300 ARC has 13 months with alternating 30 and 29 days.

The names of the first seven months are the same as the classical Greek or Roman names of the deities associated with the seven celestial bodies known to the ancients. The names of the last five are those of deities which can plausibly be associated with the three planets discovered within the last three centuries, namely, Uranus, Neptune and Pluto. The following table gives the names of the months and the number of days in each month.

 Monthnumber Monthname Numberof days Monthnumber Monthname Numberof days 1 Apollo 30 7 Chronos 30 2 Diana 29 8 Prometheus 29 3 Hermes 30 9 Orpheus 30 4 Aphrodite 29 10 Sophia 29 or 30 5 Ares 30 11 Dionysus 30 6 Zeus 29 12 Demeter 29 In a long year: 13 Persephone 30

A month consists of three consecutive tweeks. The first two always have 10 days, while the third may have 9 or 10 days, for a total of 29 or 30 days in a month.

The days of the tweek are named after the Sun and the planets of the solar system. The order of days in the tweek (corresponding to the distance of the planets from the Sun) is as follows (reading from left to right, that is, Sun Day, Mercury Day, ..., Mars Day, Jupiter Day, ..., Pluto Day):

 Sun Day Mercury Day Venus Day Earth Day Mars Day Jupiter Day Saturn Day Uranus Day Neptune Day Pluto Day

Note that not all tweeks have a Pluto Day, because some final tweeks in a month have only nine days.

This completes the definition of the Archetypes Calendar.

A note on the origin of this calendar is here.

### Correlation with the Common Era Calendar

In order to relate the Archetypes Calendar to the Gregorian (or Common Era) Calendar it is sufficient to associate a date in the Archetypes Calendar with a Julian day number. By the definition of the calendar the date 443-1-1 ARC denotes the day with Julian day number 897,474 (which is -2255-02-05 CE). This establishes a one-to-one correspondence between ARC dates and Julian day numbers and so between ARC dates and CE dates. 1-1-1 ARC denotes the day with Julian day number 736,030 (which is -2697-01-30 CE). Here are some examples of date sequences:

```    CE Calendar    ---------- Archetypes Calendar ---------    Julian day number
2010-03-07     4708-01-22  Mercury Day, Apollo 22, 4708        2,455,263
2010-03-08     4708-01-23    Venus Day, Apollo 23, 4708        2,455,264
2010-03-09     4708-01-24    Earth Day, Apollo 24, 4708        2,455,265
2010-03-10     4708-01-25     Mars Day, Apollo 25, 4708        2,455,266
2010-03-11     4708-01-26  Jupiter Day, Apollo 26, 4708        2,455,267
2010-03-12     4708-01-27   Saturn Day, Apollo 27, 4708        2,455,268
2010-03-13     4708-01-28   Uranus Day, Apollo 28, 4708        2,455,269
2010-03-14     4708-01-29  Neptune Day, Apollo 29, 4708        2,455,270
2010-03-15     4708-01-30    Pluto Day, Apollo 30, 4708        2,455,271
2010-03-16     4708-02-01      Sun Day, Diana 1, 4708          2,455,272
2010-03-17     4708-02-02  Mercury Day, Diana 2, 4708          2,455,273
2010-03-18     4708-02-03    Venus Day, Diana 3, 4708          2,455,274```

```    CE Calendar    ---------- Archetypes Calendar ---------    Julian day number
2011-01-28     4708-12-24     Earth Day, Demeter 24, 4708      2,455,590
2011-01-29     4708-12-25      Mars Day, Demeter 25, 4708      2,455,591
2011-01-30     4708-12-26   Jupiter Day, Demeter 26, 4708      2,455,592
2011-01-31     4708-12-27    Saturn Day, Demeter 27, 4708      2,455,593
2011-02-01     4708-12-28    Uranus Day, Demeter 28, 4708      2,455,594
2011-02-02     4708-12-29   Neptune Day, Demeter 29, 4708      2,455,595
2011-02-03     4709-01-01       Sun Day, Apollo 1, 4709        2,455,596
2011-02-04     4709-01-02   Mercury Day, Apollo 2, 4709        2,455,597
2011-02-05     4709-01-03     Venus Day, Apollo 3, 4709        2,455,598
2011-02-06     4709-01-04     Earth Day, Apollo 4, 4709        2,455,599
2011-02-07     4709-01-05      Mars Day, Apollo 5, 4709        2,455,600
2011-02-08     4709-01-06   Jupiter Day, Apollo 6, 4709        2,455,601```

```    CE Calendar    ---------- Archetypes Calendar ---------    Julian day number
2012-12-16     4710-12-03     Venus Day, Demeter 3, 4710       2,456,278
2012-12-17     4710-12-04     Earth Day, Demeter 4, 4710       2,456,279
2012-12-18     4710-12-05      Mars Day, Demeter 5, 4710       2,456,280
2012-12-19     4710-12-06   Jupiter Day, Demeter 6, 4710       2,456,281
2012-12-20     4710-12-07    Saturn Day, Demeter 7, 4710       2,456,282
2012-12-21     4710-12-08    Uranus Day, Demeter 8, 4710       2,456,283
2012-12-22     4710-12-09   Neptune Day, Demeter 9, 4710       2,456,284
2012-12-23     4710-12-10     Pluto Day, Demeter 10, 4710      2,456,285
2012-12-24     4710-12-11       Sun Day, Demeter 11, 4710      2,456,286
2012-12-25     4710-12-12   Mercury Day, Demeter 12, 4710      2,456,287
2012-12-26     4710-12-13     Venus Day, Demeter 13, 4710      2,456,288
2012-12-27     4710-12-14     Earth Day, Demeter 14, 4710      2,456,289```

The dates in the Common Era Calendar of new year's day in the Archetypes Calendar for ARC years 4699 through 4755 are shown below:

 ```ARC year Begins on CE Long? Leap? 4699 2001-01-24 Yes 4700 2002-02-12 Yes 4701 2003-02-02 4702 2004-01-22 Yes 4703 2005-02-09 4704 2006-01-29 Yes 4705 2007-02-17 Yes 4706 2008-02-07 4707 2009-01-26 Yes 4708 2010-02-14 4709 2011-02-03 4710 2012-01-23 Yes Yes 4711 2013-02-11 4712 2014-01-31 Yes 4713 2015-02-19 4714 2016-02-08 4715 2017-01-27 Yes 4716 2018-02-15 Yes 4717 2019-02-05``` ```ARC year Begins on CE Long? Leap? 4718 2020-01-25 Yes 4719 2021-02-12 4720 2022-02-01 4721 2023-01-21 Yes Yes 4722 2024-02-10 4723 2025-01-29 Yes 4724 2026-02-17 4725 2027-02-06 4726 2028-01-26 Yes Yes 4727 2029-02-14 4728 2030-02-03 4729 2031-01-23 Yes 4730 2032-02-11 4731 2033-01-30 Yes Yes 4732 2034-02-19 4733 2035-02-08 4734 2036-01-28 Yes 4735 2037-02-15 4736 2038-02-04 Yes``` ```ARC year Begins on CE Long? Leap? 4737 2039-01-25 Yes 4738 2040-02-13 4739 2041-02-01 4740 2042-01-21 Yes 4741 2043-02-09 Yes 4742 2044-01-30 Yes 4743 2045-02-17 4744 2046-02-06 4745 2047-01-26 Yes 4746 2048-02-14 4747 2049-02-02 Yes 4748 2050-01-23 Yes 4749 2051-02-11 4750 2052-01-31 Yes 4751 2053-02-18 4752 2054-02-07 Yes 4753 2055-01-28 Yes 4754 2056-02-16 4755 2057-02-04```

All new year's days in the Archetypes Calendar occur from January 21 to February 21. As is clear from the table above, the month and day numbers of the CE dates of new year's days in the Archetypes Calendar which are exactly 19 years apart are often the same, or if not then they differ by one day. This is an effect of the 19-year Metonic Cycle.

### Date Conversion Software

There is Windows software for converting between dates in the Common Era Calendar and dates in the Archetypes Calendar (and a variety of other dates). For further details see:

### Some Properties of the Archetypes Calendar

1. There are four types of year:

• Long years which are also leap years have 385 days.
• Long years which are not leap years have 384 days.
• Leap years which are not long years have 355 days.
• Years which are neither long years nor leap years have 354 days.

2. The pattern of year types (long, leap, both long and leap, and neither) repeats itself during each successive ARC period of 1,803 years. The years in an ARC period are symmetrical about the middle year as regards their type. In other words, for all positions p = 1, 2, ..., 901, a year with position p and and a year with position 1803+1-p are of the same type. Thus the years in an ARC period form a Helios cycle.

3. During one ARC period there are exactly 664 occurrences of long years (36.83%) and 350 occurrences of leap years (19.41%).

4. The 1,803 years consist of 22,300 months and 658,532 days.

5. Thus the mean length of a year is 365.24237 days (to 5 decimal places), which lies within the expected range during the next 1000 years of 365.24235 to 365.24245 days for the mean vernal equinox year.

6. The mean length of a month is 29.530583 days (to six decimal places), which lies at the lower limit of the expected range during the next 1000 years of 29.530583 to 29.530590 days for the observed (not TAI) synodic month.

7. The prime decomposition of 658,532 is (2^2) * 7 * 29 * 811, so there are a whole number of 7-day weeks in an ARC period (94,076 of them to be exact).

8. The day of the tweek that a given date falls on can always be known simply from the last digit of the day number, if the order of days is known (1 = Sun Day, 2 = Mercury Day, and so on).

9. This makes it possible to develop schedules which — unlike with the Common Era (a.k.a. Gregorian) Calendar — don't need to be changed from year to year to keep events (such as university lectures, annual inspections, etc.) on the same day of the tweek every year.

10. The regularity of the Archetypes Calendar, and the fact that its years overlap the years of the Common Era Calendar, make it suitable as a replacement for the CE Calendar for business purposes. Months always have either 29 or 30 days, and years may be divided into quarters consisting respectively of 89, 88, 89 and 88 days, except that the 4th quarter has either 88 days, 89 days, 118 days or 119 days depending on whether the year is a leap year, a long year, both or neither. If 4th quarters are to be compared then it may be preferable to divide the year into four quarters of three months each (with each quarter having 88 or 89 days) plus an additional month (of 30 days) in long years (which occur about every third year).

11. The year has 39 variable-length tweeks in long years and 36 in other years. These tweeks have a pattern of days of 10, 10, 9, 10, 10, 9, ... with the exception that one of the 10-10-9 triples becomes 10-10-10 in a leap year. Although — unlike with the current 7-day week — the number of days in an ARC tweek is variable, this calendar has the advantage that a year always consists of a whole number of tweeks.

### Tracking the Dark Moon

The Archetypes Calendar is intended to be an accurate lunar calendar and so we should ask how well the months stay in sync with lunations.

A dark moon occurs when the Moon is between the Earth and the Sun and the apparent centers of the Sun and the Moon are both within a plane which is perpendicular to the ecliptic. A calendar month starts at midnight at the start of the first day of the month. Calculations were made, for 1001 consecutive lunations from 1959 CE to 2040 CE, of the difference, for each lunation, between the time of the dark moon and the start of the month, assuming the calendar is being used in the GMT time zone.

The average (mean) difference was -0.0213 days, that is, about 31 minutes prior to the midnight at the start of the month. This does not mean that the dark moon usually occurs within the first hour. The times of occurrence of the dark moon vary considerably, as shown in the graph below. About 2/3rds of these dark moons occur within 12 hours of the start of the month.

The largest positive difference was 1.4288 (about 1 day 10½ hours) and the largest negative difference was -1.4693 (about 1 day 11¼ hours). The time of the dark moon differed from the start of the month by more than 24 hours in only 44 cases (4.4%). In other words, during these 1001 lunations 95.6% of dark moons occur either on the first day of the month or on the last day of the previous month (not too bad for a purely rule-based calendar, i.e., one which does not depend on observation or astronomical calculation). These results are shown graphically below (click for full size). A positive difference means that the dark moon occurred later than the start of the month.

When the same calculation was performed for 20,001 consecutive lunations over the period 1191 CE to 2808 CE the average (mean) difference was -0.0307 days (about 44 minutes). The largest positive difference was 1.4404 (about 1 day 10½ hours) and the largest negative difference was -1.5410 (about 1 day 13 hours). The start of the month differed from the time of the dark moon by more than 24 hours in only 4.95% of cases.

### Connections with the Chinese Calendar

In the Archetypes Calendar, as stated above, each month consists of two 10-day tweeks followed by a tweek of 9 or 10 days (depending on whether the month has 29 or 30 days). It seems that this scheme had already occurred long ago to the Chinese, if we can believe a contributor to Wikipedia who writes (in the article on the week):

 The Chinese 10 day week went as far back as the Shang Dynasty (1200-1045 BC). The law in the Han Dynasty (206 BC - AD 220) required officials of the empire to rest every 5 days, called "mu", while it was changed into 10 days in the Tang Dynasty (AD 618-907), called "huan" or "xún". Months were almost 3 weeks long (alternating 29 and 30 days to keep in line with the lunation). The weeks were labelled "shàng xún", "zhông xún" and "xià xún" which mean roughly "upper", "middle" and "lower" week.

In the Common Era (a.k.a. Gregorian) Calendar, Chinese new year's day almost always occurs from January 21 through February 20, and very rarely on February 21 (see Helmer Aslaksen's The Mathematics of the Chinese Calendar). It is also possible for it to occur on January 20 (this happened in 497 CE).

All new year's days in ARC years 4300 through 5200 (CE years 1602 through 2502) occur from January 21 through February 21. In ARC year 3195 (497 CE) new year's day occurred on January 20. Thus it seems very likely that the range of new year's days in the CE Calendar is the same for the Archetypes Calendar as for the Chinese Calendar, namely the 33-day period from January 20 through February 21.

The Julian day number to connect the calendar to the sequence of days was chosen so that these date ranges would be very similar (and, as it happened, the same). What is surprising is that, for any actual year, the CE date of new year in the Archetypes Calendar is usually the same as the CE date of new year in the Chinese Calendar. When they differ, they almost always differ by just one day. The following table gives the dates of new year's day in the two calendars for the CE years 2005 through 2019.

 Chinese NYD CE date ARC year CE date of ARC NYD Comparison 79-22-01-01 2005-02-09 4703 2005-02-09 Same 79-23-01-01 2006-01-29 4704 2006-01-29 Same 79-24-01-01 2007-02-18 4705 2007-02-17 One day earlier 79-25-01-01 2008-02-07 4706 2008-02-07 Same 79-26-01-01 2009-01-26 4707 2009-01-26 Same 79-27-01-01 2010-02-14 4708 2010-02-14 Same 79-28-01-01 2011-02-03 4709 2011-02-03 Same 79-29-01-01 2012-01-23 4710 2012-01-23 Same 79-30-01-01 2013-02-10 4711 2013-02-11 One day later 79-31-01-01 2014-01-31 4712 2014-01-31 Same 79-32-01-01 2015-02-19 4713 2015-02-19 Same 79-33-01-01 2016-02-08 4714 2016-02-08 Same 79-34-01-01 2017-01-28 4715 2017-01-27 One day earlier 79-35-01-01 2018-02-16 4716 2018-02-15 One day earlier 79-36-01-01 2019-02-05 4717 2019-02-05 Same

A study was made of 500 years (CE years 1900 through 2399), with these results:

• In 71.8% of years the CE dates of new year's day in the two calendars are the same.
• In 10.2% of years the ARC new year's day is a day later than the Chinese.
• In 17.4% of years it is a day earlier.
• In 0.4% of years (two cases) it is 30 days earlier.
• In 0.2% of years (one case) it is 29 days later.

Thus in these 500 years over 2/3rds of new year's days in the Archetypes Calendar occur on the same CE date as Chinese new year's days, somewhat less than 1/3rd (27.6%) occur one day earlier or later, and only 0.6% occur one month earlier or later.

This study also showed that it is almost always true that a year in the Archetypes Calendar is a long year if and only if the corresponding Chinese year has 13 months. For the 500 years studied, it was true for all but six years (98.8%), namely, 1984, 1985, 2185, 2186, 2318 and 2319 CE.

In a later study the day and month numbers of dates in the two calendars for all 365,618 days over the range 4400-01-01 ARC (= 1702-01-28 CE) through 5400-12-29 ARC (= 2703-02-07 CE) were compared. It was found that when intercalary months in the Chinese Calendar are ignored (these months have month numbers followed by an asterisk, as in 79-29-4*-11 CHL) 50.78% of the days had the same day and month numbers in both calendars, for example, 79-28-09-24 CHL = 4709-09-24 ARC (= 2011-10-20 CE).

These results are interesting because the Chinese Calendar is based on the determination of the exact times of dark moons and solar terms by means of complicated astronomical calculations, whereas the Archetypes Calendar is based on fairly simple rules.