Two Integral-Week Solar Calendars
by Peter Meyer

### 1. Definition of Terms

This article (first published 2001-02-03 CE) introduces two new solar calendars.

A solar calendar is a calendar consisting of years which accord with seasonal (or tropical) years. The new year of a solar calendar always begins at a more-or-less constant temporal offset from the time of the vernal equinox (usually 21 March in the Gregorian Calendar).

An integral-week calendar is a calendar in which the years always have an integral number of weeks, where weeks consist of a certain fixed number of days, not necessarily seven. In an integral-week calendar all days are part of some week.

In this article I will present two integral-week solar calendars, one based on a week of 5 days and one based on a week of 6 days.

### 2. Definition of the Integral Five-Day-Week Calendar

The Integral Five-Day-Week Calendar has years which consist of an integral number of 5-day weeks.

Years are of two kinds, "normal" and "long".

A normal year has seventy-three 5-day weeks, whereas a long year has seventy-four 5-day weeks. Thus the number of days in a year is 365 for normal years and 370 for long years.

Years are numbered using the astronomical convention: ..., -2, -1, 0, 1, 2, ...

By definition, a year in the Integral Five-Day-Week Calendar is a long year if and only if the year number, mod 165, is exactly divisible by 21. To explain: Suppose Y is a positive year number, then Y mod 165 is the remainder after Y is divided by 165. E.g., if Y = 1218 then Y = 7*165 + 63, so Y mod 165 is 63. In this case the year is a long year because 63 is exactly divisible by 21.

In applying this rule for negative years the mathematical definition of mod applies (not the "remainder"). E.g., for year number -281, -281 = 49 mod 165 because -281 = -2*165 + 49. In this case the year -281 is not a long year because 49 is not exactly divisible by 21.

Years in the Integral Five-Day-Week Calendar are divided into "quarters". The quarters are named simply "First Quarter", "Second Quarter", "Third Quarter" and "Fourth Quarter".

In a normal year the second quarter consists of nineteen 5-day weeks and the other quarters consists of eighteen 5-day weeks. In a long year the first and third quarters consist of eighteen 5-day weeks and the second and fourth quarters consist of nineteen 5-day weeks.

 Quarter 5-day weeks innormal year 5-day weeks inlong year First 18 18 Second 19 19 Third 18 18 Fourth 18 19

### 3. Definition of the Integral Six-Day-Week Calendar

The Integral Six-Day-Week Calendar has years which consist of an integral number of 6-day weeks.

Years are of two kinds, "normal" and "short".

A normal year has sixty-one 6-day weeks, whereas a short year has sixty 6-day weeks. Thus the number of days in a year is 366 for normal years and 360 for short years.

Years are numbered using the astronomical convention: ..., -2, -1, 0, 1, 2, ...

By definition, a year in the Integral Six-Day-Week Calendar is a short year if and only if the year number, mod 198, is exactly divisible by 8. To explain: Suppose Y is a positive year number, then Y mod 198 is the remainder after Y is divided by 198. E.g., if Y = 1236 then Y = 6*198 + 48, so Y mod 198 is 48. In this case the year is a short year because 48 is exactly divisible by 8.

As for the 5-day calendar rule, in applying this rule for negative years the mathematical definition of mod applies (not the "remainder"). E.g., for year number -553, -553 = 41 mod 198 because -553 = -3*198 + 41. In this case the year -553 is not a short year because 41 is not exactly divisible by 8.

As with the Five-Day-Week Calendar, years in the Integral Six-Day-Week Calendar are divided into "quarters" and are named "First Quarter", "Second Quarter", etc..

In a normal year the first three quarters consists of fifteen 6-day weeks and the fourth quarter consists of sixteen 6-day weeks. In a short year all four quarters consist of fifteen 6-day weeks.

 Quarter 6-day weeks innormal year 6-day weeks inshort year First 15 15 Second 15 15 Third 15 15 Fourth 16 15

### 4. Accuracy as Solar Calendars

It will now be shown that in both of these calendars the average length of the calendar year is exactly equal to the vernal equinox year, 365.2424 days (to four decimal places).

In the Integral Five-Day-Week Calendar the rule for long years implies that the structure of the calendar repeats after 165 years. In any period of 165 consecutive years there are exactly 8 long years, namely, those years with year number Y such that Y mod 165 is 0, 21, 42, 63, 84, 105, 126 or 147. In a normal year there are 365 days and in a long year there are 370 days, so in 165 years there are 8*370 + (165-8)*365 = 60,265 days, so the average length of the calendar year is 60,265/165 = 365.2424 days.

In the Integral Six-Day-Week Calendar the rule for short years implies that the structure of the calendar repeats after 198 years. In any period of 198 consecutive years there are exactly 25 short years, namely, those years with year number Y such that Y mod 198 is 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184 or 192. In a normal year there are 366 days and in a short year there are 360 days, so in 198 years there are 25*360 + (198-25)*366 = 72,318 days, so the average length of the calendar year is 72,318/198 = 365.2424 days.

### 5. Divisions of the Year in the Integral Six-Day-Week Calendar

The fact that in the Integral Six-Day-Week Calendar a normal year consists of 61 weeks and a short year of 60 weeks allows for convenient divisions of the year into approximately equal parts in several ways as follows:

• Short years may be divided into two halves of 30 6-day weeks each; normal years into halves of 30 and 31 6-day weeks.

• Short years may be divided into three thirds of 20 6-day weeks each; normal years into thirds of 20, 20 and 21 6-day weeks.

• Short years are divided into four quarters of 15 6-day weeks each; normal years into quarters of 15, 15, 15 and 16 6-day weeks.

• Short years may be divided into five fifths of 12 6-day weeks each; normal years into fifths of 12, 12, 12, 12 and 13 6-day weeks.

• Short years may be divided into six sixths of 10 6-day weeks each; normal years into sixths of 10, 10, 10, 10, 10 and 11 6-day weeks.

• Short years may be divided into ten tenths of 6 6-day weeks each; normal years into nine tenths of 6 6-day weeks each and a final tenth of 7 6-day weeks.

• Short years may be divided into 15 fifteenths of 4 6-day weeks each; normal years into 14 fifteenths of 4 6-day weeks each and a final fifteenth of 5 6-day weeks.

• Short years may be divided into 20 twentieths of 3 6-day weeks each; normal years into 19 twentieths of 3 6-day weeks each and a final twentieth of 4 6-day weeks.

• Short years may be divided into 30 thirtieths of 2 6-day weeks each; normal years into 29 thirtieths of 2 6-day weeks each and a final thirtieth of 3 6-day weeks.

The Integral Five-Day-Week Calendar does not permit such convenient divisions of the year. For this and other reasons the Integral Six-Day-Week Calendar is superior to the Integral Five-Day-Week Calendar.

### 6. Date Notation

In each of these calendars dates are written in the form year-quarter-week-day. Dates in the Integral Five-Day-Week Calendar and in the Integral Six-Day-Week Calendar are identified by "IF" and "IS" respectively. Thus, e.g., 1077-4-18-5 IF and 2911-4-15-6 IS.

For the IS Calendar an extended date notation is possible, based on the divisions of the year as given above. This takes the form year-numerator/denominator-week-day, where denominator is 2, 3, 4, 5, 6, 10, 15, 20 or 30, indicating respectively a half-year, third-year, quarter-year, ..., thirtieth-year, and numerator is the number of the unit (half-year, third-year, quarter, etc.) within this division of the year.

For the IS Calendar the notation year-quarter-week-day is equivalent to year-quarter/4-week-day, e.g., 1001-4-16-6 IS = 1001-4/4-16-6 IS. For the other divisions of the year (1001 is a normal year and so has 61 6-day weeks) this date is equivalent to 1001-2/2-31-6 IS, 1001-3/3-21-6 IS, 1001-5/5-13-6 IS, 1001-6/6-11-6 IS, 1001-10/10-07-6 IS, 1001-15/15-05-6 IS, 1001-20/20-04-6 IS and 1001-30/30-03-6 IS.

### 7. Correlation with Empirical Time

The first day of the first week of the first quarter of the first year of the first millennium in the Integral Six-Day-Week Calendar, 0-1-01-1 IS, coincides with Julian day number 969,834, which is -2057-03-20 CE. Thus the year 2001 CE corresponds to the year 4058 in this calendar. The Julian day number of 0-1-01-1 IF is still under consideration.

New Year's Day in the IS Calendar thus tends to occur in the vicinity of the vernal equinox, and First Quarter will coincide approximately with March-June, Second Quarter with June-September, Third Quarter with September-December and Fourth Quarter with December-March.