Calendar Wiki

Reference and cycles[]

This describes a method to track the solar, lunar and week cycle using a single reference.

The reference has a start date and a cycle. It advances with 30-day months but after four years (48 months), six days are added to the last month. That makes a cycle of 1446 days.

The lunar cycle can be tracked using a series of alternating 30 and 29-day lunar months. If after four years a month of 31 days is added, it makes a total of 1447 days in 49 months.

The week cycle can be tracked using 91-day trimesters (13 weeks, 3 months). Four years of four trimesters add up to 1456 days.

The solar cycle can be tracked using the familiar 365-day year of 12 months with a leap day every four years. The total number of days for four years is 1461 days.

Bar chart of the months in the cycles, shown next to each other

The reference and the three cycles

Tracking method[]

As seen above, the different calendars roughly synchronise after four years.

The lunar cycle gains one day over the reference every fourth year.

The week cycle gains ten days over the reference every fourth year.

The solar cycle gains a 30-day month over the reference every eighth year.

To convert between calendars, compute the position of the desired date in relation to the reference then convert it back to the target calendar.

Note that events such as a skipped leap day or an added leap week can complicate the calculation, so it is recommended to prepare a suitable reference start date as needed.

Proposed calendar[]

I propose a simple calendar based on the method discussed before.


The reference start date is Monday, 1 January 2001. It falls on 5 of Shawwāl in the islamic calendar and on 6 of Tevet in the jewish calendar. All months have 30 days except for the last one in a four year cycle.

Reference month lengths
01 02 03 04 05 06 07 08 09 10 11 12
Year 1 30 30
Year 2
Year 3
Year 4 36

Lunar date[]

The lunar month is about 29.5306 days long on average, although it can vary considerably. The Method One calendar simply alternates between 30-day and 29-day lunar months with a 31-day month added as the 49th in a four year cycle.

Lunar month lengths
01 02 03 04 05 06 07 08 09 10 11 12 13
Year 1 30 29 30 29 30 29 30 29 30 29 30 29
Year 2
Year 3
Year 4 31

There are six 29-day months in a year, so the months regress by six days every year compared to the reference.

Week date[]

A calendar year of 365 or 366 days does not contain an exact number of weeks. That usually means that a 53rd week is regularly added to the year to realign the week year with the solar year.

As the Method One calendar provides its own method to convert between week date and solar date, it doesn't bother with such subtlety. Every week year is exactly 52 weeks long.

A week year is divided in four trimesters of 13 weeks each. Three months are defined for each trimester as follows:

Months in a trimester
Length in days Start day End day
Month 1 33 Monday Friday
Month 2 29 Saturday Saturday
Month 3 29 Sunday Sunday

All trimesters are equal.

A date can be given as year, month and day or as year, trimester (or quarter, 1 to 4), week (1 to 13) and day of the week (Monday=1 to Sunday=7).

Since the reference start date is a Monday, every year starts on a Monday. Note that it would also be possible to choose a Sunday as start date (month lengths 29, 33, 29, Sunday=0).

There is one day added every trimester, so the calendar moves ahead by four days compared to reference each year.

Solar date[]

The solar cycle is the same as the Julian calendar except that the leap day is moved from the 29 February to the 31 December. All years have a 29 February but only one in four have a 31 December.

Solar month lengths
01 02 03 04 05 06 07 08 09 10 11 12
Year 1 31 29 31 30 31 30 31 31 30 31 30 30
Year 2
Year 3
Year 4 31

That change should make it easier to handle leap days.

There are six longer months and one shorter one, so the months advance by five days every non-leap year compared to the reference.

Example of Date Tracking[]

Now that we have an actual calendar defined we can start using the method with it.

In this example I will track the current lunar month for today's date, 22 May 2024 at the time of this writing.

Reference Position[]

First we want to locate that date in the reference.

The date we have is expressed in the Gregorian solar calendar system. The Method One solar calendar is mostly the same, but it may require a small adjustment because February has 29 days always.

In our case year 2024 is a leap year so that issue is out of the way.

The months in the solar calendar vary in length and we can use the following table to adjust the day to the 30-day a month reference:

Month length adjustment
Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
= +1 = +1 +1 +2 +2 +3 +4 +4 +5 +5

For the month of May add one to the day, which makes it 23.

Then we should look which four-year cycle it falls in. The start date of Method One calendar is 1 January 2001, so that is the 6th cycle (01-04, 05-08, 09-12, 13-16, 17-20, 21-24). There are five completed cycles before it.

The solar calendar advances by one month in eight years. In five cycles that means it got ahead of the reference by two months and 15 days.

In addition, 2024 is the 4th year in the cycle and the solar calendar advances by 5 days per year. There are 3 years completed which makes another 15 days, for a total of 3 months ahead.

May is the 5th month, add three and we find the position in the reference: Y24-M08-D23.

Lunar Calendar Month[]

In five cycles the lunar calendar got ahead of the reference by five days.

Then each year after that, it went back by six days. At this point the tally of days is a lunar year starting with 13 days deficit. If we add the missing days to 23 it gives 36, which pushes the day to the next lunar month. Note that if the target calendar was ahead instead of behind we would subtract the days, not add them.

The number "6" is small enough that the effect of 29-day lunar months versus 30-day reference months will not move the day to a different month, so we know that it falls one month greater than the reference: 9th month of 4th year in the 6th cycle.

Remember that a Method One cycle of four years includes a 49th lunar month. In the islamic calendar which has strictly 12 months per year you need to add one month per cycle: 9 + 5 = 14 ~ in this case it pushes the month to the next lunar year. The Method One calendar starts in Shawwāl which makes the second month of the lunar year Dhū al-Qaʿdah.

In the jewish calendar we already know that the date falls in Iyar or in Sivan since it follows the seasons (lunisolar calendar).

In the 19 years from the start date, the following were leap years:

Leap years
2003 2005 2008 2011 2014 2016 2019

This cycle of seven leap years repeats every 19 years which implies that any multiple of 19 years added to these years is leap.

We have 2024 = 2005 + 19, so there were eight leap months added by the jewish calendar before and 2024 itself is a leap year.

We saw that there were five completed cycles in the Method One calendar which means five additional months. The difference is three months so Tevet should correspond to the 4th Method One lunar month. The Method One lunar month we want is 9, which on a leap year means:

Tevet Shevat Adar I Adar II Nisan Iyar
4 5 6 7 8 9

The date 22 May 2024 falls in Iyar.