Quepennura Leap Week Calendar

One cycle has seven complete sub-cycles and one incomplete sub-cycle.

One cycle has exactly 479 years = 24,993 weeks = 174,951 days. The cycle has 394 common years and 85 leap years (Below is the explanation about leap years).

The averaged year length is approximately 365.24217119 days (= 174,951 days / 479 years), and this is shorter than the present solar year (approximately 365.242189 days). The solar year is getting shorter and shorter, so the averaged year length will match the solar year in the future.

One sub-cycle has exactly 62 years = 3,235 weeks = 22,645 days. The sub-cycle has 51 common years and 11 leap years (Below is the explanation about leap years).

If we know the week number when counted from the 1st week of one sub-cycle (ranged from 1 to 3,235), we can calculate the year number when counted from the 1st year of one sub-cycle (ranged from 1 to 62) with the following formula:

• Year Number = floor[ ( 62 × WN + 3205) / 3235 ] , where WN is the week number.

Leap years occurs based on the following formula:

• Leap Year = floor[ ( 479 × LYN - 140 ) / 85 ] , where LYN means the number of the leap year when counted from 1 C.E.

The above formula can be written like the following, too.

• Leap Year = ceiling( ( 479 × LYN - 224 ) / 85 )

Oppositely, when we want to know the number of the leap year in the period from 1 C.E. to the year (The period includes the year), the following formula is useful:

• LYN = floor[ ( 85 × Y + 224 ) / 479 ] , where Y is the year.

Leap years occurs based on the following formula:

• Leap Year = floor[ ( 62 × LYN - 19 ) / 11 ] , where LYN means the number of the leap year when counted from the 1st year of the sub-cycle.

The above formula can be written like the following, too.

• Leap Year = ceiling( ( 62 × LYN - 29 ) / 11 )

Oppositely, when we want to know the number of the leap year in the period from the 1st year to the year in one sub-cycle (The period includes the year), the following formula is useful:

• LYN = floor[ ( 11 × Y + 29 ) / 62 ] , where Y is the year.

The date number here is what is counted from January 1 in the year and it is ranged from 1 to 364 in one common year and from 1 to 371 in one leap year.

The following formula is valid if and only if the date number is 339 or less.

• M = floor[ ( 11 × DN + 336 ) / 339 ] , where M is the month number (January = 1, February = 2, ... , December = 12) and DN is the date number.

The months are numbered like January = 1, February = 2, ... , December = 12. Then, we can calculate from the Month to the Date Number of the Last Day of the Previous Month with the following formula.

• DN = floor[ (339 × M - 337 ) / 11 ] , where DN is the date number and M is the month.

• Let us regard January as 1, February as 2, ... , December as 12. This is treated as Month.

• The actual date is written with Capital D (initial of Date), Capital M (initial of Month) and Capital Y (initial of Year).

• The Modified Julian Day is written with Capital J (initial of Julian).

• The small alphabets mean the following.

  • d: date number (Usually it means the date number when counted from January 1, 1 C.E. However, when it is treated in one year, it is in the range of 1 to 371.)

  • w: week number (Usually it means the week number when counted from January 1, 1 C.E. However, when it is treated in one cycle, it is in the range of 1 to 24,993, and when it is treated in one sub-cycle, it is in the range of 1 to 3,235.)

  • y: year number (Usually it means the Common Era, but when it is treated in one sub-cycle, it is in the range of 1 to 62.)

  • s: the number of the elapsed sub-cycles from the beginning year of the cycle to which the day belongs.

  • c: the number of the elapsed cycles from 1 C.E.

• It is written @cycle that something is treated in one cycle. The same is true of the sub-cycle, the year and so on.

1. Calculate the date number of the last day of the previous month when counted from January 1 in the year (= d1) based on M.

   • d1 = floor[ (339 × M - 337 ) / 11 ] ( 0 ≦ d1 ≦ 339 )

2. Calculate the date number of the last day of the previous year when counted from January 1, 1 C.E. (= d2) based on Y.

   • d2 = 364 × ( Y - 1 ) + floor[ ( 85 × ( Y - 1 ) + 224 ) / 479 ] × 7

3. Calculate the Modified Julian Day (= J)based on D, d1 and d2.

   • J = D + d1 + d2 - 678576

1. Calculate the date number of the day when counted from January 1, 1 C.E. (= d) based on the Modified Julian Day (= J).

   • d = J + 678576

2. Calculate the w based on d.

   • w = ceiling( d / 7 )

3. Calculate the c based on w.

   • c = ceiling( w / 24993 ) - 1

4. Calculate the w@cycle based on w and c.

   • w@cycle = w - 24993 × c

5. Calculate the s based on w@cycle.

   • s = ceiling( w@cycle / 3235 ) - 1

6. Calculate the w@sub-cycle based on w@cycle and s.

   • w@sub-cycle = w@cycle - 3235 × s

7. Calculate the y@sub-cycle based on w@sub-cycle.

   • y@sub-cycle = floor[ ( 62 × w@sub-cycle + 3205 ) / 3235 ]

8. Calculate the year (= Y) based on c, s and y@sub-cycle.

   • Y = 479 × c + 62 × s + y@sub-cycle

9. Calculate the d@year based on d (what was calculated in Step 1) and Y.

   • d@year = d - ( 364 × ( Y - 1 ) + floor[ ( 85 × ( Y - 1 ) + 224 ) / 479 ] × 7 )

10. If d@year is 339 or less, then calculate the month (= M) based on d@year. If not, M is 12 (or December).

    • M = floor[ ( 11 × d@year + 336 ) / 339 ]

11. Calculate the date (= D) based on d@year and M.

    • D = d@year - floor[ (339 × M - 337 ) / 11 ]