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A leap cycle of 400 years contains 97 leap days or 71 leap weeks and thus overall 146097 days and 20871 weeks or 4800 twelfth-of-a-year months, spanning roughly 4947.3 lunations. Its mean year length is 365.2425 days or 365 days, 5 hours, 49 minutes and 12 seconds. It is the cycle of the Gregorian calendar and thus the International Standard Calendar from ISO 8601. Many reform proposals keep it either explicitly or implicitly, e.g. the World Calendar, the International Fixed Calendar, the Positivist Calendar, the Pax Calendar, the Hanke-Henry Permanent Calendar and even the French Republican calendar.

There are different approaches possible to define when two leap days are either five or eight years apart instead of the usual four. Alternatively, the distribution of leap weeks can be made more regular as a matter of convention. This may reduce jitter.

Rules for leap days[]

To distribute 97 additional days over a span of 400 years, one would either distribute them uniformly or define a simple algorithm. The leap day would need to occur about every 4.1237th year or roughly once every 1506 days or every 215 weeks or every 49 to 50 months.

97400 = 0.2425 =
= 1004003400 = 0.25 − 0.0075
= 143400
Gregorian calendar and derivatives
= 141100 + 1400
= 96400 + 1400 = 0.24 + 0.0025
= 24100 + 1400
= 1250 + 1400
= 625 + 1400
= 15 + 125 + 1400
= 95400 + 2400 = 0.2375 + 0.005
= 1980 + 1200
= 14180 + 1200
= 1980 + 11001200
= 90400 + 7400 = 0.225 + 0.0175
= 940 + 7400
= 14140 + 7400
= 14140 + 180 + 1200
= 15 + 17400 = 0.2 + 0.0425
= 15 + 1203400
= 15 + 1201100 + 1400
= 15 + 380 + 1200
= 15 + 120180 + 1200
97400 = 0.2425 =
modified Dee rule (33-year sub-cycle)
= (11 × 8 + 9)400 =
= (363 × 833 + 37 × 937) / 400
= (12 × 8 + 1)400 =
= (396 × 833 + 4 × 14) / 400
modified Mädler rule (128-year sub-cycle)
= (3 × 31 + 4)400 =
= (384 × 31128 + 16 × 14) / 400

4:100:400[]

The Gregorian leap rule is using Olympiads with exceptions in 3 out of 4 centennial years, leading to 8-year gaps between leap days. With general rules like the one specified in ISO 8601, this leads to an irregular pattern of leap weeks spread throughout the cycle, which may even make leap weeks seven years apart instead of the usual five or six.

There are some calendars in use that are designed to work well with the Gregorian calendar, e.g. the Thai solar calendar, the Indian national calendar and the Holocene calendar, but the modulo-4 rules for finding leap years only works transparently in them if their era is offset by an integer multiple of 400 years. This is the case with the Holocene era which starts 10000 years earlier, but it is not the case for the Buddhist eras common particularly in Thailand (+543), but also in India and Cambodia (+78, Saka) and Burma (+638).

Rules for leap weeks[]

To distribute 71 additional weeks over 400 years, one would either distribute them uniformly or define a simple algorithm. Most calendar designs have a fixed position, often at the end of the year, for the leap item, but it is also possible to intercalate it at a fixed interval, in this case c. 5.6338 years are almost 2056 days or about 294 weeks on average, or between 67 and 68 months. For an algorithmic rule, one finds a simple fraction like one fifth and adds or substracts fractions with larger numerator and small denominator as necessary; ideally the larger numerators are multiples of the smaller ones if their terms are to be substracted, otherwise they should be independent:

71400 = 0.1775 =
every eighth year with additions
= 50400 + 21400 = 0.125 + 0.0525
= 18 + 21400
= 18 + 380 + 3200
= 18 + 120 + 1400
= 18 + 3409400
= 18 + 34021001400
= 18 + 112001400
= 18 + 122003400
= 18 + 3503400
= 18 + 244001100 + 1400
= 18 + 61001100 + 1400
= 64400 + 7400 = 0.16 + 0.0175
= 16100 + 21001400
= 850 + 84001400
= 425 + 1501400
= 65400 + 6400 = 0.1625 + 0.015
= 1380 + 3200
= 1680380 + 3200
= 15380 + 3200
= 68400 + 3400 = 0.17 + 0.0075
= 17100 + 3400
Colligan Pax, YY % 6 = 0 (i.e. C00, C06, C12, C18, C24, C30, C36, C42, C48, C54, C60, C66, C72, C78, C84, C90, C96) and C99, except 400
= 17100 + 11001400
= 70400 + 1400 = 0.175 + 0.0025
= 740 + 1400
= 540 + 240 + 1400
= 18 + 120 + 1400
= 724001400 = 0.18 − 0.0025
= 181001400
= 9501400
= 754004400 = 0.1875 − 0.1
= 3161100
every fifth year with exceptions
= 804009400 = 0.2 − 0.0225
McCarty Weekdate, omits years 035, 085, 125, 170, 210, 255, 300, 345, 390
= 159400
Searle, omits C00, C50 and 375
= 152100 - 1400
Woods Jubilee, omits C25 and C75 and 000
= 151501400
= 1510400 + 1400 = 0.2 − 0.025 + 0.0025
McClenon C&T, Reich New Earth
= 15140 + 1400
= 15140 + 1801100
= 151801100
every sixth year with additions
= 2001200 + 131200 = 0.1_6 + 0.0108_3
= 16 + 131200
= 16 + 121200 + 11200
= 16 + 11001200 + 1300 + 1400
= 16 + 1100 + 13001400
= 16 + 1100 + 13001400
= 16 + 16120031200
= 16 + 1751400
= 16 + 15120021200
= 16 + 1801100 + 1120
= 16 + 1801120 + 1150
= 16 + 1801100 + 41203120
= 16 + 130140 + 1801100
= 16 + 630530140 + 1801100
every fourth year (Olympiad) with exceptions
= 10040029400 = 0.25 − 0.0725
every Julian leap year with 29 exceptions
= 1429400
every Gregorian leap year with 26 exceptions
= 14110013200 + 1400

5:40:400[]

The 5:40:400 leap week rule probably first used by McClenon for (CC)C&T and later by Reich for the New Earth Calendar is particularly remindful of the Gregorian 4:100:400 leap day rule: A year whose number is divisible by 5 has 53 weeks, except when it is also divisible by 40 unless also divisible by 400.

Considering a usual four-digit year number YMYCYDYY = CCYY,

  • for the first part, only the final digit YY needs to be inspected, because it has to be either 0 or 5,
  • for the second, exceptional part, the century and decade digits YCYD are relevant as the latter YD must be even, more specifically 2 or 6 if the century YC is odd and 0, 4 or 8 if it’s even,
  • for the final caveat, the the millennium and century digits YMYC must be treated in exactly the same way.

Therefore, this rule could even be implemented within regular expressions in text processing, without requiring actual arithmetics.

This rule is sometimes erroneously attributed to McCarty for Weekdate, perhaps due to confused names with “Mc”.

Its base cycle of 5 years has no dedicated name, unlike the 4-year Olympiad found in the Julian calendar among others.

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