Tabular Islamic Calendar - Intercalary Schemes

Intercalary Schemes

There are 11 leap years in a 30 year cycle. Noting that the average year has 354 11/30 days and a common year has 354 days, at the end of the first year of the 30-year cycle the remainder is 11/30 day. Whenever the remainder exceeds a half day (15/30 day), then a leap day is added to that year, reducing the remainder by one day. Thus at the end of the second year the remainder would be 22/30 day which is reduced to −8/30 day by a leap day. Using this rule the leap years are

2, 5, 7, 10, 13, 16, 18, 21, 24, 26 and 29

of the 30-year cycle. If leap days are added whenever the remainder equals or exceeds a half day, then all leap years are the same except 15 replaces 16.

The Ismaili Tayyebi community uses the following order of leap years in their 30-year cycle.

2, 5, 8, 10, 13, 16, 19, 21, 24, 27 and 29

Apart from these, there are two more orders for the leap years which are as follows

2, 5, 7, 10, 13, 15, 18, 21, 24, 26 and 29

2, 5, 8, 11, 13, 16, 19, 21, 24, 27 and 30

The mean month is 29 191/360 days = 29.5305555... days. This is slightly too short and so will be a day out in about 2500 years. The Tabular Islamic calendar also deviates from the observation based calendar in the short term for various reasons.