The Arithmetical or Tabular Islamic Calendar

The Islamic calendar converter on this website is based on the ḥisābi calendar, i.e. the arithmetical or tabular calendar, introduced by Muslim astronomers in the 8th century CE to predict the approximate begin of the months in the Islamic lunar calendar. This calendar is sometimes referred to as the Fātimid calendar but this is in fact one of several almost identical tabular Islamic calendars.

The months in the tabular Islamic calendar are assumed to be alternately 30 and 29 days in length resulting in a normal calendar year of 354 days (al-sanat al-basīṭa). In order to keep the calendar in step with the lunar phases every two or three years an extra day is added at the end of the year to the last month resulting in a calendar year of 355 days (al-sanat al-kabīsa).

The 30-Year Intercalation Scheme

According to the most commonly adopted method 11 intercalary days are added in every 30 years (the historical origin for this scheme is explained here).

Several slightly different intercalary schemes have been described in the literature which can be summarized as follows:

Type Intercalary years with 355 days
in each 30-year cycle
Origin/Usage
I 2, 5, 7, 10, 13, 15, 18, 21, 24, 26 & 29 Kūshyār ibn Labbān, Ulugh Beg, Taqī ad-Dīn Muḥammad ibn Maʾruf
II 2, 5, 7, 10, 13, 16, 18, 21, 24, 26 & 29 al-Fazārī, al-Khwārizmī, al-Battānī, Toledan Tables, Alfonsine Tables, MS HijriCalendar
III 2, 5, 8, 10, 13, 16, 19, 21, 24, 27 & 29 Fāṭimid calendar (also known as the Ismāʿīlī, Ṭayyibī or Bohorā calendar), Ibn al-Ajdābī
IV 2, 5, 8, 11, 13, 16, 19, 21, 24, 27 & 30 Ḥabash al-Ḥāsib, al-Bīrūnī, Elias of Nisibis

Another intercalary scheme – with intercalary years in 2, 5, 8, 10, 13, 16, 18, 21, 24, 26 & 29 – was used in a perpetual calendar inscribed on an astrolabe (CCA 127) made in 1212/13 CE (609 AH) by Muḥammad ibn Fattūḥ al-Khamāʾirī of Seville.

Of each intercalary scheme two variants are possible depending on whether the epoch of the Islamic calendar (1 Muḥarram, 1 AH) is assumed to be 15 July, 622 CE (known as the ‘astronomical’ or ‘Thursday’ epoch) or 16 July, 622 CE (the ‘civil’ or ‘Friday’ epoch).

The 8-Year Intercalation Scheme

A different scheme, based on an 8-year cycle with intercalary years in 2, 5 & 8, was used in the Ottoman Empire and in South-East Asia. Though less accurate than the 30-year cycle, this cycle was popular due to the fact that within each cycle the calendar dates fall on the same weekdays. In the Dutch East Indies the calendar was reset every 120 years by omitting the intercalary day inserted at the end of the last year.


NB: In earlier versions of this web page it was erroneously claimed that the “Kuwaiti Algorithm”, the forerunner of the present MS HijriCalendar date-conversion software, was based on type I (‘astronomical’). It was actually based on type II (‘astronomical’).

I am grateful to Yoshito Umaoka for pointing out this error.


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