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:

Scheme 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, ʿAlī al-Qūshjī, Taqī al-Dīn Muḥammad ibn Maʾrūf
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 a now lost astrolabe (IC 127) made in 1212/13 CE (609 AH) by Muḥammad ibn Fattūḥ al-Jamāʾirī 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.

Islamic Years Which Fall Within a Single Western Year

As the tabular Islamic year slowly cycles through the astronomical seasons, a given year usually partially overlaps two sequential Western years. However, as a tabular Islamic year is about 10 days shorter than a Western year, every 33 or 34 years the tabular Islamic year will completely fall within a single Western year. The following tables indicate which tabular Islamic years (computed with intercalary scheme II and adopting the ‘civil’ or ‘Friday’ epoch) fall within a single Western year.

Julian calendar
AH CE   AH CE   AH CE   AH CE
19 640   254 868   488 1095   757 1356
52 672   287 900   522 1128   790 1388
86 705   320 932   555 1160   824 1421
119 737   321 933   589 1193   857 1453
153 770   354 965   623 1226   858 1454
186 802   388 998   656 1258   891 1486
220 835   421 1030   690 1291   925 1519
253 867   455 1063   723 1323   958 1551

 

Gregorian calendar
AH CE   AH CE   AH CE   AH CE
993 1585   1127 1715   1261 1845   1396 1976
1026 1617   1161 1748   1295 1878   1429 2008
1060 1650   1194 1780   1329 1911   1463 2041
1093 1682   1228 1813   1362 1943   1496 2073

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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