Editing Tabular Islamic calendar (section)

==Intercalary schemes==

=== 30-year cycle ===
In its most common form there are 11 leap years in a 30-year cycle. Noting that the average year has {{frac|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 {{frac|11|30}} day. Whenever the remainder exceeds a half day ({{frac|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 {{frac|22|30}} day which is reduced to −{{frac|8|30}} day by a leap day. Using this rule the leap years are year number 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 as the sixth long year per cycle.

The Ismaili Tayyebi community delays three leap days by one year: the third to year 8, the seventh to year 19 and the tenth to year 27 in their 30-year cycle.
There is another version where, in addition, the fourth leap day is postponed to year 11 and the last leap day is in the last year of the 30-year cycle.

The mean number of days per month in the 30-year cycle is 29.53056 days, or {{nowrap|29d 12h 44m.}} Six months of 29 days and six with 30 days, plus 11 days of the leap years. (29 days × 6 months + 30 days × 6 months) × 30 years + 11 leap days = 10,631 days and 10,631 / 360 = 29.53056 (360 is number of months in 30 years). And this is approximately how long it takes for the moon to make full lunar cycle. 

{{anchor|Kuwaiti algorithm}}
[[Microsoft]]'s '''Kuwaiti algorithm''' is used in [[Windows]] to convert between [[Gregorian calendar]] dates and [[Islamic calendar]] dates.<ref>[http://www.microsoft.com/globaldev/DrIntl/columns/002/default.mspx#EAD Hijri Dates in SQL Server 2000] from Microsoft [https://web.archive.org/web/20080209173858/http://www.microsoft.com/globaldev/DrIntl/columns/002/default.mspx Archived Page] {{webarchive |url=https://web.archive.org/web/20100108044902/http://www.microsoft.com/globaldev/DrIntl/columns/002/default.mspx#EAD |date=January 8, 2010 }}</ref><ref>Kriegel, Alex, and Boris M. Trukhnov. SQL Bible. Indianapolis, IN: Wiley, 2008. Page 383.</ref> There is no fixed correspondence defined in advance between the algorithmic Gregorian [[solar calendar]] and the Islamic [[lunar calendar]] determined by actual observation.  As an attempt to make conversions between the calendars somewhat predictable, Microsoft claims to have created this [[algorithm]] based on statistical analysis of historical data from [[Kuwait]]. According to Rob van Gent, the so-called "Kuwaiti algorithm" is simply an implementation of the standard Tabular Islamic calendar algorithm used in Islamic astronomical tables since the 11th century.<ref name="gent">{{cite web |title= Online Calendar Converters Based on the Tabular Islamic Calendar |url = https://webspace.science.uu.nl/~gent0113/islam/islam_tabcal_others.htm | author= Robert Harry van Gent | publisher= Mathematical Institute, [[Utrecht University]] | date= December 2019 | access-date=15 November 2020 |quote=It can easily be demonstrated that the so-called 'Kuwaiti Algorithm' was based on the standard arithmetical scheme (type IIa) which has been used in Islamic astronomical tables since the 8th century CE.}}</ref>
{| class="wikitable"
|+ Overview of various 30-year leap cycles
!rowspan=2| Origin or usage
! colspan=11| Long lunar years (''n''th/30)
! colspan=11| Gap till next long year
|-
! #1 || #2 || #3 || #4 || #5 || #6 || #7 || #8 || #9 || #10 || #11
! #1 || #2 || #3 || #4 || #5 || #6 || #7 || #8 || #9 || #10 || #11
|-
| [[Kūshyār ibn Labbān]], [[Ulugh Beg]], Taqī ad-Dīn Muḥammad ibn Maʾruf
|rowspan=5| 2 ||rowspan=5| 5 ||rowspan=2| 7 ||rowspan=4| 10 ||rowspan=5| 13 || 15 ||rowspan=3| 18 ||rowspan=5| 21 ||rowspan=5| 24 ||rowspan=3| 26 ||rowspan=4| 29 
| 3 || 2 ||colspan=2| 3 || 2 ||colspan=3| 3 || 2 ||colspan=2| 3
|-
| al-Fazārī, [[al-Khwārizmī]], [[al-Battānī]], [[Toledan Tables]], [[Alfonsine Tables]], Microsoft "Kuwaiti algorithm"
|rowspan=4| 16 
| 3 || 2 ||colspan=3| 3 || 2 ||colspan=2| 3 || 2 ||colspan=2| 3
|-
| Muḥammad ibn Fattūḥ al-Jamāʾirī of Seville
|rowspan=3| 8 
|colspan=2| 3 || 2 ||colspan=2| 3 || 2 ||colspan=2| 3 || 2 ||colspan=2| 3
|-
| Fāṭimid / [[Ismaili|Ismāʿīlī]] / [[Taiyabi Ismaili|Ṭayyibī]] / Bohorā calendar, [[Ibn al-Ajdābī]]
|rowspan=2| 19 ||rowspan=2| 27 
|colspan=2| 3 || 2 ||colspan=3| 3 || 2 ||colspan=2| 3 || 2 || 3
|-
| Ḥabash al-Ḥāsib, [[al-Bīrūnī]], [[Elias of Nisibis]]
| 11 || 30 
|colspan=3| 3 || 2 ||colspan=2| 3 || 2 ||colspan=3| 3 || 2 
|}

{{anchor|Other versions}}

=== 8-year cycle ===
Tabular Islamic calendars based on an 8-year cycle (with 2, 5 and 8 as leap years) were also used in the Ottoman Empire and in South-East Asia.<ref>Ian Proudfoot, ''Old Muslim Calendars of Southeast Asia'' (Leiden: Brill, 2006 [= ''Handbook of Oriental Studies'', Section&nbsp;3, vol.&nbsp;17]).</ref> The cycle contains 96 months in 2835 days, giving a mean month length of 29.53125 days, or 29d 12h 45m.

Though less accurate than the tabular calendars based on a 30-year cycle, it was popular due to the fact that in each cycle the weekdays fall on the same calendar date. In other words, the 8-year cycle is exactly 405 weeks long, resulting in a mean of exactly 4.21875 weeks per month.

=== 120-year cycle ===
In the [[Dutch East Indies]] (now Indonesia) into the early 20th century, the 8-year cycle was reset every 120 years by omitting the intercalary day at the end of the last year, thus resulting in a mean month length equal with that used in the 30-year cycles.<ref>Gerret Pieter Rouffaer, "[https://socrates.leidenuniv.nl/view/item/1083977 Tijdrekening]", in: ''Encyclopaedie van Nederlandsch-Indië'' (The Hague/Leiden: Martinus Nijhoff/E.J.&nbsp;Brill, 1896–1905), vol.&nbsp;IV, pp.&nbsp;445–460 (in Dutch).</ref>