The Tabular Islamic Calendar: 30-Year Cycles and Leap Days

The tabular Islamic calendar (also called the arithmetical Islamic calendar) is a rule-based version of the Hijri calendar. Its months alternate between 30 and 29 days, and 11 years in every 30-year cycle get one extra day. No sighting or astronomical calculation is needed for any particular month: every date follows from simple arithmetic. Astronomers, historians and software use it to convert dates, but it is not how Ramadan or Eid are announced.

Key Takeaways

• A tabular year has 354 days, or 355 in a leap year. The extra day goes at the end of Dhu al-Hijjah.
• Each 30-year cycle has 11 leap years and exactly 10,631 days, giving an average month that is less than 3 seconds shorter than the moon's mean cycle.
• Van Gent lists four main 30-year leap-year schemes (plus rarer ones, such as an 8-year cycle), and each can be counted from a "Thursday" or a "Friday" epoch in July 622 CE.
• Tabular dates are approximations. In our comparison they often fall one day away from the calculated Umm al-Qura calendar.

How the tabular Islamic calendar works

The rules fit in a few lines. The US Naval Observatory's Islamic calendar page describes the tabular calendar as one "established for some non-religious purposes." Month lengths alternate between 29 and 30 days, and the calendar runs in a 30-year cycle in which 11 years are leap years. In a leap year, the last month, Dhu al-Hijjah, gets a 30th day.

That gives 354 days in a common year and 355 in a leap year. In a real, observed calendar, a month's length depends on the moon each time, so Ramadan can have 29 days and Sha'ban 30. The tabular calendar ignores that and fixes the pattern in advance. The sister post on Hijri year lengths covers the observed side of this.

Why 11 leap years in 30

The moon's average cycle from new moon to new moon is about 29.53 days, not 29.5. Twelve months of alternating 30 and 29 days add up to 354 days, which runs short of twelve lunations by roughly 0.367 of a day. Over 30 years that shortfall grows to about 11 days, so adding 11 days per cycle keeps the calendar in step with the moon.

R.H. van Gent of Utrecht University sets out the arithmetic on his page about the origin and mathematics of the 30-year cycle:

• One cycle has 19 years of 354 days and 11 years of 355 days, a total of 10,631 days.
• That is 1,518 weeks and five days, so weekdays only repeat exactly after seven cycles (210 years).
• The average month is 10,631 ÷ 360 = 29.530555... days, or 29 days 12 hours 44 minutes.
• The true mean lunation he gives is 29 days 12 hours 44 minutes 2.80 seconds, so the tabular month is less than 3 seconds too short.

Van Gent writes that it will take "several millennia" before the accumulated error exceeds one day. Working from his figures, 2.8 seconds per month adds up to one day after about 30,900 months, or roughly 2,570 years.

The Quran ties the moon's phases to counting time. In Surah Yunus 10:5, a translation of the meaning (The Clear Quran on quran.com) says the phases are there "so that you may know the number of years and calculation ˹of time˺." The tabular calendar is one way astronomers turned that monthly count into a fixed table. It does not replace the sighting or the authority that communities rely on for religious dates.

The four leap-year schemes

Choosing which 11 years get the extra day is where the variants come from. Van Gent's page on the variants of the arithmetical Islamic calendar lists four main patterns used by medieval astronomers and later software. In the table, bold marks the leap years that differ from scheme II, the pattern most software uses:

The differences are small. Every scheme spreads 11 leap days across 30 years, and they disagree only on whether a particular extra day falls a year earlier or later. The origin page shows that each scheme comes from the same fraction (11/30 of a day per year), with a different threshold for when the accumulated fraction becomes a leap day.

Van Gent also describes an 8-year cycle with leap years 2, 5 and 8, used in the past in the Ottoman Empire and South-East Asia. It was easier to use because dates repeat on the same weekdays every cycle, but it is less accurate than the 30-year cycle.

A worked example: is 1448 AH a leap year?

To find a year's place in the cycle, divide by 30 and take the remainder (a remainder of 0 means year 30).

• 1447 AH: 1447 = 48 × 30 + 7, so it is year 7 of its cycle. Year 7 is a leap year in schemes I and II, but not in III or IV.
• 1448 AH: 1448 = 48 × 30 + 8, so it is year 8. That is a leap year in schemes III and IV, but not in I or II.

So in the scheme II calendar used by most software, Dhu al-Hijjah 1447 had 30 days and Dhu al-Hijjah 1448 will have 29. In a scheme III calendar, it is the other way round.

Two epochs: Thursday or Friday

Every scheme also needs a starting day, and there are two. Van Gent's variants page gives 15 July 622 CE (Julian calendar) as the "astronomical" or "Thursday" epoch and 16 July 622 CE as the "civil" or "Friday" epoch.

The choice shifts every date by exactly one day: a date counted from the Thursday epoch always falls one day earlier than the same date counted from the Friday epoch. The Unicode CLDR calendar list, a shared standard for calendar data in software, shows this clearly. It defines islamic-civil and islamic-tbla with the same scheme II leap years. The only difference is that islamic-civil uses the civil epoch and islamic-tbla the astronomical one.

A full count shows how the pieces fit. From the civil epoch to 1 Muharram 1448 there are 1,447 complete years: 48 full cycles (48 × 10,631 = 510,288 days) plus 7 years containing three leap years (7 × 354 + 3 = 2,481 days). That is 512,769 days in all, which lands on 17 June 2026. With the astronomical epoch the same count lands on 16 June 2026.

How the tabular calendar compares with Umm al-Qura

The tabular calendar is not the calendar Saudi Arabia publishes. Van Gent describes the Umm al-Qura calendar as a calculated lunar calendar computed with astronomical algorithms for the location of the Ka'ba. Its month lengths follow the actual moon, so they do not alternate neatly.

We computed the first day of each month of 1448 AH with the scheme II leap years, counting days from the Julian Day Numbers of the two epochs (1948439 for the Thursday epoch, 1948440 for the Friday epoch), and checked the results against the islamic-tbla and islamic-civil calendars in the ICU library. The Umm al-Qura column comes from the Umm al-Qura tables in the open-source hijri-converter library, the calculated data this site uses. Gregorian dates are the daytime dates. Each Islamic date begins at sunset the evening before.

The astronomical-epoch column is always exactly one day earlier than the civil-epoch column, but neither matches Umm al-Qura all year. On 9 November 2026, for example, Umm al-Qura and the astronomical-epoch tabular calendar both give 29 Jumada al-Awwal 1448, while the civil-epoch tabular calendar gives the 28th.

Over a longer span the pattern holds. Using the same method and the same hijri-converter Umm al-Qura data, across the 960 months from 1420 to 1499 AH, the civil-epoch tabular calendar starts the month on the same day as Umm al-Qura 425 times, one day later 495 times, two days later 25 times and one day earlier 15 times. The astronomical-epoch version matches 495 times and is within one day in 945 of the 960 months.

Neither calendar reports a sighting. Local announcements of Ramadan and Eid can differ from both by a day, which the post on why Hijri converters disagree explains in more detail.

Where the tabular calendar is used

Software. Microsoft's .NET HijriCalendar class states that years 2, 5, 7, 10, 13, 16, 18, 21, 24, 26 and 29 of each 30-year cycle are leap years, which is scheme II. The same documentation notes that Ramadan officially starts and ends by a decree based on observation, and offers a HijriAdjustment setting to shift dates by up to two days. Systems built on Unicode CLDR offer islamic-civil and islamic-tbla alongside islamic-umalqura.

History and astronomy. Medieval astronomical tables used the 30-year cycle, and van Gent notes they were often laid out for 210 years so the weekdays would repeat. His tabular calendar converter offers most published schemes for converting historical dates and describes the results as approximate Western equivalents. That suits historical work, where the actual sighting record for a given month is usually unknown.

Community calendars. Van Gent lists scheme III under the Fatimid calendar, which he notes is also known as the Isma'ili, Tayyibi or Bohra calendar.

Not an announcement of religious dates. The USNO page notes the tabular calendar serves non-religious purposes, and Microsoft's documentation likewise says Ramadan starts and ends by decree based on observation of the moon. Communities set religious dates in different ways, by local or regional sighting or by an official calculated calendar, and that choice is a question for qualified scholars and your local mosque. The Umm al-Qura calendar explainer covers the calculated calendar that Saudi Arabia publishes.

Common questions

Is the tabular Islamic calendar the same as the Kuwaiti algorithm?

The Kuwaiti algorithm is one implementation of it. Van Gent identifies it as scheme II and describes it as the forerunner of Microsoft's HijriCalendar.

Why does my phone show a different Hijri date from the mosque?

The phone may be set to a tabular calendar (often labelled "Islamic civil" or "Islamic tabular"), to Umm al-Qura, or to an adjusted date. None of these is a local announcement. Choose the calendar that matches what you follow, and confirm key dates locally.

Check a date yourself

To see where a Gregorian date falls in the calculated Umm al-Qura calendar, use the Hijri date converter. If you are comparing it with a tabular date from a spreadsheet, phone or historical table, expect a difference of a day in many months, and check which epoch and leap-year scheme that source uses. For the names and order of the months, see the Islamic months list.