Tabular, Umm al-Qura or Sighting: Why Hijri Dates Differ

Why Do Hijri Dates Differ at All?

A Hijri month follows the moon. The moon takes about 29.53 days to go from one new moon to the next. So a month must have either 29 or 30 days.

Nobody can say in advance, by simple counting, which months get 30 days. Every calendar has to choose, and the three systems below choose in different ways.

All three agree on the month names and on the count of 12 months. They can disagree only on the day a month starts. A one-day gap at the start carries through the whole month.

What Is the Tabular Hijri Calendar?

The tabular (or arithmetical) calendar replaces the moon with a repeating pattern. Odd months always have 30 days and even months have 29. Month 12 gains a thirtieth day in a leap year.

A common year then has 354 days and a leap year has 355. Medieval astronomers built their tables on this scheme, and software still uses it.

How Does the 30-Year Cycle Work?

The pattern repeats every 30 years, and 11 of those years are leap years. The cycle total is 19 × 354 + 11 × 355 = 10,631 days.

That equals 360 months, so the average month is 10,631 ÷ 360 = 29.5306 days. The true average is 29.5306 days. The gap is tiny: the calendar slips about one day in 2,500 years.

Which Years Are Leap Years?

Several lists exist, because medieval authors rounded the last fraction of a day differently. The most common one is often called the Kuwaiti algorithm. It makes years 2, 5, 7, 10, 13, 16, 18, 21, 24, 26 and 29 long.

You can test any year with one rule. Year Y is a leap year when (11 × Y + 14) mod 30 is less than 11. For example, 1447 gives (11 × 1447 + 14) mod 30 = 1, so it is a leap year.

All five lists have 11 long years and the same 10,631-day total. They differ only in where the long years sit inside the cycle.

Why Do Tabular Tables Use Two Epochs?

The epoch is day 1 of year 1. Most software places it on Friday, 16 July 622 CE in the Julian calendar. This is Julian day number 1,948,440, which this page calls the civil epoch.

A minority of tables place the epoch one day earlier, on Thursday, 15 July 622. This shifts every date by one day. Check which epoch a converter uses before you trust its output.

How Does the Umm al-Qura Calendar Work?

The Umm al-Qura calendar is the official Hijri calendar of Saudi Arabia. It is calculated from the moon’s motion, not counted from a fixed pattern. The Kaaba in Mecca gives the place used for the calculation.

Its rule has changed over time, so the age of a source matters. The table gives the stages as described on Wikipedia.

The result is a published table of month starts, so every Hijri date maps to one Gregorian date. Lengths are irregular: months may run 29 or 30 days in any order.

This calendar is not the same as the tabular one. In 1448 AH its month starts fell one or two days earlier than the common tabular count. The table below shows this.

How Does Umm al-Qura Differ From Sighting?

Umm al-Qura is calculated, so it needs no observer. The conjunction test confirms the moon has passed the sun by sunset. This holds even if the thin crescent is too faint to see.

That is why a calculated month can start before a sighted one. The two methods ask different questions: has the moon passed the sun, and can anyone see it?

What Is Moon-Sighting Reckoning?

In sighting, a month begins when the new crescent is seen after sunset. If it is not seen after the 29th day, the month runs to 30 days. This is the oldest method.

The crescent is visible only under good conditions, often around 17 hours after conjunction. Sky, weather and place decide whether it can be seen.

Countries also use different sighting rules, and some use a moonset rule instead of a visual report. So a sighting-based date can differ between two cities on the same night.

Which System Do Old Sources Use?

It depends on who wrote the source. Astronomers working from tables, such as al-Khwarizmi and al-Battani, used the arithmetical count. Writers in places that followed sighting recorded the date their own city announced.

Some regions kept a shorter cycle. An 8-year cycle with leap years 2, 5 and 8 holds 5 × 354 + 3 × 355 = 2,835 days. That is exactly 405 weeks, so weekdays repeat on the same dates.

Wikipedia reports that the Ottoman Empire and Southeast Asia used this 8-year form. Its mean month is 2835 ÷ 96 = 29.53125 days, a little longer than the 30-year mean.

So a date in a manuscript may follow any of these rules. When the source does not say, treat the converted day as approximate.

How Big Is the Difference? A Worked Example

Take 1 Rabi II 1448 AH. We will fix it by two tabular variants and compare the result with the official Umm al-Qura table.

The Tabular Algorithm, Step by Step

1. Count the long years before year 1448. With the Kuwaiti scheme, the count is floor((11 × 1448 + 3) ÷ 30) = 531.
2. Add the days of the earlier years: 354 × 1,447 + 531 = 512,769 days.
3. Add the months before month 4: 30 + 29 + 30 = 89 days.
4. Add the epoch: 1,948,440 + 512,769 + 89 = 2,461,298. This is the Julian day number of the date.
5. Convert the number to a calendar date. 2,461,298 is 14 September 2026. With the Thursday epoch the number is 2,461,297, which is 13 September 2026.

The official Umm al-Qura table lists 12 September 2026 for the same Hijri day. So the three answers span two days.

The first five months of 1448 show the same pattern. The shift is one day in some months and two in others. Tabular months follow a strict 30-29 pattern, while Umm al-Qura months do not.

Why Does a One-Day Gap Matter for Historical Dates?

A date from an old text came from some local rule, and we seldom know which. A converter must pick one rule. So a converted date is an estimate with a margin of one or two days, not an exact day.

Does It Change the Year?

Rarely. Only the first day or two of a Hijri year, or of a Gregorian year, can cross a boundary. Most dates keep their year whichever variant you use.

Year boundaries are the exception. In the civil tabular count, 16 June 2026 is the last day of 1447 AH (30 Dhu al-Hijjah). The Umm al-Qura table calls the same day 1 Muharram 1448 AH, so the Hijri year differs by one.

What About Chronograms?

A chronogram gives a year, not a day. So its sum is usually safe from this problem. The problem appears when you turn that year into a Gregorian year, because a Hijri year spans two Gregorian years.

If a verse falls at the edge of a Hijri year, the Gregorian year can change with the variant. The chronogram decoder gives the Hijri year from the letters. The guide to reading a chronogram explains the rest of the method.

Why Does My Software Show Another Date?

Software picks one rule and applies it silently. Microsoft’s Kuwaiti algorithm is a tabular count. Its Arabic-language settings use Umm al-Qura as the default Islamic calendar.

Some web converters list only one date, without naming the variant. Two tools can then give answers a day apart, and both are following their own rule correctly.

How Should You Label a Converted Date?

Name the variant whenever you convert. A short label removes most confusion, and it lets a reader repeat your work.

1. State the calendar: tabular civil, tabular Thursday epoch, Umm al-Qura, or sighting.
2. State the leap-year scheme if it is tabular, for example the Kuwaiti scheme.
3. Give the Hijri date and the Gregorian date side by side.
4. For old dates, say that a day or two may differ in another variant.

The formula itself is covered on the Hijri to Gregorian formula page. The Hijri calendar hub lists the rest of this series. The history of the calendar explains where the era starts.

Sources and Further Reading

• Tabular Islamic calendar, Wikipedia. The 30-year cycle, 10,631 days, leap-year lists and the Kuwaiti algorithm.
• OEIS A350539. The civil epoch at Julian day number 1948440, the Thursday variant and the leap-year formula.
• Calendrical.Islamic.Civil, Hexdocs. The civil epoch of Friday, 16 July 622 (Julian) and its Thursday counterpart in software.
• Islamic calendar, Wikipedia. Umm al-Qura rule changes, sighting practice and the one to three day differences.
• Umm al-Qura calendar, Wikipedia. The 1999 and 2002 rule changes.
• The Umm al-Qura Calendar of Saudi Arabia, R.H. van Gent. Software defaults and differences of one to three days from sighted dates.
• Umm Al-Qura Calendar, official site. The month starts of 1448 AH used in the table.
• The tabular dates are computed in code from the formulas above. See the methodology.

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