Tithi: why the Hindu day is a lunar day

The definition

A Tithi is the time it takes for the angular distance between the Moon and the Sun, measured along the ecliptic, to increase by twelve degrees. At new Moon the two are together, at zero degrees. When the Moon has moved twelve degrees ahead, the first tithi ends; at twenty-four degrees the second ends; and so on until the Moon is 180 degrees from the Sun at full Moon, the end of the fifteenth tithi. The dark fortnight then runs from 180 back round to 360. Thirty tithis make one synodic month, about 29.53 days.

This is a definition in terms of an angle, not a clock. That single fact explains most of what is distinctive about the Hindu calendar.

Why a tithi varies in length

The Moon's orbit is elliptical, so its speed varies: roughly eleven to fifteen degrees a day against the stars. The Sun's apparent speed varies too, between about 0.95 and 1.02 degrees a day. The rate at which the gap between them opens is the difference, which ranges from a little over ten to a little under fourteen and a half degrees a day. A twelve-degree tithi therefore lasts anywhere from about nineteen and a half hours to about twenty-six and a half hours†. On average it is about 23.6 hours, a little shorter than a civil day.

How the civil day gets its name

The Hindu civil day runs from sunrise to sunrise and is named after the tithi current at local sunrise. Since tithis are on average shorter than days, now and then a tithi begins after one sunrise and ends before the next. It never "owns" a sunrise and so no day is named after it. This is a kshaya (lost) tithi. Less often, a long tithi covers two sunrises and two consecutive days carry the same name: a vriddhi (increased) tithi. Our research note on kshaya and vriddhi tithis across a decade describes how to count these.

Because sunrise depends on place, two cities can occasionally name the same day differently. That is not an error in either calendar; it follows directly from the definition.

Why a lunar day at all

The Vedic ritual year was organised around the phases of the Moon. The new-Moon and full-Moon sacrifices (darsha and purnamasa) were basic rites, and their timing required a way to divide the month by the Moon's progress. The Vedanga Jyotisha, the Vedāṅga of time-reckoning attributed to Lagadha, already gives rules for computing tithis within a five-year cycle. Later astronomical texts such as the Sūrya Siddhānta refined the computation with true (not mean) positions of Sun and Moon.

A day measured by the Moon's phase is also observable. Anyone can see whether the Moon is waxing or waning and estimate roughly how far it has come. The tithi turns that observation into an exact measure.

Tithi and the solar year

A purely lunar calendar drifts against the seasons by about eleven days a year. The Hindu calendar is lunisolar: months are named by the Sun's sign at the time of the new Moon, and when two new Moons fall within the same solar sign, an intercalary month (adhika masa) is inserted. In this way the tithi-based festivals stay within roughly the same season, so that Diwali, for example, always falls in autumn, while their Gregorian dates vary from year to year.

Tithi in practice

Most Hindu festivals and vratas are fixed by tithi: Janmashtami on Krishna Ashtami of Bhadrapada (in the purnimanta reckoning), Rama Navami on Shukla Navami of Chaitra, the fortnightly Ekadashi fasts. The rule books specify not only the tithi but the part of the day at which it must prevail, such as sunrise, midday or midnight. That is why the festival date in a given year is sometimes debated. The same tithi-based reasoning underlies Muhūrta, the choosing of times, where some tithis are classed as more suitable than others for particular undertakings.

Many families also keep a "tithi birthday": the lunar date of a person's birth, which falls on a different Gregorian date each year.

Tithi groups

The tradition groups the fifteen tithis of each fortnight into five classes that repeat three times: nanda (1, 6, 11), bhadra (2, 7, 12), jaya (3, 8, 13), rikta (4, 9, 14) and purna (5, 10, 15). The names are descriptive labels used in muhurta texts such as Rama Daivajna's Muhurta Chintamani.

A clean way to compute it

Take the Moon's longitude, subtract the Sun's longitude, add 360 if the result is negative, and divide by twelve. The whole number part plus one is the tithi number (1 to 30). Notice that the ayanamsa cancels out: since both longitudes are shifted by the same amount, the tithi is the same in sidereal and tropical reckoning.

Try it

The Birthday tithi tool finds the tithi of a birth date and the Gregorian date on which it falls this year. The Panchang tool shows today's tithi and when it ends, and Festivals lists observances by tithi.

† Marked dates or readings are debated among scholars or still being confirmed by our faculty.