Metonic Cycle: Does Your Tithi Return Every 19 Years?

The Metonic cycle is the near-coincidence of 235 lunar months with 19 solar years. For anyone who keeps a birthday by the tithi, it answers a practical question: when does the lunar birthday land on the same Gregorian date again? The cycle suggests “every nineteen years”, and this page tests that answer. It reads the tithi at the New Delhi sunrise for each of 55,152 consecutive days from 1 January 1950 to 31 December 2100, and for every date from 1950 to 2024 it asks whether the same date eight, eleven, nineteen and up to 76 years later opens on the same tithi. Nineteen years later it does on 56.5% of dates. The rest of the page is about why the figure is not higher, and why no other span under 76 years does as well.

The Arithmetic: 235 Months Against 19 Years

en-Wikipedia’s article on the lunar month quotes the Chapront-Touzé and Chapront figure for the mean synodic month, new moon to new moon: 29.530589 days. The Gregorian calendar’s mean year follows from its own leap rule, 97 leap days in 400 years, which gives 365.2425 days. Twelve lunar months fall about eleven days short of that year, so a lunisolar calendar has to add a thirteenth month now and then. The table shows how close three whole-number pairings come.

Eight years with 3 added months overshoot by 1.59 days; eleven years with 4 added months fall short by 1.51. Put the two together and the errors nearly cancel: 8 plus 11 is 19 years, 99 plus 136 is 235 months with 7 of them added, and the lunar months now run over by only 1.94 hours. Measured against the tropical year of 365.2422 days that the Metonic-cycle article uses, the excess is 2.08 hours; either way it is about two hours in nineteen years.

A tithi is a twelve-degree step of the Moon’s lead over the Sun, one thirtieth of a lunar month on average, so those two hours are 0.082 of a tithi. At that rate the drift builds up to a whole tithi only after about 12 cycles, roughly 231 years. It comes to roughly a quarter of a tithi after three cycles and a third after four, so on its own it is not what decides whether a tithi birthday returns on the same date.

Why the Leap Days Matter More Than the Two Hours

A birthday is a calendar date, and between two equal dates there is always a whole number of days. Nineteen Gregorian years contain either four leap days or five, so the span is 6,939 or 6,940 days, never the mean 6939.608. Against 235 mean months that is 0.69 of a day too short or 0.31 of a day too long, which is 0.70 of a tithi behind or 0.32 of a tithi ahead. That fraction-of-a-day offset swamps the two-hour drift, and the census below shows it at work.

The Census: The Same Date, N Years Later

For each of the 27,394 dates from 1950 to 2024, the sunrise tithi was compared with the sunrise tithi on the same date a set number of years later. The site’s own lunar series does the calculation, and it reproduces all 730 sunrise tithis in the panchang table for 2026 and 2027. The month name is not compared, only the tithi; near an added month or a sign boundary the names can part, and the page does not count how often.

Nineteen years is the best span in the table, and it is still only a little better than a coin toss: 56.5% the same, 42.1% one tithi away, and 376 dates, 1.4%, two away. Eight years never brings the tithi back. Every one of the 27,394 dates is one or two tithis behind, because 99 months run 1.59 days past the date. Eleven years manages 7.1%, and twenty-seven, which is 8 plus 19, manages only 0.5%.

Four Leap Days or Five: The Nineteen-Year Split

Divide the nineteen-year comparisons by span and the leap-day arithmetic shows up exactly. The 20,499 spans of 6,940 days, with five leap days, move the tithi forward by 0.31 on average, and 63.2% of them land on the same tithi. The 6,876 spans of 6,939 days, with four, move it back by 0.69, and only 36.5% match. Both averages agree within a hundredth of a tithi with the figures worked out from the mean month, 0.32 and 0.70. So the chance of a nineteenth-year return depends partly on where the birth date sits among the leap years.

Why Longer Multiples Do Worse

Thirty-eight, fifty-seven and seventy-six years are two, three and four Metonic cycles, yet each matches less often than a single cycle. The average shift is not the reason. For the 20,440 fifty-seven-year spans of 20,819 days it is almost nil, −0.06 of a tithi, and they still match only 35.9% of the time. The reason is scatter: real new moons come early or late against the mean ones, so the shift between two equal dates is spread around its average. The table gives the scatter of the shift (its standard deviation, in tithis) beside how far each span is from a whole number of anomalistic months, the Moon’s return to perigee, the nearest point of its orbit, 27.554551 days in the lunar-month article.

The two spans nearest a whole anomalistic count, twenty-seven and eight years, scatter least. The two farthest, fifty-seven and seventy-six, scatter most, at more than twice the nineteen-year figure, and nineteen, eleven and thirty-eight fall between in the same order on both columns. That is a correlation the census shows; the page does not try to separate the Moon’s orbit from other causes. Low scatter alone does not make a match. Eight and twenty-seven years scatter least, but their average shift is more than a whole tithi, so they never or hardly ever land on the same one.

Which Year Brings the Same Tithi on the Same Date First

For each date from 1950 to 2024, the census also searched every year from one to 76 later for the first one in which the date opened on its own tithi again. No date matched in fewer than eleven years. The commonest first match is 19 years later (56.5% of dates), then 46 (12.5%), 38 (11.3%) and 11 (7.1%). For 764 dates, 2.8%, no year within 76 brought the tithi back to the date at all.

One Date Across Eight Cycles: 29 October

Take the date this page was published and step it through nineteen-year cycles from 1950 to 2083. The last column shows how far through its tithi the Moon was at sunrise, from 0 at the tithi’s start to 1 at its end.

The first three steps do not hold: Krishna Tritiya in 1950, Krishna Chaturthi in 1969, Krishna Panchami in 1988. From 2007 the date settles. 29 October 2026 holds Krishna Chaturthi at sunrise, and the computation puts the same tithi on the date in 2045, 2064 and 2083. The position inside the tithi still wanders, from 0.16 to 0.45 across those cycles. That is the scatter from the section above, seen on a single date.

Born on 29 February

Nineteen is odd, so nineteen years after a leap day there is no 29 February: the nineteen leap days from 1950 to 2024 are exactly the dates the nineteen-year census had to leave out. Looking at the two neighbouring dates instead, 28 February carries the leap-day tithi in eleven of those years and 1 March in nine, both of them in one, and in every one of the nineteen at least one of the two does. Seventy-six is a multiple of four, so a leap day does have a leap day seventy-six years on, except 29 February 2024: 2100 is a century year that the Gregorian rule leaves without one. The same skipped day is why 306 of the seventy-six-year spans are a day shorter than the rest.

Seven Added Months in Every Nineteen Years

Seven added months, as in the arithmetic, show up in the Hindu calendar too. The site’s table of adhika months covers 2020 to 2045. Its generator calls a month adhika when the Sun enters no sidereal sign between the new moon that opens it and the one that closes it, and the table lists ten of them and no kshaya month. Each of the eight runs of nineteen calendar years inside that span, 2020–2038 through 2027–2045, holds exactly seven. The added months come at seven gaps of three years and two of two. And each one whose nineteenth-year partner falls inside the table returns under the same name, starting within a day of the same date:

• Adhika Ashwina: from 17 September 2020, and again from 18 September 2039
• Adhika Shravana: from 18 July 2023, and again from 17 July 2042
• Adhika Jyeshtha: from 17 May 2026, and again from 16 May 2045

The next adhika month in the table is Adhika Chaitra, from 15 March 2029; the adhik maas page explains why 2027 and 2028 have none. The Indian rule works from the Sun’s entries into the signs, not from a fixed list of years; the nineteen-year pattern is its result.

Meton, the Hebrew Calendar and Easter

The history here is reported from en-Wikipedia’s article on the Metonic cycle. The article names Meton of Athens, in the 5th century BC, as having judged the cycle to be a whole 6,940 days, and it says he introduced the cycle to the Attic calendar in 432 BC. Citing E. G. Richards’s Mapping Time, it gives the cycle as 125 full months of 30 days and 110 of 29, which the arithmetic confirms adds up to 6,940. It reports that the Babylonians used the cycle from the late sixth century BC and that, in the Babylonian and Hebrew calendars, years 3, 6, 8, 11, 14, 17 and 19 of each cycle are the long, thirteen-month ones. The Hebrew cycle (in the article’s word, a small maḥzor) is built on a fixed month of 29 13753/25920 days, and 235 of those make 6939.69 days, less than a tenth of an hour from the mean-month figure above.

The same article reports that a nineteen-year cycle is used to compute the date of Easter. It traces the first method to the Alexandrian computist Anatolius around AD 260, and says a later version, the one behind the Easter tables of Dionysius Exiguus and Bede, prevailed across Christendom at least until the Gregorian calendar came in during 1582. It also credits Callippus, in the century after Meton, with a cycle of four nineteen-year periods, 76 years with a mean year of exactly 365.25 days. That comes to 27,759 days, the same whole-day span the census finds between equal dates seventy-six years apart whenever no century year drops its leap day.

What This Page Does Not Give

Nothing here says what a nineteen-year return, a tithi or an added month means for a person. A count of sunrises can confirm whether a date and a tithi coincide and nothing more, so the page reports only the coincidence. It also does not say on which day a tithi birthday should be kept when the tithi is skipped or holds two sunrises. The tithi birthday dates page sets out the sunrise reading it uses and where family calendars differ. The census compares sunrise tithis, while a birth’s own tithi belongs to the moment of birth, and for an early-morning or late-evening birth the two need not be the same.

Related Pages

To find the tithi of a birth moment from the Moon’s and Sun’s degrees, see birth tithi from date of birth. Why a single lunar month runs longer or shorter than 29.53 days is the subject of lunar month length, and how often a calendar year holds a thirteenth full moon is counted in how many Ekadashi, Purnima and Amavasya in a year. The daily panchang pages give the sunrise tithi at New Delhi for each day of 2026 and 2027.

In Brief

• The cycle: 235 mean lunar months run 1.94 hours longer than 19 Gregorian years
• Built from: 8 years (99 months) plus 11 years (136 months); 7 added months in all
• Same date, same sunrise tithi, 19 years later: 56.5% of dates 1950–2024; 63.2% when the span holds five leap days, 36.5% when it holds four
• After 8 years: never the same tithi; after 38, 57 or 76: 42.5%, 35.0%, 37.8%
• Adhika months, 2020–2045: seven in every nineteen-year run of calendar years