Why Add a Whole Extra Month? Keeping Moons and Seasons Together
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Twelve lunar months fall eleven days short of a solar year, so a calendar following the moon drifts through the seasons unless something is inserted. The solutions adopted reveal what each society considered most important to keep fixed.
The arithmetic that causes the problem
A lunar month averages about twenty nine and a half days, so twelve of them make roughly three hundred and fifty four days, which is around eleven days short of the time the earth takes to go round the sun. A calendar counting only months therefore shifts backwards through the seasons by about eleven days each year, completing a circuit in roughly thirty three years, which means a given month falls in every season over a lifetime. That is a problem for anyone whose festivals, planting or tax collection are tied to the seasons and not for anyone whose observances are tied to the moon itself. The available responses are to ignore the sun, to ignore the moon, or to insert extra months periodically, and all three have been adopted.
The three kinds of calendar
Every system answers the same question differently:
- •Purely lunar, where months follow the moon and the year drifts through the seasons freely
- •Purely solar, where the year tracks the sun and months are administrative divisions unrelated to the moon
- •Lunisolar, where months follow the moon and extra months are inserted to keep the year aligned with the sun
- •Lunisolar systems need seven extra months in every nineteen years to stay aligned
- •That nineteen-year relationship is close enough to exact to be usable for centuries
- •Some systems used observation to decide each year rather than following a fixed cycle
How the insertion was decided
Two approaches have been used and the shift between them is historically revealing. Observational systems decided each year whether an extra month was needed, based on whether the crops were ready, whether the barley had ripened or whether the season was running late, with the decision taken by an authority who announced it, which kept the calendar tied to reality and made it unpredictable more than a year ahead. Calculated systems fix the insertions in a repeating cycle, which makes the calendar predictable indefinitely and removes the need for anyone to decide. The move from one to the other happens in several traditions and generally follows the same logic, since a dispersed community cannot rely on an announcement reaching it in time, and a fixed rule lets everyone compute the same answer independently.
Adjusting the solar year too
Purely solar calendars face a smaller version of the same problem and solve it the same way, by inserting time. The year is not a whole number of days, being about three hundred and sixty five and a quarter, so a calendar of whole days drifts unless a day is added periodically, which is the leap day. The Julian arrangement added one every four years, which slightly overcorrects, accumulating an error of about three days in four centuries, and by the sixteenth century the date of the equinox had shifted by ten days from where the church calendar assumed it. The Gregorian reform of 1582 dropped those ten days outright and refined the rule by omitting the leap day in most century years, which reduces the error to about one day in three thousand years. Adoption took until the twentieth century in some countries.
Where each solution survives
The three approaches are all still in use and the consequences are visible annually. The Islamic calendar is purely lunar and does not intercalate, so its months move through the seasons and the month of fasting occurs at every time of year across a generation, which is a deliberate feature rather than an oversight. The Hebrew calendar is lunisolar with a fixed nineteen-year cycle, adding a repeated month in seven years out of nineteen, which keeps its festivals in their seasons. The Chinese calendar is lunisolar with insertion determined by the sun's position, which is why the new year falls on different dates. The Gregorian calendar is purely solar, and its months have no relationship to the moon at all, which is why the date of Easter, computed from a lunar rule, moves.
The takeaway
Twelve lunar months fall about eleven days short of a solar year, so a purely lunar calendar circuits the seasons in roughly thirty three years. Lunisolar systems insert seven extra months every nineteen years, which is close enough to exact to hold for centuries. Deciding by observation gave way to fixed calculation wherever a dispersed community could not rely on an announcement.