Author: Saito Shin
Reviewer: Baiyan

  If my math is right, this article will be put on public display for your judgment on Lunar New Year’s Day (or sometime afterward). Happy New Year, everyone! Posting on New Year’s Day is no easy task. I’m not asking for triple pay (or any pay at all); all I ask is that you give it a careful read. If you could give it the usual like-comment-favorite combo (scratch that) and share it, even better!

Wishing everyone a happy, healthy New Year!

  You have probably heard that 2020 is a “double leap year.” The Year of the Rat has 384 days, which means an extra month of work thanks to the leap fourth lunar month. The math makes sense, but why should we have to work overtime? Today, let us take a look at intercalation in calendars.

  First, what exactly is a “calendar”?

  A calendar is a system for determining the lengths of days, months, and years and the relationships among them. In other words, it sets the order by which we measure time. Some calendar months and years take their lengths from the cycles of the Sun and Moon; others are fixed by convention. The earliest calendars paid close attention to the Moon’s waxing and waning. Later, agriculture made it important for calendars to keep the seasons regular and seasonal markers in the proper order. These rules kept the sequence of months and days aligned with the apparent positions of the Sun and Moon in the sky. Without that alignment, the seasons would eventually be turned upside down and lunar phases would fall out of step.

  Next, what is “intercalation”?

  The tropical year measures the cycle of the seasons, while the synodic month measures the cycle of the lunar phases. Calendar makers therefore need an accurate length for each. Yet neither the tropical year (365.2422 days) nor the synodic month (29.5306 days) contains a whole number of days, and the two periods are not simply commensurable. If a calendar used their exact astronomical lengths, years and months would begin at a different time of day on each cycle (and how are we supposed to spend 0.2422 of a day?). That would be highly inconvenient for work and daily life. Calendars instead assign whole numbers of days to their years and months, creating what are called calendar years and calendar months. This leaves discrepancies between the calendar and the astronomical periods. Left uncorrected, those discrepancies build up and eventually throw the calendar into disorder. Correcting for them is what calendar makers call intercalation.

  Most readers already know how intercalation works in the Gregorian calendar, the solar calendar now in common use. A solar calendar takes the tropical year as its basic unit and does not follow the synodic month. It treats the year as the primary unit and aims to keep the mean calendar year equal to the tropical year; the number of days in each month and the number of months in each year are set by convention. Today’s worldwide civil calendar developed from the Julian calendar. The Julian calendar assigned 365.25 days to a tropical year and twelve months to each year. Odd-numbered months had 31 days; even-numbered months had 30, except February, which had 29 because it was then the month in which prisoners were executed. A leap day was added to February after every three ordinary years, bringing it to 30 days in a leap year. Later misunderstandings resulted in twelve leap years over a span of 33 years, three more than intended. The ruler at the time ordered two corrections. First, leap years would be suspended for the next eleven years; once the three extra leap years had been offset, the four-year cycle would resume. Second, August, his birth month, would become a long month. The pattern of long and short months after August was reversed, and the extra day was taken from February. February consequently has 28 days in an ordinary year and 29 in a leap year.

  The Julian calendar was arguably the best calendar of its time. Several Christian countries in Europe agreed to adopt it and, based on the astronomical observations then available, fixed the vernal equinox on March 21. The Julian year, however, is 0.0078 day longer than the tropical year. Over 1,257 years, that small difference added up to an error of about ten days, so that by 1582 the Sun was observed to cross the vernal equinox on March 11. Pope Gregory XIII accepted a proposal by the Italian physician and philosopher Lilio and decreed two changes: the ten days from October 5 through October 14 that year would be dropped, and the rule of one leap year every four years would be replaced with 97 leap years every 400 years. The new rule is the one we know today. A year divisible by 100 is a century leap year only if it is also divisible by 400; every other year divisible by 4 is a leap year. The reformed calendar became known as the Gregorian calendar. It is far more accurate than the Julian calendar, although it still gains 1.2 days in 4,000 years. The years 4000 and 8000 should therefore not be leap years (unless a better calendar has been devised by then).

  To understand intercalation in the traditional Chinese calendar, we first need to know how that calendar is constructed. It is lunisolar, not purely lunar as it is often said to be. A lunar calendar makes the month its primary unit and sets the length of a calendar month by the synodic month. The length of its calendar year is fixed by convention and has no relation to the tropical year. A synodic month lasts 29.5306 days. Lunar calendars therefore give 30 days to odd-numbered months and 29 days to even-numbered months, beginning each month with the first appearance of the new Moon. Twelve months make a 354-day year. Twelve synodic months, however, total 354.3671 days, so over thirty years they exceed thirty calendar years by 11.013 days. Such a calendar stays in step with the phases of the Moon, but each ordinary year is about eleven days shorter than the tropical year. It falls behind by a month in three years, and after about seventeen years the order of its months and seasons would be reversed. That would make it a poor guide to agriculture. The traditional Chinese calendar was developed to resolve this conflict.

  A lunisolar calendar gives equal weight to the year and the month. It uses the synodic month as the length of a calendar month, then inserts leap months to keep the calendar year aligned with the tropical year. The tropical year contains about 12.368 synodic months. To keep the average calendar year close to the tropical year, an ordinary lunisolar year has twelve months and a leap year has thirteen. The additional month is the leap month. Calculations show that seven leap months in nineteen years is a reasonable arrangement, one that keeps the order of the months from drifting away from the seasons. Inserting leap months creates a large difference in year length: an ordinary year has 353 to 355 days, while a leap year has 383 to 384. So yes, you really do get another month’s salary; there is no need to feel cheated. The placement of the leap month is closely tied to the twelve principal terms among the twenty-four solar terms. Strictly speaking, the system comprises twelve principal terms and twelve sectional terms. In a lunisolar calendar, every month normally contains a designated principal term, a detail we will leave aside here. Nineteen tropical years contain 228 sectional terms and 228 principal terms, while nineteen lunisolar years contain 235 synodic months. Seven months must therefore lack a sectional term, and seven must lack a principal term. Under the rule eventually adopted, a month without a principal term becomes that year’s leap month. It takes the name of the preceding month with the word “leap” placed before it. Calendar makers have followed this method ever since.

  Why, then, do we never seem to encounter a leap first lunar month? Earth passes aphelion in early July each year, roughly around the beginning of the sixth lunar month. At that point, Earth is moving more slowly along its orbit, while successive principal terms remain 30° apart in the Sun’s apparent ecliptic longitude. The interval between them is therefore longer, making a leap month more likely. As far as I know, no leap first lunar month will occur between now and 2060. Until then, a two-month winter vacation is out of the question.