Author: 斎藤信
Reviewer: 白烟

  If I have counted correctly, this article will reach you on the first day of the Lunar New Year—or sometime afterward—making a conspicuous public appearance. Happy New Year to everyone! Publishing on the first day of the year is not easy. I will not ask for triple pay (or any pay at all); I only hope you will read carefully. A like, a comment, and a favorite—scratch that—and a share would be even better!

Happy New Year, and may everyone enjoy good health!

  You have probably heard that 2020 is a “double leap year.” The Year of the Rat has 384 days, so we have to put in an extra month’s work—an intercalary fourth lunar month. I understand the arithmetic, but why the overtime? Today, then, we will talk about intercalation in calendars.

  First, let us look at what a “calendar” is.

  A calendar is a system for calculating the lengths of days, months, and years and the relationships among them, thereby establishing the order of time. The lengths of some calendar months and years are determined by the cycles of the Sun and Moon; others are set by convention. The earliest calendars paid close attention to the waxing and waning of the Moon. Later, the needs of agriculture required calendars in which the seasons remained regular and seasonal markers followed a stable order. These rules kept the sequence of months and days in the calendar aligned with the apparent positions of the Sun and Moon in the sky. Without that alignment, seasons would be reversed and lunar phases would fall out of place.

  Next, let us examine “intercalation.”

  The tropical year is the cycle of the seasons, while the synodic month is the cycle of the lunar phases. A calendar must therefore determine both periods precisely. Yet neither the tropical year (365.2422 days) nor the synodic month (29.5306 days) is an integer, and the two are not simply commensurable. If years and months retained their exact astronomical lengths, their starting times would not fall at a fixed point within the day—and who knows what to do with 0.2422 of a day? That would be highly inconvenient for work and daily life. Calendars therefore define years and months as whole numbers of days. These are known as calendar years and calendar months. A discrepancy remains between them and the astronomical periods. If ignored, it accumulates and eventually throws the calendar into disorder. A calendar’s method for correcting that discrepancy is called intercalation.

  Most readers will know how intercalation works in the Gregorian calendar, the solar calendar in common use. A solar calendar takes the tropical year as its basic unit and has nothing to do with the synodic month. It treats the year as the primary component and aims to make the mean calendar year equal to the tropical year; the number of days in a month and the number of months in a year are conventional. 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 inserted in February after every three ordinary years, giving it 30 days in a leap year. Later misunderstandings led to twelve leap years being inserted over a span of 33 years, three too many. The ruler of the time ordered two corrections. First, there would be no leap years for the next eleven years; once the three extra leap years had been offset, the four-year cycle would resume. Second, August, the month of his birth, 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 adopted it together and, based on contemporary astronomical observations, stipulated that the vernal equinox must fall on March 21. The Julian year, however, is 0.0078 day longer than the tropical year. Over 1,257 years, this tiny difference accumulated into an error of about ten days, so that by 1582 the Sun was observed to pass the vernal equinox on March 11. Pope Gregory XIII therefore accepted a proposal by the Italian physician and philosopher Lilio and issued a decree: the ten days from October 5 through October 14 that year were removed, and the rule of one leap year every four years was replaced with 97 leap years every 400 years. This is the familiar modern rule: 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 must first understand how that calendar is constructed. It is a lunisolar calendar, not the purely lunar calendar it is often called. A lunar calendar treats the month as its primary component and tries to make the synodic month the length of a calendar month. The length of its calendar year is conventional and unrelated to the tropical year. A synodic month lasts 29.5306 days. Lunar calendars therefore assign 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, last 354.3671 days, so over thirty years they exceed thirty calendar years by 11.013 days. This calendar follows the phases of the Moon, but each ordinary calendar year is about eleven days shorter than the tropical year. It falls behind by one month in three years, and after about seventeen years its sequence of months and seasons would be reversed. Such a system would be a poor guide to agriculture. The traditional Chinese calendar was developed to solve this problem.

  A lunisolar calendar gives equal weight to the year and the month. It uses the synodic month as the length of a calendar month and inserts leap months so that the tropical year determines the length of the calendar 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. Calculation shows that seven leap months in nineteen years is a reasonable arrangement. It prevents the order of the months from becoming detached from the seasons. The insertion of 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 you really do receive another month’s salary; there is no need to feel cheated. The placement of the leap month is closely connected to the twelve principal solar terms among the twenty-four solar terms. Strictly speaking, the system contains twelve principal terms and twelve sectional terms. In a lunisolar calendar, every month normally contains a designated principal term, a point we will not explore further 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. The rule eventually adopted makes a month without a principal term the leap month of that year. It takes the name of the preceding month with the word “leap” placed before it. Later calendar makers have continued to use this method to the present day.

  Why, then, have you never encountered a leap first month? In early July each year, Earth passes aphelion, roughly around the beginning of the sixth lunar month. Earth is then moving more slowly along its orbit, while the difference in the Sun’s apparent ecliptic longitude between successive principal terms remains 30°. The interval between two principal terms is therefore longer, increasing the probability of a leap month. As far as I know, no leap first month will occur between now and 2060. Until then, forget about getting a two-month winter vacation.