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Why Did the Old Japanese Calendar Add a 13th Month, and How Did It Pick Which One?

Category: History & Culture

Imagine a calendar where New Year's Day slowly slides through the year until it lands in the middle of summer. A calendar that counts months by the moon, and does nothing else, would do exactly that in under 20 years [2]. The short answer to how people stopped it: now and then they squeezed in a 13th month, called a leap month, about seven times every 19 years, and they put it in a month that lacked a particular seasonal marker. The rest of this page asks why that works, what happens when the rule gets stuck, and how Japan's last leap month never got to happen.

1. What Happens to a Calendar That Follows Only the Moon?

Japan used to run on a moon-and-sun calendar. Its last version, the Tenpō calendar (the one most people now mean by "kyūreki", or "old calendar"), was replaced when Japan adopted the Gregorian calendar in 1873 [1]. Chinese and Korean calendars work on the same basic idea. In such a calendar, a new moon starts each month, and the gap from one new moon to the next averages about 29.5 days [1].

Here is the catch. Twelve of those months last about 354 days, which is roughly 11 days shorter than the 365 days it takes the seasons to come around [1]. Left alone, the dates slide earlier against the seasons by about 11 days a year, and in under 20 years New Year's Day would arrive around the summer solstice [2]. That would make calendar dates increasingly unreliable for planning when to sow seeds.

2. How Does a 13th Month Fix the Drift, and Why Seven Times in 19 Years?

The fix was to add an extra month whenever the calendar had fallen close to a month behind. That month is the leap month. If it follows the 3rd month, it is called the "leap 3rd month", and that year has 13 months [1]. Over the long run, such months were added about seven times every 19 years [2].

The number 19 is not random. Nineteen years of 12 months make 228 months, and seven leap months bring the total to 235. Those 235 moon months last almost exactly as long as 19 solar years, with a leftover of only about two hours [3]. The Greek astronomer Meton of Athens is credited with spotting this match in the 5th century BC, so it is called the Metonic cycle [3].

A leap month does not go into a fixed slot like "always after month 3". It goes in when the drift has grown big enough. That is why the same calendar date can land in different seasons in different years, by as much as about a month [2]. For example, the first day of the first month fell on February 1 in 1870 and on February 19 in 1871 [1].

3. Which Month Got the Extra One? The "No Mid-Term" Rule

To decide where to put it, calendar-makers used another clock: the 24 solar terms, which divide the year into 24 seasonal markers about two weeks apart. Every second one is called a mid-term, and the solstices and equinoxes are mid-terms [4]. A simple way to picture it: a mid-term is like a signpost that normally stands inside each month, and a month's number is tied to rules about particular signposts, such as the month holding the winter solstice being month 11.

The Tenpō calendar's rules, as the National Astronomical Observatory of Japan (NAOJ) describes them, were these [4]:

  • A month with no mid-term is a candidate for the leap month.
  • The month containing the winter solstice is the 11th month.
  • The month containing the spring equinox is the 2nd month.
  • The month containing the summer solstice is the 5th month.
  • The month containing the autumn equinox is the 8th month.

Why do some months hold no signpost? Dividing the 365.24-day year by 12 gives mid-terms about 30.4 days apart, while a moon month is only about 29.5 days. Because signposts are spaced a little wider than months, the signpost slips later within the month, and every so often a month ends up with none. That is the arithmetic of the two spacings, not a line quoted from the sources. The result is a calendar where the number of a month is tied to the sun's position, and the leap month shows up exactly where the moon months have fallen behind.

4. Why Does the Rule Get Stuck Around 2033?

The earth's orbit is slightly oval, so in winter the signposts come closer together, and a month can hold two of them [4]. The extra rules about the 2nd, 5th, 8th and 11th months were added to cope with this. Around 2033 and 2034, though, the Tenpō rules produce two months with two mid-terms and three with none. Only one month is left between the month holding the autumn equinox (the 8th) and the month holding the winter solstice (the 11th), and the rules cannot say which should be the 9th or the 10th [4]. NAOJ says this has never happened since the Tenpō calendar began [4].

Notice that this is not the news that a leap month comes in 2033. It is that the rule cannot decide where to put it. NAOJ lays out three possible fixes: a leap 11th month, a leap 1st month in the next year, or a leap 7th month [4]. Who decides? Since 1873, Japan has used the Gregorian calendar for all official purposes, so NAOJ notes there is no public body that issues a ruling on the old calendar [4]. Traditions like the autumn moon-viewing night or the Tanabata star festival follow old-calendar dates, so people who care will have to agree on a method themselves.

5. How Did Japan's 1872 Calendar Switch Erase a Leap Month?

The old calendar was dropped in a hurry. On the 9th day of the 11th month of Meiji year 5 (December 9, 1872), the government announced it would switch to a solar calendar with 365-day years and a leap day every four years. The 3rd day of the 12th month of the old calendar became January 1 of Meiji year 6 (1873) [5]. So the old calendar's last 12th month lasted only two days.

Had the calendar stayed, Meiji year 6 would have had a leap 6th month and 13 months. One often-repeated explanation is that the government, which had just begun paying officials monthly salaries and was short of money, would have owed 13 payments that year, and that this helped push the change. That story comes from a memoir by Ōkuma Shigenobu as summarized in Japanese Wikipedia, so treat it as one proposed motive, not the only reason [6].

6. How Can You Try the Numbers and See a Leap Month Yourself?

You need only a calculator and a few minutes. Multiply 29.5 by 12, then subtract the answer from 365 to get the yearly drift. Multiply that gap by 19 and compare it with seven times 29.5; the two come out within a few days of each other (about 209 and 206.5), which is the Metonic cycle at work. With a more precise month length, the gap shrinks to about two hours. You can also look at a printed East Asian calendar or an online table of traditional lunar dates, and watch for a month number that appears twice in a row, which usually marks a leap month. Then ask yourself: if you were the calendar-maker for a farming town, which clock would you trust more, the moon or the sun?

Sources

  1. National Astronomical Observatory of Japan (NAOJ), FAQ: "What is the 'old calendar' (kyūreki)?" https://www.nao.ac.jp/faq/a0304.html (Japanese; read in full. Used for what "old calendar" means, the 29.5-day month, the 354-day year, leap months, and the 1870 and 1871 New Year examples.)
  2. NAOJ, FAQ: "Does the 'old calendar' fit the seasons better than today's calendar?" https://www.nao.ac.jp/faq/a0305.html (Japanese; the main explanation was read, but the fetched copy was cut off near the end. Used for the 11-day drift, the summer-solstice New Year, seven leap months in 19 years, and the one-month seasonal swing.)
  3. Wikipedia, "Metonic cycle" https://en.wikipedia.org/wiki/Metonic_cycle (English; only the opening summary was read. It is the only source here for the 235-month match, the two-hour gap, and Meton in the 5th century BC, so those figures should be checked against an astronomy reference.)
  4. NAOJ blog, "Leap month undecided? When is the Mid-Autumn Moon in 2033?" (August 29, 2023) https://www.nao.ac.jp/news/blog/2023/20230829-koyomi.html (Japanese; the fetched copy was partly cut off, but the rules, the 2033 problem, the three fixes and the absence of an official body were read. Used for the mid-term rule and the 2033 puzzle.)
  5. National Archives of Japan, "Meiji 5 (1872), 11th month: Adoption of the solar calendar" https://www.archives.go.jp/ayumi/kobetsu/m05_1872_05.html (Japanese; the fetched text had character-encoding problems, so only the numbers and dates could be read. They match the NAOJ blog and Wikipedia. Used for the proclamation date and the switch from the 3rd day of the 12th month to January 1.)
  6. Japanese Wikipedia, "Meiji calendar reform" https://ja.wikipedia.org/wiki/%E6%98%8E%E6%B2%BB%E6%94%B9%E6%9A%A6 (Japanese; the opening sections were read. A secondary source that cites Ōkuma Shigenobu's memoir; used only for the salary explanation, which is stated as one proposed reason.)