The month that never existed

Japan's old lunisolar calendar: the moon-and-sun arithmetic that still fills the small print of Japanese datebooks — and autumn 2033, when the rules for naming the months break for the first time since 1844. ← Calendar Contraptions

Japan swapped its lunisolar calendar for the Gregorian one in 1873, but the old one never really left. It survives as the fine print in Japanese wall calendars and phone apps: the traditional festivals, the moon phases, and the rokuyō — the six-day fortune cycle that decides when weddings and funerals get scheduled. Every part of it is still computed the old way: a month is wherever the new moon says it starts, and the month's name is whichever official solar term it contains. That naming rule has a flaw nobody noticed for 189 years, and in autumn 2033 it finally fires: for seven months the rules cannot name the calendar at all. This page runs the whole machine — the moon, the sun, the fortune cycle — and then breaks it on schedule.

The reading

Rokuyō, the six lucks

Six names cycle through the calendar: 先勝 senshō ("win first"), 友引 tomobiki ("friends pulled in" — funerals are avoided, lest they take more friends), 先負 senpu ("lose first"), 仏滅 butsumetsu ("Buddha's death" — the worst day), 大安 taian ("great peace" — book the wedding), and 赤口 shakkō ("red mouth" — inauspicious, except the noon hour). The cycle has nothing to do with the sky. It is pure modular arithmetic on the old calendar date, and the old date is what this page computes — so the fortune machine is just:

That is genuinely all of it. The first day of each month starts a fixed name — month 1 and 7 begin with 先勝, month 2 and 8 with 友引, and so on, six months cycling twice; a leap month repeats its number's name. From the formula, seven traditional festivals always land on the same fortune: old New Year (1/1) is 先勝, the doll festival (3/3) is 大安, Boys' Day (5/5) is 先負, Tanabata (7/7) is 先勝, the harvest moon (8/15) is 仏滅, the later moon (9/13) is 先負, and old Seven-Five-Three (11/15) is 先勝. The names even encode the calendar's shape: when a fortune day is skipped on a calendar page, you are looking at a 30-day month that just ended — or a leap month beginning.

The names look Buddhist, and that is the historical joke. The cycle descends from 六壬, a Chinese time-divination system credited to the 7th-century astronomer Li Chunfeng; the day-based form reached Japan centuries later, and the modern six names first appear in print in 1747. A theory of its arrangement says the Buddha's own days were placed deliberately: the Buddha's birth (4/8) lands on 大安, his enlightenment (12/8) on 先勝, his death (2/15) on 仏滅 — all three check out in the formula. In 1873 the Meiji government banned fortune notes from official calendars as superstition, which should have been the end of it. Instead the banned notes went underground into private "ghost calendars", and by 1882 they were back. Of all the old calendar notes, the fortune cycle is the one that outlived its own ban: a 2024 poll found 71.5% of Japanese still check it for wedding dates, a city council was publicly scolded for handing out calendars with it in 2016, and crematoria in at least two cities close on 友引 days — the day that "pulls in friends" is a bad day to burn anyone.

The machine underneath

Under the fortunes, the old calendar is a three-gear mechanism. Gear one is the moon: a month begins on the day the new moon falls on, in Japan's time zone, and runs 29 or 30 days. Gear two is the sun: the solar year is cut into 24 unequal terms at exact 15° steps of the sun's apparent longitude — the system called 定気, adopted in 1844. Gear three is the naming rule: twelve of the 24 terms are "major terms", and each carries a month number — the month containing 雨水 (the sun at 330°) is month 1, 春分 is month 2 … 冬至 (270°) is month 11, 大寒 (300°) is month 12. A month containing no major term is a leap month and repeats the number before it. That's the whole constitution — and it ran, first as the Tenpō calendar (1844–1873) and then unofficially, on the honest understanding that roughly one year in three needs a leap month and the rule always knows where to put it.

The picked old year, month by month, straight from the moon and sun machinery. Leap years have thirteen chips; the leap month repeats its neighbor's number.

Why the rule breaks in 2033

Gear two is the culprit. Equal angles of the sun are not equal times: Earth runs fastest near perihelion in January, so the major terms bunch up — in late autumn and winter, two of them can squeeze into a single 30-day month, and somewhere another month gets none. Kepler's second law, wearing a calendar. The classical rule is usually described as "a month is named for the major term it contains" — but in the years with a double-term month, that description contradicts itself, and practice has always followed the other formulation, written down by the astronomer Hirayama Kiyotsugu in 1912: count the months from the solstice — the winter-solstice month is month 11, the spring-equinox month is month 2, summer-solstice month 5, autumn-equinox month 8 — and in a 13-month stretch, the first termless month becomes the leap month. The count and the contained-term description agree in ordinary years. In 1965 and 1985 they disagreed, and the printed calendars follow the count. Which leaves exactly one way the machine can fail: the count itself contradicting the anchor rule. For the autumn of 2033 through spring of 2034, it does — the machinery produces this, computed live, instants in Japan's time zone:

Seven months — the table's B through H — must carry just six names (8, 9, 10, 11, 12, 1) plus one leap month, and the count has to break a promise: the autumn-equinox month would have to be month 9, but rule one fixes it as month 8. That is the whole crash: not a bug, a contradiction — the first since the rule was adopted in 1844. Autumn 2033 is a pile-up of edge cases: the autumn equinox, Frost Falls and Light Snow each fall on the same day as a new moon (21 h, 6 h and 1.5 h before it — they count into the new month), while the winter solstice misses its new moon by 5 h — but across midnight, so it stays behind. One month (E) holds two major terms; a second (G) does it again in February. The reference argument over what to do has four official candidates:

The calendar-industry association (Nittenkyō) recommended the leap-11 option in 2015 — it honors the tradition that the solstice month must be month 11, nearly every perpetual calendar already prints it, and every serious candidate rule lands on it. But the association also notes that no official body will ever decide: the calendar was abolished in 1873, and the National Astronomical Observatory's position is that there is no one left to rule. The stakes are real and oddly concrete. The harvest moon of 2033 is either September 8 or October 7 depending on the reading; the 2034 old New Year is February 19 under every mainstream option — which is also, not coincidentally, exactly when China's Spring Festival falls that year, because China's rule cannot break (see below). A Buddhist trade journal warned of "big confusion" in funeral scheduling, and the observatory's own chief of calendar computation conceded in 2024 that both harvest-moon readings have a claim: "celebrate twice."

This is not a one-off — but the next one is far off. The reference literature flags 2147 as the next breakdown (a rule-2 tangle this page's counting rule resolves, which is exactly the kind of choice no abolished calendar can settle) and 2223 as the one after. The page's own computation, run over 1873–2100, finds the 2033 contradiction and one more thing the references don't flag: in 2053 the Great Cold term lands at 23:59 on the winter-solstice month's last day — one minute from sliding into the next month and rearranging the year's count. The count absorbs that minute without complaint today, but no printed rule can settle a minute the model itself promises only to a quarter hour. The honest answer is that 2053 is fine until it isn't, and the difference is smaller than the machinery's own error bar.

China's version can't break

The same lunisolar family runs Chinese New Year, and its rule is built so this cannot happen: the solstice month is always month 11, and if a solstice-to-solstice stretch holds thirteen months, the first termless one is the leap month — evaluated with priority on the solstice, exactly the tie-break Japan's rule lacks. It was stress-tested by accident: in 1851–52, China's own calendar put the spring-equinox month as New Year (no month 2 that year) and the machinery still worked, because the rule only ever leans on the solstice. Japan borrowed a variant of the Chinese rule in 1844 but kept the unequal-term naming front and center — which is why one country's calendar has a scheduled software crash and the other doesn't. India's traditional calendars take a third way out entirely: when the rules would fail, a month is simply skipped (a "ksaya" month — it last happened in 1982, when month 10 didn't exist).

Running the machine over 1873–2100

Walk the whole span, apply the count, and check the four anchor promises every year. The machine finds one contradiction:

The "old calendar" isn't the old calendar

Everything this page computes is a modern reconstruction. The real Tenpō calendar was computed against Kyoto's true solar time (longitude 135°46′E, plus up to ±15 minutes of equation of time), with 1840s constants; today's "old calendar" runs on Japan Standard Time (135°E mean solar time) and modern celestial mechanics. The seam is real: shift the clock by minutes and occasional dates move by a day, and a moved term can move a month name. The irony the history keeps: the Tenpō reformers measured a solar year of 365.24223 days against the true 365.24219 — a better figure than the Gregorian calendar's 365.2425. The calendar Japan abandoned was more accurate than the one it adopted; the switch happened anyway, for reasons that had nothing to do with astronomy.

The official machinery did not quite die either. The National Astronomical Observatory still announces next year's solar terms, moon phases and month-lengths in the government gazette every February — the "30-day months and 29-day months" arithmetic is still published by the state, just without an official name. The Observatory will answer old-calendar conversion questions only back to the 1873 switch; for later years, the tables this page's machinery is validated against were published (unofficially) by the Japan Coast Guard's hydrographic department, which kept printing a full old-calendar conversion table through 2010.

The 1873 switch

On December 9, 1872 (old calendar: month 11, day 9) the government announced that the next day's month — old December, which had run 29 or 30 days for centuries — would instead last exactly 2 days: old 12/3 became January 1, 1873. The speed had a motive a modern auditor would respect: government salaries had just switched to monthly payments, and 1873 contained a leap month — thirteen paydays. Under the new calendar, twelve; and the two-day December meant that year cost the treasury only eleven. Okuma Shigenobu's memoir says the quiet part out loud. Calendar sellers, who had already been shipping 1873 lunisolar calendars since October, were ruined overnight; Fukuzawa Yukichi, feverish with flu, dashed off a defense of the switch that reportedly sold 100,000 copies, arguing that anyone baffled by the new calendar was simply revealing whether they had learned to think. The old system's last full day was old 12/2 = December 31, 1872 — 28 days that existed in no year ever since.

The lineage

Senmyō ran for 823 years — longer than every other Japanese calendar combined. Its replacement came from Shibukawa Harumi, a Go master turned astronomer who had watched the Chinese-derived Senmyō miss a 1672 lunar eclipse and then watched his own draft miss a 1675 solar eclipse — two humbling data points that taught him the 13th-century Chinese constants no longer fit Japan's sky. His fix accounted for the longitude difference between China and Japan, and the court adopted his Jōkyō calendar in 1684 after a faction fight over a rival draft; Saikaku and Chikamatsu both wrote about the 800-year reform. The line from Shibukawa to the Meiji switch runs through four more reforms, each tuned — and the last of them, Tenpō, is the machine this page reimplements with 21st-century gears.

Old calendar → Gregorian

What this page doesn't do

This is a reconstruction of the modern "old calendar" (the one printed in Japanese shops today), not a historical ephemeris: pre-1874 dates computed here are projections, and historical Tenpō records can differ by a day or a month name — the Observatory answers pre-1873 conversions from the actual Tenpō tables, and this page deliberately stays on the modern side of the seam. The machinery covers 1873–2100. New-moon instants are good to a couple of minutes and solar terms to roughly a quarter hour, which is why the 2053 verdict above is reported as a knife-edge instead of a fact; month starts and names, which only need the day, are validated against the Coast Guard's published 1997–2010 tables — all 5,113 days of them — with zero mismatches. And the rokuyō is folklore: the machine computes it perfectly and the machine is not entitled to an opinion about your wedding.