Moon, seasons and the leap month: how a traditional calendar stitches two clocks together
Qingming usually falls on 4 or 5 April, but the Gregorian date of the Mid-Autumn Festival changes each year and may fall in September or October. Qingming is set by the Sun's position. Mid-Autumn is lunar month eight, day fifteen, so its month follows the lunar phases. A traditional lunisolar calendar has to reconcile these unequal cycles.
What is the traditional Chinese calendar?
The traditional Chinese calendar is a lunisolar calendar: months follow the Moon's phases, each starting at new moon and averaging 29.53059 days, while the year is kept in step with the Sun-defined seasons by inserting leap months. It is neither a purely lunar calendar nor a solar one, but a scheme that stitches the two cycles together.
Why does the lunar calendar need leap months?
Because twelve lunar months add up to only 354.37 days, about 10.88 days short of the 365.2422-day tropical year — after three years the gap exceeds a whole month. Without leap months the months would creep earlier every year and drift through the seasons; the calendar inserts seven leap months every nineteen years to pull them back into line, so leap months do not simply come every third year.
How are the 24 solar terms defined?
The 24 solar terms are defined by the Sun's position along the ecliptic, not by the Moon: one term every 15° of solar longitude, twenty-four in a full circle, with the twelve at multiples of 30° called major terms (zhongqi). Because they follow the Sun, their Gregorian dates are almost fixed — Qingming always falls around 4 or 5 April — and a lunar month that contains no major term is the candidate for a leap month.
Lunar phases set the months; the Sun sets the seasons; leap months correct the difference
The Gregorian date of Qingming barely moves, while Mid-Autumn changes place every year. This lesson puts the Moon’s months and the Sun’s seasons on one timeline, to see how they come apart and how a leap month pulls them back.
Start with Qingming and Mid-Autumn
Qingming keeps almost the same Gregorian date every year, while Mid-Autumn can slide from the middle of September to the start of October. They are printed in the same almanac, but they were not scheduled by the same rule.
Qingming is set by the Sun reaching a specified ecliptic longitude. Mid-Autumn is lunar month eight, day fifteen, which normally falls near the full Moon. The lunar-phase cycle and the solar year have different lengths, so the dates do not move together.
Three calendars use different references
A solar calendar matches the seasons and divides months administratively; the civil calendar Taiwan calls 國曆 is one of these. A lunar calendar matches lunar phase, twelve synodic months to a year, and lets the seasons slip away.
The Chinese agricultural calendar is lunisolar: months set by conjunction, years tied to the Sun by the solar terms. So the first falls near new Moon, the fifteenth near full Moon, and Lunar New Year never slides into summer.
Twelve months fall eleven days short
A synodic month averages 29.53059 days, so twelve make 354.37; the tropical year is 365.2422. That is 10.88 days a year, and three years of it is more than a whole month.
A purely lunar calendar does not correct the gap, so its festivals pass through the seasons. Ramadan in the Islamic calendar is a familiar example. This follows directly from arranging months by lunar phase without tying the year to a season.
The twenty-four terms measure the Sun
A solar term is not a forecast; it is the Sun’s position on the ecliptic. One term every 15° gives exactly twenty-four in a circuit, and the twelve at multiples of 30° are the principal terms.
Terms and the Gregorian calendar both follow the tropical year, so a term’s Gregorian date moves only two or three days. Its lunisolar date jumps around instead, because those months are set by conjunction.
A month without a principal term is a leap-month clue
Principal terms come about every 30.44 days and a synodic month is 29.53. Sooner or later one month runs from start to finish without a principal term in it, and that month is the leap-month candidate.
The judgement belongs to a whole calendar year: count from the month holding the winter solstice to the next such month, and if there are thirteen, the first without a principal term becomes the leap month. Real scheduling also weighs the exact conjunction instant and the time zone in use.
Calendar rules, almanacs and the limits of this model
農曆 is the set of astronomical rules; 農民曆, the printed almanac, adds sexagenary cycles, zodiac animals and auspicious notes on top. This lesson covers only the astronomical layer.
The interactive model uses a constant-rate Sun and a mean month length. It can show you the reasoning, but it will not give you the official lunisolar date of any year. For real dates, use an ephemeris from an authoritative source.
Nineteen-year calendar drift lab
Drag the timeline, or pick a solar term directly. The term wheel, the three-calendar table and the month track all read the same instant, so switching calendars shows you one day under three sets of rules rather than three unrelated scenes.
The wheel shows solar position and term category. Use the buttons below to select a term.
A month without a principal term is only a candidate. A real calendar also depends on which month holds the winter solstice, on the exact instant of conjunction, and on the civil time zone the calculation is stated in.
This is an idealised teaching model: the Sun moves at a constant rate, every month is exactly 29.53059 days, and day zero puts a winter solstice and a new-moon conjunction at the same instant. It demonstrates the reasoning, not the official calendar date of any real year.
Two limits of this model
The names of the terms record the climate and farming experience of the ancient Central Plains. The astronomical instants can be computed precisely, but the names are not a weather forecast for any part of Taiwan on that day — no snow on Xiaoxue is entirely normal.
The model on this page demonstrates a principle. It is not an official lunisolar-date converter and not a perpetual calendar. For the real lunisolar date, solar-term instant or almanac content of a given day, use an ephemeris with a stated time zone from an authoritative source.
Qingming and Mid-Autumn follow different cycles
Qingming usually falls around 4 or 5 April. Mid-Autumn is fixed at lunar month eight, day fifteen, but its Gregorian date can fall in September or October. Both appear in the same almanac, although their dates are set by different references.
Qingming is the instant when the Sun reaches a specified ecliptic longitude. The Mid-Autumn date comes from a calendar whose months follow lunar phases, and day fifteen is normally near the full Moon. The lunar-phase cycle and seasonal cycle have different lengths, so their dates cannot stay aligned without a correction.
What each calendar is actually watching
A solar calendar follows the Sun alone: the year matches the seasons, and month lengths are an administrative division with no link to lunar phase. The everyday Taiwanese word 國曆 (guoli) means exactly this — the civil Gregorian calendar. A lunar calendar follows the Moon alone: each month opens at conjunction, twelve months make a year, and the seasons are left to fall where they will.
The Chinese agricultural calendar, 農曆, belongs to a third family: it is lunisolar. Months are still set by conjunction, but the year is tied back to the Sun through the twenty-four solar terms. That is why the first of the month sits near new Moon and the fifteenth near full Moon, while Lunar New Year never drifts out of late winter. Calling it simply a lunar calendar drops the half of it that has been watching the Sun all along.
Twelve months do not fill a year
A synodic month averages 29.53059 days, so twelve of them come to 354.37 days. One orbit of Earth around the Sun, the tropical year, is 365.2422 days. The gap of 10.88 days is where this whole lesson starts.
The 29.53 days run from one new moon to the next. The Moon's actual trip around Earth takes only 27.3 days; the extra two are there because Earth has moved along its own orbit in the meantime. The moon-phase lesson has the diagram for why the two periods differ.
The gap does not stop after one year. It is 21.75 days in the second and 32.63 days in the third — already more than a full month. Switch the model to the lunar calendar, drag to year three, and you can watch the start of the year march forward while the season underneath it moves on.
Left alone, a new year that began in late winter walks into autumn, then into summer, and comes back round after roughly thirty-three years. To keep months tied to the Moon and years tied to the seasons, there is one move left: occasionally give a year one extra month.
A purely lunar calendar moves through the seasons
The Islamic calendar is a purely lunar calendar still in use. Its year has twelve synodic months and no leap month, so Ramadan arrives about eleven days earlier on a solar calendar each year and passes through every season over a few decades. This is the intended result of a system that aligns months with lunar phases without fixing them to a season.
Set the model to the lunar calendar and let it run. The start of the year moves earlier and the seasonal offset grows. Switch back to the lunisolar calendar to compare how inserting leap months keeps the year within a similar seasonal range.
The twenty-four terms follow solar longitude
A solar term is defined by the position of the Sun, not by the weather. The path the Sun appears to trace across a year is the ecliptic, and the angle measured along it is ecliptic longitude. Under the true-qi rule in force today, a term is set every 15°, which divides the full 360° into exactly twenty-four.
The ecliptic is Earth's orbital plane projected onto the sky. Why the Sun, Moon and planets all keep close to it is where the ecliptic-plane lesson starts, back at how the solar system formed.
Those twenty-four split into two kinds. The twelve at whole multiples of 30° are the principal terms, including both equinoxes and both solstices. The twelve halfway between them are sectional terms, including Lichun, Qingming and Bailu. Leap-month placement uses the twelve principal terms.
Solstices mark day-length extremes, but the earliest sunrise and latest sunset fall on other dates. The sunrise and sunset times lesson separates the two clocks.
A term is an instant, not a whole day. The moment the Sun crosses that longitude is the term; the date printed in an almanac is simply the day that moment falls on. It is also why the same term can land on different dates in different time zones.
Why solar terms stay near the same Gregorian dates
Solar terms are set by the Sun's position, while the Gregorian calendar approximates the tropical year. Their dates therefore stay close: Qingming usually falls from 4 to 6 April, and the winter solstice from 21 to 23 December.
What movement is left comes from the Gregorian calendar approximating 365.2422 days with whole days and leap days: one extra day every four years, skipped in century years, restored every fourth century. The same term in the lunisolar calendar is a different story — its months are set by conjunction, so a term lands on a different lunisolar date every year.
Principal terms and leap-month placement
The lunisolar answer is to make the whole year thirteen months when that is needed, rather than padding the end of it. The clue lives in the principal terms: they come about every 30.44 days while a synodic month is only 29.53, so sooner or later a month passes from start to finish without one.
The normalised rule this model applies is: start from the month containing the winter solstice and count up to, but not including, the next such month. If that run holds thirteen months, the first month in it with no principal term is the leap-month candidate. The solstice month always contains Dongzhi, so it can never be the candidate itself.
A published calendar must also handle the exact instants of conjunction and winter solstice and the civil time zone used. In rare cases, two consecutive months can contain no principal term, so the complete calendar rules are needed to choose between them. The on-screen label says candidate because this model demonstrates the main rule rather than calculating an official calendar year.
Nineteen tropical years come to 6939.6018 days and 235 synodic months to 6939.68865 — less than 0.09 days apart. Take away the 228 months that nineteen ordinary years already hold and seven are left over, which is all the familiar "seven leap months in nineteen years" claims. It tells you roughly how often a leap month turns up; it is not a table you can copy, and in practice the gap between two leap months is sometimes two years and sometimes three.
235 synodic months = 6939.68865 days · 19 tropical years = 6939.6018 days · difference 0.08685 days
From mean qi to true qi
Han-dynasty sources already record the complete set of twenty-four terms. The Taichu calendar adopted them in an official calendar system, and later calendars continued to revise the observations and calculations.
Early practice used mean qi: divide the length of a year into twenty-four equal parts of about 15.22 days each. It is easy to compute, but Earth does not orbit at a constant speed — faster near perihelion, slower near aphelion — so the mean-qi instants drift away from where the Sun actually is.
The Qing-dynasty Shixian calendar moved to true qi, setting a term at every 15° of solar longitude. Adjacent solar terms are about 14.7 to 15.7 days apart; adjacent principal terms are about 29.4 to 31.4 days apart. This page uses a constant-rate Sun to keep the model easy to inspect, so it is closer to mean qi. Modern calendars use true qi.
Calendar rules and almanac content are different layers
農曆 names the astronomical rules: conjunction fixes the first of the month, principal terms decide the leap month, and solar terms are marked alongside. 農民曆, the printed almanac, carries those dates plus sexagenary cycles, zodiac animals, auspicious-and-inauspicious notes and directions.
Both layers print on one page, which makes them easy to mistake for one thing. The first can be checked against astronomical calculation; the second belongs to folk tradition. This site makes no claim about its efficacy and offers no date-selection advice.
Compare the Gregorian dates of Mid-Autumn and Qingming over three consecutive years. Mid-Autumn moves noticeably, while Qingming usually changes by only a day or two. The first reflects changes in lunisolar months; the second follows the solar year.
Four things to try
Drag the time back to year 0. All three calendars open their year on the same day and all three seasonal offsets read 0 days.
Switch to the lunar calendar and drag through to year three, watching the seasonal drift go from about −10.9 days to about −32.6 days.
Switch back to the lunisolar calendar and find the block on the month track with no principal term, marked as the leap-month candidate. Notice that the months on either side of it each hold one.
Select Dongzhi, then Xiazhi. Solar longitude jumps from 270° to 90° and the month track moves along with it to half a year later.
Three calendars side by side
This table states the division of labour once; the comparison rows in the lab are the live version of it at whatever instant you have selected.
Calendar
What it follows
Do months track lunar phase?
Do seasons stay put?
Does it need intercalation?
Solar calendar
The Sun’s annual position
No; month length is an administrative division
Yes
A leap day, never a leap month
Lunar calendar
The lunar phase cycle
Yes; the first falls near conjunction
No; about 10.88 days earlier each year
None
Lunisolar calendar
Months by the Moon, years by the solar terms
Yes; the first falls near conjunction
Yes, within a month either way
A leap month when one is needed
FAQ
Is 農曆 the same thing as a lunar calendar?
A solar calendar follows the Sun alone: the year matches the seasons, and month lengths are an administrative division with no link to lunar phase. The everyday Taiwanese word 國曆 (guoli) means exactly this — the civil Gregorian calendar. A lunar calendar follows the Moon alone: each month opens at conjunction, twelve months make a year, and the seasons are left to fall where they will. The Chinese agricultural calendar, 農曆, belongs to a third family: it is lunisolar. Months are still set by conjunction, but the year is tied back to the Sun through the twenty-four solar terms. That is why the first of the month sits near new Moon and the fifteenth near full Moon, while Lunar New Year never drifts out of late winter. Calling it simply a lunar calendar drops the half of it that has been watching the Sun all along. 農曆 names the astronomical rules: conjunction fixes the first of the month, principal terms decide the leap month, and solar terms are marked alongside. 農民曆, the printed almanac, carries those dates plus sexagenary cycles, zodiac animals, auspicious-and-inauspicious notes and directions.
Why do most of the twenty-four solar terms land on nearly fixed Gregorian dates?
Solar terms are set by the Sun's position, while the Gregorian calendar approximates the tropical year. Their dates therefore stay close: Qingming usually falls from 4 to 6 April, and the winter solstice from 21 to 23 December. What movement is left comes from the Gregorian calendar approximating 365.2422 days with whole days and leap days: one extra day every four years, skipped in century years, restored every fourth century. The same term in the lunisolar calendar is a different story — its months are set by conjunction, so a term lands on a different lunisolar date every year.
How is a leap month decided — is it really every three years?
The lunisolar answer is to make the whole year thirteen months when that is needed, rather than padding the end of it. The clue lives in the principal terms: they come about every 30.44 days while a synodic month is only 29.53, so sooner or later a month passes from start to finish without one. The normalised rule this model applies is: start from the month containing the winter solstice and count up to, but not including, the next such month. If that run holds thirteen months, the first month in it with no principal term is the leap-month candidate. The solstice month always contains Dongzhi, so it can never be the candidate itself. A published calendar must also handle the exact instants of conjunction and winter solstice and the civil time zone used. In rare cases, two consecutive months can contain no principal term, so the complete calendar rules are needed to choose between them. The on-screen label says candidate because this model demonstrates the main rule rather than calculating an official calendar year.
The astronomical definitions, term rules and intercalation account come from the official pages above; the interactive model, code and wording are this site’s own. The model uses a constant-rate Sun and a mean synodic month, which is closer to mean qi than to current practice, and it cannot replace a published almanac, a calendar conversion, or any date-selection use.