How to calculate when Lunar Leap Years are?
The Lunisolar/Lunar Calendar is familiar in the lives of Vietnamese people, but not everyone fully understands how this traditional calendar is calculated, how leap years are determined, and which month is chosen as the leap month in a leap year. For the most thorough understanding, everyone can review the basic concepts in the calculation of the Vietnamese Lunar Calendar (which is actually a branch of lunisolar calendars) here.
In general, the core reason of the Chinese/Vietnamese Lunar Calendar is that the Lunar Month cycle (or synodic month, or lunation) appears on average once every 29.53059 days. Meanwhile, the solar cycle from one Vernal Equinox to the next — the basis for calculating the solar calendar we commonly use — lasts 365.24219 days. Since a normal lunar year has 12 months, it totals only 354.36708 days, falling short by 10.87511 days compared to a solar year.
After 3 years, the deficit accumulates to about 32.62533 days, equivalent to more than one lunar month. At that point, a leap month is added to the lunar year. Thus, a lunar leap year consists of 13 months, equivalent to approximately 383.89767 days.
Specifically, the calculation of the Vietnamese Lunar Calendar includes the following principles:
- The first day of a Lunar Month is the day that includes New Moon. For example, there is a New Moon at 16:57 April 21st 2031 (UTC+0), equivalent to 23:57 the same day in Vietnam (UTC+7). So with Vietnamese, April 21st 2031 is the first day of a Lunar Month. If by some reasons, this New Moon is shifted some minutes to 0:00 April 22nd 2031 (UTC+7), then Vietnamese will consider April 22nd 2031 is the first day of a Lunar Month instead. A Lunar Month can only have an integer number of days, so it will be 29 or 30 on a case-by-case basis.
- Similarly, the day that includes the beginning moment of a Solar Term will be considerd the beginning day of this Solar Term. It's true even if the beginning moment is 23:59:59.999. The Lunar Month of the beginning day of a Solar Term is considered "owning" this Solar Term.
- Đông Chí (or Dongzhi), the last Solar Term in a common solar year with Ecliptic Longitude 270° and the beginning marked by the December Solstice, must be scheduled in Lunar Eleventh Month.
- Normally there are 12 lunar months between an Lunar Eleventh Month and the next Lunar Eleventh Month. But if there are 13 lunar months in this duration, it's when a lunar month will be considered the leap month. The leap month will be the first one in this duration that not own any Major Solar Term.
- Vietnamese Lunisolar Calendar is based on UTC+7 (GMT+7), it's actually slightly different with Chinese Lunisolar Calendar which based on UTC+8 (although all other rules are the same). According to the above example, there is a New Moon at 16:57 April 21st 2031 (UTC+0). With Vietnamese, April 21st 2031 is the first day of a Lunar Month because it's based on UTC+7. But with Chinese, the first day of a Lunar Month is April 22nd instead because based on UTC+8, the New Moon is at 0:57 this day.
We can say that this calculation method is quite flexible and varies depending on astronomical developments at any given time.
Does the Vietnamese Lunar Calendar follow the Metonic cycle?
Currently, some enthusiasts have discovered a regular "phenomenon" of the Vietnamese Lunar Calendar: if you take the corresponding solar year and divide it by 19, the remainder can help determine whether that year is a leap year. If the remainder is 0, 3, 6, 9, 11, 14, or 17, that year is a leap year. For example, if you divide 2025 by 19, the remainder is 11, and the lunar year of 2025 is a leap year.
This rule is none other than the Metonic cycle, named after the 5th-century BCE Greek researcher, which states that there will be 7 leap years in every 19 lunisolar years. This rule was adopted as a rigid regulation in calendars of the ancient Babylonians, modern Jews, and even within the Buddhist calendar.
Of course, there are discrepancies between different calendars. For instance, according to the Hebrew calendar, their year 5785 began on October 2, 2024 (Gregorian Calendar), and the remainder rule when dividing their year number by 19 differs. For them, if the remainder is 0, 3, 6, 8, 11, 14, or 17, that year is a leap year. Overall, however, it remains an alternating cycle of leap year intervals: three 3-year intervals followed by one 2-year interval, then two 3-year intervals followed by one 2-year interval, and repeating...
The principles for calculating the Vietnamese Lunar Calendar do not mention the Metonic cycle at all, but why does calculating by the Metonic cycle still match? To clarify that question, we shall just examine a 19-year cycle.
First, it can be affirmed that if a New Moon is assigned as the 1st day of 11th lunar month, it must occur before the Winter Solstice by no more than one lunar month which is equivalent to 29.53059 days. A normal lunar year has only 354.36708 days, so after one year, the same lunar date shifts 10.87511 days earlier compared to the solar calendar. Thus, the new moon of 11th lunar month in the preceding year must occur no more than 18.65548 days (29.53059 - 10.87511) before the Winter Solstice of the corresponding year. If it exceeds this threshold, the period from the 1st of 11th lunar month of that year to the 1st of 11th lunar month of the following year will contain an additional leap month.
Now let us take an ideal case in that the lunar cycles occur evenly and the time interval between the New Moon of 11th lunar month and the Winter Solstice of year 0, denoted as T0, is 1 day. Thus, the corresponding time interval T1 of year 1 will be 11.87511 days after a normal year (1 + 10.87511). We call the period from the 1st of 11th lunar month in year 0 to the 1st of 11th lunar month in year 1 as Phase 1.
- Phase 1: Normal phase - T1 = 11.87511
- Phase 2: Normal phase - T2 = 22.75022 (11.87511 + 10.87511). At this point, the 18.65548 threshold has been crossed
- Phase 3: Leap phase - T3 = 4.09474 (22.75022 + 10.87511 - 29.53059). After accounting for the leap month, T3 falls back below the 18.65548 threshold
- Phase 4: Normal phase - T4 = 14.96985
- Phase 5: Normal phase - T5 = 25.84496
- Phase 6: Leap phase - T6 = 7.18948
- Phase 7: Normal phase - T7 = 18.06459
- Phase 8: Normal phase - T8 = 28.9397
- Phase 9: Leap phase - T9 = 10.28422
- Phase 10: Normal phase - T10 = 21.15933
- Phase 11: Leap phase - T11 = 2.50385
- Phase 12: Normal phase - T12 = 13.37896
- Phase 13: Normal phase - T13 = 24.25407
- Phase 14: Leap phase - T14 = 5.59859
- Phase 15: Normal phase - T15 = 16.4737
- Phase 16: Normal phase - T16 = 27.34881
- Phase 17: Leap phase - T17 = 8.69333
- Phase 18: Normal phase - T18 = 19.56844
- Phase 19 (Phase 0): Leap phase - T19 = 0.91295999999999
- Phase 20 (Phase 1): Normal phase - T20 = 11.78807
The above hypothetical calculations match the Metonic cycle. The concept of "phase" here differs from a lunar year, but the occurrence of lunar leap years will also match if the leap months within a phase are all taken from the 1st month onwards. However, one may notice that T0 decreases slightly after a 19-year cycle, hinting that the Metonic cycle may deviate in the future. Theoretically, when T0 equals 0 (T1 = 0 + 10.87511), the next phase 0 will not be a leap year. Let us also examine this hypothetical scenario:
- Phase 1: Normal phase - T1 = 10.87511
- Phase 2: Normal phase - T2 = 21.75022
- Phase 3: Leap phase - T3 = 3.09474
- Phase 4: Normal phase - T4 = 13.96985
- Phase 5: Normal phase - T5 = 24.84496
- Phase 6: Leap phase - T6 = 6.18948
- Phase 7: Normal phase - T7 = 17.06459
- Phase 8: Normal phase - T8 = 27.9397
- Phase 9: Leap phase - T9 = 9.28422
- Phase 10: Normal phase - T10 = 20.15933
- Phase 11: Leap phase - T11 = 1.50385
- Phase 12: Normal phase - T12 = 12.37896
- Phase 13: Normal phase - T13 = 23.25407
- Phase 14: Leap phase - T14 = 4.59859
- Phase 15: Normal phase - T15 = 15.4737
- Phase 16: Normal phase - T16 = 26.34881
- Phase 17: Leap phase - T17 = 7.69333
- Phase 18: Normal phase - T18 = 18.56844
- Phase 19 (Phase 0): Normal phase - T19 = 29.44355
- Phase 20 (Phase 1): Leap phase - T20 = 10.78807
The time span from phase 0 of the Year Giáp Ngọ 2014 to phase 0 of the Year Quý Sửu 2033 is precisely such a case as calculated above, where T2014 approximately equaling 0 day leads to T2033 equaling 29. The year 2033 could have deviated from the Metonic cycle were it not for a coincidence: the leap month of the subsequent phase 1 fell into the 11th lunar month so 2033 remains a leap year. Below is the actual recorded timing of New Moon occurring before the Winter Solstice during this period, with New Moon data sourced from AstroPixels:
- Year 2014 (Phase 0): T2014 = 0 days
- Year 2015 (Phase 1): T2015 = 11 days
- Year 2016 (Phase 2): T2016 = 22 days
- Year 2017 (Phase 3): T2017 = 3 days
- Year 2018 (Phase 4): T2018 = 15 days
- Year 2019 (Phase 5): T2019 = 26 days
- Year 2020 (Phase 6): T2020 = 7 days
- Year 2021 (Phase 7): T2021 = 17 days
- Year 2022 (Phase 8): T2022 = 28 days
- Year 2023 (Phase 9): T2023 = 9 days
- Year 2024 (Phase 10): T2024 = 20 days
- Year 2025 (Phase 11): T2025 = 1 day
- Year 2026 (Phase 12): T2026 = 13 days
- Year 2027 (Phase 13): T2027 = 24 days
- Year 2028 (Phase 14): T2028 = 5 days
- Year 2029 (Phase 15): T2029 = 16 days
- Year 2030 (Phase 16): T2030 = 27 days
- Year 2031 (Phase 17): T2031 = 8 days
- Year 2032 (Phase 18): T2032 = 18 days
- Year 2033 (Phase 0): T2033 = 29 days
- Year 2034 (Phase 1): T2034 = 11 days
Of course, not every case where phase 0 has T0 = 0 (or nearly) results in the next phase 0 deviating from the Metonic cycle, because intervals still retain decimal fractions and cycles fluctuate in reality. "Phase 0" years such as 1881, 1900, 1938, 1995, or 2052 are all proofs of this. With natural variations, even calendars rigidly regulated by the fixed Metonic cycle require adding or removing days to compensate for discrepancies.
To summarize, we can see that the calculating method of Chinese/Vietnamese Lunar Calendar and the Metonic cycle both solve the nonperfect relationship between lunar month and solar year cycles, with the same fundamental principles of compensating when the difference is too far. So they match most of time by nature, but sometimes the Metonic cycle is just not as flexible.
The advantage of the Metonic cycle is perhaps that it allows anyone to calculate leap years on their own, making it as common as the leap year rule of the solar calendar. Calculating the lunar calendar like the Chinese/Vietnamese method might be slightly complex for an average person, but nowadays calendar calculation apps like the Lunisolar Calendar app can easily handle it with any astronomical forecast source.
Anyway, according to HAHTech's calculations, the Vietnamese Lunar Calendar matches the Metonic cycle from as early as 1872 all the way to 2146 (outside this period, it occasionally deviates). Thus, during the era we live in, Vietnamese people can safely take the corresponding solar year, divide it by 19, and determine leap years with remainders of 0, 3, 6, 9, 11, 14, or 17.
Leap 12th Month, Leap 1st Month: Which years do they fall in?
Another misconception about the Vietnamese Lunar Calendar is that it does not take the 12th month (Tháng Chạp) or 1st month (Tháng Giêng) as a leap month. However, no such rule exists. Calendars with rigid Metonic regulations usually fix which month becomes the leap month during a leap year, whereas the Chinese/Vietnamese Lunar Calendar is more flexible according to the principle that whichever month lacks a Major Solar Term (Trung Khí) becomes the leap month.
We can imagine that in a year with a leap 12th month, the 30th of the 12th month would not yet be the final day before Tet, as another leap 12th month would need to follow. Or in a year with a leap 1st month, people would finish celebrating Tet only to find that after the 1st month ends, another 1st month begins. Should we avoid this scenario by ruling out the 12th or 1st months as leap months? The problem is that if a 12th or 1st month without a Major Solar Term is found, no other month will satisfy this condition during the same period.
According to experts' metaphorical explanations, the Major Solar Terms from one Winter Solstice to the next form the partitions of 12 boxes, while the 13 New Moons of a leap year act like 13 grains of rice (a normal year has only 12 grains). When distributing 13 grains of rice into 12 boxes, at least one box will contain 2 grains of rice, and between these 2 grains of rice, there is no Major Solar Term partition.
Yet it can also be asserted that a second "box" containing 2 "grains of rice" is unlikely, because if it did, one "box" would be left with no "grains of rice" at all. The average distance between 2 "partitions" (meaning 2 Major Solar Terms) is about 30.43685 days (365.24219 / 12); meanwhile, the distance between 2 "grains of rice" is only 29.53059 days (the lunar cycle), which is nearly 1 day shorter. That is why we can say that if a 12th or 1st month (or any month) without a Major Solar Term is found, no other month will meet this condition during the same period.
Of course, the distance between 2 Major Solar Terms is not fixed either. The time from the Vernal Equinox to the Autumnal Equinox is about 185 days, whereas the time from the Autumnal Equinox to the following Vernal Equinox is only about 180 days. Thus, boxes numbered 1, 2, and 3 after each Winter Solstice will be shorter than the following 6 boxes. Surely for this reason, the probability of having a leap 11th month, leap 12th month, or leap 1st month is lower than other months.
According to HAHTech's calculations, there was a leap 1st month in 1651, and there will be another leap 1st month in 2148. Meanwhile, there was a leap 12th month in 1498, and the next leap 12th month will be in... 3358. As mentioned above, the year 2033 will have a leap 11th lunar month.
Look up which month is leap month in Lunisolar Calendar
Extracted from the Lunisolar Calendar app, HAHTech has built an additional mini-tool to help look up a series of lunar years to see whether they are leap years and, if so, which month is leap. This allows everyone to review the most recent lunar leap years, find out when is the next lunar leap year is, or study the leap year cycle over a long period.
- Year: It has to be a positive integer. If you are willing to look up a Before Common Era year, do select the BCE option below
- BCE or CE: It is the option for Before Common Era or Common Era
- New Moon soure: It is the data source for New Moon moments. This option will be explained more detail here
- UTC offset: It is your timezone's time offset. For example, Vietnamese use UTC+7
- Language: This tool will use the chosen language, Vietnamese of English. The default language is English
After entering the input data, everyone can click the "Search" button. If there are any incorrect inputs, the system will prompt you to re-enter them.
The results returned will be a list of lunar years (calculated based on the corresponding solar year) within a 100-year span, starting from 50 years prior to the selected year up to 50 years after.
If any year lacks lunar calendar data, the result will display 0 for that year.
You can check the Vietnamese version of this mini-tool here.