Lunisolar calendar

A lunisolar calendar is a calendar whose date indicates both the moon phase and the time of the solar year. If the solar year is defined as a tropical year then a lunisolar calendar will give an indication of the season; if it is taken as a sidereal year then the calendar will predict the constellation near which the full moon may occur. Usually there is an additional requirement that the year has a whole number of months, in which case most years have 12 months but every second or third year has 13 months.

Examples

The Hebrew, Hindu lunisolar, Buddhist, Tibetan calendars, Chinese calendar used alone until 1912 (and then used along with the Gregorian calendar) and Korean calendar (used alone until 1894 and since used along with the Gregorian calendar) are all lunisolar, as was the Japanese calendar until 1873, the pre-Islamic calendar, the republican Roman calendar until 45 BC (in fact earlier, because the synchronization to the moon was lost as well as the synchronization to the sun), the first century Gaulish Coligny calendar and the second millennium BC Babylonian calendar. The Hebrew, Chinese and Coligny lunisolar calendars track the tropical year whereas the Buddhist and Hindu lunisolar calendars track the sidereal year. Therefore the first three give an idea of the seasons whereas the last two give an idea of the position among the constellations of the full moon. The Tibetan calendar was influenced by both the Chinese and Hindu calendars. The Islamic calendar is a lunar, but not lunisolar calendar because its date is not related to the sun. The Julian and Gregorian Calendars are solar, not lunisolar, because their dates do not indicate the moon phase — however, without realising it, most Christians do use a lunisolar calendar in the determination of Easter.

There are some lunisolar calendar reform proposals: One is the Hermetic Lunar Week Calendar which normally consists of 12 lunar months and a leap month every 2 or 3 years, and with a year that always starts near the vernal equinox. Another is The Simple Lunisolar Calendar whose year always begins between Gregorian December 3 and January 1. Also there is the Meyer-Palmen Solilunar Calendar whose year always begins near the vernal equinox by using a 2498258 days in 84599 months in a 6840-year-cycle rule.

Determining leap months

To determine when an embolismic month needs to be inserted, some calendars rely on direct observations of the state of vegetation, while others compare the ecliptic longitude of the sun and the phase of the moon.

On the other hand, in arithmetical lunisolar calendars, an integral number of synodic months is fitted into some integral number of years by a fixed rule. To construct such a calendar, the average length of the tropical year is divided by the average length of the synodic month, which gives the number of average synodic months in a year as 12.368266...

Continued fractions of this decimal value give optimal approximations for this value. So in the list below, after the number of synodic months listed in the numerator, an integer number of tropical years as listed in the denominator have been completed:

12	/	1	=	12	(error = -0.368266... synodic months/year)
25	/	2	=	12.5	(error = 0.131734... synodic months/year)
37	/	3	=	12.333333...	(error = 0.034933... synodic months/year)
99	/	8	=	12.375	(error = 0.006734... synodic months/year)
136	/	11	=	12.363636...	(error = -0.004630... synodic months/year)
235	/	19	=	12.368421...	(error = 0.000155... synodic months/year)
4131	/	334	=	12.368263...	(error = -0.000003... synodic months/year)

The 8-year cycle (99 synodic months, including 3 embolismic months) was used in the ancient Athenian calendar. The 8-year cycle was also used in early third-century Easter calculations (or old Computus) in Rome and Alexandria. It is called an Octaeteris (plural Octaeterides).

The 19-year cycle (235 synodic months, including 7 embolismic months) is the classic Metonic cycle, which is used in most arithmetical lunisolar calendars. It is a combination of the 8- and 11-year period, and whenever the error of the 19-year approximation has built up to a full day, a cycle can be truncated to 8 or 11 years, after which 19-year cycles can start anew. Meton's cycle had an integer number of days, although Metonic cycle often means its use without an integer number of days. It was adapted to a mean year of 365.25 days by means of the 4×19 year Callipic cycle (used in the Easter calculations of the Julian calendar).

Rome used an 84-year cycle from the late third century until 457. Early Christians in Britain and Ireland also used an 84-year cycle until the Synod of Whitby in 664. The 84-year cycle is equivalent to a Callipic 4×19-year cycle (including 4×7 embolismic months) plus an octaeteris and so has a total of 1039 synodic months (including 31 embolismic months). This gives an average of 12.369047... synodic months per year (with error=0.011123... synodic months/year, a less good approximation than the Metonic 19-year cycle).

The last listed approximation with the 334-years cycle (4131 synodic months, including 123 embolismic months) is very sensitive to the adopted values for the lunation and year, especially the year. There are different possible definitions of the year, other approximations may be more accurate. For example (4366/353) is more accurate for a vernal equinox tropical year and (1979/160) is more accurate for a sidereal year.

Calculating a "leap month"

A rough idea of the frequency of the intercalary or leap month in all lunisolar calendars can be obtained by the following calculation, using approximate lengths of months and years in days:

• Year: 365.25, Month: 29.53
• 365.25/(12 × 29.53) = 1.0307
• 1/0.0307 = 32.57 common months between leap months
• 32.57/12 − 1 = 1.7 common years between leap years

A representative sequence of common and leap years is ccLccLcLccLccLccLcL, which is the classic nineteen-year Metonic cycle. The Hebrew and Buddhist calendars restrict the leap month to a single month of the year, so the number of common months between leap months is usually 36 months but occasionally only 24 months elapse. The Chinese and Hindu lunisolar calendars allow the leap month to occur after or before (respectively) any month but use the true motion of the sun, so their leap months do not usually occur within a couple of months of perihelion, when the apparent speed of the sun along the ecliptic is fastest (now about 3 January). This increases the usual number of common months between leap months to roughly 34 months when a doublet of common years occurs while reducing the number to about 29 months when only a common singleton occurs.

Cycles

Complete lunisolar cycles with a whole number of days:

Years	Months	Days	Leap Months	Solar Leap Days	Mean Year	Mean Month	Comments
19	235	6940	7	5	365.2631...	29.531914...	Metonic Cycle
76	940	27759	28	19	365.25	29.530851...	Callippic cycle
304	3760	111035	112	75	365.2467...	29.530585...	Hipparchic cycle
334	4131	121991	123	81	365.2425...	29.530622...
372	4601	135870	137	90	365.2419...	29.530536...	Gregoriana
649	8027	237042	239	157	365.2419...	29.530584...	Old Lufkan Calendar
706	8732	257861	260	171	365.2422...	29.530577...	Two 353-year cycles
1689	20890	616894	622	409	365.2421...	29.530588...	Goddess Lunar Calendar
1803	22300	658532	664	437	365.24237...	29.530582...	Archetypes Calendar

The 372-year cycle is equivalent to six 62-year leap week cycles and the 1803-year cycle also has a whole number of weeks.

See also

• Month
• Solar calendar
• Lunar calendar

External links

• Perpetual Chinese Lunar Program
• Lunisolar Calendar
• Lunisolar calendar cycles