A lunisolar calendar is a calendar in many cultures 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 have a whole number of months, in which case most years have 12 months but every second or third year has 13 months.
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The Hebrew, Buddhist, Hellenic, Hindu lunisolar, Tibetan, Chinese (used alone until 1912 and then used along with the Gregorian), and Korean (used alone until 1894 and since used along with the Gregorian) calendars are all lunisolar, as was the Japanese calendar until 1873, the Hawaiian calendar, the preIslamic 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, the Byzantine Calendar, and the second millennium BC Babylonian calendar. The Chinese, Coligny and Hebrew^{[1]} lunisolar calendars track more or less 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 English also used a lunisolar calendar before their conversion to Christianity^{[citation needed]}.The twelvemonth, thirtyday Hawaiian calendar incorporates the same sophisticated astronomy used by the ancient Polynesians to navigate throughout the Pacific and details when plants should be sown and fish caught. ^{[2]}
The Islamic calendar is a lunar, but not a 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, a lunisolar calendar is used in the determination of the Christian celebration of Easter.
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. The Hawaiians observe the movement of specific stars and insert months accordingly.
On the other hand, in arithmetical lunisolar calendars, an integral number of months is fitted into some integral number of years by a fixed rule. To construct such a calendar (in principle), 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 tropical 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, approximately 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)
Note however that in none of the arithmetic calendars is the average year length exactly equal to a true tropical year. Different calendars have different average year lengths and different average month lengths, so the discrepancy between the calendar months and moon is not equal to the values given above.
The 8year cycle (99 synodic months, including 3 embolismic months) was used in the ancient Athenian calendar. The 8year cycle was also used in early thirdcentury Easter calculations (or old Computus) in Rome and Alexandria.
The 19year 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 11year period, and whenever the error of the 19year approximation has built up to a full day, a cycle can be truncated to 8 or 11 years, after which 19year 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 84year cycle for Easter calculations from the late third century until 457. Early Christians in Britain and Ireland also used an 84year cycle until the Synod of Whitby in 664. The 84year cycle is equivalent to a Callipic 4×19year cycle (including 4×7 embolismic months) plus an 8year cycle (including 3 embolismic months) and so has a total of 1039 months (including 31 embolismic months). This gives an average of 12.3690476... months per year. One cycle was 30681 days, which is about 1.28 days short of 1039 synodic months, 0.66 days more than 84 tropical years, and 0.53 days short of 84 sidereal years.
The next approximation (arising from continued fractions) after the Metonic cycle (such as a 334year cycle) is very sensitive to the values one adopts for the lunation (synodic month) and the year, especially the year. There are different possible definitions of the year so other approximations may be more accurate. For example (4366/353) is more accurate for a tropical year whereas (1979/160) is more accurate for a sidereal year.
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:
A representative sequence of common and leap years is ccLccLcLccLccLccLcL, which is the classic nineteenyear Metonic cycle. The Buddhist and Hebrew calendars restrict the leap month to a single month of the year; the number of common months between leap months is, therefore, usually 36, but occasionally only 24 months. Because 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, 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.


