# Lunisolar calendar

A lunisolar calendar ( Latin: luna 'moon' and sol 'sun') or a bound lunar calendar , like every lunar calendar, primarily contains 12 lunar months (→ lunation ) as calendar months . To approximate the solar year (→ tropical year ), a thirteenth lunar month is switched on on average every three years .

## Applications

The lunisolar calendars include

Most folks use solar calendars because they allow accurate synchronization with the seasons . Pure lunar calendars are known only to a handful.

## purpose

The older calendars were lunar calendars because they were based on surely observable celestial phenomena, namely the phases of the moon . For a solar calendar , the solar phases, which are much more difficult to determine, for example the equinoxes or the solstices , must be known.

A pure lunar calendar has no connection to the solar year or the seasons. It shifts backwards by about eleven days in each solar year. A lunisolar calendar, on the other hand, creates an approximate adjustment to the seasons, which determine religious ( seasonal feast dates) and economic (sowing and harvesting dates) life. It follows the solar year with a maximum deviation of ± 2 weeks.

## Astronomical basics

The long-term synchronization in a lunisolar calendar between months and years is possible every 19 years, because 19 solar years are roughly the same as 235 lunar months. This period of time, which is equal to 6940 days, is the Meton period , the resulting cycle is the Meton cycle .

When it turned out that 6940 days for 19 solar years is about a quarter of a day too many, the period was increased to four times its duration and this was set equal to 27,759 days. The Callippian period arose on which the Callippic cycle is based.

In lunisolar calendars, in which the average calendar year is kept at 365.25 days by a leap day every four years, the callipic period divided by four is applicable. It is the corrected Meton period to 6,939.75 days (6,939.75 ÷ 19 = 365.25).

## Construction of a lunisolar calendar

The construction of a lunisolar calendar is based on the lunar calendar . Calendar months are still either full months of 30 days or hollow months of 29 days. The previous lunar calendar years of 12 months and 354 days each (with a leap day of 355 days) remain as common calendar years and are only supplemented by occasional leap years. A 13th calendar month is appended to leap years.

It was already known in antiquity that, analogous to the Meton cycle, 19 calendar years consisted of 235 calendar months. 110 of them are hollow months, 125 are full months. That makes 6940 days, the length of the meton period. The composition of calendar years in antiquity is not known. The following construction could have been possible:

8 common years of 6 hollow and 6 full months each = 48 hollow months and 48 full months (354 days each)
4 common years of 5 hollow and 7 full months = 20 hollow months and 28 full months (each 355 days, with switching Day to adapt to the lunar year )
7 leap years with 6 hollow and 7 full months each = 42 hollow months and 49 full months (each 384 days)

This construction can be seen in the Jewish calendar , although years with 353, 383 and 385 days appear there due to religious traditions. The order of leap years, which has not been handed down for antiquity, consists in the Jewish calendar of the years 3, 6, 8, 11, 14, 17 and 19.

There is also an ancient description according to which hollow and full months do not follow each other by law:
every 235 months are set as full months. However, one day is omitted (switched off) every 64 days. This happens almost regularly 110 times in the 6940-day period, which indirectly turns full months into hollow months. However, the canceled day is usually not the 30th day of a full month. It is believed that this complicated rule was only applied in an astronomical calendar, not in a civil calendar.

In a Callipean lunisolar calendar, three 19-year periods of 6940 days each were followed by a 19-year period of 6,939 days, in which one day was omitted compared to the scheme described. Nothing is known about this detail either.

The difficulties in calculating the Easter date stem from the fact that, in contrast to the Jewish calendar, neither the Julian nor the Gregorian calendar are lunisolar calendars. In order to determine the spring full moon that determines Easter , a calendar calculation with months from a lunar calendar must be made . First, as there, one forms years of 354 days each. If the 13th full moon falls before March 22nd, the year is extended by a lunar calendar month ( moon jump ). That happens seven times in a meton period. Six moon jumps are given 30 days, the seventh with 29 days. Since the leap day, which is added every four years in the Julian calendar, extends the lunar calendar months with a share of 4.75 days to 19 years, the balance for 19 years is:

19 x 354 days + 6 x 30 days + 29 days + 4.75 days = 6939.75 days = corrected Meton period

The three leap days left out of the Gregorian calendar in 400 years do not change the procedure. The above balance remains, the “lost days” indirectly shift the calculated day of the spring full moon ( solar equation ).

### example

If you were to create a lunisolar calendar today, you can use the continued fraction for high accuracy :

 12 / 1 = 12 = [12] (Error = −0.368266 ... synodic months / year) 25 / 2 = 12.5 = [12; 2] (Error = 0.131734 ... synodic months / year) 37 / 3 = 12.333333 ... = [12; 2, 1] (Error = −0.034933 ... synodic months / year) 99 / 8 = 12.375 = [12; 2, 1, 2] (Error = 0.006734 ... synodic months / year) 136 / 11 = 12.363636 ... = [12; 2, 1, 2, 1] (Error = −0.004630 ... synodic months / year) 235 / 19 = 12.368421 ... = [12; 2, 1, 2, 1, 1] (Error = 0.000155 ... synodic months / year) 4131 / 334 = 12.368263 ... = [12; 2, 1, 2, 1, 1, 17] (Error = −0.000003 ... synodic months / year)

A cycle of 334 years can be divided into 17 19-year cycles with 235 months each and an 11-year block with 136 months (4131 - 17 * 235 = 136). First, the 19-year cycle is created, which can be based on the Easter calculation in the Gregorian calendar: each year is first provided with 12 months, which alternate between 30 and 29 days, which results in 354 days. For a (solar) year you first take 365 days; the correction is made in a later step. This gives a difference of 11 days per year - 209 in total (11 * 19). 228 (= 19 * 12) months have already been distributed and 7 are still missing. 209 days can be spread over the 7 months so that 6 months have 30 days and a month only 29 days. The leap months are distributed evenly in the 19-year block, so that the first month of the year always occurs after the new year of the solar year: Years 1, 3, 6, 9, 11, 14 and 17 (the 29-day leap month on year 17 ). This 19-year cycle can be run through 17 times. This is followed by an 11 year block that is constructed in a similar way (leap months in year 1, 3, 6 and 9). But there is one day too many (11 * 11 - 4 * 30 = 1) for the leap months. So that the length of the leap month does not have 3 different values, this day is added to the last leap month of the seventeenth 19-year cycle, which was initially determined to be 29 days long. Now all that remains is to correct the solar year. For the sake of accuracy, this is done analogously to the Iranian calendar : 8 leap days in 33 years. This leap day is simply added to the twelfth month of the (solar) leap year, as this has 29 days and is then 30 days long. If you want to synchronize this with the 334 years, you don't start with a new 33-year cycle immediately after 330 years, but insert a 4-year block with a leap day. However, this reduces the accuracy of the (solar) year from 1 day difference in 4269 years to 1 day in 3077 years (corresponds almost exactly to the inaccuracy of the Gregorian calendar of 1 day in 3225 years). For the months, there would be an offset of one day after 2441 years.
The advantage of this calendar construction is the consistently same length of eleven months in a year - only the twelfth month and the leap month vary in length - and every 19 years the (sun) year begins at the beginning of the first month. The disadvantage is the uneven distribution of the 30-day months. Sometimes 4 of these months follow one another (month 11, month 12 in a sunny year, leap month with 30 days, first month of the following year), which then sometimes leads to the beginning of the month deviating from the new moon by 1 to 2 days. This deviation can only be avoided by uncoupling the months from the year or an astronomical calculation of the beginning of the month and year. This means that the first day of the first month only falls on the first day of the (solar) year in 3.386% of the years, instead of 5.389% of the years. It can also take up to 57 years for the beginning of the first month of the year to coincide with the beginning of the solar year. In addition, there is no longer a fixed length of the individual months - in a maximum of 4 consecutive years a month has the same number of days (refers to the uncoupling of the months from the year so that they run more synchronously with the moon and not on an exact astronomical calculation).

## literature

• LE Dogett: Calendars. In: Explanatory Supplement to the Astronomical Almanac. University Science Books, Sausalito CA ( English ), online .
• BL van der Waerden : Greek Astronomical Calendars. II. Callippos and his Calendar. In: Archive for History of Exact Sciences. 29, 2, 1984, , pp. 115-124.

Wiktionary: Lunisolar calendar  - explanations of meanings, word origins, synonyms, translations

## Notes and individual references

1. The length of the solar year was very well known in ancient times. The Solar-Lunar concept comes about because the farmers had to orientate themselves on the solar year, while it was practical for the appointments in everyday life to e.g. B. to arrange "three days after the new moon".
2. ^ Evans, J. and Berggren, JL: Geminus, Introduction to the Phenomena , Princeton University Press 2006, VIII 52, p. 184
3. ^ Evans, J. and Berggren, JL: Geminus, Introduction to the Phenomena , Princeton University Press 2006, VIII 53-55, p. 184
4. BL van der Waerden: Greek Astronomical Calendars, II. Callippos and his Calendar , Archive for History of Exact Sciences 29 (2), 1984, pp. 122-123