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The best known of these is the Tabular Islamic calendar: in brief, it has a 30-year cycle with 11 leap years of 355 days and 19 years of 354 days. In the long term, it is accurate to one day in about 2,500 solar years or 2,570 lunar years. It also deviates from observation by up to about one or two days in the short term.
Because of these two variations in angular rate, the actual time between lunations may vary from about 29.274 days (or 29 d 6 h 35 min) to about 29.829 days (or 29 d 19 h 54 min). [6] The average duration in modern times is 29.53059 days with up to seven hours variation about the mean in any given year.
Meton of Athens, in the 5th century BC, judged the cycle to be a whole number of days, 6,940. [3] Using these whole numbers facilitates the construction of a lunisolar calendar. A tropical year (about 365.24 days) is longer than 12 lunar months (about 354.36 days) and shorter than 13 of them (about 383.90 days).
Each complete cycle of phases is called a "lunation". [7] The approximate age of the Moon, and hence the approximate phase, can be calculated for any date by calculating the number of days since a known new moon (such as 1 January 1900 or 11 August 1999) and reducing this modulo 29.53059 days (the mean length of a synodic month).
A calendar year is an approximation of the number of days of the Earth's orbital period, as counted in a given calendar. The Gregorian calendar, or modern calendar, presents its calendar year to be either a common year of 365 days or a leap year of 366 days, as do the Julian calendars. For the Gregorian calendar, the average length of the ...
Years began on a jiǎzǐ (甲子) day (and a 72-day wood phase), followed by a bǐngzǐ day (丙子) and a 72-day fire phase; a wùzǐ (戊子) day and a 72-day earth phase; a gēngzǐ (庚子) day and a 72-day metal phase, and a rénzǐ day (壬子) followed by a water phase. [15] Other days were tracked using the Yellow River Map (He Tu).
An eclipse cycle where the number of solar years (365.371) closely matches (by chance) the number of days in 1 solar year (365.242). Lasting 365 years 4.5 months or 4519 lunations. It is the eighth convergent in the continued fractions development of the ratio between the eclipse year and the synodic month, giving a series of eclipses one ...
So after 19 years the epact must be corrected by +1 in order for the cycle to repeat over 19 years. This is the saltus lunae ("leap of the moon"). The sequence number of the year in the 19-year cycle is called the golden number. The extra 209 days fill 7 embolismic months, for a total of 19 × 12 + 7 = 235 lunations.