Editing Epact (section)

==Lunar calendar==

Epacts can also be used to relate dates in the lunar calendar to dates in the common solar calendar.

===Solar and lunar years===

A [[tropical year|solar calendar year]] has 365 days (366 days in [[leap year]]s).  A [[lunar year|lunar calendar year]] has 12&nbsp;lunar months which alternate between 30 and 29&nbsp;days for a total of 354&nbsp;days (in leap years, one of the lunar months has a day added; since a lunar year lasts a little over {{sfrac|354|1|3}}&nbsp;days, a leap year arises every second or third year rather than every fourth.)

If a solar and lunar year start on the same day, then after one year the start of the solar year is 11&nbsp;days after the start of the lunar year. These excess days are epacts, and have to be added to the lunar year to complete the solar year; or from the complementary perspective they are added to the day of the solar year to determine the day in the lunar year.

After two years the difference is 22&nbsp;days, and after 3&nbsp;years, 33&nbsp;days. Whenever the epact reaches or exceeds 30&nbsp;days, an extra (embolismic or [[Intercalation (timekeeping)#Lunisolar calendars|intercalary]]) lunar month is inserted into the lunar calendar, and the epact is reduced by 30&nbsp;days.

Leap days extend both the solar and lunar year, so they do not affect epact calculations for any other dates.{{explain|date=May 2023|reason=Solar leap days occur in different years than lunar leap days; how could they not (erratically) throw the calculations one day off?}}

===19-year cycle===
The [[tropical year|solar calendar year]] is slightly shorter than {{sfrac|365| 1  |4}} days, while the [[synodic month]], on average, is slightly longer than {{sfrac|29| 1 |2}} days meaning both are non-integers. This gets corrected in the following way. Nineteen tropical years are deemed to be as long as 235 synodic months ([[Metonic cycle]]). A cycle can last 6939 or 6940 full days, depending on whether there are 4 or 5 leap days in this 19-year period.

After 19 years the lunations should fall the same way in the solar years, so the epact should repeat after 19 years. However, {{nobr|19 × 11 {{=}} 209,}} and this is not an integer multiple of the full cycle of 30 epact numbers ({{nobr|209 [[modular arithmetic|modulo]] 30 {{=}} 29,}} not&nbsp;0). So after 19 years the epact must be corrected by +1 in order for the cycle to repeat over 19 years. This is the {{lang|la|saltus lunae}} ("leap of the moon"). The sequence number of the year in the 19-year cycle is called the [[golden number (time)|golden number]]. The extra 209 days fill 7 embolismic months, for a total of {{nobr|19 × 12 + 7 {{=}} 235 lunations.}}

===Lilian (Gregorian) epacts===
When the [[Gregorian calendar]] reform was [[#refpbull|instituted in 1582]], the lunar cycle previously used with the Julian calendar to complete the calculation of Easter dates was adjusted also, in accordance with a (modification of the) scheme devised by [[Aloysius Lilius]].<ref>
{{cite conference
 |last = Moyer |first = G.
 |year = 1983
 |title = Aloysius Lilius and the ''Compendium novae rationis restituendi kalendarium''
 |editor1-last = Coyne    |editor1-first = George V. |editor1-link = George V. Coyne
 |editor2-last = Hoskin   |editor2-first = Michael A.
 |editor3-last = Pedersen |editor3-first = O.        |editor3-link = Olaf Pedersen
 |book-title = Gregorian Reform of the Calendar
 |conference = Vatican Conference to commemorate [the Gregorian Reform's] 400th&nbsp;Anniversary, 1582–1982
 |publisher = [[Pontifical Academy of Sciences]], [[Vatican Observatory]]
 |place = Vatican City, IT
 |page = 171
 |url = http://www.casinapioiv.va/content/dam/accademia/pdf/es3.pdf <!-- actually, URL for whole ''Proceedings of ...''. -->
 |conference-url = http://www.casinapioiv.va/content/dam/accademia/pdf/es3.pdf
 |access-date= 2024-01-20
 |archive-url = https://web.archive.org/web/20191116143633/http://www.casinapioiv.va/content/dam/accademia/pdf/es3.pdf
 |archive-date = 2019-11-16 |df = dmy-all
}}
</ref>
There were two adjustments to the old lunar cycle:
* a "solar equation", decrementing the epact by 1, whenever the Gregorian calendar drops a leap day (3&nbsp;times in 400&nbsp;calendar years), and
* a "lunar equation", incrementing the epact by 1, 8&nbsp;times in 2500&nbsp;calendar years (7&nbsp;times after an interval of 300&nbsp;years, and the 8th&nbsp;time after an interval of 400&nbsp;years).

The revised "solar equation" was intended to adjust for the Gregorian change in the solar calendar, if they were applied at 1&nbsp;January of the Julian calendar instead of the Gregorian calendar as the reformers implemented it; moreover the corrections to the solar calendar are leap days, whereas there are 30&nbsp;epact values for a mean lunar month of {{nobr|{{sfrac|29|1|2}} days}} and a bit: Therefore changing the epact by 1&nbsp;day does not exactly compensate for a dropped leap day. The "lunar equation" only approximately adjusts for what had (by 1582) been seen after many centuries of recording, that the Moon moves a little faster than the expectation of the rate used for it in the old lunar cycle. By 1582 it was noted (for example, in the text of the bull ''[[Inter gravissimas]]'' itself) that the new and full moons were at that point occurring "four days and something more" sooner than the old lunar cycle indicated.