The Ideal Metonic Epacts for the Next
4K Vernal-Equinox Calendar-Years

Simon Cassidy, November 2006

ABSTRACT / INTRODUCTION

The astronomically calculated, mean number of synodic lunations in a Vernal-Equinox year will, during the next four millenia, vary between a ratio of 12.368273 and 12.368276 (irrespective of changes in earthrotation calendar day-length). A solar calendar based on maintaining the Vernal Equinox on the same calendar date for some static, or very gradually sliding, longitude, can take advantage of this astronomical ratio to maintain a very accurate and rational, parallel Metonic lunar calendar with minimum jitter, by using an individual Epact shift, for each of the Golden Numbers in their natural order (instead of all at once, as in the Gregorian system, every 100-300 years), occurring in every twelfth year, and thus maintain a ratio of 12.36827485 calendar lunations per calendar year.

For example: By decrementing the epact value of the next (in natural order from the previously decremented) Golden Number, at the start of every A.D. year that includes the start of a Chinese "Year of the Monkey" (which we will refer to as MoONKEY years, defined as every A.D. year-number that is evenly divisible by 12), and by using the Dee-Cecil** solar calendar, with the same table of Epact-dates as used in the Gregorian lunar calendar (see FIG.1 below), we can find better*** starting dates for Dee-Cecil lunations for any A.D. year (with year number Y), by using a MoONKEY-based Dee-Cecil Epact number, E(Y), defined by the following equation:

E(Y) = ((Ymodulo19)*11-floor((Y-1584-12*(Ymodulo19))/228))modulo30

(where floor(x) truncates to the signed integer at, or next below, x).

The lunations of A.D. year Y start on the dates below that contain the Epact number E(Y) of that A.D. year (unless there is a property* of that year* that requires using the date with the starred Epact)*.

 ```FIG.1_____Gregorian_and_MoONKEY-based_Dee-Cecil_Epact_dates______ DATE |JAN |FEB |MAR |APR |MAY |JUN |JUL |AUG |SEP |OCT |NOV |DEC | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 1ST. | 0 | 29 | 0 | 29 | 28 | 27 | 26 | 25 | 23 | 22 | 21 | 20 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_24_|_ _|_ _|_ _|_ _| 2ND. | 29 | 28 | 29 | 28 | 27 |*25 |*25 | 23 | 22 | 21 | 20 | 19 | _____|_ _|_ _|_ _|_ _|_ _|_26_|_25_|_ _|_ _|_ _|_ _|_ _| 3RD. | 28 | 27 | 28 | 27 | 26 | 25 | 24 | 22 | 21 | 20 | 19 | 18 | _____|_ _|_ _|_ _|_ _|_ _|_24_|_ _|_ _|_ _|_ _|_ _|_ _| 4TH. | 27 |*25 | 27 |*25 |*25 | 23 | 23 | 21 | 20 | 19 | 18 | 17 | _____|_ _|_26_|_ _|_26_|_25_|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 5TH. | 26 | 25 | 26 | 25 | 24 | 22 | 22 | 20 | 19 | 18 | 17 | 16 | _____|_ _|_24_|_ _|_24_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 6TH. |*25 | 23 |*25 | 23 | 23 | 21 | 21 | 19 | 18 | 17 | 16 | 15 | _____|_25_|_ _|_25_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 7TH. | 24 | 22 | 24 | 22 | 22 | 20 | 20 | 18 | 17 | 16 | 15 | 14 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 8TH. | 23 | 21 | 23 | 21 | 21 | 19 | 19 | 17 | 16 | 15 | 14 | 13 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 9TH. | 22 | 20 | 22 | 20 | 20 | 18 | 18 | 16 | 15 | 14 | 13 | 12 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 10TH.| 21 | 19 | 21 | 19 | 19 | 17 | 17 | 15 | 14 | 13 | 12 | 11 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 11TH.| 20 | 18 | 20 | 18 | 18 | 16 | 16 | 14 | 13 | 12 | 11 | 10 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 12TH.| 19 | 17 | 19 | 17 | 17 | 15 | 15 | 13 | 12 | 11 | 10 | 9 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 13TH.| 18 | 16 | 18 | 16 | 16 | 14 | 14 | 12 | 11 | 10 | 9 | 8 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 14TH.| 17 | 15 | 17 | 15 | 15 | 13 | 13 | 11 | 10 | 9 | 8 | 7 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 15TH.| 16 | 14 | 16 | 14 | 14 | 12 | 12 | 10 | 9 | 8 | 7 | 6 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 16TH.| 15 | 13 | 15 | 13 | 13 | 11 | 11 | 9 | 8 | 7 | 6 | 5 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 17TH.| 14 | 12 | 14 | 12 | 12 | 10 | 10 | 8 | 7 | 6 | 5 | 4 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 18TH.| 13 | 11 | 13 | 11 | 11 | 9 | 9 | 7 | 6 | 5 | 4 | 3 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 19TH.| 12 | 10 | 12 | 10 | 10 | 8 | 8 | 6 | 5 | 4 | 3 | 2 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 20TH.| 11 | 9 | 11 | 9 | 9 | 7 | 7 | 5 | 4 | 3 | 2 | 1 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 21ST.| 10 | 8 | 10 | 8 | 8 | 6 | 6 | 4 | 3 | 2 | 1 | 0 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 22ND.| 9 | 7 | 9 | 7 | 7 | 5 | 5 | 3 | 2 | 1 | 0 | 29 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 23RD.| 8 | 6 | 8 | 6 | 6 | 4 | 4 | 2 | 1 | 0 | 29 | 28 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 24TH.| 7 | 5 | 7 | 5 | 5 | 3 | 3 | 1 | 0 | 29 | 28 | 27 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 25TH.| 6 | 4 | 6 | 4 | 4 | 2 | 2 | 0 | 29 | 28 | 27 | 26 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 26TH.| 5 | 3 | 5 | 3 | 3 | 1 | 1 | 29 | 28 | 27 |*25 |*25 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_26_|_25_| 27TH.| 4 | 2 | 4 | 2 | 2 | 0 | 0 | 28 | 27 | 26 | 25 | 24 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_24_|_ _| 28TH.| 3 | 1 | 3 | 1 | 1 | 29 | 29 | 27 |*25 |*25 | 23 | 23 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_26_|_25_|_ _|_ _| 29TH.| 2 |LEAP| 2 | 0 | 0 | 28 | 28 | 26 | 25 | 24 | 22 | 22 | _____|_ _|_DY_|_ _|_ _|_ _|_ _|_ _|_ _|_24_|_ _|_ _|_ _| 30TH.| 1 | | 1 | 29 | 29 | 27 | 27 |*25 | 23 | 23 | 21 | 21 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_25_|_ _|_ _|_ _|_ _| 31ST.| 0 | | 0 | | 28 | |*25 | 24 | | 22 | |*19 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_26_|_ _|_ _|_ _|_ _|_20_|```

* Both the Gregorian and this congruent version, of the Dee-Cecil MoONKEY lunar calendar, use *19, on the last day of the years with Epact 19, if there is a "saltus" at the end of that year. A "saltus" occurs at the end of year Y if and only if E(Y+1) = (E(Y) + 12) modulo 30, in a MoONKEY calendar. Alternatively we can determine if a "saltus" occurs at the end of a given year Y, by satisfying both of two simple conditions:
(a) [Y modulo12 < 11] i.e. the year must not immediately precede a MoONKEY year and;
(b) [(floor((Y-1800)/12)modulo19 = (Ymodulo19 +1)modulo19] i.e. the number of MoONKEY years since 1800, for the given year, is the same modulo 19 as its Golden Number.
The Gregorian calendar (but maybe not a MoONKEY calendar) would use *25 instead of 25 whenever Y modulo19 is greater than 10, i.e. whenever the year Y's Golden Number, GN, is greater than 11.

** The Dee-Cecil solar calendar differs from the Gregorian solely due to differing leap years. A Dee-Cecil year Y is a leap year whenever Y modulo33 is positive and evenly divisible by 4. The Gregorian and Dee-Cecil solar calendars intersected with coincidental dates and identical longitudes for their calendric meridians in the fall of A.D. 1799 (noon September 19th. near longitude 77 degs. west of Greenwich). For this example the Dee-Cecil yearly Epacts E(Y), and thus the Epact dates (but not necessarily the days since leap days are different), are coordinated to the Gregorian yearly Epacts for the years A.D. 1786 to A.D. 1822 centered on a common "saltus" at the end of A.D. 1804.

*** "better starting dates" means better than a non-MoONKEY-based Epact (such as the previously suggested epact shift, of all Golden Numbers at once, every 231 Dee-Cecil years) and much better than the Gregorian Epact dates, in terms of the jitter in the lunar calendar and in the occurrence of anomalous 28, 31, 59 or 1 day lunations, since the only anomalies remaining are the 31-day lunations that straddle the leap day, Feb 29th. (in leap years with positive Epact number less than 25). These remaining occasional leap-day generated 31-day lunations are unavoidable without altering the Gregorian table above, to something like the following, where *24 through *1 are used (instead of 24 through 1) in leap years:

 ```FIG.2____Ideal_Luna1_dates for MoONKEY-based_Dee-Cecil_epacts____ DATE |JAN |FEB |MAR |APR |MAY |JUN |JUL |AUG |SEP |OCT |NOV |DEC | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 1ST. | 0 | 29 | 0 | 29 | 28 | 27 | 26 | 25 | 23 | 22 | 21 | 20 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_24_|_ _|_ _|_ _|_ _| 2ND. | 29 | 28 | 29 | 28 | 27 | 26 | 25 | 23 | 22 | 21 | 20 | 19 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 3RD. | 28 | 27 | 28 | 27 | 26 | 25 | 24 | 22 | 21 | 20 | 19 | 18 | _____|_ _|_ _|_ _|_ _|_ _|_24_|_ _|_ _|_ _|_ _|_ _|_ _| 4TH. | 27 | 26 | 27 | 26 | 25 | 23 | 23 | 21 | 20 | 19 | 18 | 17 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 5TH. | 26 | 25 | 26 | 25 | 24 | 22 | 22 | 20 | 19 | 18 | 17 | 16 | _____|_ _|_24_|_ _|_24_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 6TH. | 25 |*24 | 25 | 23 | 23 | 21 | 21 | 19 | 18 | 17 | 16 | 15 | _____|_ _|_23_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 7TH. | 24 |*23 | 24 | 22 | 22 | 20 | 20 | 18 | 17 | 16 | 15 | 14 | _____|_ _|_22_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 8TH. | 23 |*22 | 23 | 21 | 21 | 19 | 19 | 17 | 16 | 15 | 14 | 13 | _____|_ _|_21_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 9TH. | 22 |*21 | 22 | 20 | 20 | 18 | 18 | 16 | 15 | 14 | 13 | 12 | _____|_ _|_20_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 10TH.| 21 |*20 | 21 | 19 | 19 | 17 | 17 | 15 | 14 | 13 | 12 | 11 | _____|_ _|_19_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 11TH.| 20 |*19 | 20 | 18 | 18 | 16 | 16 | 14 | 13 | 12 | 11 | 10 | _____|_ _|_18_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 12TH.| 19 |*18 | 19 | 17 | 17 | 15 | 15 | 13 | 12 | 11 | 10 | 9 | _____|_ _|_17_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 13TH.| 18 |*17 | 18 | 16 | 16 | 14 | 14 | 12 | 11 | 10 | 9 | 8 | _____|_ _|_16_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 14TH.| 17 |*16 | 17 | 15 | 15 | 13 | 13 | 11 | 10 | 9 | 8 | 7 | _____|_ _|_15_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 15TH.| 16 |*15 | 16 | 14 | 14 | 12 | 12 | 10 | 9 | 8 | 7 | 6 | _____|_ _|_14_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 16TH.| 15 |*14 | 15 | 13 | 13 | 11 | 11 | 9 | 8 | 7 | 6 | 5 | _____|_ _|_13_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 17TH.| 14 |*13 | 14 | 12 | 12 | 10 | 10 | 8 | 7 | 6 | 5 | 4 | _____|_ _|_12_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 18TH.| 13 |*12 | 13 | 11 | 11 | 9 | 9 | 7 | 6 | 5 | 4 | 3 | _____|_ _|_11_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 19TH.| 12 |*11 | 12 | 10 | 10 | 8 | 8 | 6 | 5 | 4 | 3 | 2 | _____|_ _|_10_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 20TH.| 11 |*10 | 11 | 9 | 9 | 7 | 7 | 5 | 4 | 3 | 2 | 1 | _____|_ _|_ 9_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 21ST.| 10 | *9 | 10 | 8 | 8 | 6 | 6 | 4 | 3 | 2 | 1 | 0 | _____|_ _|_ 8_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 22ND.| 9 | *8 | 9 | 7 | 7 | 5 | 5 | 3 | 2 | 1 | 0 | 29 | _____|_ _|_ 7_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 23RD.| 8 | *7 | 8 | 6 | 6 | 4 | 4 | 2 | 1 | 0 | 29 | 28 | _____|_ _|_ 6_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 24TH.| 7 | *6 | 7 | 5 | 5 | 3 | 3 | 1 | 0 | 29 | 28 | 27 | _____|_ _|_ 5_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 25TH.| 6 | *5 | 6 | 4 | 4 | 2 | 2 | 0 | 29 | 28 | 27 | 26 | _____|_ _|_ 4_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 26TH.| 5 | *4 | 5 | 3 | 3 | 1 | 1 | 29 | 28 | 27 | 26 | 25 | _____|_ _|_ 3_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 27TH.| 4 | *3 | 4 | 2 | 2 | 0 | 0 | 28 | 27 | 26 | 25 | 24 | _____|_ _|_ 2_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_24_|_ _| 28TH.| 3 | *2 | 3 | 1 | 1 | 29 | 29 | 27 | 26 | 25 | 23 | 23 | _____|_ _|_ 1_|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 29TH.| 2 |LEAP| 2 | 0 | 0 | 28 | 28 | 26 | 25 | 24 | 22 | 22 | _____|_ _|_*1_|_ _|_ _|_ _|_ _|_ _|_ _|_24_|_ _|_ _|_ _| 30TH.| 1 | | 1 | 29 | 29 | 27 | 27 | 25 | 23 | 23 | 21 | 21 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _| 31ST.| 0 | | 0 | | 28 | | 26 | 24 | | 22 | |*19 | _____|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_ _|_20_|```

Explanation of the Equation for the Epact E(Y) of Year Y A.D.

E(Y)=((Ymodulo19)*11-floor((Y-1584-12*(Ymodulo19))/228))modulo30

where (Ymodulo19)*11 is just (when taken modulo30) the Gregorian epact for the period AD 1700-1899, and in the MoONKEY system this epact (here synched to the Gregorian epacts for the dozen years 1800 through 1811 AD), is decremented every 228 years (instead of at certain century years in the Gregorian system, where it is also sometimes incremented!).

But this subtraction of 1, every 228 years, is not carried out simultaneously for every Golden Number (as in the Gregorian system), but instead it is done for each Golden Number separately and in turn (taken in their natural order from 1 to 19) so that one Golden Number (the next in turn) has its epact decremented every 12th year. This explains the subtraction of floor((Y-1584-12*(Ymodulo19))/228).

The floor() function is used instead of int() in order that the result is sensible for all values of Y (not just when Y > 1799).

The year 1584 appears here as the base year in the count of 228-year periods in order that, when Y is 1812 AD (1584+228) there occurs a decrementing of the MoONKEY epact (wrt Gregorian ca. 1800) associated with years having Golden Number 1 (i.e. years having Ymodulo19 = zero).

Quite by coincidence, 1584 is the first year of continuous English occupation of the North American mainland at the calendric meridian of the Dee or Dee-Cecil calendar (initiated by the first, recorded as merely reconnaisance, expedition sent by Sir Walter Raleigh to 77 degrees West which secretly left two Englishmen to observe the lunar eclipse of November 1584 and to winter over until the next overtly colonizing and massive miltary expedition, which included Thomas Harriot, in 1585)!

-- Dee's and Franklin's Years,
Simon Cassidy,
1053 47th. St.
Emeryville Ca.
U.S.A. 94608.