# Dominical letter

Dominical letters are letters A, B, C, D, E, F and G assigned to days in a cycle of seven with the letter A always set against 1 January as an aid for finding the day of the week of a given calendar date and in calculating Easter.

A common year is assigned a single dominical letter, indicating which letter is Sunday (hence the name, from Latin dominica for Sunday). Thus, 2011 is B, indicating that B days are Sunday. Leap years are given two letters, the first indicating the dominical letter for January 1 - February 28 (or February 24, see below), the second indicating the dominical letter for the rest of the year.

In leap years, the leap day may or may not have a dominical letter. In the original 1582 Catholic version, it did, but in the 1752 Anglican version it did not. The Catholic version caused February to have 29 days by doubling the sixth day before 1 March, inclusive, because 24 February in a common year is marked "duplex", thus both halves of the doubled day had a dominical letter of F.[1][2][3] The Anglican version added a day to February that did not exist in common years, 29 February, thus it did not have a dominical letter of its own.[4][5]

In either case, all other dates have the same dominical letter every year, but the days of the weeks of the dominical letters change within a leap year before and after the intercalary day, 24 February or 29 February.

## Examples

• 2000 BA
• 2001 G
• 2002 F
• 2003 E
• 2004 DC
• 2005 B
• 2006 A
• 2007 G
• 2008 FE
• 2009 D
• 2010 C
• 2011 B
• 2012 AG
• 2013 F
• 2014 E
• 2015 D
• 2016 CB
• 2017 A
• 2018 G
• 2019 F
• 2020 ED
• 2021 C
• 2022 B
• 2023 A
• 2024 GF
• 2025 E
• 2026 D
• 2027 C
• 2028 BA
• 2029 G
• 2030 F
• 2031 E
• 2032 DC

The dominical letter of a year determines the days of week in its calendar:

## History

Dominical letters were a device adopted from the Romans by chronologers to aid them in finding the day of the week corresponding to any given date, and indirectly to facilitate the adjustment of the "Proprium de Tempore" to the "Proprium Sanctorum" when constructing the ecclesiastical calendar for any year.[6] The Christian Church, due to its complicated system of movable and immovable feasts, has long been concerned with the regulation and measurement of time. To secure uniformity in the observance of feasts and fasts, it began, even in the patristic age, to supply a system of reckoning (computus) by which the relation of the solar and lunar years might be accommodated and the celebration of Easter determined.[6] It adopted the astronomical methods that were available at the time, and these methods and their methodology have become traditional and are perpetuated in a measure to this day, even the reform of the calendar, in the prolegomena to the Breviary and Missal.

The Romans were accustomed to dividing the year into nundinæ, periods of eight days; and in their marble calendars (fasti), of which numerous specimens remain, they used the first eight letters of the alphabet (A to H) to mark the days of which each period was composed. When the Oriental seven-day period (week) was introduced in the time of Cæsar Augustus, the first seven letters of the alphabet were employed in the same way to indicate the days of the new division of time. Some surviving (albeit fragmentary) marble calendars show both cycles side by side (see "Corpus Inscriptionum Latinarum", 2nd ed., I, 220; the same peculiarity occurs in the Philocalian Calendar of A.D. 356, ibid., p. 256). This device was imitated by the Christians.

## Dominical letter cycle

The days of the year from 1 January to 31 December are marked with a continuous recurring cycle of seven letters: A, B, C, D, E, F, G. The letter A is always set against 1 January, B against 2 January, C against 3 January and so on, so that G falls to 7 January; A falls again to 8 January and also recurs consequently on 15 January, 22 January and 29 January. Continuing in this way, 30 January is marked with a B, 31 January with a C, and 1 February with a D. When this is carried on through all the days of a common year (i.e. non-leap year) then D corresponds to 1 March, G to 1 April, B to 1 May, E to 1 June, G to 1 July, C to 1 August, F to 1 September, A to 1 October, D to 1 November, and F to 1 December — a resulting ADDGBEGCFADF sequence which Durandus recalled by the following distich:

Alta Domat Dominus, Gratis Beat Equa Gerentes
Contemnit Fictos, Augebit Dona Fideli.

Another one is:

Yet another is:

At Dover dwell George Brown, Esquire; Good Christopher Finch; and David Fryer.

Clearly, if 1 January is a Sunday, all the other days marked by A will be Sundays; If 1 January is a Saturday, then Sunday will fall on 2 January which is a B, and all the other days marked B will be Sundays; if 1 January is a Monday, then Sunday will not come until 7 January, a G, and all the days marked G will be Sundays.

A complication arises with leap years, which have an extra day. Traditionally, the Catholic ecclesiastical calendar treats 24 February as the day added, as this was the Roman leap day, with events normally occurring on 24–28 February moved to 25–29 February. The Anglican and civil calendars treat 29 February as the day added, and do not shift events in this way. But in either case[6] with leap years, 1 March occurs one day later in the week than 1 February, or, in other words, from leap day onwards for the rest of the year the Sundays come a day earlier than in a common year. Thus a leap year has two Dominical Letters, the second being the letter which precedes that with which the year started. For example in 2012 (= AG), Sundays preceding leap day were A days, and G days for the rest of the year.

## Dominical letter of a year

The dominical letter of a year is defined as the letter of the cycle corresponding to the day upon which the first Sunday (and thus every subsequent Sunday) falls. Leap years have two Dominical Letters, the second of which is the letter of the cycle preceding the first; the second letter describes the portion of the year after the leap day.

The Gregorian calendar repeats every four hundred years. Of the four hundred years in a Gregorian cycle, there are 43 common years with Dominical letter A, 43 common years with B, 43 with C, 43 with E, and 43 with G. Of common years with Dominical letter D and F, there are 44 for each. Of the leap years, there are 13 each with BA, DC, and GF; 14 each with ED and FE, and 15 each with AG and CB.

In the Julian calendar, the cycle is 28 years, 7 of which are leap years, and the remaining 21 are common years. Each of the seven Dominical letters is split evenly among the 21 common years, and each of the seven double letters for leap years, BA, CB, DC, ED, FE, GF, and AG, occur once in every 28-year cycle.

## Calculation

The dominical letter of a year can be calculated based on any method for calculating the day of the week, with letters in reverse order compared to numbers indicating the day of the week.

For example:

• ignore periods of 400 years
• considering the second letter in the case of a leap year:
• for one century within two multiples of 400, go forward two letters from BA for 2000, hence C, E, G.
• for remaining years, go back one letter every year, two for leap years (this corresponds to writing two letters, no letter is skipped).
• to avoid up to 99 steps within a century, there is a choice of several shortcuts, e.g.:
• go back one letter for every 12 years
• ignore multiples of 28 years (note that when jumping from e.g. 1900 to 1928 the last letter of 1928 is the same as the letter of 1900)
• apply steps between multiples of 10, writing from right to left:
2000 1990 1980 1970 1960 1950 1940 1930 1920 1910 1900
BA   G    FE   D    CB   A    GF   E    DC   B    .G

• Note the dummy step (we skip A between 1900 and 1910) because 1900 is not a leap year.

For example, to find the Dominical Letter of the year 1913:

• 1900 is G
• 1910 is B
• count B A GF E, 1913 is E

Similarly, for 2007:

• 2000 is BA
• count BA G F E DC B A G, 2007 is G

For 2065:

• 2000 is BA
• 2012 is AG, 2024 is GF, 2036 is FE, 2048 is ED, 2060 is DC, then B A G FE D, 2065 is D
• or from 2000 to 2060 in steps of 10, written backward: DC B AG F ED C BA, starting from 2000 is BA we get 2060 is DC, then again B A G FE D, 2065 is D (or, writing the last part backward too: D FE G A B <DC> B AG F ED C BA)
• or ignore 56 years, 2056 is BA, count G F E DC B A G FE D, 2065 is D

### The Odd+11 method

A simpler method suitable for finding the year's dominical letter was discovered in 2010. It is called the Odd+11 method.[7]

The procedure accumulates a running total T as follows:

1. Let T be the year's last two digits.
2. If T is odd, add 11.
3. Let T = T/2.
4. If T is odd, add 11.
5. Let T = T mod 7.
6. Count forward T letters from the century's dominical letter (A, C, E or G see above) to get the year's dominical letter.

The formula is $[ \frac{y+11(y \bmod 2)}{2} + 11 (\frac{y+11(y \bmod 2)}{2}\bmod 2)] \bmod 7$.

### Formula derived from Gauss's algorithm

For the Gregorian calendar, the formula is[8]

$\mbox{DL} = (y\bmod 4\times 2 + y\bmod 7\times 4 + c \bmod 4\times 2) \bmod 7,$

for the Julian calendar it is

$\mbox{DL} = (y\bmod 4\times 2 + y\bmod 7\times4 + c \bmod 7 + 2) \bmod 7,$

where

• DL = dominical letter (A = 1,..., G = 0)
• y = (year - 1) mod 100
• c = [(year - 1)/100].

Note: For a leap year, the second letter = the first letter - 1.
For 2013, where y = 12, c = 20, DL = (0 + 20 + 0) mod 7 = 6 = F.
For 1582, where y = 81, c = 15, DL = (2 + 16 + 1 + 2) mod 7 = 0 = G.

### Rule of De Morgan's

1. Add 1 to the given year.
2. Take the quotient found by dividing the given year by 4 (neglecting the remainder).
3. Take 16 from the centurial figures of the given year if that can be done.
4. Take the quotient of III divided by 4 (neglecting the remainder).
5. From the sum of I, II and IV, subtract III.
6. Find the remainder of V divided by 7: this is the number of the Dominical Letter, supposing A, B, C, D, E, F, G to be equivalent respectively to 6, 5, 4, 3, 2, 1, 0.[6]

So the formula is

$(1 + year + [year/4] + [(year - 1600)/400] - [(year - 1600)/100]) \mod 7$.

It is equivalent to

$(year + [year/4] + [year/400] - [year/100] - 1) \mod 7$

and

$(y + [y/4] + 5(c\mod4) -1) \mod 7$.

For example, to find the Dominical Letter of the year 1913:

• (1 + 1913 + 478 + 0 - 3) mod 7 = 2
• (1913 + 478 + 4 - 19 -1) mod 7 = 2
• (13 + 3 + 15 -1) mod 7 = 2

Therefore, the Dominical Letter is E.

### Dominical letter in relation to the Doomsday Rule

The "doomsday" concept in the doomsday algorithm is mathematically related to the Dominical letter. Because the dominical letter of a date = the dominical letter of a year (DL) + the day of the week (DW) and the dominical letter for the doomsday is C, we have

$\mbox{C} = (\mbox{DL} + \mbox{DW}) \bmod 7$
$\mbox{DL} = (\mbox{C} - \mbox{DW}) \bmod 7$
$\mbox{DW} = (\mbox{C} - \mbox{DL}) \bmod 7$

Note: A = 1 = Mon, B = 2 = Tue, C = 3 = Wed, D = 4 = Thurs, ..., G = 0 = Sun. i.e. in our context, C is mathematically identical to 3.

The doomsday of the year 2013 is Thursday, so DL = (3 - 4) mod 7 = 6 = F. The Dominical Letter of the year 1913 is E, so DW = (3 - 5) mod 7 = 5 = Friday.

## Complete tables

### Table of dominical letters for years

For years outside the range of this table, use the fact that the dominical letters repeat exactly every 400 years.

            ┌────┬────┬────┬────┐
│1600│1700│1800│1900│
│2000│2100│2200│2300│
┌───────────┼────┼────┼────┼────┤
│         00│ BA │ C  │ E  │ G  │
├───────────┼────┼────┼────┼────┤
│85 57 29 01│ G  │ B  │ D  │ F  │
│86 58 30 02│ F  │ A  │ C  │ E  │
│87 59 31 03│ E  │ G  │ B  │ D  │
│88 60 32 04│ DC │ FE │ AG │ CB │
├───────────┼────┼────┼────┼────┤
│89 61 33 05│ B  │ D  │ F  │ A  │
│90 62 34 06│ A  │ C  │ E  │ G  │
│91 63 35 07│ G  │ B  │ D  │ F  │
│92 64 36 08│ FE │ AG │ CB │ ED │
├───────────┼────┼────┼────┼────┤
│93 65 37 09│ D  │ F  │ A  │ C  │
│94 66 38 10│ C  │ E  │ G  │ B  │
│95 67 39 11│ B  │ D  │ F  │ A  │
│96 68 40 12│ AG │ CB │ ED │ GF │
├───────────┼────┼────┼────┼────┤
│97 69 41 13│ F  │ A  │ C  │ E  │
│98 70 42 14│ E  │ G  │ B  │ D  │
│99 71 43 15│ D  │ F  │ A  │ C  │
│   72 44 16│ CB │ ED │ GF │ BA │
├───────────┼────┼────┼────┼────┤
│   73 45 17│ A  │ C  │ E  │ G  │
│   74 46 18│ G  │ B  │ D  │ F  │
│   75 47 19│ F  │ A  │ C  │ E  │
│   76 48 20│ ED │ GF │ BA │ DC │
├───────────┼────┼────┼────┼────┤
│   77 49 21│ C  │ E  │ G  │ B  │
│   78 50 22│ B  │ D  │ F  │ A  │
│   79 51 23│ A  │ C  │ E  │ G  │
│   80 52 24│ GF │ BA │ DC │ FE │
├───────────┼────┼────┼────┼────┤
│   81 53 25│ E  │ G  │ B  │ D  │
│   82 54 26│ D  │ F  │ A  │ C  │
│   83 55 27│ C  │ E  │ G  │ B  │
│   84 56 28│ BA │ DC │ FE │ AG │
└───────────┼────┼────┼────┼────┤
│1600│1700│1800│1900│
│2000│2100│2200│2300│
└────┴────┴────┴────┘


### Table for days of the year

                ┌───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┐
│Jan│Feb│Mar│Apr│May│Jun│Jul│Aug│Sep│Oct│Nov│Dec│
┌───────────────┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┤
│(29) 22 15  8 1│ A │ D │ D │ G │ B │ E │ G │ C │ F │ A │ D │ F │
├───────────────┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┤
│(30) 23 16  9 2│ B │ E │ E │ A │ C │ F │ A │ D │ G │ B │ E │ G │
│(31) 24 17 10 3│ C │ F │ F │ B │ D │ G │ B │ E │ A │ C │ F │ A │
│     25 18 11 4│ D │ G │ G │ C │ E │ A │ C │ F │ B │ D │ G │ B │
│     26 19 12 5│ E │ A │ A │ D │ F │ B │ D │ G │ C │ E │ A │ C │
│     27 20 13 6│ F │ B │ B │ E │ G │ C │ E │ A │ D │ F │ B │ D │
├───────────────┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┤
│     28 21 14 7│ G │ C │ C │ F │ A │ D │ F │ B │ E │ G │ C │ E │
└───────────────┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┘


## Practical use for the clergy

The Dominical Letter had another practical use in the days before the Ordo divini officii recitandi was printed annually (thus often requiring priests to determine the Ordo on their own). Easter Sunday may be as early as 22 March or as late as 25 April, and there are consequently 35 possible days on which it may fall; each Dominical Letter allows five of these dates, so there are five possible calendars for each letter. The Pye or directorium which preceded the present Ordo took advantage of this principle, including all 35 calendars and labeling them primum A, secundum A, tertium A, and so on. Hence, based on the Dominical Letter of the year and the epact, the Pye identified the correct calendar to use. A similar table, but adapted to the reformed calendar and in more convenient shape, is found at the beginning of every Breviary and Missal under the heading "Tabula Paschalis nova reformata".

The Dominical Letter does not seem to have been familiar to Bede in his "De temporum ratione", but in its place he adopts a similar device of seven numbers which he calls concurrentes (De Temp. Rat., cap. liii), of Greek origin. The Concurrents are numbers denoting the days of the week on which 24 March falls in the successive years of the solar cycle, 1 standing for Sunday, 2 (feria secunda) for Monday, 3 for Tuesday, and so on; these correspond to Dominical Letters F, E, D, C, B, A, and G, respectively.

## Use for mental calculation

There exist patterns in the dominical letters, which are very useful for mental calculation.

Patterns for years:

To use these patterns, choose and remember a year to use as a starting point, such as 2000=BA.

Note that because of the complicated Gregorian leap-year rules, these patterns break near some century changes. Note the reverse alphabetical order.

1992 3 4 5 96 7 8 9 2000 1 2 3 04 5 6 7 08 9 0 1 2012 3
ED  C B A GF E D C  BA  G F E DC B A G FE D C B  AG  F


and

(note the reversed order of the years
as well as of the letters)

2040 2030 2020 2010 2000 1990 1980 1970 1960 1950
AG   F    ED   C    BA   G    FE   D    CB   A
|   |     |   |     |   |     |   |     |   |
G   FE    D   CB    A   GF    E   DC    B   AG
2046 2036 2026 2016 2006 1996 1986 1976 1966 1956


Patterns for days of the month:

The dominical letters for the first day of each month form the nonsense mnemonic phrase "Add G, beg C, fad F".

The following dates, given in day/month form, all have dominical letter C: 4/4, 6/6, 8/8, 10/10, 12/12, 9/5, 5/9, 11/7, 7/11 (see also the Doomsday rule).

We are able to calculate the Dominical letter in this way (function in C), where:

• m = month
• y = year
• s = "style"; 0 for Julian, otherwise Gregorian.
 char dominical(int m,int y,int s){
int leap;
int a,b;
leap=(s==0&&y%4==0)||(s!=0&&(y%4==0&&y%100!=0||y%400==0));
a=(y%100)%28;
b=(s==0)*(4+(y%700)/100+2*(a/4)+6*((!leap)*(1+(a%4))+(leap)*((9+m)/12)))%7+
(s!=0)*(2*(1+(y%400)/100+(a/4))+6*((!leap)*(1+(a%4))+(leap)*((9+m)/12)))%7;
b=(b==0)*(b+7)+(b!=0)*b;
return (char)(64+b);
}