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# Dominical letter

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### 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.

## Contents

• History 1
• Dominical letter cycle 2
• Further examples 3
• Dominical letter of a year 4
• Calculation 5
• The Odd+11 method 5.1
• Formula derived from Gauss's algorithm 5.2
• Rule of De Morgan's 5.3
• Dominical letter in relation to the Doomsday Rule 5.4
• Julian calendar 5.5
• Complete tables 6
• Table of dominical letters for years 6.1
• Julian calendar 6.1.1
• Gregorian calendar 6.1.2
• Table for days of the year 6.2
• Practical use for the clergy 7
• Use for mental calculation 8
• References 10

## History

Per Thurston, (1909) dominical letters were:

Thurston continues that the Christian Church, with its "complicated system of movable and immovable feasts" has long been concerned with the regulation and measurement of time; he states: "To secure uniformity in the observance of feasts and fasts, she began, even in the patristic age, to supply a computus, or system of reckoning, by which the relation of the solar and lunar years might be accommodated and the celebration of Easter determined."[6] He continues, that naturally she "adopted the astronomical methods then available, and these methods and the methodology belonging to them having become traditional, are perpetuated in a measure to this day, even the reform of the calendar, in the prolegomena to the Breviary and Missal."[6]

He then goes on to note that:

and that this device was imitated by the Christians.[6]

## Dominical letter cycle

Thurston (1909) goes on to note that "the days of the year from 1 January to 31 December were marked with a continuous recurring cycle of seven letters: A, B, C, D, E, F, G... [and that 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.[6]

He notes that A falls again on "8 January, and also, 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."[6]

When this is carried on through all the days of a common year (i.e. ordinary, or 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"; the resulting ADDGBEGCFADF sequence Thurston observes, is one "which Durandus recalled by the following distich:

Another one is "Add G, beg C, fad F," and yet another is "At Dover dwell George Brown, Esquire; Good Christopher Finch; and David Fryer."

Clearly, Thurston continues, "if 1 January is a Sunday, all the other days marked by A will be Sundays; [i]f 1 January is a Saturday, Sunday will fall on 2 January which is a B, and all the other days marked B will be Sundays; [i]f 1 January is a Monday, then Sunday will not come until 7 January, a G, and all the days marked G will be Sundays."[6]

Thurston then notes that a complication arises with leap years, which have an extra day.[6] 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, with leap years, Thurston explains, "1 March is then one day later in the week than 1 February, or, in other words, for the rest of the year [from leap day onward] the Sundays come a day earlier than they would in a common year."[6]

Thus a leap year is given two Dominical Letters, as Thurston explains, "the second being the letter which precedes that with which the year started."[6] For example, in 2012 (= AG), Sundays preceding leap day were A days, and G days for the rest of the year.

## Further examples

The following are sample of years, and their assigned dominical letters:

• 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
• 2033 B
• 2034 A
• 2035 G
• 2036 FE
• 2037 D
• 2038 C
• 2039 B
• 2040 AG

The dominical letter of a year determines the days of week in its calendar. The following are the correspondences between dominical letters, and the day of the week on which their corresponding common and leap years begin:

## 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 400 years. Of the 400 years in a single Gregorian cycle, there are :

• 44 common years for each single Dominical letter in D and F;
• 43 common years for each single Dominical letter in A, B, C, E, and G;
• 15 leap years for each double Dominical letters in AG and CB;
• 14 leap years for each double Dominical letters in ED and FE;
• 13 leap years for each double Dominical letters in BA, DC, and GF.

The Julian calendar repeats every 28 years. Of the 28 years in a single Julian cycle, there are

• 3 common years for each single Dominical letter in A, B, C, D, E, F, and G;
• 1 leap year for each double Dominical letters in BA, CB, DC, ED, FE, GF, and AG.

## 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:
2020 2010 2000 1990 1980 1970 1960 1950 1940 1930 1920 1910 1900
ED   C    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 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 equals the dominical letter of a year (DL) plus the day of the week (DW), and the dominical letter for the doomsday is C except for the portion of leap years before February 29 in which it is D, 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: G = 0 = Sunday, A = 1 = Monday, B = 2 = Tuesday, C = 3 = Wednesday, D = 4 = Thursday, E = 5 = Friday, and F = 6 = Saturday, i.e. in our context, C is mathematically identical to 3.

Hence, for instance, 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.

Doomsday Dominical letter
Sunday C, DC
Monday B, CB
Tuesday A, BA
Wednesday G, AG
Thursday F, GF
Friday E, FE
Saturday D, ED

### Julian calendar

To find the dominical letter in the Julian calendar, find the remainder of the year mod 28, and look it up in the following table.

Year mod 28 Dominical letter
0 DC
1 B
2 A
3 G
4 FE
5 D
6 C
7 B
8 AG
9 F
10 E
11 D
12 CB
13 A
14 G
15 F
16 ED
17 C
18 B
19 A
20 GF
21 E
22 D
23 C
24 BA
25 G
26 F
27 E

## Complete tables

### Table of dominical letters for years

#### Julian calendar

Years 0
700
1400
100
800
1500
200
900
300
1000
400
1100
500
1200
600
1300
00 28 56 84 DC ED FE GF AG BA CB
01 29 57 85 B C D E F G A
02 30 58 86 A B C D E F G
03 31 59 87 G A B C D E F
04 32 60 88 FE GF AG BA CB DC ED
05 33 61 89 D E F G A B C
06 34 62 90 C D E F G A B
07 35 63 91 B C D E F G A
08 36 64 92 AG BA CB DC ED FE GF
09 37 65 93 F G A B C D E
10 38 66 94 E F G A B C D
11 39 67 95 D E F G A B C
12 40 68 96 CB DC ED FE GF AG BA
13 41 69 97 A B C D E F G
14 42 70 98 G A B C D E F
15 43 71 99 F G A B C D E
16 44 72 ED FE GF AG BA CB DC
17 45 73 C D E F G A B
18 46 74 B C D E F G A
19 47 75 A B C D E F G
20 48 76 GF AG BA CB DC ED FE
21 49 77 E F G A B C D
22 50 78 D E F G A B C
23 51 79 C D E F G A B
24 52 80 BA CB DC ED FE GF AG
25 53 81 G A B C D E F
26 54 82 F G A B C D E
27 55 83 E F G A B C D

#### Gregorian calendar

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

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

### Table for days of the year

Days Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
1 8 15 22 (29) A D D G B E G C F A D F
2 9 16 23 (30) B E E A C F A D G B E G
3 10 17 24 (31) C F F B D G B E A C F A
4 11 18 25 D G G C E A C F B D G B
5 12 19 26 E A A D F B D G C E A C
6 13 20 27 F B B E G C E A D F B D
7 14 21 28 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 4 5 16
ED  C B A GF E D C  BA  G F E DC B A G FE D C B  AG  F E D CB

and

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

2090 2080 2070 2060 2050 2040 2030 2020 2010 2000 1990 1980 1970 1960 1950 1940 1930 1920 1910
A    GF   E    DC   B    AG   F    ED   C    BA   G    FE   D    CB   A    GF   E    DC   B
|     |   |     |   |     |   |     |   |     |   |     |   |     |   |     |   |     |   |
AG    F   ED    C   BA    G   FE    D   CB    A   GF    E   DC    B   AG    F   ED    C   BA
2096 2086 2076 2066 2056 2046 2036 2026 2016 2006 1996 1986 1976 1966 1956 1946 1936 1926 1916

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);
}

## References

1. ^ Peter Archer, The Christian Calendar and the Gregorian Reform (New York: Fordham University Press, 1941) p.5
2. ^ Bonnie Blackburn, Leofranc Holford-Strevens, The Oxford Companion to the Year (Oxford: Oxford University Press, 1999), p.829
3. ^ Calendarium (Calendar attached to the papal bull "Inter gravissimas")
4. ^ "Anno vicesimo quarto Georgii II. c.23" (1751), The Statutes at Large, from Magna Charta to the end of the Eleventh Parliament of Great Britain, Anno 1761, ed. Danby Pickering, p.194.
5. ^ J. K. Fotheringham, "Explanation: The Calendar", The Nautical Almanac and Astronomical Ephemeris for the year 1931, pp.735-747, p.745, ... 1938, pp.790-806, p.803.
6. ^ a b c d e f g h i j k l Thurston, H. (1909). Dominical Letter. In The Catholic Encyclopedia. New York: Robert Appleton Company. see New Advent at [1], accessed 27 January 2015.
7. ^ Chamberlain Fong, Michael K. Walters: "Methods for Accelerating Conway's Doomsday Algorithm (part 2)", 7th International Congress of Industrial and Applied Mathematics (2011).
8. ^ Gauss, Carl F. (1981). "Den Wochentag des 1. Januar eines Jahres zu finden. Güldene Zahl. Epakte. Ostergrenze.". Werke. herausgegeben von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. (2. Nachdruckaufl. ed.). Hildesheim: Georg Olms Verlag. pp. 206–207.