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. 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.

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.

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 (four centuries). 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.

Here's the following table that the Dominical letter of a year determines the days of week in its calendar:

Sunday  letter	Date January  1st Sunday	number of perpetual	calendar	frequency  Gregorian
A	1	7	common year starting on Sunday	43
AG	1	14	leap year starting on Sunday	15
B	2	6	common year starting on Saturday	43
BA	2	13	leap year starting on Saturday	13
C	3	5	common year starting on Friday	43
CB	3	12	leap year starting on Friday	15
D	4	4	common year starting on Thursday	44
DC	4	11	leap year starting on Thursday	13
E	5	3	common year starting on Wednesday	43
ED	5	10	leap year starting on Wednesday	14
F	6	2	common year starting on Tuesday	44
FE	6	9	leap year starting on Tuesday	14
G	7	1	common year starting on Monday	43
GF	7	8	leap year starting on Monday	13
			total	400

Examples

1894 G 1895 F 1896 ED 1897 C 1898 B 1899 A

1900 G 1901 F 1902 E 1903 D 1904 CB 1905 A

1906 G 1907 F 1908 ED 1909 C 1910 B 1911 A

1912 GF 1913 E 1914 D 1915 C 1916 BA

1917 G 1918 F 1919 E 1920 DC 1921 B 1922 A

1923 G 1924 FE 1925 D 1926 C 1927 B 1928 AG

1929 F 1930 E 1931 D 1932 CB 1933 A

1934 G 1935 F 1936 ED 1937 C 1938 B 1939 A

1940 GF 1941 E 1942 D 1943 C 1944 BA

1945 G 1946 F 1947 E 1948 DC 1949 B 1950 A

1951 G 1952 FE 1953 D 1954 C 1955 B 1956 AG

1957 F 1958 E 1959 D 1960 CB 1961 A

1962 G 1963 F 1964 ED 1965 C 1966 B 1967 A

1968 GF 1969 E 1970 D 1971 C 1972 BA

1973 G 1974 F 1975 E 1976 DC 1977 B 1978 A

1979 G 1980 FE 1981 D 1982 C 1983 B 1984 AG

1985 F 1986 E 1987 D 1988 CB 1989 A

1990 G 1991 F 1992 ED 1993 C 1994 B 1995 A

1996 GF 1997 E 1998 D 1999 C 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

Relationship between the Sunday letters and perpetual calendar

Dominical Letter(s)	A	B	C	D	E	F	G	AG	BA	CB	DC	ED	FE	GF
Number of perpetual calendar	7	6	5	4	3	2	1	14	13	12	11	10	9	8

Number of perpetual calendar for Julian calendar and Gregorian calendar and solar cycles

SC	0	1 *	2	3	4	5 *	6	7	8	9 *	10	11	12	13 *
#J	7	8	3	4	5	13	1	2	3	11	6	7	1	9
#G	1	9	4	5	6	14	2	3	4	12	7	1	2	10
SC	14	15	16	17 *	18	19	20	21 *	22	23	24	25 *	26	27
#J	4	5	6	14	2	3	4	12	7	1	2	10	5	6
#G	5	6	7	8	3	4	5	13	1	2	3	11	6	7

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. 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. 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 Augustus Cæsar, 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 of a date

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. A is always set against 1 January, B against 2 January, C against 3 January, and so on. Thus F falls to 6 January, G to 7 January; A again recurs 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. This is carried on through all the days of an ordinary year (i. e. not a leap year). Thus 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 result which Durandus recalled by the following distich:

Alta Domat Dominus, Gratis Beat Equa Gerentes

Contemnit Fictos, Augebit Dona Fideli.

Another one is:

Add G, beg C, fad F.

Yet another:

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

Clearly, if 1 January is a Sunday, all the days marked by A will also be Sundays; If 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; 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.

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 to leap years, and do not shift events in this way.

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 in decades, writing from right to left (future to past):

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.

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:

da

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.^{[citation needed]}

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

Determination of the Sunday letter 

The allocation of the Sunday letters to the calendar years repeated in the Julian calendar every 28 years. This cycle is called the sun compass. Each of these 28 years j a sun compass is SC -called number in the following Computus assigned:

SZ = (j + 9)  mod 28; Result: SC = 0 *), 1, ..., 26, 27 

• The constant association between Julian Solar Cycle SC and Dominical Letter DL following table shows:

SC

0

1 *

2

3

4

5 *

6

7

8

9 *

10

11

12

13 *

DL

A

GF

E

D

C

BA

G

F

E

DC

B

A

G

FE

SC

14

15

16

17 *

18

19

20

21 *

22

23

24

25 *

26

27

DL

D

C

B

AG

F

E

D

CB

A

G

F

ED

C

B

• Every 4 years, the Dominical Letter DL moves by 2 letters in the alphabet (second shift after the leap day).
• The italicized letter is January and February in leap years.

The Gregorian calendar is the allocation of the Sunday letters to Sonnenzirkel no longer constant. It changes whenever the full centuries that are not divisible by 400, the leap fails. Here, move every time. Every Sunday letters for a position in the alphabet forwards.

The association between Gregorian Solar Cycle SC and Dominical Letter DL shows for the years 1900 to 2099 the following table:

SC

0

1 *

2

3

4

5 *

6

7

8

9 *

10

11

12

13 *

DL

G

FE

D

C

B

AG

F

E

D

CB

A

G

F

ED

SC

14

15

16

17 *

18

19

20

21 *

22

23

24

25 *

26

27

DL

C

B

A

GF

E

D

C

BA

G

F

E

DC

B

A

Both the Julian and the Gregorian calendar, the Sunday letters can quite easily with the calendar by W. Bogatyrev determine - can in leap years there are only the second Sunday letter read (valid after the leap day).

External links

• Catholic Encyclopedia article on Dominical letter
• Wikisource article on ecclesiastical calendars