Dominical letter

Dominical letters are a method used to determine the day of the week for particular dates.

They are derived from the Roman practice of marking the repeating sequence of eight letters A–H (commencing with A on 1 January) on stone calendars to indicate each day's position in the eight-day market week (nundinae). The word is derived from the number nine due to their practice of inclusive counting. After the introduction of Christianity a similar sequence of seven letters A–G was added alongside, again commencing with 1 January. The dominical letter marked the Sundays. Nowadays they are only used as part of the computus, which is the method of calculating the date of Easter.

A common year is assigned a single dominical letter, indicating which lettered days are Sundays in that particular year (hence the name, from Latin dominica for Sunday). Thus, 2017 is A, indicating that all A days are Sunday, and by inference, 1 January 2017 is a Sunday. Leap years are given two letters, the first valid for January 1 – February 28 (or February 24, see below), the second for the remainder of the year.

In leap years, the leap day may or may not have a dominical letter. In the Catholic version it does, but in the 1662 and subsequent Anglican versions it does not. The Catholic version causes 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 have a dominical letter of F. The Anglican version adds a day to February that did not exist in common years, 29 February, thus it does not have a dominical letter of its own.

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

History

Per Thurston (1909), 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."

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." 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."

He then goes on to note that:

The Romans were accustomed to divide the year into nundinæ, periods of eight days; and in their marble fasti, or calendars, 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, or week, was introduced in the time of Augustus, the first seven letters of the alphabet were employed in the same way to indicate the days of the new division of time… [noting as well that] fragmentary calendars on marble still survive in which both a cycle of eight letters — A to H — indicating nundinae, and a cycle of seven letters — A to G — indicating weeks, are used side by side (see "Corpus Inscriptionum Latinarum", 2nd ed., I, 220… [where the] same peculiarity occurs in the Philocalian Calendar of A.D. 356, ibid., p. 256)...

and that this device was imitated by the Christians.

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.

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."

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:

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

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."

Thurston then notes that a complication arises with leap years, which have an extra day. Traditionally, the Catholic ecclesiastical calendar treats 24 February (the "bissextus") as the day added, as this was the Roman leap day (bis sextus ante Kalendas Martii), 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."

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." For example, in 2016 (= CB), all C days preceding the leap day were Sundays, and all B days for the rest of the year.

Dominical letter of a year

The dominical letter of a year provides the link between the date and the day of the week on which it falls. The following are the correspondences between dominical letters and the day of the week on which their corresponding common and leap years begin:

• A common year starting on Sunday
• B common year starting on Saturday
• C common year starting on Friday
• D common year starting on Thursday
• E common year starting on Wednesday
• F common year starting on Tuesday
• G common year starting on Monday
• AG leap year starting on Sunday
• BA leap year starting on Saturday
• CB leap year starting on Friday
• DC leap year starting on Thursday
• ED leap year starting on Wednesday
• FE leap year starting on Tuesday
• GF leap year starting on Monday

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 D and F;
• 43 common years for each single Dominical letter A, B, C, E, and G;
• 15 leap years for each double Dominical letter AG and CB;
• 14 leap years for each double Dominical letter ED and FE;
• 13 leap years for each double Dominical letter 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 A, B, C, D, E, F, and G;
• 1 leap year for each double Dominical letter 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 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 plus 11 method

A simpler method suitable for finding the year's dominical letter was discovered in 2010. It is called the "odd plus 11" method.

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

$\left({\frac {y+11(y{\bmod {2}})}{2}}+11\left({\frac {y+11(y{\bmod {2}})}{2}}{\bmod {2}}\right)\right){\bmod {7}}.$

De Morgan's rule

This rule was stated by Augustus de Morgan:

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.

So the formula is

$\left(1+{\text{year}}+{\frac {\text{year}}{4}}+{\frac {{\text{year}}-1600}{400}}-{\frac {{\text{year}}-1600}{100}}\right)\mod 7$.

It is equivalent to

$\left({\text{year}}+{\frac {\text{year}}{4}}+{\frac {\text{year}}{400}}-{\frac {\text{year}}{100}}-1\right)\mod 7$

and

$\left(y+{\frac {y}{4}}+5(c{\bmod {4}})-1\right){\bmod {7}}.$

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 letter of a date equals the dominical letter of a year (DL) plus the day of the week (DW), and the letter for the doomsday is C except for the portion of leap years before February 29 in which it is D, we have:

${\begin{aligned}{\text{C}}&=({\text{DL}}+{\text{DW}}){\bmod {7}}\\{\text{DL}}&=({\text{C}}-{\text{DW}}){\bmod {7}}\\{\text{DW}}&=({\text{C}}-{\text{DL}}){\bmod {7}}\end{aligned}}$

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
	Non-leap year	Leap year
Sunday	C	DC
Monday	B	CB
Tuesday	A	BA
Wednesday	G	AG
Thursday	F	GF
Friday	E	FE
Saturday	D	ED

All in one table

If the year of interest is not within the table, use a tabular year which gives the same remainder when divided by 400 (Gregorian calendar) or 700 (Julian calendar). In the case of the Revised Julian calendar, find the date of Easter (see section "Calculating Easter Sunday", heading "Revised Julian calendar") and enter it into the "Table for days of the year" below. If the year is leap, the dominical letter for January and February is found by inputting the date of Easter Monday. Note the different rules for leap year:

• Gregorian calendar every year which divides exactly by four, but of century years only those which divide exactly by 400 (so ignore the left-hand letter given for a century year which is not leap).
• Julian calendar every year which divides exactly by four
• Revised Julian calendar every year which divides exactly by four, but of century years only those which give remainder 200 or 600 when divided by 900.

Julian calendar	700 1400	100 800 1500	200 900 1600	300 1000 1700	400 1100 1800	500 1200 1900	600 1300 2000	Paschal full moon date Year mod 19
Gregorian calendar	1700 2100		1800 2200		1900 2300	1600 2000		Gregorian (1900–2199)	Julian calendar
00 28 56 84	DC	ED	FE	GF	AG	BA	CB	14 Apr	5 Apr
01 29 57 85	B	C	D	E	F	G	A	3 Apr	25 Mar
02 30 58 86	A	B	C	D	E	F	G	23 Mar	13 Apr
03 31 59 87	G	A	B	C	D	E	F	11 Apr	2 Apr
04 32 60 88	FE	GF	AG	BA	CB	DC	ED	31 Mar	22 Mar
05 33 61 89	D	E	F	G	A	B	C	18 Apr	10 Apr
06 34 62 90	C	D	E	F	G	A	B	8 Apr	30 Mar
07 35 63 91	B	C	D	E	F	G	A	28 Mar	18 Apr
08 36 64 92	AG	BA	CB	DC	ED	FE	GF	16 Apr	7 Apr
09 37 65 93	F	G	A	B	C	D	E	5 Apr	27 Mar
10 38 66 94	E	F	G	A	B	C	D	25 Mar	15 Apr
11 39 67 95	D	E	F	G	A	B	C	13 Apr	4 Apr
12 40 68 96	CB	DC	ED	FE	GF	AG	BA	2 Apr	24 Mar
13 41 69 97	A	B	C	D	E	F	G	22 Mar	12 Apr
14 42 70 98	G	A	B	C	D	E	F	10 Apr	1 Apr
15 43 71 99	F	G	A	B	C	D	E	30 Mar	21 Mar
16 44 72	ED	FE	GF	AG	BA	CB	DC	17 Apr	9 Apr
17 45 73	C	D	E	F	G	A	B	7 Apr	29 Mar
18 46 74	B	C	D	E	F	G	A	27 Mar	17 Apr
19 47 75	A	B	C	D	E	F	G	Month
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	Feb Mar Nov
23 51 79	C	D	E	F	G	A	B	Aug
24 52 80	BA	CB	DC	ED	FE	GF	AG	Jan May Oct
25 53 81	G	A	B	C	D	E	F	Apr Jul
26 54 82	F	G	A	B	C	D	E	Sep Dec
27 55 83	E	F	G	A	B	C	D	Jun
Day	1	2	3	4	5	6	7	Table of letters for the days of the year
	8	9	10	11	12	13	14
	15	16	17	18	19	20	21
	22	23	24	25	26	27	28
	29	30	31

Calculating Easter Sunday

Golden number		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Paschal full moon date	Gregorian (1900–2199)	14 Apr	3 Apr	23 Mar	11 Apr	31 Mar	18 Apr	8 Apr	28 Mar	16 Apr	5 Apr	25 Mar	13 Apr	2 Apr	22 Mar	10 Apr	30 Mar	17 Apr	7 Apr	27 Mar
	Julian	5 Apr	25 Mar	13 Apr	2 Apr	22 Mar	10 Apr	30 Mar	18 Apr	7 Apr	27 Mar	15 Apr	4 Apr	24 Mar	12 Apr	1 Apr	21 Mar	9 Apr	29 Mar	17 Apr

To find the golden number, add 1 to the year and divide by 19. The remainder (if any) is the golden number, and if there is no remainder the golden number is 19. Obtain the date of the paschal full moon from the table, then use the "week table" below to find the day of the week on which it falls. Easter is the following Sunday.

Week table: Julian and Gregorian calendars

For Julian dates before 1300 and after 1999 the year in the table which differs by an exact multiple of 700 years should be used. For Gregorian dates after 2299,the year in the table which differs by an exact multiple of 400 years should be used. The values "r0" through "r6" indicate the remainder when the Hundreds value is divided by 7 and 4 respectively, indicating how the series extend in either direction. Both Julian and Gregorian values are shown 1500–1999 for convenience.

The corresponding numbers in the far right hand column on the same line as each component of the date (the hundreds, remaining digits and month) and the day of the month are added together. This total is then divided by 7 and the remainder from this division located in the far right hand column. The day of the week is beside it. Bold figures (e.g., 04) denote leap year. If a year ends in 00 and its hundreds are in bold it is a leap year. Thus 19 indicates that 1900 is not a Gregorian leap year, (but 19 in the Julian column indicates that it is a Julian leap year, as are all Julian x00 years). 20 indicates that 2000 is a leap year. Use Jan and Feb only in leap years.

Century digits		Remaining year digits				Month					Day of week	Number
Julian (r ÷ 7)	Gregorian (r ÷ 4)
r5 19	16 20 r0	00 06 00 17 23	28 34 00 45 51	56 62 00 73 79	84 90	Jan				Oct	Sat	0
r4 18	15 19 r3	01 07 12 18	29 35 40 46	57 63 68 74	85 91 96			May			Sun	1
r3 17	—	02 00 13 19 24	30 00 41 47 52	58 00 69 75 80	86 00 97	Feb			Aug		Mon	2
r2 16	18 22 r2	03 08 14 00 25	31 36 42 00 53	59 64 70 00 81	87 92 98	Feb	Mar			Nov	Tue	3
r1 15	—	00 09 15 20 26	00 37 43 48 54	00 65 71 76 82	00 93 99			Jun			Wed	4
r0 14	17 21 r1	04 10 00 21 27	32 38 00 49 55	60 66 00 77 83	88 94				Sep	Dec	Thu	5
r6 13	—	05 11 16 22	33 39 44 50	61 67 72 78	89 95	Jan	Apr	Jul			Fri	6

For determination of the day of the week (1 January 2000, Saturday)

• the day of the month: 1
• the month: 6
• the year: 0
• the century mod 4 for the Gregorian calendar and mod 7 for the Julian calendar 0
• adding 1 + 6 + 0 + 0 = 7. Dividing by 7 leaves a remainder of 0, so the day of the week is Saturday.

Revised Julian calendar

• Use the Julian portion of the table of paschal full moons. Use the "week table" (remembering to use the "Julian" side) to find the day of the week on which the paschal full moon falls. Easter is the following Sunday and it is a Julian date. Call this date JD.
• Subtract 100 from the year.
• Divide the result by 100. Call the number obtained (omitting fractions) N.
• Evaluate ⁠7N/9⁠. Call the result (omitting fractions) S.
• The Revised Julian calendar date of Easter is JD + S − 1.

Example. What is the date of Easter in 2017?

2017 + 1 = 2018. 2018 ÷ 19 = 106 remainder 4. Golden number is 4. Date of paschal full moon is 2 April (Julian). From "week table" 2 April 2017 (Julian) is Saturday. JD = 3 April. 2017 − 100 = 1917. 1917 ÷ 100 = 19 remainder 17. N = 19. 19 × 7 = 133. 133 ÷ 9 = 14 remainder 7. S = 14. Easter Sunday in the Revised Julian calendar is April 3 + 14 − 1 = April 16.

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	93	94	95	96	97	98	99	2000	01	02	03	04	05	06	07	08	09	10	11	12	13	14	15	16	17	18	19	2020
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	A	G	F	ED

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 or month/day 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).

Use for computer calculation

Computers 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);
 }

See also

• Lectionary#3 year cycle

Bibliography

•  This article incorporates text from a publication now in the public domain: Herbermann, Charles, ed. (1913). Catholic Encyclopedia. New York: Robert Appleton Company. {{cite encyclopedia}}: Missing or empty |title= (help)

External links

• Chisholm, Hugh, ed. (1911). "Calendar s.v. Ecclesiastical Calendar" . Encyclopædia Britannica. Vol. 4 (11th ed.). Cambridge University Press. p. 992.
• Catholic Encyclopedia article on Dominical letter