# Dominical letter

Dominical letters or Sunday letters are a method used to determine the day of the week for particular dates. When using this method, each year is assigned a letter (or pair of letters for leap years) depending on which day of the week the year starts on.

Dominical letters 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.[1][2][3] 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.[4][5]

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

## HistoryEdit

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."[6]

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, [the Church] 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 it "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:

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.[6]

## Dominical letter cycleEdit

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:

Alta Domat Dominus, Gratis Beat Equa Gerentes

Contemnit Fictos, Augebit Dona Fideli."[6]

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."

Months L
Jan Oct A
May B
Aug C
Feb Mar Nov D
Jun E
Sept Dec F
Apr July G
• If the letter (L) of the first day of a month is the dominical letter of the year, the month will have a Friday the 13th. That is to say, if the first day is Sunday, the 13th day will be Friday.

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 (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."[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 2020 (= ED), all E days preceding the leap day will be Sundays, and all D days for the rest of the year.

## Dominical letters of the yearsEdit

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:

The Gregorian calendar repeats every 400 years, i. e., every 4 centuries. Of the 400 years in one 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 one Julian cycle, there are/is:

• 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.

## CalculationEdit

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, the table below can be used.
Year mod 28 #
00 06 12 17 23 0
01 07 12 18 24 6
02 08 13 19 24 5
03 08 14 20 25 4
04 09 15 20 26 3
04 10 16 21 27 2
05 11 16 22 00 1

Red for the first two months of leap years.

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

• 1900 is G and 13 corresponds to 5
• G + 5 = G − 2 = E, 1913 is E

Similarly, for 2007:

• 2000 is BA and 7 corresponds to 6
• A + 6 = A − 1 = G, 2007 is G

For 2065:

• 2000 is BA and 65 mod 28 = 9 corresponds to 3
• A + 3 = A − 4 = D, 2065 is D

### The odd plus 11 methodEdit

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

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

### De Morgan's ruleEdit

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.[6]

So the formulae (using the floor function) for the Gregorian calendar is

${\displaystyle 1.\left(1+{\text{year}}+{\Big \lfloor }{\frac {\text{year}}{4}}{\Big \rfloor }+{\Big \lfloor }{\frac {{\text{year}}-1600}{400}}{\Big \rfloor }-{\Big \lfloor }{\frac {{\text{year}}-1600}{100}}{\Big \rfloor }\right){\bmod {7}}.}$

It is equivalent to

${\displaystyle 2.\left({\text{year}}+{\Big \lfloor }{\frac {\text{year}}{4}}{\Big \rfloor }+{\Big \lfloor }{\frac {\text{year}}{400}}{\Big \rfloor }-{\Big \lfloor }{\frac {\text{year}}{100}}{\Big \rfloor }-1\right){\bmod {7}}}$

and

${\displaystyle 3.\left(y+{\Big \lfloor }{\frac {y}{4}}{\Big \rfloor }+5(c{\bmod {4}})-1\right){\bmod {7}}}$      (where ${\displaystyle {\text{y}}}$  = last two digits of the year, ${\displaystyle {\text{c}}}$  = century part of the year).

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.

De Morgan's rules no. 1 and 2 for the Julian calendar:

${\displaystyle 1.}$  and ${\displaystyle 2.\left({\text{year}}+{\Big \lfloor }{\frac {\text{year}}{4}}{\Big \rfloor }-3\right){\bmod {7}}}$

To find the Dominical Letter of the year 1913 in the Julian calendar:

• (1913 + 478 − 3) mod 7 = 1

Therefore, the Dominical Letter is F in the Julian calendar.

In leap years the formulae above give the Dominical Letter for the period from the leap day to the end of the year. To find the Dominical Letter for the period from the beginning of the year to the leap day (inclusive) subtract 1 from the number representing the original Dominical Letter.

### Dominical letter in relation to the Doomsday RuleEdit

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:

{\displaystyle {\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 tableEdit

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 the section "Calculating Easter Sunday", subsection "Revised Julian calendar" below) and enter it into the "Table for days of the year" below. If the year is a leap year, the dominical letter for January and February is found by inputting the date of Easter Monday. Note the different rules for leap years:

• Gregorian calendar: every year which divides exactly by 4, but of century years only those which divide exactly by 400; therefore ignore the left-hand letter given for a century year which is not a leap year.
• Julian calendar: every year which divides exactly by 4.
• Revised Julian calendar: every year which divides exactly by 4, but of century years only those which give the 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 SundayEdit

 Golden number Paschal Gregorian Julian full moon (1900–2199) date 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 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 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 calendarsEdit

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 N/A 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 N/A 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 N/A 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 calendarEdit

• 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.

## Clerical utilityEdit

The dominical letter had another practical utility in the period prior to the annual printing of the Ordo divini officii recitandi, in which period, therefore, Christian clergy were often required to determine the Ordo independently. Easter Sunday may be as early as 22 March or as late as 25 April, and consequently there are 35 possible days on which it may occur; each dominical letter includes 5 potential dates of these 35, and thus there are 5 possible ecclesiastical calendars for each letter. The Pye or Directorium which preceded the present Ordo took advantage of this principle by delineating all 35 possible calendars and denoting them by the formula "primum A", "secundum A", "tertium A", et cetera. Hence, based on the dominical letter of the year and the epact, the Pye identified the correct calendar to use. A similar table, adapted to the reformed calendar and in more convenient form, is included in the beginning of every breviary and missal under the heading "Tabula Paschalis nova reformata".

Saint Bede does not seem to have been familiar with dominical letters, given his "De temporum ratione"; in its place he adopted a similar device of Greek origin consisting of seven numbers, which he denominated "concurrentes" (De Temp. Rat., Chapter LIII). The "concurrents" are numbers that denote the days of the week on which 24 March occurs in the successive years of the solar cycle, 1 denoting Sunday, 2 (feria secunda) for Monday, 3 for Tuesday, et cetera; these correspond to dominical letters F, E, D, C, B, A, and G, respectively.

## Use for mental calculationEdit

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

### Patterns for yearsEdit

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 21 22 23 24 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 C B A GF

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 monthEdit

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 calculationEdit

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);
}


## ReferencesEdit

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 Archived February 15, 2005, at the Wayback Machine. (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. 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).