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.

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."[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 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:

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."

MonthsL
Jan OctA
MayB
AugC
Feb Mar NovD
JunE
Sept DecF
Apr JulyG
  • 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 years

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 four 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:

  • 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, the table below can be used.
Year mod 28#
00 06 12 17 230
01 07 12 18 246
02 08 13 19 245
03 08 14 20 254
04 09 15 20 263
04 10 16 21 272
05 11 16 22 001

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 method

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

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.[6]

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

It is equivalent to

and

    (where = last two digits of the year, = 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:

and

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 last ten months of the year. To find the Dominical Letter for the two first months of the year to the leap day (inclusive) subtract 1 from the calculated number representing the original Dominical Letter; if the new number is less than 0, it must be changed to 6.

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:

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.

DoomsdayDominical letter
Common yearLeap year
SundayCDC
MondayBCB
TuesdayABA
WednesdayGAG
ThursdayFGF
FridayEFE
SaturdayDED

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 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 DCEDFEGFAGBACB14 Apr5 Apr
01 29 57 85 BCDEFGA3 Apr25 Mar
02 30 58 86 ABCDEFG23 Mar13 Apr
03 31 59 87 GABCDEF11 Apr2 Apr
04 32 60 88 FEGFAGBACBDCED31 Mar22 Mar
05 33 61 89 DEFGABC18 Apr10 Apr
06 34 62 90 CDEFGAB8 Apr30 Mar
07 35 63 91 BCDEFGA28 Mar18 Apr
08 36 64 92 AGBACBDCEDFEGF16 Apr7 Apr
09 37 65 93 FGABCDE5 Apr27 Mar
10 38 66 94 EFGABCD25 Mar15 Apr
11 39 67 95 DEFGABC13 Apr4 Apr
12 40 68 96 CBDCEDFEGFAGBA2 Apr24 Mar
13 41 69 97 ABCDEFG22 Mar12 Apr
14 42 70 98 GABCDEF10 Apr1 Apr
15 43 71 99 FGABCDE30 Mar21 Mar
16 44 72 EDFEGFAGBACBDC17 Apr9 Apr
17 45 73 CDEFGAB7 Apr29 Mar
18 46 74 BCDEFGA27 Mar17 Apr
19 47 75 ABCDEFG Month
20 48 76 GFAGBACBDCEDFE
21 49 77 EFGABCD
22 50 78 DEFGABC Feb Mar Nov
23 51 79 CDEFGAB Aug
24 52 80 BACBDCEDFEGFAG Jan May Oct
25 53 81 GABCDEF Apr Jul
26 54 82 FGABCDE Sep Dec
27 55 83 EFGABCD Jun
Day1234567 Table of letters for
the days of the year
891011121314
15161718192021
22232425262728
293031

Calculating Easter Sunday

Golden number 12345678910111213141516171819
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 digitsRemaining year digitsMonthDay of
week		Number
Julian
(r ÷ 7)		 Gregorian
(r ÷ 4)
r5 1916 20 r000 06 17 2328 34 45 5156 62 73 7984 90JanOctSat0
r4 1815 19 r301 07 12 1829 35 40 4657 63 68 7485 91 96MaySun1
r3 17N/A02 13 19 2430 41 47 5258 69 75 8086 97FebAugMon2
r2 1618 22 r203 08 14 2531 36 42 5359 64 70 8187 92 98FebMarNovTue3
r1 15N/A 09 15 20 26 37 43 48 54 65 71 76 82 93 99JunWed4
r0 1417 21 r104 10 21 2732 38 49 5560 66 77 8388 94SepDecThu5
r6 13N/A05 11 16 2233 39 44 5061 67 72 7889 95JanAprJulFri6

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.

Clerical utility

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 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.

199293949596979899200001020304050607080910111213141516171819202021222324
EDCBAGFEDCBAGFEDCBAGFEDCBAGFEDCBAGFEDCBAGF

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

2090208020702060205020402030202020102000199019801970196019501940193019201910
AGFEDCBAGFEDCBAGFEDCBAGFEDCB
AGFEDCBAGFEDCBAGFEDCBAGFEDCBA
2096208620762066205620462036202620162006199619861976196619561946193619261916

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

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 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. 1 2 3 4 5 6 7 8 9 10 11 12 Thurston, H. (1909). Dominical Letter. In The Catholic Encyclopedia. New York: Robert Appleton Company. see New Advent at , 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).

Bibliography

  •  This article incorporates text from a publication now in the public domain: Herbermann, Charles, ed. (1913). "article name needed". Catholic Encyclopedia. New York: Robert Appleton.
  • Wikisource Chisholm, Hugh, ed. (1911). "Calendar s.v. Ecclesiastical Calendar". Encyclopædia Britannica. 4 (11th ed.). Cambridge University Press. p. 992.
  • Catholic Encyclopedia article on Dominical letter
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