Dominical letter

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:

The Gregorian calendar repeats every 400 years (four centuries). Of the 400 years in a single Gregorian cycle, there are:

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

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:

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

Similarly, for 2007:

For 2065:

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

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

It is equivalent to

and

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

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:

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.

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:

Calculating Easter Sunday

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.

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

Revised Julian calendar

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.

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

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: