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 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 on stone calendars to indicate each day's position in the eight-day market week. 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 marks the Sundays. Nowadays they are used primarily 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. 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, the second for the remainder of the year.
In leap years, the leap day may or may not have a 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. This is coincidental: until 1970 24 February was the feast day of St Matthias, a "Double of Second Class", hence duplex. 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. After the 1662 reform there was correspondence between the Archbishop of Canterbury and the printer of the Book of Common Prayer, in which it was explained that the feast day of St Matthias now fell on 24 February every year.
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 and arrangement

Per Thurston, dominical letters are:
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."

Dominical letter cycle

Months
Jan OctA
MayB
AugC
Feb Mar NovD
JunE
Sept DecF
Apr JulyG

Thurston continues:

Of course, "24 February" is not "counted twice". The 23rd is ante diem vii kalendas Martias, the next day in a leap year is a.d. bis sextum kal. Mart., the next day is the regular a.d.vi kal. Mart., and so to the end of the month. For example, in 2020, all days preceding the leap day will correspond to a common-year E calendar, and all days afterward will correspond to a common-year D calendar. The relevant line of the Februarius page in the Kalendarium of a 1913 Breviarium Romanum reads:



The first column is the epact, a replacement for the golden number, from which the age of the moon was computed and announced in some English cathedrals prior to the Reformation. The second column is the letter, the third the Roman date and the fourth the modern date. A note at the foot of the page reads:

In anno bissextili mensis Februarius est dierum 29. et Festum S. Mathiae celebratur die 25. Februarii et bis dicitur sexto Kalendas, id est die 24. et die 25. et littera Dominicalis, quae assumpta fuit in mense Januario, mutatur in praecedentem; ut si in Januario littera Dominicalis fuerit A, mutatur in praecedentem, quae est g. etc.; et littera f bis servit, 24. et 25.

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 years is beginning and ending:
The Gregorian calendar repeats every 400 years. Of the 400 years in one Gregorian cycle, there are:
The Julian calendar repeats every 28 years. Of the 28 years in one Julian cycle, there are:
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:
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:
Similarly, for 2007:
For 2065:
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 =.
  4. If T is odd, add 11.
  5. Let T = T mod 7.
  6. Count forward T letters from the century's dominical letter 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.
  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.
  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 formulae for the Gregorian calendar is
It is equivalent to
and
For example, to find the Dominical Letter of the year 1913:
Therefore, the Dominical Letter is E.
De Morgan's rules no. 1 and 2 for the Julian calendar:
To find the Dominical Letter of the year 1913 in the Julian calendar:
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 first two months of the year to the leap day 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 plus the day of the week, 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 = mod 7 = 6 = F. The dominical letter of the year 1913 is E, so DW = 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 or 700. In the case of the Revised Julian calendar, find the date of Easter Sunday 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:
When a country switched to the Gregorian calendar, there could be some unusual combinations of dominical letters.

Some examples

Enter the "all in one table" to find the date of the paschal full moon, 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 AD years since AD 42

Note that this table does not work for AD years at the early stage of the real Julian calendar before year AD 42 or for any BC year, except when using the Julian calendar rules for proleptic dates. The duration of months, and the number and placement of intercalated days in February also changed inconsistently before AD 42 in the early Julian calendars, depending on places and years, causing finally lot of confusion in the population.
In these early AD years and in all BC years, with the effective Julian calendars used locally to align the counting of years, a variable number of days at end of the months were also still counted relatively from the start of the next named month, and years were theoretically starting on 1 March. As well, all these early years were effectively counted inclusively and positively from a different, much earlier epoch in other eras, such as the supposed foundation of Rome, or the accession to power of a local ruler.
Instructions
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 left hand column on the same line as each component of the date 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 left hand column. The day of the week is beside it. Bold figures 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,. 20 indicates that 2000 is a leap year. Use bold Jan and Feb only in leap years.
For determination of the day of the week
Example. What is the date of Easter in 2017?
.. Golden number is 4. Date of paschal full moon is 2 April. From "week table" 2 April 2017 is Saturday........ Easter Sunday in the Revised Julian calendar is.

Calculate the day of the week in the Revised Julian calendar

Note that the date in the Revised Julian and Gregorian calendars is the same up until 28 February 2800, and that for large years it may be possible to subtract 6300 or a multiple thereof before starting so as to reach a year within or closer to the table.
To look up the weekday of any date for any year using the table, subtract 100 from the year, divide the number obtained by 100, multiply the resulting quotient by seven and divide the product by nine. Note the quotient. Enter the table with the Julian year, and just before the final division add 50 and subtract the quotient noted above.
Example: What is the day of the week of 27 January 8315?
,,,,. 2015 is 700 years ahead of 1315, so 1315 is used. From the table: for hundreds : 6. For remaining digits : 4. For month : 0. For date : 27... Day of week = Tuesday.

Dominical letter

To find the dominical letter, calculate the day of the week for either 1 January or 1 October. If it is Sunday, the Sunday Letter is A, if Saturday B, and similarly backwards through the week and forwards through the alphabet to Monday, which is G.
Leap years have two letters, so for January and February calculate the day of the week for 1 January and for March to December calculate the day of the week for 1 October.
Leap years are all years that divide exactly by four, with the following exceptions:
Gregorian calendar – all years divisible by 100, except those that divide exactly by 400.
Revised Julian calendar – all years divisible by 100, except those with a remainder of 200 or 600 when divided by 900.

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". 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 for Monday, 3 for Tuesday, et cetera; these correspond to dominical letters F, E, D, C, B, A, and G, respectively.

Use for computer calculation

Computers are able to calculate the Dominical letter for the first day of a given month in this way, where:

char dominical

Years are also given a dominical letter or pair of dominical letters according to the first day in January and last day in December: when they are equal, only the first letter is given. The dominical letter of the last day of December just precedes in the ordered cycle, the dominical letter of the first day in January for the next year.