# Dominical letter

To create the list of Dominical letters, a cycle was composed of the first seven letters of the alphabet (A - G).
These were established according to their position in the first seven days of the calendar.
So for example, if the year starts on a Sunday, all Sundays will have the Dominical letter of A, every Monday will have the letter B, and so on till you reach Saturday, which are all given the letter G.
This cycle loops and repeats itself throughout the entirety of the year with each day being accounted for by one of the seven letters.
In this example, in the case of a leap year, 31 December would be given the letter A.
These letters of the day (Latin: *literae calendarum*).
Of the seven letters, the one that falls on the first Sunday of the year is called the “Dominical letter” of the year (Latin: *literae dominicales*),
its name derived from the word “Sunday” in Latin (*dies Dominica*); some sources simply call it the “Sunday letter”.

Januar 1 day of week | dominical letter |
---|---|

Monday | G |

Tuesday | F |

Wednesday | E |

Thuesday | D |

Friday | C |

Saturday | B |

Sunday | A |

There were problems that arose in the leap year during the Middle Ages, because the 24th of February (VI Kalendas Martias bis) was considered a leap day back then. In order for the table of dominical letters to be able to apply to leap years, the 24th and the 25th of February needed to have the same letter, all the while maintaining the pattern where no more than seven letters were being used in each week. T o solve this, after every leap day, the next Dominical letter was moved back one place. For example, the leap year 1516 had a Sunday with the letter F before the leap day, and after the leap day it was a Sunday with the letter E. Because of this, leap years were assigned two dominical letters: before and after the leap day. In this case, it was FE, then in the following leap year of 1520, the Dominical letter AG was used. Each leap year therefore received a pair of Dominical letters that could be applied to both calendars.

In the Julian calendar, you can assign a Dominical letter to the solar cycle according to the table below. The first year of the solar cycle had a leap year and began on a Monday.

solar cycle | dominical letter |
---|---|

1 | GF |

2 | E |

3 | D |

4 | C |

5 | BA |

6 | G |

7 | F |

8 | E |

9 | DC |

10 | B |

11 | A |

12 | G |

13 | FE |

14 | D |

15 | C |

16 | B |

17 | AG |

18 | F |

19 | E |

20 | D |

21 | CB |

22 | A |

23 | G |

24 | F |

25 | ED |

26 | C |

27 | B |

28 | A |

**i = 7 - (year + year / 4 - 3) % 7**

i | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

dominical letter | A | B | C | D | E | F | G |

Only integers are counted, so the '%' character (the modulo) serves to indicate that the remainder of the division is the only part to be used. In the table below, you can find the Dominical letter next to the calculated one. In the case of a leap year, this formula calculates the second letter, with its first letter being the next letter in the alphabet as usual.

In the Gregorian calendar, this assignment is no longer possible, so each century has one of the four rows of Dominical letters. This is due to the impenetrable centenary years, where the 28-year cycle gets disrupted (centenary years means years such as 1700, 1800, 1900, 2100). In these non leap years, the cycle of rotation is disrupted and gives us an irregular or “anomalous cycle”. In the table below we can find the Dominical letter for any Gregorian year.

year \ century | (c % 4) = 1 1700 2100 | (c % 4) = 2 1800 2200 | (c % 4) = 3 1500 1900 2300 | (c % 4) = 0 1600 2000 2400 |
---|---|---|---|---|

00 | C | E | G | BA |

1 29 57 85 | B | D | F | G |

2 30 58 86 | A | C | E | F |

3 31 59 87 | G | B | D | E |

4 32 60 88 | FE | AG | CB | DC |

5 33 61 89 | D | F | A | B |

6 34 62 90 | C | E | G | A |

7 35 63 91 | B | D | F | G |

8 36 64 92 | AG | CB | ED | FE |

9 37 65 93 | F | A | C | D |

10 38 66 94 | E | G | B | C |

11 39 67 95 | D | F | A | B |

12 40 68 96 | CB | ED | GF | AG |

13 41 69 97 | A | C | E | F |

14 42 70 98 | G | B | D | E |

15 43 71 99 | F | A | C | D |

16 44 72 | ED | GF | BA | CB |

17 45 73 | C | E | G | A |

18 46 74 | B | D | F | G |

19 47 75 | A | C | E | F |

20 48 76 | GF | BA | DC | ED |

21 49 77 | E | G | B | C |

22 50 78 | D | F | A | B |

23 51 79 | C | E | G | A |

24 52 80 | BA | DC | FE | GF |

25 53 81 | G | B | D | E |

26 54 82 | F | A | C | D |

27 55 83 | E | G | B | C |

28 56 84 | DC | FE | AG | BA |

Instructions for use: from the year in question we first determine the century (in this case the century is defined as the year starting with 00 and ending with 99). From there, take our whole number and divide it by four to determine the column. Among the rows, we find the last two digits of the year. From these two points we find the intersection, and get our Dominical letter. An interesting thing of note in the Gregorian calendar is that years which end with the same double digit can only appear in one of four possible days of the week. For example, years ending with 00 can only have 1 January on a Friday (Dominical letter C), Wednesday (E), Monday (G), or Saturday (B), regardless of which century it falls in.

Unsurprisingly, there are several different tables floating around that have been used to find the Dominical letter for a given year. Below, I present a more compact table from 1764, which works with a four-hundred-year cycle of the Gregorian calendar:

BA gfe | DC bag | FE dcb | AG fed | CB agf | ED cba | GF edc | |
---|---|---|---|---|---|---|---|

0 | 0 | 4 | 8 | 12 | 16 | 20 | 24 |

28 | 32 | 36 | 40 | 44 | 48 | 52 | |

56 | 60 | 64 | 68 | 72 | 76 | 80 | |

84 | 88 | 92 | 96 | - - | - - | - - | |

100c | - - | - - | 4 | 8 | 12 | 16 | 20 |

24 | 28 | 32 | 36 | 40 | 44 | 48 | |

52 | 56 | 60 | 64 | 68 | 72 | 76 | |

80 | 84 | 88 | 92 | 96 | - - | - - | |

200e | - - | - - | - - | 4 | 8 | 12 | 16 |

20 | 24 | 28 | 32 | 36 | 40 | 44 | |

48 | 52 | 56 | 60 | 64 | 68 | 72 | |

76 | 80 | 84 | 88 | 92 | 96 | - - | |

300g | - - | - - | - - | - - | 4 | 8 | 12 |

16 | 20 | 24 | 28 | 32 | 36 | 40 | |

44 | 48 | 52 | 56 | 60 | 64 | 68 | |

72 | 76 | 80 | 84 | 88 | 92 | 96 |

Instructions for use: first, we need to determine which century in the four-hundred-year cycle the year we’re searching for belongs to. If you look at the cycles, you may have noticed how they cycle every 400 years (e.g. 1600, 2000, 2400, and so on). Remove as many multiples of 400 as possible, and let the remainder be your guide as to whether you will use 0, 100, 200, or 300. So, if we’re looking for a Dominical letter for the year 2020, the year belongs to the zero century, and has the Dominical letter of ED. Next, let’s try to find the Dominical letter for 2011. 2011 also uses the zero century, but because 2011 is an ordinary year (not a leap year), we need to find the most recent previous leap year in order to determine the Dominical letter. In this case, that would be the year 2008. So, starting in the 0 row, we will go out to the 8, where the column reads FE/dcb. Since 2011 is not a leap year, we won’t use FE, but instead we’ll calculate the difference between 2011 and 2008, and use this number to determine which of the lower-case letters is our dominical letter. Since 11 - 8 = 3, we will look for the third lowercase letter, which gives us 'b'. If we were looking for the Dominical letter of 1989, we would get the letter 'a.' Try it yourself. The first thing we do is find how many multiples of 400 fit into 1989, and use this number to choose our row. In this case, 400 fits into 1989 a total of 4 times (400 * 4 = 1600). We then subtract this number from the year in question (1989 - 1600 = 389) to find which century row we should be using. In this case, that’s the 300 row. Now, take the year where the most recent leap year had previously occurred (1988), and subtract this number from the year we’re trying to find the dominical letter for (1989 - 1988 = 1). So, in the 300 row, above the number 88, we find the first lower-case letter (upper-case is only for leap years), and the first letter of ‘agf’ is the letter ‘a’. The Dominical letter of impenetrable centenary years (for example, 1700, 1800, 1900, 2100) is just written below the century in the previous table.

**i = 7 - (year + year / 4 - 3 - gregorian_repair) % 7**

**gregorian_repair = 10 + 3 * ((year / 100) - 15) / 4**

which can be conveniently combined to form:

**i = 7 - (year + year / 4 + 1 - 3 * ((year / 100) - 15) / 4) % 7**

Only integers are counted; the '%' character (the modulo) indicates that the remainder is to be used. With this, we can assign a Dominical letter from the Julian calculations table, using the number we get from our own calculation. The same instructions for determining the Dominical letter during leap years also applies. The Gregorian correction is the difference in days between the Julian and Gregorian calendars. In the 20th and 21st centuries it was 13 days, with the correction increased by 3 days every 400 years. The constant 10 in the Gregorian correction formula is due to the ten days that were omitted during the Gregorian calendar reform.