Dominical letter

A simpler method suitable for finding the year's dominical letter was discovered in 2010. It is called the "odd plus 11" method.[6]

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

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

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.[7]

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. (1 + 1913 + 478 + 0 − 3) mod 7 = 2

2. (1913 + 478 + 4 − 19 − 1) mod 7 = 2

3. (13 + 3 + 15 -1) mod 7 = 2

Hence, the Dominical Letter is E in the Gregorian calendar.

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

Hence, 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 (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.

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.

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 Sunday (see the section "Calculating Easter Sunday", subsection "Revised Julian calendar" below) and enter it into the "Table of letters for the 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.[8]

When a country switched to the Gregorian calendar, there could be some unusual combinations of dominical letters.

• 1582: Many Catholic countries switched to the Gregorian calendar Friday October 15. The table above indicates that year 1582 had the dominical letter G in the Julian calendar and C in the Gregorian one. So the dominical letters for 1582 in these Catholic countries became GC for mixing the two calendars used in this legal year, a special combination not seen before and after with a single calendar used in the same legal year.
• 1752: The British Empire and its colonies switched to the Gregorian calendar Thursday September 14. 1752, a leap year, had in the Julian calendar dominical letters ED and in the Gregorian one dominical letters BA, so the dominical letters for 1752 in Britain were EDA, a very special combination which also only applies to this legal year.

Note that this table does not work for AD years at the early stage of the real Julian calendar before March 1 AD 4[9] or for any BC year, except when using the Julian calendar rules for proleptic dates (which are different from effective historic dates, whose effective calendar in use depended on the location of dated events or the location of the person using the calendar, sometimes differently between political/civil or religious purposes in places where both calendars still coexisted). The duration of months, and the number and placement of intercalated days also changed inconsistently before AD 42 in the early local Julian calendars which used native names for the months, depending on places and years, causing finally a lot of confusion in the population (so dating events precisely in that period is often difficult, unless they are correlated with observed lunar cycles, or with days of the week, or with another calendar).

In these early AD years and in all BC years, with the effective Julian calendars used locally to align the counting of years (but still with the tradition inherited from the earlier Roman calendar for noting days in each year), a variable number of days at end of the months (after the last day of its ides but before the last day of calends which started the next month) were also still counted relatively from the start of the next named month (on the last day of its calends), and years were theoretically starting on March 1 (but with the last days of the year in February also counted from the New Year's Day in 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 (and still not relatively to the supposed date of birth of Christ, which was fixed later arbitrarily by a Christian reform for the modern Julian calendar so that this epoch for the Christian era starts now on January 1 in proleptic year AD 1 of the modern Julian calendar, but the real date of birth of Christ is still not known precisely but certainly falls before, somewhere in the last few BC years).

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 (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 left 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 bold 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 bold Jan and Feb only in leap years.

For determination of the day of the week (January 1, 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.

• 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 April 2 (Julian). From "week table" April 2, 2017 (Julian) is Saturday. JD = April 3. 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.

Note that the date (and hence the day of the week) in the Revised Julian and Gregorian calendars is the same up until February 28, 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 (omitting fractions) by seven and divide the product by nine. Note the quotient (omitting fractions). 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?

8315 − 6300 = 2015, 2015 − 100 = 1915, 1915 ÷ 100 = 19 remainder 15, 19 × 7 = 133, 133 ÷ 9 = 14 remainder 7. 2015 is 700 years ahead of 1315, so 1315 is used. From the table: for hundreds (13): 6. For remaining digits (15): 4. For month (January): 0. For date (27): 27. 6 + 4 + 0 + 27 + 50 − 14 = 73. 73 ÷ 7 = 10 remainder 3. Day of week = Tuesday.

To find the dominical letter, calculate the day of the week for either January 1 or October 1. 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 January 1 and for March to December calculate the day of the week for October 1.

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.

• Chisholm, Hugh, ed. (1911). "Calendar s.v. Ecclesiastical Calendar" . Encyclopædia Britannica. Vol. 4 (11th ed.). Cambridge University Press. p. 992.