The Sunday letters

Dominical letters: the medieval machine that wrote the weekdays into the calendar itself — the doomsday trick's quiet twin. ← Calendar Contraptions

The doomsday trick — the previous piece in this corner — is a machine that computes a weekday in your head. This one is its older, quieter sibling: a machine that records them. Write the letters A to G against the days of the year, A on January 1, B on January 2, and round they go. Exactly one letter lands on every Sunday, and that letter is the year's dominical letter — its Sunday letter. 2026 is a D year: every D-day in it is a Sunday, from January 4 onward, and a clerk who knew that needed no arithmetic at all to plan the working year.

For centuries this was how weekdays were read, not computed: the letter sat at the top of almanac pages and carved calendar staves, the Church's Easter tables were sorted by it, and it survives in the tables still printed in breviaries. It is also the same seven-spoked wheel as Conway's doomsday, read from the opposite side — this page builds the bridge between them.

The ruler and the poem

The letters are nailed to the year, not to the weekdays: January 1 is always an A-day, February 1 always a D-day, and the first day of every month keeps its own letter for life. A medieval reader did not memorize that table — he memorized a poem. The twelve month-start letters spell out of a couplet credited to the liturgist Durandus, where the first letters of the twelve words are January through December:

An English mnemonic kept the same letters: "At Dover dwell George Brown, Esquire; Good Christopher Finch; and David Fryer." And the letters are older than the weeks they mark: the Romans wrote their eight-day market week as A–H on stone calendars — the word nundinae comes from novem, nine, because Romans counted inclusively — and when the seven-day week arrived, the first Christians added the A–G row alongside it. Both cycles survive together on marble fragments and on the Philocalian calendar of AD 356. The device that preserves most of what we know about them (Thurston's 1909 account) calls it plainly "a device adopted from the Romans by the old chronologers" to find the day of the week for any date.

The leap splice

A common year is 52 weeks and one day, so each new year's letter slides one step back down the alphabet. A leap year moves two — so it gets two letters: the first rules January and February, the second (always the alphabetical predecessor of the first) rules from March on. 2040 is AG: A-days are Sundays through February, G-days from March.

Where the leap day itself sits split the two halves of the western church. Rome's Inter gravissimas (1582) kept the old Roman habit of doubling the sixth day before the Kalends of March — February 24 — so the letter f serves twice, on the 24th and the 25th, and Saint Matthias's feast slides to the 25th in leap years. The Anglican settlement of 1662 (and Britain's calendar act of 1751) simply let February 29 exist as a day with no letter at all. Both splices do the same job: from March 1, the Sundays sit one letter earlier than they did in January.

The doomsday bridge

The doomsday dates — 4/4, 6/6, 8/8, Pi Day, the last day of February — are always C-days (in January and February of leap years, D-days). That makes the year's letter and its doomsday one subtraction apart, in the bridge's own numbering where the letters run G = 0 and the weekdays count Sunday = 0: C ≡ letter + doomsday (mod 7). 2013 is an F-year with a Thursday doomsday: 6 + 4 = 10 ≡ 3 = C. The table is that subtraction, done once and for all — the doomsday piece reaches the same rows from the other side.

Three routes to the letter

The first route is the definition: find the first Sunday, read its letter. The second is the odd + 11 machine — Fong and Walters' 2010 shortcut, the same one the doomsday trick runs, but counted forward along the alphabet from the century's letter (BA for 1600 and 2000, then C, E, G). De Morgan's rule is the Victorian one-liner: add the year, its quarter, and the centuries' correction, then read the remainder against A = 6 … G = 0. All three land on the same letter every time; this page computes all three and shows them agreeing.

The letter and Easter

Why the Church cared: Easter is the Sunday after a full-moon date that can fall anywhere from March 22 to April 25 — 35 possible days. Sundays come every seven days, so each letter owns exactly five of the 35. That arithmetic is why the old tables could promise to name a whole calendar from two keys — the letter and the epact (the moon-age number). The medieval Pye shelved its 35 possible Easter calendars as "primum A, secundum A, tertium A" — the first, second, third of the five A-Easters. Below: your year's five candidates, with the real Easter marked and its Pye name.

Bede, working a century before the letters reached his books, ran the same machine with numbers instead: his concurrents, 1 to 7, name the weekday of March 24 — a date that is always an F-day in the letter chain, which is what makes the trick work. Concurrent 1 (March 24 a Sunday) means the year is F, and 2 through 7 walk backward through the week to E, D, C, B, A, G.

The 400-year ledger

The Gregorian cycle closes every 400 years — 146,097 days, exactly 20,871 weeks — so the letter census of any 400-year span is the census of all of them. It is not a fair census. Counted live over :

Years that broke the letters

The letters assume one calendar runs the whole year. When countries switched, some years ran on two — and the letters kept the receipts. The full cast of switch nights lives in Ten days that never happened; here are the three years the letters remember.

The Julian twin

On the Julian calendar the letters run a clean 28-year cycle — 28 years are 10,227 days, exactly 1,461 weeks — with three common years and one leap year for every letter. No century reset: each Julian century slides the letters one step further, forever. The Gregorian calendar paid for its smoother centuries with the 400-year cycle and the lopsided census above. In 1913 the two systems sat one letter apart — E on the Gregorian calendar, F on the Julian — and the gap shifts by one letter at each century turn.

Honest limits

The letters belong to a calendar, not to history. This page reads proleptic Gregorian (and, in the twin section, proleptic Julian) across 1600–2999, while real countries switched at different times or never — the three special years above are the exceptions that make the point. The civil and Catholic leap-day conventions differ only in the back half of February, and both are conventions; the Revised Julian calendar used by some Orthodox churches splits the difference with its own 900-year leap rule and agrees with Gregorian weekdays until 2800. And a letter alone never names a weekday — it names the Sundays. The rest is counting, which the doomsday piece teaches with a drill, and this page declines to compete with.