poetic form · the Prosody Workshop
The Seventh Stanza Would Be the First
A sestina does not rhyme. It ends its lines on the same six words over and over, and shuffles them by one fixed rule between stanzas. Everyone teaches the rule. Almost nobody says why the poem stops at six stanzas, and the answer is not taste: apply the shuffle a seventh time and every word is standing exactly where it started. The form closes because the arithmetic underneath it closes. Below you can watch that happen, write one with the machinery keeping score, and turn a dial to the numbers where it does not work at all.
The rule, and the thing the rule does
Six words. Stanza one ends its lines on them in order. To get the next stanza you read the previous one from the outside in, alternating bottom and top: last, first, second-last, second, third-last, third. 6 1 5 2 4 3. The troubadours called it retrogradatio cruciata, the crosswise backward step.
Here it is running on the six words Arnaut Daniel used in about 1200, in the poem that invented the form. Press step. Watch the seventh row.
The seventh stanza is the first stanza. Not approximately: the same word in every one of the six positions. A poet who wrote a seventh stanza would be writing the first one again, so the form stops at six and adds a three-line envoi instead. Six is not a choice about length. It is the order of a permutation.
And notice what the six stanzas together have done. Each of the six words has stood in each of the six positions exactly once, and no combination has happened twice. The grid is a Latin square. That is what the form is actually for, and the shuffle is a machine for producing it.
The studio
So write one. Put six words in the top row and the machine will tell you, line by line, which one you have to land on, and whether you landed on it. It keeps no other score: the metre, the sense and the argument are yours. Nothing is sent anywhere, and your draft survives a reload.
The check is exact-word and case-insensitive, and it ignores punctuation at the end of the line. It is deliberately strict, because the interesting thing about the form is where you have to fight it. Real poets bend the rule constantly, which is measured further down.
Why six, and what happens at the other numbers
Nothing above used the number six except by arriving at it. The shuffle is defined for any number of end-words, so ask the same question of all of them: start from 1, 2, 3 and so on, apply the shuffle over and over, and see whether every word visits every position before anything repeats.
Drag the dial. The wheel shows the positions, and an arc from each position to the one its word moves to next. When the form works, the arcs make a single closed thread through every point. When it fails, they fall into separate loops that never meet, and each loop is a set of words condemned to circulate among themselves.
Every number from 2 to 60. Highlighted ones close.
Four end-words do not work, and the failure is easy to feel: the third word never moves. It ends the third line of every stanza forever. Seven does not work either, and eight splits the words into two families of four that never trade places. The numbers that do work start 1, 2, 3, 5, 6, 9, 11, 14, 18, 23, and they thin out. They are called Queneau numbers, after Raymond Queneau, who asked which lengths admit the form, and they are A054639 in the encyclopedia of integer sequences. Six is one of the small handful a poet could plausibly have stumbled on.
The shuffle is doubling
Here is the reason the good numbers look so arbitrary. Number the positions 1 to 6 and ask where the word in position k goes. It goes to 2k, unless 2k runs off the end, in which case it goes to 13 − 2k. One rule, two branches, and both branches are the same thing: multiply by two, modulo thirteen, and fold the answer back into 1 to 6 by ignoring the sign.
| position k | 2k | 2k mod 13 | folded into 1..6 | the spiral says |
|---|
Thirteen because six words means 2n + 1 = 13. So the question "does the sestina close?" is the question "does repeatedly doubling reach every number from 1 to 6 before returning?", which is a question about the multiplicative order of 2 modulo 13, and that is why the answer is a thin, irregular-looking set. The form is a fact about small primes wearing a Provençal costume.
It also explains the failures exactly. With four end-words the modulus is 9, which is not prime, and 3 divides it: doubling can never move 3 anywhere, so the third word is stuck. With eight the modulus is 17, doubling has order 8, but it passes through 16, which is −1, so the fold sends it home halfway and the words split into two orbits of four.
An error you can reproduce
Jacques Roubaud, the Oulipo mathematician and poet who did more than anyone to popularise this question, published 141 as a length that works. It does not. The shuffle on 141 end-words falls into three separate loops of 47, so a 141-word "sestina" would return to its opening arrangement after 47 stanzas with two thirds of the arrangements never reached. The encyclopedia records the correction; the dial above reproduces it if you have the patience to extend the range, and the offline verifier checks it directly.
What nine poets actually did
All of that is about the rule. Whether poems obey it is a different question, and an empirical one, so here are nine of them: the sestinas I could reach in a machine-readable text that is public domain worldwide, from the first one ever written to 1896. No claim that this is every such poem, only that these nine were chosen before any of them was measured. Their end-words were pulled out of the raw source files by a program, and the permutation between each pair of consecutive stanzas was measured.
| poem | lang | date | words | stanzas | spiral | latin square |
|---|
Six of the nine use the spiral and nothing but the spiral, across seven hundred years and three languages, including both of the poems that double the length. That is a stronger result than I expected from a rule transmitted by imitation for that long. It also settles what a "double sestina" of the older kind is: Petrarch and Sidney both run twelve stanzas on six words with the ordinary spiral, and since the spiral closes at six, their stanza 7 reproduces their stanza 1 exactly, which the measurement confirms for both. The second half of a double sestina is the first half again, in a new set of sentences.
The three poems that depart are the interesting ones, and each departs differently.
Spenser rotates
Swinburne trades the spiral for rhyme, and can prove he had to
The twelve-word double sestina cannot be done, and is not
And they bend the words themselves
| poem | wrote | for | the line |
|---|
One question that is still open in the wrong place
A footnote for anyone who follows the sequence link. A054639 carries a question posted by David Wasserman on 30 August 2011 and never answered there: is the order of this permutation always equal to the length of the orbit of 1? He reported checking it up to 9450. It is a fair thing to ask, because it is false for permutations in general, and the encyclopedia's own entry says "of order n" where a commenter says "is an n-cycle", which are not the same claim.
The answer has existed since 1969 and is one click from the question, in a paper the entry already links. M. Bringer, in Mathématiques et sciences humaines, proves as Lemme 1 that the length of every cycle divides the length of the cycle containing the first element, and draws the corollary directly:
Corollaire. L’ordre de δn est égal à l’ordre de C1 ; en effet l’ordre de δn étant le ppcm des ordres de Ci, est égal à l’ordre de C1.
M. Bringer, "Sur un problème de R. Queneau", Mathématiques et sciences humaines 27 (1969), pp. 13–20, p. 16. numdam.org. Quoted from the Numdam scan, whose OCR renders δ as 8 and C1 as Cx; the symbols are restored here and the sentence is otherwise verbatim.
The argument is one line once you see the map as doubling. If the cycle through 1 has length p, then 2p ≡ ±1 modulo 2n+1, so for any position x, applying the shuffle p times sends x to ±x, and the fold into 1..n makes that x itself. Every cycle length divides p, so the least common multiple is p. Nothing is claimed as new here: this is Bringer's, and the point is only that the question and its answer have been sitting one hyperlink apart for fifteen years. What this page adds is a wider check. The scan below walks the actual cycle structure using no number theory at all, so that it cannot accidentally assume what it is testing.
The check
Everything above is recomputed, not typed in. One module, research/sestina-studio/spiral.mjs, defines the permutation; this page loads that exact file and so does the verifier, which also asserts the two copies are byte-identical. verify-the-seventh-stanza.mjs runs 147 assertions offline: the spiral table against the canonical sestina order printed in the reference works, the arithmetic formula against the permutation for every n up to 2000, the Queneau numbers against the published terms of A054639, Asveld's three-case characterisation against the directly computed cycle structure, Roubaud's 141, and every number this page prints about the nine poems, recomputed from the raw source files whose hashes are in research/sestina-studio/sources/MANIFEST.md.
Where this ends, and what it does not know
- One character of one source was changed. The Wikisource transcription of Arnaut has d’ncle where the word is oncle. That is a typing slip in a modern transcription, not a reading, and it is the only edit made to any text here. The measurement is run twice, with and without it, and both numbers are in the committed output.
- "Did the poet use this word" is not always a yes or no. Sixteen line-ends in the corpus are not exactly one of the poem's words but share its tail, and they are listed above rather than counted silently. Every one of them is resolvable that way, which is itself a finding: no poet here ends a line on something unrelated. But whether "along" discharges the obligation to end on "long" is a judgement about poetry, and the page shows both readings rather than picking.
- Arnaut rests on one modern transcription with no named print source. I went looking for a critical edition to collate against and the only one reachable in the public domain, Canello 1883, has OCR damage inside the sestina itself: its oncle reads as onde throughout, and scan boilerplate is interleaved between the poem's lines. Measuring it would have taken a dozen character-level repairs, and the repair list has one entry for a reason, so the collation was not done and is not claimed. The oldest poem here therefore has the weakest provenance of the nine, which is the opposite of what you would want.
- The corpus is small and it is not a sample. Nine poems chosen because their texts are free worldwide and machine-readable, which biases hard toward the canonical. It says nothing about the thousands of sestinas written since 1900, and the twentieth century is exactly where the form loosened most.
- One obvious poem is missing on purpose. Pound's "Sestina: Altaforte" (1909) is public domain in the United States and under copyright in the United Kingdom and the European Union until 2043. This site is served in all three, so the poem is not measured and its text is not in the repository.
- Spenser is read but not quoted. The 1579 quarto opens his sestina with a drop capital that the transcription encodes as separate elements, so read as plain text the first line loses its initial letter. No line-end is affected, so the measurement is sound, but no line of that poem is reproduced here.
- Stanza boundaries in the Spenser source are inferred. That source marks verse lines but not stanzas, so breaks are read off the edition's own indentation and then checked against the requirement that every stanza be exactly six lines. A wrong rule would fail that check.
- The studio checks the form, not the poem. It knows nothing about metre, syllables or sense. The workshop's scansion bench next door does the metrical half, and joining them is the obvious next bench.
Part of the Prosody Workshop. Texts from Wikisource and Project Gutenberg, all public domain worldwide; each file's URL, retrieval date and SHA-256 are in the source manifest. Sequence data cross-checked against the On-Line Encyclopedia of Integer Sequences, A054639.