A Poetry Sporadical of Repeating Forms

A Numbers Theory of the Sestina and Similar Repeating Forms

Variations on Retrogradatio Cruciata

by Sara Gwen Weaver

The intrinsic nature of a sestina1 comes from the order given to teleutons by the notorious retrogradatio cruciata procedurethe “cross.” For each full stanza after the first stanza, teleutons weave together from the preceding stanza from either outside edge inward2:

Two Primary Traits of Retrogradatio Cruciata. The cross produces a characteristic teleuton pattern that exhibits two primary traits:3

  1. Recycling – If the cross were to be carried to a new stanza beyond the final main stanza (referred to in this essay as a “shadow” stanza, of course never actually written), the order of the teleutons in that shadow stanza would match the order of the teleutons in the sestina’s initial stanza. Furthermore, no stanza prior to that shadow stanza has a teleuton order matching that of the first stanza.4

  2. Uniqueness – The particular line taken by any given teleuton in one stanza is not duplicated in any other stanza. Consequently, each teleuton occupies each line position once and only once during the course of the poem.

Examination of variations of the cross can enable an understanding of the crucial role these two primary traits play in formulating the personality of the sestina. Furthermore, cross variations can help isolate secondary traits of the procedure, features that are part of the nature of the sestina while not present in most other poems with patterns that carry the two primary traits (for instance, the sestina’s notorious immediate repetition of teleutons from one stanza to the next). Even if a poet were to never write a poem that follows any of the derivatives produced by cross variations, knowing how those derivatives would differ from traditional cross patterns can illuminate the classical sestina itself, deepening our appreciation for how any poem speaks through form.5

Traditional Retrogradatio Cruciata in the Sestina. If the teleutons are assigned letters A through F, then the cross produces the following order for the lines (L) and main stanzas (S) of a sestina —

Sestina – Traditional Retrogradatio Cruciata
  S1 S2 S3 S4 S5 S6
L1 A F C E D B
L2 B A F C E D
L3 C E D B A F
L4 D B A F C E
L5 E D B A F C
L6 F C E D B A

Inspection of this table confirms that the sestina exhibits both primary traits: (1) If the cross is applied to the 6th stanza, the order of the teleutons in the shadow stanza would be ABCDEF, recycling to the initial stanza (while no other stanza in the sestina after the first has that order); and (2) The position for each teleuton is unique: present once and only once in any particular column or row.

Basic Retrogradatio Cruciata Variations – Different Numbers. The simplest sestina derivatives retain the cross while altering the line and stanza count of the poem, using a “magnitude” number higher or lower than the sestina’s characteristic 6. Using the cross on a magnitude of 5 – five stanzas of five lines each – produces the derivative form known as a pentina. Inspection of the teleuton table for a pentina confirms that like the sestina, it too exhibits both primary traits —

Pentina – Traditional Retrogradatio Cruciata
  S1 S2 S3 S4 S5
L1 A E C D B
L2 B A E C D
L3 C D B A E
L4 D B A E C
L5 E C D B A

Using the cross on magnitude of 3 – three stanzas of three lines each – produces the derivative form known as a tritina.6 Once again, inspection of the teleuton table for a tritina confirms that like the sestina and pentina, it also exhibits both primary traits —

Tritina – Traditional Retrogradatio Cruciata
  S1 S2 S3
L1 A C B
L2 B A C
L3 C B A

What about other numbers? Ignoring the trite one-liner produced by the number 1, against which the cross has nothing to act, we’ll refer to sestina-like derivatives with magnitude numbers other than 3 or 5 as an n-tina,7 with n as the magnitude number indicating how many stanzas and how many lines per stanza comprise the poem (ignoring the envoy), leaving any latin nomenclature for anyone wishing to claim creation of any specific rendition. Using the cross on magnitude 2 – two stanzas of two lines each – produces a derivative form that in its brevity does still exhibit both primary traits (it being mathematically impossible for it not to do so), with the simple teleuton switch AB / BA.

But quickly we find that not all magnitude numbers produce a teleuton pattern that exhibits both primary traits. Using the cross on magnitude 4 – four stanzas of four lines each – produces a derivative form with the following teleuton order —

4-tina – Traditional Retrogradatio Cruciata
  S1 S2 S3 S4
L1 A D B A
L2 B A D B
L3 C C C C
L4 D B A D

First, the recycling trait is not present, since applying the cross to the 4th stanza would produce a shadow stanza with teleuton order DACB, which fails to match the order of the poem’s initial stanza. Second, the uniqueness trait is not present, since each row has at least one duplication of a teleuton, with the 3rd line of each stanza using only the 3rd teleuton throughout (which of course also means that the 3rd teleuton fails to occupy each line position during the course of the poem). Of course, the absence of the two primary traits does not mean that a poet ought abstain from writing a 4-tina constructed in this manner; in fact, the constant repetition of the 3rd teleuton through each 3rd line could give such a poem a hint of the feel of a kyrielle. However, here we can start to see how the two primary traits can be used to understand some of the nature of the sestina, since the absence of those two primary traits by this 4-tina produce a poem that lacks a certain balance boasted by the sestina.

Some numbers produce a teleuton pattern that partially exhibits the recycling trait in the sense of returning to the origin in the shadow stanza (although usually also returning to the first stanza in one or more earlier stanzas within the poem itself), but fails to exhibit the uniqueness trait.8 For example, the following pattern is produced for 8 —

8-tina – Traditional Retrogradatio Cruciata
  S1 S2 S3 S4 S5 S6 S7 S8
L1 A H D B A H D B
L2 B A H D B A H D
L3 C G E F C G E F
L4 D B A H D B A H
L5 E F C G E F C G
L6 F C G E F C G E
L7 G E F C G E F C
L8 H D B A H D B A

Applying the cross to the final stanza here would produce a shadow stanza matching the poem’s first stanza. However, the poem also cycles back to its original teleuton pattern halfway through, which of course also means that some of the teleutons will not be found in all of the line positions (for instance, the first teleuton A never appears in lines 3, 5, 6, or 7 for any stanza).

The incidence of magnitude numbers for which the cross will exhibit both primary traitsthat is, magnitude numbers that produce a poem with a balanced integrity to the teleuton order that reflects that of the sestinais as odd as the incidence of numbers that are prime, as evidenced by the inventory for numbers through 32 given in a later section of this treatise.9 Line counts for a poem constructed using any given number (ignoring additional lines for an envoy) are shown as one indication of the difficultyboth for the poet and the reader!of attempting the cross for any number much higher than the mid-20’s (if even that far).

The teleuton table for a 9-tina constructed via the traditonal cross is given in a later section of this essay. But let’s do one more teleuton table before leaving this section, the teleuton table for the traditional cross applied to magnitude 14, one of the numbers for which the cross does exhibit both primary traits —

14-tina – Traditional Retrogradatio Cruciata
  S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14
L1 A N G K I J E L F C M H D B
L2 B A N G K I J E L F C M H D
L3 C M H D B A N G K I J E L F
L4 D B A N G K I J E L F C M H
L5 E L F C M H D B A N G K I J
L6 F C M H D B A N G K I J E L
L7 G K I J E L F C M H D B A N
L8 H D B A N G K I J E L F C M
L9 I J E L F C M H D B A N G K
L10 J E L F C M H D B A N G K I
L11 K I J E L F C M H D B A N G
L12 L F C M H D B A N G K I J E
L13 M H D B A N G K I J E L F C
L14 N G K I J E L F C M H D B A

Why a 14=tina? Write one, and you’ve got a sonnet crown composed on the basis of a structure determined by retrogradatio cruciata! In fact, since a typical sonnet crown picks up the last line of one sonnet as the first line of the next, the crown structure blends in quite nicely with the manner in which the cross moves from one stanza to the next. Of course, unless you use one of the few rhyme groups with 14 good rhyming words, at best you can hope to have a classical rhyming scheme (e.g., Petrarchan) for only one of the 14 stanzas. But for the sake of writing a double repeating form, could any Tilt-a-Whirler seriously complain?

Beyond the lure of something like a "crossed crown," however, does it really matter which magnitude numbers “work” – that is, which numbers produce a teleuton pattern that exhibits both primary traits? Perhaps not. However, a poet who does attempt a tritina, a pentina, and a 9-tina will frequently notice quite quickly that although the the two primary traits are present for those alternate magnitudes, the feel given to the cross’ weave by having an odd number of stanzas and an odd number of lines per stanza can differ markedly from the feel one finds in the sestina with its even-numbered magnitude of 6 or for similar even-numbered magnitudes such as 14 or 18. The odd-numbered magnitude poems can seem to fold differently around their middle lines and middle stanzas than the even-numbered magnitude poems. Similarly, then, knowing why the some magnitude numbers fail to exhibit one or both primary traits can deepen appreciation for the particular way the cross weaves its threads through the sestina.

Reverse Retrogradatio Cruciata. Another relatively simple variation on the cross merely reverses the procedure. That is, rather than setting the teleuton for the first line of a new stanza as the teleuton of the last line of the preceding stanza, instead the procedure would be launched by taking the teleuton of the first line of the preceding stanza as the teleuton for the last line of the new stanza, then filling out the new stanza’s teleutons in reverse order from the penultimate line back to the first line in a weave similar to that familiar to the traditional cross devotee.10 The “reverse cross” process would create this teleuton table for an alternative sestina-esque poem —

Sestina-esque – Reverse Retrogradatio Cruciata
  S1 S2 S3 S4 S5 S6
L1 A D B C E F
L2 B C E F A D
L3 C E F A D B
L4 D B C E F A
L5 E F A D B C
L6 F A D B C E

Close observation reveals that this is the same teleuton pattern that would emerge by filling out each new stanza top down starting from the center of the preceding stanza weaving back and forth back out to the preceding stanza’s either end, instead of from the outside in as done by the traditional procedure. As ought not be surprising by now, this reverse cross process creates a teleuton pattern for a 6-line 6-stanza poem that ehibits both primary traits: all teleutons are used uniquely in each line number through each of the stanzas; and if the procedure were to be applied to a shadow stanza beyond the final full stanza, the next stanza would recycle back to the teleuton sequence of the initial stanza.

Obviously, a reverse cross process can also be used for any of the other numbers for which the traditional cross works. The same findings presented above continue to hold: if the traditional cross works for a particular number (e.g., 2, 3, 5, 6, 9, 11, 14, et cetera), then a reverse cross procedure will work. Conversely, for numbers for which the traditional cross fails to exhibit one or both of the primary traits (e.g., 4, 7, 8, 10, 12, 13, et cetera), the same one or both primary traits will be absent under a reverse cross procedure.

While both the traditional cross and a reverse cross create quite similar weaves of teleutons, one difference is immediately apparent, perhaps obvious even to a novice to the sestina. In the traditional sestina, both the writer and reader must deal not only with the heavy echo of teleutons throughout the poem, but very specifically with the very instant echo that occurs as one stanza passes to the next: the first line of a new stanza picking up the teleuton of the previous stanza’s last line. Various devices for dealing with the weight of this immediate echo include (but of course not remotely limited to): modifying the part of speech for the teleuton (e.g., noun in one line, verb in the next); use of homonyms; addition of prefixes or inclusion within a longer word; or even simply repeating most or all of the exact same phrase with the teleuton but with another meaning (e.g., taking the statement of one line as a question in the next). Even so, no matter how creative the poet or understanding the reader, this “echo fold” must be crafted five times in a sestina (and of course far more so in any n-tina of any numbers higher than six). While completely retaining the feel of the rest of the sestina’s weave,11 a reverse cross procedure avoids the echo fold at each stanza break. This observation is not intended to suggest that a reverse cross procedure is better nor worse than the traditional cross, but merely to use the variation as one way to see and understand one important aspect of the traditional cross. For the poet or reader who uses and appreciates the echo fold,12 the traditional cross does something that gives the form a special edge. Conversely, if the echo fold seems to cripple the form for a poet or a reader, a reverse cross procedure offers very much the same feel of the usual sestina weave without what might be seen by those parties as a drawback.

Double Retrogradatio Cruciata. What emerges if the cross is used to weave threads together into strands, then if those strands are themselves woven together using the cross again? For one thing, we could use such a double cross approach to fill in some of the numbers which fail to exhibit one or both traits under the traditional cross procedure.13 The following table, for example, shows what happens if a 2-tina with teleutons A and B is woven using the cross with a second 2-tina having teleutons C and D. (In this and any other teleuton table where multiple application of the cross is used, shaded borders inside the table are present only to aid visualizing how the table has been constructed. For the poem itself, the stanzas and lines would remain intact as indicated in the column and row headings.) —

2x2-tina – Double Retrogradatio Cruciata
  S1 S2 S3 S4

Inspection of this teleuton table quickly finds that in contrast with the 4-tina table developed using the traditional cross, the double cross applied to two 2-tina poems produces a 4-line 4-stanza poem that exhibits both primary traits: it recycles back to itself; and all teleutons are used uniquely throughout the poem.

We can similarly “multiply” crosses, where one mutiplier is a suitable magnitude number for which we have previously established a workable pattern (even if established via such multiplication), and each successive multiplier is itself a number for which there is a workable pattern. For example, the following table presents the pattern for a 6-line 6-stanza alternative to the traditional sestina, in this instance first composing two tritinas, one tritina with teleutons A-B-C and another tritina with teleutons D-E-F, then using the 2-tina style of the cross weave to combine those two tritinas, what we will refer to as a [3x2] poem

[3x2]-tina – Double Retrogradatio Cruciata
  S1 S2 S3 S4 S5 S6

As with our [2x2] double cross and as would be seen by any other combination using workable numbers, this produces a pattern that does exhibit both primary traits.14 To some perhaps even mostauthors and readers, the result of this pattern may seem weaker than the weave produced for a traditional sestina; but again, such judgments are a matter of taste and the needs of any particular poem. In a 6-line 6-stanza poem that more clearly wishes to set apart two contrasting perspectives, the manner in which this poem segregates each trio of teleutons into the first half or second half of each stanza and in the poem itself might have a more dramatic effect than via the full weave of the traditional cross.15

Notice that we get a different double weave table if we first compose three 2-tina blocks, then weave those three blocks together like a tritina, thereby composing what we will refer to as a [2x3] poem. As with the [3x2] table, this exhibits both primary traits

[2X3]-tina – Double Retrogradatio Cruciata
  S1 S2 S3 S4 S5 S6

Of course, a reverse cross could be used for the tritinas in either the [3x2] or the [2x3] double.

Thus, multiple application of the traditional cross can embelish a sestina mtutation writer’s repertoire in two ways: (a) First, it can to the list of “workable” numbers for which an n-tina will satisfy both conditions, such as 4 ([2x2]), 8 ([2x2x2]), 10 ([2x5] or [5x2], each of which produce different teleuton tables), and so on. (b) Second, it can add variations for n-tinas for which the traditional cross already gives a basic workable teleuton pattern, such as 6 (in lieu of the classical sestina, using [2x3] or [3x2]), the 9-tina (via [3x3]), and so on.

Note, however, that two classes of numbers remain for which even multiple cross weaves (using the traditional weave, without resorting to alternative weave sequences as presented later in this essay) still fails to provide us with a workable number for which the n-tina exhibits both primary conditions: (a) If the number is a prime number for which the traditional cross fails to work, then multiple cross techniques will also fail to provide a solution that exhibits both primary traits. For example, since the traditional cross fails to work for a 7-tina, and since 7 is a prime number, multiple cross techniques can’t help us out. (b) If the number is the product of two or more prime numbers, and if one or more of those prime numbers is a number for which the traditional cross fails to work, then multiple cross techniques will fail to produce a workable result for the larger version. Unless the traditional cross works directly for that larger number, a satisfactory result is not attainable via ordinary application of the cross. For example, since no acceptable traditional result can be obtained for a 7-tina, a workable table via multiple cross techniques is not possible for a 14-tina via either [2x7] or [7x2]; however, since the traditional cross works directly for a 14-tina without multiple application, that would be the way to go. However, since no acceptable result can be obtained for a 7-tina, making a good result via [3x7] or [7x3] unavailable, and since no direct result is obtained for a 21-tina via the traditional cross, a 21-tina remains unworkable via any ordinary application of the cross, including multiple techniques. (Of course, a 21-tina that exhibits both primary traits can be composed via sequence methods described later in this essay.)

As suggested previously in this treatise, there is nothing inherently “wrong” with composing any n-tina (for example, a 21-tina) that fails to exhibit one or both primary traits. The absense of either or both primary traits merely means that such a poem will not exhibit the full formal elegance given to a sestina by the the traditional cross weave; but in some instances (e.g., such as was seen with the repetition of teleutons in the 3rd line of each stanza of a standard 4-tina), that “flaw” may produce an effect desired for a particular poem.

We can now develop an inventory of traditional n-tina solutions through n=32. The “lines” column indicates the total number of lines for the entire poem, excluding the envoy, providing a general indication of the difficulty and unlikelihood of application of the cross for magnitudes greater than several dozen, if that. For the versions involving multiple cross application, the multiplication pairs indicate the order in which the cross is applied, as previously presented.

n lines Traditional Retrogradatio Cruciata Multiple Retrogradatio Cruciata
2 4   BOTH primary traits present prime – multiple application unavailable
3 9   BOTH primary traits present prime – multiple application unavailable
4 16   NEITHER primary trait present [2x2]
5 25   BOTH primary traits present prime – multiple application unavailable
6 36   BOTH primary traits present [2x3] [3x2]
7 49   NEITHER primary trait present prime – multiple application unavailable
8 64   ONLY partial recycling trait present [2x2x2]
9 81   BOTH primary traits present [3x3]
10 100   NEITHER primary trait present [2x5] [5x2]
11 121   BOTH primary traits present prime – multiple application unavailable
12 144   NEITHER primary trait present [2x2x3] [2x3x2] [3x2x2] [2x6] [6x2]
13 169   NEITHER primary trait present prime – multiple application unavailable
14 196   BOTH primary traits present no suitable multiple combo
15 225   ONLY partial recycling trait present [3x5] [5x3]
16 256   NEITHER primary trait present [2x2x2x2]
17 289   NEITHER primary trait present prime – multiple application unavailable
18 324   BOTH primary traits present [2x3x3] [3x2x3] [3x3x2] [2x9] [9x2]
19 361   NEITHER primary trait present prime – multiple application unavailable
20 400   ONLY partial recycling trait present [2x2x5] [2x5x2] [5x2x2]
21 441   ONLY partial recycling trait present no suitable multiple combo
22 484   NEITHER primary trait present [2x11] [11/2]
23 529   BOTH primary traits present prime – multiple application unavailable
24 576   NEITHER primary trait present [2x2x2x3] [2x2x3x2] [2x3x2x2] [3x2x2x2] [2x6x2] [4x6] [6x4]
25 625   NEITHER primary trait present [5x5]
26 676   BOTH primary traits present no suitable multiple combo
27 729   NEITHER primary trait present [3x3x3] [3x9] [9x3]
28 784   NEITHER primary trait present [2x14] [14x2]
29 841   BOTH primary traits present prime – multiple application unavailable
30 900   BOTH primary traits present [2x3x5] [2x5x3] [3x2x5] [3x5x2] [5x2x3] [5x3x2] [5x6] [6x5]
31 961   NEITHER primary trait present prime – multiple application unavailable
32 1024   NEITHER primary trait present [2x2x2x2x2]

Obviously, for any n-tina with a workable number, mutations can be developed via reverse cross, either on the application within each individual block, on the way the blocks are then woven together on any subsequent multiple weaving, or on both.

Sequences versus Patterns – Opening the Door to More Variations. Both the traditional retrogradatio cruciata and the reverse retrogradatio cruciata exhibit both primary traits for certain magnitude numbers (2, 3, 5, 6, 9, 11, 14, et cetera). Can teleutons be rearranged in other patterns for those magnitude numbers, and still retain both primary traits? And are there any patterns that would work for the magnitude numbers for which traditional retrogradatio cruciata does not exhibit those primary traits? The short answer to both questions: yes.

The key to developing a weave method that produces a teleuton table that will be guaranteed to exhibit both primary traits is to focus not on the pattern for each successive stanza in the poem, but rather on the line sequence that any teleuton takes stanza by stanza through the course of the poem. For instance, the traditional way to describe the sestina’s pattern via retrogradatio cruciata is as was described at the very beginning of this essay, a process frequently tagged with numbers representing the positions of a teleuton in a preceding stanza, rearranged in the order to be used for the next succeeding stanza: 615243. Now, look instead at the teleuton table produced for a sestina, and this time look not at the positions of all teleutons in any one stanza, but rather at the line positions of one teleuton through all of the stanzas. For the first teleuton in a classical sestina, for instance, this sequence would be 124536: the first teleuton occupies line 1 in stanza 1, line 2 in stanza 2, line 4 in stanza 3, line 5 in stanza 4, line 3 in stanza 5, and line 6 in stanza 6.

Quick inspection of any other teleuton in a sestina confirms an obvious but crucial fact: all teleutons follow the same wandering path through the poem, simply each starting out from a different position.16 Since all teleutons take the same path, we shall always look to the sequence beginning with 1, that is, the path taken by the first teleuton of the initial stanza. Notice that for the classical sestina, that sequences ends with the sestina’s magnitude, 6. In fact, if the recycling trait is possessed by an n-tina composed using the traditional cross, the sequence will end with n. In contrast, for a reverse cross sestina-esque poem, the sequence of the first teleuton is 165324.

For a 6-line 6-stanza sestina-esque poem, there are 120 different such threaded sequences in which all 6 numbers are present. It can be easily demonstrated that the teleuton tables produced by any of those 120 different sequences exhibit both primary traits: recycling and uniqueness. Conversely, it can be shown that any of the other 600 possible patterns17 for weaving teleutons from one stanza to the next fail one or both of the primary traits. Following is the teleuton table produced by the sequence 135246 —

Sestina-esque – Teleuton Sequence 135246
  S1 S2 S3 S4 S5 S6
L1 A F D B E C
L2 B E C A F D
L3 C A F D B E
L4 D B E C A F
L5 E C A F D B
L6 F D B E C A

This sestina-esque mutation retains the echo fold of the traditional sestina (characteristic of any sequence tag that has the magnitude number in its final place, here 6), but holds pairs of lines with teleutons intact throughout the poem (with switched order for half of those occurrences, i.e., leading half the time, following the other half). By contrast, the traditional sestina includes a single repetition for two pairs with respect to any given teleuton. Again, the mutation weave obtained via the sequence 135246 is not inherently better or worse than the classical sestina weave; but its different character may help better distinguish the particular weave produced by the traditional cross.

Whether or not the poet writes a sestina-esque mutation, teleuton sequence can produce something that traditional cross procedure fails to achieve: for any magnitude number 2 or higher, a teleuton table that exhibits both primary traits! For instance, the following teleuton table for a 7-tina is produced using a sequence of 1245637 —

7-tina – Teleuton Sequence 1245637
  S1 S2 S3 S4 S5 S6 S7
L1 A G C F E D B
L2 B A G C F E D
L3 C F E D B A G
L4 D B A G C F E
L5 E D B A G C F
L6 F E D B A G C
L7 G C F E D B A

Quick inspection confirms that this teleuton table exhibits both primary traits. Moreover, through the first four lines of each stanza it looks and sounds remarkably like a sestina; in fact, since usually a reader’s ear would not keep very close track past those initial four lines, a 7-tina constructed according to this table could easily fool a reader who hasn’t stopped to count lines and stanzas (or to double-check the full weave pattern) into thinking a classical sestina is being presented.

An understanding of sequence behavior also leads us as close to a traditional sestina as we can get for a 12-tina. Although the traditional cross produces a 12-tina18 that does not possess the two primary traits, the sequence 124895[10]7[11]36[12] produces the following teleuton table, where the 9th and 10th lines of the preceding stanza are switched before each application of the traditional cross —

12-tina – Teleuton Sequence 124895[10]7[11]36[12]
  S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12
L1 A L F C K G J E I H D B
L2 B A L F C K G J E I H D
L3 C K G J E I H D B A L F
L4 D B A L F C K G J E I H
L5 E I H D B A L F C K G J
L6 F C K G J E I H D B A L
L7 G J E I H D B A L F C K
L8 H D B A L F C K G J E I
L9 I H D B A L F C K G J E
L10 J E I H D B A L F C K G
L11 K G J E I H D B A L F C
L12 L F C K G J E I H D B A

More Sequences – A Return to Traditional Retrogradatio Cruciata. For any n-tina that works like a sestina, the sequence ends with the magnitude number, the sequence’s expression of the recycling trait. We can use that fact as the starting point for expressing a methodology for development of the sequence for a working n-tina. For any magnitude n for which the traditional cross does not work, this methodology will demonstrate that failure directly.

In the following, n is the order of the n-tina; y is set as the number at the beginning of a step; and x is calculated as the number at the end of a step. The function mod (y;2) is a simple modulus function set equal to 1 for any odd number y and 0 for any even number y; also, note that (-1)^0=1.

  1. Start with n. Set y = n.

  2. To take a step, calculate x = n*mod(y;2) + ( y – mod(y;2) )*(-1)^mod(y;2) / 2.

  3. If x = 1, end the steps. If x > 1, reset y = x and return to repeat the preceding stage.

Count the steps – the number of times the second stage of this process is complete. If that count is equal to n-1 that is, one less than the magnitude orderthen the n-tina will exhibit both primary traits, and the progression of the x values derived by the successive steps will provide the teleuton sequence in reverse order. For any n-tina that arrives at x=1 in less steps than n-1, the associated n-tina will not exhibit both primary traits.

For instance, for the classical sestina, this procedure would be as follows:

  1. For a sestina, n=6. So to start the first step, we set y=6.

  2. The steps we take are these:

    1. Step 1 – Since mod(6;2)=0, x = 6*0 + (6-0) * 1/2 = 3.

    2. Step 2 – Resetting y=3, note that mod(3;2)=1. So x = 6*1 + (3-1) * (-1)/2 = 5.

    3. Step 3 – Resetting y=5, note that mod(5;2)=1. So x = 6*1 + (5-1) * (-1)/2 = 4.

    4. Step 4 – Resetting y=4, note that mod(4;0)=0. So x = 6*0 + (4-0) * 1/2 = 2.

    5. Step 5 – Resetting y=2, note that mod(2;0)=0. So x = 6*0 + (2-0) * 1/2 = 1.

  3. Since x = 1, the process is now completed after taking 5 steps.

Since the number of steps is 5, one less than the magnitude order of 6, this demonstrates that the traditional cross produces a 6-tina—our classical sestina—which will exhibit both primary traits. In reverse order, the steps have developed the sequence that produces the sestina: 124536.

Quick calculation shows that for n=4 (a 4-tina), the number of steps in this process is only 2, less than the 3 (i.e., 4-1) needed to ensure presence of the primary traits. In fact, whenever n is a power of 2 greater than or equal to 4 (e.g., 4, 8, 16, 32, 64, et cetera), the associated n-tina will not exhibit the primary traits: set m equal to the power of 2; the number of steps in the procedure will then simply be equal to m; and for m>1, m < 2^m-1.

Similarly, the step-counting procedure can be used to demonstrate that any n-tina of magnitude 2^m-1 where m>2 (e.g., 7, 15, 31, 63, et cetera) will also fail to exhibit the primary traits, since the first step will produce a number that is a power of 2, whereupon the result found in the preceding paragraph emerges. Other sub-classes of magnitudes for which the primary traits are absent can be similarly derived from the step-counting procedure.

There is no numerological magic to this step-counting procedure; it is nothing more than a mathematical expression of the cross. However, rather than complete an entire teleuton table, then perform the check of the shadow stanza and the check of all lines and columns for uniqueness, this procedure provides a single, direct calculation. Moreover, for larger magnitude numbers that fail with short cycles (for instance, n=63), the steps to this calculation can demonstrate that failure much quicker than by filling out a teleuton table (even if that table were only filled out through the first short cycle), as well as carving out entire sub-classes of magnitudes that fail to exhibit the primary traits.

Line Pairings – The Sestina’s Dance Cards. Consider the sestina’s stanzas as comprising six separate dances danced by pairs of partners. Let each of those six dances be danced twice, where if one partner was the lead in one of those two dances, then that partner will be a follow with a different partner the other time around. Each dancer will sit out two dances, so that each dancer will dance ten times, five times as lead and five times as follow. What we’ve described here essentially gives us a way of looking at pairs of lines in each stanza of a sestina, where one teleuton is lead to another if the other is in a succeeding line of the same stanza, and vice versa.

Returning to the teleuton table for a traditional sestina, let’s look at A’s dance card. A is lead to B in the first dance, lead to E in the second dance, lead again to B in the third dance, lead to D in the fourth dance, lead to C in the fifth dance, and sits out the sixth dance having never had a dance leading to F. The second time each of those dances is danced in our dance analogy, A sits out the first dance, then is follow to F in the second dance, follow to D in the third dance, follow to F again in the fourth, follow to E in the fifth, then finally follow to C in the sixth, never having a dance in which A is follow to B. Summarizing, A leads B twice but never follows to B; and A follows F twice but never leads to B. Quick inspection of the dance cards for the other five teleutons shows a similar pattern: each leads twice to the partner to whom each does not follow; and each follows twice the partner to whom each does not lead.

The sequence 123456 produces a teleuton table that exhibits both primary traits, but has each of the teleuton dancers “married” to one other teleuton dancers through all five dances in which it leads, then to a different partner through all five dances in which it follows. The rotating wheels of this teleuton table may seem relatively trite to the diehard sestina devotee, but could be of interest to a poet wishing to retain the recycling and uniqueness traits (and the sestina’s echo bend stanza to stanza) while adding a stronger “togetherness” of the teleuton connections with each other.

Sestina-esque – Teleuton Sequence 123456
  S1 S2 S3 S4 S5 S6
L1 A F E D C B
L2 B A F E D C
L3 C B A F E D
L4 D C B A F E
L5 E D C B A F
L6 F E D C B A

The sestina-esque teleuton table presented near the beginning of this essay’s section on sequences—produced via the sequence 135246 —holds dance cards that almost seem cozier! For instance, in that mutation, A leads to B three times and follows to B three times—six of the ten “dances” versus only five in the 123456 sequence we just left.

In contrast, consider the following teleuton table, one of the author’s personal favorite mutations, produced by the sequence 154623 —

Sestina-esque – Teleuton Sequence 154623
  S1 S2 S3 S4 S5 S6
L1 A C B F D E
L2 B F D E A C
L3 C B F D E A
L4 D E A C B F
L5 E A C B F D
L6 F D E A C B

As with any sestina-esque poem created by the sequence method, this poem exhibits both primary traits, recycling and uniqueness. It does not have the hard, immediate echo bend of the classical sestina, but does have a softer feeling of the echo bend since the final teleuton of a preceding stanza does find its place in the second line of each succeeding stanza. This teleuton table can even be constructed via a procedure that very closely resembles the traditional cross: instead of starting the weave from the bottom half of the preceding stanza, one starts with the top half; and for selecting the teleutons for the top half’s contribution to the weave, one starts from the center of the preceding stanza instead of from the outer edge; otherwise, the familiar back-forth weave of the cross is used.

However, in contrast with the classical sestina – in fact, in contrast with the vast majority of the teleuton tables produced by the 120 sequences for a 6-line 6-stanza sestina-esque form – this teleuton table has no repetitions of “dance pairings” for any teleutons dance card. In the dances we have envisioned, each teleuton is lead to each other teleuton once and only once, and likewise is follow to each other teleuton once and only once. Some may find in these full dance cards a more balanced feel and sound to the resulting form. Those who would still prefer the classical sestina, the presence of this alternative can help emphasize the additional inner connections created by the duplicate pairings that are present via the traditional cross.

9-Tina and Sudoku. When the uniqueness trait was spelled out at the beginning of this essay, there was a temptation to call it the “sudoku” trait, since each teleuton’s letter appears in only one row and one column of a poem’s teleuton table, temptingly similarly to a sudoku puzzle. However, all traditional sudoku puzzles exhibit one additional trait that is not produced by cross patterns, which in this essay will be referred to as the “sudoku trait”: each letter appearing once and only once in each sectional block of a table. For example, in a 9x9 sudoku puzzle that is segregated into nine separate 3x3 blocks, each letter appears once and only once in each of those separate blocks.

Let’s return to what we earlier called the 9-tina, a 9-line 9-stanza poem using the traditional cross. Although both of our cross primary traits are present, the resulting table would not make a good sudoku pattern, since each 3x3 block does not uniquely contain all 9 teleutons (for instance, A appears twice in the northwest block of 9):

9-tina – Traditional Retrogradatio Cruciata
  S1 S2 S3 S4 S5 S6 S7 S8 S9
L1 A I E G F C H D B
L2 B A I E G F C H D
L3 C H D B A I E G F
L4 D B A I E G F C H
L5 E G F C H D B A I
L6 F C H D B A I E G
L7 G F C H D B A I E
L8 H D B A I E G F C
L9 I E G F C H D B A

However, we can achieve a good sudoku pattern and still exhibit both cross primary traits by a three-step process: (1) using the traditional cross on three separate tritinas; then (2) wrapping those three blocks like a tritina (that is, what in this treatise has been discussed as a multiple cross [3x3] table19); then (3) by suitably shifting columns among various blocks (via a third level of the cross!), achieving the following table20

9-tina – [3x3x3] Multiple Retrogradatio Cruciata
  S1 S2 S3 S4 S5 S6 S7 S8 S9
L1 A I E G F B D C H
L2 B G F H D C E A I
L3 C H D I E A F B G
L4 D C H A I E G F B
L5 E A I B G F H D C
L6 F B G C H D I E A
L7 G F B D C H A I E
L8 H D C E A I B G F
L9 I E A F B G C H D

A final note on the 9-tina: if one is willing to do without the recycling trait (which few but the most savvy readers would notice anyway), but retain the uniqueness trait and pick up full sudoku discipline along in the bargain, then an almost inexhaustible supply of unique potential 9-tina patterns (certainly more than even the most prolific poet could ever write in a lifetime) can be produced by purchasing any sudoku puzzle book, completing any of the puzzles (or cheating by going directly to the solutions pages), and using the completed puzzle to determine the teleuton table.21

Concluding Remarks. We can summarize the traits of a classical sestina’s main stanzas seen during the course of this essay’s exploration of cross variations. These traits help shape the form within which the sestina poet conveys the poem’s voice and image:

Any or all of these traits can be modified or even eliminated by applying the variations presented in this essay. None of the variations – not even those producing the established tritina or pentina mutations – can ever hope to unseat the sestina as the crowning glory of retrogradatio cruciata. However, poems that exhibit some of the splendor of the sestina can be written by varying the number of lines and stanzas, the order in which the cross is applied (e.g., reversing), by defining an alternative sequence for the way each teleuton weaves through the stanzas, by applying the cross multiple times to portions of a poem, or even by using some altogether different method for reorganizing teleutons (e.g., reference to sudoku solutions). To some degree, such poems will exhibit a character that is similar or that differs from the sestina, either providing a different vehicle for portraying a different poetic voice or image or helping to illuminate the way in which a sestina exhibits its own character.

1     A selection of sestinas published at Tilt-a-Whirl: Buster and Etta, by Michael Cantor; Grammaticum Vitae, by Margery Hauser; and Invocation to the Dawn, by Juleigh Howard-Hobson.

2     After six stanzas – six lines each – have proceeded in this manner, a sestina typically ends with an envoy. In the English language, the sestina has traditionally been written in decasyllabic meter. Although they are sestina “traits” in their own right, neither the sestina envoy nor sestina meter will be addressed in this essay.

3     These two traits are closely related. In fact, for any sestina-like poem for which the teleuton pattern is constructed via sequences, such as are examined later in this essay, any poem that exhibits the uniqueness trait will automatically exhibit the recycling trait. However, as will be seen in this essay’s brief examination of “sudoku” patterns, it is possible to construct non-sequenced teleuton patterns that exhibit the uniqueness trait without necessarily satisfying the recycling trait. In any event, even when the two traits are so closely linked as to be completely interdependent, they are distinguishable in the sense of character given to the poetic form. That is, both the poet and the reader can separately sense the uniqueness of teleuton position separately from sensing the return to the poem’s starting position, even if the two traits are flip sides of the same coin.

4     Contrast this single-cycle aspect of the sestina’s recycling primary trait with the teleuton pattern of the 21-tina, which repeats the first stanza’s teleuton order in the 8th and 15th stanzas before ending a third loop in time for the first stanza to again repeat in the shadow 22nd stanza (of course, not having each teleuton occupy each and every line position through the course of the full 21-tina, since 3 quick partial cycles are made). For purposes of this essay, any n-tina which recycles precisely for the shadow stanza, but which also recycles prior to that shadow stanza, will be referred to as having a “partial” recycling trait.

5     The author would like to take this opportunity to point out that although this essay will involve a fair degree of mathematical presentation, expression of poetic form as a mathematical formula is most definitely not the goal of this exercise. Rather, I hope to bring mathematics as simply another language to the question of how repeating form affects the expression of a poetic idea.

6     The tritina happens to be the sole non-trivial “working” member of a certain class. For any magnitude equal to 2^n – 1, where n > 2, the cross cycles too early, with the (n + 2) stanza’s teleutons duplicating the pattern of the first stanza. But for n = 2, giving us the tritina since 2^2 – 1 = 3, the (2 + 2) stanza doesn’t exist, since the tritina has only three main stanzas. As frequently suggested in this essay, do note that the observation that members of the class with magnitude 2^n – 1 do not possess one or both of the primary traits does not mean that any poem in that class is necessarily bad form. For instance, a 63-tina cycles back to the first stanza every seventh stanza, completing nine full cycles just in time to actually satisfy the cycling primary trait (although obviously failing the uniqueness primary trait). For any poet interested in taking on the daunting task of a 3,969-line poem (not counting envoy), the nines and sevens in a 63-tina may prove any bit as elegant as the weave of a classical sestina.

7     To distinguish from the classical tritina, pentina, and sestina – all three of those forms developed using the traditional cross – this essay will generally refer to any mutations (e.g., using a reverse cross) of those particular magnitudes respectively as a tritina-esque, a pentina-esque, or a sestina-esque form.

8     The converse is not true. That is, using retrogradatio cruciata, there are no numbers that exhibit the uniqueness trait, but fail to exhibit the recycling trait.

9     Inspection of numbers higher than 30 continues to defy any comprehensive orderly incidence of numbers that “work” by satisfying both conditions. For instance, the fact that 9, 39, 69, and 99 suggests some affinity of the cross with cycles of 30, until one finds both primary traits absent for a 129-tina. Certain sub-classes can be carved out. For instance, it can be algebraically demonstrated that for every third magnitude number beginning with 4 – that is, 4, 7, 10, 13, 16, et cetera – the traditional cross will always produce a teleuton pattern that includes one line number for which the same teleuton is used throughout the poem in each stanza, thereby failing to exhibit the uniqueness trait. However, generalization from the rules applicable to such sub-classes is generally not possible.

10     This reverse procedure produces a teleuton table different from that produced by simply reversing the order of the stanzas, that is, beginning with the implicit seventh stanza of ABCDEF, but then using the usual sestina’s sixth stanza as the mutant’s second stanza, the usual fifth stanza as the third, and so on until the usual second stanza is taken as the mutant’s sixth. Jumping ahead to this essay’s section on alternatives produced by sequences, this “backward”-stanza version can be produced by the sequence 163542, accordingly does possess both primary traits. In this essay, “ retrogradatio cruciata” will be taken to indicate a reversal of the process itself, rather than a reversal of the stanzas produced by the traditional cross procedure, although obviously either is a legitimate variation.

11     For example, notice that the reverse cross method produces a teleuton pattern with teleuton pairs in consecutive lines that mirrors the pairing pattern of the traditional cross.

12     In the earliest development of the sestina, the echo fold may have been an intended carryover from the chanso redonda, an old troubadour form wherein the rhyme order of each new stanza was a direct inversion of its predecessor stanza’s rhyme order.

13     As noted in a later section of this essay, a poem that exhibits both primary traits can easily be produced for any and all magnitude numbers by using teleuton sequences to produce the necessary patterns.

14     The section of this essay presenting telueton sequences notes that reference to any of the 120 possible teleuton sequences for a 6-tina would produce a sestina-esque poem that exhibits both primary traits, but that any of the other possible 600 weave patterns representing the arrangement of teleutons in one stanza relative to the preceding stanza would not exhibit one or both primary traits. Notice that in both instances – reference to teleuton sequence or refence to teleuton patterns – the tag is of the same order of magnitude as the poem (for a sestina, 6) and remains intact for the entire body of each stanza throughout the entire poem. Multiple applications of the cross produces teleuton tables that can not be created by comprehensive application of either sequences or patterns across the entire stanza or poem, instead using those sequences or patterns within segments of a poem, then combining those segments via additional applications of the cross.

15     Advanced devices can be developed to switch columns in a teleuton table in order to introduce more diversity should a particular table – such as the one constructed here for the double cross performed first on tritinas, then by weaving those together like a 2-tina – exhibit too much “chunkiness,” with teleutons seemingly stuck in portions of a stanza for too many successive stanzas. If any columns are switched, for example, the uniqueness trait will still be present. If the entire procedure is taken into account – the multiple application of cross together with column switching – then a recycling procedure can be worked out for any such advanced device.

16     The fact that each teleuton of a sestina follows the same path as the path taken by the first teleuton reflects the weaving nature of the cross. The section of this essay addressing sudoku-like patterns will allude to the existence of teleuton tables (e.g., as drawn from sudoku puzzle solutions) which possess the uniqueness primary trait, but which do not recycle, in part because each teleuton follows a different path through the table than any other teleuton.

17     The 720 possible weaving patterns of a 6-line 6-stanza poem – 120 threaded sequence patterns that exhibit the two primary traits and another 600 that do not exhibit the two primary traits – are a subset of the 193,491,763,200,000 possible arrangements of the six teleutons randomly distributed – as opposed to weaving according to sequence or patterns – where each teleuton appears only once per stanza. If the 6 teleutons are completely randomized, so that 2 or more of a particular teleuton can appear in a particular stanza, then the number of possible arrangements is even larger. As with all numerical expressions in this essay, the point is not to delve into combinatorial mathematics, but rather to use such numbers to help emphasize the special nature of an aspect of the process or structure of the sestina, in this instance focusing on how patterns carve out a relatively small subset of the huge number of possible teleuton arrangements, then how sequences with all 6 numbers further carve that subset into an even smaller subset, then finally how the particular arrangement dictated by the cross procedure chooses the one particular teleuton arrangement characteristic of the classical sestina.

18     As presented in this essay, the 12-tina is 12 stanzas of 12 lines each, the usual expectation being that the cross would operate fully on all 12 lines. This is of course different from what has typically been referred to as a “double sestina,” comprised of 12 stanzas with only 6 lines each, composed simply by proceding through the main stanzas of a classical sestina twice in succession.

19     Since a [3x3] multiple cross 9-tina is composed by weaving three tritinas, that process also fails to exhibit the sudoku trait unless the third step of this special three-step procedure is taken, since each 3x3 block will contain only 3 of the 9 teleutons, instead of uniquely containing each of the 9 teleutons.

20     Additional teleuton tables different from this result – but possessing the same character of exhibiting both cross primary traits, as well as the sudoku trait – can be developed by using the reverse cross on one or more of the basic tritinas, by using the reverse cross on the weaving together of those blocks, or by using the reverse cross within the final step of the three-step process described here.

21     Note that teleuton patterns borrowed from a sudoku puzzle solution usually can not be developed from or expressed as a single sequence such as is described in this essay, since the path followed by any one teleuton through the puzzle is not the same path followed by any other teleuton.

Sara Gwen Weaver, in correspondence with the editor, described herself as a young novitiate of repeating forms. Sara kept a poetry blog until she became ill in December 2010.