Sunday, April 28, 2013


While writing the next post which will fit parallel processing into this thread, I took a break to work on a section of a piano piece I may or may not complete some day. The reason I am posting it here is that this segment is an algorithmic expansion of a single through-composed measure of music, about two and a half seconds, into 22 measures lasting around 50 seconds. (The title, "se," is the Proto-Indo-European root meaning "seed" or "sow.") I'm not putting this passage out as great music certainly (it's out of its context here anyway), but as an example of the potential for the "spiral group" of permutations for music composition. More specifics on the creation of this brief passage later when I get to applications of these permutations by three well-known contemporary composers.  For now, see if you can hear the spirals at work in this passage. And whether you can or not (I think not), does it matter to the music? In other words, what's the connection, if any, between construction and perception?

Saturday, April 20, 2013

Jeu de Cartes

Mathematicians do not deal in objects, but in relations between objects; thus, they are free to replace some objects by others so long as the relations remain unchanged.

The idea that someone with a bit of natural dexterity and a lot of practice can trace any card's path as it travels through several riffle shuffles of a deck is amazing to most of us (and a good reason we amateurs should stay away from Blackjack tables and Poker games). The above video by Kevin Houston gives an excellent demonstration of how card tracking is possible using a "perfect shuffle" and then goes on to explain the math behind why this particular shuffle works as it does. In passing, musicians will recognize one of the techniques since it's based on modular arithmetic––the same elementary principle as the one at work behind the "octave equivalence" and "pitch class" concepts. But what we're interested in here is still mostly the "geometries of permutations."

Another interesting card shuffle is often associated with the mathematician Gaspard Monge (1746-1818), a colorful character active during and after the French Revolution and usually thought of as the inventor of descriptive geometry. What is often called the "Mongean shuffle" works like this. . . .

We'll select only six cards from the deck to make the principle clear, keeping in mind that, once understood, it can be applied to the entire deck of 52 cards. Let's say we have, arranged from top to bottom, the Ace, 2, 3, 4, 5, and 6 of clubs. Now, ignoring the mechanics of how this shuffle might be performed in practice by a dealer or magician (it's rather clumsy compared to the riffle illustrated in the video), we lay out a single Mongean shuffle of these six cards face up. Begin by laying down the top card, the Ace, then place the next card, a deuce, on the table to the left of the Ace, then the three to the right of the Ace, the four to the left of the deuce, the five to the right, and finally the six to the left. The result is the re-ordering 6, 4, 2, Ace, 3, 5:

Flipping this map diagonally and repeating the procedure, we can follow any card through six shuffles until it returns to its original position.

Why does this map look familiar? (As if you haven't guessed already.) But "looks familiar" isn't enough. Let's try applying a procedure that, once again, will be familiar to most musicians – or at least to those who have played/looked at Bach's Musical Offering.[1]

Here again is the basic Mongean shuffle:

Let's get the cards out of the way to follow the moves more easily. This gives us a sort of placeless map .... (sorry) ...

Start by flipping this map over horizontally, resulting in a retrograde map.

Then flip that result over vertically, resulting in an inversion of the retrograde.

Then lay the cards back on the table using the new map to see what this retrograde-inversion shuffle results in.

The retrograde-inversion of the Mongean shuffle produces precisely the same map we encountered in the sestina.

But we're not finished quite yet. Trying to visualize the Mongean shuffle as a transformed version of a spiral (the way we initially described the sestina) is a bit of a stretch. Still, following a curvy path, it almost makes sense to see how the spiral:

might be seen as a tangled/untangled version of Monge, which itself can be seen to resemble a snake twisting back on itself or the action of a wave rebounding from the side of a pool:

If we want to get away from continuous "curving actions" but still stay with some sort of helpful visualization that gives the same results, we might think of these two procedures as things to do with two three-pronged combs. Making one comb out of the left side and one out of the right, we reverse one of them, stretch them both a bit, and then interleave the result. Then the spiraling looks like this in the corresponding comb version:

Starting back at the consecutive numbers once more, this time dividing the positions into odd and even, we separate them again, turn one of them backward again, and then recombine by concatenation.  So the wave in its corresponding comb version looks like this:


Ideomorphically, of course, the two worlds can't be mixed––playing cards are not words and card shuffling is not poetry. And neither, again ideomorphically, is a snake or a spiral a card trick or a poem. And that's just the point, because we are searching for sub-surface isomorphic actions–beyond metaphor and analogy–that connect otherwise unrelatable worlds. In the isomorphic underworld, not only can a hair comb be a metaphorically useful object-symbol within a poem, it can also provide a translucent structural basis for the poem. In fact, the first of the two "comb moves" above is precisely the way the sestina was often described, rather than the spiral description I chose to begin with. The technique in medieval times was known as retrogradatio cruciata–"backward crossing."[2]

Next: Just what is multi-tasking parallel processing, anyway? ––––>

[1] I mention Bach in this passing reference rather than Schoenberg, whom many would prefer to focus on as the more obvious culprit. I did this to keep my background emphasis that all these tricks are ancient & didn't just spring into being starting in the early 20th century. It's relatively easy to identify the inventor of the steam engine, but not so easy to identify the inventor of the wheel. By the time JS Bach came on the scene, many of these tricks/transformations had already long been in use. For a fun time with the Bach canons in the Musical Offering, there are two interesting web sites – one in a math version and one in a non-math version.
[2] Treated as religious symbolism, this term might also suggest anything from dark word play to heresy to outright satanism. But I have never read any suggestion of this elsewhere, so this must remain my personal fantasy. (Still, an interesting potential plot twist for a novelist or playwright, though?)

Tuesday, April 9, 2013

A Proper Sestina

. . . how much time is lost in invention, internal arrangement, and combination! for which nobody thanks us, even supposing our work happily accomplished.
––Goethe, Conversations with Eckermann

The sestina is a poetic form that I, at least, don't hear much about these days. Its invention is commonly attributed to Arnaut Daniel (fl. c.1180-c.1210), a Provençal troubadour of the 12th century. It pre-dates the invention of the initial canonic form of the sonnet, attributed to Giacomo da Lentini (fl. c.1233-c.1248). In contrast to the sonnet which has had a relatively unbroken history, the sestina keeps dropping out of sight and being rediscovered (by poets, at least), a discontinuity that appeals to me and fits in nicely with what's yet to come in this thread.  But now we need an example.

Dante Alighieri was a big admirer of the poetry of Arnaut Daniel, famously paying tribute to him in Purgatorio XXVI.139-48. Dante also occasionally used the sestina form himself.  One of the better known of these sestinas is "Al poco giorno, ed al gran cerchio d'ombra" which, for those fluent enough in Italian to appreciate fully, can be found >here<.  Later, one of Dante Alighieri's great admirers, Dante Gabriel Rossetti (1828-1882), translated this sestina into English, adding his own title.[1] This translation is where I will jump back into my story–––

Rossetti, Madonna Pietra

Sestina of the Lady Pietra degli Scrovigni

To the dim light and the large circle of shade
I have clomb, and to the whitening of the hills,
There where we see no colour in the grass.
Nathless my longing loses not its green,
It has so taken root in the hard stone
Which talk and hears as though it were a lady.

Utterly frozen is this youthful lady,
Even as the snow that lies within the shade;
For she is no more moved than is the stone
By the sweet season which makes warm the hills
And alter them afresh from white to green,
Covering their sides again with flowers and grass.

When on her hair she sets a crown of grass
The thought has no more room for other lady;
Because she weaves the yellow with the green
So well that Love sits down there in the shade,––
Love who has shut me in among low hills
Faster than between walls of granite-stone.

She is more bright than is a precious stone;
The wound she gives may not be healed with grass;
I therefore have fled far o'er plains and hills
For refuge from so dangerous a lady;
But from her sunshine nothing can give shade,––
Not any hill, nor wall, nor summer-green.

A while ago, I saw her dressed in green,––
So fair, she might have wakened in a stone
This love which I do feel even for her shade;
And therefore, as one woos a graceful lady,
I wooed her in a field that was all grass
Girdled about with very lofty hills.

Yet shall the streams turn back and climb the hills
Before Love's flame in this damp wood and green
Burn, as it burns within a youthful lady,
For my sake, who would sleep away in stone
My life, or feed like beasts upon the grass,
Only to see her garments cast a shade.

How dark soe'er the hills throw out their shade,
Under her summer-green the beautiful lady
Covers it, like a stone covered in grass.

–––Dante Alighieri
         (Tr., Dante Gabriel Rossetti)

Taking up now where my previous sestina entry, "On the Advice," left off, let's try to do this mostly without reference to any spirals. Go to the first stanza in Dante's poem, note the order of the final word in each line, and assign a letter or number:

                                                shade    hills    grass    green    stone    lady
                                                   A         B          C           D           E           F
                                                   1          2           3          4            5           6

Now, using the same letters or numbers, do the same with the second stanza: 

                                                lady    shade    stone    hills    green    grass
                                                   F         A           E          B          D         C
                                                   6         1            5          2           4         3

Going through six complete stanzas in this way, stopping before the final three-line envoi[2], the linear "rhyme" scheme[3] for Dante's poem emerges whether you know the spiral trick or not. Using alphabetic notation for now:


If you follow the pattern by connecting the dots (words) through six verses, you come up with a "map" that looks like this:

This is one possibly helpful way of representing "what happens" structurally in the background just barely below the surface, not only for the poet making a sestina, but also––in some way––for the poet's audience hearing or reading it.

We don't have to squint very hard to see that the paths between successive verses are not drawn at random.  There is a repeating pattern here whether or not the reader is aware of it in-time.  And this leads to the more usual descriptions of "how to" compose a sestina.

The repetition scheme above read, say, in this way:
ABCDEFF AEB    DCCFD ABEE       CB FA DD    EACFB        BDFEC           A
is how the reader/listener experiences the form in time.  (Spaces indicate one possible reading –– an "interpretation" which is outside the poet's control and further helps to mask the structure.)  It is doubtful, to me at least, that anyone, especially those unfamiliar with the sestina form, upon hearing a sestina for the first time, would "cognize" (entrap by the conscious) the form that produced this string of repeated words.  That there were repeated words, yes, of course; but not a/the compositional pattern.

But the question at this point is not the analytically easy factual question, "What is happening structurally in this or any sestina?" but: How does one "experience" the defining structure of a sestina?––the thing that makes it a sestina and not a sonnet or rap[4] or prose.  At the risk of drawing opprobrium from a sphere of poesy experts and amateurs, here is what I believe is happening (maybe even meant to happen in some sense) and not happening, experientially . . .

First of all, here's what doesn't happen in experiencing the poem.  Remember the initial, and surely the simplest, explanation for the spiral algorithm is to think first of a string of things in one dimension. Then to rearrange them by going into two dimensions and applying a spiral to select the things into a new order. And then carrying that newly ordered string back into one dimension.  That works fine for the carpenter building a sestina, the poet. But no way on earth does anyone experience a sestina by following the changing positions of the terminal words from one stanza to the next as a spiral. So what possibly does happen then?

Each of the six repeated words (A through F) carries some subjective (first-person) "weight" for the reader[5] that interacts with the poet's "narrative" beginning in the first stanza.  This may be at a conscious or subconscious level, but the main point is that the reader takes a word––like taking a card from a deck offered by a magician (NB!)––and is drawn in to follow it from verse to verse through the sestina's contrapuntal (NB!) maze.  The next time through, the reader might take a different word leading to a different path through the exact same surface narrative.  [Why am I reminded of Bach?!] And at any point in the middle of a reading, there is the possibility that attention might be drawn from the initially chosen word to a different one, subverting a consistent "correct" path; so instead of following, say, (D-D-D-D-D-D), the poem's structure and the reader's "attention status" might encourage a reading that reflects the path (D-D-D-D-E-E) or (D-D-F-D-C-D)––an uncountable number of ways to experience (interpret?? analyze???) the poem.

I realize that "uncountable" may seem like an overstatement, but remember that, if you wish to defeat this claim by actually calculating the number of possible paths as a fun exercise in combinatorics, you must include "attention flagging" moments––paths which leave a hole in the structural fabric. E.g. (still concentrating only on the word repetition scheme), we must include paths such as (A-B-B-x-A-C), (C-C-e-k-F-x), (p-q-E-q-E-E), and so on, where the lower case letters indicate undefinable variables ("stray thoughts") that happen to pop into the mind during a reading and that may or may not have anything to do with the poem but nevertheless do have something to do with the reader's experience.[6] One possible reading in this view––and maybe the most common and "human" one––might be, for example, (a-b-c-d-x-f) which might mostly follow the surface narrative but doesn't (can't or won't) follow the sestina's characteristic word repetition pattern at all.  (This is a version of what might be called the just-let-it-wash-over-you experience––an approach not to be scoffed at in approaching a new work.) Being quasi-engaged in something while remaining oblivious to how or why it works is not uncommon.

And now for a pop quiz.

You may not have noticed anything unusual about the fact that I chose, as an example to work from, an English translation of a sestina by Dante Alighieri rather than finding a sestina originally written in English.  Finding this Rossetti translation was a bit of serendipity, because it now allows me to ask you all a question that, if you answer it honestly to yourself, may prove a wee bit embarrassing... (It was embarrassing for me when I made the discovery.) –– Now that you know the sestina's defining form,  whether you think of it as spiral or linear, did you notice, when reading this poem, that the form was violated?  It wasn't Dante, but the translator Rossetti, who made a "mistake." The "mistake" occurs in verse 5 where Rossetti has reversed Dante's (proper sestina) terminal word order for lines 4 and 5.  We might try to "correct" the mistake by naively reversing lines 4 and 5 of the translation:

"Corrected" verse:

A while ago, I saw her dressed in green,––
So fair, she might have wakened in a stone
This love which I do feel even for her shade;
I wooed her in a field that was all grass
And therefore, as one woos a graceful lady,
Girdled about with very lofty hills.

This would technically fix the form, but now we have a real conundrum about what the verse says, or is supposed to say––how it scans between two poets––one poetizing directly in Italian and the other translating poetically into English.  Here is the original for comparison:
  • Io l'ho veduta già vestita a verde
  • Sì fatta, ch'ella avrebbe messo in pietra
  • L'amor, ch'io porto pure alla sua ombra;
  • Ond'io l'ho chiesta in un bel prato d'erba
  • Innamorata, come anco fu donna,
  • E chiusa intorno d'altissimi colli.

Even to attempt to discuss this form-content anomaly (I certainly hesitate to accuse Rossetti of making a mistake!), let alone resolve it, is well beyond my abilities in both poetry and translation. But that isn't the point here at any rate. My point is to demonstrate that if you disrupt the highly complex rhythm below the surface set up by a rigid structural scheme such as the spiral algorithm, it may or may not affect the content or message at the surface––that is, the experience.

This idea of sub-surface structure vs surface perception will return in a more violent form when I discuss music applications later. But for now I want to explore not mistakes and violations in the form, but surprising logical extensions.

(By the way, how did you do on the pop quiz?
Did you spot the mistake before I told you about it? 
And if so, did you discover it by hearing it
or analyzing your way into it?)

[1] This is not meant to be a full, or even fully accurate, tracing of a small piece of poetry's history, nor to give the sestina undue importance in that history. But in setting it up I did realize that it illustrates nicely how poets, possibly more than other artists, seem to talk with one another across centuries.
[2] I am purposely leaving out the envoi to make this demonstration more clear. I am aware that the envoi as a culmination or destination or turn may even be the point of a given sestina, so leaving it out of the discussion misses the full "poetry." However, I have noted that there are many variant "acceptable" patterns for the envoi that would better be discussed in light of the pattern found in those first six stanzas which are my focus here.  In other words: another time, maybe.
[3] More properly called a repetition scheme.
[4] . . . but rapping within the confines of the canonical sestina form would be an interesting challenge, wouldn't it.
[5] "Weight" is meant to convey more than "meaning" and refers not only to acceptable dictionary definitions, but to incorrect understandings, past associations, contexts, and even the subjectivized sound or "flavor" of the word.  Also, "read" may also be read as "hear" or "listen to."
[6] My contention (better: my guess), which I doubt I could ever adequately defend, is that a "perfect attention"-reading is either impossible or rarely achieved.  But as far as I know, either to demonstrate or disprove my contention would take defeating the first-person avowal problem––or worse, asking the reader to pay attention to what she is paying attention to, and then accurately report back.  (This is also not a good strategy for making love, but I digress (or perhaps it wouldn't be a digression––this particular sestina, after all, is about . . . . ) . . . . )

Art's Dueling Problematics

"I don't know if you have the same experience, but the snag I come up against when I'm telling a story is the dashed difficult problem of where to begin it." (Bertie Wooster)
–– P. G. Wodehouse

If the reader has not guessed by now, it should be obvious at the end of the next sestina entry that the subject of this thread is not the sestina, nor is it poetry or poetry's underlying/defining forms.  Neither, in the final analysis, is it about the "spiral algorithm" per se, an example of [––abusing mathematical terminology:] an isomorph that pervades  [––now stealing from crystallography:] the characteristic idiomorphs that define and sort the specific arts as-we-know-them and guarantee to keep them separate from (or blissfully ignorant of) one another.

I will be skipping between "fields" willy-nilly from poetry to math to card tricks to computers to music, and possibly a few more. Along the way I will string them all on the same necklace, their common thread in this case being the spiral algorithm. But also along the way I will try (I can't see any QEDs ahead) to gain some insight into a deeper ancient issue.

I am looking for oblique ways (I know of no direct routes) to approach what I consider to be art's dual problematic: the place of structure, or form, in the act of creation versus[!] structure's myriad, unpredictable and seemingly inscrutable effects in perception of the object created. In my own mind, this is not unrelated to the deeper dual problematic that can be traced back at least to the pre-Socratics: If the world is one, why does it appear to be many? But then if the world is many, why does it appear to be one?[1]

This is not to suggest there is a puzzle here that can be solved given a puzzle-solver of sufficient genius provided with sufficient information. I'm using the more convoluted noun "problematic" to distinguish it from the way we normally view problems as puzzles. In the way I am using it here, what a problematic describes is a special kind of problem (whether it has actual "solutions" or not is irrelevant) that, by the simple act of working hard at it, drives us to deeper levels of understanding.

[1] "Dual problematics" idea extrapolated from David L. Hall, Eros and Irony; Thinking through Confucius; The Uncertain Phoenix; and others.

Sunday, April 7, 2013

Sestina On the Advice of Anthony Hecht

As any form becomes canonical,
it virtually invites
experiment, variation, violation, alteration.

Logic is an arrow––to the point, singular, literal . . . no detours. 
Metaphor is a map, full of blind alleys, irrelevancies, absurdities, ambiguities, multiple meanings, choices . . . interesting things. It lets you begin your story where you please––middle, end, wherever, it doesn't matter.
Taking you to the edge and convincing you not to jump, logic keeps you safely within the world you are given.
Subverting reality, distracting you to a world that may never exist, metaphor takes you to the edge, tells you to fly, and pushes you off.
Logic is a necessary connection.
Metaphor is an imaginary bridge.
Premise and consequence: metaphor and logic are inseparable––together they create art and science.

Internal Critic:

OK, then.  You know quite well that a proper sestina consists of 6 verses of 6 lines each, followed by a closing 3-line envoi. "On the Advice of Anthony Hecht" either is a single 6-line verse followed by a 1-line envoi, or it is six 1-line verses followed by a 1-line envoi. Or is it just prose –– notes for a future poem perhaps? Another W.I.P.-mask you so like to hide behind to avoid completing anything. In any case, it fails as a sestina.

Furthermore, each verse of a proper sestina must contain six distinct words that are repeated from one verse to the next; also, these repeated words must appear as the terminal word of each line. "On the Advice" has only 2 repeated words, and these never appear at the end of a line/verse.

Here's a suggestion: Let's generously accept "On the Advice" as a bizarre 2-verse "binary sestina." If we label "logic"=0 and "metaphor"=1, then the 6-line stanza begins this obviously unfinished poem with the word repetition order 010101.  This might be the beginning of a more reasonable structure getting back to something more closely resembling the sestina form we have come to expect.  So we keep what you have as verse A and work out the word repetitions as the structural basis for five more verses B through F.  Then you will only have to compose out from this scheme:
A   010101
B   100110
C   011010
D   001101
E   100011
F   111000


That's very helpful of you, but that's a lot more 0's and 1's than I had planned.  I'm not sure I have that much to say about logic and metaphor.

Internal Critic:

Well . . .


And seriously––... didn't you catch the recap of all six key words in the envoi?

Internal Critic:

Well, no. I really didn't hear the six key words you think you "repeated" in the envoi.  But still I'd certainly have to pan any attempt to keep my attention through 36 lines with 18 logics and 18 metaphors, plus another three of each for a proper envoi.  But I have another suggestion for you.  Why not try actual, proper rhyming?  Leave the first stanza as it is, then in stanza B use, oh, I don't know, 0="pedagogic" and 1="petits-fours."  Then maybe verse C could use, let's see ... how about 0="trick" and 1="door."  And since we now have six distinct words in the pattern, why not go back and change verse A so it uses all six of these words––label them 1 through 6––one in each of verse A's six lines? Then we, I mean you, could riff on this pattern:
A   123456
B   615243
C   364125
D   532614
E   451362
F   246531

"New" and "different" are so overrated. You're brilliant. You can do this! Trust me.



Internal Critic:

And it would be a nice touch––a tribute, if you will––if you could try iambic pentameter rather than the frankly quite ugly prose you seem to like. The public, after all, has come to expect the beauty of stacked iambs.  After all, they are the feet we've come to love.



So you'll finally be happy if I write a poem using the exact formula concocted by a troubadour to win a poetry slam in some tavern in Provence 800 years ago?

Internal Critic:

Well . . .  But it will be your own voice, of course.  There's still plenty of good poetry to be written in ....


. . .

Thanks for all this. Really. But I sort of like it just the way it is. Whatever it is.

Internal Critic:

OK. Your call. But one last thing–––.

It's not all that important, but . . .

       what were you trying to say about logic and metaphor?

The Form

After many false starts, I've finally realized that the best way in to the convoluted story I want to tell is by simply drawing your attention to one of the most unexceptional connections imaginable between nature and human design. The connection could have been made in a variety of ways, so I'll imagine one to stand for all possibilities.

One morning, well over two millennia ago, a man strolled out onto the beach as he had done thousands of times before. He leaned down and picked up an object that had washed ashore.  But this time, rather than just looking at this object which was so familiar to him, he "saw" it. This was the object:

And today we have proof of what he saw thousands of years ago, because he left a record of his in-sight:

The volute or scroll or spiral form is ubiquitous in nature and design, and has been thoroughly studied and remarked upon. We know its equations. We know the logarithmic spiral whose arms get further apart with every turn as in the examples above––and we know the relationship of the general logarithmic spiral to the golden mean and Fibonacci spirals.  Simplest of all is the arithmetic or Archimedean spiral:

All of this is in two dimensions. If we ask what a spiral looks like in three dimensions, we are led to the helix, another well-studied form. But the seed question I wish to pose here is: What does a spiral "look like" in one dimension?

Imagine a collection of eight "things"––we'll label them A through H. 

It doesn't matter what the things are––letters, numbers, words, colors, pitches, durations, mangoes, rabbits, playing cards, shot glasses, walnut shells, dried peas––, but what does matter is how they are arranged and how we are going to REarrange them. (Think, if you will, of the old shell game.) We order the original things along a line (1 dimension) and then we temporarily "go into" 2 dimensions and superimpose a spiral: 

Then, beginning with H (we could also begin with D, or we could rotate the spiral and begin with or A or E, with similar but not identical results) we follow the spiral around and collect the letters in a new ordering: 

We then superimpose the same spiral and, beginning in the same place  on the spiral, which is now D, repeat the process (which we can now call an "algorithm").

This results in the new order:

If we continue to do this, boring as it may seem to the reader at this point, we notice that we arrive back at the original ordering after only four "moves":

If we reduce the number of "things" to 4, then we find it only takes three "moves" to return to the original order:

We also observe that in this case the "C" is stationary.  This quirk as well as the shortcut return to original order in both of these examples is in contrast to a maximal run of the spiral algorithm such as this one with six things that takes six moves to return the original order:

There is one more interesting feature of the spiral algorithm––its inverse: how it works backward––but I will save that for the two target applications of the spiral algorithm that I am now headed for, one very old one in poetry and one several relatively recent one in music.