Monday, June 18, 2012
Phi Day: A Post about the Golden Section
And what, may you ask, is that?
Not that I'm a math geek – math, in fact, was my least favorite subject in school – but I am fascinated by “Phi” and its relation to the arts, especially music.
Phi, basically, relates to the Golden Section, a proportional ratio that is not two equal halves but rather two slightly different sections, not quite two-thirds to one-third. It occurs frequently in nature – the whorl of a nautilus shell, the divisions of a leaf – and may be found in architecture (for instance, one could argue, the Parthenon) or in the placement of objects in a painting (rather than being stuck squarely in the center, no pun intended, is the rectangular shape subdivided by its focal point into something other than equal halves?).
In mathematics, there's a famous sequence of numbers called the Fibbonacci Series where certain numbers can be divided by phi. In other words, 8 could be divided in half as 4+4 or, according to phi, as 5+3. Likewise, 8 could be half of 16 or it could be the “phi” part of 13 – as in 8+5. The interlocking sequence thus becomes
(3+2 = [5)+3 = (8]+5 = [13)+8 = (21]+13 = 34) etc.
and so on, multiplying like Mr. Fibbonacci's rabbits (which, curiously, is where the sequence originated, theoretically).
When I was in graduate school, I often heard how Bela Bartók's music frequently divided along the lines of the Golden Section, especially his Music for Strings, Percussion and Celeste and “very likely” other works from the same period like the 3rd String Quartet.
Usually, someone would point out the climax of the piece didn't happen ½ way through or ¾'s of the way or even 87% of the way through but at the Golden Section or 61.8% of the way through. Take the number of measures and multiply by .618... – that's what we call “phi” just as we call .314... “pi” – and there it was.
Or, in other pieces, wasn't.
Maybe it was a measure off, or even several measures off. Oops.
Now, we don't perceive music the way we do a painting or a building which we see in self-contained spaces in front of us.
Music unfolds over time.
If I'm listening to Bartók's 3rd Quartet, one of my favorite pieces, it may go by so quickly, I'm disappointed when I realize it's over already, while the person usually sitting in front of me is so bored, he thinks it will never end.
(That's another issue, completely, but hey...)
I remember asking my professor “if we listen to music in time, wouldn't the Golden Section be figured according to its duration rather than the number of measures, since the meters and tempos change?”
Everybody looked at me like I was... “one of those” and the topic was dropped.
The question is, since this happened occasionally in certain other pieces even by composers like Beethoven, was it something these composers consciously plotted or was it, like the way that leaf in nature divides, something that happened naturally – accidentally?
Did Bartók use it consciously in determining where the music's climax should be?
Bartók never wrote anything about his use of the Golden Section anywhere, that I'm aware of. But he was particularly close-mouthed about almost any technical aspects of his style and very few of his sketches survive. Who's to say he didn't consider it consciously?
A couple seasons ago, when I got to interview Peter Bartók, the composer's son, who was a toddler when the Music for String, Percussion & Celeste was written, I asked him if his father ever talked about how he composed music and he said “No, I was just a child – he didn't even talk about that with his colleagues!”
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It began as an experiment. Remembering the discussions about the Golden Section in Bartók's music, I started working it into my own music (which I've discussed elsewhere in this blog.)
Then I read a novel by English author, David Mitchell, Cloud Atlas, to my thinking one of the most innovative novels since James Joyce's Ulysses (oh yes, and Saturday was Bloomsday, the celebration of the day on which Joyce's novel takes place: you can read a post about that, here.)
Anyway, Cloud Atlas has a rather unique structure, six nested stories that, once interrupted, resume in a mirror pattern – that is, picking up where the stories left off in reverse order. I wrote about possible structural issues, particularly pointing to the Mirror-Form Golden Section.
It occurred to me later that, like Bartók's music being listened to “in time” rather than measure-by-measure, Mitchell's novel wouldn't be divisible by pages but rather by the number of words.
Thinking of Georges Perec's novel Life: A User's Manual which the author describes as “novels” in the plural – it also is a collection of stories interwoven in a particularly highly structured way, using a “Knight's Tour” to order the chapters,I came across his use of creative constraints, especially in another novel, The Void as it translates into English, which is written without the letter “e.”
Without getting into how the Golden Section and Phi apply to the music I've been writing over the past decade, suffice it to say when I started writing novels, it seemed logical to want to figure out where, in the overall scheme of things, my major climactic points would fall.
Then I just kept subdividing the proportions further and further down from the larger viewpoint (the macro-structure) to the smaller details (the micro-structure), down to the chapters, sections, paragraphs, even sentences.
Like I said, it was an experiment.
Composing aside, I've been enjoying writing “music appreciation thrillers” the past few years, starting with Dr. Dick and his sidekick Buzz Blogster in a parody of a mega-hit by Dan Brown which I called The Schoenberg Code and then continued with a thoroughly original story (insofar as stories are 'original') which I'm in the process of posting now, The Doomsday Symphony.
At the moment, I'm working on another one, The Lost Chord which began life a few years ago as a parody of Dan Brown's next thriller but which I abandoned because I didn't like it. However, I did like the essence of my version of the plot and most of the characters' names (no way I was going to abandon Yoda Leahy-Hu, Iobba Dhabbodhú and LauraLynn Hardy, but that's another matter...).
I am over half-way through it – in fact I'm well on my way to the PHI POINT, the Golden Section of the entire novel.
Considering what I said before – about how we listen to music unfolding over time – how do I know this?
We read the same way we listen to music, whether you're a speed reader or you can only take in a page or two before you fall asleep.
In this case, I made the conscious decision to say “okay, The Lost Chord will be 180,000 words.”
Here's a section of Chapter 27 which I wrote yesterday as an example. It's only 905 words, so read through it before you continue.
To explain, there is Dr. Richard Kerr (a thinly-veiled manifestation of myself) with his assistant Cameron Pierce and LauraLynn Hardy whose cousin, the composer Robertson Sullivan (the son of Gilbert N. Sullivan), was just murdered on the eve of the first rehearsal for his new opera, Faustus, Inc., at the Schweinwald Festival in Bavaria. She herself has been involved in scientific research into the study of musical creativity, by the way. Her and her cousin's great-grandfather, Harrison Harty, was a composer who left behind a journal about his studies one summer at the legendary Schweinwald Academy in 1880 where a fellow student was Gustav Mahler and where Brahms was a visitor. For some reason, at one point the journal breaks into code and somebody else (known to them as musicologist Rothbart Girdlestone) tried to kill LauraLynn to obtain it. The “artifact” mentioned briefly is something that also, presumably, has some important clues.
Now, the presumed killer (known to them as psychiatrist Dr. Iobba Dhabbodhú) needs Kerr's help in finding some specific fountain and while there are other plot-threads twisting around here, suffice it to say that Kerr and Company have found their way to the home of composer Howard Zender, pushing 90, who presumably is going to have some answers for them. (By the way, he lives in an elaborate late-19th Century chalet added on to what was originally a 14th Century castle outside Garmisch-Partkenkirchen, okay?) Oh, the Will mentioned briefly is Zender's nephew, Will Schlegel.
(It is unnecessary to know that Girdlestone and Dhabbodhú are one and the same, two disguises of the composer who calls himself Tr'iTone who is only one of the villains. The other one is a SHMRG operative named Garth Widor, but I digress.)
In this scene, Howard Zender and LauraLynn Harty have gotten into a discussion about creativity.
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Curiously, Zender had twelve different kinds of tea in his kitchen cupboard, making the choice, reaching any consensus, more challenging. I decided, given the hour, something decaffeinated, less exotic, would be preferable. The kettle (a watched pot) took forever to come to a boil – Zender detested boiling water in a microwave. It surprised me he'd even given into the modern convenience of tea-bags, ugly tails hanging over the cup's rim, but maybe Will had made the substitution and just never told him.
For living in a century-old German chalet appended to a 14th-Century castle, Zender's kitchen was like any suburban home's, even if considerably more organized and efficient, whether his doing or Will's. I let the tea steep for three minutes before removing the bags, preparing a tray with cookies and crackers.
Cameron had been burying his nose deep in Harrison Harty's mysterious notebook, carefully writing out a translation of the code, progressing more quickly as he gradually became more accustomed to the substitutions. It would be a slow, daunting task under the best of circumstances: who knew what secrets it might yield?
He barely looked up, nodding, as I placed the teacup beside him: I'm sure he would have preferred coffee. He took a sip, then bent over the page, back to work.
"So far, the problem is dealing with something objectively that's primarily subjective," LauraLynn said, continuing her conversation with Zender, "deriving facts from various series of data, eventually distilling certain common patterns, not like applying inductive reasoning where scientific observation yields consistent facts over frequent repetitions to something that's more deductive. We've spent the past few years accumulating lots of data in interviews with various composers from around the world, only starting on ways of filtering that data to discern certain thought-patterns."
"You're talking about codifying inspiration?" Zender asked, sounding more amused than argumentative, something of a twinkle in his eye. "Sounds kind of contradictory, applying science to something as mystical as inspiration."
Knowing better than to get involved here, I picked up the artifact, checking out the scratchings on its back.
"What I'm hoping," she said, "is, if we can replicate the process, can we trigger creative responses in someone who's..."
"Perhaps suffering from writer's block? Or," Zender responded, "uncreative to begin with?"
He nibbled on a cookie before continuing, apologizing for this guilty pleasure. I only assumed he meant the cookie.
"How can they create something without any knowledge of what they're creating? Wouldn't they need some kind of set-skills? The language of music is little different than that of any science."
He also asked if there's a distinction between blindly following these set-skills or breaking the rules to create art, wondering if any of her scientific variables could take that into account.
"There's something else Schoenberg said," Zender added. "A craftsman creates because he can – an artist creates because he must."
It started again, always falling – floating, a flying squirrel not yet Icarus – this dream that invariably began the same way, coming to him unbidden in that indefinable state between consciousness and unconsciousness, where he was most open to inspiration, suspended between Reality and Art: Time stopped, unimportant, what he had most. Lips in the palm of his hand, lips he wanted to kiss but was afraid would bite his face. He splashed through a mirror, past the firing squad, the armless statue...
"No," LauraLynn said, carefully sipping her tea, looking into the valley below, "that's like having one hundred chimps writing Hamlet. We can't take people off the street and say 'Write a symphony!' I'd so much rather have a term we could use that's not quite as loaded as 'inspiration' is, frankly."
She thought a moment before she continued, "Though maybe it's like this: let's say you hand someone a violin, train them to hold it and, eventually, to play a simple tune."
Turning around, she hadn't noticed Zender was in some kind of trance, not that he was inattentive or rude.
“There're thousands of hours of hard work before making her professional debut.”
Still, teaching someone how to write a simple tune or a symphony wasn't the point, just a starting place.
"But teaching someone set-skills," Zender resumed, "and having them compose by rote is not like turning them instantly into composers, though the world's already full of com-poseurs who genuinely think they are. Science," he admitted, "has never explained the difference how creative artists work, defining what made Mozart different from Beethoven.
"Copland told me he saw everything in a flash, everything whole, complete, but not every composer works that way. I can't find the light till I'm well into the tunnel, sometimes.
"There is, if you'll pardon the [Socratic?] argument, something each of us possesses that resonates within – but in different ways. Our inner child not withstanding, it's something I call our Inner Chord. For people who are tone-deaf or for whom all music sounds alike, it is silent though perhaps only impaired."
I found it impossible not to listen, deciding to put the artifact down on the table for the moment just as Cameron chewed on his pen after pushing the journal aside.
"How it resonates, creating overtones, determines our becoming performers, composers, or music-lovers, how we respond, form our stylistic preferences. But we can also lose that connection with it after a time.
"It becomes this thing, then, that we will always be searching for, what has now become our 'Lost' Chord."
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If you look at the Fibbonacci Series (see above), you'll notice a pattern: alternating segments of larger and smaller numbers. Translated to musical beats or a novel's words, they create a kind of rhythm. Now, when I subdivided my 180,000 words down to the chapter level, Chapter 27 turned out to be 2,368 words long, its internal Golden Section being 1,463 words. The second 'half' (or section) of this chapter, quoted above, is 905 words long.
This further subdivides into two segments of 559 words and 346 words. Each of these can be divided by Phi into (345+214) words and (214+132) words.
You can see that there is a rhythm at work here, longer segments followed by shorter segments, an internal lower-level climax (a short segment) then followed by a longer segment and so on.
Since not everything in a thriller needs to be “Action” (though some seem to be almost all action) there is often some contrast which might be created through longer paragraphs or sentences building up to shorter and shorter paragraphs and sentences.
Not every paragraph or sentence is completely self-contained. Dialogue, for instance, doesn't often move this way, but the topics around the dialogue or the focus of each segment might fit within a section's “boundaries.”
So, the “Golden Section” for this excerpt occurs after the surreal sequence about the lips on the hand biting his face, the firing squad, the armless statue (incidentally, a reference to Jean Cocteau's famous film, Blood of the Poet a kind of quantum challenge to the standard perception of time and chronology). That opening portion is 559 words long. The remaining portion is 346 words.
Why did I do this?
For one thing, the chicken climbing Mount Everest aside, I wanted to see where these points of arrival might fall and to space them proportionally, from the major climax-points down to the subtler changes that might happen within a given scene.
Have you ever read something and felt the pacing was terrific or that section just went on too long? One of the problems I had with my initial run-through with The Lost Chord was that, like its model, it had this enormous epilogue, actually epilogues within epilogues and I felt the pacing was all wrong – actually, it was all “out of proportion.”
Have you noticed, when you're reading a mystery or a thriller, after it builds to a climax, the next scene is often a contrast, more low-key, often longer, perhaps more slowly paced? But that as you get closer to the ending, these segments get shorter, are faster-paced and start tying what before seemed like unrelated elements.
This is why chapters turn into page turners, leaving you at a cliff-hanger; then you turn the page and start reading about Aunt Katie. But as you progress toward the conclusion (and your own thoughts about how it's going to work out), you realize there's something about Aunt Katie that's going to be important.
But if the author resolved the cliff-hanger, it would be over too soon and the sense of anticipation weakened: you want to find out what's going to happen, now you have to keep reading.
So when I started writing this fully revised version of The Lost Chord, I wanted to extend this sense of proportion down to smaller and smaller elements.
When I wanted to figure out where my climaxes should be, according to the Golden Section, I multiplied 180,000, the pre-determined length, by .618 and found that the chapter ending with the novel's “turning point” (not the actual dramatic climax, but a point where the action takes a definite turn toward the climax) would be at 111,240 words.
Taking it from there, there would be “sub-points” building in the tension that would be slightly lower high-points, so I divided each of the two segments on either side of “Big Phi” by .618 and placed the ends of chapters there.
I would begin with a general outline of the plot – basic premises, major occurrences, specific dramatic high-points. These, I stretched out over my word/time-line.
As I filled in the chapters, I began writing summaries of the action and generic views of expository scenes. Very often I found these broke quite readily into four sections. I could figure out, by constantly subdividing it – sort of like deconstructing the plot down to the molecular level – similar proportions at not just the chapter and section level but at the paragraph and sentence level as well.
Like I said, it was an experiment: I wanted to see if I could make it work. Curiously I found myself having to do very little editing to come up with the right number of words, or the balance of phrases.
It felt odd writing it almost sentence by sentence rather than full-steam-ahead from one chapter to the next. But I'd already taken care of most of that in the outline: this was like creating a painting with pencil outlines in a sketch, then figuring out what colors went where before filling in the individual brushstrokes that make up the textures and details you would only notice on closer inspection.
In fact, I discovered, in order to pace it toward the focal point, I often would begin writing at the end of the section, then work my way, sentence by sentence, backward to make sure I ended up where I needed to be when I needed to be there. This also helped me be, in one sense, more precise but also at times showed me I needed more time to let this idea expand: the biggest problem, working within the outline, was realizing if I wasn't careful it come off reading like a series of bullet-points rather than continuous, unfolding prose.
So I created an outline template (each chapter having four sections, each subdivided further and further into four subsections) and filled in the necessary word-count according to Phi, the Golden Section.
So, going back to Chapter 27 (the above example), I broke the 346 words of the 4th section into two sub-segments of 214 and 132 words, down to four smaller segments each to get
436 = (132+82) + (82+50)
then took the 132-word segment and divided it into (82+50) words or [(51+31)+(31+19)] words. And so on.
So if I take that last section from my example, rather than going through the whole thing, and look at it in my sketches, it looks kind of like this: unfortunately, the outline format and indentations is all screwed up, transferring from a Word Document to blogspot, so you'll have to bear with the profusion of numbers – but basically the larger numbers break down into pairs of proportionally divided numbers and so on from 346 words for the D-Section of Chapter 27 all the way down to clauses of 7 or 5 words.
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Chapter 27: D-1 = 214 = ((i.) 132+ (ii.) 82) = [(82+50)+(51+31)] =
i. = 82 = (51+31) = [(32+19) + (19+12)] =
a''. = 51 = (32+19) =
32 = (20+12) = [(12+8)+(7+5)]
[20 = (12+8) = [(7+5)+8]] "No," LauraLynn said, carefully sipping her tea,  / looking into the valley below, / "that's like having one hundred chimps writing Hamlet. 
[12 = (7+5)] = We can't take people off the street  / and say 'Write a symphony!' 
 = I'd so much rather have a term we could use that's not quite as loaded as 'inspiration' is, frankly." (this should break into (12+7) but it doesn't...)
(Ah, so on second thought, how about this revision which fits the pattern?)
I'd so much rather have a less loaded term we could use  / without quite the baggage 'inspiration' has, frankly."  (actually, I like that better)
b''. = 31 = (19+12) = [([7+5]+7) + (5+7)]
[19 = [(12+7)+] She thought a moment before she continued , "Though maybe it's like this:  / let's say you hand someone a violin, 
[+12] ...train them to hold it  / and, eventually, to play a simple tune."  (In this case, the 7+5 is reversed...)
ii. = 50 = (31+19) = [(12+7)+(7+5)] + [(7+5)+7)]
[31 = (19+12)] Turning around, she hadn't noticed Zender was in some kind of trance,  not that he was inattentive or rude. 
 “There're thousands of hours of hard work  / before making her professional debut.”  (I'm still not satisfied with this one: doesn't flow conversationally...)
[19 = (12+7)] Still, teaching someone how to write a simple tune or a symphony  / isn't the point, just a starting place. 
1b. = 82 words = (51+31)
i'. = 51 = (32+19) = [(20+12)+(12+7)]
[32 = [(12+8) + 12)] =  "But teaching someone set-skills," Zender resumed, "and having them compose by rote  / is not like turning them instantly into composers,  though the world's already full of com-poseurs  / who genuinely think they are. 
[19 = (12+7)] Science," he admitted, "has never explained the difference how creative artists work,  / defining what made Mozart different from Beethoven. 
ii'. = 31 = (19+12)
[19 = (12+7)] "Copland told me he saw everything in a flash, everything whole, complete,  / but not every composer works that way. 
 I can't find the light  till I'm well into the tunnel, sometimes.  (Originally, this was 'Sometimes, I can't find the light till the end of the tunnel' but that breaks into 6+6; I fixed the wording but also placed the 'sometimes' at the end, balancing the proportions.]
2. = 132 words = (82+50)
a. = 82 words = (51+31)
i'. = 51 = (32+19) = [(20+12)+(12+7)]
[32 = (20+12) = [(8+12)+12]] "There is, if you'll pardon the (Socratic?) argument,  / something each of us possesses that resonates within, but in different ways.  / Our inner child not withstanding, it's something I call our Inner Chord. 
[19 = (12+7)] For people who are tone-deaf or for whom all music sounds alike,  / it is silent or perhaps only impaired." 
ii'. = [31 = (19+12)] I found it impossible not to listen, deciding to put the artifact down on the table for the moment,  / just as Cameron chewed on his pen after pushing the journal aside. 
b. = 50 words = (31+19)
i'. = 31 = (19+12) = [(12+7)...] = [19 = (12+7) = [(5+7)+7]
"How it resonates, creating overtones,  / determines our becoming performers, composers, or music-lovers,  / how we respond, form our stylistic fingerprints.  /
But we can also lose that connection with it after a time. 
ii'. = [19 = [(5+7)+7]
"It becomes this thing, then,  / that we will always be searching for,  /
what has now become our 'Lost' Chord." 
*** ***** ******** ***** ***
This excerpt was chosen more or less randomly – largely because I wrote it yesterday, so it's still fresh and editable – and it only demonstrates the method behind my phi-oriented madness. I gave you the whole 905-word excerpt so you could follow the rhythmic pacing through a longer span (I could demonstrate the structure of a Beethoven symphony movement by playing a couple phrases (a few seconds of music) but instead gave you, say, part of the exposition (several minutes of music).
If you're wondering how it works in the long run, if you've been following the progress of The Doomsday Symphony on the installment plan, here, you will already have an idea. Earlier today, I posted the PHI POINT of The Doomsday Symphony, a 130,000 word novel, on PHI DAY, June 18th (6-18) – if you haven't already noticed, each chapter is posted at 6:18am – and the same structure applies here, as it does in The Lost Chord.
Except, being 130,000 words as opposed to 180,000 words like The Lost Chord, the proportions are all different and the numbers will be adjusted accordingly (the sketch files are all locked in a dead computer, for what it's worth). I didn't always break it down to the sentence level, sometimes just the paragraphs were enough, but the idea is basically the same.
You can read an earlier post that “analyzes” the opening of The Doomsday Symphony, here.
Yes, it's very abstract and terribly left-brained and I may seem like a terminally OCD geek to anyone reading this (though I assure you my house is a mess, my life is terribly disorganized and I'm basically a slob).
The whole point for an artist is to create something out of these patterns we have around us and find ways of challenging ourselves to perhaps do something different (if not original) with them that somehow combine elements of structure (all language is structure) with elements of spontaneity (telling a story that engages both the mind and the heart).
"Phi Day" may not be as catchy as "Pi Day" because we can't bake pies with the Greek letter 'pi' on it, but it seemed an appropriate day to post the "Phi Chapter" of my novel and write about its significance in the way I think about writing.
Do you need to understand that to appreciate it?
No, of course not, just as you don't have to understand the implications of 19th Century Tonality to enjoy the really cool music Richard Wagner composed for his operas, The Ring of the Nibelung.
If you do understand it, then you can appreciate the music on an entirely different level.
I should mention, the name for my 90-year-old composer Howard Zender (loosely based on Elliott Carter), who describes the "Lost Chord" in this excerpt, comes from one of my favorite novels of all time, Howards End by E.M. Forster, not just because of its famous scene describing listening to Beethoven's 5th Symphony but because of its recurrent theme: "Only connect."
It is the artist's job to "connect" with the audience whether it's a single listener or millions of readers. How the artist does that, how the audience responds to it is one of the great mysteries of art if not life.
The great thing about Art is you can keep coming back to it and discovering new things about it each time, things that very often you had no idea were there the first time you encountered them.
- Dick Strawser