Friday, February 7, 2020

Searching for gold

Last week I participated in a little teaching day focused on the so-called "Golden Ratio." Students were introduced to the basics of the math behind the ratio as well as some creative exercises related to drawing Fibonacci squares, measuring various rectangles and faces, looking at math in nature, etc. Naturally, my job was to bring in some activities related to music.

I've always harbored a fairly typical skepticism about just how important this ratio is when applied across time to musical structures, but it's certainly an interesting topic. The skepticism, for me, comes from whether a 1.62/1 ratio (as applied to dividing up a musical structure) is really significant because of the mathematical underpinnings or just because having something really interesting/dramatic happen about 2/3 of the way through just makes sense for a lot of practical reasons. [A true "Golden Ratio" moment should happen about 62% of the way into a section, although I suppose it could also happen 62% from the end (38% from the beginning).*] 

It would be odd to have the most interesting thing happen early on (I've written about this!), and it also seems a little odd to finish at the point of highest stress, whether it's because it's satisfying to recapitulate or provide a sense of denouement or whatever. Two-thirds has that nice, "we're more than halfway there, and we're ready for something big that will then require some wrapping up" kind of ideal. Still, my own instinct would be to guess that the best time for a "significant" event would be a little past the 2/3 point (66%), so if there's evidence that the most special moment works better just before the 2/3 mark, that would be interesting and maybe an argument that the mathematical principle really does have some important role in our perception of what balance is best.*

I know there's a lot of speculation about Bartók's use of Golden Ratios at multiple levels in his intricately structured works, but for my purposes, I decided it would make more sense to go with the very familiar strains of Beethoven's 5th, a work my students will be studying later anyway. I'd found this recap of Derek Haylock's speculations about the Golden Ratio in the symphony's first movement, and his concept seemed legit enough to present as an example of how a Golden Ratio event might work, whether coincidental or not. But as the day got closer, I still had reservations.

Actually, for starters, I think it's worth mentioning a basic issue with applying the Golden Ratio to music. There is ample evidence that "golden proportions" are appealing in the visual dimension, although there are plenty of confounding factors as to why that might be. But it is a big analogical step to say that the same principles that spatially define how we view the balanced design of, say, the Parthenon will automatically be felt across time. When one looks at the Parthenon, one can see start point, golden division, and endpoint all at once. In a musical work, one would only be able to feel the rightness of a golden division in retrospect. Which, of course, would be possible, but that's a still a significant perceptual difference.

But, let's say dividing a work or a subset of a work into golden sections does have some perceivable value. Derek Haylock's BIG IDEA about Beethoven's 5th goes like this: [I'm going to refer to the famous duh-duh-duh-DUHHHH as "Motive X."]
  • We hear Motive X immediately in m.1. 
  • Motive X makes its big Recapitulation reappearance at m. 372
    • This is actually assuming the 124-bar Exposition section is repeated. So if you look at a score, the Recapitulation begins at m. 248 [248 + 124 = 372].
  • The final appearance of Motive X occurs at m.602, which is to say right after 601 full bars.
  • Divide 601/372 and you get: 1.615, which is very close to THE Golden Ratio, 1.618.
Haylock actually does the calculations a little differently. As far as I can tell, he counts the last Motive X appearance as starting at m.477 (the 601st bar with repeat), which doesn't quite make sense. Beethoven, as it happens, extends the normal 3 pickup notes of Motive X (G-G-G-Eb) by three extra bars, which gives us 15 repeated G's before the arrival on E-flat (see below, using Liszt's piano reduction). That starts in m.475, so the final occurrence of Motive X should either be identified as a stretched-out X starting at m.475 or, more to my liking, at m.478 where we actually get three G's which land on an E-flat in m.479. Anyway, according to Haylock's calculation, there are 600 measures before the last occurrence of X. He multiples that by the approximate Golden Ratio division of 62% and gets 372 exactly, which sure does seem nice. 

I go through all of those details in part to make the point that there are lots of ways to crunch the numbers to get what you want. Notice that Haylock gets the perfect m.372 by applying the imperfect 62%, rounded up from the more truly golden 61.8%. Using my method, I could take the fact that the final Motive X starts on m.602 and when I divide 602/372, I get a perfect 1.618. But that's really cheating because the arrival at m.602 signals the end of only 601 complete measures. From a timeline perspective, the beginning of m.1 should be labeled as 0!

And if we're thinking in terms of timelines, it's worth noting that this movement includes multiple fermatas and an extended, out-of-time oboe cadenza in m.268 [m. 392 with repeat]. All of these factors would change the timing proportions in terms of what we actually hear. And if we're going to discuss number crunching, we must stop to ask why Haylock decided the important division was between the first and last instances of the famous motive as opposed to the beginning and ending of the whole movement. Don't you think he would've been just as excited or moreso if the Recapitulation had happened exactly 61.8% of the way into the piece? Spin the numbers and proportions around enough and you're gonna find some fun relationships.

Speaking of which! As I thought about this and whether the golden division should really occur 61.8% of the way through the first movement, which would be a little beyond the start of the Recapitulation, it suddenly dawned on me that this moment might coincide with the surprising, time-stopping oboe cadenza I mentioned above. In a movement and symphony full of dazzling, innovative ideas and ceaseless energy, this sudden stoppage, where the ceaseless energy is temporarily suspended and the sound of full orchestra is reduced to a single, lost voice, this might be the most unexpected and singular moment. 

So, I started doing the research ("ratio shopping," as I've come to call it) and found right away that the Paavo Jarvi version I often use for teaching has this cadenza begin 4:14 (254 seconds) into a 6:51 (411) performance. 411/254 = 1.618 !!!!! 

[NOTE: Because this video annoyingly begins with 16 seconds of the conductor entering the stage to applause, the cadenza actually begins at 4:30 instead of 4:14.]

I think that's pretty cool! Of course, performance decisions regarding consistency of tempo, length of fermatas and length of this very cadenza will alter those proportions. I did a quick sampling of other versions, most of which placed the cadenza just a bit earlier. 
Still, I think a strong case can be made that this golden division is more significant than the "statements of motive X" divisions. Just to bring this back to the visual, here's what the sound wave proportions of the entire movement look like when split at the "golden oboe" moment.

I imagine someone else must have observed that this cadenza moment has golden ratio qualities, though the only half-baked-research citation I found is in this book, but Google doesn't show me all the pages! Also, note that you may investigate the structure of this movement with my own little interactive score/video/outline page here.

As for "ratio shopping," it is a fun game. The morning I was presenting, I woke up early and found myself wondering where the Golden Ratio might occur in some well-known bits of pop culture. What about in Star Wars? What about the guitar solo in "Don't stop believin'?" Quick Answers: In the original Star Wars (admittedly, the somewhat altered-from-original version on Disney Plus), the golden ratio, counting from the opening scroll to the end credits, occurs just as Luke, Han, and Chewbacca arrive at the prison block to rescue Princess Leia. (I was hoping for the quiet moment when Ben Kenobi turns off the tractor beam, which turns out to be about 68% of the way in). The "Don't stop" guitar solo happens about 75% through, though a song with a fade-out at the end is hard to calculate for sure. For some reason Richard Strauss's perfect song, Morgen, came to mind later, and I found that its frozen-in-time moment happens about 68% of the way in.

All of this confirms my initial suspicion that "magic moments" (golden or otherwise) will often happen between 2/3 and 3/4 of the way in, beyond the gold. But I'm sure if I spent enough time ratio shopping, I'd find more golden things.

And, if YOU would like to try some ratio shopping, I've made this handy little Google Spreadsheet. You'll need to save a copy to your own Google account, and then you can enter in timings for various works to see where the Golden Ratio would occur and to check out other timings percentage-wise.

Having said all this, it turned out I never made it as far as Beethoven in my class presentation! [I will get a chance to return to the topic with the students later.] The Fibonacci music things I ended up doing that day...will have to wait for another day here.


* There's probably a good argument to be made that the Golden Ratio better suits music that is focused on ideals of balance and elegance, so the Classical Era sonata form structures of Mozart probably fit this best. The three-part structure of Exposition/Development/Recapitulation in this period tends to feature shorter Development sections than in later, more Romanticized 19th-century structures, so that the expansion created by Development isn't in a 1:1:1 proportion with the opening and closing sections. (Remember, 1:1:1 would tend to create proportions related by thirds (2 to 1)  rather than the more golden 1.62 to 1.)

In fact, having thought I was done with this post, I only just realized that some "golden" analyses of sonata structures actually put the smaller part of the division first, dividing as follows: Exposition || Development + Recapitulation. This actually fits well with the binary form origins of early Classical sonata forms (and the fact that some of these sonatas have repeats for both Exposition and for Development/Recap). In this formulation, the Exposition is a basic structure which is then expanded on in a longer (perhaps 62% longer!) second section. See this table of Mozart piano sonatas. The idea would be that one presents ideas and then expands them by golden means. A Fibonacci Sequence, pairs from which can be used to approximate the Golden Ratio, represents just this sort of expansion as each number is greater than the previous by a distance of the previous previous number: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. A movement with a first section of 55 measures and a following section of 89 measures would, by definition, end up with golden proportions.

Note that my discussions of Beethoven's 5th and other "golden moments" are looking for a big event to happen about 62% way in, so the focus here is on satisfactory position of the most climactic or unstable moment. The focus is on the placement of what's most compelling/interesting. But now that I think about it, this might be a less satisfactory way to realize this particular principle in music than simply focusing on balancing of section lengths. I've even added a new "Early Gold" column to my spreadsheet so it will compute the perfect time for a golden division early or late.

However, this was only supposed to be a footnote about a casual encounter with the Golden Ratio, so I'll leave that question...for now.

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