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. 

• Szell, 1.612
• Bernstein, 1.609
• Gardiner, 1.606
• Kleiber, 1.605
• Karajan, 1.602
• Norrington, 1.599
• Toscanini, 1.576

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.

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* 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.

Hindemirth Day

Following on my previous post, I'm now three days beyond Hindemith's actual birthday (Nov. 16) and a day past my own, but a friend's Facebook greeting inspired me to add to my slowly growing series of "Happy Birthday" settings. Just wish I'd thought to do it on Saturday.

This post will be brief because this is a very simple setting, though I'm pleased with how well it works. The accompaniment is simply the opening four measures of the exhausting and brutal piano part to Hindemith's trombone sonata. It's a work which I played a few times back in the day and which, to be honest, had a lot to do with my own negative attitudes about the composer, although I've softened a bit on that. Anyway, I simply replaced the original trombone part with the more familiar birthday tune, added a final F Major chord, and there you have it. The piano's dotted rhythms work well, there are some pitch connections that make sense, and there's something satisfying about the piano left hand arriving at m.4 as the birthday tune is held before its final phrase.



This is already more Hindemith content on the blog than I'd have ever anticipated, but maybe it will lead to more. After all, a little less than two years ago, I finished up an unexpected series of encounters with Barber.

Here are links to more re-imaginings of the famous tune. [YouTube Playlist]

• Bruch
• Messiaen
• 12-tone
• Sondheim
• Bach (fragment)
• Ravel
• Prokofiev (see footnote)

The Man beHind a Myth

It's been awhile since I posted here for reasons I might get around to writing about some time. But I still find occasional inspiration in the odd Facebook conversation. And many of my Facebook conversations are odd. Also, many of my Facebook conversations involve me making fun of violas, Haydn, and Hindemith. I recently proposed that some of my friends (including a violist) who think highly of these two H-men should look to reconfigure Boston's venerable Handel and Haydn Society as the Hindemith and Haydn Society, replacing the annual Messiah performances with the crowd-pleasing tones of Das Unaufhörliche. Who would turn down a chance to see a German oratorio with a title that translates as "The Incessant?"

And the truth is, I appreciate plenty of things about Haydn, Hindemith, and even - horrible as it is to have to hear - the viola. But it's more fun to make fun, so when my violist friend tried to say his deep admiration for Hindemith is not just a function of being a violist (Hindemith was a violist who wrote significantly in alto clef), I said "Ha," and also created this useful graphic to illustrate the Hindemithian hypnosis that is likely at play:


A few days and comments down the road, friend violist offered this amusing graphic which alludes to the composer's intensely critical personality:

The idea of Virtual Hindemith judging immediately brought to mind a memorable Tom Cruise scene from one of my three favorite movies of all time. (Any chance we can get Cruise to star in a Hindemith biopic?) And thus, it wasn't long before I was doing my own video mashup of a stern Hindemith photo with Tom's hyper-focused delivery. My interest in musical mashups should be well-known to anyone who's read 0.3% of this blog, but it was fun to explore the mashup idea in the visual realm. I think it really works!

Impossible Peace = Impossible Piece?

In my previous post, I wrote about saturating "Amazing Grace" with accidentals as a response to a colleague's call for a hymn harmonization "rife with secondary dominants." [← What a sentence!] When I posted my response (in even cruder form than here), this colleague wrote back:

"Perfect. And then they shall read Friede auf Erden and we will be done."

I'm proud to say I knew exactly what he meant because, about 20 years ago, I was the accompanist for a chorus that was rehearsing Arnold Schoenberg's Xtreme motet Friede auf Erden. The original version is approximately ten minutes of 8-part a cappella choral writing moving back and forth between rich Romantic harmony and intense chromaticism. The music was proving to be a big challenge for this chorus (as it would be for just about any group of humans) and there was a really important rehearsal which fell on a day of heavy snow.

When word went out that that evening's rehearsal was cancelled, I decided to see if I could use the free time to create some home practice aids to help choristers learn these challenging parts; so I entered all the notes into MIDI and posted those files online. (In those pre-broadband days, posting actual audio files [like mp3s] would've required more bandwidth than was practical, but standard web browsers could play back MIDI files, which basically just provide instructions about which pitches to play (using awkward, clinky sounds).) I believe the practice files proved to be helpful, and the final performance went well as best as I recall. In the performance, the director actually opted to have the choir perform the motet twice, once a cappella, and once with the orchestral accompaniment Schoenberg had created when he realized how difficult this music was for singers.

For purposes of demonstration in this post, it actually took me a while to find a recording that actually "works" for my ears, though there are several admirable live performances that have their virtues (such as this one). Then I came upon this recording by the English choir Tenebrae - a performance which strikes me as nothing less than miraculous:

Somehow this group manages to make even the thorniest sections sound logical and transparent, and the often skyscraping soprano part never sounds strained. Based on many other recordings I've sampled, I'm sure there are listeners who prefer a heavier, richer choral sound for this repertoire, but though the "British Light" sonority isn't always my cup of tea for Romantic works, it really works for me here. [By contrast, here's a wonderful "British Light" recording of an absolutely perfect German Romantic motet which just leaves me wanting that extra bit more of overwhelming sound for the final cadence at 2:59.]

It's also worth hearing this version, by Boston University student forces, with the orchestral accompaniment. The instrumental parts really help to focus the harmonies, and in this case support a full and sumptuous choral sound that might not tune so well on its own, but which pairs well with the style.

ANY....way, what brings me here today is that my colleague's comment made me remember that I've just had these Schoenberg MIDI files sitting on a hard drive for two decades. So much potential energy!

I made a stupid joke about combining the chromatic counterpoint with a samba beat - and I haven't ruled that out!* - but I decided that what really interested me the most was simply s-l-o-w-i-n-g things down so that the ear could have more time to process some of the fast moving brain-benders. Although Schoenberg is best-known for writing fully atonal music in which traditional harmony doesn't play a role, he also was able to work within a gorgeous Romantic canvas. But the extreme demands this music makes on the singer and listener can obscure that at times. Friede auf Erden is the kind of music that almost fights against itself - which is actually kind of cool, but also problematic.

So for now, mostly all I've done is "record" this with strings (synth strings, alas) at an almost impossibly slow tempo (synth string players have infinitely long bows), about three times more slowly than it would be performed. I chose to bathe the admittedly unsatisfying string sound in a lot of reverb so that what emerges is kind of a 30,000-foot view of the piece. I also made the choice, admittedly mostly for practical reasons, to remove tempo changes and dynamics, so what's left behind is just pure counterpoint swimming in reverb - which is kind of a fun contradiction.

Obviously, the result is NOT Friede auf Erden - it's missing the poetic language, the correct proportions, the highs and lows. But I do find it to be really beautiful, beyond just as an ambient haze phenomenon. More than anything, it tends to sound a lot like Mahler, though at various points it also reminds me of Wagner, the Barber Adagio (and a general American kind of string sound), certain kinds of film soundtracks; and I do think there's some value in - pardon me for saying it - smoothing out some of the edges of Schoenberg's work. I know that kind of thinking goes against a lot of aesthetic thinking, but I'm just being honest. Part of me actually enjoys listening to this more than the original, which admittedly would sound better in a live space than it does over laptop speakers.

What it doesn't remind me of so much is Minimalism, because the harmonies do change quite regularly, but it might be fair to say that what's going on is the application of a Minimalist time-scale to music that is otherwise quite dense and boundary-pushing from a tonal perspective. I love being able to settle into each harmony and let it unfold, and though this is obviously an enormous distortion of the composer's intent, I do think it makes a case for how beautiful this music is. (Not as beautiful as THIS Schoenberg, which is perfect as it is.)

I think it's possible that Schoenberg's original falls into the world of masterpieces which are impossible to realize perfectly, which is interesting given that the title translates as "Peace on Earth" - another ideal that can seem impossible to realize. Stretching it out to absurd lengths is not a true solution, but rather an interesting way to open up the soundworld and let the listener indulge more mindfully in each passing moment.

Oh, and unrelated to anything else, this is my 600th post here at MMmusing!

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* I haven't tried adding a samba beat, but for fun, I did create this 100-second version with pizzicato strings. I'll offer no defense except that...I like it! [Yes, the impossibly fast pizzicato 8th notes are particularly surreal.]

There are no accidents - just accidentals!

Again I find myself at the blog following on some Facebook-inspired digressions.*

A colleague casually mentioned, as a topical aside, that he was listening to sight-reading exams. Another colleague, gamely avoiding the topic at hand, asked:

"Can we please focus on the important issues though... how did the sightreading tests go?" 

I couldn't agree more that this qualified as the more important issue! Colleague #1 replied:

"I need more hymn harmonizations that are rife with secondary dominants. I often let [students] choose which voice to read and there aren't many with an accidental in every voice."

And we're off!

If it should turn out that I am nothing more than a robot, the first clue might be how predictably I react to this sort of stimulus. You want a hymn harmonization that's rife with secondary dominants?!? I cannot resist such a siren song. Basically, secondary dominants are chords which include accidentals as a way of strengthening the approach to the next chord. So a harmonization saturated with secondary dominants would have lots of pitches outside the given key, which of course would make sight-singing more difficult. (A harmonization with no accidentals means all the notes would fall into the "diatonic" Do-Re-Mi-Fa-So-La-Ti pattern, a stepladder of pitches to which our ears are conditioned to relate; adding accidentals is like hiding steps where not expected...which could certainly lead to accidents.)

I'm not sure why I chose "Amazing Grace," but I suppose I was drawn both to its familiarity and the challenge it presents as a fairly simple, pentatonic tune (which means there aren't so many different pitches to harmonize). It's easy to add one or two secondary dominants to any harmonization, but as they start to pile up, each accidental pointing to a different key, the center of gravity gets wonky. The familiarity of the "Amazing Grace" tune helps in that regard, but it was fun to work on balancing these excesses in such a way that there still seems to be some direction. I especially like the bass line from m.4-8, but I find that all of it hangs together...ish. The voice-leading has some issues, but this is where I say, "hey, this was only a rainy day diversion."



Of course, there are tons of chromatic harmonizations of Amazing Grace that use various extended jazz harmonies (see here, for example**), but those are not necessarily conceived with the idea of strict four-part harmony in mind, and anyway, my inspiration for doing this was to stick to secondary dominants, so mostly that's what I did. With the exception of m.7, there are diversionary accidentals in every measure, mostly doing secondary dominant kinds of things.

Oh, and as follow-up to my previous post, the final cadence gets lost in a Tristan haze....

AND, this harmonization sparked a comment which sparked another, very different creative concept which I will reveal...in a day or two!

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* And here's a Facebook-inspired digression: I think Facebook is fantastic, and though it's certainly caused its share of problems, I tend to blame users at least as much as the company. I have wonderful interactions and conversations there and don't find it all that difficult to avoid the negativity. The secret trick is to avoid the negativity. And to seek out interesting conversations. Because of the far-flung connections it enables, there are topics I might otherwise never get to engage were it not for Facebook. (This is not a defense of its policies or motivations, just a suggestion that one doesn't have to get sucked into its darker aspects.)