Saturday, February 29, 2020

Notes that float

As the Internet, social media, and blogs have evolved over the years, it's likely that more people now see this blog (if they see it at all) on a mobile device than in its true webpage form. This is a little sad for me since I once spent a lot of time tweaking the Blogger templates to create a look I really liked. A typical mobile trip to MMmusing won't even show cool things like my should-be-patented "Multimedia Musing Machine," nor will it show my homemade Bachground wallpaper.

I also recently remembered that the web version of the blog has long featured a link to one of my favorite recordings of my own playing. I opened a 2012 recital (wow, so long ago!) with the Allemande from Bach's Partita No. 4 in D Major. I'd fallen in love with this piece after reading Jeremy Denk's wonderful 7-part blog series about it back in Aught-Seven. (Denk's series begins here.) 

I continue to find it unique among Bach's works for its meandering qualities, melodic but not in a particularly memorable way (meaning there's not really a singable tune). Rather, the highly ornamented right hand rolls along with a wide variety of rhythmic figurations and a sense of harmony that seems more free-floating than "typical" Bach. It manages to be improvisatory and intricate. And inspired.There's nothing else quite like it.

I have nothing against the other six movements of this suite, but this Allemande definitely stands on its own. (It is not as out-of-scale as the mighty Chaconne is in the D Minor violin partita, but it is unusually long and winding for an Allemande.) I also have nothing in particular against other recordings of this piece, but my own performance happens to say just what I would want this deeply personal music to say, so I figured it was time I posted it on YouTube. As it is a live performance played in front of an audience from memory, it is certainly not perfect, but it's really not the kind of thing that needs to be perfect.

To accompany the recording, I prepared my own engraving of this ornate score in Lilypond, a rewarding creative challenge in its own right. Because this music has such a linear quality, as if Bach is inventing each new melodic diversion on the spot, I liked the idea of a continuously scrolling visual. Lilypond has a function which makes it easy to create a score that extends horizontally for exactly as long as needed. Imagine a short sheet of paper (only needs to accommodate the height of two staves) which stretches out across the room. 

I also allowed Lilypond to let the music spacing breath a bit more than I did when preparing a companion "normal-sized" copy. When printing to paper, all of the elaborate rhythms (some measures include as many as 24 notes across) have to be made to fit within orderly staff systems and with some degree of logic across multiple pages. For a long scrolling video score, it's actually better if the measures are relatively uniform in length so that the scrolling speed doesn't have to vary too much, whereas normal engraving will usually take more advantage of less dense bars by compressing their spacing.

In some respects this is a work-in-progress, but I was happy to develop some more techniques both working with Lilypond and in creating scrolling animation. I have some other projects up my sleeve that will require more of the latter. 

But for now, I'll end this rather discursive post with this sublimely discursive bit of Bach:

A few postscripts:
  • I am aware that the notes in the video are rather small. This is in part because using bigger notes would mean faster scrolling which can get a little dizzying, but also because I'm more interested in the visual than in the specific details. As I've written many times before, although there are plenty of nice ways to accompany music with visuals, I don't think anything beats the beauty of a musical score. Obviously, it provides a very close analog to the sounds being heard, but I find that listening while watching notes often helps to sharpen the ears. [To be totally honest, I wish the notes were a little bigger, so I might re-do this, but it takes time since quite a few synch points need to be entered manually to make the music flow properly.]
  • Though still a work-in-progress, you can see what my engraving looks like on paper via this download.

Monday, February 24, 2020


Today just happens to mark the 13th anniversary of this blog. (MMmusing is a teenager!) And 13 just happens to be a Fibonacci number! So what better way to mark this occasion (and finish up this little blog series) than with a bit of Fibonacci fun?

In my last two posts, I talked about encounters I had with the Golden Ratio and the Fibonacci Sequence while preparing to lead some musical sessions as part of a broader academic day focused on those topics. I've already written about looking for the Golden Ratio in Beethoven and about writing a little vocal warm-up using Fibonacci numbers.

The latter is based on a simple diatonic pattern in which the primary notes of a scale (the "white notes" in C Major) are used as stepping tones for scalar ascents of 1, 2, 3, 5, and 8. Naturally, I was also interested in exploring larger numbers in the series, and though it's not practical to sing a range covering 55 notes, the piano keyboard can easily accommodate that. EXCEPT, I was actually a little surprised to realize that the 88 notes of a piano include only 52 white notes.

So, although I like the way the diatonic Fibonacci patterns emphasize the 3rd, 5th, and octave of a major scale, I set to work organizing the full chromatic complement of white AND black notes, which easily accommodates the Fibonacci 55. Sadly, the next number in the series is 89, and that's one note too far, but I also feel there might be diminishing returns if the number stretches out too far. (See postscript #1 to test this out.) It becomes difficult beyond 55 (or maybe beyond 21 or 34?) to feel the relationship to the groups preceding.

Anyway, the skeleton of my piece-in-progress is this very simple expression of Fibonacci numbers through expanding meters (Remember, the Fibonacci Sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55):

There's not a lot of real "composing" going on there, but I then added a right hand part to emphasize some of those metrical tricks, build suspense, and also help support a vibrant ending in which a big C Major chord settles out into longer and longer notes values based on...well, you can figure it out. This little etude is called "Fibonacci Frenzy." (Yeah, yeah, yeah, I should record it like a real pianist and not subject you to this robo-performance, but I am appreciating the way Noteflight allows for the embedding of scrolling notation.)

 Since there are only a couple of hours left on this blogday, I'll just add two quick postscripts.
  • In response to my Fibonacci vocal warm-up, Twitter friend Dan introduced me to a very entertaining and ever-expanding piece based on a Fibonacci-esque sequence: Narayana's Cows. This music definitely pushes the limits of how far out we can perceive these expanding connections.
  • Rather coincidentally, this Sound Field video crossed my path a couple of days ago. Early on [0:21-0:40], one of the hosts indulges in some of that mystical "some music sounds right yada yada because...well...Golden Ratio!" As often happens with such discussions, there's not much substantive probing of what's going on and how valid this is aesthetically, though I appreciated some acknowledgement [3:28 - ] that Golden Ratios might just be the kind of thing you can always find if you look in the right place. But, most notably for me, the same host, Nahre Sol, ends the video with a little composition she wrote using both the Golden Ratio to govern the structure and some Fibonacci numbers to generate some riffs. (Oddly enough, the complete composition is not played in this video, so we don't get a chance to experience the Golden Ratio!) If you like hearing numbers dance, you might enjoy this.

    Happy MMmusing Day!

Saturday, February 22, 2020

Warming up with math

In our last episode, I recounted some experiences thinking about the use of the Golden Ratio and the Fibonacci Sequence in musical structures. I focused in that post on the question of finding a mathematically "golden moment" in a work like Beethoven's Symphony No. 5, but for the teaching task that inspired all of this, I never even got around to Beethoven. (My students will get to hear me talk about the Golden Ratio in the Beethoven later in the semester.)

On the special "Golden Ratio" day in question, I was more focused on providing some direct musical encounters with the ratio that students could feel for themselves. We actually first experimented with seating the students in Fibonacci rows as follows:

X    X
X    X    X
X    X    X    X    X
X    X    X    X    X    X    X    X

Then we tried some counting-off games to help them feel how each row was the sum of the two rows in front. 

But I had also started playing around with composing some music that used these relationships, specifically: 1, 1, 2, 3, 5, 8, 13, etc. It wasn't too long before I'd come up with the following:

It actually had begun with just numbers as lyrics (1, 1, 1 2, 1 2 3, 1 2 3 4 5, 1 2 3 4 5 6 7 8), but I soon realized it might make an interesting vocal warm-up, so there you go. Of course, in real life, I probably would just notate the first three bars as one 3/8 bar, but these meters and the beaming patterns help to reinforce the additive process that's going on. Here's what it might sound like sung by creepy robots (but I have found it to be a fun and useful choral warm-up with humans):

And here's what it looks like notated fully. (If the embedded score doesn't work well, go here.)

One of the surprisingly rewarding things about doing this was realizing that the last three measures ascend to the 3rd, the 5th, and the octave, and of course, a major triad is constructed from root, 3rd, and 5th. That is pretty cool, although if you include the 2/8 bar, that one ascends to the 2nd scale degree which doesn't fit so nicely into the triad. 

Notice that, unlike the subtler principle of putting a structurally important musical moment 55/89 (61.8%) of the way into a piece, this use of the Fibonacci Sequence is right there on the surface. For better or for worse, the performer and listener can directly feel the asymmetry caused by the additive process. When a composer, intentionally or not, puts a surprisingly subdued oboe cadenza 254 seconds into a fiery 411-second movement (see end of previous post), it's highly unlikely that anyone could experience that golden relationship so consciously, though we may at some level feel its rightness. Of course, some would say that is part of the magic.

I have one more composition that came out of this whole experimenterience...and I'll reveal that soon.

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.