By: Keith Devlin @profkeithdevlin

Regular readers, brace yourself: I am about to tell you about a new musical interpretation of the Fibonacci Sequence. Yes, I am going to describe how a group of musicians—two musicians in particular—set out to capture in music the perceived beauty in the Fibonacci Sequence (and the asymptotic Golden Ratio). 

That’s right, you just heard that from one of those spoil-sport mathematicians who keeps pointing out that most of the claims you find in popular articles, blogposts, and videos about the Golden Ratio (or the Fibonacci Sequence) and beauty are pure fiction—urban myths, whose origin can be traced back to a German author called Adolf Zeising, who in 1855 published a book titled:

A New Theory of the proportions of the human body, developed from a basic morphological law which stayed hitherto unknown, and which permeates the whole nature and art, accompanied by a complete summary of the prevailing systems.

That wretched, fanciful book is where the claim first appears that the proportions of the human body are based on the Golden Ratio. For example, taking the height from a person's naval to their toes and dividing it by the person's total height yields the Golden Ratio. So, Zeising claims, does dividing height of the face by its width. 

Not content with postulating connections of the Golden Ratio to the human body, Zeising went on to imagine connections of those human-body proportions to ancient and Renaissance architecture. 

All of which would have been justified, had he or anyone else then or since come up with a shred of evidence to support his claims. But that has not occurred. On the contrary, what has happened is an endless stream of classroom studies of first-year psychology undergraduates being subjected to tests demonstrating that (modern) humans absolutely do not find the Golden Ratio a particularly attractive proportion. (As to the ancient Greeks, say, there is no evidence they did either. And while I am at it, let me point out yet again that the Parthenon does not exhibit the Golden Ratio. Yes, people have measured it. Thickly drawn lines on photographs can hide a multitude of false beliefs.)

The trouble is, measuring anything as complex as the human body, it's easy to come up with examples of ratios that are very near to 1.6 or 5/3. It requires scientific justification to jump from a measurement close to 1.6 to a conclusion about the Golden Ratio. In the case of Fibonacci numbers occurring in flowers and plants, there is such an explanation. In that sense, you can claim the Fibonacci numbers and the Golden Ratio “describe some of the beauty in nature”, though it is more accurate to say they describe aspects of the way flowers and plants grow. In the case of the Nautilus shell and spiral galaxies, however, measurements of the spirals shows that, while they do indeed approximate constant-angle spirals, that angle is not even close to the golden ratio, so there is no connection to Fibonacci numbers.

I presented some of the evidence against the better known Golden Ratio claims in a Devlin’s Angle post in May 2007 titled The Myth That Will Not Go Away, where I also recommend an excellent book by Mario Livio that lays out a much more extensive array of damning evidence.

So what made me throw my weight behind a new album by New York City-based composer, producer, and audio engineer E Scott Lindner, titled In Flowers Through Space? 

It’s an experimental concept album based on the Fibonacci sequence, recorded at Lindner’s Pinch Recording studio, and released on February 28 under Ropeadope Records’ Infinity Gritty imprint. What makes this relevant to MAA members is that the entire development and production was based on the Fibonacci numbers.

For instance—and this is just one of many examples—the number of classical and jazz musicians on each of the nine tracks (the first being, appropriately, an empty track) increases to mirror the Fibonacci sequence through to 21, namely 0, 1, 1, 2, 3, 5, 8, 13, and 21. 

But the number of instruments on each track is just one example. Lindner and his partner in the album’s development, Adam Ahuja, a keyboard/synthesizer player and the owner of the Infinity Gritty records label, based everything they did on Fibonacci numbers.

To start the composition process, synthesizer player Jim Panels wrote code for a “Fibonacci calculator” that took frequencies in the human hearing range (20hz-20khz) and translated them into musical notes, which they used to convert the Fibonacci numbers into musical notes.

They then used that sequence of notes to mark corresponding keys on the piano with tape, and used those notes as the basis of the first track, “1(n1)”, which Ahuja performed, and again in melodies and solos throughout the entire album.

For the composition “8” they wrote the parts for the different instruments on index cards, put those cards into a piñata, and had the musicians break the piñata to determine the order in which they would be sequenced, thereby introducing an element of randomness on top of the underlying numerical sequence. (Read on to see what was going on here.) You can listen to “8” here.

They even went as far as choosing the recording dates according to the Fibonacci sequence, recording on the 5th of the month, the 8th, the 13th, and the 21st.

The result is a diverse collection of tracks that blend classically-inspired musical forms with contemporary jazz performance. 

 “This is the most complex and emotional album I've made, and it's completely guided by this mathematical construct [the Fibonacci sequence]," says Lindner. 

Lindner’s publicist, Justin Schmidt of the Missing Piece Group, based in Seattle, says it is “a bewitching composition that melds themes of mystery, nature and science into a spiritual celestial journey,” and describes it as “creating an evolving and cinematic listening experience that calls to mind the work similar innovators such as Kamasi Washington or Johnny Greenwood.”

Readers can of course listen to, or sample, the album and decide for themselves whether they like it. I do, but tastes vary.

But why am I writing about it here, and not dismissing the Fibonacci claims as yet another instance of “Fibonacci Folly.” Fortunately, I was prevented from jumping to that conclusion by watching a (not yet released) 66 minute documentary film about Lindner having the same title as the album. What I found was a fascinating attempt (successful to my mind) to base creative art on mathematics. [Pending release of the documentary, you can view the trailer here.]

I note that Lindner is not the first to try to create beautiful experiences from the Fibonacci numbers. Indeed, in ancient times, Sanskrit poets played with the sequence in connection with prosody, though since they looked only at the first few numbers in the sequence, it is perhaps more accurate to say they were looking at the rule for generating the sequence, rather than the numbers themselves. In any event, the Sanskrit writings appear to be the first written reference to the rule.

Again, inspired by the relatively recent notion (debunked as indicated above, until/unless someone provides solid evidence to the contrary) that the Fibonacci numbers embody some kind of human aesthetics, in the 1940s the famous French architect Corbusier advocated, and allegedly used, the golden ratio in architecture.

Likewise, there is some evidence to suggest that the musician Claude Debussy used the golden ratio in some of his compositions. (There are some patterns that can be discerned as embodying the golden ratio, though it is not clear if they are intended or spurious.)

There are also examples of Fibonacci numbers being embodied in Twentieth Century painting, though frequently-cited claims that works by earlier, famous artists are Fibonacci-inspired are all wishful thinking.

What is different about Lindner, however, is the way he—more accurately, he and Adam Ahuja, working together—used the Fibonacci sequence. 

Ahuja describes the process in the video. Lindner was, he says, casting around for ideas for his next album. Ahuja, who through his on-camera comments reveals himself to be a math and science geek, thought they should start with a single, well-defined concept, and suggested the Fibonacci sequence, which, he says, always fascinated him.

As might be expected, given all those golden ratio and beauty claims that permeate the Internet, he had bought into the myth, as did Lindner. (As did I too, for the early part of my secondary career as a mathematics evangelist!) For instance, in the documentary Lindner claims that the Fibonacci numbers capture some of the beauty of flowers, which is scientifically correct, but he also claims that they are observable in spiral galaxies in space, which, as I noted earlier, is not the case.

So the mathematical and scientific grounds on which Lindner’s project rests range all the way from the good, through the decidedly shaky, to the just plain false. But this is where, at least for me, the story gets really fascinating.

Listening to the music and watching the documentary video, what becomes clear is the way the album derives from the entire golden ratio myth. Whether this or that particular claim about the golden ratio is valid or not is not important; what drives the process is the musicians’ shared belief in the myth. (And remember, there definitely is truth in some of the claims, including the one about flowers, which is the key term in the title of Linder’s album.)

What gripped me was seeing how a group of highly talented, experienced musicians came together to create music intended to embody the Fibonacci numbers. Their success (listen yourself and check it out) was surely dependent on their belief that the Fibonacci numbers capture the beauty found in nature, and indeed in Creation itself. But the reality, I suggest, is that the musical beauty in the final compositions came from their creativity as musicians, not some underlying power of a particular sequence of numbers. It is a story of collaborative human creativity, with a mathematical concept playing a catalytic, guiding role. The beauty comes from the musicians.

Ahuja, in fact, comes close to saying this himself. He observes (8:40 in the video) that the creative process they embarked on could be “something cheesy. Or it can be something real … and interesting … something you learn something from.”

The question they asked themselves was, “How can we take this mathematical construct and turn it into music?” he reported.

“Take the beauty of nature, do we get the same kind of beauty when we turn in into music?”

He compares the creative process they developed and followed to the creation of the universe from the Big Bang, guided and ruled by the mathematical laws of nature. Complexity emerges over time, occasionally shaken up by randomness. (Hence the piñata exercise, shown in the video.)

“Everything follows this principle [the Fibonacci rule],” he observes. “This is that special formula, this is that source code that we are working with.”

“Our basic laws of nature are fixed in this particular universe, so we are saying ‘This is the law. Now that you have the law, everything follows from there.’ ”

As you watch the evolution of the album in the documentary and listen to the commentaries of Ahuja and some of the musicians, you recognize the major role played by the Fibonacci numbers. They (and the formula that generates them) are the laws of nature in the musical universe they seek to develop, the glue—the only glue—that holds everything together. But the creativity, and the true beauty of their final compositions, comes not from a formula but from a group of talented humans, looking for way to embody a formula.

In a sense, any mathematical formula or concept would have worked just as well—provided everyone involved believed it was special and embodied some hidden secret of the universe. For this is a story not about how mathematics captures aspects of the universe and of life, but about human creativity and the role played by mathematics in our culture. As Justin Schmidt commented, the development process Lindner and Ahuja followed resulted in “a bewitching composition that melds themes of mystery, nature and science into a spiritual celestial journey.”  All the crucial key words in one sentence.

[A somewhat similar phenomenon can be observed in the financial world with the success of the “Fibonacci Trading” method, which is driven by the shared belief that searching for appearances of the Fibonacci sequence in the financial markets can lead to insights into how to trade successfully. Insofar as this approach leads to any success, it surely comes down to the fact that enough traders buy into the story. In a system with tight feedback loops, like the financial markets, that can result in an edge for those who follow the rule.]

From the perspective of mathematics, this is then, as I remarked above, not the familiar story of how mathematics can capture structure in the universe, in life, and in human experience. Nor is it the usual story of how we use mathematics to design and build artifacts and systems that somehow reflect those captured structures. It is, rather, a completely new take on both of those stories.

Most uses of mathematics are outside-in. We take some aspect of reality and strip away everything until all we are left with is an underlying abstract structure that we can analyze and manipulate until we get to a stage where we can deduce results and feed them back into the world in the form of action. A classic case are the very counting numbers where the Fibonacci numbers can be found. Those are the result of an abstraction from the human ability to recognize two discrete collections as having the same “size,” first carried out in Sumeria around 10,000 years ago. Numbers arose as the what that equivalent collections have in common.

In Lindner’s case, however, mathematics is used inside-out. He starts with a mathematical concept, an abstraction living in his mind, and uses it to scaffold the creation of music—music that emerges from the collective minds of the musicians.

As I said earlier, I recommend you to listen to, or at least sample, the album and see what you think. You may or may not like it; that comes down to taste. But you should definitely watch the documentary video to see how the album was put together, and how a group of highly talented professional musicians interact with a mathematical concept and use it to guide a composition. The similarities with, and the contrasts to, the way professional mathematicians interact with mathematics in their creative work is fascinating—and occasionally revealing.

FOOTNOTE: Regular readers will know I worked on a project (somewhat) similar to Lindner and Ahuja some years ago, teaming up with a Santa Cruz, CA based choral group called Zambra to interpret mathematical equations in song. I  devoted two Devlin’s Angle posts to that project: January 2012 and April 2012  and published the entire album, Harmonious Equations (prose plus singing) on my Stanford homepage. As explained in the Harmonious Equations Performance Guide, we likewise looked for different ways to embody the mathematics in the music and the lyrics. [We also collaborated with the dance group MoveSpeakSpin, led by Karl Schaffer and Erik Stern, to add dance interpretation, but unfortunately did not have the funds to make a video recording.] With that project in my background, I was clearly well disposed to take a close look at In Flowers Through Space. I’m glad I did.