This article explores the mathematical and more specifically the musical products of a very simple equation. In that exploration, we touch not only mathematics and music, but art, architecture, nature and philosophy; so those who are usually squeamish about mathematics are encouraged to read on.
Most readers who made it through high school algebra should be familiar with quadratic functions and the parabolas described by these functions on the x-y plane. For those who have forgotten, a parabola looks like this:
Parabolas are seen not only in high-school math classes, but often in nature as well. Among the most exquisite uses of parabolae can be found in the architecture of Antoni Gaudí. I had the priveledge of seeing many of his buildings and spaces in Barcelona, including this magnificent example of parabolic architecture:
But (as usual), I digress. For the remainder of this article, we will focus on a particular class of these functions, called logistic functions:
f(x) = ax(x-1)
Logistic functions have roots and 0 and 1, and describe a downward facing parabola (or “water-shedding parabola” in the parlance of my high-school pre-calculus teacher). The peak of this parabola depends on the value of a, and as we will soon see, this is the least of the interesting properties dependent on a.
Now, instead of simply graphing the function on an x-y plane, apply the output of the function back as the next input value in a process known as iteration:
xn+1 = axn(xn–1)
This is a fancy way of saying “do the function over and over again.” What is interesting is that for different values of a, one will get different results. For low values (where a is less than one), repeated iterations get closer and closer to zero. If a is between 1 and 3, the it will end up at some value between zero and one. Above 3, things get more interesting. The first range bounces around between two values, as characterized below:
As a increases, eventually the results start bouncing among four values, and then eight, then sixteen, and so on. These “doubling periods” get closer and closer together (those interested in this part of the story are encouraged to look up the Feigenbaum constant). Beyond about 3.57 or so, things get a little crazy, and rather than settling into a period behavior around a few points, we obsserve what is best described as “chaotic behavior,” where the succession of points on the logistic function varies unpredictably.
It is not random in the same way that we usually think of (like rolling dice or using the random-number generators on our computers), but has rather intricate patterns within – those interested in learning more are encouraged to look up “chaos.” This chatoic behavior can be musically interesting, and I have used the chaotic range of the logistic function in compositions, such as the following except from my 2000 piece Spin Cycle/Control Freak.
One can more vividly observe the behavior I describe above as a graph called a bifurcation diagram. As the values is a increase (a is labelled as “r” in this graphic I shamelessly but legally ripped off from wikipedia), one can oberve vertically the period doubling where the logistic map converges on a single value, then bounces between two points, then four, then eight, and so on, until the onset of chaos at approximatley 3.57.
There are tons of books and online articles on chaos, the logistic function, and its bifurcation diagram. Thus, it’s probably best that interested readers simply google those phrases rather than suffer through more of my own writing on the topics. However, I do have more to say on my musical interpretations of these concepts.
Given my experience in additive synthesis and frequency-domain processing (if I have lost you, then skip to the musical excerpt at the end, it’s pretty cool), I of course viewed this map as a series of frequency spectra that grow more or less complex based on a. I implemented this idea in Open Sound World. using the logstic function and its bifurcation diagram to drive OSW’s additive synthesizer functions. The results were quite interesting, and have been used in several of my live performances. I use my graphics tablet to sweep through different values of a on the horizontal axis as in the bifurcation diagram:
Photo by Tiffany Worthington
The resulting sound is the synthesis of frequences based on the verticle slice through the diagram.
Click here to listen to an example.
In the periods of chaos, the sound is extremely complex and rich. Below 3.57 and in the observable “calm periods,” the sound is simpler, containing on a few components forming somethin akin to an inharmonic chord. In true chaotic fashion, small movements along the horizontal axis result in dramatic differences in the spectrum and the timbre. The leads to a certain “glitchy” quality in the sound – one can practice control over time to make smooth transitions and find interesting “islands of stability” within the timbral space.
I have used this simple but evocative computer instrument in several performances, including my 2006 Skronkathon performance as well as my work last year with the Electron SAlon series. I have really only scratched the surface the possibilities with this concept, and hope to have more examples int the future.
As many readers are probably aware, cows are considered sacred in Hindu culture, and a family cow is treated in many ways as a prized pet. Cats in India don't quite have the same status. Not only do most Indians not consider them as appropriate pets, but some also regard them in a suspicious, even superstitious manner (as have many cultures throughout history, sadly). Thus, most cats are street animals that generally must fend for themselves. As many of us know, cats are supurb hunters and scavengers, so many do manage. I tended to see cats near food establishments, this near-white cat was wandering the dining area of a dormitory in Delhi we stayed at for a couple of nights, begging for scraps (which I of course provided). The grey-striped fellow in the photo below managed to find a good location at the cantina in one of heavily touristed palaces in the city of Jaipur. That probably accounts for its more healthy appearance:
I've been a little under the weather the last couple of days. Actually, more than a little. Mostly just resting and recuperation, with Luna?s help, of course. That is, when she?s not too busy looking out the window or hunting bugs, which she spent a lot of time doing yesterday and today. There was one large fly in the window that she became quite obsessed with, tracking it and swatting. Took a while for the mighty hunter to finally catch her quarry. 
I arrived in Lower Manhattan at about 8:38. Walking up Dey Street I decided to stop for coffee and walked across Church Street onto the Plaza of the World Trade Center. I called my father on my cell phone and we were talking which kept me from entering the building. I was on a bench right in front of the WTC 1 and turned slowly for no real reason and saw the entire plane hit the tower. I saw the wing extended from the building on the south side and a large explosion. Then smoke. Then everything was frozen, very still, with a perfect New York blue sky framing the backdrop of explosions. I ran when the glass and metal begin to fall from the sky hitting all around me…At that point I began running north. I got to East 4th Street completely dazed. On Houston St. I saw the WTC 1's needle crumble to the ground and heard on blasting radios that both of the towers had fallen. From there I made my way to the Williamsburg Bridge and made my way back home. Tens of thousands of people were walking across the bridge, a surreal exodus no one was quite prepared to cope with…
Having followed the redevelopment story over the past few years, I am used to the new design, and I am mindful of the controversies around the original Twin Towers in the late 1960s as being too “boring” and somehow an affront to New York's skyline, and in the long run nothing could have been further from the truth. The building designs have attracted a certain amount of controversy, but nearly as much as the design for