Modernism

Weekend Cat Blogging #67

Luna blends in perfectly with her dad's decor while getting warm on a table above our main heater. The temperature this morning was 49F (10C)! That is a travesty in September, usually the hottest month of the year along the California coast. It's not helping my recovery from whatever ailment has befallen me this week, either.

Anyhow, being stuck home gives me plenty of time to host Weekend Cat Blogging #67, taking over from last week's host, chefsarahjane. Assuming I'm feeling better over the weekend and things warm up a bit, I'll be spending as much time as I can outside – but that shouldn't stop me from posting updates.

So send us your feline articles and photos; either leave a comment on this post, or send me a message. Bonus points if you can fit into the cats-with-music-and-art theme of this forum.







The Logistic Function and its Discontents

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.

A perfectly clear day…

Today was a warm, perfectly clear day, notably absent of the fog, cold and wind of the past week…

It seems like September 11 is always a bright, clear day, both here in northern California as well as back in New York. It was certainly a clear day five years ago as well…

There are certainly a lot being written online on this fifth anniverary of 9-11 – some of it genuine (particularly from New Yorkers, though not limited to them), a lot of it propaganda, or phony sentimentality or “patriotism.” I would stay clear of the whole thing if I didn't think I had something unique to contibute…

I begin with quote from this view from the WTC plaza that I read this afternoon:

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…
I never thought I would work in the World Trade Center. But I enjoyed my work there. Like all artists, I needed the money, but I also made many friends and learned a good deal about other parts of life, other skills that I never knew I possessed.
The views of the harbor were magnificent and inspirational. I remember ending long days by looking out the windows and feeling so very refreshed and glad to be in such a beautiful city like New York.
But that view is gone and so much has changed. Though I have felt such anger and frustration, more that ever in my life, I know I will survive. I made it out alive for a reason.

At the same time, I was 3,000 miles away in the totally boring and pathetic Bay Area town of Fremont, arising from bed an hour or so later to face an ordinary day at work 40 miles away along with daily challenges of my unravelling relationship. That was all, at least temporarily, swept away by what I saw and heard on CNN (only after receiving an email exhorting everyone to turn on the news). I didn't get into work until sometime in the afternoon, and then spent much of the day online looking at photos; I continued to wade through photos well into the night after returning home.

Things like this were not supposed to happen, not to New York, not to its iconic and familiar skyscrapers, not to the indistructable momunments of the modern world. Whether it was during my happy years in Berkeley or the miserable year in Fremont that was only beginning, I'd always be able to go back to New York – and suddenly that was no longer something I could count on, everything was mortal, and I was “guilty” somehow for not being there…

I did make it back to New York in Novemember, 2001, and of course had to see ground zero for myself. This is a photo I took looking into the site, from the north (Varick Street, I think):

In the longer view, the thought I have kept coming back to was the idea that something so beautiful (the World Trade Center complex, the New York City skyline, the emblems of 20th Century modernism) were destroyed by something so ugly (Islamic fundamentalism, the tribal and primitive past). Having not lost anyone personally on 9-11, I am able to think about things like this, and how what I want in return is the restoration of modern ideals. I will leave the political and cultural aspects aside for another post (I'm sure I'll get back to it again), and focus for this anniversary on the aesthetic and architectural.

The Skyscraper Museum in New York, which is hosting a commemoration of what it labels the “original World Trade Center,” lends its support the idea that the age of the vertical metropolis is far from over. In that spirit, the Lower Manhattan Development Corporation (LMDC) unveiled the latest design for the new World Trade Center complex, including the new major buildings and the memorial:

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 the memorial. Any design certainly has it's imperfections, but as it stands it is perfectly in harmony with the modernism of both the original and new World Trade Center complexes, and it includes the most important generally-agreed-upon elements such as preserving the footprints of the original towers, and even incorporates a piece of the original retaining wall (illustrated in the picture to the right). Certainly, the criticms have their place, notably the concern by 9-11 families for the victims' names. But that's where their “rights” end, at least as far as the restoration is concerned. I mean no disrespect to the people who lost loved ones, but if we followed the advice of some outspoken victims' relatives and the opinions of many unaffected people in Middle America, such as the author of this article (he can't even get his dueling memorial-advocacy websites straight!), lower Manhattan would be overrun with kitch, over-sentimentality and insipid tributes to phoney patrotism. That may be fine for Oklahoma City, but not for New York! It's bad enough that we have to live with middle America's patron saints of mediocrity as our leaders along with their policies that make our great cities targets for more terrorism, let alone have to deal with their ideas of what makes a good memorial…But, I did say I'll save the politics for later. So for now, I leave with with another quote from that original site I references that discusses what not only it's author (Sami Plotkin) but indeed many New Yorkers might have wanted:

None of us wanted to see [the twisted structure of the 500 foot wall] go. We have been comforted over the past two weeks by its presence. We have marvelled at its strange beauty, as it rose from the ruins, an intricate lacy network of sheer strength; a stoic remnant that survived the blast and struck us with its splendor in the golden light of sunset, as bright rays glanced off the building behind, glinting between its metal beams and through the great accidental window which was rent through at just the right height. It was an image of alarming beauty, and the serendipity of its perfect composition was significant to us. In the most simple and sentimental way, the evening light shining through that accidental window was for us a ray of hope. Although we all understood that for now it must come down, whispered questions eddied through the crowd as we wondered, would the pieces be saved? Does the Mayor know that we have already begun to see this as our monument? For we have. That mangled and beautiful skelet! on was a monument that spoke more poignantly of New York's pain and resilience than any artwork we could have devised.







Thoughts on last night's performance

In this article I review my performance last night at the plug:dos headphone festival in San Francisco.

First, the venue itself. 5lowershop is in a warehouse near the junction of highways 280 and 101 in San Francisco. It’s at the edge of the Bernal Heights neighborhood.


The venue and its surroundings have that seedy edge-of-the-city feel that I probably wouldn’t want to live in but nonetheless often find intriguing and romantic. It’s just another part of the quintessentially “modern” world.

The interior matches the exterior, a jumble of areas within the warehouse, including the main performance area. The space is quite porous with the outside, and I noticed several cats wander though, including the grey fellow and a small black-and-white kitten. They were presumably feral cats attracted by the warmth, activity and possibility of food. Feral cats are an inevitable part of urban environments, but it’s still heartbreaking to see them this way. I was also concerned for them because of the dogs that were present, fortunately the dogs seemed to be pets and quite mellow.

The atmosphere of people crowded in a warehouse listening to headphones was quite unusual to say the least. Some of the performances were quite interesting, including a serinate for voice and hammer-dulcimer, and of course several acts mixing guitar, analog synthesizers and turntable. The analog synths didn’t strike me as a good fit for headphone performance, and thus avoided them myself (as described in my article on the preparation), but they did a good job of keeping the sound within a reasonable range.

Despite the best efforts of the organizers, whom I liked and thought did a good job overall, things tended to run rather late, and I ended up going on 9:40PM, two hours after my scheduled performance. But I think it went well musically, pretty much meeting my expectations for mixing ambient and rhythmic/punctuated material while keeping things mellow for the headphones. I did bounce around and repeat elements more than I expected, but such is the nature of improvisation, reacting as things unfold.

The equipment (Dell Laptop, Emulator X, E-MU 1616m, E-MU Xboard 25) performed flawlessly. I did make a direct recording on the laptop, and will be posting that shortly. I am also planning to make that the first release in my planned podcast series.

UPDATE: you can now listen to the audio from this performance. Enjoy!

Professors, Monks, Imbalance, Pattern, Harmony and Noise

A fun, far-reaching flight of fancy for this evening's post.

I opted to enjoy a quiet day off in my yard rather than fight the inevitably nasty Santa-Cruz-area traffic. It's actually been quite productive, a lot of cleaning in the garden as well as some much needed maintenance work on the outdoor sculptures. In particular, rust management on the metalworks, and cleaning off the accumulated grime from my own fountain sculpture entitled Imbalance. I don't use a lot of chemical treatments in the water because a lot of local critters wander through and drink from the surface, notably neighborhood cats and the hummingbird that is flittering about the fountain as I write this – or rather, was around the fountain until I pulled out the camera. Anyhow, here is a post-cleaning photo (I do need to figure out something to hide that electrical cord):

In keeping with the work's title, the various columns have shifted and tilted in relation to the ground below and the weight of the stone elements.

After a mid-afternoon's hard work, I settled down to relax, enjoy a refreshing beverage and read for a bit. I am currently reading Metamagical Themas: Questing for the Essence of Mind and Pattern by Douglas Hofstadter, who is best known for his earlier book Gödel, Escher, Bach. It's actually not as heavy as the name implies. It's a series of pieces Hofstadter did for Scientific American in the early 80s, covering a wide variety of issues including patterns, creativity, language, etc. The two articles a read this afternoon dealt with the pattern and aesthetics of the music of Chopin, and transformations on simple “parquet floor” patterns as a form of visual music, respectively. While the latter was more interesting to me personally, it is the former that I wish to write about. While I admire the musicality and technical skill of Chopin as both a composer and pianist, I can't say that I've ever been a “fan.” Indeed, his music is about 180 degrees from my own aesthetically. However, I was struck in particular by one passage Hofstadter wrote:

That there are semantic patterns in music is as undeniable as that there are courses in the theory of harmony. Yet harmony theory has no more succeeded in explaining such patterns than any set of rules has yet succeeded in capturing the essence of artistic crfeativity. To be sure, there are words to decribe well-formed patterns and progressions, but no theory yet invented has even come close to creating a semantic sieve so fine as to let all bad compositions fall through and to retain all good ones. Theories of musical quality are still descriptive and not generative; to some extent, they can explain in hindsight why a piece seems goodm, but they are not sufficient to allow someone to create new peices of quality and interest.

I was reminded of an article that I read last week entitled A Monk's Musical Musings: Musical Philosophy. The author, Huchbald, attempts to argue (with all the style and sophistication usually found in right-wing political bloggers) that everything right and good in music derives from the “god-given” harmonic series, and anything that eschews baroque-era diatonic voice leading rules is somehow not music at all. In the process, he dismisses atonal music (and probably a lot of other music) as “noise.”

There are numerous ways to refute his claims (other than simply celebrating noise as music), perhaps the simplest being the rather casual way he dismisses everything other than his voice-leading rules as “simply rules based on taste which can be left to the discresson [sic] of the composer.” Well, as Hofstadter eloquently points out, this discretion and not the rules is precisely what makes for the best music. It was what separates a genius like J.S. Bach (admired by both authors discussed here) from a typical student in a first-year class on music theory. The sieve is simply too coarse, and “accepts” both the good and bad equally. One need only consider what Bach was able to do contrapuntally with the chromatic theme of A Musical Offering to see how much more there is to even baroque music than basic harmony. There is something in Bach's music that can be described and informed by harmonic theory, but it doesn't tell nearly the whole story, nor explain how he can work with both harmonicity and chromaticism with such ease.

But back to the god-given harmonic series. Simply put, the harmonic series as a set of frequencies that are all integer multiples of the lowest, or fundamental frequency. That is, for fundamental f, the harmonic series is (1)f, 2f, 3f, 4f and so on. Starting on a really low C, i.e., the bottom C of a piano, one can approximate the corresponding harmonic series as follows:

Note the use of “approximate”, we'll get back to that in a moment. The harmonic series does indeed play an important role in acoustics, the timbre of musical instruments and are perception of musical harmonies. For those who would like play with the harmonic series, a good example can be found in the “additive_synthesis” tutorial of Open Sound World – in OSW, simply go to Help:Browse Tutorials, select the “audio” subfolder and open “additive_synthesis.osw”. You can increase or decrease the contribution of different harmonics and hear the effect on the timbre of a sound. The low harmonics (2,3,4, etc.) do indeed contribute to a constant timbre, though some of the higher harmonics start to get a little “squirrelly.” As one gets into harmonics that are not simple powers of two or multiples of three and a power of two (e.g., 6, 12, etc.), the harmonics appear to play less of a role, even when they can be approximated by notes in the western diatonic scale. Moreover, these are approximates that differ from the standard note degrees in western music, the divergence is illustrated in in this chart and elsewhere. One can preserve harmonic relationships using so-called “just intonation”, which is easily to do on electronic instruments, but would require our friend to retune his guitar whenever he changed keys.

Even if one accepts the harmonic series as central to making music, there are numerous ways to use it besides diatonic voice leading. Consider the first few harmonics, which form octaves and perfect fifths. Octaves and perfect fifths are the most consonant intervals – any popular or contemporary musician will immediately recognize them as “power chords.” Prior to the baroque era, such power chords were used quite often in western music, both serious and popular, as the consonances and cadences. In serious music, there were also the Greek modes, which initially did not include the Ionic mode corresponding to our modern notion of a major scale. Indeed, one of the more common modes was the Dorian mode, which can be found on the piano by playing the white keys starting on D. It is a minor mode that can be found in some of my favorite pre-baroque music such as Josquin Des Prez's Missa Mater Patris, and is the foundation for the blues scale that informs American jazz and popular music. Despite violating most of the rules Huchbald puts forth as inherent in music, minor modes sound quite “natural” and moving to most people.

And what of music beyond the harmonic series? Many (most?) acoustic instrument timbres have overtones outside the harmonic series, and indeed some instruments (e.g., bells) can be very inharmonic. Such inharmonicity can lend itself to different ideal tunings and scales than western just intonation, and indeed we see different tunings in other musical traditiions, such as Middle Eastern, South Asian and Southeast Asian (i.e., gamelan) music. Even where we don't hear the western diatonic scale and direct allusions to the harmonic series, we can nonetheless recognize the music as music, and appreciate it in many levels, from simple enjoyment to deep spiritual understanding.

As modern composers and musicians, we often work to subvert these traditions, and indeed I found myself experimental with alternate tunings, such as 19-tone and Bohlen 833 (Golden Ratio). They have tonicities that can be quite different from what we are used to, but a good composer should be able to immerse himself or herself in them and use knowledge from other musical experiences to produce something interesting.

Well, that's enough on the Monk's philosophy and my opinions to the contrary. In subsequent articles, I would like to touch more upon alternative tunings as well as some more of Hofstadter's writings, which certainly deserve more time.