Art

Worthless Kitty Redux: Pyramid

Pyramid
September 19, 2004

Red
It was a red triangle
That formed on the side of the black pyramid
It was the red light curving from the source
Bent around the invisible smoke

It is hard enough to keep things straight
Without adding an extra dimension
But I string them all together
In hopes of something beautiful, something fantastic
For which I will find myself forever grateful
I thank my geometric farm
And turn all of my eyes elsewhere

It is maybe five in the morning,
I look over towards the window of our New York apartment
The dawn of early summer meeting the waning glow of city lights
And see her rising from the bed
In her T-shirt and boxers
Her indefinite-length brown hair irrelevant against her own shadow
This is a safe place
It is us against them
And I think we might still stand a chance

© 2004 Amar Chaudhary

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.







Webs on an autumn afternoon

It's been a rather pleasant October afternoon, warm, breezy, with a clear sky. The mobile sculpture Airborne catches both the wind and the waning October sun:

The garden plants are doing about as well as they have all year. Admist a recent burst of flowers, I noticed this rather impressive spider web:

…not to mention the rather impressive spider that inhabits it:

The peace of the backyard was briefly interrupted by the sound of cats fighting. More worrisome was the sound of an angry dog barking in response. After peeking over the fence to investigate, I was assured by a neighbor that it was “just some crazy cats.” One of the “crazy cats” wandered into view and I immediately recognized him as the friendly grey tabby that often visits my yard (I jokingly refer to him for a while as Luna's “boyfriend”). Foruntately, he seemed to be none the worse for wear.

Cats, or more specifically, cat allergies, have been much in the news this weekend. The New York Times featured an article on a California biotech company that is breeding hyperallergenic “no sneeze” kitties, two of which are pictured to the right. The market for the hypoallergenic cats, which the company says will cost about $4000 USD each, is people who love cats in spite of their allergies. It is certainly a high price tag, but I gather so are the medications for the most severe allergies. Those who seek a more affordable feline companion and want to continue to adopt shelter cats can take heart in a study supporting the theory that having pets cuts allergy risks. Finally, there is this story from Wales about a hospital fighting to keep their cat Tibs, who has chearing up patients for years. While I do my best to avoid hospitals, I know having a cat around would help me during a health crisis.

I had an opportunity last night to jam with some friends and acquaintances I have not seen in a while. I played keyboard, with primarily piano, electric piano and organ sounds, though I did add a Moogerfooger pedal to the mix. Musically, we did a mixture of jazz standards, some 12-bar and 16-bar “headless” jams, and several trippy free-jazz experiments with keyboard, guitar, bass and drums. The latter reminded me of how I would like to get together a standard “quartet” at some point that freely moves back and forther between jazz/funk and experimental improvisation. It would be quite a contrast to my recent performances, but still consistent with my musical vision and sensibilities…

…in another example of slipping back and forth between disparate musical styles, I was listening earlier to alternating tracks from Ethiopiques, which I described in an earlier article, and the rather dark, political, and vaguely Middle-Eastern electronic music of Muslimgauze. The two albums could not be more different in geography, style, production and social context, yet they seemed to work well together. The dark electronica of Muslimgauze worked for me, dispite an implicit political view I probably don't share, and the gritty funk of Ethiopiques brought me back to reality. Perhaps here is the seed of another musical project…

…or just idle thoughts on a warn autumn day…










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.

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.







Weekend Cat Blogging #65: Luna in stained glass then and now

WCB this week is being hosted by Bonnie Loves Cats, featuring Darlin' Darla, a purebred Himalayan kitty who is available for adoption in Charlottesville, Virginia.

Last August (2005), I got a great shot of Luna basking in the light from our stained glass window:

This is one of the images featured on my Art Photography page, which definitely needs to be updated with some recent (and not so recent) selections.

This August, I managed to get a shot of her in nearly the same spot by the window again:

It's amazing to see how she's grown up in a year. But she'll always be a kitten to me.




Green Kitties

Green kitties need green love!

This was an early rendering I created with Poser, combining cats and geometric elements into a surrealist image. It brings together my interests in cats and modernist art of the 20th century.