Music & Fractals

"Why do rhythms and melodies, which are composed of sound, resemble the feelings; while this is not the case for tastes, colors, or smells?"---Aristotle

I claim no originality for the following idea, which could possibly answer Aristotle's question (footnote 1); I first saw it expounded in a letter from one John Lambshead, published in the English magazine Hi-Fi Review (footnote 2). Nevertheless, it ties in so closely with my own experience of digital audio that it struck me like a satori.

Before continuing, I should explain what is meant by the word "fractal." In a nutshell, although formal geometry deals with objects that are one-, two-, or three-dimensional in nature---mathematicians and astrophysicists are happy with objects that have more than the traditional three dimensions; just be glad you don't find one of these weird beings sitting next to you in a bar---there are objects that turn out to have a fractional dimensionality. Clouds, coastlines, the branch structure of trees, the body's blood vessels, for example, are objects or systems which are defined by between one and two or between two and three dimensions. The thing these mathematically peculiar things have in common is a self-referring nature, wherein their macroscopic features are echoed in their fine structure, which is further echoed in their microscopic features, and so on ad infinitum, a kind of infinite crinkliness. Without external reference, it is therefore impossible to estimate their size or how far away they are. Such systems or objects also appear "natural" to the eye, and the adoption of fractal-producing algorithms has greatly enhanced the believability of computer-generated images.

Mr. Lambshead's letter pointed out the fact that traditionally styled buildings are pleasing to the eye because they are pseudo-fractal in nature; the closer the onlooker gets to them, the more detail there is to see. By contrast, a "modern" structure appears identical, only larger, as the onlooker draws closer. The legacy of the Bauhaus school and the activities of popular modernists such as Mies van der Rohe and Cesar Pelli result in city centers that no longer feature human-scaled detail to add interest and friendliness to the buildings, only large-scale, impersonal statements devoid of anything that might speak on a human level. As the Prince of Wales once put it, modern architecture produces buildings "with no character except arrogance" (footnote 3).

As with architecture, so with music. Mr. Lambshead conjectured in his letter that naturally originating sounds were pseudo-fractal in character; that is, their waveforms have a wealth of fine detail, and that detail itself has an even finer-structured wealth of fine detail, and so on, until the crinkliness of the waveform is finally enveloped in the analog noise that accompanies every sound we hear. I agree that "pseudo-fractal" is the correct description, as any sound other than noise can't be wholly fractal; that would imply an absence of bandwidth limiting. Nevertheless, a casual study of music waveforms reveals the fact that without having a time or amplitude scale attached, it is impossible to assign any kind of loudness or frequency value to them, which does suggest a fractal-like nature, at least within the audio band. I would also add to Mr. Lambshead's hypothesis the statement that music itself is also pseudo-fractal. The large structure of a work tends to be echoed on a smaller scale within that work, right down to the individual notes that go to make up its essence. And when the work is played, those notes are realized in the form of sounds that continue the fractal character.

To respond to Aristotle, it is the capability of music to contain within itself, both intellectually and sonically, layers of ever-increasing detail, a character that echoes the fractal nature of reality, that I feel allows it to communicate in such a fundamental yet complex way with the listener. To understand and appreciate visual art, you need to have a cultural context; to understand music, you need only be a human being.

As well as allowing one to score points on long-dead Greek philosophers, Mr. Lambshead's hypothesis is relevant to the ever-enduring analog vs digital debate. Think about it. An analog recording and playback process will preserve the natural, pseudo-fractal character of the sound at the expense of an increase in the level of the already present noise, which is itself fractal. By contrast, a digital recording and playback process is fundamentally unnatural in that it fails to preserve the pseudo-fractal nature of real sounds below a certain threshold. Instead, that fine detail in the original signal fails to be recorded at all.

In addition, the noise that the quantizing process (if undithered) adds is also unnatural in that it is directly related to the input waveform rather than having a random nature; it is therefore a distortion. However, as long as the digital system's cutoff of detail is at a low enough level---if the system has enough bits---the listener will not be aware of the fundamental change that has taken place in the music and sound's character. And as Stanley Lipshitz has pointed out many times to the professional community, the appropriate use of dither at the primary A/D conversion allows the preservation of signal detail that would otherwise occur below the digital system's nominal cutoff threshold.

But the comparison between architecture and sound recording is apt. With an analog signal, the closer you get to the sound---the higher the level you choose to listen---the more detail there is to be heard, the only tradeoff being an increase in the level of the signal's intrinsic background noise. With a digital recording, this is not true. No matter how close you get, how high you set the playback level, there is no more detail to be discerned than what was captured at the time of A/D conversion. In fact, digital audio is like a television image: you can only get so close; if you get any closer than that optimum distance, you just become more aware of the "joins" in the image.

Footnote 1: From the 29th Problem, as quoted by Hermann Helmholtz in his 1885 classic On the Sensations of Tone (p.251 of the Dover edition).

Footnote 2: Hi-Fi Review, January 1990, p.5.

Footnote 3: In A Vision of Britain: A Personal View of Architecture (Doubleday, 1989), p.35.