Colossal Gardner, ch. 27


I’m not sure I would consider Fractal Music to be part of Infinity, but that’s how Martin chose to categorize it in this book. On the other hand, the concept mixes two of my favorite things – fractals and synthesizers – so I’m willing to be forgiving.

Gardner starts out by discussing music’s relationship to the other arts within Plato’s thought that all art imitates nature. This may be true for paintings and written fiction, but in what way does someone singing opera imitate a bird call or burbling water? He then introduces a discovery by Minnesotan physicist Richard F. Voss, who joined IBM’s Watson Research Center. This discovery involves spectral density, AKA the power spectrum, and autocorrelation. Mandelbrot, who developed the field of fractals and also worked at Watson, suggested an easy way to understand these concepts. Say you record a sound and play it back at different speeds – you’d expect something like a violin to sound very different when played slower. But, there is a class of sounds where, if you change the frequency, all you have to do is adjust the volume to make it sound exactly the same again. He called these sounds “scaling noises.”


(All rights belong to their owners. Images used here for review purposes only.)

The simplest of the scaling noises is “white noise,” or Johnson noise, which is a random collection of all frequencies. It’s what you get when you listen to a detuned radio. The correlation, which measures how fluctuations at any given moment are related to previous fluctuations, for white noise is 0, except right at the beginning, when it is 1. You can use spinners (above) for generating a “white tune,” where there’s no correlation between any two notes. The note names are for a piano keyboard, and you can pick “fa” for middle C. The sizes of the segments for the notes can be completely arbitrary. With the above spinner, the notes have been weighted to give precedence to those around fa. A second spinner can be used to assign the lengths of the notes as 1/8th beats, 1/4 beats, 1/2 beats or a full beat. Alternatively, you can use dice, random number tables, or the digits of an irrational number (such as PI or e).

The next step is to use Brownian noise, which mimics the random thermal movements of small particles in a liquid. To create Brownian music, add one more spinner divided into segments marked 0, +1, -1, +2 and -2. So, you start by spinning the Do, Re, Me spinner for the first note, then use the offset spinner to step to the subsequent notes based on the first one (up one or two notes, or down one or two notes each time). The duration of the notes could also be controlled by the offset spinner to make it Brownian. You may note that Brownian music is going to have a higher correlation factor.

If you want to find a halfway point between White and Brownian music, you can go stochastic. In this case, you set rules for changing to a note based on the previous 3 or 4 notes. One way to do this is to analyze Bach’s music and look at how often a particular note follows another triplet. If certain transitions never happen in Bach’s music, we can introduce rejection rules to prevent our using them. The result would be Bach-like at the beginning, but eventually it would still become random.

So, what Voss did was to find a compromise between White and Brownian music by selecting a scaling noise halfway between. This is called 1/f in spectral terminology. Music based on 1/f is mildly correlated, and it repeats over both short and long runs. 1/f noise is sometimes referred to as flicker noise, and Mandelbrot found that it’s very widespread in nature. The annual flood levels of the Nile is a 1/f fluctuation. From here, Gardner goes into a brief excursion into fractals, and how they relate the Peano curve to the Koch snowflake, and Bill Gosper’s flowsnake. Other 1/f fluctuations include variations in sunspots, the wobble of the Earth’s axis, membrane currents in the nervous systems of animals, and the uncertainties in time measured by an atomic clock. So yeah, we’re now getting to the idea of music imitating nature, if we use a 1/f spectrum. Most classical, jazz, and rock is 1/f.


(Comparing white, 1/f and Brownian noise.)

Voss devised a way to simulate a computer program for generating 1/f music. Pick 8 notes and a scale of 16 tones, and write 0-7 on a chart as below. Write the numbers out in binary form and label the columns “Red,” “Blue,” and “Green.” Take three 6-sided dice, one of each color. Roll all three to get a number from 3 to 18 for your starting tone. Now, look at the chart, and see that when you go from 000 to 001, only the bit for the red die changes. So, pick up the red die, leaving the blue and green as-is. Roll the red die, and the total of the 3 dice gives you your next tone. Going from 001 to 010, both the red and green bits change, so roll those dice, and continue until you have your full song, 8 notes at a time. Gardner comments that this process is not exactly 1/f, but it’s close enough for jazz. You can also use 4 dice for a range of 21 tones. Note lengths are generally fixed, but that’s tweakable with a similar chart and dice system. Gardner adds that 1/f music is fractal, in that if you compare a short note string to a much longer passage, they are similar. “The tune never forgets where it has been.” He also comments that a melody can be changed to something new by playing it backwards, to turn it upside-down, or both, and they still retain their 1/f spectral densities. He specifically mentions Mozart’s Mirror Canon as an example of the style (which was not actually written by Mozart).


(Dice chart for making music.)

Now, the synthesizer part that I mentioned above is not directly described in this chapter, but there is a lead-in with “mountain music.” The idea is that mountain ranges look fractal or random, so you go out, photograph a mountain range, and then use the skyline to translate the heights into notes. Villa Lobos did this using the mountain skylines around Rio de Janeiro, and Sergei Prokofiev did the same for Sergei Eisenstein’s film “Alexander Nevsky” (1938). Terahiko Terada (1878-1935), a Japanese physicist, discussed doing something similar, imprinting skylines or people’s profiles on records to create new sounds. And, this is where things get fun with synths. If you digitize said skyline, or someone’s profile, and normalize that for a 5V control signal, you could use it for sound envelopes in a synth, either replacing the ADSR, or looping it as an audio waveform, or feeding it into a VCO to control pitch or filtering. The possibilities are endless for experimental music – all you need is a video camera, facial recognition software, and a breakout board… Anyway, Gardner returns to mountain music in chapter 47.

Sidenote: As I was writing this blog entry, I happened by the Scientific American magazine website, and found an article that had just been posted a few minutes earlier about Betty Shannon, wife of computer pioneer Claude Shannon. In the article, the author mentioned that one of Betty’s technical papers was on “Composing Music by a Stochastic Process.”

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