"Cantor's Dust" is one of the first songs Luis invented for this project—I performed it with him as part of a Columbia College Chicago colloquium before I was hired to work there. Andy Tillotson (our longtime friend and bandmate in Sonus Umbra and Might Could before that) also performed. Although the version we have recorded as The Devil's Staircase rocks it out more now, originally the arrangement was amenable to playing with a small set of acoustic instruments.
I may be inspired to write a longer introduction to fractals later, but for this post I'll explain enough to get at some of the ideas used in the song.
A fractal, as described by Benoit Mandelbrot (hear a favorite fractal-related song of mine related to him here), is a "shape made of parts similar to the whole in some way." The term self-similar is also used a lot in writings.
Fractals are a frequent source of inspiration in the arts, and people usually think of visual art when picturing fractals. A computer image of the Mandelbrot set mentioned in the song linked above artistically shows a fractal. The finger-like projections on the wave tips in Hokusai's famous print shown below have fractal properties.
A representation of the Mandelbrot set (Wolfram MathWorld) and The Great Wave off Kanagawa by Hokusai
The Cantor set
The fractal inspiring our song is the Cantor ternary set, named after mathematician Georg Cantor. It's easy to picture that set. Proceed in steps. First, take a line segment. Then, remove the middle third. Then, remove the middle third of the two pieces left. Then, keep going ad infinitum. What's left is the Cantor ternary set.
Generating the Cantor ternary set (source: Wikimedia commons)
It may seem like if one keeps doing this, there will be nothing left. Picturing some numbers will help if I am more precise. Suppose the first segment represents the interval from zero to one on a number line (including zero and one). Mathematicians write this set as [0, 1], with square brackets used to remind us that the endpoints are included.
The Cantor ternary set instructions have us take out the set (1/3, 2/3), with round parentheses used to remind us that 1/3 and 2/3 are NOT taken out. These are two examples of points in the set that are never taken out. Then the sets (1/9, 2/9) and (7/9, 8/9) are removed, basically preserving the 1/9, 2/9, 7/9, and 8/9 points forever. Other points that aren't endpoints also survive this process.
A weird property of this set is that if you add up the lengths of the pieces removed after an infinite number of steps, you get the length of the whole segment. But there are infinitely many points in the set.
One further clarification: the term ternary is used in the description to refer to ternary (base-3) numerals. Any point in the Cantor ternary set from [0,1] does not require the digit 1 to be expressed as a fraction in base 3. For example, we'd write the normal base-10 fraction 1/3 in base 3 as 0.1 (that is, 1(1/3)), but we can also write it as 0.0222... because 1/3 = 0(1/3) + 2(1/9) + 2(1/27) + 2(1/81)...
Phew, so what's that got to do with the song?
The theme of removing the middle third happens in at least two ways in the song Cantor's Dust. The introduction of the bass and guitar in the song uses no thirds as the introduction gets underway. The bass plays the root and fifth note of the C scale starting around 0:33 of the song, whereas the guitar arpeggiates over some kind of suspended chord featuring the root, fourth, and fifth note around 0:42. It isn't until later in the song where a flatted third shows up, confirming that the song is in C minor.
The first chord I play on acoustic guitar in the song. Note the lack of third. That would be an E or E-flat (a 1 or 2 on the second note of the tablature).
The song is inspired by the Cantor set structurally as well—the first and last third of the song are similar, and the middle sounds completely different. The arpeggios I wrote above appear in the first and last third, but not the middle.
Georg Cantor was an influential thinker in number theory and set theory—for example, his work helped determine how countable certain sets of numbers are. However, he dealt with trouble getting the professorships he wanted and fought philosophically with other thinkers of his time. He suffered from serious depression and died poor in a sanatorium. A minor key was an informed choice for this piece.
Something else cool and related - the Sierpinski carpet
An iteration of the Sierpinski carpet (Wikimedia commons)
The Sierpinski carpet shown above is a fractal that can be seen in light of the Cantor ternary set. In it, start with a solid square, and take out the central square. Repeat for several steps. If you do this with a 3D object, you get a Menger sponge.
Sierpinski carpets are used as antenna shapes. If you stop (like the photo above) after a finite number of steps, you can construct the thing and it contains many different "mini-lengths" of antennae. That makes an antenna shape like this useful for picking up a variety of wavelengths and frequencies.
This is one way to turn the Cantor set 2-D. Another (actually called "Cantor's Dust") is what you get if you take whole middle-third strips out of both the horizontal and vertical dimensions instead of central squares.
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