So I'm sitting here reading about superstrings, and how it, because of using an abstract "string" as its basis, is very much like music. Then I think, hmmm, let's take scientific organicism to the extreme. Mathematically, we get what you see above: a fractal. The iterations of the Koch Snowflake take each line, separates it into three segments, deletes the middle segment and adjoins the ends of the two remaining segments with two new lines so that it forms a triangular "hump" in the middle of the line. The first iteration you see all the way to the left is this process applied to the starting point, which is a triangle (right side up or upside down give you the same results).

So what if you took this to music? Look at only the top line of the triangle (ignore the rest of the snowflake). The shape it takes through successive iterations is much like the path of a lot of organic music. It rises to a point and comes back down, though perhaps not as symmetrically as a fractal. But for kicks, let's take the greatest triangle known in music: I-V-I and embellish it to match Koch's snowflake. So let's say we have I-IV-V-I. The 2 I's correspond to the beginning and end of the line segment while IV and V correspond to the hump in the middle. So applying fractal rules to music we have:

Iteration 0: I
Iteration 1: I-IV-V-I
Iteration 2: I-[IV/IV-IV/V]-IV-[IV/V-V/V]-V-[IV-V]-I
Iteration 3: I-[[IV/IV/IV-V/IV/IV]-IV/IV-[IV/IV/V-V/IV/V]-IV/V]]-...

Obviously, this gets rapidly complex and I am using the IV and V chords functionally relative to their ending chord. But despite the caveats of turning four lines into four music harmonic "points" and using a teleological endpoint (the I at the end), the basic idea of extreme organicism is carried out: wonderful complexity can come from simple, seemingly staid algorithms. Let's concretize the example and Start with C Major. We then have:

Iteration 0: C
Iteration 1: C-F-G-C
Iteration 2: C-[Bb-C]-F-[C-D]-G-[F-G-]-C
Iteration 3: C-[[Eb-F]-Bb-[F-G]-C]-etc...

In math, the shape generated by infinite iterations of the factal seed algorithm is neither a direct flat shape in the second dimension nor a three dimensional image. I think that the kock snowflake as above is supposed to have finite area but infinite diameter, a property which, when combined with certain definitions of dimensionality, yields a result of an irrational dimensionality above 2 but below three (think radical five).[1] But what about finite result in music? Math is infinite but western music is modulo 12. What would a fractal piece of music sound like? What about fractal frequencies? Timbres?

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[1] The figure's area can be bound by any shape that totally surrounds it: a circle, a box. Easy to see why the area is finite. But its perimeter always increases. Since everything is equilateral, the two new segments of each iteration is 1/3 the length of the original line. If we start with a unit length segment in the 0th iteration, we get 4/3 for the 1st, 16/9 for the 2nd, and 64/27 for the third iteration. In general, the length of the nth iteration is (4/3)^n times whatever length you start out with. This obviously grows without bounds, and yet the area remains completely within a large circle you could draw.