Structure from Chaos: Fractals Generated on the Circle of Fifths via the Chaos Game

Running the chaos game on the circle of fifths, producing music to accompany the beautiful fractals that result.

We start using a triangle on the notes C, E, and A♭ (forming an augmented chord), and play the chaos game to generate the Sierpiński triangle.

Then we use a hexagon on the notes C, D, E, G♭, A♭, and B♭ (a whole-tone scale). For optimal packing, the ratio used to divide the lines in the chaos game is 0.667 for a hexagon.

Next we use all 12 notes (the chromatic scale) to form a dodecagon fractal. The dodecagon is optimally packed with a ratio of 0.789 to divide the lines.

Finally, we use a square (i.e. a diamond) on the notes C, E♭, G♭, and A (forming a diminished 7th chord). Playing the normal chaos game on a square, however, doesn't yield a fractal. It only produces uniform noise within the square. When a simple restriction is added: not allowing any corner to be repeated twice in a row, a beautiful fractal results.

0:00 Sierpiński Triangle
3:54 Hexagon Chaos Game
7:05 Dodecagon Chaos Game
10:27 Square Chaos Game
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[Vindilax]

4:52 its amazing how similar this sounds to Chamber of Sages from Legend of Zelda.

[dragonfire_]

So um, for the square: Yes it will not work if the line ratio is 0.5. Cuz in order to the sierpinski carpet to be made, you need the line ratio to be 2/3 at a random vertex AND MIDPOINT. So restricting the process is the only best way if you run the chaos game to the circle of fifths

[BA-webv]

Me trying to do every combination lock to get in my brothers phone:

[CiscoWes]

Played the video too loud and my furniture started to levitate. And I think I opened a portal.

[watchtowerdragon]

Chaos Within Structure
=
Structure Within Chaos

[MrEthanhines]

Must be what Spielberg was thinking b4 he made close encounters

[blobby-urvn]

it would be interesting to hear what a different selection of notes assigned for the octaves or halving of the distances between the two points instead of repeating the same pitch. eg run through the harmonic series using that note that is being halved as the fundamental, eg at 0.49 E, E, B, E, G#B, D and based over a C fundamental C, C, G, C, E, G, Bb and etc

[WildStar]

7:05 The dodecagon game is the *scariest* because of that ratio: Seven *ate* Nine!! 😮

[marceloislas]

That is more the result of the chosen method rather than something to do with notes and music.

[twbvqc]

i just listened to fractals for so long

[holyelephantmg]

2:25 *insert Zelda chest opening theme here*

[jacpa]

Its nice you doing other sort of content!

[alexaleman]

Sounds like hitting the jackpot on a slot machine, or so ive heard 😂

[SanityInAnAmazonBoxShorts]

Me at the doorbell forgetting I have the house to myself for an hour:

[jaydensanimations]

7:01 Now *that* is what I call, the Koch Hexagram!

[juanpabloaraujotome]

Sumamente interesante, saludos desde uruguay

[stanley]

Nice video! I had no idea about "chaos games".

I'm curious, though, what language and libraries are you using for visualizing these fractals and playing the notes?

[andrecole]

The C minor note sounds like a coin getting collected in a game when it is speed up

[Nivaleia]

Why is 0.789 used on the dodecahedron fractal?

[LonestarVeganShow]

Simple: By introducing an algorithm(ie pattern), a structure will always proceed