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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▶ “Discrete Algorithmic Mathematics” by Stephen B. Maurer & Anthony Ralston: amzn.to/4bmsOvG
#fractal #math #music #beauty #art #mathematics #code #programming #computerscience #processing #java #visualization #algorithmicmusic #computermusic #experimental #hypnotic #randomness

[@CiscoWes]

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

[@Vindilax]

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

[@grishhung2828]

It's the whole tone scale! That specific series of six notes, do, re, mi, fi, si, and li (which are, in this case, C, D, E, F#, G#, and A#) are often used for making mysterious, dreamlike songs in movies and video games. Each note is spaced two semitones apart. It sounds particularly like the Chamber of Sages here (which also uses this note set) because we're in the same key (C).

[@TylrVncnt]

Haha thought the same thing g

[@baefaux]

It even looks a bit like the chamber of sages from top down too

[@DF28.mashups]

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-we6bv]

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

[@channalbert]

9:00: Coltrane is jealous.

[@MrEthanhines]

Must be what Spielberg was thinking b4 he made close encounters

[@BlackMoonMaster]

i love fractals

[@marceloislas1413]

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

[@watchtowerdragon7098]

Chaos Within Structure = Structure Within Chaos

[@blobby-ur7vn]

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

[@Nivaleia541]

Why is 0.789 used on the dodecahedron fractal?

[@jacpa2011]

Its nice you doing other sort of content!

[@reid_makes_art]

Super cool channel man!

[@sistajoseph]

Beautiful

[@stanley9252]

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?

[@TZG-cr1uc]

The sounds are satisfying

[@maxdoubt5219]

So cool! Like a treasure hunt.

[@morgunstyles7253]

Great video

[@DaveLopez575]

3:25 fractal sounds.. awesome 😎

[@holyelephantmg8838]

2:25 *insert Zelda chest opening theme here*

[@LukesTheBomb]

THE TRIFORCE IS REAL!

[@D4N15H_]

1:35 The sound is like c418 - cat

[@jaydensanimations7022]

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

[@deathhzrd]

The dodecagon is super cool

[@EntergeticalakaBot]

Algomotion voice jumpscare???

[@OLIVE_MANN]

When are you releasing on spotify

[@juanpabloaraujotome5272]

Sumamente interesante, saludos desde uruguay

[@Poetateoe]

Why was 0.789 used for the Dodecahedron? Curious to look further into it

[@FantomMC2]

Summon the Triforce

[@alexaleman16]

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

[@davidknopp1208]

It’s the trifor

[@HalfBreadOrder]

y

[@legendari-on-yt-you-mfers]

Debating using this for music inspo

[@lyanporto]

on the first one if you started with a point that doesn't lie on the triangle, or if the first line's middle point doesn't fall in the triangle, would it still work?

[@Rubygal]

Look at all the little triangle people

[@Mareloko41]

If we could use this to find of song of stars in the night sky that would be awesome

[@rvsamsung77]

C augmented C-E-A ❤

[@danielkeller6610]

C E Ab

[@juanpalml7]

What if you select a dot inside of the void

[@shikaishik]

何でこうなるのですかね？ あと、中東のマカームをベースに作図したらどうなりますかね？

[@aaroncornelison5229]

What happens in the sierpinski triangle if the initial point selected is in the largest "blank" triangle in the middle somewhere? Would it be the same but with an extra dot where you started?

[@emeraldnickel]

Yes - this is demonstrated in a short on this channel.

[@arvydas86]

Run first triangle game again, but first random point start in center. What you'll get?

[@pagezmusic5013]

Make a mandelbrodt set next!!!

[@vyrezbeatz1159]

What software is this and can it be used in a DAW like FL Studios. Thank you!

[@96thelycan]

i belive if you changed tthe note to be jazz bass samples it would sound really fire

[@majorse203]

wowie

[@attaulwasay]

Which software did you use to simulate?

[@loveandoneness.n.e.t]

Is there any correlation between chaos game and cymatics? Seems the organization is similar, in a way

[@PupdudePwns]

Sierpinski did the Triforce before it was cool

[@jonatandjurachkovitch460]

But this is true for any triangle, why the circle of fifths?

[@thecloudwyrm7966]

Interesting. Does the last shape have any sort of name? Great video

[@AlgoMotion]

Not that I know of, just a cool looking fractal! Thanks for watching.

[@jeffsmith9384]

@@AlgoMotion it looks like a fps map

[@durimmiziraj4815]

P5.js? Tell me pls.

[@chry003]

I tried doing it, but it doesn't work.. Maybe I didn't do it correctly idk.

[@joaoericocaldasdelima7735]

1º - Suavemente

[@mcasualjacques]

maybe the 3 circle points and the midway process are just using the chaos dots to display 'their' structure/ so it's not as much the chaos initially containing any structure. if the rules were for example non-equilateral placement, the chaos would 'display' the structure of that arrangement. But but maybe we could consider randomness a rule like the others, what happens next you won't believe! Just kidding i don't know what would come out of it. Oh you could use digits of pi or phi to replace random().

[@mcasualjacques]

i started writing this in Daz Studio (Qt/Javascript1) charged particles interacting through time so motion. ballz = Scene.findNodeByLabel("ballz"); nodes = ballz.getNodeChildren(); f = new Array() for( i = 0; i < n; i++ ) { node = nodes[i]; f[i] = Math.random() * 2 - 1; obj = node.getObject() if( obj ) shp = obj.getShape(0) mat = shp.getMaterial( 0 ); if( f[i] < 0 ) mat.setDiffuseColor( Color( "blue" ) ); else mat.setDiffuseColor( Color( "red" ) ); } dsg

[@mcasualjacques]

we could say but not really understand "it's not order out of chaos, it's chaos into order or constrained by order, or marshalling chaos or classifying random things like positions (or distances) in the game's domain.

[@alexrayder1]

That's interesting but, what is the point of using notes if you're not doing it for scales ?

[@floretionguru2977]

Hey AlgoMotion, I also work with music, quaternions and the Sierpiński triangle. We should do a project together.

[@facundomonaco2733]

Does anyone know what kind of app or program is he using for this?

[@drdca8263]

Presumably code he wrote?

[@Fromyoutube640]

1.Sierpiñski triangle 2.Hexagon Chaos Game ...

[@gbyt034]

at ~0:25 the inside out theme's motif plays

[@gbyt034]

0:23 allows one to hear it in full

[@azid_hsb]

Do more bogosort live please?

[@AlgoMotion]

It's live right now!

[@azid_hsb]

Okay

[@pistol._.alexxx]

my teacher showed this video in class

[@elbonnieto8929]

7:55 minecraft music disc

[@Finity2010-ud2rl]

You're really restrcting yourself by doing these in the Circle Of Fiths. There are plenty of chaos games that are interesting that aren't mutliples of 12.

[@Have79e]

The Pythagoreans were right 😨

[@Bozpot]

It sounds like Arkenoid.

[@matthewgrabowski9212]

If you want a headache watch this video

[@titaniumbone5979]

voice reveal?

[@neopalm2050]

chaos game is my least favourite way of rendering an IFS, but I guess the better ways can't be easily turned into music.

[@shredhappy101]

The second one sounds almost exactly like Zelda OoT

[@bantri256]

It sounds, like the Duran Hungry Wolf arpeggio: kzbin.info/www/bejne/roOyfKuDl9h8iKM

[@Mareloko41]

Did you purposely stayed away from the center?

[@gfghdgfghd6391]

The triangle cannot be in the center because of the way the points are oriented

[@gfghdgfghd6391]

If you start in the center, there will be 1 dot in the center of the serpenski triangle

[@gfghdgfghd6391]

The aspect ratio prevents it from being in the center for the rest of the

[@defitsch]

Has nothing to do with the circle of fifths though. It just depends if you start with an equilateral polygon like triangle, hexagon, etc. It's not the magic of music, as one might think when reading the title ;)

[@AlgoMotion]

The circle of fifths is just being used here as a vehicle for adding *audio* to these well-known fractal visualizations. It facilitates a basic form of *algorithmic composition* of music to accompany the chaos game visuals (algorithmic composition is a common theme on this channel). It should be clear that the circle of fifths is not "producing" the fractals, but rather the fractals are just being superimposed on top of the circle of fifths, merely to add a musical element.

[@savonliquide7677]

What is the connections with circle of fifth? Except the circle? The notes are not even related to the position or angle inside but only to the random note (and not even the good one) i dont t deny the beauty of maths here (that are not a scoop but ok it is nice) but why underlying lies like wrong links

[@chaosme1ster]

The 3 points on a circle producing the Sierpinski triangle (the 1st example in the video) is a well-known mathematical phenomena. The circle of fifths really has nothing to do with it, just as the "halfway" notes are in no way "halfway" between anything and anything. I don't mind though -- I think the fractals still are cool and (for most people) unexpected, and the notes add a fun dimension to watching the video.

[@Vlow52]

Basically, a complicated cherry picking, it works only because of restrictions of an angles you start with. If you would really take it randomly, it would be just a mess.

[@Titus2.3-5]

Pretty snowflakes

[@littleguy-ij8jf]

Has absolutely nothing to do with circle of fifths but still cool

[@isohumulone]

1, Your first point cant be randomly placed. 2. This has nothing to do with the circle of fifths.

[@themichaelconnor42]

This isn't chaos at all. Your initial limitations are where the structure comes from. It's like dumping a bunch of sand on top of a bucket, then blowing away all the sand until only the bucket remains and saying... Look! Structure from chaos!

[@Naseem384]

This is kinda cool but it has literally nothing to do with the circle of fifths.