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Hi,

here are two pieces of music with fractal structure. I think they might be of interest here. 

During my studies in 90's,  I took a course on electronic music by Pauli Laine in Helsinki University. He plaid there some fractal music pieces, and discussed about fractals and music. I composed then a piece inspired by Koch snowflake, but simplified into form that could be utilized in a multi-voice composition with clear "traditional" music structures.
The piece is here:
https://www.youtube.com/watch?v=GVK5N7HQf8Y

The basic form of the fractal can be described as three notes with fundamental frequencies p k1*p p, where k1 is a coefficient, with value of e.g. 1.5.
The next layer is made by replacing each note with notes p k2*p p, resulting in note sequence p k2* p  k2*(p k1*p p) p k2*p p.
which makes them to appear with three times higher tempo than the first layer. You then continue this, and you get more and more layers. The idea is to select the coefficients k1, k2, k3... in such a way that you get some musical chords and melodies, when you play back the layers. In this version I selected the coefficients to produce well-known intervals and chords. 

The basic idea of the second piece is even simpler, and the original idea was not to make music with fractal nature, but now when I made the graphics for it I realized that actually it shows quite strongly that. The idea was simply to count binary integers from 000000000 to 111111111 one by one in tempo.
Here is the piece:
https://www.youtube.com/watch?v=UXKUWLJEfgE

Each bit is mapped to a note, and zero means "note off" and one means "note on". I have three mappings in the piece, where first one takes whole note scale in succession, second uses distributed diatonic scale, and the third one is a chord. The graphics follow strictly the music, and the fractal nature is best shown in the third variation. You can understand the fractal nature also if you consider each bit as a separate voice. The melody in one voice is monotonic, as only a single note is repeated. However, the rhythm has fractal characteristics, as the same rhythmic structure (a rest followed by a note) is repeated at all time scales. Actually, it would be cool to generate this with infinite number of layers, with possibility to zoom the tempo, similarly as you do with e.g., mandelbrot fractals.

The music has been created as midifiles and sheet music with unix shell scripts, max patches, and midi sequencing,  and realized by synthesizers and real musicians.

I hope you enjoy! All comments are welcome.

-Ville Pulkki

Nice, I liked the Koch curve one best :)


I had a go at this, approximating infinite layers by 7 octaves with the outermost fading in/out as the tempo/pitch doubles:
http://mathr.co.uk/misc/2016-03-31_counting_accelerando.ogg

Source code for Pure-data at: http://code.mathr.co.uk/bandlimited/blob/HEAD:/bl-countings.pd (though I post-processed the resulting file in Audacity and the waveform you draw will be different from mine).

Thanks!

>I had a go at this, approximating infinite layers by 7 octaves with the outermost fading in/out as the tempo/pitch doubles:
>http://mathr.co.uk/misc/2016-03-31_counting_accelerando.ogg

That is cool!

Hi Ville & welcome to the forum!

I too like the koch-curve very much!
Especially that you took the time to 'convert' it into real music for real instruments. All attempts of fractal music I heard until now sounded pretty bad because of the used sounds.
Yours is the first one that I as a full-time musician actually like to listen to!
(except of course psytrance-music that in it's whole rhythmic structure is extremely close to the cantor set as I'm convinced)

this is cool stuff!

very interesting and nice

>Especially that you took the time to 'convert' it into real music for real instruments. All attempts of fractal music I heard until now sounded pretty bad because of the used sounds.
>Yours is the first one that I as a full-time musician actually like to listen to!

Thanks!

I made the recording as normal instrument-by-instrument studio recording, which I then found not a very good idea. There are many bars where they are off beat, and there are no global expression, such as crescendo etc. I then thought that I should have gathered an ensemble, and recorded it in a hall. Never had time, funding and stamina to do that. If someone would like to play this with their orchestra (strings+some horns+few percussions + synth/acoustic organ), I am willing to make the sheet music for your ensemble. I would love to conduct that also, if any possibility.

Talking about the composition itself. I realized in the course (1991), that most of the fractal music sounded like postmodernist atonal compositions, which is of course ok as it is. However, I just thought make something different. I thought that it could be possible to find a way that I could utilize western classical music structures with fractal nature, and this was my finding.

Bravo! encore!
very "pleasing to the ear" fractal pieces .