This track has been created using an L-system, which is a type of fractal where large parts are comprised of smaller parts that look like reduced copies of the large parts.

As an example, a melody going up by a Just-Intonation perfect fifth (3/2 frequency ratio) could be comprised of three smaller copies going up 3/2, down 3/2, then up 3/2. This simple example would not be musically very interesting, however the idea is there.

In this particular track, the following mappings were used:

1 beat 1/1 --> 1/2 beat 4/3, then 1/8 beat 1/1, then 1/8 beat 9/8, then 1/4 beat 3/2 (inverted)
1 beat 3/2 --> 5/4, then 6/5 (both 1/2 beat)
1 beat 4/3 --> 10/9, then 6/5 (reversed)
1 beat 5/4 --> 9/8, then 10/9 (reversed)
1 beat 6/5 --> 4/3, then 10/9 (inverted)
1 beat 9/8 --> 1/1, then 9/8
1 beat 10/9 --> 1/1, then 10/9

Reversed means the smaller copy is played back in i) the reverse order and ii) the smaller copies are themselves reversed
Inverted means that the frequency shifts are inverted, e.g. f --> 1/f. So 10/9 inverted means ‘go down by 10/9’ or ‘shift by 9/10’

So an L-system melody has a set of melodies (in this case one each for 1/1, 3/2, 4/3, 5/4, 6/5, 9/8, 10/9) and each has its own set of ‘daughter’ melodies.

Each daughter has:
- an associated frequency shift
- a timescale, here they are all 1/2, 1/4 or 1/8 beat
- optional inversion and reversing operations, giving 4 possible orientations in total.

Rules include
- product of daughter frequency shifts must equal parent frequency shift
- sum of daughter timescales must equal parent timescale of 1 beat
- continue iterating until maximum timescale of (iterated) daughters is 1/2^n.

In this particular track, the three parts were:
Bass: reference freq 100 Hz, max timescale 1/32
Alto: ref freq 200 Hz, max timescale 1/64
Treble ref freq 400 Hz, max timescale 1/128

This gave a 3 part invention! Due to the fractal melody/harmony, the different parts were consonant with each other.

One extra option is pre-tuning or post-tuning.
- Pre-tuning means ‘take the note frequency to be from the start of the fractal block’
- Post-tuning means ‘take the note frequency to be from the end of the block’

In this track, post-tuning was used, which means that at the end of blocks the tuning tends to be in 1:2:4 (octaves) ratio. Pre-tuning could just as easily be used, meaning that blocks start off on octaves and change to other chords during each block.

Enjoy! Dave

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See also:
- https://arxiv.org/a/ryan_d_1 (arXiv) for a list of my academic paper pre-prints, including my writing on music theory and Just Intonation
- http://orcid.org/0000-0002-4785-9766 (ORC ID) for my academic profile
- https://www.linkedin.com/in/davidryan59 for my LinkedIn profile

Example Just Intonation Fractal 3-part Invention

Composer, contrapuntist.

Website: https://www.piershudson.com
Social Media: https://linktr.ee/piershudsoncomposer
Email: piershudson@mail.com

miscellaneous Chunes grown here from Loopyfruit crossbreeds. i also comment way too much if ur into that

we hate soundcloud!!!

most of the music is straight from tiktok.  the other music is from a friend

keyboardist + producer, https://soundcloud.com/gatorcomingforyou
synths for http://soundcloud.com/extraextramusic
guest keyboards for https://soundcloud.com/hoodedyouth
curator for https://soundcloud.com/untrusted

also do some classical shit

❤️the https://soundcloud.com/burncartel

I'm into Just Intonation (JI), which is using harmonics (or undertones) to generate music. Over time, my profile will fill up with wonderful and melodious New Harmony, fresh from the infinite vaults of the original and best source of harmony, Just Intonation.

I’ve written some papers on the music theory of Just Intonation. You can check them out via the arXiv link below. Topics include the mathematical analysis of harmony (in Just Intonation), and a notation system to cover all of Just Intonation.

Enjoy listening!

I'm a Norwegian guy who loves to compose music. I am one half of "Sound Picture Studios" a little company that writes music for film productions. In the songs I try to capture different emotions I feel in my life and translate it into music. I hope you like it!

Example Just Intonation Fractal 3-part Invention

David Ryan