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

I'm a one-man music laboratory -- mostly centered around the electric and acoustic mountain dulcimer, which I make myself in my workshop in South East Ohio, and more often these days in Wooster, which is Northeast Ohio.

I'm currently working with the aspects of the mountain dulcimer which tend to calm and soothe -- though I go through strong reactionary cycles now and again, when I have had enough of the meditative stuff. Sometimes this takes the form of wild filter-synched electronica, and sometimes it is more bluesy.

My music contains many influences, with Celtic music being the primary one most of the time. I've always been an improviser, and I have had some formal jazz training (Ohio State WI-89 BA Music), though I'm not a jazz musician by any stretch. 

I love working with extremely simple, circular chord progressions: those progressions that seem to go on for hours and are hard to stop playing. There is a strong minimalist vibe to this.

I also love the DRONE which is one of the most appealing aspects of the mountain dulcimer. These days, I tend more toward arpeggiating the open drone strings of the dulcimer, rather than the more strident strummed drone typical of the more traditional styles of playing.

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!

Example Just Intonation Fractal 3-part Invention

David Ryan