Music Composers Unite!
Alright here goes. Recently the question of "fractal" music arose on the forum here, and it incited me to revisit the topic.
Applying fractal concepts to music could be interpreted many ways, and with no precedent by which to proceed (my favorite conditions by which to work incidentally), I decided to work out this first piece, Ode to Earth, in accordance with my own musical interpretation of fractal structure.
Keep in mind, the "fractalization" was only carried out through the thread of subject entries, and the remainder of the composition is free counterpoint. The next logical progression would be to work out a fugue in which the subject is tranformed fractally combined with countersubject treated likewise, but when you start playing with this process, you'll undoubtedly conclude that this might be a practically impossible ambition. I don't know yet - doing is knowing.
The linked image shows in the bottom staff the elemental subject, and hence "algorithm", but I've reduced the note values for compactness. This basic shape is a note, followed by a note of half its duration two scales steps above, followed by a note of half its duration one scale step above. Each staff above shows the next generation of fractal transformation (this is not the score of the piece, these entries follow one another). In the composition itself, the subject is always in the top voice, except the initial subject entry, which I chose to accompany with the alto voice, though not out of necessity.
Now the concept of fractal design is that as a structure is built in complexity, each stage of that building process produces a form which mimicks the elemental shape, hence at any stage the elemental shape can be traced back through all nested levels.
I've labeled the intervals, to a depth before they became quite messy and illegible, to demonstrate how each progressive stage of fractalization mimicks the basic form, ie, one unit, followed by one scaled to half duration a third above, followed by one scaled to half duration a second above. Following the A1, A2, A3 labels et al will demostrate the inheritance of the basic shape through progressive stages.
While it might appear structurally complex at first glance, the procedural algorithm is quite simple and easy. Take any subject, and using the first note as a reference value, for instance a whole note, at each stage of fractalization, write (insert) the subject starting from each note in the previous subject, scaled to the relative duration of the present note and to its present pitch. When you stand back, you'll observe that this process always results in an overall shape, not to mention each more primitive nested level, which is congruent to the original, elemental subject.
Beware: Keep your vertical steps moderate, and your notes in the elemental subject few, or it'll spiral out of control (and practical range) with surprising speed. In Ode to Earth, you'll observe each fractalization doubled the phrase length, and halved the minimum note value.