Music Composers Unite!
I want to have a good ponder over this, it looks amazing....
At the time this was written, I'd already gotten a pretty firm grasp on the process of fractalization of a subject, but the entire picture had not yet come into clear focus. Often times, multiple realizations are simultaneously converging on one great epiphany, but their interrelationship not fully understood until integrated.
For instance I'd already both learned and to some extent, departed, from Bach's signature treatment of a motif through linear sequences, and come to prefer sequences built off an underpinning of tones taken from another subject, instead of a series of scale-wise steps, a superimposing of one subject over another, if you will. The realization was gradual, but inevitable, that this sort of superimposition of subjects was a general case, and that ordinary prolation, fractalization, and just about any contrapuntal imitative device one can imagine including ordinary answer, are all merely special casesand subsets of this overarching technique, and can all be described/defined as such.
I have referred to this greater case as distributive expansion of two or more subjects (self included), and nearly every single fugal device can be shown to be a special case within.
The process of fractalization, for instance, can be demonstrated to be the special case of the distributive expansion of a subject across itself. More trivially, a typical real answer might be described a the distributive expansion of a subject across a single note whose pitch is a fifth higher, and whose duration is the same as the initial note of the first subject. If you'd take a bit of time to look over the example in the fractalization thread, it will become readily apparent that this distributive expansion approach inevitably leads to quite a bit of prolation, except in the most trivial of special cases.
I've neglected to discuss much of the matter here, as it is of little or no concern to the vast majority of membership, nor terribly relevant to their writing aspirations, but have in the past continued into deeper waters with a few through personal correspondence.
I'm compelled to agree with Dave. This is the secret level - and I found the bonus box!
By Jove, I think you've got it. Unfortunately, the functionality of this site (for my account at least) has dwindled to the extent that I can hardly post simple text, and attachments have become impossible, without being met with a flurry of captcha tests, which themselves then crash and glitch in a continuous circle. I've provided an example, which you can view here:
May your fugal endeavors be met with ample reward.
ps - just to get you started in the right direction, one of those elemental subjects is the one off which Geodesic For Horns was built, entirely.
H. S. Teoh said:
I remember your fractal fugue, and that at the time, my reaction was one of skepticism.
However, revisiting the subject (ha!) again now, I'm starting to get a glimpse of something awesome. In the original thread, of course, you only used a very simple 3-note seed for constructing the entire fugue, which belies the potency of the method, but now that you state the insight the way you did, I'm starting to appreciate its true power. If we start with two different subjects, for example, we can recursively expand each subject against the other and obtain a kind of deep correspondence between them in the outworking of the fugue. The same could be done for a subject/countersubject pair.
Or we could analyse a single subject into its constituent motifs, and expand the motifs w.r.t. each other, thereby producing a vast amount of potential material that can be used for further development.
Another avenue of exploration is to "stretto fractalization": subject A could be expanded wrt subject B (not necessarily distinct from A), such that the occurrences of A are in stretto, i.e., the length of A corresponding to each note of B "exceeds" the length of the note, so that its tail end overlaps with the A corresponding to the next note of B. If carefully planned, the result could be made harmonious, and could result in a very interesting kind of texture indeed. It would be a kind of stretto where the entries aren't merely positioned that way just because they happen to harmonize, but the overarching arc of the stretto entries exhibits a correspondence with B (which may be the same as A itself).
Kristofer Emerig said:
[...] I've provided an example, which you can view here:
Whoa. Whoa. At first I didn't understand the example, but when it finally dawned on me... whoa. My mind is completely blown! This is a whole new level of counterpoint than I've ever imagined possible. Now I shall have to wipe the slate of my current grasp of fugue clean and rethink the entire concept in a fundamental way!
@Kris and HS,
Kris, when I saw your attachment, I instinctively felt I could use this technique as it feels very suitable for my not so tonal way of thinking and I immediately thought of HS's stretto fractalization, although my lateral thinking couched it in terms of quantum mechanics! (don't ask). Fractal technique has some of the spirit of Messiaen to me, especially in his use of non-retrogradable rhythm which seems particularly suited as a method of building towards a central point and the reversing out again symmetrically.
My thoughts have turned to applying this technique in 2 ways. Firstly, in an orchestral context, the use of delicate or blatant timbres and combinations thereof - either as foreground or background - with fractal movement is going to be a rich seam of resource. Antiphonal usage of a fractal cells in motivic development might well be fun, or using it to create a soft bustling accompaniment - creating a lattice of sound over time.
Secondly, I thought about linearity and perception of the flowering principle inherent in fractalisation and came up with a vertical solution which is not really original, but could be used as a slightly different way to motivate composing or generate new material. This idea in performance, could be presented as fast or slow as you like and could just be an element of the accompaniment behind prominent lines or could be featured. I have used 4ths, but any notes will do. This is reminiscent of Hansons' mirror harmony and is perhaps a bit too simplistic, but it could be utilised in so many ways. (see attached).
I can see why you've got into this Kris, I am going to have to learn about mapping non-linear equations here, is that something you do when starting out with a new piece? Have you tried anything like mirrored fractals (is that even possible), or overlapping fractals (like a lattice)?
As an organising principle, it is exciting and I will probably try it out for one of my remaining preludes and fugues.
Love the two subject reciprocal jiggery pokery....profound predictive organisation this deep is only a calculation or two away from another Austrians approach, although he decided to sacrifice concord, which some might argue made his task a little easier.