Post by fuguestate on Aug 28, 2017 21:32:44 GMT
For those interested in a level of counterpoint that possibly goes beyond even Bach, try figuring out Kristofer Emerig's "fractal fugue" concept. Here are some pertinent information:
Basic fractal fugue
While the basic fractal structure outlined in the above link may look trivial and obvious to anyone who has heard about fractals, Kristofer's insight goes far deeper than that. To quote the man himself:
To summarize the concept, consider the traditional Bach fugue with subject entry in the home key (I), followed by an answer in the dominant (V), followed by more entries, usually starting with the 3rd voice in the home key (I). This sequence of keys, I-V-I, can be alternatively interpreted as scale degrees, i.e., in the key of C, for instance, as the sequence of notes C-G-C. Thus, the traditional fugue exposition with the key structure I-V-I can be considered to be a fractal expansion of the subject against the 3-note motif C-G-C. Now if the subject itself consists of these 3 notes, then we can say that it is a fractal fugue: one where the subject is expanded against itself. But it goes beyond this.
Kristofer posted an example of 2 bars of music wherein the top voice is expanded in this way relative to the bottom voice, and the bottom voice simultaneously is expanded relative to the top voice, and the resulting voices harmonize with each other seamlessly. This represents a level of counterpoint far beyond anything I'd imagined possible, and will probably be a good challenge for any would-be counterpoint masters.
Basic fractal fugue
While the basic fractal structure outlined in the above link may look trivial and obvious to anyone who has heard about fractals, Kristofer's insight goes far deeper than that. To quote the man himself:
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 cases and 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.
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 cases and 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.
To summarize the concept, consider the traditional Bach fugue with subject entry in the home key (I), followed by an answer in the dominant (V), followed by more entries, usually starting with the 3rd voice in the home key (I). This sequence of keys, I-V-I, can be alternatively interpreted as scale degrees, i.e., in the key of C, for instance, as the sequence of notes C-G-C. Thus, the traditional fugue exposition with the key structure I-V-I can be considered to be a fractal expansion of the subject against the 3-note motif C-G-C. Now if the subject itself consists of these 3 notes, then we can say that it is a fractal fugue: one where the subject is expanded against itself. But it goes beyond this.
Kristofer posted an example of 2 bars of music wherein the top voice is expanded in this way relative to the bottom voice, and the bottom voice simultaneously is expanded relative to the top voice, and the resulting voices harmonize with each other seamlessly. This represents a level of counterpoint far beyond anything I'd imagined possible, and will probably be a good challenge for any would-be counterpoint masters.