Music Composers Unite!
Kepler 62, like our investigations into the plantery solar system associated with the same name, is an attempt at a glimpse into other spaces, and specifically in this instance, other tonal spaces. The high probability of misinterpretation must be averted immediately, lest Kepler 62 be interpreted as a foray into the mythological realm of atonality, or presented as some alternative to tonality. It is not, but rather an excursion into one possible parallel tonal hierarchy alongside and in inferior position to conventional diatonicism. This is no attempt to build a better mouse trap, nor one which does else but catches mice, but only to build a different mouse trap.
It is something best described as paratonality, because it does not eschew the fundamental concept of tonality, ie, a hierarchy of relative consonance and dissonance imbuing musical language with meaning, but rather exploits those same tonal mechanics through a different pallette of modality, intervals, and ratios. Neither does Kepler 62 abandon the conventional protocols of counterpuntal practice. Those same techniques, or ones analogous to them, are applied to a new tonal space but remain intact in principle. Simpler ratios (greater consonance) are approached from more complex, dissonant ones by step. Leaps are less frequent the greater the interval, balanced by contrary motion, and out of relative dissonance are avoided altogether. The principles of counterpoint concern themselves chiefly with the mangement of consonance and dissonance across the vertical (pitch) and horizontal (time) polyphonic plane, and with a little bit of considered thought, can be extrapolated to any various collection of pitches and resulting intervals.
Tonal answer is somewhat of a misleading term I've never favoured, as real fugal answers are no more real than tonal answers, which are in turn no more tonal than the other. A more insightful term for "tonal" answer would be…Continue
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,…Continue
The other day, while chewing my morning bagel and listening to some JS Bach, the Crab Canon, no less, my thoughts drifted from my toroidal breakfast morsel to the lemniscate of Bernoulli specifically and the concept of symmetries nested within symmetric isometries in general, and its ramifications within the discipline of counterpoint.
This was not a novel chain of association for me, as the thought has been crowding my mind with ever growing urgency and frequency for the last two years, and served as a central principle of design and impetus in much of the music I've produced in that span. Nick Capocci recently remarked to me, and I quote without permission,
"It [symmetry] is a powerful intellectual and philosophical concept, and, naturally, finds a ready medium in free counterpoint".
I quote the above because I can not find more apt words than Nick's.
So what of it? Forays into the exploration of symmetric form within counterpoint are nothing new, and imitative counterpoint could be viewed as fundamentally grounded in that pursuit. The methods of textural inversion and imitation (transformational symmetry, mathematically speaking) have been well established and documented for four centuries.
There is nevertheless a glaring hole, both theoretical and practical, where melodic inversion and retrogradation is concerned, particularly of entire polyphonic textures. The smattering of extant works left by Bach in the mirror fugues of Kunst der Fuge and the Crab Cannon are hardly a large enough body of work from which to derive a rigorous contrapuntal methodology for mirror techniques.
Until I experienced a recent epiphany, just a year or two old now, it was apparent to me that such mirror forms in counterpoint could only be improvised haphazardly case by case through a blend of intuition and trial and error alone. Viewed in light of the realization that music, as an abstract…Continue