Music Composers Unite!
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 contrapuntal construct at least, is an isometry of a frequency axis (v) against a time axis (h), it became apparent to me that it is indeed possible to derive from the conventional (or any) protocols of voice leading a consequent set of protocols for melodic invertibility and retrogradibility.
The epiphanic realization was in understanding that melodic invertibility and retrogradability are integrally related, in fact, for phrases which are mutually symmetric across (v) and (h), their results are precisely identical. Moreover, simultaneous melodic inversion and retrogradation of such a phrase is identical to the original phrase itself. And so a methodical analysis of any sample of music with respect to melodic invertibility and retrogradability must begin with parsing that phrase into the largest elements which are symmetric to either or both axes. The inherent guarantee in this approach is that the smallest of possible elements would be merely two consecutive notes, a configuration which will invariably be mutually symmetric to both (v) and (h).
Well, I'm still assembling the specific details of my methodology for writing melodically invertible and retrogradable counterpoint, a system which might be viewed as an extension and modification of the principles outlined by Fux, with focused scrutiny on considerations of symmetry, yet there's a great deal to be learned by practice and sometimes failure. I hope the Mass Mysteria to be a proving ground for the application of my theoretical observations.
Edit 9/30/10: Still working on the mass (in between four other projects) which despite being a sincerely devotional and religious effort, shares also the purpose of a proving ground for the aforementioned techniques. Upoaded to my page are two motets "In Diebus Vocis Septimi Angeli", I and II, which are strict vertical mirror images of eachother. I suppose it doesn't really matter which is designated as "Inversus" and "Rectus".