Michael Gogins

Several years of experience composing in PTV space demonstrates beyond a doubt (to me, at least) that designing score generators to operate in voice-leading and chord spaces has extremely significant long-term artistic potential. I would say that the gamble of work invested has more than paid off as hoped. I would even go so far as to say that, upon it, probably rest hopes for algorithmically generating music of historic significance, or for the future appeal of generative music to wider audiences.

But experience also demonstrates that there are fundamental problems in current geometric representations of music. The main problem is that while composers easily produce voice-leadings from one arity of pitch-class set to another, modeling this in geometric music theory is not trivial and has not been done in a satisfactory manner. It’s not a show-stopper, but a solution would greatly magnifiy the significance of the approach.

The reason for preferring geometric music theory (as in the work of Tymoczko, Fiore, or Chew) over pattern-based or heuristic approaches (such as that of David Cope) is that the geometric approach seems more mathematically concise, more scientifically fundamental, and more stylistically flexible. I believe it is increasingly evident that the geometric approach is grounded in deep, cross-cultural structures of musical perception. I hope that musicologists will get even more scientific and do more real psychological experiments to empirically validate the extent of this grounding and to identify the theories that best fit the evidence and are most productive for future research.

However, of course, composing is not musicology. For me, the path forward lies in staying current with the musicological research, especially with respect to geometrically modeling progressions in functional harmony and of arbitrary arity, while trying to come up with computer implementations that work for real composing. It may be necessary to move to spaces that do not so directly map voices to dimensions. The spaces also need to incorporate other dimensions of voices in addition to just pitch-class or pitch.

I can and will proceed to investigate simultaneously both generative algorithms, and the fundamental spaces involved in adequate representations of compositional operations. With respect to algorithms, the two basic algorithms will probably remain rewriting systems (such as Lindenmayer systems) and attractors (such as iterated function systems). Mapping onto musical spaces is certainly easier with rewriting systems, but the highly desirable goal of enabling parametric composition seems to be easier to achieve with attractors.

Defining spaces that represent chords of arbitrary arity with additional attributes of each voice such as loudness, duration, and instrument assignment, and finding convincing mappings to such spaces from the attractors of iterated function systems or other parametrically mappable dynamical systems, would hit the ball miles out of the park and that is what I am trying to do.

I have indeed been trying to do this since 1984, when I got the idea after hearing Benoit Mandelbrot lecture on the Mandelbrot Set at the University of Washington during the period I was taking a seminar in computer music with John Rahn. That’s a long time to work on a single problem! But the thing is, I definitely am far closer to a solution than I was at the beginning. I know how to do the parametric mapping. I can prove that some such systems are musically universal. I now know how to model and generate voice-leadings and chord progressions, instead of just sprays of notes.  I have preliminary ideas on functional progressions and perhaps even prolongations. The only remaining problems are the arity problem and the attractor mapping problem.

The mapping problem has several potential solutions, such as fractal interpolation, or simply collapsing the dimensions of attractors down to a few. The arity problem is the only remaining fundamental problem.