Maximal Chaos and Complexity Measures

I was glad to see that my analogy between musical "spaces" and abstract mathematical spaces made it into the discussion in Chapter 4 which I just finished reading. Apparently I was one step ahead of the game.

An interesting notion in the reading that paralleled my current research in my Complex Analysis class involved the "complexity measure." In trying to quantify the amount of complexity in an improvised piece one may choose to analyze many different features or even the piece as a whole to see what elements change the song as a whole. In much the same way, I am currently working with a couple fellow students to identify what causes maximal chaos in a complex system.

Borgo makes the point that throughout the course of an improvised piece, if the performers choose to branch into unfamiliar playing styles or note sequences, the listeners may become uncomfortable or feel distant to the music being played. With increasing amounts of chaos in the music, this could spell trouble for the performer-audience relationship. Likewise, when creating music with The John Fox Company, one of the things we have explicitly tried to avoid while performing music is an extreme amount of chaos. The nature of the music is already unconventional as it is, with the conglomeration of odd sounds being layered on top of each other in a sea of electronic chaos. Too many strange sounds on top of each other may begin to sound like a drone or may become unintelligible, and the surprises we are able to create in the music become drowned out and thus the musical experience is less effective.

In the mathematical research in complex analysis, we take a similar approach to the problem of achieving maximal chaos. Without going into too much detail, we have essentially picked specific families of functions that are generalized and we can easily study the behavior of under iteration, and we play with different methods of iteration in order to produce the most "picture" which translates into maximal chaos. The more stuff going on in our graph, the more chaos is present. By restricting ourselves to such familiar functions, we are doing what many improvisers do in their music; that is, the improviser of music will choose phrases wisely or stick to a particular chord progression in order to maintain the element of surprise as well as the audience's attention. The element of familiarity avoids alienating the audience, in much the same way that well-understood functions give us a starting point in research and allow us to draw sensible conclusions rather than give us information that could be interpreted many ways if we are not familiar with the dynamics of the functions.