Okay, I'm not quite done with complexity yet. As indicated in the last post, complexity, for better or worse or (mostly) in-between, can describe a number of musical qualities or characteristics. When used in connection with something very specific, especially countable items (notes on a page, for example), the aspect of complexity described can acquire a quantitative dimension and make comparisons within and without a piece of work possible; also, when a clear process is used in composition, one might speak usefully in terms of a measurable computational complexity (which, of course, has its own complications, indeed uncertainties, which many others online can describe much more credibly). However, some of the most important forms of musical complexity are far less easily reducible to the quantitative, and some methods, while quantifiable, as often not particularly relevant when quantified to any actual music. Among these forms (incomplete, I'm sure, off the top of my head and jotted on the back of a restaurant napkin) are:
*Linear complexity — How variegated or detailed is a single line of music? Variety of materials, density and speed of their deployment, differentiation into principle, subordinate, and ornamental parts. Which may be related to
* Textural complexity — Is a complex or ensemble heard as individual parts or a single gestalt? An ensemble texture may be found to be full of activity, indeed layers of activity — when analyzed into individual components — and there may well be considerable change over shorter or longer periods of time, but the net effect may instead tend rather towards coherence and continuity. Which may be related to
* Pre-compositional complexity. Like textural complexity, this is more about forests than trees: in much serial music, for example, the actual set-, row-, and array-work, while possibly comprising the greater part of the compositional time and effort, is actually the lesser part of the final composition. The pre-compositional design can make it possible to guarantee that the final composition has particular characteristics — for example a uniform (or weighted, as the case may well be) distribution of elements — which make the end result, the sound itself, possible, without making the particular methods used at all necessary for a listener to know. Which may be closely related to
* Mathematical complexity. This is used probably more as a term of deprecation than estimation. However, in itself, using mathematics in composition shouldn't automatically raise objections, and the variety of mathematics used shouldn't automatically lead to a particular evaluation. Music theory journals happen to be filled with articles about a rather narrow variety of musically-applied mathematics (12-tone so-called "set theory" for example), but this reflects more on a heavily encouraged area of research than on the extents and limits of mathematical applications in music. There are numerous lines of mathematics-related musical research and composition that have both received less academic attention and are no less deep or complex in their musical applications or mathematical sophistication. I think, for example, that my recent work with Beckett-Gray codes is such an example, and as applied to scoring patterns in an ensemble work, has an acoustical immediacy that demonstrates perfectly that the mathematically complex can not only be completely accessible but musically compelling.
*Material complexity. Literally, how much and how wide a variety of stuff is in a piece of music? There should be abundant evidence out there that simply adding more stuff can make a work messier but does not automatically make a piece of music more complex, while the reduction of some material density can often facilitate the perception of complex musical phenomena that would otherwise go unheard. Many works of classic early minimalism have this characteristic.
*Associative complexity: To what degree does a work depend upon its connections to other works or repertoires? To what degree is does a work depend upon connections to extra-musical phenomena? Which may be closely related to
*Systematic complexity. How does a work or group of works cohere and create networks of internal or external relationships? The systematic complexity of Wagner's Ring or Tolkien's Middle Earth books have encouraged generations of geeks to commit large portions of their lives, memories, and imaginations to the acquisition of ever more details, terms, lists, characters, genealogies, maps, languages, motives etc. in the gradual assimilation of artworks that aspire to evoke alternative worlds. The material is simply rich enough in its structures and connections — many of them fragmentary, incomplete, ambiguous, or illogical, but hey, what else do you expect from a world? — to sustain indefinitely ethusiastic — geekily enthusiastic — extended examination. (One common effect of such engagement is its sublimation into gaming, aform of play in which a fictional world can be permutated and extended well beyond the boundaries of a completed piece of fiction.) While neither of these examples are particularly attractive to me, I do think that they exemplify the potential for an artwork enbracing an encyclopedic form of complexity to be successful and even popular. (I am much more taken in by the of-this-world-ness and flowing diction of Finnegans Wake or the erotic systematics of Duchamp's art and writings than Tolkien's stubbornly wooden diction and prudishness (to be fair, Tolkien suffered from being born into the most deadly wooden age of philology and translation; it shows up on every page of his work); likewise, I find the inter-penetrating worlds of musical and historical experience in Ives's Fourth or Second Orchestral Set or Cage's Songbooks simply more musically attractive and personally relevant than Wagner, but maybe that's just my own geekhood).