Once upon a time, I was inclined to imagine the development of musical thought and expression as taking place in a strictly linear fashion. It was irresistible to conceive of the Renaissance masters as primitive musical actors in relation to those of the Classical period, and likewise to construe the Classic school as worthy of merit, sure, but technically inferior to that of the late Romantics. The argument was simple: harmonic complexity seemed to me to reach its peak at the turn of the 20th century (we’ve all heard this claim in one guise or another) and given both this purported pinnacle and the correlate “development” of civilisation, it was no great step to extrapolate a continuum of decreasing complexity right back to the primordial element of western art music, chant.
The rich harmonic palette of Byrd’s music for instrumental consort and Purcell’s Fantasias and In Nomines, when contrasted against the foursquare harmonic forms of late Haydn and early Mozart, wrested me from the grasp of this fallacy. A recomplexification takes place at the close of the Classic period, culminating in the most parsimonious of voice-leading of the school of ‘extended tonality’ and ultimately disintegrating following the schematisation of Viennese twelve-tone composition. Reactionary or not, American minimalism and European so-called ‘sacred’ minimalism ushered in a return to less complex harmonic models, while simultaneously the old guard of the avant-garde pushed levels of complexity in musical expression through the roof. Such curves of complexity vs. time are not limited to the parameter of harmony; a case study in the transformation rhythmic interest with respect to time is the wane in the density of production of immensely complex isorhythmic choral writing of the 16th and 17th century.
It might be observed that complexity as a parameter for comparison is subjective, but if we inherit the definition of Kolmogorov complexity from the field of algorithmic information theory and modify it for our needs we arrive at a plausible definition of musical complexity, at least for the purposes of this discussion. The K-complexity of a string of characters is, roughly, the length of the smallest computer (Turing machine) program that theoretically will reproduce that string. Modifying for the language of musical analysis, we claim that the complexity of a parameter of a musical work is given by the length of the most efficient description of that parameter in a functional sense.
I think that information theory and musical thought ought to operate in tandem to arrive at more axiomatic descriptions of complexity for musical entities so that historicist theses such as that presented at the outset of this post can be refuted by argument from the non-linearity of the time/complexity correspondence.