Good point about the lack of functional structure these two styles suffer from.I'm working on a string quartet these days so I am thinking a lot about structure. Most of what I write could be called "character pieces" which means that they are small forms. When you try to compose a larger form, the problem of structure becomes critical. Or so I think.
But for much of the last century, like taxes, the problem of structure has been avoided or evaded! There have been several strategies to avoid the problem, some obvious, others not so obvious. Moment form is one of the most obvious. I talked about this method here. In moment form you write a number of self-contained sections that can be played in any order. One of the most famous examples is Stockhausen's Klavierstücke XI. There are 19 fragments, chosen to be played in a random order and the performance is over when any fragment has been played three times. Every performance is therefore different. Here is one performance:
Does this piece have a structure? Yes, of course it does. Here, taken from the Wikipedia article I linked above, is a description of the structure:
So that should be easy to hear! Right! It is one of the principles of high modernism that serious music must have a high degree of complexity. But this kind of structure, while certainly complex, is not audible to an ordinary listener. Or probably to any listener. The analyses of this piece have been made by professional theorists specializing in contemporary music and with unlimited time to study the score. And even then, sometimes, they made mistakes in their analyses. This is why I say the structure is not audible to the listener. From the ordinary listener's point of view, this is probably no different from someone playing random notes and rhythms on the piano. In fact, the director Peter Weir did a wonderful satire on this in the film Green Card with Gerard Depardieu at the piano. The 'piece' starts at the 1:53 mark in this clip:Though composed with a complex serial plan, the pitches have nothing to do with twelve-tone technique but instead are derived from the proportions of the previously composed rhythms (Truelove 1984, 103–25; Truelove 1998, 190).The durations are founded on a set of matrices all of which have six rows, but with numbers of columns varying from two to seven. These matrices "amount to sets of two-dimensional 'scales'" (Truelove 1998, 190). The first row of each of these rhythm matrices consists of a sequence of simple arithmetic duration values: two columns of + , three columns of + + . , four columns of + + . + , etc., up to seven columns; each successive row after the first consists of increasingly finer, irregular subdivisions of that value (Truelove 1998, 192–97). These "two-dimensional scales" are then permuted systematically (Truelove 1998, 190, 202–204), and the six resulting, increasingly larger matrices were combined together to form the columns of a new, complex Final Rhythm Matrix of six columns and six rows (Truelove 1998, 190, 198–201). Stockhausen then selected nineteen out of the thirty-six available rhythmic structures to compose out into the fragments of Klavierstück XI (Pereira 1999, 121; Truelove 1998, 206; Truelove 1984, 97)...
Structure for someone like Haydn, is something for the listener to hear. Sometimes it is absolutely obvious as when the main theme returns in a first movement recapitulation, or the rondo theme returns in the last movement. Variation form is also very easily heard. But Haydn also engaged in some more subtle structures like a minuet in palindrome form or a false reprise. The false reprise is like a false entry in a Bach fugue: it sounds like it will be an entry, but evaporates and a different voice comes in with the subject. Haydn could even be more subtle with structure. In the first movement of his String Quartet op. 64, no. 4 in G major, he begins the recapitulation with the main theme in major, just as in the exposition, but almost immediately slips into the minor so you are sure this is a false reprise. But it stays in minor and brings back material from the second theme, which was in minor in the exposition. So this was a real reprise presented as a false reprise! You don't get much subtler than that. Here is that first movement:
But what we have in composers like Stockhausen is a high degree of inaudible complexity which is, from a listener's point of view, of no structural function because you can't hear it. Not as structure, at least.
John Cage, a really fascinating figure, gave up all pretense of structure and used coin-tossing methods to create pieces by chance. This can be interesting music to listen to, but I doubt anyone would claim that it has any kind of structure at all, even inaudible.
For me, the challenge as a composer is to discover ways of structuring music, an art in time. There are lots of models from Gregorian chant up to the 19th century. There are even 20th century models like Debussy, Stravinsky, Shostakovich and Steve Reich, all of whom were interested in creating audible musical structures. Here is the first movement of Stravinsky's Symphony in Three Movements with Georg Solti conducting the Chicago Symphony: