Recursion and Proportion in Music

The composer Alan Bush wrote an introduction to Erno Lendvai's book "Bela Bartok: an analysis of his music" which is particularly assertive. Welcoming the analysis of Lendvai, and situating it with the analytical work of Asaviev (about whom I will post soon - he's important), Hindemith, Cooke and Ansermet, he forcefully rejects the inclusion of Schoenberg into this canon of music theorists. Citing Schoenberg in Style and Idea, where he says:

"the term emancipation of the dissonance refers to its comprehensibility, which is considered equivalent to the consonances's comprehensibility. A style based on this treats dissonances like consonances and renounces a tonal centre"

Bush argues that this is nonsense. He says:

"dissonance is not the same as consonance; it has different acoustical and physiological effects. Therefore dissonance ought not to be treated as if it were identical with consonance. And in any case the renunciation of a tonal centre does not follow from an previously stated proposition and is merely a dogmatic assertion of the composer's [Schoenberg's] belief."

Bush's objection to this is on scientific grounds. There is no proof for the assertion that Schoenberg makes. Indeed there may be more scientific evidence for the opposite.

I find myself in a curious position here, because I love Schoenberg's music. I also think that Schoenberg was one of the more sensible theorists of tonal music. To me, it feels like Brahms (big fist-fulls of notes) and has the same passionate intensity. Although not serial, the Op.11 piano pieces are as great as any Brahms intermezzo.

But Bush's point is made in the context of supporting Lendvai, who has a thesis about proportion, and in particular Golden Section, Fibonacci proportion, in the music of Bartok. 

The Fibonacci numbers are clearly audible in Bartok - and indeed all those 3, 5, 8, 13s  all create the sense of driving vitality which is the hallmark of his music. But I don't really find Schoenberg's piano pieces any less well-proportioned. Is it Fibonacci? Are there Golden Sections in those wild cadenza-like passages in the opening? Probably not. But there is something which in the hands of a skilled performer, gives the things passionate consistency.

Recalling my last three posts on metagames and absence and conviviality,  the Golden Section analysis of Lendvai has been fascinating me. Since I have made so much of the point of the recursion of concepts which help to simplify the metagame tree and so lead to understanding, I have begun to think that Bartok's Golden Section rhythms and harmonies are also recursive in a similar way. The music explains itself by revealing the recursive principles which underlie it. As those principles are grasped by the listener, and the composer plays around with expectations, so a sense of finality and satisfaction is gradually generated. 

But the recursions in music are much deeper that simple numerical sequences. I might be tempted to argue that in the Schoenberg, the phrasing and the contour of the melodies (yes, they are melodies - and rather beautiful ones at that), sets up its own pattern. Little motifs are suggested in the opening, only to become important in the music's close (for example, the motif below)

But then there are also the levels of recursion between the melodic form, the phrasing and articulation, and the overall structure. Remembering that I have argued that new concepts (recursions) arise out of perceived absences, I wonder how it is that those moments of melodic suggestion and sudden bursts of emotion in the opening give rise to absences in the listener, which the composer then suggests simple formulae (like the  one above) that might serve to simply the complex web of metagame possibilities that the listener is grappling with.

I don't yet have a more precise mathematical analysis of this. But I am not unconvinced of the possibility that one might be found.