Meta-complexity in Brahms’s Clarinet Quintet

A nthony Burton’s program notes for the London Phil’s ‘Chamber Contrasts’ series at Wigmore Hall emphasize how impressed Brahms was with the potential of clarinet, as a result of his hearing clarinettist Richard Mühlfeld perform in 1891. But Brahms composed this Op. 115 clarinet quintet within a few years of his death from cancer, a period when many of his friends and family died (his sister Elise, his brother Fritz, Elizabeth von Herzogenberg, Hermione Spies, Theodor Broth, Clara Schumann, etc.). This makes me think that the recursive, ‘re-considering’ structures in this quintet maybe derive more from the emotional impact of those events on Brahms—than from any scheme Brahms may have had, to create in Op. 115 a virtuosic clarinet showpiece for the great Mühlfeld.

L arge-scale variation comes from relationships between variations (the theme included) and how these relationships are united into a pervasive algorithm that encapsulates the entire set—that is, an operating system governing the entire set. In other words, these are not a mere series of independent objects and methods (scripts) that are expected derivatives of a parent object or class. They are a ‘network-based object-oriented OS’.

W hat I mean is, Brahms’s variations feel to me ‘network-based’, not ‘hierarchical’. It seems (to me) that Brahms was exposing larger-scale connections, and transforming his idea into a multi-domain network rather than a hierarchical structure with conventional ‘parent’ exposition classes and ‘child’ instantiations. The arrangement of variations involves pairs of diatonic pitch-class cells derived from the theme. But the textural differences that follow are like remote procedure-calls that cross-link the tension-producing structures within the variations’ designs. This gives the piece an attractive sort of ‘meta-complexity’.

I n other people’s analyses (links below), variation sets in Brahms’s multi-movement instrumental compositions are usually only moderately complex. Variation sets in interior movements often do not have full ‘closure’. In the variations in this Clarinet Quintet in B minor, Brahms suggests or ‘projects’ closure by recapitulating first-movement phrases. But the promised closure doesn’t come until the very end. It is as though there is a latent, suspended complexity that is irreducible until the very end.

T his gets me to thinking about how Brahms does this, and how we might devise similar structures and mechanics in new music. The ‘Kolmogorov complexity’ (links below) of a software program or script, or a piece of text or a musical composition, is a measure of the computational resources needed to specify the object. Kolmogorov complexity is also known as descriptive complexity, Kolmogorov-Chaitin complexity, stochastic complexity, algorithmic entropy, or program-size complexity.

O ne of the appealing, intuitively plausible qualities of Kolmogorov complexity as a metric to characterize what I was hearing last night in the account of the Brahms clarinet quintet given by Robert Hill and his colleagues has to do with how the aggregate complexity goes, for large composite or ‘compound’ or ‘network’ or ‘cloud’ objects that are constituted from multiple smaller objects. For example, consider a compound object built up from objects X and Y. The Kolmogorov complexity of the two of them together K(X,Y) is:

K(X,Y) = K(X) + K(Y|X) + O(ln(K(X,Y)))

T his says that the shortest program that reproduces X and Y is no more than a logarithmic term larger than a program to reproduce X and a program to reproduce Y given X. This basically puts a quantitative ‘bound’ on the amount of mutual information there is between X and Y in terms of the Kolmogorov complexity.

T he amount of unresolved tension we feel in the inner movements in this Brahms quintet—and the intensity of the desire we feel for the eventual resolution of these tensions at the end—feels, to me, logarithmic. Maybe some future music theory PhD candidate will explore this notion properly and critically. For now, it’s happy enough for me to give you a few interesting links below and the glimmer of a potential reason why the complexities and proportions of this quintet seem so ‘right’ and so beautiful.

• Robert Hill page at Royal College of Music
  
• Robert Hill page at LPO
  
• Brahms clarinet quintet in B minor score (free download at IMSLP)
  
• Chain Rule for Kolmogorov complexity (page at Wikipedia)
  
• Wodowski clarinet mouthpieces (used by Robert Hill and others)
  
• Calude C. Information and Randomness: An Algorithmic Perspective. 2e. Springer, 2010.
  
• Dovern W. Polyphony & Complexity. Wolke, 2002.
  
• Downey R, Hirschfeldt D. Algorithmic Randomness and Complexity. Springer, 2010.
  
• Horne W. Review: The variations of Johannes Brahms. Music and Lett 2006;87;334-7
  
• Katz M. Templets and the Explanation of Complex Patterns. Cambridge Univ, 2009.
  
• Leister K. Brahms Clarinet Quintet. (Brilliant, 2007.)
  
• Li M, Vitanyi P. An Introduction to Kolmogorov Complexity and Its Applications. 2e. Springer, 1997.
  
• Littlewood J. The Variations of Johannes Brahms. Plumbago, 2004.
  
• Manno R. Brahms Clarinet Quintet. (Oehms, 2009.)
  
• Miller F, Vandome A, McBrewster J, eds. Chain Rule for Kolmogorov Complexity. AlphaScript, 2010.
  
• Narmour E. The Analysis and Cognition of Melodic Complexity: The Implication-Realization Model. Univ Chicago, 1992.
  
• Pacun D. Large-scale form in selected variation sets of Johannes Brahms. PhD Dissertation, Univ Chicago, 1998.
  
• Walton B. Brahms Clarinet Quintet. (Testament, 2007.)