Repetition and the Apophatic in Music: An Information Theoretical Approach

All analysis involves the application of constraint on experience. “Constraint” itself can be variously defined as background, absence or context – it is the domain of the “not there”. In Information Theory, it is measureable as Shannon’s ‘redundancy’, the inverse of information. Recent scholarship in Information Theory has borrowed a term from theology, labelling the broader domain of “not information” (beyond Shannon) as Apophatic.

At the heart of the Shannon notion of redundancy are concepts of repetition, similarity, identity and analogy. Since Hume, the identification of similarity and analogy has introduced questions about human reasoning which cast doubt on assumptions about expectation, induction, causation and probability. Hume famously considered the likeness of eggs, but the likeness of melodies, themes, harmonic patterns, rhythms and so forth is, I argue, more compelling because it carries with it the visceral dimension that is shared between musicians and intrigues analysts.

In considering Bach’s fugue in Ab major, I start from the perspective of an early champion of Shannon’s work, the cybernetician and psychiatrist Ross Ashby. Ashby argued that:

“The principle of analogy is founded upon the assumption that a degree of likeness between two objects in respect of their known qualities is some reason for expecting a degree of likeness between them in respect of their unknown qualities also, and that the probability with which unascertained similarities are to be expected depends upon the amount of likeness already known.”

(b)

(a)

The Bach fugue presents a variety descriptions of different aspects of the music, where each description considers the counting of and distribution of particular features (rhythm, melody, intervals, etc). For example, how is (a) the same as (b)? Each description exists in the context of other descriptions; each description constrains other descriptions (e.g. the description of rhythm is constrained by the description of dynamics or pitch). Moreover, the identification of similarity within each description entails assumptions about the degree of likeness in “unknown qualities”. As the music unfolds, some of these assumptions about unknown qualities will be revealed to be errors, causing continuous reassessment of what counts as the same and what doesn’t.