The Logical Structure of an Algorithmic Theory of Tonal Music

for The Logical Structure of an Algorithmic Theory of Tonal Music David Meredith St. Anne’s College, University of Oxford Thesis submitted for degree of D.Phil. in Trinity term 1996 A study of the metatheoretical problems suffered by theories that aim to characterize musical styles leads to certain conclusions as to the logical structure that should be possessed by a style theory and the methodology that should be employed in a style theory research programme. Some of the research that has led to formal style theories has been based upon Chomskyan generative linguistics. But there are serious flaws in Chomsky’s methodology that have been transmitted to the musicological research influenced by it. Chomsky believes that a grammar must not only weakly generate the sentences of a language, but also strongly generate correct structural descriptions of these sentences. I believe weak generation is a sufficient condition on the adequacy of a grammar because strong generation cannot be tested. In Part 1, I propose that a theory for a musical style should be an algorithmic style theory which is a hypothesis that a composing algorithm generates all and only those pieces that are either members of a corpus or in the style of that corpus, as determined by an acceptability algorithm resembling a Turing test. An algorithmic style theory is always either true or false, and either refuted or unrefuted. It can never be verified but it should be tested for overgeneration and undergeneration until it is refuted. Chomsky believes that a linguistic performance theory should consist of a competence theory that generates grammatical sentences, supplemented by ‘performance filters’ that take grammatical sentences as input and generate acceptable sentences as output. Because acceptability can be operationally defined, a performance theory can be refutable. But grammaticalness cannot be operationally defined, so a competence theory cannot be tested. A competence theory becomes partially testable when it forms part of a performance theory, but competence should not be studied in isolation from performance. Some style theorists who study competence in isolation from performance are more concerned with the elegance of their theories than with weak generation. Others believe that avoiding overgeneration is more important than avoiding undergeneration. I believe style theorists should be more concerned with accounting for existing pieces in a style than with avoiding overgeneration or with keeping their theories as simple as possible. Many different theories might successfully approximate a style, but the complexity of acceptable music severely limits the class of possibly correct algorithmic style theories for a style. Weak generation is therefore a sufficient condition on the adequacy of an algorithmic style theory. In Part 2, I define a formal system for representing pitch in tonal music and describe how pitch relations can be represented using digraphs. I prove a theorem that relates circuits in the thirds relation digraphs with the traditional major and minor scales. Finally, I describe the progress I have made towards developing an algorithmic style theory for the style of a subset of Bach’s chorale harmonizations. (Approximate total number of words in thesis : 97000)