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The Blues and the Abstract Truth: Music and Mental Models

The idea that there is a grammar of music is probably as old as the idea of grammar itself, and the idea that there should be formal grammars of music followed equally hard upon the Chomskean application to natural languages of the formal techniques used to analyse logical and mathematical languages – see Winograd 1968, Lindblom and Sundberg 1972, Steedman 1973, 1977, Longuet-Higgins 1978, Johnson-Laird 1991, and many others reviewed by Sundberg and Lindblom 1991. Nevertheless, it is probably fair to say that such musical formal grammars have lagged behind the linguistic ones in terms of descriptive adequacy, constraint on formal generative power, psychological plausibility, and computational practicality, especially as far as harmonic analysis goes. For example, Steedman 1984 offers a generative grammar for chord progressions in jazz twelve-bar blues. The grammar is used to account for the fact that the chord sequences shown in Figure 1 – the last of which shares no bars in common with the first, other than the fifth and final bars – are perceived by jazz musicans as being in some sense “paraphrases”. That is, they are all instances of the twelve bar form, one of the basic forms of the music. (The notation, which is for the most part standard, is explained in an appendix to the present chapter.) The 1984 paper attempts to capture this fact in a small number of rules modeled on the linguist’s device of a “rewriting rule”. The principal rules are shown in Figure 2, in which is a variable over relative chord roots , , etc, and is to be read as “the chord that is ’s (so that if is , is ). I will briefly summarise the motivation for these rules, but the reader is directed to the earlier paper for a fuller account. Rule 0 simply defines the skeleton of the twelve bar as two two bar units of tonic harmony, followed by two two bar units constituting a “plagal cadence” – a progression