Musical Syntax I: Theoretical Perspectives

Abstract

The understanding of musical syntax is a topic of fundamental importance for systematic musicology and lies at the core intersection of music theory and analysis, music psychology, and computational modeling. This chapter discusses the notion of musical syntax and its potential foundations based on notions such as sequence grammaticality, expressive unboundedness, generative capacity, sequence compression and stability. Subsequently, it discusses problems concerning the choice of musical building blocks to be modeled as well as the underlying principles of sequential structure building. The remainder of the chapter reviews the main theoretical proposals that can be characterized under different mechanisms of structure building, in particular approaches using finite-context or finite-state models as well as tree-based models of context-free complexity (including the Generative Theory of Tonal Music) and beyond. The chapter concludes with a discussion of the main issues and questions driving current research and a preparation for the subsequent empirical chapter Musical Syntax II.

Appendix: The Chomsky Hierarchy

Noam Chomsky introduced a containment hierarchy of four classes of formal grammar in terms of increasing restrictions placed on the form of valid rewrite rules [25.52]. A formal grammar consists of a set of nonterminal symbols (variables), terminal symbols (elements of the surface), production rules, and a starting symbol to derive productions. In the following description, $a\in T^{\mathrm{*}}$ denotes a (possibly empty) sequence of terminal symbols, $A,B\in V$ denote nonterminal symbols, $\alpha\in(V\cup T)^{\mathrm{+}}$ denotes a nonempty sequence of terminal and nonterminal symbols, and $\beta,\beta^{\prime}\in(V\cup T)^{\mathrm{*}}$ denote (possibly empty) sequences of terminal and nonterminal symbols. The difference between different formal grammars in the Chomsky hierarchy relates to different possible production rules.

Every grammar in the Chomsky hierarchy corresponds with an associated automaton: while formal grammars generate the string language, formal automata specify constraints on processing or generating mechanisms that characterize the formal language. Automata provide an equivalent characterization of formal languages as formal grammars.

1.1 Type 3 (Regular)

Grammars in this class feature restricted rules allowing only a single terminal symbol, optionally accompanied by a single nonterminal, on the right-hand side of their productions

$$\begin{aligned}\displaystyle A&\displaystyle\rightarrow a\\ \displaystyle A&\displaystyle\rightarrow aB\quad\text{(right linear grammar) or}\\ \displaystyle A&\displaystyle\rightarrow Ba\quad\text{(left linear grammar)}\;.\end{aligned}$$

Regular grammars generate all languages which can be recognized by a finite-state automaton, which requires no memory other than a representation of its current state.

It is essential to note that regular grammars are not equivalent to Markov models or k-factor languages (see Sect. 25.4.1 above).

1.2 Type 2 (Context Free)

Grammars in this class only restrict the left-hand side of their rewrite rules to a single nonterminal symbol – i. e., the right-hand side can be any string of nonterminal symbols

$$A\rightarrow\alpha\;.$$

The equivalent automata characterization of a context-free language is a nondeterministic pushdown automaton, which is an extension of finite-state automata that employs memory using a stack and state transitions may depend on the current state as well as the top symbol in the stack.

1.3 Type 1 (Context Sensitive)

Grammars in this class are restricted only in that there must be at least one nonterminal symbol on the left-hand side of the rewrite rule and the right-hand side must contain at least as many symbols as the left-hand side, i. e., string length increases monotonically in the production sequence.

$$\beta A\beta^{\prime}\rightarrow\alpha,\quad\left|\beta A\beta^{\prime}\right|\leq\alpha$$

Context-sensitive languages are equivalently characterized by a linear bounded automaton, that is a state-machine extended by a linear bounded random access memory band, whose transitions depend on the state, the symbol on the memory band.

1.4 Type 0 (Unrestricted)

Grammars in this class place no restrictions on their rewrite rules

$$\alpha\rightarrow\beta$$

and generate all languages which can be equivalently characterized by a universal Turing machine (the recursively enumerable languages), which is the same as the linear bounded automaton for context-sensitive languages without bounds on the memory tape.