Composing Music with Grammars

grammar the whole system and structure of a language or of languages in general, usually taken as consisting of syntax and morphology (including inflections) and sometimes also phonology and semantics. a particular analysis of the system and structure of language or of a specific language. a set of actual or presumed prescriptive notions about correct use of a language Computing: a set of rules governing what strings are valid or allowable in a language or text.

syntax - the arrangement of words and phrases to create well-formed sentences in a language. morphology - the study of the forms of things, in particular Linguistics: the study of the forms of words. semantics - the branch of linguistics and logic concerned with meaning. the meaning of a word, phrase, sentence, or text

Definitions token - grammatical symbol root token – root of parse tree vocabulary (V)– all the tokens plus the null token alphabet of terminals (T)– subset of vocabulary language (L) – subset of finite concatenations of tokens of T alphabet of non-terminals (N) – categories of macrostructures in V sentential form – sequence of non-terminals production (rewrtiting) rule – LHS  RHS

Generative Grammar expressed as a quadruple G = (N,T,∑,P) G - generative grammar N - alphabet of non-terminals T - alphabet of terminals ∑ - root token P - collection of production rules of the form: α → β, where α, β ∈ (N ∪ T) ∗, and (N ∩ T) = Ø. (“ ∗ ”denotes the powerset or set of all subsets)

Chomsky’s four types of grammars Free (type 0) – imposes no restrictions on the form of the production rule. Intermediate strings can expand and contract in length. Allows for infinite strings and null strings. Context-sensitive (type 1) – A α B → A β B. alpha produces beta in the context of A and B. α → Ø is forbidden. Context-free (type 2) – very useful with regards to natural language and programming. Good for representing multi-leveled syntactic formations of music. Any non-terminal (representing a macrostructural category like motive, phrase, sentence, section, movement, entire piece, etc.) may generate a string of tokens at a lower level. Finite-state (type 3) – no more than one non-terminal token may appear on each side of any production rule. Type 2 does not have this limitation.

Production/Rewrite Rules Random selection Serial selection – Useful for musical purposes. No object is selected again until all the others are selected. (Compare to serialism) Finite-state transition matrix – successive productions depend on previous productions. For example: LHS-> (A ) (B ) (C ) (D ) Selection functions – higher level complex functions that return an integer value that corresponds to an object index. Meta-production rule – a single generation by a rule is to be substituted for all occurrences of the LHS of the rule. An example of horizontal dependency.

The Bol Processor (Bel and Kippen) A tabla-oriented generative grammar based on machne learning. Acquires knowledge through parsing expert input, and attempted generation, followed by expert approval or disapproval.

EMI – experiments in Musical Intelligence (David Cope, UC Santa Cruz) Arguably the most successful and developed automated grammar for music making. Analyzes the style of a composer through batch input of a multitude of pieces. Produces pieces of music that are highly faithful to the composer’s style. Developed in response to writer’s block occurring after the spending of a large commission for an opera. Composed in 10 days.

David Cope discusses EMI in a Radiolab interview

SenGen Production Rules Sentence -> [PrepositionalPhrase], + NounPhrase1 + VerbPhrase VerbPhrase1 -> [AdverbPhrase] + Verb1 + [PrepositionalPhrase || AdverbPhrase] VerbPhrase2 -> [AdverbPhrase] + Verb2 + NounPhrase + [PrepositionalPhrase || AdverbPhrase] NounPhrase -> [Determiner,90%] + [AdjectivePhrase] + Noun + [PrepositionalPhrase] NounPhrase1 -> [Determiner,90%] + [AdjectivePhrase] + agent AdverbPhrase -> [AdverbPhrase,25%] + Adverb AdjectivePhrase -> [AdverbPhrase,15%] + Adjective + [PrepositionalPhrase] PrepositionalPhrase -> Preposition + NounPhrase + [, + PrepositionalPhrase] (last stacked PrepPhrase preceeded by and)

Note: credit must be given to Curtis Roads and Paul Wieneke from whose paper Grammars as Representations for Music, I quote directly at times in this presentation as well as the New Oxford American Dictionary application bundled with Mac OS X. S.R. Holtzman deserves credit for the section on production rules taken from Using Generative Grammars for Music Composition. For a proper bibliography, see my accompanying paper, Composing Music with Grammars.