Chapter 11
Musical Design II: Grammars

The typescript for chapter 11 is provided in AutomatedComposition11.pdf.

The typescript defines a grammar as “a scheme of relationships which explains how the elements of a structure join together into the whole.” Grammars are thus suitable for explaining both hierarchical structures like the organization of the Catholic church, and non-hierarchical structures like family trees.

Next follows a series of definitions from Noam Chomsky's Syntactic Structures. Relationships between grammatical units are distinguished between phrase-structures and transformations. Among phrase-structures, furthur distinction is drawn between context-free grammars and context-sensitive grammars:

Synthesis, or derivation, is seen as a process of taking a fairly general concept called the archetype (or axiom) and applying a series of productions to deduce specifics from generalities, ultimately obtaining a specific instance, or statement of the archetype. The inverse of synthesis is analysis or parsing, which begins with a statement and applies reductions to obtain an archytype.

Markov chains have been categorized as context free with productions of the form X → aX, but this typescript asserts that Markov chains are actually leftward context sensitive, with productions of the form aX → abX.

An illustration is provided whereby measures 41-43 from Stravinsky's Variations: Aldous Huxley in Memorium are text-encoded using Alan Ashton's linear music code. This text is then charted as a nested structure with the composition (the archetype), elaborating out into sections, measures, parts, durations & pitches, and ultimately into textual units.

The concept of self similarity is introduced in connection with recursive grammars that apply the same small, efficient set of productions to nested levels of organization.

A survey of hierarchical systems of musical analysis distinguishes between functional approaches where discrete notes serve as grammatical units at all levels of generality and architectural (also architectonic) approaches which identify single notes only on their most specific levels. Functional approaches include Gottfried Weber's 1817 Theory of Composition and Heinrich Schenkers 1935 system of graphic reduction. Archetectural approaches include the 1924 system devised by Alfred Lorenz to analyze the music of Richard Wagner, the procedures for rhythmic analysis devised in 1960 by Leonard Meyer and Grosvenor Cooper, and James Tenney's method of analysis based upon principles of Gestalt psychology. Work published in 1983 by Ray Jackendoff and Fred Lerdahl has applied non-hierarchical models to account for dualities of function.

James Tenney used nested structures in early computer composed pieces including his 1963 Phases for computer-generated tape and his 1964 Music for Player Piano. Tenney's 1984 Bridge for two pianos employed a four-tier self-similar organization in which “sequences” divided into “clangs”, which divided in turn into “elements” (chords), and which ultimately divided into individual notes. Tenney's approach, motivated by Gestalt psychology, was the direct inspiration for my own 1980 Crystals.

A second historical thread has seen composing programs directly influenced by Chomsky's linguistic model. These began with a 1978 document by Curtis Roads describing a utility which accepted an archetype expressed as a sequence of “tokens”, a set of context-free productions causing “non-terminal” tokens to be rewritten as strings of “non-terminal” and/or “terminal” tokens, and a set of mappings from “terminal” tokens to musical objects. This context-free implementation was extended to context-sensitive and weighted random scenarios with Steven Holtzman's 1980 “generative grammar definitional language for music”, and with Bernard Bel's “bol processor”, versions of which existed as early as 1982.

Kevin Jones's 1981 paper “Compositional Applications of Stochastic Processes” describes an approach which carves chunks of musical space into progressively smaller pieces until the program finally obtains descriptions of individual notes. The approach closely resembles what I did in Crystals, but was arrived at along an independent path with explicit Chomskian influences.

Another of my compositions, Transitions, was one of the suite of projects created for the U.S. Pavilion at Expo '85 in Tsukuba, Japan. By this time I had become aware of the Chomskian model. My composing program used a recursive grammar to generate a newtwork of similarities and contrasts. To present Transitions, Bob Franki developed animated graphics which actively displayed the process beginning from a single block representing the entire piece as an initial origin transitioning to a final goal, then carving this block out while inserting recursive sub-goals and sub-origins. A snapshot of this graphic was reproduced in Perspectives of New Music. This network was subsequently fleshed out using the constrained search method described in Chapter 14. Further implementation details for this project appear in the 1985 ICMC Proceedings. An Oberheim eXpander realization ofTransitions is available on this site in MP3 format.

The practical programming content of Chapter 8 was provided by two examples: Demonstration 8: A Functional Grammar and Demonstration 9: An Architectural Grammer. Both process illustrated the principle of top-down design in that general facts about the form as a whole were progressively refined to produce specific facts about content. Demonstration 8 progressively elaborates a simple melody with escaping tones, reaching tones, and changing-note groups. Demonstration 9 carves a region of musical space into progressively smaller regions. Both examples employ self-similar productions to deduce attributes of components from their containers; however, both qualify as context sensitive because they employ statistical feedback to influence selections of chromatic material.

© Charles Ames Page created: 2017-03-12 Last updated: 2017-03-12