The generative theory of tonal music (GTTM) is a system of music analysis developed by music theorist Fred Lerdahl and linguist Ray Jackendoff. First presented in their 1983 book of the same title, it constitutes a "formal description of the musical intuitions of a listener who is experienced in a musical idiom" with the aim of illuminating the unique human capacity for musical understanding. The musical collaboration between Lerdahl and Jackendoff was inspired by Leonard Bernstein's 1973 Charles Eliot Norton Lectures at Harvard University, wherein he called for researchers to uncover a musical grammar that could explain the human musical mind in a scientific manner comparable to Noam Chomsky's revolutionary transformational or generative grammar. Unlike the major methodologies of music analysis that preceded it, GTTM construes the mental procedures under which the listener constructs an unconscious understanding of music, and uses these tools to illuminate the structure of individual compositions. The theory has been influential, spurring further work by its authors and other researchers in the fields of music theory, music cognition and cognitive musicology.
Theory GTTM focuses on four hierarchical systems that shape our musical intuitions. Each of these systems is expressed in a strict hierarchical structure where dominant regions contain smaller subordinate elements and equal elements exist contiguously within a particular and explicit hierarchical level. In GTTM any level can be small-scale or large-scale depending on the size of its elements.
Structures
I. Grouping structure GTTM considers grouping analysis to be the most basic component of musical understanding. It expresses a hierarchical segmentation of a piece into motives, phrases, periods, and still larger sections.
II. Metrical structure Metrical structure expresses the intuition that the events of a piece are related to a regular alternation of strong and weak beats at a number of hierarchical levels. It is a crucial basis for all the structures and reductions of GTTM.
III. Time-span reduction Time-span reductions (TSRs) are based on information gleaned from metrical and grouping structures. They establish tree structure-style hierarchical organizations uniting time-spans at all temporal levels of a work. The TSR analysis begins at the smallest levels, where metrical structure marks off the music into beats of equal length (or more precisely into attack points separated by uniform time-spans) and moves through all larger levels where grouping structure divides the music into motives, phrases, periods, theme groups, and still greater divisions. It further specifies a “head” (or most structurally important event) for each time-span at all hierarchical levels of the analysis. A completed TSR analysis is often called a time-span tree.
IV. Prolongational reduction Prolongational reduction (PR) provides our "psychological" awareness of tensing and relaxing patterns in a given piece with precise structural terms. In time-span reduction, the hierarchy of less and more important events is established according to rhythmic stability. In prolongational reduction, hierarchy is concerned with relative stability expressed in terms of continuity and progression, the movement toward tension or relaxation, and the degree of closure or non-closure. A PR analysis also produces a tree-structure style hierarchical analysis, but this information is often conveyed in a visually condensed modified "slur" notation. The need for prolongational reduction mainly arises from two limitations of time-span reductions. The first is that time-span reduction fails to express the sense of continuity produced by harmonic rhythm. The second is that time-span reduction—even though it establishes that particular pitch-events are heard in relation to a particular beat, within a particular group—fails to say anything about how music flows across these segments.
More on TSR vs PR It is helpful to note some basic differences between a time-span tree produced by TSR and a prolongational tree produced by PR. First, though the basic branching divisions produced by the two trees are often the same or similar at high structural levels, branching variations between the two trees often occur as one travels further down towards the musical surface. A second and equally important differentiation is that a prolongational tree carries three types of branching: strong prolongation (represented by an open node at the branching point), weak prolongation (a filled node at the branching point) and progression (simple branching, with no node). Time-span trees do not make this distinction. All time-span tree branches are simple branches without nodes (though time-span tree branches are often annotated with other helpful comments).
Rules Each of the four major hierarchical organizations (grouping structure, metrical structure, time-span reduction and prolongational reduction) is established through rules, which are in three categories:
The well-formedness rules, which specify possible structural descriptions. The preference rules, which draw on possible structural descriptions eliciting those descriptions that correspond to experienced listeners’ hearings of any particular piece. The transformational rules, which provide a means of associating distorted structures with well-formed descriptions.
I. Grouping structure rules
Grouping well-formedness rules (G~WFRs)
"Any contiguous sequence of pitch-events, drum beats, or the like can constitute a group, and only contiguous sequences can constitute a group." "A piece constitutes a group." "A group may contain smaller groups." "If a group G1 contains part of a group G2, it must contain all of G2." 'If a group G1 contains a smaller group G2, then G1 must be exhaustively partitioned into smaller groups."
Grouping preference rules (G~PRs)
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