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.Chomsky, Noam (1957). Syntactic Structures. The Hague: Mouton; Chomsky, Noam (1965). Aspects of the Theory of Syntax. Cambridge, Massachusetts: MIT Press; Chomsky, Noam (1966). Topics in the Theory of Generative Grammar. The Hague: Mouton.
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.Jackendoff, Ray (1987). Consciousness and the Computational Mind. Cambridge, Massachusetts: MIT Press; Temperley, David (2001). The Cognition of Basic Musical Structures. Cambridge, Massachusetts: MIT Press; Lerdahl, Fred (2001). Tonal Pitch Space. New York: Oxford University Press; Lerdahl, F., & R. Jackendoff (2006). "The Capacity for Music: What Is It, and What's Special About It?" Cognition, 100.1, 33–72.
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. They have two functions: to establish tree-structure relations (time-span trees), and to provide rhythmic criteria to supplement pitch criteria that determine the structural importance of events (p. 119). 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-spansA time-span is a length of time spanning from one metrical event up to, but not including, the next event. (This is the minimal condition on 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.Harmonic rhythm is the pattern of durations produced by changes in the harmony at the musical surface. 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)
alternative form: "Avoid analyses with very small groups – the smaller, the less preferable." (Proximity) Consider a sequence of four notes, n1–n4, the transition n2–n3 may be heard as a group boundary if: (slur/rest) the interval of time from the end of n2 is greater than that from the end of n1 to the beginning of n2 and that from the end of n3 to the beginning of n4 or if (attack/point) the interval of time between the attack points of n2 and n3 is greater than between those of n1 and n2 and between those of n3 and n4.
(Change) Consider a sequence of four notes, n1–n4. The transition n2–n3 may be heard as a group boundary if marked by (Register) the transition n2-n3 involves a greater intervallic distance than both n1-n2 and n3-n4, or if (Dynamics) the transition n2-n3 involves a change in dynamics and n1-n2 and n3-n4 do not, or if (Articulation) the transition n2-n3 involves a change in articulation and n1-n2 and n3-n4 do not, or if (Length) n2 and n3 are of different length and both pairs n1,n2 and n3,n4 do not differ in length.
(Intensification) A larger-level group may be placed where the effects picked out by GPRs 2 and 3 are more pronounced. (Symmetry) "Prefer grouping analyses that most closely approach the ideal subdivision of groups into two parts of equal length." (Parallelism) "Where two or more segments of music can be construed as parallel, they preferably form parallel parts of groups." (Time-span and prolongational stability) "Prefer a grouping structure that results in more stable time-span and/or prolongational reductions."
Transformational grouping rules
Grouping overlap (p. 60) Given a well-formed underlying grouping structure G as described by GWFRs 1-5, containing two adjacent groups g1 and g2 such that g1 ends with event e1, g2 begins with event e2, and e1 = e2
a well-formed surface grouping structure G' may be formed that is identical to G except that it contains one event e' where G had the sequence e1e2, e'=e1=e2 all groups ending with e1 in G end with e' in G', and all groups beginning with e2 in G begin with e' in G'.
Grouping elision (p. 61). Given a well-formed underlying grouping structure G as described by GWFRs 1-5, containing two adjacent group g1 and g2 such that g1 ends with event e1, g2 begins with event e2, and (for left elision) e1 is harmonically identical to e2 and less than e2 in dynamics and pitch range or (for right elision) e2 is harmonically identical to e1 and less than e1 in dynamics and pitch range,
a well-formed surface grouping structure G' may be formed that is identical to G except that it contains one event e' where G had the sequence e1e2, (for left elision) e'=e2, (for right elision) e'=e1,
all groups ending with e1 in G end with e' in G', and all groups beginning with e2 in G begin with e' in G'.
II. Metrical structure rules
Metrical well-formedness rules (M~WFRs)
"Every attack point must be associated with a beat at the smallest metrical level present at that point in the piece." "Every beat at a given level must also be a beat at all smaller levels present at that point in that piece." "At each metrical level, strong beats are spaced either two or three beats apart." "The tactus and immediately larger metrical levels must consist of beats equally spaced throughout the piece. At subtactus metrical levels, weak beats must be equally spaced between the surrounding strong beats."
Metrical preference rules (M~PRs)
(Parallelism) "Where two or more groups or parts of groups can be construed as parallel, they preferably receive parallel metrical structure." (Strong beat early) "Weakly prefer a metrical structure in which the strongest beat in a group appears relatively early in the group." (Event) "Prefer a metrical structure in which beats of level Li that coincide with the inception of pitch-events are strong beats of Li." (Stress) "Prefer a metrical structure in which beats of level Li that are stressed are strong beats of Li." (Length) Prefer a metrical structure in which a relatively strong beat occurs at the inception of either a relatively long pitch-event; a relatively long duration of a dynamic; a relatively long slur; a relatively long pattern of articulation; a relatively long duration of a pitch in the relevant levels of the time-span reduction; a relatively long duration of a harmony in the relevant levels of the time-span reduction (harmonic rhythm).
(Bass) "Prefer a metrically stable bass." (Cadence) "Strongly prefer a metrical structure in which cadences are metrically stable; that is, strongly avoid violations of local preference rules within cadences." (Suspension) "Strongly prefer a metrical structure in which a suspension is on a stronger beat than its resolution." (Time-span interaction) "Prefer a metrical analysis that minimizes conflict in the time-span reduction."