A Generative Theory of Tonal Music (GTTM) is a system of music analysis

Musical analysis

Musical analysis is the attempt to answer the question how does this music work?. The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst, and according to the purpose of the analysis. According to Ian Bent , analysis is "an...

 written by Fred Lerdahl

Fred Lerdahl

Alfred Whitford Lerdahl is the Fritz Reiner Professor of Musical Composition at Columbia University, and a composer and music theorist best known for his work on pitch space and cognitive constraints on compositional systems or "musical grammar[s]." He has written many orchestral and chamber...

 (American composer and theorist) and Ray Jackendoff

Ray Jackendoff

Ray Jackendoff is an American linguist. He is professor of philosophy, Seth Merrin Chair in the Humanities and, with Daniel Dennett, Co-director of the Center for Cognitive Studies at Tufts University...

 (American linguist and clarinetist) in 1983 that strives to illuminate the unique human capacity for musical understanding.

The collaboration between Lerdahl and Jackendoff was inspired by Leonard Bernstein

Leonard Bernstein

Leonard Bernstein August 25, 1918 – October 14, 1990) was an American conductor, composer, author, music lecturer and pianist. He was among the first conductors born and educated in the United States of America to receive worldwide acclaim...

’s 1973 Charles Eliot Norton Lectures

Charles Eliot Norton Lectures

The Charles Eliot Norton Professorship of Poetry at Harvard University was established in 1925 as an annual lectureship in "poetry in the broadest sense" and named for the university's former professor of fine arts. Distinguished creative figures and scholars in the arts, including painting,...

 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

Noam Chomsky

Avram Noam Chomsky is an American linguist, philosopher, cognitive scientist, and activist. He is an Institute Professor and Professor in the Department of Linguistics & Philosophy at MIT, where he has worked for over 50 years. Chomsky has been described as the "father of modern linguistics" and...

’s revolutionary transformational

Transformational grammar

In linguistics, a transformational grammar or transformational-generative grammar is a generative grammar, especially of a natural language, that has been developed in the Chomskyan tradition of phrase structure grammars...

 or generative grammar

Generative grammar

In theoretical linguistics, generative grammar refers to a particular approach to the study of syntax. A generative grammar of a language attempts to give a set of rules that will correctly predict which combinations of words will form grammatical sentences...

.

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 it 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 theory

Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...

, music cognition

Music cognition

Music cognition is an interdisciplinary approach to understanding the mental processes that support musical behaviors, including perception, comprehension, memory, attention, and performance...

 and cognitive musicology

Cognitive musicology

Cognitive musicology is a branch of Cognitive Science concerned with computationally modeling musical knowledge with the goal of understanding both music and cognition. More broadly, it can be considered the set of all phenomena surrounding computational modeling of musical thought and action...

.

The 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.

I. Grouping structure

GTTM considers grouping analysis to be the most basic component of musical understanding. It expresses a hierarchical segmentation of the 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

Tree structure

A tree structure is a way of representing the hierarchical nature of a structure in a graphical form. It is named a "tree structure" because the classic representation resembles a tree, even though the chart is generally upside down compared to an actual tree, with the "root" at the top and the...

-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 are often observed 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 (represented by a filled node at the branching point) and progression (represented by simple branching, with no node). Time-span trees do not make this distinction. All time-span trees branches are simple branches without nodes. (Although time-span tree branches are often annotated with other helpful comments.)

The Rules

Each of the four major hierarchical organizations (grouping structure, metrical structure, time-span reduction and prolongational reduction) is established through rules, which are divided into three categories:

1. The Well-formedness rules, which specify possible structural descriptions.
2. The Preference rules, which draw on possible structural descriptions eliciting those descriptions that correspond to experienced listeners’ hearings of any particular piece.
3. The Transformational rules, which provide a means of associating distorted structures with well-formed descriptions.

Grouping Well-Formedness Rules (G~WFRs)

1. "Any contiguous sequence of pitch-events, drum beats, or the like can constitute a group, and only contiguous sequences can constitute a group."
2. "A piece constitutes a group."
3. "A group may contain smaller groups."
4. "If a group G₁ contains part of a group G₂, it must contain all of G₂."
5. 'If a group G₁ contains a smaller group G₂, then G₁ must be exhaustively partitioned into smaller groups."

Grouping Preference Rules (G~PRs)

1. "Avoid analyses with very small groups- the smaller, the less preferable."
2. (Proximity) Consider a sequence of four notes, n₁ - n₄, the transition n₂ - n₃ may be heard as a group boundary if: a.(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 b.(attack/point) the interval of time between the attack point of n₂ and n₁3 is greater than that between the attack points of n₁ and n₂ and that between the attack points of n₃ and n₄.
3. (Change) Consider a sequence of four notes, n₁ - n₄, the transition n₂-n₃ may be heard as a group boundary if marked by a. register, b. dynamics, c. articulation, d. length.
4. (Intensification) "Where the effects pick out by a GPRs 2 and 3 are relatively more pronounced, a larger-level group may be placed."
5. (Symmetry) "Prefer grouping analyses that most closely approach the ideal subdivision of groups into two parts of equal length."
6. (Parallelism) "Where two or more segments of music can be construed as parallel, they preferably form parallel parts of groups."
7. (Time-span and prolongational stability) "Prefer a grouping structure that results in more stable time-span and/or prolongational reductions."

Transformational Grouping Rules

1. Grouping overlap (p. 60).
2. Grouping elision (p. 61).

Metrical Well-Formedness Rules (M~WFRs)

1. "Every attack point must be associated with a beat at the smallest metrical level present at that point in the piece."
2. "Every beat at a given level must also be a beat at all smaller levels present at that point in that piece."
3. "At each metrical level, strong beats are spaced either two or three beats apart."
4. "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)

1. (Parallelism) "Where two or more groups or parts of groups can be construed as parallel, they preferably receive parallel metrical structure."
2. (Strong beat early) "Weakly prefer a metrical structure in which the strongest beat in a group appears relatively early in the group."
3. (Event) "Prefer a metrical structure in which betas of level L_i that coincide with the inception of pitch-events are strong beats of L_i."
4. (Stress) "Prefer a metrical structure in which betas of level L_i that are stressed are strong beats of L_i."
5. (Length) Prefer a metrical structure in which a relatively strong beat occurs at the inception of either relatively long: a. pitch-event; b. duration of a dynamic; c. slur; d. pattern of articulation; e. duration of a pitch in the relevant levels of the time-span reduction; f. duration of a harmony in the relevant levels of the time-span reduction (harmonic rhythm).
6. (Bass) "Prefer a metrically stable bass."
7. (Cadence) "Strongly prefer a metrical structure in which cadences are metrically stable; that is, strongly avoid violations of local preference rules within cadences."
8. (Suspension) "Strongly prefer a metrical structure in which a suspension is on a stronger beat than its resolution."
9. (Time-span Interaction) "Prefer a metrical analysis that minimizes conflict in the time-span reduction."
10. (Binary Regularity) "Prefer metrical structures in which at each level every other beat is strong."

III. Time-Span Reduction Rules

Time-span reduction rules begin with two segmentation rules and proceed to the standard WFRs, PRs and TRs.

Time-Span Segmentation Rules

1. "Every group in a piece is a time-span in the time-span segmentation of the piece."
2. "In underlying grouping structure: a. each beat B of the smallest metrical level determines a time-span T_B extending from B up to but not including the next beat of the smallest level; b. each beat B of metrical level Li determines a regular time-span of all beats of level L_i-1 from B up to but not including (i) the next beat B’ of level L_i or (ii) a group boundary, whichever comes sooner; and c. if a group boundary G intervenes between B and the preceding beat of the same level, B determines an augmented time-span T’_B , which is the interval from G to the end of the regular time-span T_B."

Time-Span Reduction Well-Formedness Rules (TSR~WFRs)

1. "For every time-span T there is an event e (or a sequence of events e₁ - e₂) that is the head of T."
2. "If T does not contain any other time-span (that is, if T is the smallest level of time-spans), there e is whatever event occurs in T."
3. If T contains other time-spans, let T₁,…,T_n be the (regular or augmented) time-spans immediately contained in T and let e₁,…,e_n be their respective heads. Then the head is defined depending on: a. ordinary reduction; b. fusion; c. transformation; d. cadential retention (p. 159).
4. "If a two-element cadence is directly subordinate to the head e of a time-span T, the final is directly subordinate to e and the penult is directly subordinate to the final."

Time-Span Reduction Preference Rules (TSR~PRs)

1. (Metrical Position) "Of the possible choices for head of time-span T, prefer that is in a relatively strong metrical position."
2. (Local Harmony) "Of the possible choices for head of time-span T, prefer that is: a. relatively intrinsically consonant, b. relatively closely related to the local tonic."
3. (Registral Extremes) "Of the possible choices for head of time-span T, weakly prefer a choice that has: a. a higher melodic pitch; b. a lower bass pitch."
4. (Parallelism) "If two or more tome-spans can be construed as motivically and/or rhythmically parallel, preferably assign them parallel heads."
5. (Metrical Stability) "In choosing the head of a time-span T, prefer a choice that results in more stable choice of metrical structure."
6. (Prolongational Stability) "In choosing the head of a time-span T, prefer a choice that results in more stable choice of prolongational structure."
7. (Cadential retention) (p. 170).
8. (Structural Beginning) "If for a time-span T there is a larger group G containing T for which the head of T can function as the structural beginning, then prefer as head of T an event relatively close to the beginning of T (and hence to the beginning of G as well)."
9. "In choosing the head of a piece, prefer the structural ending to the structural beginning."

Prolongational Reduction Well-Formedness Rules (PR~WFRs)

1. "There is a single event in the underlying grouping structure of every piece that functions as prolongational head."
2. "An event e_i can be a direct elaboration of another pitch ei in any of the following ways: a. e_i is a strong prolongation of e_j if the roots, bass notes, and melodic notes of the two events are identical; b. e_i is a weak prolongation of e_j if the roots of the two events are identical but the bass and/or melodic notes differ; c. e_i is a progression to or from e_j if the harmonic roots of the two events are different."
3. "Every event in the underlying grouping structure is either the prolongational head or a recursive elaboration of the prolongational head."
4. (No crossing branches) "If an event e_i is a direct elaboration of an event e_j, every event between e_i and e_j must be a direct elaboration of either e_i, e_j, or some event between them."

Prolongational Reduction Preference Rules (PR~PRs)

1. (Time-span Importance) "In choosing the prolongational most important event e_k of a prolongational region (e_i - e_j), strongly prefer a choice in which e_k is relatively time-span important."
2. (Time-span Segmentation) "Let e_k be the prolongationally most important region (e_i - e_j). If there is a time-span that contains e_i and e_k but not e_j, prefer a prolongational reduction in which e_k is an elaboration of e_i; similarly with the roles of e_i and e_j reversed."
3. (Prolongational Connection) "In choosing the prolongationally most important region (e_i - e_j), prefer an e_k that attaches to as to form a maximally stable prolongational connections with one of the endpoints of the region."
4. (Prolongational importance) "Let e_k be the prolongationally most important region (e_i - e_j). Prefer a prolongational reduction in which e_k is an elaboration of the prolongationally more important of the endpoints."
5. (Parallelism) "Prefer a prolongational reduction in which parallel passages receive parallel analyses."
6. (Normative Prolongational Structure) "A cadenced group preferably contains four (five) elements in its prolongational structure: a. a prolongational beginning; b. a prolongational ending consisting of one element of the cadences; (c. a right-branching prolongational as the most important direct elaboration direct of the prolongational beginning); d. a right-branching progression as the (next) most important direct elaboration of the prolongational beginning; e. a left-branching ‘subdominant’ progression as the most important elaboration of the first element of the cadence."

Prolongational Reduction Transformational Rules

1. Stability Conditions for Prolongational Connection (p. 224): a. Branching condition; b. Pitch-collection condition; c. Melodic Condition; d. Harmonic Condition.
2. Interaction Principle: "In order to make a sufficiently stable prolongational connection e_k must be chosen from the events in the two most important levels of time-span reduction represented in (e_i - e_j)."

Source

• Lerdahl, Fred and Ray Jackendoff (1983).A generative theory of tonal music. Cambridge, MA: MIT Press.

Lerdahl

• Lerdahl, Fred (1987). Timbral Hierarchies. Contemporary Music Review 2, no.1 , p. 135-60.
• Lerdahl, Fred (1989). Atonal Prolongational Structure. Contemporary Music Review 3, no. 2., p. 65-87.
• Lerdahl, Fred (1992). Cognitive Constraints on Compositional Systems. Contemporary Music Review 6, no. 2, p. 97-121.
• Lerdahl, Fred (Fall 1997). Spatial and Psychoacoustic Factors in Atonal Prolongation.Current Musicology 63, p. 7-26.
• Lerdahl, Fred (1998). Prolongational Structure and Schematic Form in Tristan's Alte Weise.Musicae Scientiae, p. 27-41.
• Lerdahl, Fred (1999). Composing Notes. Current Musicology 67-68, p. 243- 251.
• Lerdahl, Fred (Autumn 2003). Two Ways in Which Music Relates to the World. Music Theory Spectrum 25, no. 2, p. 367-73.
• Lerdahl, Fred (2001). Tonal Pitch Space. New York: Oxford University Press. 391 pages. (This volume includes integrated and expanded versions of these articles: Lerdahl, Fred (Spring/Fall, 1988). Tonal Pitch Space. Music Perception 5, no. 3, p. 315-50; and Lerdahl, Fred (1996). Calculating Tonal Tension. Music Perception 13, no. 3, p. 319-363.

Lerdahl and Jackendoff

• (Autumn 1979-Summer 1980). Discovery Procedures vs. Rules of Musical Grammar in a Generative Music Theory. Perspectives of New Music 18, no. ½, p. 503-10.
• (Spring 1981). Generative Music Theory and Its Relation to Psychology. In Journal of Music Theory (25th Anniversary Issue) 25, no. 1, p. 45-90.
• (October 1981). On the Theory of Grouping and Meter. The Musical Quarterly 67, no. 4, p. 479-506.
• (1983). An Overview of Hierarchical Structure in Music.Music Perception 1, no. 2.

Reviews of GTTM

• Child, Peter (Winter 1984). Review of A Generative Theory of Tonal Music, by Fred Lerdahl and Ray Jackendoff. Computer Music Journal 8, no. 4, p. 56-64.
• Feld, Steven (March 1984). Review of A Generative Theory of Tonal Music, by Fred Lerdahl and Ray Jackendoff. Language in Society 13, no. 1, p. 133-35.
• Hantz, Edwin (Spring 1985). Review of A Generative Theory of Tonal Music, by Fred Lerdahl and Ray Jackendoff. Music Theory Spectrum 1, p. 190-202.

Further reading

• Sunberg, J. and B. Lindblom (1976). Generative theories in language and music description. Cognition, 4, 99-122.
• Temperley, D. (2001). The cognition of basic musical structures. Cambridge, MA: MIT Press.
• Palme C. and C.L. Krumhansl (1987). Independent temporal and pitch structures in determination of musical phrases. Journal of Experimental Psychology: Human Perception and Performance, 13, 116-126.
• Palme C. and C.L. Krumhansl (1990). Mental representations for musical meter. Journal of Experimental Psychology: Human Perception and Performance, 16, 728-741.
• Boros, James (Winter 1996). A Response to Lerdahl. Perspectives of New Music 34, no. 1, p. 252-58.
• Foulkes-Levy, Lauradella (1996). A Synthesis of Recent Theories of Tonal Melody, Contour, and the Diatonic Scale: Implications for Aural Perception and Cognition. Ph.D. diss., State University of New York at Buffalo.
• David Temperley (2007). Music and Probability. Cambridge, MA: MIT Press.