Generative theory of tonal music

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.

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.

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.[5] 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[6]) 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.

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.[7] 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.[8]

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).

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."

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 points of n₂ and n₃ is greater than between those of n₁ and n₂ and between those 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, or d. length.
4. (Intensification) A larger-level group may be placed where the effects picked out by GPRs 2 and 3 are more pronounced.
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."

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

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."

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 beats 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 beats 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."

1. Metrical deletion (p. 101).

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

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 L_i 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."

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."

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 time-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."

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 e_j 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."

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."

1. Stability conditions for prolongational connection (p. 224): a. Branching condition; b. Pitch-collection condition; c. Melodic condition; d. Harmonic condition.
2. Interaction principle: "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)."

• Lerdahl, Fred (1987). "Timbral Hierarchies". Contemporary Music Review 2, no. 1, p. 135–160.
• 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. 367–373.
• 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. 315–350; and Lerdahl, Fred (1996). "Calculating Tonal Tension". Music Perception 13, no. 3, p. 319–363.)
• Lerdahl, Fred (2009): "Genesis and Architecture of the GTTM Project". Music Perception 26(3), doi:10.1525/MP.2009.26.3.187, pp. 187–194.

• Jackendoff, Ray (1987): Consciousness and the Computational Mind. Cambridge: MIT Press. Chapter 11: "Levels of Musical Structure".
• Jackendoff, Ray (2009): "Parallels and Nonparallels Between Language and Music". Music Perception 26(3), pp. 195–204.

• (Autumn 1979 – Summer 1980). "Discovery Procedures vs. Rules of Musical Grammar in a Generative Music Theory". Perspectives of New Music 18, no. ½, p. 503–510.
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• (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.

• 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.
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• 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.