Generative theory of tonal music

Last updated

The generative theory of tonal music (GTTM) is a system of music analysis developed by music theorist Fred Lerdahl and linguist Ray Jackendoff. [1] 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" [1] with the aim of illuminating the unique human capacity for musical understanding. [2]

Contents

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. [3]

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. [4]

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. [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.

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. [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]

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:

  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.


I. Grouping structure rules

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. alternative form: "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:
    1. (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
    2. (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
    1. (Register) the transition n2-n3 involves a greater intervallic distance than both n1-n2 and n3-n4, or if
    2. (Dynamics) the transition n2-n3 involves a change in dynamics and n1-n2 and n3-n4 do not, or if
    3. (Articulation) the transition n2-n3 involves a change in articulation and n1-n2 and n3-n4 do not, or if
    4. (Length) n2 and n3 are of different length and both pairs n1,n2 and n3,n4 do not differ in 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."

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)

  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 beats of level Li that coincide with the inception of pitch-events are strong beats of Li."
  4. (Stress) "Prefer a metrical structure in which beats of level Li that are stressed are strong beats of Li."
  5. (Length) Prefer a metrical structure in which a relatively strong beat occurs at the inception of either
    1. a relatively long pitch-event;
    2. a relatively long duration of a dynamic;
    3. a relatively long slur;
    4. a relatively long pattern of articulation;
    5. a relatively long duration of a pitch in the relevant levels of the time-span reduction;
    6. a relatively long 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."

Transformational metrical rule

  • Metrical deletion (p. 101).
  • Given a well-formed metrical structure M in which
    1. B1, B2 and B3 are adjacent beats of M at level L1, and B2 is also a beat at level Li+1,
    2. T1 is the time-span from B1 to B2 and T2 is the time-span from B2 to B3, and
    3. M is associated with and underlying grouping structure G in such a way that both T1 and T2 are related to a surface time-span T' by the grouping transformation performed on G of
      1. left elision or
      2. overlap,

    then a well-formed metrical structure M' can be formed from M and associated with the surface grouping structure by

    1. deleting B1 and all beats at all levels between B1 and B2 and associating B2 with the onset of T', or
    2. deleting B2 and all beats at all levels between B2 and B3 and associating B1 with the onset of T'.

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 TB 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 Li-1 from B up to but not including (i) the next beat B’ of level Li 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 TB."

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,...,Tn be the (regular or augmented) time-spans immediately contained in T and let e1,...,en 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 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."

IV. Prolongational reduction rules

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 ei can be a direct elaboration of another pitch ej in any of the following ways: a. ei is a strong prolongation of ej if the roots, bass notes, and melodic notes of the two events are identical; b. ei is a weak prolongation of ej if the roots of the two events are identical but the bass and/or melodic notes differ; c. ei is a progression to or from ej 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 ei is a direct elaboration of an event ej, every event between ei and ej must be a direct elaboration of either ei, ej, or some event between them."

Prolongational reduction preference rules (PR~PRs)

  1. (Time-span importance) "In choosing the prolongational most important event ek of a prolongational region (ei – ej), strongly prefer a choice in which ek is relatively time-span important."
  2. (Time-span segmentation) "Let ek be the prolongationally most important region (ei – ej). If there is a time-span that contains ei and ek but not ej, prefer a prolongational reduction in which ek is an elaboration of ei; similarly with the roles of ei and ej reversed."
  3. (Prolongational connection) "In choosing the prolongationally most important region (ei – ej), prefer an ek that attaches to as to form a maximally stable prolongational connections with one of the endpoints of the region."
  4. (Prolongational importance) "Let ek be the prolongationally most important region (ei – ej). Prefer a prolongational reduction in which ek 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: "to make a sufficiently stable prolongational connection ek must be chosen from the events in the two most important levels of time-span reduction represented in (ei – ej)."

Related Research Articles

Rhythm generally means a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular recurrence or pattern in time can apply to a wide variety of cyclical natural phenomena having a periodicity or frequency of anything from microseconds to several seconds ; to several minutes or hours, or, at the most extreme, even over many years.

<span class="mw-page-title-main">Music theory</span> Study of the practices and possibilities of music

Music theory is the study of the practices and possibilities of music. The Oxford Companion to Music describes three interrelated uses of the term "music theory": The first is the "rudiments", that are needed to understand music notation ; the second is learning scholars' views on music from antiquity to the present; the third is a sub-topic of musicology that "seeks to define processes and general principles in music". The musicological approach to theory differs from music analysis "in that it takes as its starting-point not the individual work or performance but the fundamental materials from which it is built."

<span class="mw-page-title-main">Metre (music)</span> Aspect of music

In music, metre or meter refers to regularly recurring patterns and accents such as bars and beats. Unlike rhythm, metric onsets are not necessarily sounded, but are nevertheless implied by the performer and expected by the listener.

<span class="mw-page-title-main">Musical analysis</span>

Musical analysis is the study of musical structure in either compositions or performances. According to music theorist Ian Bent, music analysis "is the means of answering directly the question 'How does it 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 Bent, "its emergence as an approach and method can be traced back to the 1750s. However it existed as a scholarly tool, albeit an auxiliary one, from the Middle Ages onwards."

Schenkerian analysis is a method of analyzing tonal music based on the theories of Heinrich Schenker (1868–1935). The goal is to demonstrate the organic coherence of the work by showing how the "foreground" relates to an abstracted deep structure, the Ursatz. This primal structure is roughly the same for any tonal work, but a Schenkerian analysis shows how, in each individual case, that structure develops into a unique work at the foreground. A key theoretical concept is "tonal space". The intervals between the notes of the tonic triad in the background form a tonal space that is filled with passing and neighbour tones, producing new triads and new tonal spaces that are open for further elaborations until the "surface" of the work is reached.

In poetic and musical meter, and by analogy in publishing, an anacrusis is a brief introduction. In music, it is also known as a pickup beat, or fractional pick-up, i.e. a note or sequence of notes, a motif, which precedes the first downbeat in a bar in a musical phrase.

<span class="mw-page-title-main">Generative grammar</span> Theory in linguistics

Generative grammar is a theoretical approach in linguistics that regards grammar as a domain-specific system of rules that generates all and only the grammatical sentences of a given language. In light of poverty of the stimulus arguments, grammar is regarded as being partly innate, the innate portion of the system being referred to as universal grammar. The generative approach has focused on the study of syntax while addressing other aspects of language including semantics, morphology, phonology, and psycholinguistics.

Generative music is a term popularized by Brian Eno to describe music that is ever-different and changing, and that is created by a system.

Alfred Whitford (Fred) Lerdahl is an American music theorist and composer. Best known for his work on musical grammar, cognition, rhythmic theory and pitch space, he developed the Chomsky-inspired Generative theory of tonal music alongside the linguist Ray Jackendoff.

<span class="mw-page-title-main">Ray Jackendoff</span> American linguist and philosophy professor

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. He has always straddled the boundary between generative linguistics and cognitive linguistics, committed to both the existence of an innate universal grammar and to giving an account of language that is consistent with the current understanding of the human mind and cognition.

In music theory, prolongation is the process in tonal music through which a pitch, interval, or consonant triad is considered to govern spans of music when not physically sounding. It is a central principle in the music-analytic methodology of Schenkerian analysis, conceived by Austrian theorist Heinrich Schenker. The English term usually translates Schenker's Auskomponierung. According to Fred Lerdahl, "The term 'prolongation' [...] usually means 'composing out' ."

"Cognitive Constraints on Compositional Systems" is an essay by Fred Lerdahl that cites Pierre Boulez's Le Marteau sans maître (1955) as an example of "a huge gap between compositional system and cognized result," though he "could have illustrated just as well with works by Milton Babbitt, Elliott Carter, Luigi Nono, Karlheinz Stockhausen, or Iannis Xenakis". To explain this gap, and in hopes of bridging it, Lerdahl proposes the concept of a musical grammar, "a limited set of rules that can generate indefinitely large sets of musical events and/or their structural descriptions". He divides this further into compositional grammar and listening grammar, the latter being one "more or less unconsciously employed by auditors, that generates mental representations of the music". He divides the former into natural and artificial compositional grammars. While the two have historically been fruitfully mixed, a natural grammar arises spontaneously in a culture while an artificial one is a conscious invention of an individual or group in a culture; the gap can arise only between listening grammar and artificial grammars. To begin to understand the listening grammar, Lerdahl and Ray Jackendoff created a theory of musical cognition, A Generative Theory of Tonal Music. That theory is outlined in the essay.

Music semiology (semiotics) is the study of signs as they pertain to music on a variety of levels.

In music cognition and musical analysis, the study of melodic expectation considers the engagement of the brain's predictive mechanisms in response to music. For example, if the ascending musical partial octave "do-re-mi-fa-sol-la-ti-..." is heard, listeners familiar with Western music will have a strong expectation to hear or provide one more note, "do", to complete the octave.

<span class="mw-page-title-main">Pitch class space</span>

In music theory, pitch-class space is the circular space representing all the notes in a musical octave. In this space, there is no distinction between tones that are separated by an integral number of octaves. For example, C4, C5, and C6, though different pitches, are represented by the same point in pitch class space.

Cognitive musicology is a branch of cognitive science concerned with computationally modeling musical knowledge with the goal of understanding both music and cognition.

<span class="mw-page-title-main">Bell pattern</span> Rhythmic pattern of striking a hand-held bell or other instrument

A bell pattern is a rhythmic pattern of striking a hand-held bell or other instrument of the idiophone family, to make it emit a sound at desired intervals. It is often a key pattern, in most cases it is a metal bell, such as an agogô, gankoqui, or cowbell, or a hollowed piece of wood, or wooden claves. In band music, bell patterns are also played on the metal shell of the timbales, and drum kit cymbals.

In music, a cross-beat or cross-rhythm is a specific form of polyrhythm. The term cross rhythm was introduced in 1934 by the musicologist Arthur Morris Jones (1889–1980). It refers to a situation where the rhythmic conflict found in polyrhythms is the basis of an entire musical piece.

Irène Deliège is a Belgian musician and cognitive scientist. She was born in January 1933 in Flanders, but has spent most of her life in French-speaking Brussels and Liège, Belgium. She is noted for her theory of Cue Abstraction, and for her work in establishing the European Society for the Cognitive Sciences of Music.

This is a glossary of Schenkerian analysis, a method of musical analysis of tonal music based on the theories of Heinrich Schenker (1868–1935). The method is discussed in the concerned article and no attempt is made here to summarize it. Similarly, the entries below whenever possible link to other articles where the concepts are described with more details, and the definitions are kept here to a minimum.

References

  1. 1 2 Lerdahl & Jackendoff 1983, p. 1.
  2. Lerdahl & Jackendoff 1983.
  3. 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.
  4. 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.
  5. 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).[ full citation needed ]
  6. A 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.)
  7. Harmonic rhythm is the pattern of durations produced by changes in the harmony at the musical surface.
  8. Lerdahl & Jackendoff 1983, p. 122.

Sources

Further reading by the authors

Lerdahl

Jackendoff

Lerdahl and Jackendoff

Reviews of GTTM

Further reading

Bibliography on automation of GTTM