Generative Theory of Tonal Music

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Generative Theory of Tonal Music (GTTM) is a system for the musical analysis of tonal music , which was published in 1983 in the book of the same name by the American composer and music theorist Fred Lerdahl (* 1943 ) and the American linguist and clarinetist Ray Jackendoff (* 1945 ).

The collaboration between Lehrdahl and Jackendoff was stimulated by Leonard Bernstein's Charles Eliot Norton Lectures at Harvard University in 1973 , in which he challenged participants to develop a "musical grammar" that scientifically explains the unique human ability to understand music, comparable to Noam Chomsky's revolutionary generative transformation grammar . In contrast to analysis methods that had been developed up to then, the GTTM tries to orientate itself on the mental processes through which the listener develops an unconscious understanding of the music (focus on hierarchical structures). They orientate themselves towards an "ideal" listener with musical understanding and the ability to recognize musical objects and to abstract structures. An attempt was made to achieve a synthesis of the views and methods of contemporary linguistics with the insights of modern music theory.

The GTTM had a great influence on the further work of the authors as well as other researchers in the fields of music theory , music perception and cognitive musicology .

Influences and basics

Giver

Heinrich Schenker , an Austrian music theorist and composer, founded the reduction analysis. According to his hypothesis , some tones are perceived as more important than others. The "unimportant" (e.g. secondary notes) are decorations, decorations, continuations and connections of the main notes. Reductions at all levels create a hierarchical structure of the work. The reduction analysis tries to trace the superficial note image back to a basic sentence - "original sentence" - in the background.

Generative linguistics

Generative linguistics is an attempt to describe the human understanding of language and the associated ability to understand and form an infinite number of sentences, including those that have never been heard before. This skill is learned so early and is so ingrained in our minds that it cannot be fully learned by instruction. In linguistics an attempt is made to control this entire ability, its conscious and unconscious parts, through a formal system of rules, i.e. i.e. through the grammar to describe.

The parallel relevant to the development of the GTTM is the combination of psychological knowledge with a formal structure of the theory. The term “generative” should not be misunderstood. It does not mean that one can develop an algorithm for the generation of language with this theory , but rather it is meant more in a mathematical sense to describe a - mostly infinite - set with finite formalisms. The GTTM is not about listing what works are basically possible, but about being able to create a structural description or analysis for existing works. Linguistics sees itself as a branch of psychology that tries to make empirically verifiable statements about language, a complex aspect of human life. Likewise, the ultimate goal of GTTM is an understanding of musical perception, a psychological phenomenon.

Theory

The GTTM focuses on four systems that are supposed to map our musical perceptions. Each of these systems consists of a strict hierarchical structure in which dominant regions contain smaller sub-regions and similar elements are adjacent to one another within a level.

The structures

Grouping structure

The analysis of the groupings is seen as the most basic component of musical understanding. A hierarchical segmentation of the piece into motifs , phrases , periods and even larger areas is carried out.

Metrical structure

The metric structure represents the regular, hierarchical alternation of (heavy and light) beats that the listener associates with musical events. Basic distinction: beat / beat = infinitesimal point in time, time span / time span = time from one beat to the next. The metric structure is about beats / beats. Strong Beat: Beat on a level L (level) is also a beat on a higher level. Otherwise: weak beat.

Humans don't perceive too many metric levels at the same time. Usually there is a dominant level in which the conductor moves his hands, the listener rocks his foot, etc. - the "tactus". (Renaissance expression, mostly strongly determined by harmonic rhythm)

Time-span reduction (TSR)

The time span reductions are based on the information from the group and metric structure. A tree diagram is created that hierarchically connects the time spans on all temporal levels of the work. You start at the lowest level, where the metrical structure divides the music into beats of equal length (or, more precisely, into strokes, between which there is the same time span). From there you go up through all higher levels, which are divided into motifs, phrases, periods and even larger units by the grouping structure. In addition, a head (the structurally most important element) is determined for each time span at all hierarchical levels. The complete TSR analysis is called a time span tree .

Prolongational reduction (PR)

The prolongation reduction (PR) maps our "psychological" experience of tension and relaxation on precise structural forms. In TSR, the hierarchy of elements is created according to the rhythmic stability. In PR, on the other hand, the hierarchy is built on the basis of relative stability, which is represented in the form of continuity and progression, the movement towards tension or relaxation and the degree of completion. The PR is mainly needed because the TSR has two limitations:

  1. The TSR cannot reproduce the feeling of continuity created by the harmonic rhythm.
  2. The TSR can - even if it relates individual tones to certain beats of a certain group - say nothing about the flow of the music beyond the boundaries of these temporal segments.

During PR, a tree diagram is also created or the connections are shown in a visually condensed, "slur notation". There are three types of branches for the tree diagram:

  • strong prolongation (represented by a circle at the branch point),
→ Fundamental tones, bass tones and melody tones are identical.
  • weak prolongation (represented by a filled circle) and
→ Fundamental tones are the same, but bass and / or melody tones are different.
  • Progression (represented by a simple branch without a circle).
→ Harmonic basis is different.

In contrast to all other structures of the theory, the PR is created from top to bottom, i.e. from the highest level (whole work) to the smallest (individual notes). This is due to the fact that functions such as building up and relieving tension only arise through the context of a musical object. There is no such distinction in TSR tree diagrams. However, there are often other helpful comments for this. On the higher structural levels, the two tree diagrams are often very similar, but the further down you move towards the musical surface, the more often you will find differences in the branches.

The rules

Each of the four hierarchical systems is created by rules that can be divided into three categories:

  1. Well-formedness rules - the well -formedness rules indicate the possible structural representations.
  2. Preference rules - the preference rules select from the possible structural representations those that correspond to the hearing impressions of a trained listener. Sometimes the GPRs conflict with each other, hence the term “preference”. Basically, more stable variants are to be preferred.
  3. The transformation rules offer possibilities to combine distorted structures (= exceptions such as group overlaps of an otherwise regular structure) with well-formed representations. These play a major role in linguistics, where they are more of a marginal phenomenon.

The system takes a musical surface as input and uses it to generate the structure that the listener perceives as output.

Grouping Structure Rules

Grouping Well-Formedness Rules (GWFR)

In summary: The grouping structure is hierarchical, non-overlapping, recursive and each group must consist of related elements.

  1. Any related sequence of tones, beats, or the like. can form a group, and only contiguous sequences can form groups.
  2. A piece represents a group.
  3. A group can contain subgroups.
  4. If a group G1 contains part of group G2, it must contain the entire group G2.
  5. If a group G1 contains a smaller subgroup G2, then G1 must be completely divided into subgroups.
Grouping Preference Rules (GPR)

There are basically two types of clues as to how the listener perceives the grouping:

  1. Local details such as attack, articulation , dynamics and registration . Elements are more likely to be perceived as a group if they are close to one another in terms of time or pitch, are similar (see gestalt theory ) or are linked by game instructions.
  2. More global things like symmetrical, motivic, thematic, rhythmic or harmonic parallels. There absolutely doesn't have to be perfect parallels to be perceived as parallel.
  3. Avoid analyzes with very small groups: the smaller, the less preference.
  4. Proximity - given a sequence of four notes n1 - n4, the transition n2 - n3 can be heard as a group boundary if:
    1. slur / rest - the time interval from the end of n2 to the beginning of n3 is greater than from the end of n1 to the beginning of n2 and that from the end of n3 to the beginning of n4.
    2. attack / point - the time interval between the attacks of n2 and n3 is greater than between the attacks of n1 and n2, and between the attacks of n3 and n4.
  5. Change - with a given sequence of four notes n1 - n4, the transition n2 - n3 can be heard as a group boundary if it is characterized by register, dynamics, articulation or length.
  6. Intensification - where the effects of 2nd and 3rd appear particularly strong, a superordinate group should be created.
  7. Symmetry - prefer groupings that come closest to an ideal division into elements of equal length.
  8. Parallelism - where two or more segments can be viewed as parallel, they preferably form parallel parts of the groups.
  9. Time-span and prolongational stability - prefer groupings that lead to a more stable TSR and / or PR.

Metrical Structure Rules

Metrical Well-Formedness Rules (MWFRs)
  1. Each attack must be linked to a beat on the lowest metric level available in that part of the piece.
  2. Every hit at a certain level must also be present at all lower levels.
  3. On each metric level, strong beats are either two or three beats apart.
  4. The “tactus” and the metric planes above must consist of beats at regular intervals. On the lower metric levels, weak beats must be evenly spaced from the surrounding strong beats.
Metrical Preference Rules (MPRs)
  1. Parallelism - where two or more groups or parts of groups can be viewed as parallel, they are preferably given a parallel metric structure.
  2. Strong beat early - prefer a metric structure in which the strongest beat in a group appears relatively early within the group.
  3. Event - prefer a metric structure in which beats that coincide with the beginning of a tone are strong beats of the respective level Li.
  4. Stress - prefer a metric structure in which stresses are strong beats at each level.
  5. Length - prefer a metric structure in which a relatively strong beat occurs simultaneously with the beginning
    1. a relatively long tone
    2. a relatively long dynamic curve
    3. a relatively long slur
    4. a relatively long articulation pattern
    5. a long tone in the relevant levels of the TSR
    6. a relatively long harmony in the relevant level of the TSR (harmonic rhythm)
  6. (Bass) Prefer a metrically stable bass.
  7. Cadence - definitely prefer a metric structure in which cadences are metrically stable; d. H. In any case, avoid violating local rules of preference within cadences.
  8. Suspension definitely prefer a metric structure where a suspension is on a stronger count than its resolution.
  9. Time-span Interaction - prefer a metric analysis that minimizes TSR conflicts.

Time-Span Reduction Rules

Time-Span Segmentation Rules (TSRSRs)
  1. Each group in a piece corresponds to a period of the TSR of the piece.
  2. In the basic grouping,
    1. each beat B (beat) of the smallest metric level defines a time span TB that extends from B to the next beat of the same level (excluding this next beat, of course).
    2. each beat B of the metric level Li (level) determines a regular period of time, which is the sum of all beats of level Li-1 (= level below), i.e. from B to the next beat B 'of the same level, or to the group boundary, depending on what comes earlier.
    3. if a group boundary G lies between B and the previous beat of the same level, B determines an excessive time period T'B which corresponds to the distance from G to the end of the regular time period TB.
Time-Span Reduction Well-Formedness Rules (TSRWFRs)

Every time span has a head . The TSRWFRs describe how to determine the head. (And with it what the tree structure looks like → left / right branch)

  1. For every time span T there is an object e (event) or a sequence of objects e1 - e2 that is the "head" of T.
  2. If T does not contain any smaller time spans (i.e., if T is at the lowest level of time spans) then "e" corresponds to whatever happens in T.
  3. If T contains other time periods T1, ..., Tn (regular or excessive) with heads e1, ..., en, then the head of T is defined by
    1. Normal reduction
    2. fusion
    3. transformation
    4. Cadential return
  4. If a cadence of two elements is directly subordinate to the head e of a period T, ...
Time-Span Reduction Preference Rules (TSRPRs)

Since the TSRWFR leave several options open in some cases, preference rules are needed again. There are three categories:

  • Local rules - only deal with rhythmic and tonal contents of the time span. (1,2,3)
  • Nonlocal rules - reference to other time periods. (4,5,6)
  • Structural accent rules - refer to the group transitions / boundaries. (7.8)
  1. Metrical Position - Of the possibilities for the head of a period T, prefer the one who is in a relatively strong metric position.
  2. Local Harmony - of the possibilities for the head of a period T, prefer the one
    1. is in itself relatively consonant .
    2. is relatively closely related to the local tonic .
  3. Registral Extremes - of the possibilities for the head of a period T give a slight preference to the possibility
    1. has a higher melody
    2. has a deeper bass
  4. Parallelism - if one or more periods of time can be viewed motivically or rhythmically in parallel, try to assign them parallel heads .
  5. Metrical Stability - prefer a choice of heads for a time period T that results in a more stable metric structure.
  6. Prolongational Stability - prefer a choice of heads for a time T, which leads to a more stable extension structure.
  7. Cadential Retention - if the following conditions hold for a period of time T, then denote the progression as cadence and give it strong preference as head :
    1. There is an object or a sequence of two objects that form a half or full cadence or a fallacy.
    2. The last element of the progression is at the end of T or extended to the end of T.
    3. There is a supergroup G that includes T for which the progression can function as a structural end.
  8. Structural Beginning - if there is a time span T in a group G, the head of which can be regarded as the structural beginning of the group, then prefer an object as the head of T that is relatively close to the beginning of T and thus also close to the beginning of G.
  9. When choosing the heads for a piece of the structural end give preference to the structural beginning.

Prolongational Reduction Rules

There are two possible notations: tree diagram and slur notation (based on Schenker). The line from the more important element always continues.

Prolongational Reduction Well-Formedness Rules (PRWFRs)

Short form: 1. There has to be a head . 2. Defined continuations: strong and weak prolongation, as well as progression. 3. Every musical object must be connected to the tree. 4. Branches must not cross one another.

  1. The tree diagram must contain a single main object that acts as the rollover head .
  2. An object ei can be a continuation of another tone in the following ways:
    1. ei is a strong prolongation of ej when the root, bass and melody of the two objects are identical.
    2. ei is a weak prolongation of ej if the root note is the same, but the bass curve or the melody are different.
    3. ei is a progression of ej when the harmonic fundamental tones of the two objects are different.
  3. Each object of the basic group structure is either a prolongation head or a recursive continuation of the prolongation head.
  4. No crossing branches - if ei is a direct continuation of an object ej, then every object between ei and ej must also be a direct continuation of ei, ej or an object in between.
Prolongational Reduction Preference Rules (PRPRs)

Tasks: Determine the most important object for the prolongation in a region (ei-ej) and determine whether it is a prolongation of ei or ej.

  1. When choosing the most important element ek of a prolongation region (ei-ej), an ek is to be preferred, which is relatively important for the TSR.
  2. Time-span segmentation - be ek the most important element in a prolongation region (ei-ej). If there is a period of time that includes ei and ek but does not include ej, then a PR is preferable in which ek is a continuation of ei; and analogously if ei and ej are exchanged.
  3. Prolongational Connection - when choosing the most important element ek in a prolongation region (ei-ej), an ek is to be preferred that is connected to one of the endpoints of the region in such a way that the most stable possible prolongation connection is created.
  4. Prolongational Importance - be ek the most important element in a prolongation region (ei-ej). Prefer a PR in which ek is a continuation of the endpoint that is more important for the prolongation.
  5. Parallelism - prefer a PR where parallel passages get parallel analysis.
  6. Normative Prolongational Structure - a cadence group should preferably contain four (five) elements in its prolongational structure:
    1. A prolongation as a start
    2. A prolongation as a conclusion that consists of an element of cadence.
    3. A right-branching prolongation as the most important direct continuation of the beginning of the prolongation.
    4. A progression branching off to the right as the most important direct continuation of the initial prolongation.
    5. A left branching subdominant progression as the most important continuation of the first element of the cadence.

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literature

  • Fred Lerdahl, Ray S. Jackendoff: A Generative Theory of Tonal Music (= MIT Press Series On Cognitive Theory and Mental Representation ). MIT Press, Cambridge Mass. 1996, ISBN 0-262-62107-X , ISBN 978-0-585-37588-5 (e-book).

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