Grouping

David MeredithAalborg University

Musical grouping structure

• Listeners automatically chunk or “segment” music into structural units of various sizes– e.g., motives, themes, phrases, sections, movements

• Such a structural unit is called a group (Lerdahl and Jackendoff, 1983, p.12)• Some types of group indicated in scores

– e.g., by breath marks, phrase marks, slurs• Wind players often breathe at group boundaries• Performers often slow down at the ends of larger groups• Grouping structure is way that a piece is perceived to be segmented into

structural units at various time scales

Musical grouping structure

• “Grouping can be viewed as the most basic component of musical understanding [because once a listener has] construed a grouping structure [he or she] has gone a long way towards ‘making sense’ of it” (Lerdahl and Jackendoff, 1983, p.13)

Musical grouping structure

• Listen to the fragment above• The whole fragment forms a structural unit or group• Naturally divides into two, equal-length phrases• Each phrase divides into three ‘motives’ which are groups

at a smaller level than the phrase• How can we infer the grouping structure of a musical

passage from its ‘surface’?

Lerdahl and Jackendoff’s (1983) Generative Theory of Tonal Music

• Four interacting modules– Grouping structure: motives, themes, phrases, sections– Metrical structure: “hierarchical pattern of beats”– Time-span reduction: how some events elaborate or depend on other events– Prolongational reduction: the “ebb-and-flow of tension”

• Each module contains well-formedness rules and preference rules

Musical surface

Grouping structurerules

Prolongationalreduction

Metrical structurerules

Time-spanreduction

rules

Prolongationalreduction

rules

Grouping structure Metrical structureTime-spanreduction

Lerdahl and Jackendoff’s (1983) theory of grouping structure

• Musical grouping is an “auditory analog of the partitioning of the visual field into objects, parts of objects, and parts of parts of objects” (p.36)

• Theory of grouping “seems to consist largely of general conditions for auditory pattern perception that have far broader application than for music alone” (p.36)

• “A listener needs to know relatively little about a musical idiom in order to assign grouping structure to pieces in that idiom” (p.36)

• Theory can only cope with homorhythmic music

Lerdahl and Jackendoff’s grouping well-formedness rules

• Theory consists of 5 well-formedness rules, 7 preference rules and 2 transformational rules

• GWFR 1: “Any contiguous sequence of pitch-events, drum beats, or the like can constitute a group, and only contiguous sequences can constitute a group.” (p.37)– Means we can represent a group with a slur

• GWFR 2: “A piece constitutes a group.” (p.38)• GWFR 3: “A group may contain smaller groups.” (p.38)

– e.g., sections are split into phrases, which are split into motives

Lerdahl and Jackendoff’s theory of grouping structure

• GWFR 4: “If a group G1 contains part of a group G2, then it must contain all of G2” (p.38)– A higher-level group cannot start or end in the middle of a

lower-level group• A boundary between adjacent groups must also be a boundary

between groups at lower structural levels

– Adjacent groups in the same level cannot overlap• In fact they can in certain special cases covered by the

transformational rules

Lerdahl and Jackendoff’s theory of grouping structure

• GWFR 5: “If a group G1 contains a smaller group G2, then G1 must be exhaustively partitioned into smaller groups.” (p. 38)– The situation shown above cannot happen– If both A and B are group boundaries, then there

must be a group that starts at A and ends at B

AB

Well-formedness is not enough

• GWFRs are not enough to predict the grouping structure that we hear

• Need preference rules to isolate the well-formed analyses that correspond best to what we hear

• Both structures above are well-formed, but the lower one is certainly not what we hear– Can check this by playing with gaps at the group boundaries

Gestalt principles of proximity and similarity

• Principles governing musical grouping seem to be similar to those governing grouping in vision

• Gestalt theory (Wertheimer, 1938; Köhler, 1947; Koffka, 1935) proposes that grouping of elements in a visual scene governed by a small number of principles

• The principles of proximity and similarity seem to be operating in the perception of musical grouping structure

Gestalt principle of proximity

• In the left figure, we see a group of two circles on the left and 1 circle on the right

• In the middle figure, we see a single circle on the left and two circles on the right

• In the right figure, we see a group of three circles, with none separated more from the others

• In the left image, we do not, for example, see a single circle on the left and then a group of two circles with a large gap between them

• This suggests that elements are grouped according to proximity

Gestalt principle of similarity

• In the left figure, we see a group of three squares and a group of two circles

• In the right figure, we see a group of two squares and three circles

• In the left image, we do not, for example, see a group of two squares and a group containing two circles and a square

• Suggests that we group elements by similarity

Similarity and proximity in music

• Wertheimer (1938) himself observed that similarity and proximity seem to operate in music

• In A and B, proximity in time is operating– In A, the third note is heard to be grouped with the first

two, whereas in B it is heard to be grouped with last two• In C and D, similarity in pitch is operating– C is a group of 2 Fs followed by a group of 4 Cs– D is a group of 4 Fs followed by a group of 2 Cs

A B C D

Gestalt principles can compete or reinforce each other

• On the left, proximity and similarity reinforce each other

• On the right, the principles compete, giving rise to ambiguity

Lerdahl and Jackendoff’s Local Detail Grouping Preference Rules

• GPR 1: “Avoid analyses with very small groups – the smaller, the less preferable.” (p.43)– A single event can only act as a group if it is strongly

isolated or functions as a motive by itself– Example above might be a rare exception to GPR 1

(from Liszt’s Sonata in B minor)

Lerdahl and Jackendoff’s Local Detail Grouping Preference Rules

• GPR 2 (Proximity): “Consider a sequence of four notes n1, n2, n3, n4. All else being equal, the transition n2-n3 may be heard as a group boundary if– a. (Slur/Rest) the interval of time from the end of n2 to the beginning of n3 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 that between the attack points of n1 and n2 and that between the attack points of n3 and n4.”

• This is L&J’s version of the Gestalt principle of proximity• GPR 2 does NOT apply in the cases below. Why?

Lerdahl and Jackendoff’s Local Detail Grouping Preference Rules

• GPR 3 (Change): “Consider a sequence of four notes n1, n2, n3, n4. All else being equal, the transition n2-n3 may be heard as a group boundary if– a. (Register) the transition n2-n3 involves a greater intervallic distance than both n1-n2 and n3-n4,

or if– b. (Dynamics) the transition n2-n3 involves a change in dynamics and n1-n2 and n3-n4 do not, or if

– c. (Articulation) the transition n2-n3 involves a change in articulation and n1-n2 and n3-n4 do not, or if

– d. (Length) n2 and n3 are of different lengths and both pairs n1, n2 and n3, n4 do not differ in length.”

• “One might add further cases to deal with such things as change in timbre or instrumentation.” (p.46)

• GPR 3 does not predict any grouping boundaries in the examples below

GPR 3a GPR 3b GPR 3c GPR 3d

Applying the local detail rules

• All places where GPRs 1-3 apply are indeed group boundaries, except– transition from 8 to 9– transition from 9 to 10

• do not hear because 10-11 is more salient and both would lead to a 1-note group (conflicting with GPR 1)

– transition from 18 to 19

Lerdahl and Jackendoff’sLarger-level grouping rules

• GPR 4 (Intensification): “Where the effects picked out by GPRs 2 and 3 are relatively more pronounced, a larger-level group boundary may be placed.” (p.50)– In example above, GPR 2 predicts boundaries between

each set of triplets– Gap between 3rd and 4th set of triplets much bigger, so GPR

4 predicts a higher-level boundary here

Lerdahl and Jackendoff’sLarger-level grouping rules

• GPR 5 (Symmetry): “Prefer grouping analyses that most closely approach the ideal subdivision of groups into two parts of equal length.” (p.49)– GPR 5 applies in example above between notes 10

and 11– Also operates at 6-7 and 16-17

Lerdahl and Jackendoff’sLarger-level grouping rules

• GPR 6 (Parallelism): “Where two or more segments of music can be construed as parallel, they preferably form parallel parts of groups.” (p.51)– Any two groups that begin in a similar way can be

construed as parallel (hence “parts of groups”)• e.g., bars 1-2, 3-4, 5-6 in example above from

Beethoven’s Quartet Op.18, No.1

Deliège’s (1987) Experiments on L&J’s grouping theory: Experiment 1

• Deliège carried out two experiments on L&J’s grouping theory

• In Experiment 1, subjects listened to extracts from instrumental works and marked perceived groups on paper

• Allowed to listen to examples as many times as necessary – number of times noted by experimenter

• Compared subjects’ analyses with those predicted by GPRs 2 and 3

Deliège’s (1987) Experiments on L&J’s grouping theory: Experiment 1 Results

• Results of experiment 1 were:– Both musicians and non-musicians were

significantly in agreement with rules– Musicians were significantly more in agreement

with rules than non-musicians– Non-musicians did not need significantly more

repetitions than musicians– All subjects needed significantly more repetitions

when responses not in agreement with rules

Deliège’s (1987) Experiments on L&J’s grouping theory: Experiment 1 Results

• Recall that L&J suggested addition of further part to GPR3 to account for change in timbre

• Deliège found three rules to be particularly important– GPR 2b, Attack-Point– GPR 3b, Change in dynamics– a version of GPR 3 for change in timbre

• Other rules used less by everyone, but much less by non-musicians

• Results imply that all subjects use primarily the relative lengths of events and changes in loudness and timbre to infer grouping structure

• Musical training makes people sensitive to other cues too

Deliège (1987): GPR 3d or GPR 2b?

• GPR 3d predicts a group boundary between end of sequence of short notes and start of sequence of longer notes

• But listener only knows first long note is first in a sequence of long notes when he hears the beginning of the 3rd long note

• GPR 3d therefore implies that listener retrospectively hears a group boundary before the first long note when he hears the 3rd long note

• Deliège actually found listeners hear a boundary after the first long note– Seems to be a modified form of GPR 2b

Deliège’s second experiment

• L&J do not provide any measure of rule strength to allow for resolution of conflict between preference rules

• Deliège presented subjects with scales modified to investigate effects of GPRS conflicting with each other

• Listeners were forced to choose one group boundary when rules predicted two (see example above)

• Found 93% of non-musicians and 97% musicians responded in accordance with rules (i.e., chose one of the two predicted boundaries)

• Musicians responded significantly more in agreement with rules• Change in timbre was the most frequently used cue

References

• Deliège, I. (1987). Grouping conditions in listening to music: An approach to Lerdahl and Jackendoff’s Grouping Preference Rules. Music Perception, 4(4), 325–360.

• Koffka, K. (1935). Principles of Gestalt Psychology. Harcourt, Brace and World, New York.

• Köhler, W. (1947). Gestalt Psychology. Liveright, New York.• Lerdahl, F. and Jackendoff, R. (1983). A Generative Theory of

Tonal Music. MIT Press, Cambridge, MA.• Wertheimer, M. (1938). Laws of organization in perceptual

forms. In W. D. Ellis, editor, A Source Book of Gestalt Psychology. Routledge and Keegan Paul, London. Originally published in 1923.