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Grouping
David Meredith
Aalborg 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
Musical surface
•
Grouping structure
rules
Metrical structure
rules
Time-span
reduction
rules
Prolongational
reduction
rules
Grouping structure
Metrical structure
Time-span
reduction
Prolongational
reduction
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
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
A
B
• 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
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
A
B
C
D
• 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
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 3a
GPR 3b
GPR 3c
GPR 3d
• 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
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’s
Larger-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’s
Larger-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’s
Larger-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 nonmusicians
• 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.