The Effects of Manipulation of Pitch Distributional - T-Space
60
The Effects of Manipulation of Pitch Distributional Properties of Melodies on Listeners' Perceptions of Tondity by Nichol as Alexander Smith A thesis submitted in conformity with the requirements for the degree of Master of Arts Graduate Department of Psychology University of Toronto O Copyright by Nicholas Alexander Smith 1998
The Effects of Manipulation of Pitch Distributional - T-Space
Properties of Melodies on Listeners' Perceptions of Tondity
by
for the degree of Master of Arts
Graduate Department of Psychology
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The Effects of Manipulation of Pitch Distributional
Properties of Melodies on Listeners' Perceptions of Tonaiity
Nicholas Alexander Smith
Graduate Department of Psychology
Abstract
The role of pitch distributionai information in listenen perception
of tonality in
algorithmically composed melodies was investigated. In a probe-tone
study, listeners heard
random orderings of the notes of the chromatic scaie in which the
durations of the notes were
systematicaily manipuiated so as to express different degrees of
tonal structure (tonal
magnitude). It was found that Iisteners o d y perceive tonality in
melodies at certain levels of
tonal magnitude, and when the pattern of note durations matches a
hierarchical representation
of tonal structure. It was also found that the tonal magnitudes of
Bach fugues corresponds to
the tonal magnitude required for listeners' perception of tonality
in the algorithmically
composed melodies. The results are discussed in terms of a process
of matching pitch
distributional information to cognitive representations of tonal
structure.
Acknowledgments
1 wodd first like to thank Dr. Mark Schmuckler for his guidance and
support in the
development of this projet, and for the meticdous comments made on
the written product. 1
would also iike to thank Dr. Bruce Schneider, for his helpfd
comments on experimental
design. Thanks also go out to Dr. David Huron. for the much needed
H u d m m tutonal. Last,
but definitely not least, I would like to thank my parents for theu
support and encouragement.
Table of Contents
Evidence for the Objective and Subjective Structure of Tonality,
5
Objective structure, 5
Subjective structure, 7
Key-finding algorithms, 9
to tonality perception, 14
Experirnent 1, 18
Design and Procedure, 21
List of Tables
1. Mean probe-tone ratings for listeners in the hierarchical
condition. 25
2. Mean probe-tone ratings for listeners in the random condition,
26
3. Tonal magnitude of 12 major key fugues, 36
4. Percent correct scores for sarne-different ratings for rnelody
comparisons. 41
List of Figures
structure of tonality, 4
2. Two duration profiles that differ greatly in their absolute
values but correlate highly
with standard key profile, 16
3. Mean intercorrelations between listeners ' probe-tone ratings in
the hierarchicd and
randorn conditions as a function of tond magnitude, 24
4. Three measures of correspondence between listeners' probe-tone
ratings and
representation of tonal structure, 28
5. Absolute deviations of the duration profiles for the C major and
Ab major fugues
from tonal magnitude profiles as a function of tonal magnitude,
35
vii
Properties of Melodies on Listeners' Perceptions of Tonality
Music has both an objective physical sinichire and a subjective
psychological
structure. There is objective structure in the notes written on a
sheet of music and in the
sound waves traveling towards the listener's ear during a musical
performance. Music has
subjective structure within the mind of the listener. It is musical
structure, in both its
objective and subjective foms, that pennits a collection of pitches
to be understood as music.
and to be understood meaningfully.
There are many different types of musical structure, several of
which have been
studied by various researchers. It has metrical structure, such
that musical events tend to
occw at periodic time intervals. It also has phrase structure
meaning that musical events can
be grouped together into larger units (Stoffer, 1985; Deliège,
1987; K d a n s l , 1996).
Musical events also follow one another in such a way that some
events are more expected
than others (Meyer, 1956; Namour, 1990; Schmuckler, 1989,1990;
Cuddy & Lunney, 1995;
Bigand & Pineau, 1997)
P iich Structure
One main property of music is its pitch structure. Music generally
makes use of a
lirnited number of discrete pitches dong the pitch continuum. This
subset of possible pitches
is represented by ail the black and white keys on a piano keyboard.
Within this subset is a
repeating pattern of 12 pitches called the chromatic scale. The
pitches of the chromatic scale
are given the names C, C#, D, D#, E, F, F#, G, G$, A, A#, and B.
Musical pitch is
perceived both in a monotonic and circular way. It is monotonic in
that playing adjacent keys
2
h m left to right on the piano will produce successively higher
pitches, or tones that increase
systematically in frequency. But it is circular in that there is a
percephial similarity between a
tone and a tone with a pitch 12 keyboard steps (semitones). or an
octave, above (Shepard,
1964). The names of pitches in the chrornatic scaie are assigned in
a cyclic way. The pitch
one semitone above B is C. Because there is a cyclic structure to
the chromatic scale two
notes that are an octave apart are given the same name. The term
'pitch class" refers to the
class of pitches that share the same name regardless of the octave
to which they belong.
This circdar pattern of the chromatic scale is not the only type of
pitch structure.
Music also has tonality. Piston (1969). in his classic rext,
defuies tonality as "the organized
relationship of tones in music. This relationship ... implies a
central tone with dl other tones
supporting it or tending toward it, in one way or anotherw (p. 30).
In tonality some pitches of
the chromatic s a l e are more important than others. Tonality
refers to the organization of
pitches on the basis of their importance, cenaality and stability.
One description of the
importance of certain pitches in tonality is the diatonic scale.
The diatonic scaie is a subset of
pitches in the chromatic scaie that are important in a tonality.
Most people are familiar with
the diatonic scale in the iorm of *do, re. mi, fa, so. la. ti, do".
One of the many ways of
playing this scale is to play the pitches C, D, El F, G, A, B. C
(the white keys on the piano
key board).
Music theorists (Meyer, 1956; Lerdahl, 1988) have also descnbed the
pitch
organization of tondity in terms of a tonal hierarchy. Pitches that
occupy higher position in
the hierarchy are generally perceived to be more stable within the
tonal context than those
occupying lower positions. A schematic representation of the tonal
hierarchy is shown in the
top part of Figure 1.
The most important pitch, the one at the top of the tonal
hierarchy, is the tonic. In
Figure 1 the tonic is C. Immediately below the tonic, on the second
highest level of the
hierarchy are pitches that are 4 and 7 steps above the tonic; in
this case the pitches E and G.
On the next level down, are pitches that are 2, 5, 9 and 11 steps
above the tonic; D. F, A and
B. The pitches that occupy the lowest level of the hierarchy are
those that are 1.3.6. 8 and 10
steps above the tonic; C # , D P. F # , G # and A $ .
One important point about the tonal hierarchy, and tonality in
generai, is that any
pitch from the chromatic scale can potentially occupy the top
position in the hierarchy. In this
sense, it is possible to talk about many tonalities. each with a
different pitch as the tonic.
Different tonalities, or keys (not to be confused with keys on a
piano keyboard), are
identified by the narne of the pitch which acts as the target. The
are also two types of
hierarchical organization, called modes. These are cailed the major
mode and the minor
mode. Though they are both used frrquently in music, the present
thesis will focus on the
major mode. It is customary to identify the name of a key by the
name of the pitch the serves
as the tonic, and by the mode of the key. The hierarchy shown at
the top of Figure 1
represents the key of C major.
One consequence of the fact that there are different possible
musical keys is that the
importance of any given pitch is determined relative to a
particular context, or key. For
exarnple, the pitch C is the most important pitch in the context of
the key of C major, but is
one of the least important pitches in the context of the key of F$
major. Accordingly, the
given local tonality of a passage of music plays an extremely
important role in how listeners
Figwe 1. A music-theoretical (top), objective (rniddle; Knunhawl,
1990) and psychological
(bottom; Knrmhansl& Kessler, 1982) description of the
hierarchicd structure of tonality.
Pitches occupying higher positions on the hierarchy occur more
frequently in music and
receive higher probe-tone ratings.
C C # D û # E F F # G G # A A # B Pitch
C C # D D # E F F # G G # A A # B Pitch of Probe Tone
understand musical pitch.
Given that tonafity perfoms the crucial function of organizing
musical pitch by
theoretically establishing a hierarchical framework of importance,
the question irnmediately
follows as to how tonality is established in the listener's mind.
As mentioned above. any of
the 12 pitches of the chromatic scale can serve as the tonic of a
key in one of two modes.
major or minor. Thus, there are 24 possible keys. or 24 different
ways in which the listener
c m organize the incoming pitches. This is a formidable task
because the listener must
identiS the key solely on the basis of information present in the
music.
Evidence for the Objective and Subjective Stmcture of
Tonality
Given the theoretical importance of tonaiity as an organizer of the
relations between
musical pitches, one obvious question involves the evidence for. on
both an objective and
subjective level, the existence of tonality. In the case of
objective structure, how is this
pattem of relations represented in the actual notated musical
score, or in the sounded musical
events? In the case of subjective structure, how is the pattern of
relations represented
psychologically, and what implications might such representations
have for the perception of
music?
Objective Siructure
What evidence is there for the reality of this hierarchy in the
objective structure of
music? Some of the most compellîng evidence for the objective
structure of tonality cornes
from analyses of distributional information, or information about
how frequent are the
different pitches that typically occur in music. Such interest in
distributional information in
music has roots in antiquity. Aristotle remarked that, "al1 the
best mes make frequent use of
the mese (a Greek classical equivaient to the tonic), and al1 good
musicians employ it
€requently, and quickiy revert to it. if they leave it, but not to
any other note to the sarne
extentw. He continues to Say that, "the mese is a kind of
conjunction especially in good
music, because its note most often underlies the tune" (Aristotle,
Problem XU(, tram. 1970).
One implication of this early observation is that the nurnber of
times a pitch occurs relates to
its importance.
More contemporary studies of distributional information have
applied a more
systemtic method of observation to Western tonal music. Much of
this work grew out of an
interest in applying information theory to music (Pinkerton, 1956;
Youngblood, 1958;
Knopoff & Hutchinson, 1983), and has observed similar effects
with regard to tonal ity. In
this work distributional information is often characterized in
ternis of how often the 12
chromatic pitches occur in various musical contexts. Kmhansl (1990)
sumrnarizes many of
these studies, and fin& that when different pieces of music are
aansposed into a common key
and consolidated into a single octave, the distribution of the
frequencies of occurrence for
each pitch (middle of Figure 1) mimics the music-theoretical
description of hierarchical
structure (top of Figure 1). Thus, pitches that are theoretically
important for a given key, such
as the tonic and the pitches 4 and 7 steps above the tonic. are
played most frequently, and the
less important pitches, the remaining pitches of the diatonic scale
(those that are 2,5,9 and
11 steps above the tonic) and non-diatonic pitches (those that are
1 ,3 ,6 ,8 and 10 steps above
the tonic), are piayed l a s frequently. S h i l a r fidings were
observed for the distribution of
dwations, or the total time each of the pitch classes were played,
in a short piano piece by
Schubert (Hughes, 1977). Thus, analyses of both frequency of
occurrence and the total
duration of each pf the notes of the chromatic scale provide
compelling evidence for the
existence of objective tonal structure in music.
Subjective Stnictwe
What evidence is there for hierarchical structure of tonality in
the subjective.
psychological representations of musical pitch? K m h a n s l and
colleagues have conducted
numerous investigations of the subjective structure of tonality;
these studies are summaïized
in Krumhansl (1990). Early work by Knimhansl and Shepard (1979)
provided the First
evidence for the psychological existence of the hierarchical
representation of musical pitch.
These experiments used a probe-tone paradigm in which listeners
heard a key-defining
context (either an incomplete ascending or descending scaie)
followed by a probe tone, with
the pitch of this probe tone changing on successive trials.
Listeners were asked to rate how
well the probe tone completed the context. Analyses of the pattern
of listeners' responses
reflected a few different parameters: (a) the difference in
frequency between the pitch of the
probe tone and the finai pitch of the context, and (b) the relative
importance of the probe tone
in the theoretically hierarchical structure of a tonality.
Again employing the probe-tone procedure, Knimhansl & Kessler
(1982) replicated
and extendeci these f'mdings, using a variety of tonality defining
contexts (e.g. scales. chords
and chord cadences) and testing different patterns of hierarchical
structure (i.e. major and
minor tonalities using different notes as the tonics). These
authors also found that listeners
rated the probe tones baseci on their position within the tonal
hierarchy of the key of the
context. For example, high ratings were given to different probe
tones that were the tonics of
the respective contexts, despite the fact that these probe tones
had different absolute
8
frequencies. More generally, the relative hierarchical pattem of
responses was simila. for al1
the major key contexts. with an equally reliable (albeit different)
pattern of response for the
minor key contexts. T ~ u s , two basic hierarchical structures
were found: one for major keys,
and one for minor keys.
These two basic hierarchical stmctures will be referred to as the
standardized key
profiles. The standardized major key profile. relative to the key
of C, is shown at the bottom
of Figure 1. Note the correspondence between the theoretical
hierarchical description of
tonality and the standard key profile. The highest rated probe tone
is the pitch at the highest
level of the hierarchy. The second and third highest ratings were
given to the second and
third highest pitches on the hierarchy. and so on.
Finally. it is important to note that the probe-tone ratings have a
comparable
hierarchical structure to the distributional information: described
above. This fit can be seen
by matching the outlines of the middle and bottom panels of Figure
1. Thus, not only is it
possible to quantify both the objective and subjective smicmal
properties of tonality, it
appears that there is a close correspondence between these IWO
descriptions. Accordingly,
music theoretic. information-theoretic and psychological
characterizations of musical tonality
are in close congruence (For a discussion of the relations between
these different approaches
see Cuddy, 1997).
The observed congruence between the objective and subjective
hierarchical structure
of tonality has led researchers in at l e s t two different
directions. One outgrowth of this
match has been in the development of key-fmding algorithms, which
attempt to model both
9
objective (i. e., music-theoretic) analyses and subjective (i.e..
Listeners ' perceptions) musical
tonality. A second outgrowth of these ideas has k e n in the
developrnent of algorithrnic
compositionai procedures. based on the distributionai properties of
the occurrence of musical
notes in various keys.
Key-finding Aigorithm
The goal of work on key-finding is to output an accurate
representation of the musical
tonality of a piece or passage of music based on some form of
analysis of the musical score.
Several different key-finding algorithms have b e n proposed (e.g.,
Longuet-Higgins &
Steedrnan, 1971: Hoitzman, 1 977), which employ different
strategies with varying degrees of
success. One key-fincihg algorithrn, developed by Krurnhansl and
Schmuckler (1986;
described in Krumhansl. 1990), is of special interest here because
it incorporates both the
distribution information and the psychological data collected by
Knimhansl and Kessler
(1982).
The key-aigorithm operates by correlating duration distributions
for musical passages
with standardized key profiles for the 24 major and minor keys.
derived by Knimhansl and
Kessler (1982). The aigorithm outputs 24 correlation coefficients
describing the degree of
relatedness between the distributional information and the
standardized profiIes for the 24
major and rninor keys. One impiication of this key-fmding aigorithm
is that listeners
detemine the tonality of music by mentally "correlating"
distributional information with
psychological representations of different keys, with the strongest
positive correlation
detemiuung the key of the passage.
Krumhansl and Schmuckler provided a variety of tests of this
key-fuding algorithm
10
(see Knimhansl, 1990, for description). One application foliowed on
work by Cohen (1991),
in which listenen heard the f i t four musical events (notes or
chords) of twelve preludes
from J. S. Bach's The Weil-Tempered Clavier, and were asked to sing
the scale that they
thought corresponded to the key of the piece. On approximately 75%
of the trials, iisteners
sang the scale of the key in which Bach composed the prelude. For
cornparison. Knvnhansl
and Schrnuckier applied their key-finding algorithm to the first
four notes (including notes
played simultaneusly) of the prelude; the algorithm identified the
correct key in al1 cases,
suggesting that the correlation-style matching process might make a
good model for human
tonali ty perception.
The key-finding algorithm has also been shown to be able to predict
listeners'
ongoing perceptions of tonality. Smith (1996) applied the
key-finding algorithm to
successive sections of the first nine bars of the second movement
of Beethoven's Waldstein
Sonata. This piece of music was chosen because it exhibits a large
amount of key movement
within a relatively shon (nine bars) section. and theoretical
predictions of the piece's tonality
have been made (Lerdahl, 1988). Listeners who were unfamiliar with
the piece performed in
a probe-tone study that assesseci their perceptions of tonality at
the end of each of nine
consecutive bars. The algonthm's first and second predictions
(highest positive correlations)
rnatched listeners' perceptions of the tonality for six of the nine
bars. The possible rote of
expectancy and farniliarity in listeners' perception of tonality of
the Waldstein Sonata is
discussed by Cuddy and Smith (1997a, 1997b, in press).
In another test of the key-finding algorithm. Schmuckler and
Tomovski (1997) found
that the algorithm was able to model listeners' perceptions of
tonality in Chopin's E minor
prelude. Overall. these applications suggest that pattern matchlng
of distributional
information to perceived hierarchies of stability (Knimhansl &
Shepard. 1979; Knimhansl &
Kessler, 1982) provides a reasonably good mode1 for listeners'
perceptions of musical
tonality.
A [go n*th mica& Composed Music
Algo&hmically composed music is music created through the
application of a set of
rules. and random processes. Although algorithmic composition has
its roots in information
theory (e.g. Pinkerton, 1956), it has aiso become a powexfui
experimentai tool in music
perception research. This is because the process of creation allows
the researcher to specib
certain characteristics of the music, while allowing the rest to be
determineci by chance. Thus
the researcher is able to define causal relations between the
structure of music and how it is
(potentially) perceived by the listener. Several snidies have used
this technique to identify the
types of musical structure necessary to elicit perceptions of
tonality in the listener.
West and Fryer (1990) created melodies composed of the seven notes
of the diatonic
scaie (C, D, EI F, G, A. B). The notes in these melodies were
played in a randorn order, and
(presumably) for equal duratiom. In this case, the distributionai
information inherent in these
melodies can be expressed in a two level hierarchy, with the top
level conüuning the notes of
the major scale, and the bottom level containing the remaining
notes of the chromatic scale.
These random melodies were foiiowed by a probe tone belonging to
the diatonic scde of the
melody, and listeners rated whether the probe tone would make a
suitable tonic for this
melody. West and Fryer found that tonic was not always correctiy
identifiai, and that
listeners often confused the fifth (G), fourth (F) and third (E)
scale degree as the tonic of the
melody. They concluded that the distributional information present
in the melody (the
diatonic scale) was not suffcient to uniquely identi€y the tonic,
and that the order in which
notes were played was important for the perception of
tonality.
A few cornments can be made on West and Fryer's (1990) findings.
Interestingly, the
fact that their listeners often confused the fifth, fourth and
third scale degree with the tonic is
actually in agreement with predictions based on Knimhansl and
Schrnuckler's key-finding
algorithrn. Men the algorithrn is applied to the distributional
infomation in West and
Fryers's melodies (using a duration profile made up of 1's and
0's)' the highest positive
correlations with major keys are with C major G = -76.p c .01). G
major G. = -68. p < .OS)
and F major (r = .55, ns). Based on the sparse durational
infomation in the melody. and no
note order information, the key-finding algorithm outperforms West
and Fryer's listeners,
and even corroborates listeners' second and third guesses. In
fairness, however, it should be
noted that the key-fmding algorithm did produce a high correlation
with the key of A rninor
(r = -71, p c .01), the tonic of which was rated as not
particularly suitable for the tonic of the
melody.
Work by Oram and Cuddy (1995) has also found listeners to be
sensitive to the
distributional properties of algorithmically composed melodies. In
this probe-tone study.
listeners heard four different types of rnelodies. One melody was
comprised of the seven
notes of the major scale (diatonic sequence). The other three types
of melodies were made up
of notes that did not belong to any single major or minor scale
(nondiatonic sequence). Al1 of
the meIodies had twenty notes that each sounded for 200 ms, with
one note occurring eight
times, two notes occurring 4 times, and four notes occurring once
each. These frequency of
occurrence values were assigneci to the notes in a way that
contravened the conventions of
Western tonal music. It was found that iistenen' probe-tone ratings
were positively related to
the number of tirnes that the probe tone was heard in the melody,
but more so for the diatonic
melodies than the nondiatonic melodies. Orarn and Cuddy concluded
that the increased
relation between Frequency of occurrence and probe-tone rating for
the diatonic melodies
reflects the influence of an intemalized representation of tonal
structure.
Finally, a study by Lantz (1995) also investigated the use of
distributional
information in algorithmically composed melodies. This work started
frorn the observation
that the frequency of occurrence and the total duration of
different pitches are naturally
correlated in musical passages. with highly frequent notes having
greater total durations that
infrequent ones. Lantz attempted to tease apart duration and
frequency of occurrence on
listeners' perceptions of tonality, using rnelodies composed of the
six notes of the C major
(C, E and G) and F# major (F#, A# and C#) triads. The
distributional properties of the
melodies were manipulated in two ways. First, the notes of one
triad were made longer while
keeping the notes of both triads equal in frequency of occurrence.
Second. the notes of one
aiad were made more frequent while keeping the durations of both
triads constant Thus each
triad was emphasized via different means, with the pitches of one
triad longer in duration,
whereas the pitches of the other triad were played more frequently.
Probe-tone ratings
revealed that pitches with longer duration received higher
goodness-of-fit ratings than pitches
that were played more frequently. Accordingly. listeners'
perceptions of tonaiity appear to
depend on the duration profile of a piece of music, rather than its
frequency of occurrence
profile.
Overail, these fmdings demonstrate the utility of algorithmic
composition as an
experimental tool in music perception research. They also motivate
the research of the
present thesis.
Cn.ticisms of Attempts îo Relate DLstn*buh'oml h f o m n u n to
Tonality Perception
Before proceeding to the aim of the present thesis, it is
worthwhile addressing the
critics of work discussed above. There has been a rather lively
debate in the recent music
perception Literature (see Butler, 1989; Krwnhansl 1990; Butler,
1990 for an animated
interchange) regarding the adequacy of pitch distributional
information as a basis for the
perception of tonality. Some researchers (e.g. Brown & Butler.
1981; Brown, 1988: Butler,
1990) have argued that pitch distributional information is of
Limited value because it fails to
preserve information about the temporal ordering of notes,
something that affects listenew'
perception of tonality. It has also been argued (Buder, 1989, 1990)
that the resdts of probe-
tone studies (e.g.. Krumhansl, 1979; Knimhansl& Kessler, 1982;
Cuddy and Badertscher,
1987) may not reflect an intemal representation of tonal structure,
but rather short-term
memory effects for recentl y heard pitches.
The Aim of the Ptesent Thesis
The present thesis continues the investigation into the relation
between the
distributional properties of melody and listeners* perceptions of
tonality. Revious research
has found that both the dishibutional properties of tonal music and
listeners' perceptions of
musical pitch evidence a common hierarchical structure. and are
thus highly correlateci. In
addition, experimental manipulation of the distributional
properties of music in
algorithmically cornposed melodies has demonstrated that listeners
are sensitive to
distributional information in musical passages and can use such
information in perceiving
tonality. This thesis investigates whether a simple correlation
between music's distributional
information and this subjective/objective hierarchical structure is
sufficient for the perception
of tonality in listeners.
As describeci earlier, one implication of the work on k e y - r i n
g and algorithmically
composed music is that listeners' sense of key may develop by
somehow matching the
distributional information inherent in the objective structure of
music to psychological
representations of the structure of tonality, and nibsequenîiy base
judgements of anality on
the best match. This idea has been most explicitly formalized by
KNmhansl and Schuckler
in their key-fiding algondun (Krumhansl, 1990).
One concem with this correlation style matching process as a mode1
for tonal
perception stems from the fact that in computing relatedness,
correlations convert boch
variables into standard units prior to their cornparison. One
consequence of this
transformation is that, although it preserves the relative
magnitude of the variation between
individual values of a variable relative to the mean, it elimuiates
the absolute magnitude. As
a consequence, models of tonality perception based on correlationai
measures are insensitive
to the absolute magnitude of values in the duration profile, as
well as the ratios berween the
values. A s shown in Figure 2, an equaliy good match can be made
between Knunhansl and
Kessler's (1982) standard key proNe and two duration profiles that
Vary drastically in the
absolute magnitude of their components. If tond perception is auly
based on correlating
duration profiles with cognitive representations of musical keys,
then listeners ' perceptions of
tonality should be relatively unaffecteci by variation in the
absolute magnitude of values in
16
Figure 2. An example of two duration profiles that differ greatly
in ternis of their absolute
values, but that both correlate highiy with the standard key
profile (Knimhansl& Kessler.
1982). The correlation for the profiles for tonal magnitude of 0.5
and 4.5 are -997 and -897
respectively.
C C # D D # E F F # G G # A A # B Pitch
the duration profile, so long as the correlation remains
strong.
One way of evaluating this correlation-style matching process is to
manipulate the
distributional properties of music such that the relative
magnitudes of the profiles values
remain highly similar, whiie having the absolute magnitudes of the
profiles Vary. An
appropriate starting point for such an approach is to create a
range of melodies whose
duration profiles al1 correlate highly with the standardized key
profile ( h h a n s l & Kessler,
1982), but nevertheless Vary in the absolute duration of the
different pitches. Such duration
profiles can be created via any nurnber of transformations of the
original profile, including
adding, subtracting, multiplying or dividing these profiles by a
constant. One particularly
intriguing transformation involves raising the values in the
profile to an exponent; this
transformation has some interesting (and occasionally unique)
properties:
1. Exponentiai transformations systematically change the absolute
magnitude of the
disaibutional information while consistently producing high
correlations with the
tonal hierarchy .
2. They are bounded at the lower end of the exponent continuum. AH
values in the
duration profile equal 1 when raised to the power of 0, resulting
in a uniform
distribution (no variation in the distributional information, and
hence no objective
tonal s tmcture) .
3. They are practically bounded at the upper end of the exponent
continuum. With
very high exponents the ratio between the tonic and the rest of the
pitches approaches
1:O. Thus, the tonic becomes the one and only pitch.
4. Exponentid transformations provide a continuous variable that is
wefl suited for
psychophysicai study.
In the present thesis, this exponent dimension will be referred to
as "tonal
magnitude." with the tonal magnitude of a duration profile
expressed in terms of the
exponent to which the standardized key profile ( K m h a n s l
& Kessler. 1982) is raised.
Examples of the duration profiles with different tonal magnitudes
are given in the appendix.
This thesis will examine a nurnber of specific research
questions:
1. What degree of tonal magnitude is necessary for listeners to
perceive the tonaiity of
a melody? In psychophysicai terms. what is the absolute threshold
of tonality as a
function of tonal magnitude? (Experiment 1)
2. What is the psychophysical function relating tond magnitude and
perceived of
tonaiity? Does the perception of tonaiity systematically increase
with increases in
tonal magnitude, or is tonaiity perceived categorically, with
systematic increases in
tonal magnitude causing a discontinuous shift into the perception
of tonality?
(Experiment 1)
3. What is the tonal magnitude of composed tonal music? Does the
tonal magnitude
of composed music correspond to the threshold of tonality.
(Experiment 2)
4. Does tond magnitude influence the encoding of musical matenals?
Specifically,
are melodies with greater tonal magnitudes encoded and remembered
more easily
than those with lower tond magnitudes. (Experiment 3).
Experiment 1
The airn of Experiment 1 was to determine the characteristics of
disaibutionai
information necessary for the perception of tonality in listeners.
Towards this end, the tonal
magnitude of the duration profiles of melodies were systematically
manipulated
exponentiaily, and listeners' perceptions of tonality in response
to ihis manipulation was
exarnined.
Participantr
Forty students at the University of Toronto at Scarborough
participated in this
experiment in exchange for payment or course credit. They ail met
the training prerequisite of
three years of music training. They reported an average of 9.1
years of music training on
their ptimary instrument, and reported listening to music for an
average 13.3 hours per week.
Stimulus Materials and Experimentaf Apparatus
The stimuli consisted of a series of algorithmically composed
melodies with varying
levels of tonal magnitude. Al1 melodies were 10 seconds in total
length, and contained 24
notes in d l (two occurrences of each pitch of the chromatic
scale). The method of calculating
pitch duration is shown in Equation 1. The duration (D) of a given
pitch @) is calculateci by
taking Knimhansl and Kessler's (1982) standardized major key
profile value 0 for that
pitch (p) and raising it to a given tonal magnitude exponent (m).
This duration is then
expressed as a percentage by dividing it by the surn of ai i
transformed key profile values
(twelve values in dl , one for each chromatic pitch). Finaiiy, the
percentage value of this pitch
is then multiplied by 10000 (the duration of the melody in
milliseconds) and divided by 2
(the number of occurrences of each pitch); thus. a given pitch had
the sarne duration for both
its occurrences. The note durations used in these melodies are
shown in the appendix. The
melodies were then created by randomly permuting the order of the
24 notes. with the onset
of a note imrnediately following the offset of the previous note.
AU notes were sounded with
the same loudness.
The experiment contained one between-subjects factor. with 20 lis
teners each
participating in one of two conditions. In the first, or
"hierarchical" condition, melodies were
constructed as just descnbed, with the distribution of tone
durations for the notes of the
chromatic scale based on the standardized key profile, as modified
by the tonal magnitude
value. In the second, or "random" condition, after the durations
for the 12 pitches were
detemined, the assignrnent of durations to specific pitches was
then scrarnbled nius,
although the set of durations for melodies in the random condition
were the sarne as in the
hierarchicai condition. the pattern of distributionai information
no longer coincided with the
hierarchical organization described by M m 1 and Kessler (1982).
Put another way,
whereas both hierarchical and random melodies contained equivalent
numbers of long and
short notes, they differed in whether or not the durational pattern
of these notes conformed to
hierarchical key structure.
Al1 melodies were played on a Yamaha DX7 synthesizer, set to a
preset electric piano
timbre (E. Pno 6.2). The synthesizer was connected to a Macintosh
8100 AV computer via
MIDI interface. and was contmlled by the MAX programming language.
Audio output from
the synthesizer was fed into a Mackie Micro Series 1202 mixer, and
was ultimately presented
to listeners through two Boss MA-12 micro monitors.
Design and Procedure
The study employed the probe-tone method developed by Krumhansl and
Shepard
(1979). Each trial consisted of a presentation of a melody. a 1 s
silent interval, and a 2 s probe
tone. The probe tone was one of the 12 pitches of the chromatic
scale, and was played with
the same timbre, pitch height (Le. octave) and loudness as the
melody. After each probe tone
listeners rated on a 7-point scde how well they felt the probe tone
fit into the context of the
melody they had just heard. It was stressed that they were to judge
how well this probe tone
fit into the melody in general; not as a continuation of the
melody.
in each block of the experiment there were 14 triais, with each
listener hearing the
same melody on al1 trials in a given block. The first two trials in
the block were considered
practice, and were intended to farniliarize the listener with the
melody for ba t block. Trials 3
to 14 contained the 12 probe pitches presented in a random order
for each listener. There
were ten blocks in al1 in this experiment. with each block
corresponding to a different tonal
magnitude value. The 10 values of tonal magnitude examined were
0.0, 0.5, 1.0, 1.5, 2.0. 2.5,
3.0, 3.5, 4.0 and 4.5. The order of these 10 blocks was randomized
for each listener.
Each listener heard a different random order of notes in a melody
for a given tonal
magnitude. In order to avoid carry-over effects between blocks of
triais, hence between tonal
magnitudes, the melodies were randornly tramposeci to a different
tonic for each block. This
random transposition was different for each listener. The entire
experimental session lasted
approximately 45 minutes, after which each listener filled out a
subject information fom, and
was debriefed.
Prior to any analysis. there was a need to organize the collected
data into a common
framework. Because the melodies presented to listeners were
randomly tmnsposed to
different keys, transposing iisteners' probe-tone ratings back to a
cornmon key resulted in
ratings that represent the goodness-of-fit of a probe-tone of a
given scale degree, rather than
that of a given absolute pitch. Listeners' probe-tone ratings are
presented here with reference
to the key of C major. For the random condition. in which the
calculated note durations were
randomly assigned to different chromatic pitches, the probe-tone
ratings were organized in
terms of the scale degree of that pitch's duration. For exarnple,
the rating for the probe tone
whose pitch had the longest duration in the melody was assigned to
the pitch C. The rating
for the probe-tone whose pitch had the second longest duration was
assigned to the pitch G,
and so forth. Thus. the hierarchical organization of pitch
durations that was destroyed in the
random condition has been salvaged here to be able to analyze and
present the data in an
orderly way.
A preliminary analysis of data investigated the degree to which
listeners ' probe-tone
ratings were intercorrelated for each different level of tonal
magnitude in the hierarchical and
random conditions. For each Level of tonal magnitude, within each
condition (random vs.
hierarchical) the mean correlation between each listener's
probe-tone ratings and those of al1
the other listeners was calculated, and averaged across listeners.
The mean intercorrelation
was found to Vary both as a function of condition (random vs.
hierarchical) , and as a function
of tonal magnitude. An analysis of variance found highly
significant main effects for
condition (F ,,,, = 70.00,~ < .001), tond magnitude (F ,,, =
15-61, p < .001). as well as a
significant interaction effect for condition and tonal magnitude (F
= 10.19, p c .001). As
23
shown in Figure 3, the increased degree of intercorrelation was
much more pronounced in
the hierarchical than in the random condition. The increase in
listener agreement at the higher
tonal magnitude levels of the hierarchical condition suggest that
there is some common
structure guiding listener responses.
Subsequently, analyses further explored the probe-tone ratings as a
function of tonal
magnitude in the hierarchicd and randorn conditions. The mean
probe-tone ratings for the 12
notes of the chromatic scale as a function of the different levels
of tonal magnitude are
presented in Table 1 for the hierarchical condition, and in Table 2
for the random condition.
A preliminary inspection of the data reveals that differences in
the mean probe-tone ratings
become more pronounced at higher levels of tond magnitude in the
hierarchical condition.
Listenerd probe-tone ratings were then submitted to a one-way
ANOVA. to assess
these intuitions concerning differentiations between the probe tone
ratings as a function of
tond magnitude and condition. The results for the hierarchical
condition are shown at the
bottom of Table 1. For tonal magnitudes 0.0 and 0.5, no significant
differences were found in
the ratings listeners gave for probe tones of different pitches.
However, for tonal magnitudes
of 1.0 and greater, significant differences were found, suggesting
that listenen did indeed
differentiate between the probe tones at these higher levels of
tond magnitude. When the
same analysis of variance was performed on the data from the random
condition, shown in
Table 2, no significant effect of pitch of the probe tone was found
at any level of tonal
magnitude, sugges ting that listeners were unable to differentiate
between the probe tones. In
terms of goodness-of-fit, listenen are only able to differentiate
between probe tones when
context melodies had a hierarchically organized pattern of
durations, with this pattern having
24
Figure 3. Mean intercorrelations between listeners' probe-tone
ratings in the hierarchicai and
random conditions as a function of tonai magnitude.
O 0.5 4 1.5 2 2.5 3 3.5 4 4.5 Tonal Magnitude
- Hierarchical -mm--11. Random
Table 1. Mean probe-tone ratings for listeners in the hierarchical
condition.
Tonal Magnitude
rK&K - Correlation between mean probe-tone ratings and
Krurnhansl& KessIer's (1982) standardized tonal hierarchy
ratings
rDur - Correlation between the duration profiles for the different
tond magnitudes
absK&K - The toiai absolute deviation of the mean probe-tone
ratings frorn Knimhansl& Kessler's (1982) standardized tond
hierarchy ratings
rTM - The correlation between the rK&K, rDur, or absK&K
values and their conesponding tond magnitude values
Table 2. Mean probe-tone ratings for listeners in the random
condition.
Tond Magnitude
rK&K - Correlation between mean probe-tone ratings and
KrurnhansI & Kessler's (1982) standardized tond hierarchy
ratings
F(11209, 0.96 0.81 0.84 0.35 1.37 1.09 1.33 1.08 1.42 1.04
rK&K -.20 .12 -.58 .23 -46 .O8 .35 .59' .48 -.O8
rDur na .13 -.58 -20 .51 -.O2 .30 -53 .51 -.28
absK&K 17.00 14.94 17.95 15.89 14-58 16.52 12-70 14.49 12.98
14.50
rDur - Correlation between the duration profiles for the different
tonal magnitudes
rTM
-48
-25
-.67"
absK&K - The totai absoIute deviation of the mean probe-tone
ratings from Knrmhansl& Kessler's (1982) standardized tonal
hierarchy ratings
fIM- The correlation between the rK&K, rDm, or absK&K vdues
and their corresponding tonal magnitude values-
tonal magnitudes of 1 .O or higher.
Both of these analyses suggest that higher levels of tonal
magnitude influence
listeners' ratings, but oniy when these levels of tonal magnitude
are hierarchically organized.
The next step in the analysis was to determine whether or not these
ratings were thernselves
organized in a way matching the hierarchical structure of the
melodies. The results of various
attempts at assessing this match are shown in the bottom rows of
Tables 1 and 2 for the
hierarchical and random groups, respectively, as well as being
presented graphically in
Figure 4.
The first measure of correspondence was the correlation between the
mean probe-tone
ratings and Knunhansl and Kersler's (1982) standard key profile ( s
e Figure 1). Because the
standard key-profile represents the hierarchical structure used to
create the melodies, strong
positive correlations with the standard key profile indicate the
presence of hierarchical
structure in the probe-tone ratings. For the hierarchical
condition, there was a general
increase in this correlation coefficient with increasing tonal
magnitude (see Table 1 and top
panel of Figure 4). This shows that the degree of hierarchical
structure in the probe-tone
ratings increases as a function of tonal magnitude. M e n these
correlation coefficient were
themselves correlated with their corresponding tonal magnitude they
were found to be
significantly linearly related ( r = 38, p c .01).
For the random condition, there was no systematic increase in the
correlation between
probe-tone ratings and the standard key profile as a function of
tonal magnitude (see Table 2
and top panel of Figure 4). Confïming this lack of difference,
there was no relation between
the correlation coefficients and their correspondhg tonal magnitude
value ( r = .48,p > .05).
Figure 4. Three measure of correspondence between listeners'
probe-tone ratings and
representations of tonal structure.
Combîions with Tonal Hierarchy
O 0.5 1 1.5 2 2 5 3 3.5 4 4.5 Tonal Magnitude
Correlations with Duration Profile 1 -
-0.8
O 0.5 1 1.5 2 25 3 3.5 4 4.5 Tonal Magnitude
Absolute Oeviation from Tonal Hieraithy 20 -
O 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Tonal Magnitude
One point of interest in the preceding analysis is that, for the
frst measure of
correspondence, a tonal magnitude value of O (al1 durations of
equal value) actually produced
a reasonably good fit with the standard key profile. On the basis
of this correlation alone, one
might be tempted to conclude that some degree of tond structure is
present in such melodies.
However, it is important to note that the actual values for the
various probe-tone ratings for
this tonal magnitude. were. in fact, very similar (see Table l),
and did not differ significantly.
The fact, then, that a strong correlation was observed between the
ratings and the standard
key profile despite there being no reliable difference between the
probe-tone ratings
themselves underscores the danger of using correlations as a
measure of correspondence; it
emphasizes relative patterns at the expense of absolute magnitude
information.
A second rneasure of correspondence was the correlation of the mean
probe-tone
ratings and the durations of the 12 pitches of the melodies. For
the hierarchical condition
there was a general increase in the strength of correlation as a
function of tonal magnitude
(see Table 1 and the middle panel of Figure 4), with the increase
in correlation as a function
of tonal magnitude approaching significance ( r = .65,p = .06). In
contrast. there was no
relation between the average rating and duration profiles as a
function of tonal magnitude for
the randorn condition ( r = -25. n-S.).
To avoid this problematic insensitivity of correlation coefficients
to absolute
differences, a third measure of correspondence was used --the
absolute deviation between the
average probe-tone ratings (as a function of condition and tonal
magnitude) and the standard
key profile. Both the hierarchical and random conditions evidenced
a better fit between
average ratings and the standard key profile with increasing tond
magnitude. However. this
trend was more pronounced in the hierarchicai ( r = -.91,p <
.01) than in the random
condition ( r = -.67. p < .OS). In addition, a one-way ANOVA
using different tonal
magnitudes as the randorn variable revealed a significant
difference in absolute deviations in
hierarchical relative to random condition VCfs = 13.06 vs. 15.16.
respectively; F,,, = 9.05,
MSE = 2 .43 ,~ c.05). Regarding the relation between tonal
magnitude and perceived
tonality, the psychophysical function, as shown in aii three panels
of Figure 4. appears to be
more or Less continuous. There are no categorical shifts into the
perception of tonality
in sumrnary, the results of Experiment 1 show that listeners are
sensitive to the
distributional properties of melodies, pnmarily when the
distributional information has
hierarchical structure at certain levels of tonal magnitude.
Discussion
The results of Experiment 1 support the claim that listeners'
perceptions of tonality
can be affected by the manipulation of the pitch distributional
properties of music. Even
when the duration profiles of melodies correlateci highly with a
standardized key profile
(Knimhansl& Kessler, 1982). Listeners failed to discriminate
between different probe tone in
te- of their goodness-of-fit (as in the hierarchical tonal
magnitude 0.5 condition), nor did
these correlate significantly with the standardized key profile for
tonal magnitudes of 1.5 and
less. These fuidings suggest that the absolute duration values of
pitches. expressed here as
tond magnitude, are important for the perception of tonality and
that pattern matching by
itself is not a sufficient mode1 for the rnatching process
performed by listeners.
The fact that listeners in the random condition failed to
discriminate between different
probe tones in terms of their goodness-of-fit is a very significant
fiding. It demonstrates that
31
the probe-tone method does not sùnply measure some type of echoic
or sensory memory, as
suggested by Butler (1989, 1990). If listeners' probe-tone ratings
were merely driven by the
contents of a sensory buffer, one would expect no differences
between the ratings given in
the hierarchical and random conditions. given that the elements
stored in this buffer in no
way differed. Because differences were found between groups whose
stimuli differed oniy in
that one group heard hierarchically organized stimuli and the other
heard randornly ordered
stimuli, such differences are likely a product of listeners'
sensitivity to the organization of
elements. rather than the elements alone.
Another important point should be made in reference to Butler's
(1989) criticism that
disûibutional information provides an inadequate ba i s for the
perception of tonali ty because
it ignores information about the temporal ordering of pitches.
Although Brown and Butler
(1981) and Brown (1 988) have found that the temporal ordering of
pitches effect listeners '
perceptions of tonaiity. the resdts of Experirnent 1 demonstrate
that the perception of
tonality is not criticaily dependent on specific orderings of
pitches. Listeners were able to
perceive the tonal structure of melodies, despite the fact that the
ordering of pitches in these
melodies was randomly determineci. This fmding runs contrary to
West and Fryer's (1990)
result that a random ordering of notes of the diatonic scale
provides insuffkient cues to
tonality. Experiment 1 has shown that random o r d e ~ g s of notes
of the chromatic scale is a
sufficient cue to tonality, provided that the note durations
represent the hierarchical structure
of tonality.
In general, these fmdings support the view that the perception of
tonality is not
directiy reliant on a sensory representation of pitch. but instead
that the perception of tonality
is mediated by a cognitive representation of pitch. one that
represents the organization of
pitch elements and their relations between one another. This view
would gain even stronger
support by a demonstration that the cognitive representation of
pitch could be evoked in the
presence of only sparse sensory information. A future experiment
might try presenting
listeners with melodies in which one pitch (the tonic) is played
for a relatively long duration,
while the other eleven pitches of the chromatic scale are played
for much shorter, but
identicaï amounts of time. If a probe-tone study were to find that
listenen show a hierarchical
pattern of ratings, one where the probe tones a perfect fifth or
major third above the longest
sounded note received higher ratings than probe tones a minor
second or augmented fourth
above the longest sound note, we wouid have evidence for a top-down
process in tonality
perception. somewhat like phonemic restoration in speech and
Gestalt grouping principies in
vision Research with relative less sparse distributional
infoxmation (Cuddy & Badertscher,
1987; Cuddy, 1991) has dernonstrateci the recovery of the tond
hierarchy with simple three
note melodies.
Experiment 2
Experiment 1 demonstrated the psychological importance of tonal
magnitude for the
perception of tonality. Experiment 2 atternpted to show the reality
and importance of tonal
magnitude in the objective structure of tonal music. The resuhs of
Experiment
ldemonstrated the importance of the absolute durations of different
pitches in the perception
of tonality. These durations were manipulated according to the
tonal magnitude method
described in Equation 1, to determine the tonal magnitudes at which
listeners perceive
tonality in melodies. In Experiment 2, pieces of music from the
Western tonal music
33
repertoire were assesseci in tenns of their tonal magnitude. It was
hypothesized that the tonal
magnitude of composed music would correspond to the tonal magnitude
at which listeners
were able to perceive tonality algorithmically composed
melodies.
Method
The musical exarnples for this anaiysis were the 12 major key
fugues from J. S.
Bach's The Well-Tempemd Clavier (Book 1). The musical scores were
encoded as kem files,
a format that can be analyzed using the Humdncm Toolkit. The fugues
were considered to be
suitable for this analysis because they are clear examples of tonal
music. In fact, Bach wrote
me Well- Tempered Clavier to demonstrate the ability of an
equal-tempered tuning system to
allow musiciaw to play pieces in several tonalities, or keys,
without have to retune their
key board.
For each fugue, the total duration, in beats, for each pitch of the
chromatic scale was
determineci. These duration profiles were then transposed to a
common key (C major), and
were converted to percentage scores by dividing the duration for
each pitch by the total
duration for dl 12 pitches. These percent duration profiles for
each of the higues represent
the primary data for subsequent analysis.
To measure the tonal magnitude of each fugue, the percent duration
profile for the
fugue was compared with percentage values for different tonal
magnitudes. These percent
tonal magnitude profiles were created by raising the value of the
standard key profile
(Knunhansl& Kessler, 1982) to an exponent (the tonal
magnitude), and then dividing each
value by the surn of al1 of the aansformed values. Tonal magnitude
profiles were created for
magnitude values from 0.0 to 5.0, in increments of 0.05.
34
The profiles for each fugue were cornpareci to tonal magnitude
profiles for al1 levels
using a simple measure of similarity. The absolute deviation in the
percentage score between
the fugue and the tonal magnitude profile was calculated for each
pitch, then summed across
al1 12 pitches. For this measure. the cornparisons yielding smaller
total absolute deviation
scores were considered to be better matches than cornparisons
yielding a larger scores. The
tonal magnitude profile that yielded the lowest absolute deviation
was considered to be the
tonal magnitude of the fugue. Figure 5 illustrates the results of
these calculations for two
fugues, C major and Ab major. The lowest point on the hinction
represents the tonal
magnitude of the fugue. The tonal magnitude of each fugue is shown
in Table 3. The values
range from .80 to 2.05, with a mean tonal magnitude of 1.28. Also
shown in Table 3 are the
correlations between each fugue's duration profile and the standard
key profile (Knimhansl
& Kessler, 1982). The magnitude of these correlation varies
across fugues, and is positively
correlated with the fugue's tonal magnitude ( r = -7347 <
.01)
A Monte Car10 simulation was conducted to provide a basis of
cornparison for these
analyses. For each fugue. the duration profile was randomly pemuted
and the tonal
magnitude of this randomized profile was then deterrnined. This
procedure was perforrned
100 times for each fugue. The means and standard deviations for
these 100 cornparisons of
each fugue are also shown in Table 3, dong with the distance (in
standard units) of the
fugue's tonal magnitude from the mean of the control distribution,
and the probability
associateci with tonal magnitudes of that size or greater.
Discussion
Experiment 2 provides support for the claim that certain absolute
relations between
35
Figwe 5. The absolute deviation of the duration profiles for the C
major (top) and A b major
(bottom) fugues fiom tonal magnitude profiles as a function of tond
magnitude. The lowest
point on the function corresponds to the tonal magnitude of the
fugue.
C Major Fugue
O 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Tonal Magnitude
Ab Major Fugue
O 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Tonal Magnitude
36
Table 3. The tonal magnitude of the 1 2 major key fugues from J. S.
Bach's WeIl- Tempered
Clavier Book 1. Also shown are correlations between the fugues'
duration profiles and
Krumhansl and Kessler's (1982) standardized key profile, as well as
the means and standard
deviations of control distributions created by calcuiating the
tonal magnitudes of 100
randomized versions of the fugues' duration profiles. The z scores
give the distance. in
standard units, of the fugue's tonal magnitude from the mean of the
control distribution.
Tonal Correlation Control Distniution Fugue z P
Magnitude with K&K Mean SD
C major
C# major
D major
B major
the durations of different pitches are important for the perception
of tonality. There is a
general congruence between the mean tonal magnitude of the fugues
(1.28), and the tond
magnitude at which listeners began to differentiate between the
probe tones (1.0). as well as
when listeners' probe-tone ratings began to correlate significantly
with the standardized key
profile (2.0).
The results of the Monte Carlo simulation indicate that the tond
magnitudes of the
fugues are not typical to any duration profile. The fact that the
obtained tonal magnitudes
were well above the means of their control distributions
demonstrate that they are highly
specific to the ordering of values in the fugue's duration
profile.
The strong positive correlation between a fugue's tonal magnitude
and its correlation
with the standard key profile is interesting. It may be that the
fugues with lower values
contain more modulations (changes to different tonalities) than
those with higher values.
Because the duration profiles for the fugue were calculated by
surnming note durations across
the fugue as a whole. this analysis is insensitive to local changes
in tonality over the course
of the piece. In fugues with more modulations, the tonic changes
more often, thus distributing
the durational emphasis of the tonic to severai pitches. The
pitches in the duration profile
becorne less differentiated in absolute terms. As a result the
duration profiie has both a lower
tonal magnitude and a lower correlation with the standard key
profile (Knimhansl& Kessler,
1982).
In general, Experirnent 2 shows a close correspondence between the
objective
hierarchical structure in the distributionai information of
composed music and the subjective
hierarchical structures in listeners' representations of musical
pitch.
Experiment 3
Experiment 1 demonstrated that pitch distributional information is
important in the
perception of tonality. It also provided evidence for cognitive
representations of the structure
of tonality. The aim of Experiment 3 was to determine whether p a t
e r degrees of pitch
organization would heip listeners encode and remember melodies.
Based on the results of
Experiment 1, it was assurneci that higher Ievels of tond magnitude
would evoke a more
salient representation of the hierarchical relations arnong
pitches. Given the limited arnount
of information that we can process and remember, and that this
Iimit can be increased by
organizing the information (Miller, 1956). it was hypothesized that
the increases in pitch
organization at higher levels of tonal magnitude would facilitate
the encoding and memory
for melodies.
Much research (e.g.. Dowling, 1978, 1991; Cuddy, Cohen &
Mewhort. 1981:
Croonen & Kop. 1989; Croonen 1994, 1995) has suggested that
pitch structure plays an
important role in memory for melodies. In Experiment 3, the pitch
structure of melodies was
manipulated in terms of tonal magnitude. In an experimental design
common to research in
melodic memory, listeners were asked to judge whether two
consecutive melodies were the
same or different. This type of cornparison was made for melodies
at different levels of tonal
magnitude and when the melodies were either hierarchically or
randomly organized.
Purtrrtrc@un&
Ten students at the University of Toronto at Scarbomugh
participated in this
experiment in exchange for course credit. They ail met the training
prerequisite of three years
of music training. The participants reported an average 8.11 years
of iraining on their prirnary
39
instrument, and reporteci listening to an average 14.06 hours of
music per week. None of the
listeners had participated in Experiment 1.
Design and Prucedure
The melodies used in this experiment were identicai to those of
Experiment 1. Each
melody had 24 notes. 2 occurrences of each note of the chromatic
scale. The order of these 24
notes was random. The note durations were determined according to
Equation 1, and
assigned to the corresponding pitches for the hierarchical
melodies, and randomly assigned to
pitches for the random melodies. The meiodies were presented using
the same equipment and
settings as those describeci in Experiment 1.
The trials in this experiment were comprised of pairs of melodies.
The first melody in
each pair was called the 'standard', and the second melody was the
'comparison'. There were
two types of standard and comparison meIodies-- hierarchical and
random. Thus there were
four types of standard-comparison pairings:
hierarchical-hierarchical, hierarchical-random,
random-random and randorn-hierarchicai. Each of these four painngs
were presented at five
different tonal mapinides: 0.5, 1.5,2.5,3.5 and 4.5, resulting in
20 different types of mals.
Each trial type occurred 5 times in the experiment producing 100
triais in d l ; these 100 triais
occurred in a different random order for each subject. On any given
aid. the ordenng of the
notes and tond magnitude were identical for both the standard and
comparison rnelody.
When the melodies differed, they only differed in the durations for
which the notes were
sounded.
Listeners were told that on each trial they would hear a pair of
melodies. They were
told to listen closely to the two melodies, and to judge whether
the second (comparison)
melody was different irom the first (standard) melody. The
experiment was self-paced, and
lasted about 45 minutes. after which each listener filled out a
subject information form and
was debriefed.
Results
Given the structure of the different types of trials, "correct"
responses varied as a
function of the trial type. Specifically, the correct responses
were "same" on the Hierarchicai-
Hierarchical and Random-Random trials, and "different" on the
Hierarchical-Random and
Random-Hierarchical trials. The percent correct across subjects for
each type of trial is
shown in Table 4, and were analyzed using a the-way ANOVA with the
within-subjects
factors of tonal magnitude (0.5, 1.5,2.5.3.5.4.5), standard type
(hierarchical vs. random)
and cornparison type (hierarchical vs. random). This analysis
failed to reveal any main effects
for tonal magnitude, (F,,, = .85, p > .05). standard type (F,,,,
= .75, p > .05), or comparison
type (F,,n = .ï8. p > .Os). The only result was the interaction
between standard type and
comparison type (F,,,, = 116.41. p < .001). Listeners were
bet?er able to judge the two
melodies as different in the hierarchicai-random (A4 = 64.0) and
random-hierarchical (M =
64.0) conditions, than judge the two melodies as being the same in
the hierarchical-
hierarchical (M = 32.4) and random-random (A4 = 38-0) conditions.
This interaction effect
Iikely reflects a bias towards responding "different*.
In an atternpt to take response bias into account, d' prime scores
were caiculated as a
mesure of Listeners' sensitivity to differences between the
melodies. D' prime scores for
each standard type at each level of tonal magnitude were calcuiated
using the percent correct
score for the correspondhg comparison type as the hit rate, and 1
minus the percent correct
41
Table 4. The percent correct scores for Listeners' same-different
ratings for different types of
cornparisons. Also shown are d' scores that describe listenen'
sensitivity to differences
between the standard and cornparison melodies.
- p. . - - - - - - - - - p- - -
Random
score for the other cornparison type as the false a l m rate. As
shown in Table 4, al1 the
resulting d* prime values are srnail. This indicates that listenen
were unable to identify
differences between the hierarchical and random melodies under any
conditions.
Discussion
in Experiment 3, listeners failed to show the expected increase in
performance at
higher levels of tonal magnitude. Listeners' performance was
essentially at chance. Although
a nul1 result, the most obvious interpretation of this finding is
that the presence of higher
levels of tonal magnitude failed to facilitate encoding and memory
of these melodies. Before
accepting such an interpretation, however, it should be pointed out
that these melodies were
quite long and complex. Accordingly, the amount of information that
listeners were required
to remember may have still exceeded general processing limitations,
producing poor
performance which would then mask any differences as a hinction of
pitch organization. For
exarnple, the melodies of this study were comprised of 24 notes. In
cornparison, Dowling
(1978), who employed a similar paradigrn, used melodies containing
only five notes, whereas
Croonen's (1995) study on melodic mernory employed melodies with
only seven notes.
Perhaps using melodies with fewer notes would enable listeners to
benefit kom the increased
pitch organization in melodies on higher tonal magnitudes.
General Discussion
In summary, the present thesis has uncovered several interesting
results. First,
listeners were found to be sensitive to hierarchical structure in
distributional information. The
differences between listeners' probe-tone rating in the
hierarchical and random conditions of
Experiment 1, reflect a sensitivity to the specific hierarchical
structure of pitches fomd in
tonal music.
Second. random orderings of notes were able to produce the
perception of tonality.
In Expenment 1. Iisteners' probe-tone ratings reflected the
hierarchical struchire of tonaiity
despite the fact that ordering of pitches was randomly detemined.
This finding nins contrary
to other work (e-g., West & Fryer, 1990) which suggests that
random orderings of pitches are
unable to produce the perception of tonaiity.
Third, the absolute properties of distributional information (tonal
magnitude) were
found to be important. Melodies whose duration profiles correlated
strongly with the
standardized key profile differed in their ability to produce the
perception of tonality
depending on the absolute values of elements within the duration
profile.
Fourth, a correspondence was found between the absolute properties
of duration
profiles required for the perception of tonality in
algorithrnically composed melodies
(Expetiment 1) and those of fugues by Bach (Experiment 2). This
supports the claim that
tonal magnitude is a useful and valid description of the absolute
distributional properties of
music.
The present thesis investigated both the objective and subjective
structure of tonality
in an attempt to describe how they might be relateci. Following on
previous work in this area
(e.g. Krumhansl, 1990), this thesis exarnined the possibility that
the perception of tonality
involves a process of matching the objective structure to the
subjective psychological
structure of tonality. What can be said about this matching
process? Experirnent 1 provideci
strong support for the claim that correlation is not an adequate
mode1 for this matching
process because it is insensitive to the absolute values of pitch
durations. Thus. the rnatching
44
process that relates the objective and subjective structure of
tonality must be one that is
sensitive to the absolute durations of pitches.
Such a process c m be viewed from at least two different
perspectives. The fint
involves a sensitivity to the proportion of the total duration that
a given pitch is sounded. For
example, the perception of tonality rnay be dependent on the tonic
of a meIody being sounded
for a certain percentage of the total duration of the melody. A
second. and closely related,
way of looking at the process involves a sensitivity to the ratios
between the durations of
different pitches. What may be critical for the perception of
tonality is that the tonic pitch is
sounded for three times the duration of one pitch, and twice the
duration of another. These
two perspectives essentially deal with the sarne properties of the
duration profile, but differ in
an important way. The f i t deals with elements, the durations of
pitches, more or Iess in
isolation, while the second deais with the relations or ratios
between elements. This leads to a
very interesting question: Does the matching process operate on
relations between elements
in the duration profile, or does it operate on individual elements
within the profile?
One direction for future research involves the investigation of
musical performance.
The present thesis has investigated both the perception of tonal
structure and the tonal
stmcture in musical scores. But how is tonal structure expressed in
musical performance?
Recent research (e.g., Thompson & Cuddy, 1997) has demonstrated
that the perception of
tonality is increased in performed music relative to music lacking
performance expression.
Other research (Thompson. Sundberg, Friberg & Fryden, 1989) has
found that Iisteners find
mechanical performances of melodies to sound more musical when the
certain pitches are
systematically emphasized by making them longer in duration. Might
one dimension of
d
musical performance and the expression of tonal stmcture be the
amplification or
exaggeration of the durational basis of tonal structure? Does a
human performance of a Bach
fugue have a higher tonal magnitude than a mechanicd
performance?
The results presented here help to more clearly define the nature
of the stmctures and
processes involved in the perception of tonality. The objective and
subjective structure of
tonality has been descnbed, as well as the influence of the
objective structure on the
subjective stnxcture of tonality. Research into musical performance
could help to complete
the picture by describing how subjective tonal structure is
expressed, and thus made
objective.
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Appendix
Duration profiles for melodies of different tond magnitudes used in
Experirnent 1 and 2. The
duration values shown are in milliseconds.
Tonal Magnitude
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