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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