254
Jon B. Prince & Mark A. Schmuckler
T H E T O NA L -M E T R I C H I E R A R C H Y: A C O R P U S A NA LYS I S
J O N B. P R I N C E
Murdoch University, Perth, Australia
M A R K A. S C H M U C K L E R
University of Toronto Scarborough, Toronto, Canada
DESPITE THE PLETHORA OF RESEARCH ON THE ROLE
of tonality and meter in music perception, there is little
work on how these fundamental properties function
together. The most basic question is whether the two
hierarchical structures are correlated – that is, do metrically stable positions in the measure preferentially feature tonally stable pitches, and do tonally stable pitches
occur more often than not at metrically stable locations?
To answer this question, we analyzed a corpus of compositions by Bach, Mozart, Beethoven, and Chopin, tabulating the frequency of occurrence of each of the 12
pitch classes at all possible temporal positions in the bar.
There was a reliable relation between the tonal and
metric hierarchies, such that tonally stable pitch classes
and metrically stable temporal positions co-occurred
beyond their simple joint probability. Further, the pitch
class distribution at stable metric temporal positions
agreed more with the tonal hierarchy than at less metrically stable locations. This tonal-metric hierarchy was
largely consistent across composers, time signatures,
and modes. The existence, profile, and constancy of the
tonal-metric hierarchy is relevant to several areas of
music cognition research, including pitch-time integration, statistical learning, and global effects of tonality.
Received: October 18, 2012, accepted March 17, 2013.
Key words: pitch, time, tonality, meter, hierarchy
A
NEAR- UNIVERSAL CHARACTERISTIC OF
Western music is the use of tonality (Krumhansl, 1990) and meter (Lerdahl & Jackendoff,
1983). Accordingly, there is abundant research in music
cognition on these topics (cf. Krumhansl & Cuddy, 2010;
London, 2004). For instance, tonality influences memory
for musical materials (Dowling, 1978, 1991; Krumhansl,
1991), discrimination of musical chords (Beal, 1985),
change detection (Cohen, Trehub, & Thorpe, 1989;
Lebrun-Guillaud & Tillmann, 2007; Trainor & Trehub,
Music Perception, VOLUM E 31, ISSU E 3,
R IG HTS RES E RV E D . PLEASE DIR E CT ALL
PP.
254–270,
IS S N
0730-7829,
1994; Trehub, Cohen, Thorpe, & Morrongiello, 1986),
ratings of completion and goodness (Boltz, 1989a,
1989b; Cuddy, 1991; Prince, 2011), melodic recognition
(Bigand & Pineau, 1996; Schmuckler, 1997), recall (Boltz,
1991), and expectation (Schmuckler, 1989, 1990;
Tillmann, Janata, Birk, & Bharucha, 2008). Similarly,
meter plays a pivotal role in the perception of musical
sequences, guiding attention and providing a framework
for organizing the events in a melody (Jones & Boltz,
1989; Large & Jones, 1999; Palmer & Krumhansl,
1990). Its importance as a central organizational factor
is evident in its influence on recognition memory (Jones
& Ralston, 1991; Keller & Burnham, 2005), complexity
(Povel & Essens, 1985), categorization (Bigand, 1990;
Desain & Honing, 2003; Hannon & Johnson, 2005), similarity (Creel, 2012; Monahan & Carterette, 1985),
change detection (Bergeson & Trehub, 2006; Hannon
& Trehub, 2005; Jones, Johnston, & Puente, 2006; Repp,
2010; Smith & Cuddy, 1989), and expectancy (Huron,
2006; Large & Palmer, 2002; Repp, 1992). The role of
meter in performance deserves its own volume (including Drake & Botte, 1993; Essens & Povel, 1985; Palmer,
1997; Povel & Essens, 1985; Repp, Iversen, & Patel, 2008;
Shaffer, Clarke, & Todd, 1985; Sloboda, 1983), but is
beyond the purview of the current study.
Literature on how tonality and meter work together,
however, remains surprisingly small. Experimental
work manipulating the alignment of the tonal and metric hierarchies supports the notion that listeners are
sensitive to both hierarchies when listening to melodies
(Bigand, 1997; Palmer & Krumhansl, 1987a, 1987b; Serafine, Glassman, & Overbeeke, 1989). Interestingly,
these authors reported that the tonal and metric hierarchies functioned independently, implying that these
hierarchies need not necessarily co-occur. Other
researchers report interactive relations between tonality
and meter (Boltz, 1989b, 1991; Dawe, Platt, & Racine,
1993, 1994, 1995; Schmuckler, 1990; Schmuckler &
Boltz, 1994). A substantial related literature has examined the independence versus interaction of pitch and
time (see for reviews, Ellis & Jones, 2009; Krumhansl,
2000; Prince, Thompson, & Schmuckler, 2009),
although these investigations have been confined to
aspects of pitch and temporal information other than
tonality and meter per se (but see Prince, Thompson,
et al., 2009, for an exception). Deutsch (1980) found
ELEC TR ONIC I SSN
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DOI: 10.1525/ M P.2014.31.3.254
Tonal-Metric Hierarchy
that using temporal grouping in musical sequences
influences the encoding of tonal information (see also
Laden, 1994). She found that recall accuracy (melodic
dictation) declined when temporal segmentation cues
conflicted with pitch sequencing cues, which decreased
further when sequences did not conform to a simple
hierarchical structure. However, more recent work in
the perception of probe events (Prince, Thompson,
et al., 2009) found mixed evidence, specifically an asymmetric interaction between pitch and time. When
reporting if probe events occurred on or off the beat,
tonally stable pitches were more likely to be reported as
being on the beat, showing an interference of pitch
structure (tonality) on temporal perception. However
there was no complementary effect of temporal position
when reporting if the probe was in or out of the key.
The music theory literature also explores the relation
between tonality and meter. Combining concepts from
cognitive psychology and Schenkerian analysis, Lerdahl
and Jackendoff (1983) created a model that uses wellformedness and preference rules to describe how listeners apprehend the structure in a musical sequence. Wellformedness rules provide a number of structural
descriptions of the musical piece in question, whereas
the preference rules function to determine which
description most accurately corresponds to the listener’s
experience. Their model includes rules for deriving metrical structure from the musical surface; this information then feeds into a time-span reduction that focuses
on the pitch structure. Interestingly, in this model pitch
and temporal structures are not treated entirely independently – for example, metric preference rule 7 specifically incorporates tonal function into deriving the
metrical structure (‘‘Strongly prefer a metrical structure
in which cadences are metrically stable,’’ p. 88). Similarly, time-span reduction preference rules incorporate
rhythmic structure (rule 1: ‘‘prefer a choice that is in
a relatively strong metrical position,’’ p. 160) and meter
(rule 5: ‘‘prefer a choice that results in more stable
choice of metrical structure,’’ p. 165). Benjamin (1984)
also acknowledges the importance of tonal structure in
his theory of musical meter, arguing that tonal structure
influences the derivation of the meter at several levels of
metrical analysis. Temperley (2001) discusses a number
of possible factors in key identification, but does not
consider metrical stability; the success of his model notwithstanding casts doubt on the idea that metrical
strength and tonal stability are aligned in any consistent
way.
There are several algorithms that derive musical
meter based on coding melodic (pitch) information,
although these models rely on pitch repetition rather
255
than tonal stability. For instance, Collard, Vos, and
Leeuwenberg (1981) determine the meter by compiling
distance between pitch pattern repetitions. The
approach of Steedman (1977) is similar, in that he takes
repetitions in melodies to indicate points of metric
accent (expanding on Longuet-Higgins, 1976). The
period of these accents then forms the metric grouping
and thus indicates the metrical framework. It is noteworthy that Steedman asserts that key identification is
independent from knowledge of meter, and that scale
has no influence on metric structure. The success of
these techniques suggests that composers (at least Bach)
use melodic repetition – but not necessarily tonality –
and metric hierarchies in a congruent manner. Other
algorithms extract metrical structure with purely temporal factors (particularly onsets and grouping), without including any parameters related to pitch (Desain,
1992; Desain & Honing, 2003; Povel & Essens, 1985).
Similarly, the most common keyfinding algorithm does
remarkably well despite ignoring all temporal information (Krumhansl, 1990; Krumhansl & Schmuckler,
1986; Schmuckler & Tomovski, 2005; but see Schmuckler, 2009); however, others contend that note order,
recency, and possibly other temporal factors are critically important (Brown, 1988; Brown, Butler, & Jones,
1994; Huron & Parncutt, 1993; Temperley & Marvin,
2008). For example, Aarden (2003), using reaction times
to the seven diatonic tones of the major scale, found that
stability of these tones varied as a function of placement
in a musical passage. Specifically, he observed that the
diatonic major tonal hierarchy values as defined by
Krumhansl and Kessler (1982) best characterized listeners’ responses at points of closure (i.e., cadences), but
were less descriptive of tonal percepts at other points in
a passage.
Overall, the body of theoretical literature that specifically examines the relation between tonality and meter
is miniscule. Similarly, the extant empirical literature on
how tonality and meter combine focuses principally on
perceptual processes, and not on determining whether
tonal and metric structures actually co-occur in music.
Thus, there appears to be virtually no information as to
whether or not tonality and meter are systematically
aligned. The only related research on this topic comes
from Järvinen’s (1995) work examining 18 examples of
bebop style jazz improvisation. He calculated a weighted
average of the number of times each pitch class occurred
on a given temporal position (down to the eighth note
level). Järvinen found that tonally stable pitches
occurred more often at metrically stable temporal positions, and tonally unstable pitches (i.e., nondiatonic)
occurred more often at less stable temporal positions.
256
Jon B. Prince & Mark A. Schmuckler
In this study we examine this issue with regard to
the common practice period of Western music by
asking the following questions. First and foremost,
is there an alignment of tonal and metric information
in typical tonal music, such that tonally stable notes
occur on metrically stable temporal positions, and
vice versa? Alternatively, tonally important pitches
might be equally prevalent across temporal position
(i.e., not selective for metric stability), or similarly,
metrically stable positions may not favor any particular pitch. To investigate this question, the most obvious starting point is to examine the tonal and metric
hierarchies in corpus data. With occasional deviations, perceptual studies of both hierarchies concur
with frequency of usage and theoretical predictions
(Krumhansl, 1990; Palmer & Krumhansl, 1990, but
see Aarden, 2003); similar hierarchies also arise in
music production studies (Palmer & Pfordresher,
2003; Schmuckler, 1989, 1990). Accordingly, it is of
interest to determine whether the frequency of cooccurrence of each pitch class and temporal position
(within a measure) conforms to perceptual studies of
the tonal-metric hierarchy (e.g., Prince, Thompson,
et al., 2009).
The next most fundamental question is whether the
relationship between tonality and meter varies across
musical genre, musical mode, and/or time signature.
For instance, this hypothetical relationship may vary
across musical genre, particularly between styles that
are largely diatonic (e.g., Baroque music) relative to
styles containing increasing levels of chromaticism
(e.g., Classical to Romantic and beyond). Similarly,
musical mode is of interest, given the evidence that
minor keys are somewhat more perceptually ambiguous
than major keys (Delzell, Rohwer, & Ballard, 1999; Harris, 1985; Vuvan, Prince, & Schmuckler, 2011; Vuvan &
Schmuckler, 2011). Lastly, research examining in detail
the frequency of occurrence of tones across temporal
positions within different time signatures is virtually
nonexistent; therefore, it is unknown whether any existing alignment between tonal and metric information
varies for different metric structures.
Answering these questions is the main goal of the
present research. Towards this end, we quantified the
frequency of occurrence of each scale degree at each
temporal position in 365 works from representative
composers of tonal Western music (Bach, Mozart,
Beethoven, and Chopin), totaling 721,293 notes. Within
a given piece, we assumed one metric hierarchy and one
tonal hierarchy (i.e., ignoring modulation) throughout;
we return to the implications of this latter assumption in
the General Discussion.
Method
MATERIALS
Sequenced (i.e., not performed) MIDI files of a selection
of works by Bach, Mozart, Beethoven, and Chopin were
downloaded from http://kern.ccarh.org/ (CCARH,
2001) and http://www.piano-midi.de/ (maintained by
Bernd Krueger; see Appendix A for full list of pieces).
The corpus consists of works for piano, with the exception of Beethoven’s Symphony 5, 1st movement.
All pieces were transposed to have a tonic of C major
or A minor (as appropriate), and sorted into time signature categories of 2/4, 3/4, 4/4, 6/8, 9/8, and 12/8. The
key indicated in the title of each piece was used to
classify the mode and key signature. For pieces whose
title provided no key information, the key signature of
the score, in addition to a visual inspection of the first
and last eight measures of the piece, were used to determine the key (tonic and mode). Again, one key was
assigned for the entire piece, ignoring modulations.
Note onset times were used to determine the position
of each note within the metric hierarchy of the appropriate time signature, correcting for any pickup beats.
Custom scripts written in MATLAB 7.0 were used to
analyze the corpus.
PROCEDURE
The first step in analyzing the tonal-metric hierarchy
was to examine the independent tonal and metric hierarchies of these pieces. Accordingly, we tabulated how
often each scale degree occurred, separately for the
major and minor mode pieces.1 Quantification of the
metric hierarchy was accomplished by counting how
often a note was sounded at each temporal position;
this count was done separately for different time signatures. The tonal and metric counts were then compared
to perceptual measures of tonal and metric stability,
taken from Krumhansl and Kessler (1982) and Palmer
and Krumhansl (1990), respectively. The purpose of
these comparisons was to determine if the tonal and
metric information of these pieces conformed to the
perceptually derived profiles for these two dimensions.
The second step of the analysis was to determine the
frequency of occurrence of each pitch class (after transposition) at each temporal position, separately for all
pieces in the corpus. These tonal-metric distributions
were grouped within composer, time signature, and
1
In addition to measuring frequency of occurrence, the cumulative
duration of each pitch class was also calculated. These two calculations
gave identical results (r > .99). For continuity of calculation techniques
across dimensions, frequency of occurrence was used.
Tonal-Metric Hierarchy
257
TABLE 1. Correlations Between the Pitch Class Distributions of the Current Corpus and the Tonal Hierarchy Values of Krumhansl and Kessler
(1982).
Major modality
K&K 1982
Bach
Mozart
Beethoven
MEAN
Minor modality
Bach
Mozart
Beethoven
Chopin
Bach
Mozart
Beethoven
Chopin
.83
.91
.96
.91
.92
.99
.82
.87
.95
.86
.94
.99
.90
.96
.95
.95
.90
.98
.99
.95
.92
.95
.95
.84
.96
.99
.99
.95
TABLE 2. Mean Intercorrelation of the Metric Distributions of the Current Corpus (for Each Composer) and the Matching Metric Hierarchy
Values of Palmer and Krumhansl (1990).
2/4
3/4
4/4
6/8
9/8
12/8
MEAN
P&K 1990
Bach
Mozart
Beethoven
Chopin
.83
.95
.88
.82
.86
.94
.95
.92
.96
.88
.92
.89
.94
.97
.93
.88
.94
.97
.93
.96
.88
.93
.79
.90
.96
.90
.87
.93
.89
Note: All hierarchies are restricted to the sixteenth note level.
mode, then compared with the perceptual profiles and
across composers.
Results
DISTRIBUTION OF PITCH CLASSES AND TEMPORAL POSITIONS
Composers’ use of the 12 pitch classes in the corpus
correlated highly with the perceptually derived major
and minor tonal hierarchies (see Table 1). However,
correlations between composers were higher (mean
r ¼ .94) than those with the tonal hierarchies (mean
r ¼ .87). Similarly, the observed metric distributions
within the measure correlated well with the PalmerKrumhansl metric hierarchies (mean r ¼ .87), but again
were lower than the inter-composer correlations (mean
r ¼ .92, see Table 2). Overall, these tables reveal that
there was a remarkable agreement across composers in
their use of the tonal and metric hierarchies, despite
differences in mode, musical genre, and time signature.
Figure 1 shows three different versions of both the
major and minor tonal hierarchies – the frequency of
occurrence observed in the corpus (limited to the downbeat metric position), the overall total frequency of
occurrence (summing across all metric positions), and
the Krumhansl-Kessler (1982) goodness of fit ratings.
Figure 2 compares the metric hierarchies in the corpus
to the Palmer and Krumhansl (1990) goodness of fit
ratings (for time signatures 2/4, 3/4, 4/4, and 6/8).
TONAL-METRIC DISTRIBUTION ANALYSIS
For ease of depiction, the tonal-metric hierarchy counts
present pitch classes and temporal positions by decreasing levels of frequency of occurrence instead of the conventional method of sorting these dimensions
chromatically and chronologically, respectively (as in Figures 1 and 2). For example, Table 3 depicts the tonalmetric frequency of occurrence values for the corpus
pieces in a major tonality (all transposed to C major) in
the most common time signature of 4/4. The pitch classes
(rows) are sorted as C, G, E, D, F, A, B, G#, D#, F#, A#, and
C#, an ordering based on the observed pitch class distribution occurring on the downbeat (not across all metric
positions) for all major-mode pieces in the corpus.2
2
It is important to note that this ordering of pitch classes does not
correspond exactly with their perceived psychological stability as
indicated by the classic tonal hierarchy values of Krumhansl and
Kessler (1982). Probably the most significant point of divergence in this
regard arises in the ordering of the diatonic scale tones, which ranked D,
F, A, and B in frequency of occurrence in our corpus, as opposed to F, A,
D, and B in the Krumhansl and Kessler tonal hierarchy values. Of lesser
note, the non-diatonic tones in our corpus appear in the order G#, D#, F#,
A#, and C#, as opposed to F#, G#, D#, A#, and C#, in Krumhansl and
Kessler. Although the convergence with the Krumhansl and Kessler major
profile is strong (r ¼ .96), our data also mirror Aarden’s (2003) rank
ordering of the seven diatonic major scale tones (based on reaction times)
more strongly at points of closure (cf. p. 52; r ¼ .71) than for continuations (cf. p. 75; r ¼ .29). Separately, for the minor pieces the pitch class
distribution on the downbeat correlates with the Krumhansl and Kessler
minor tonal hierarchy at r ¼ .91.
258
Jon B. Prince & Mark A. Schmuckler
FIGURE 1. Tonal hierarchy values: frequency of occurrence (on downbeat, and all temporal positions) and Krumhansl-Kessler (1982) goodness of fit
ratings, all standardized to maximum of 1. (a) Major mode hierarchy, (b) Minor mode.
The columns of Table 3 represent temporal positions
within the measure, sorted as beats 1 (downbeat), 3,
2þ4, eighth notes, sixteenth notes, and Other, grouped
in accordance with theoretical predictions of beat
strength (e.g., Lerdahl & Jackendoff, 1983). That is, the
frequency of occurrence of temporal positions is averaged (not summed) within a category of equal metric
strength, such that beats 2 and 4 are averaged, becoming
2þ4, then the remaining 4 eighth note (off-beat) positions, followed by the next lower metric hierarchy level
(off-beat sixteenth notes), then ‘‘Other’’ (designating
thirty-second notes, sixteenth note triplets, and any
non-quantized metric positions).
Finally, Table 4 provides the same data for pieces in
the minor mode (4/4 time signature, all composers),
also with the rows sorted according to the pitch class
distribution of the minor mode on the downbeat. The
full data set of all time signatures, modalities, and composers is available upon request from the first author.
Figures 3-8 depict the tonal-metric hierarchies of each
time signature as 3-dimensional histograms, separately
for major and minor (collapsing across composer). The
y axis shows the frequency of occurrence of each combination of pitch class (x axis) and temporal position
(z axis). These graphs use the same arrangement as
Tables 3 and 4; namely, they sort the axes by frequency
of occurrence on the downbeat, as well as averaging
across temporal positions with equal metric stability.
These graphs demonstrate that the most common
occurrence is a tonally stable pitch (e.g., the tonic) at
a metrically stable temporal position (e.g., the downbeat), and that the frequency of occurrence decreases
at lower levels of tonal and metric stability. At first
glance, the overall tonal-metric hierarchy looks like
a replication of the tonal hierarchy at different temporal
positions. Closer inspection, however, reveals that the
favoring of tonally stable pitch classes is more discernible at the downbeat than at lower levels of metric
stability. Indeed, the correlation of the KrumhanslKessler major tonal hierarchy with the pitch class distribution decreases across metric category stability:
r ¼ .96 for the downbeat, then r ¼ .92, .90, .88, and
.87 down to the ‘‘Other’’ level. For the same metric
categories in the minor mode, the pattern is r ¼ .91 for
the downbeat, then r ¼ .83, .84, .85, and .79. This
change is also evident in Figure 1, as it compares the
tonal hierarchy for only downbeat occurrences to the
overall total (summing all metric positions).
These figures also demonstrate the remarkable consistency of the tonal-metric hierarchy across time signature and mode. Despite these changes in the metrical
framework and major/minor modality, the profile of the
tonal-metric hierarchy remains largely the same. This
shape is slightly less reliable for Figures 7 and 8, likely
due to the small sample size (note the markedly lower
scale of the y axis compared to other figures).
Because Figures 3-8 show that notes rarely occurred
at fine subdivisions, and in the interest of simplicity,
further analyses of the tonal-metric hierarchy used the
16 most common temporal positions (e.g., sixteenth note
Tonal-Metric Hierarchy
259
FIGURE 2. Metric hierarchy values: frequency of occurrence and Palmer-Krumhansl (1990) goodness of fit ratings, all standardized to maximum of 1.
(a) 2/4 time signature, (b) 3/4 time signature, (c) 4/4 time signature, (d) 6/8 time signature.
level for 4/4).3 Note that the exclusion of the least common temporal positions represents a more conservative
analysis – the uniformly rare use of the positions of lower
metric stability might otherwise artificially inflate correlations between the tonal and metric hierarchies.
Table 5 shows the results of correlating the resulting
12 (pitch class) by 16 (metric stability) tonal-metric
matrices across composer, separately for each time
3
We used the 16 most common temporal positions from all time
signatures even if it did not correspond to the sixteenth note level (e.g.,
3/4) because we were sorting by stability as indexed by frequency of
occurrence rather than beat strength.
signature and modality. Values are the average correlation coefficients of each composer with all other
composers. All Table 5 values below .70 are the result
of cells with the minimum possible small sample size
(N pieces ¼ 2).
The strikingly high inter-composer tonal-metric correlations of Table 5 motivated including additional
composers, including some from more modern compositional periods; we chose Schubert, Brahms, Liszt, and
Scriabin (see Appendix B for the list of included pieces
of these composers). These data are not included in the
earlier analyses because our corpus had too few pieces of
these composers for valid inter-composer comparisons.
260
Jon B. Prince & Mark A. Schmuckler
TABLE 3. Tonal-metric Hierarchy of Major Tonality 4/4 Pieces, Summed Across Composer.
Major mode, Time signature 4/4, all composers
Pitch
C
G
E
D
F
A
B
G#
D#
F#
A#
C#
Downbeat
Beat 3
Beats 2þ4
8th level
16th level
Other
3356
3365
2149
1859
1588
1586
1598
517
401
531
307
320
2562
2681
1877
1978
1475
1474
1468
500
359
601
376
374
2154
2161
1650
1723
1358
1248
1302
451
334
614
319
339
1322
1528
1140
1123
813
895
819
254
176
384
215
214
530
612
423
481
308
344
356
77
75
149
80
70
36
41
30
31
22
26
26
13
12
13
11
10
Note: The four rightmost columns represent the average frequency of occurrence at the relevant temporal positions. Pitches are sorted by frequency of occurrence at the
downbeat across all major mode pieces in the corpus, transposed to C major.
TABLE 4. Tonal-metric Hierarchy of Minor Tonality 4/4 Pieces, Summed Across Composer.
Minor mode, Time signature 4/4, all composers
Pitch
A
E
C
D
B
G
F
G#
C#
F#
D#
A#
Downbeat
Beat 3
Beats 2þ4
8th level
16th level
Other
2134
2280
1462
1079
1108
1000
942
521
255
290
367
230
1591
1804
1239
1106
1169
775
922
633
261
314
354
256
1489
1721
1280
1084
1097
870
922
524
259
325
340
269
1020
1160
738
716
721
567
529
313
184
220
172
150
333
422
257
279
291
164
194
105
60
84
63
66
22
29
20
17
19
14
15
11
5
8
7
4
Note: The four rightmost columns represent the average frequency of occurrence at the relevant temporal positions. Pitches are sorted by frequency of occurrence at the
downbeat across all minor mode pieces in the corpus, transposed to A minor.
Smaller sample size can lead to unstable relationships in
the data (Knopoff & Hutchinson, 1983), as observed
for isolated cases in Table 5. However, collapsing across
time signature before testing inter-composer correlations allows a more reliable comparison, and is justified given the similarity of the tonal-metric hierarchy
across time signature (cf. Figures 3-8). Table 6 shows
the average inter-composer correlations of these eight
composers, after collapsing across time signatures
before correlating. Major and minor modes remain
separate.
Although intriguing, none of the aforementioned
analyses establish definitively whether the tonal and
metric hierarchies are truly correlated. Combining any
pitch class distribution with a set of temporal positions
would result in the commonest pitch class occurring
most frequently at the commonest temporal location
(i.e., approximately the shape of Figures 3-8), regardless
of if there is any true connection between the two. One
way to establish quantitative proof of a link between the
two hierarchies is to compare the average metric stability of each pitch class with the tonal hierarchy, and
compare the average tonal stability of each temporal
position with the metric hierarchy. The first procedure
tests if tonally stable pitches are more likely to have high
metric stability (and tonally unstable pitches have lower
metric stability). The second approach tests if metrically
stable positions have high tonal stability (and metrically
unstable positions are more likely to have lower tonal
stability). Thus, although these two procedures are
Tonal-Metric Hierarchy
261
FIGURE 3. Tonal-metric hierarchy for 2/4 time signature, for major (a) and minor (b) mode.
FIGURE 4. Tonal-metric hierarchy for 3/4 time signature, for major (a) and minor (b) mode.
conceptually related, they test separate hypotheses.
More importantly, by using the average tonal or metric
stability, these calculations are not biased by the simple
joint probability of the two independent distributions.
The average metric stability of each pitch class has
a strong positive correlation with the tonal hierarchy
(r ¼ .77 when collapsing across mode; r ¼ .68 for major
pieces, and .58 for minor). There is a similar relationship between the average tonal stability of each temporal position and the metric hierarchy (r ¼ .68; .70 for
major pieces, and .55 for minor). Thus, the tonal and
metric hierarchies are indeed connected beyond the
extent predicted by their joint probability, yet slightly
less so for minor than major.
Additional inter-composer analyses repeated these
calculations (comparing the metric stability of each
pitch class and the tonal stability of each metric position) for each composer separately, collapsing further
across mode (see Table 7). Even though the correlations
were always positive, they varied across composer. In
262
Jon B. Prince & Mark A. Schmuckler
FIGURE 5. Tonal-metric hierarchy for 4/4 time signature, for major (a) and minor (b) mode.
FIGURE 6. Tonal-metric hierarchy for 6/8 time signature, for major (a) and minor (b) mode.
Bach’s compositions, all pitch classes had similar levels
of metric stability, but metrically stable temporal positions had higher tonal stability. One way to conceptualize this result is to say that in these pieces, tonally
stable pitch classes could occur at any time in the measure, but on the downbeat, only pitches with high tonal
stability occurred. The Mozart pieces showed the opposite pattern – metrically stable locations could have
pitches of any tonal stability, while tonally stable pitches
were more limited to stable metric locations. Chopin
moved further in this direction, whereas Beethoven
respected both relationships more equally.
Discussion
The current study analyzed how central composers
from a variety of Western tonal music periods used
pitch classes and temporal positions, and in particular,
if this information was used congruently (i.e., were the
tonal and metric hierarchies positively correlated). In
Tonal-Metric Hierarchy
263
FIGURE 7. Tonal-metric hierarchy for 9/8 time signature, for major (a) and minor (b) mode.
FIGURE 8. Tonal-metric hierarchy for 12/8 time signature, for major (a) and minor (b) mode.
general, both distributions were strongly – but not perfectly – related to the perceptually derived tonal and
metric hierarchies of Krumhansl and Kessler (1982) and
Palmer and Krumhansl (1990). Although this finding is
hardly surprising on one level, it is notable in that it
emerged consistently in a corpus of music that covers
a wide range of musical styles, including styles that contain high levels of chromaticism and rhythmic deviation.
As such, the demonstration that the distribution of tonal
and metric information in such works nevertheless
conforms to a prototypical hierarchy of stability is a striking finding.
The primary goal of this study was to test if tonal and
metric information aligned in Western tonal music.
Indeed, they correlated strongly throughout this corpus,
such that tonally stable pitch classes were more likely to
occur on metrically stable temporal positions. Moreover, and mirroring the findings with the independent
tonal and metric distributions, this relation was relatively stable across time signature and modality.
264
Jon B. Prince & Mark A. Schmuckler
TABLE 5. Average Inter-composer Correlation.
Major modality
2/4
3/4
4/4
6/8
9/8
12/8
MEAN
Minor modality
Bach
Mozart
Beethoven
Chopin
.86
.82
.86
.69
.82
.91
.91
.92
.87
.92
.91
.90
.84
.82
.85
.83
.83
.81
.90
.88
.84
Bach
Mozart
Beethoven
Chopin
.72
.79
.88
.74
.78
.72
.77
.80
.87
.90
.65
.79
.88
.89
.75
.78
.72
.80
.63
.85
.89
.74
.80
.78
Note: Each value represents the average correlation of that composer with all other composers for that time signature and modality. Missing values mean no data were available.
TABLE 6. Inter-composer Correlations Including More Modern Composers, Collapsed Across Time Signature and Mode.
Major
Minor
Bach
Mozart
Beethoven
Schubert
Chopin
Brahms
Liszt
Scriabin
.84
.86
.92
.93
.93
.93
.92
.89
.88
.87
.91
.90
.80
.86
.86
.85
TABLE 7. Variations Across Composer in the Correlations of Pitch Class Average Metric Stability with the Tonal Hierarchy, and Temporal
Position Average Tonal Stability with the Metric Hierarchy.
Calculation
Bach
Mozart
Beethoven
Chopin
r(pitch class metric stability, tonal hierarchy)
r(temporal position tonal stability, metric hierarchy)
.35
.67
.66
.37
.71
.67
.78
.44
The reality of a correlated tonal-metric hierarchy may
not be much of a revelation to some, yet it is far from
a trivial finding. By themselves, tonally strong pitches
and metrically stable temporal positions function as
important cognitive reference points for the listener
when constructing a mental representation of the music
they are hearing (Krumhansl, 1990; Palmer & Krumhansl, 1990). But regardless of its intuitive nature, demonstrating this alignment between hierarchies is
significant, particularly given the frequent assumptions
of its existence. For example, Serafine et al. (1989) explicitly state that ‘‘all theorists would agree’’ (p. 404) with the
notion that tonally stable pitches occur on metrically
strong positions. Similarly, in the related (yet arguably
separate) context of harmony in melodic anchoring,
Bharucha (1984) simply asserts that ‘‘chord tones tend
to occur in metrically stressed positions more often than
nonchord tones’’ (p. 492). Both of these authors make
these assumptions explicitly, yet based on no actual evidence for this relation. More cautious approaches cite an
alignment between tonal and metric hierarchies as
a ‘‘generally held belief ’’ (Palmer & Krumhansl, 1987a).
The unique profile of the tonal-metric hierarchy
derives in part from changes in the use of pitch classes
across temporal position. Indeed, pitch class distributions correlated more highly across composers than
they did with the perceptually derived tonal hierarchy
(particularly for the minor mode, likely because of our
treating all three minor modes as one). The fact that the
tonal hierarchy did best at metrically stable locations
concurs with Aarden’s (2003) findings – that the diatonic members of the original major tonal hierarchy are
most representative of pitch expectations at points of
closure. Because the most expected closure is a cadence
on a metrically stable location, it follows that the shape
of the tonal hierarchy would change slightly based on
metric position. This interpretation accords with Huron’s (2006) proposal of four pitch schemas in Western
music (major/minor modes both for cadences as well as
continuations). Additionally, the fact that composers
deliberately violate schematic expectations (Meyer,
1957) means that their use of pitch classes and temporal
positions should not be a perfect match to the schematic
hierarchies.
The current findings also converge with the only (to
our knowledge) other existing corpus analysis of the
relation between the tonal and metric information: Järvinen (1995). The extension and generalization provided
Tonal-Metric Hierarchy
by the current findings is critical given that the metric
hierarchies in jazz may function differently than in other
musical genres. Indeed, Palmer and Pfordresher’s (2003)
distribution analysis of 1930’s swing jazz (well before the
bebop period analyzed by Järvinen) showed that in
binary meter, events occurring on beat 4 were most common, followed by beats 2 and 1. One of the more intriguing aspects of Järvinen’s study is that his weighting
technique found that non-diatonic (i.e., tonally unstable)
pitches were actually more common off the beat than on
the beat, thus observing a crossover effect. His weighting
technique involved multiplying the frequency of occurrence of notes at the quarter-note level by 2, those at the
half-note level by 4, and the whole-note level by 8. This
procedure corrected the imbalance in how often a given
metric position could occur within one bar (i.e., eighthnotes could occur at 8 different points, whole notes only
at one). Using Järvinen’s weighting technique with the
current data did not yield a similar crossover, possibly
indicating how a major stylistic difference (classical
music to bebop jazz) may actually change the tonalmetric hierarchy. Nevertheless, the advantage of tonally
stable pitches decreased at lower levels of metric stability,
thus the pattern in the current findings is qualitatively
similar to Järvinen’s, only not to the point of observing
a full crossover.
Comparisons of the tonal-metric hierarchy across
composer yielded particularly intriguing results. The
overall shape of the tonal-metric hierarchy is surprisingly consistent across composer, even when including
more modern composers still within the tonal tradition
(Brahms, Liszt, Schubert, Scriabin). Despite all intercomposer correlations remaining above .8, the more
modern composers tended to show relatively lower
coefficients, but the fact that Bach is comparable to
Scriabin in this analysis undermines this interpretation.
Despite this commonality, there were variations in how
the composers used the tonal-metric hierarchy. Specifically, correlations of the average metric stability of pitch
classes with the tonal hierarchy, as well as the average
tonal stability of temporal positions with the metric
hierarchy, did not remain entirely constant across composer. Table 7 suggests that Bach placed tonally stable
pitches in multiple places throughout the measure,
while reserving metrically stable positions for tonally
stable pitches. Conversely, Chopin was more likely to
limit tonally stable pitches to metrically stable points,
but was less restrictive on what pitch classes occurred at
these times (i.e., tonally unstable pitches could also occur
at strong metric locations). One possible mechanism for
this shift in the use of the tonal-metric hierarchy could
involve changes in composers’ use of appoggiatura and
265
grace notes, but such interpretations are speculative at
this point.
The fact that composers correlate the tonal and metric
hierarchies in their compositions raises the question of
why they would do so in the first place. One possibility
is that a correlated tonal-metric hierarchy offers a way
to maximize the strength of musical expectancies, which
are of central importance to communicating musical
emotion (Huron, 2006; Meyer, 1957). By themselves,
the tonal hierarchy influences expectations of what
(pitch) event might occur, and the metric hierarchy concerns when. Used independently (uncorrelated), they
both offer a means of creating and fulfilling/denying
expectancies. But when used in a correlated manner, the
strength of the resulting expectancies go beyond the simple joint probability of what and when towards exactly
what happens at exactly when. In other words, the tonalmetric hierarchy may provide a mechanism of strengthening the communicated musical emotion, analogous to
making the whole (expectancy) greater than the sum of
the parts (tonal and metric hierarchies).
Demonstrating the existence of a tonal-metric hierarchy is relevant to research on how pitch and time combine in music perception. This question has received
considerable treatment over the years, yet there has been
little consensus (for reviews, see Ellis & Jones, 2009; Jones
& Boltz, 1989; Krumhansl, 2000; Prince, 2011). Perceptual work examining the relation between tonality and
meter (as opposed to pitch and time) found an asymmetric relationship: tonality affected metric judgments,
but meter had no influence on key membership judgments (Prince, Thompson, et al., 2009). In fact, an effect
of rhythmic regularity on pitch height comparisons only
occurred in the context of atonal sequences (Prince,
Schmuckler, & Thompson, 2009). Moreover, Prince,
Thompson, et al. (2009) found that goodness of fit ratings for probe events consisting of any one of 12 pitch
classes at any one of eight temporal positions (eighth
note level) were predictable based on a linear combination of both the tonal and metric hierarchies. These hierarchies explained 89% of the variance in one experiment,
and 86% in another, but a multiplicative interaction term
never accounted for any additional variance.
In contrast, the shape of the tonal hierarchy varied
across levels of metric stability in the current study. The
most obvious explanation of the divergence between
these findings and earlier research is that it involves comparing a corpus analyses to perceptually derived results
(goodness of fit ratings to probe events) from perceptual
experiments. Accordingly, this result might represent
a divergence between perceptual and theoretical levels
of analysis. However, humans demonstrate exquisite
266
Jon B. Prince & Mark A. Schmuckler
sensitivity to the statistical regularities of the environment across an array of domains and modalities (Goldstone, 1998), with music being no exception (for
examples specifically in pitch and time, see Loui, Wessel,
& Kam, 2010; Tillmann, Stevens, & Keller, 2011). Consequently, this discrepancy between theoretical and perceptual findings on the tonal-metric hierarchy is
unexpected. Future research should explore this issue,
as it provides a possible exception to this principle of
perceptual learning of statistical information.
Another interpretation of this discrepancy between
perceptual results and our corpus analysis stems from
the largest limitation of this study; namely, assigning
one tonal centre to an entire piece ignores the fact that
pieces often modulate to new keys. Previous corpus
analyses take the same approach (Knopoff & Hutchinson, 1983; Krumhansl, 1990; Youngblood, 1958), with
the notable exception of Temperley (2007). Despite this
limitation, it is worth noting that composers typically
modulate to tonally proximate keys, and then return to
the original key. Given the similarity in the use of pitch
classes between related keys, short-term modulations
should not influence the overall shape of the tonal hierarchy when compiled across the entire piece. In fact,
effects of modulation would primarily serve to flatten
the distribution of the tonal hierarchy; if this flattening
has occurred in our data it would only reduce the
changes of the tonal hierarchy across metric position.
Overall, the consistent shape of the tonal-metric hierarchy demonstrates that on the aggregate, modulation is
a local effect, and does not radically influence the distribution of frequency of occurrences.
Nevertheless, determining how modulation may
affect this corpus is the most urgent next step in this
research. Methodological approaches could be to examine only pieces that do not modulate, or extract
excerpts that stay in the original key, or interpret modulated sections in terms of their new key (such that
during a modulation from C to G, all G pitches would
count as instances of the tonic). We are currently
exploring these approaches; in tandem we are planning
behavioral work to test if see if listeners’ perceptions of
modulation are similarly local. For shorter chorale
excerpts, at least, listeners are indeed sensitive to the
extent of modulation (Thompson & Cuddy, 1989,
1992); thus, once established, the tonal hierarchy may
function as a robust schema that persists through even
longer time spans.
Overall, the analyses reported in this paper substantiate the correlation of tonal and metric information in
typical tonal music, thus providing evidence for the
existence of a tonal-metric hierarchy, similar to the previously identified and well-studied tonal and metric
hierarchies. The observed tonal-metric hierarchy
remained remarkably consistent across compositional
period, time signature, and mode. Thus it is indeed true
that these composers used tonally stable pitches at
points of high metric stability. Importantly, the tonalmetric hierarchy is not a simple linear combination of
the separate hierarchies of tonality and meter. Instead, it
represents an interactive relationship between the two
hierarchies: the tonal hierarchy is most pronounced on
metrically stable temporal positions, and less distinct at
positions of lower metric stability. The current findings
therefore offer several useful contributions to the music
cognition literature. We have validated and elaborated
on a critical yet tacit assumption regarding the use of
the tonal and metric hierarchies, extended theoretical
research on hierarchical organization in musical pitch
and time, and revealed an interesting discrepancy
between perceptual and theoretical research on tonality
and meter. Hopefully these findings will stimulate future
research on the complex issue of pitch, time, and structural hierarchies in music.
Author Notes
Preparation of this manuscript was supported by the
Walter Murdoch Distinguished Collaborator Scheme
and a grant from the Natural Sciences and Engineering
Research Council of Canada to the second author. The
authors wish to thank Carol Krumhansl for her encouragement of this line of inquiry, and Davy Temperley for
his insightful comments.
Correspondence concerning this article should be
addressed to Jon B. Prince, School of Psychology, Murdoch University, 90 South Street, Murdoch, WA 6150.
E-mail: j.prince@murdoch.edu.au
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Jon B. Prince & Mark A. Schmuckler
Appendix A
Bach
Well-Tempered Clavier
Mozart
Piano sonatas
Beethoven
Piano Sonatas
Chopin
Preludes, Waltzes, Mazurkas
24 Preludes and
Fugues, Book 1
24 Preludes and
Fugues, Book 2
K.279
K.280
K.281
K.282
K.283
K.284
K.309
K.310
K.311
K.330
K.331
K.332
K.333
K.357
K.545
K.570
K.576
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
Op.
24 Preludes Op. 28, No. 1-24
Op. 6, No. 1-4
Op. 7, No. 1-5
Op. 10, No. 2, 5, 9
Op. 17, No. 1-4
Op. 24, No. 1-4
Op. 30, No. 1-4
Op. 33, No. 1-4
Op. 41, No. 1-4
Op. 50, No. 1-3
Op. 56, No. 1-3
Op. 59, No. 1-3
Op. 63, No. 1-3
Op. 64, No. 1-2
Op. 67, No. 1-4
Op. 68, No. 1-4
Op. 69, No. 2
B. 82
B. 134
B. 140
B. 150
2, No. 1-3
7
10, No. 1-3
13
14, No. 1-2
22
26
27, No. 1-2
28
31, No. 1-3
49, No. 1-2
53
54
57
78
79
81
90
101
106
109
110
111
Scherzo, Op. 81
Ballade, Op. 52, No. 4
Piano Rondo Op. 129
Symphony 5, Op. 67, 1st Mvmt.
Appendix B
Schubert
Brahms
Liszt
Scriabin
13 Variations on a Theme by
Anselm Hüttenbrenne
String Quartet No. 10 in Eb Major,
Op. 125, No. 1, Mvmt. 1-4
Trio for violin, viola and cello,
Sept. 1816
String Quartet D 804, Mvmt. 2
Ballade No. 1 in D minor, Op. 10
Hungarian Rhapsodies,
No. 12, 15
Grandes Etudes de
Paganini, No. 3, 4
Liebestraume Notturno,
No. 3
Transcendental Etudes,
No. 5 Feux Follets
Etude, Op. 2, No. 1
Impromptu in Gb major
View publication stats
String Quartet in C minor, Op. 51,
No. 1 (1873)
Waltz No. 1 in B major
Waltz No. 2 in E major
Waltz No. 8 in Bb Minor
Waltz No. 9 in D minor
Scherzo in Eb minor for piano, Op. 4
Piano Sonata No. 1 in C major, Op. 1
(2.) Andante
Fantasia, Op. 116, No. 2, 5, 6
Intermezzi, Op. 117, No. 1, 2
Rhapsodie from 4 Piano pieces,
Op. 119
12 Etudes, Op. 8,
No. 1-6
24 Preludes, Op.
11, No. 4, 15