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Interval-Class Content in Equally Tempered Pitch-Class Sets: Common Scales Exhibit Optimum
Tonal Consonance
Author(s): David Huron
Reviewed work(s):
Source:
Music Perception: An Interdisciplinary Journal,
Vol. 11, No. 3 (Spring, 1994), pp. 289-
305
Published by: University of California Press
Stable URL: http://www.jstor.org/stable/40285624 .
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Music Perception © 1994 by the regents of the
Spring 1994, Vol. 1
1, No. 3, 289-305 university of California
Interval-Class
Content
in Equally Tempered
Pitch-Class
Sets:
Common
Scales
Exhibit
Optimum
Tonal
Consonance
DAVID HURON
Conrad
Grebel
College,
University
of Waterloo
Pitch-class sets (such as scales) can be characterized according to the
inventory of possible intervals that can be formed by pairing all pitches
in the set. The frequency of occurrence of various interval classes in a
given pitch-class set can be correlated with corresponding measures of
perceived consonance for each interval class. If a goal of music-making
is to promote a euphonious effect, then those sets that exhibit a pleth-
ora of consonant intervals and a paucity of dissonant intervals might be
of particular interest to musicians. In this paper, it is shown that the
pitch-class sets that provide the most consonant interval-class invento-
ries are the major diatonic scale, the harmonic and melodic minor
scales, and equally tempered equivalents of the Japanese Ritsu mode,
the common penta
tonic scale, and the common "blues" scale. Conso-
nant harmonic intervals are more readily available in these sets than in
other possible sets that can be drawn from the 12 equally tempered
pitch chromas.
Introduction
All known musical
cultures
make use of a select repertoire
of pitches
from which musical
works are assembled.
In Western
tonal music, such
sets of pitches are typically arranged
in ascending/descending
pitch-
orderings
called
scales. Ignoring
pitch orderings,
we can borrow the term
"scale"
to denote more broadly
any pitch-class
repertoire
used in music-
making.
In some cultures,
certain
scale tones are intentionally
varied or
inflected
in pitch (Wade,
1979). However,
in most cultures,
scale
tones are
of fixed
or semistable
pitch,
and
pitch
perceptions
appear
to be dominated
by culturally
normative
categorical
perceptions
(although
see Serafine,
1988). Hundreds
of scales are known to have been used or explored
at
various
times in various parts of the world. Although
many scales have
been explored by Western
composers,
the number
of scales in Western
musical practice
remains
quite small; the diatonic major and harmonic
Requests for reprints may be sent to David Huron, Conrad Grebel College, University
of Waterloo, Waterloo, Ontario, Canada N2L 3G6.
289
290 David Huron
minor scales account for the vast majority of Western music. The common
pentatonic scale accounts for a significant portion of music in the Far East
(especially China).
As pointed out by the Gestalt psychologists, the specific pitches used in
a scale are less important than the intervallic relationships between the
tones. Provided the log frequency distances remain the same, a musical
work can be transposed in pitch without changing the essential perceptual
experience. This fact attests to the perceptual preeminence in music of
pitch intervals over pitches per se. A scale can thus be conveniently charac-
terized according to a recipe of successive intervals rather than as a set of
actual pitches.
Scales exhibit a bewildering variety of properties. Some scales have
variant ascending and descending forms (as in the case of the melodic
minor scale). Most scales can be characterized according to a primary or
final pitch
- as in the Western tonic or Indian sa. A single set of pitches can
be claimed by more than one scale, depending on what tone is deemed to
be the final or tonic pitch
- as in the case of the medieval modes.
In this paper, the focus will be on scales construed as unordered pitch-
class sets that act as pitch repertoires for music-making activities. No
account will be taken of the tonality or key-related implications of such
sets. Scales with variant ascending or descending forms will be analyzed
both as a single set and as two independent pitch-class sets. Our sole
interest will be on the interval structure
of pitch-class sets and the possibili-
ties afforded by each set for the creation of consonant harmonic intervals.
Finally, for reasons that will become apparent, we must unfortunately
limit this investigation to sets drawn from the 12 pitch chromas of the
Western equally tempered system of tuning. In principle, however, the
hypothesis tested and the method of investigation could be applied to any
fixed-pitch system of tuning in any musical culture.
Hypothesis
Musical intervals can be constructed by using either successions of
pitches (melodic intervals) or concurrent pitches (harmonic intervals).
Much of the world's music (such as the music of Asia) appears to stress the
melodic aspects of music. In Western music by contrast, there appears to
have been a comparatively greater emphasis on the vertical or harmonic
dimension of music-making. An important phenomenon related to the
sounding of concurrent pitches is the experience of sensory consonance or
dissonance (Greenwood, 1991; Helmholtz, 1863/1954; Hutchinson &
Knopoff, 1979; Kaestner, 1909; Kameoka & Kuriyagawa, 1969a, 1969b;
Malmberg, 1918; Plomp & Levelt, 1965; Seashore, 1938; Vos, 1986; and
Pitch-Class Sets and Tonal Consonance 291
others). As static isolated entities, different harmonic intervals evoke differ-
ent degrees of euphoniousness
- although such perceptions depend on the
spectral content of the participating tones, their sound pressure levels, and
their pitch register (Kameoka &c
Kuriyagawa, 1969a, 1969b; Plomp &
Levelt, 1965).
In the creation of music, musicians are undoubtedly motivated by a
wide variety of goals. If it is hypothesized that most musicians prefer to
create a euphonious sonic effect or music whose sensory dissonance is
somehow controlled or limited, one might suppose that these musicians
would seek to promote the occurrence of those intervals that exhibit a high
degree of tonal consonance. By itself, this suggestion may seem musically
unsophisticated. Cazden (1945), for example, has cautioned against con-
struing consonance and dissonance in terms of "tonal isolates" (brief verti-
cal moments). Cazden has suggested that the preeminent factor in the
perception of consonance and dissonance is the cultural context - pri-
marily culture-specific expectations of resolution. This important caveat
notwithstanding, it remains an appropriate task to investigate the possible
influence of simple sensory consonance on music-making.
Some pitch-class sets may better supply consonant intervals than other
pitch-class sets. Krumhansl
(1990) has suggested
that there may be a correla-
tion between the perceived consonance of harmonic intervals and the avail-
ability of such intervals in commonly used scales. Specifically,
Krumhansl
has noted informally that the inventory of possible intervals in the common
pentatonic and diatonic major scales roughly correspond to the ranking of
consonant and dissonant intervals (p. 276). Intervals
judged as more conso-
nant by both Western and non-Western listeners appear to be more readily
available in these scales. In this paper,
we propose to test formally this obser-
vation. Two different interpretations of "optimum consonance" will be
identified, and all possible pitch-class sets drawn from the equally tempered
scale will be characterized
according to these interpretations. To anticipate
our conclusions, it will be shown that among the chroma sets that provide
the greatest opportunity for the generation of consonant intervals are the
major diatonic scale, the harmonic and melodic minor scales, the common
"blues" scale, the common pentatonic scale, and the Japanese Ritsu mode.
Procedure
In order to explore the relationship between pitch-class sets and perceived consonance,
two indices are required: (1) a way of characterizing
the frequency of occurrence of various
intervals for any given pitch-class set, and (2) an index of tonal consonance for intervals of
various sizes. By relating the number of possible intervals of each size with the correspond-
ing tonal consonance for each interval, it is possible to determine the degree to which a
given pitch-class set maximizes the potential for consonant-sounding intervals.
292 David Huron
INTERVAL-CLASS INVENTORIES FOR PITCH-CLASS SETS
Each tone in a given pitch set can be paired with all other tones in the set. This means
that any set of fixed pitches establishes an inventory of possible pitch intervals. Consider,
by way of example, a six-note whole-tone scale in which successive pitches are two semi-
tones apart: C-D-E-Ftt -Gtt -A ft
. Pitches from this set can be readily paired together to
construct various intervals: major seconds (two semitones), major thirds (four semitones),
tritones (six semitones), etc. However, other intervals (interval sizes spanning odd-
numbered semitones) are impossible to form by selecting pitches from this set.
In the case of pitch sets with octave equivalence (i.e., pitch-class or chroma sets),
interval inventories show certain regularities. For any given pair of pitch classes (such as C
and D), a large number of intervals can be formed (e.g., M2, m7, M9, ml4). All possible
intervals that can be formed by using two pitch classes are said to belong to the same
interval class (Forte, 1973). An interval class holds all intervals that are related by octave
transpositions, including complementary intervals (i.e., intervals related by inversion). Be-
cause of octave duplication, if a pitch-class set permits the creation of major thirds, it
necessarily permits the creation of an equivalent number of minor sixths as well. In princi-
ple, a given pair of pitch classes cannot provide more intervals of one size than its comple-
mentary (inverted) interval. In Western set theory, all intervals generated by a pitch-class
set can be classified as one of just six interval classes.
Interval-class content for a given pitch-class set can be expressed by using a six-element
interval vector (Forte, 1973). In the case of the whole- tone scale, for example, the associ-
ated interval vector is [0, 6, 0, 6, 0, 3]. Each value in the interval vector indicates the
relative abundance of intervals of a given interval class that can be constructed from the
given pitch-class set. All pitch sets provide unisons for every scale tone; in the case of pitch-
class (octave-equivalent) sets, each tone also produces an octave interval. Consequently, the
number of possible unisons or octaves in a scale is uninformative. The first value in the
interval vector therefore pertains to the number of 1
-semitone (or 11
-semitone . . .) inter-
vals that can be generated from the pitch-class set. In the case of the whole-tone scale [0, 6,
0, 6, 0, 3], no minor seconds (or major sevenths . . .) are possible. Subsequent values in the
interval vector identify the frequency of occurrence for interval classes increasing in size up
to that of a tritone (6 semitones). Because intervals wider than a tritone are inversions of
intervals smaller than a tritone, larger intervals are already accounted for in the interval
vector. The six-element interval vector thus provides a complete characterization of the
potential interval content for a pitch-class set.
A CONSONANCE INDEX
In addition to the interval-class inventories, a second index is needed that characterizes
the degree of tonal consonance for various interval sizes. Of course interval size alone does
not entirely account for the perceived consonance of two concurrent tones. As mentioned
earlier, spectral content, sound pressure level, and pitch register are known to affect the
perception of tonal consonance. In order to compare interval-class inventories with tonal
consonance, we must assume that spectral content, sound pressure level, and pitch register
do not vary systematically with the type of pitch-class set. Expressed more informally, we
must assume that all music generated from the various pitch-class sets are played by similar
instruments, at roughly the same loudness, in approximately the same pitch region.
Although major strides have been made in understanding
the perception of tonal conso-
nance, some controversy remains regarding existing theories (Vos, 1986).1 Because we must
1. Vos has identified several deficiencies in the theory of consonance proposed by
Plomp and Levelt (1965) and in the theory proposed by Kameoka and Kuriyagawa
(1969a). In particular, Vos has demonstrated that, at least in the case of musician subjects,
perceived consonance ("purity" in Vos' terminology) depends on interval size in addition
to beats and roughness.
Pitch-Class Sets and Tonal Consonance 293
TABLE 1
Measures of Tonal Consonance and Dissonance for Various
Harmonic Intervals
Dissonance
Consonance
Hutchinson
& Kameoka
&
Interval Malmberg Knopoff Kuriyagawa
m2 0.00 .4886 285
M2 1.50 .2690 275
m3 4.35 .1109 255
M3 6.85 .0551 250
P4 7.00 .0451 245
TT 3.85 .0930 265
P5 9.50 .0221 215
m6 6.15 .0843 260
M6 8.00 .0477 230
m7 3.30 .0998 250
M7 1.50 .2312 255
discount spectral content, sound pressure level, and pitch register in this study, we can base
our consonance index directly on empirical records of listeners' judgments
- where only
interval size has been manipulated. In this way, we avoid having to rely on an existing
theoretical model of tonal consonance. For this study, an index of tonal consonance was
constructed by amalgamating experimental data from three well-known studies: Malmberg
(1918), Kameoka and Kuriyagawa (1969a), and Hutchinson and Knopoff (1979) - as cited
in Krumhansl
(1990) (see Table 1). Only the experimental data reported in these studies have
been used; calculations of theoretical or predicted consonance or dissonance have been ex-
cluded. In relying on these experimental results, we recognize that the perceptual data might
reflect cultural conditioning via exposure to music generated within commonly used scales.
In comparing the perceptual data with interval-class content, differences between com-
plementary intervals must be discarded. Recall that it is not possible for a given pitch-class
set to provide more opportunities for the creation of (say) major thirds than for minor
sixths. By definition, for any given pitch-class set, the opportunities for creating a given
interval must be identical to those for the complement of that interval. Although all three
perceptual studies show the minor sixth to be significantly more consonant than the major
third, this difference is immaterial when the data are compared with interval-class invento-
ries. We must, therefore, collapse the perceptual data for complementary intervals so that
the data are commensurate with the six-element interval-class content.
A single composite index was calculated
from the three
perceptual
studies as follows. First,
the consonance data for complementary intervals within each study were pooled (i.e.,
m2 + M7, M2 + m7, m3 + M6, etc.). The resulting values for each of the three empirical
sources
were normalized
so that each had a mean of zero and a standard
deviation of one. In ad-
dition, the signs were brought into agreement (the Malmberg data pertain to consonance,
whereas the other two studies measured
dissonance). The three normalized sets of data were
then averaged
together to produce a single composite consonance index shown in Table 2. This
index can be regarded
as a rough approximation
of the perceived
consonance of typical
equally
tempered interval classes constructed by using complex tones in the central pitch region.
GENERATION OF PITCH-CLASS SETS
Because the empirical measures of tonal consonance pertain to equally tempered pitches
only, it is possible to investigate only those sets that can be drawn from the set of 12 equally
294 David Huron
TABLE 2
Interval Class Index of Tonal
Consonance
Interval Class Consonance
m2/M7 -1.428
M2/m7 -0.582
m3/M6 +0.594
M3 / m6 +0.386
P4/P5 +1.240
A4 / d5 -0.453
note. Based on normalized data from
Malmberg (1918), Kameoka & Kuriya-
gawa (1969a), and Hutchinson & Knop-
off (1979). The data have been pooled for
complementary intervals.
tempered pitch classes. Pitch-class sets (such as scales) can be classified according to the
number of tones used
- the most well-known being the class of heptatonic (i.e., seven-tone)
scales or sets. In this study, all set classes were investigated
- with the exception of sets
consisting of just one or two tones. Sets consisting of a single pitch class are of little interest,
whereas sets consisting of two pitch classes would merely reproduce the rank ordering of
the interval-class consonances given in Table 2.
By using a computer program, it was possible to generate all possible sets that can be
drawn from the set of 12 pitch classes. The number of possible pitch-class sets consisting of
three or more tones is given by the following expression:
In total some 4,017 pitch-class sets were generated; however, most of these sets are simple
transpositions of each other. There are also a number of symmetries in pitch-class sets that
considerably reduces the total number of unique sets (Balzano, 1982; Forte, 1973). For
each set generated, the corresponding interval vector was determined.
OPTIMUM CONSONANCE MEASURES
In relating interval-class inventories to tonal consonance measures, one might distin-
guish two different notions of "optimum consonance." According to the first conception,
the pursuit of consonance may mean the complete absence of dissonant intervals and an
overwhelming preponderance of consonant intervals. In this case, the presence of even a
single dissonant interval would necessarily weaken the overall consonance
- and so
would be "undesirable." Accordingly, we might define an aggregate dyadic consonance
value that is calculated by multiplying the number of intervals of a given size by the
associated consonance values given in Table 2, and summing the results together for all
interval classes (i.e., cross-product). For example, in the case of the whole-tone scale [0,
6, 0, 6, 0, 3], the aggregate dyadic consonance would be 0(-1.428) + 6(-0.582) +
0(+0.594) + 6(+0.386) + 0(+ 1.240) + 3(-0.453) = -7.167. Those pitch-class sets
that provide many consonant intervals and few dissonant intervals would tend to have
higher aggregate dyadic consonance values. Large negative scores would indicate a set
that provides many dissonant intervals and relatively few consonant intervals.
Pitch-Class Sets and Tonal Consonance 295
An alternative (or weaker) notion of "optimum consonance" might recognize that
dissonances ought not to be entirely excluded from the composer's palette. Rather, the
presence of dissonant intervals ought merely to be controlled or limited. Specifically, we
might look for a rank ordering of the prevalence of various interval classes that is identical
to the rank ordering of their respective perceived consonances. According to this second
view, an optimum pitch-class set would have fewer minor seconds/major sevenths than
major seconds/minor sevenths, fewer major seconds/minor sevenths than tri
tones, fewer
tritones than major thirds/minor sixths, and so on. To this end, a Pearson's coefficient of
correlation can be calculated by comparing the interval vector values for each pitch-class
set with the perceived consonance data given in Table 2 (i.e., product-moment). For exam-
ple, in the case of the whole-tone scale, we would measure the correlation between [0, 6, 0,
6, 0, 3] and [-1.428, -0.582, 0.594, 0.386, 1.240, -0.453] (Pearson's
r = -0.14). A high
positive correlation would indicate that the frequencies of occurrence of the various inter-
vals that can be formed within the set are directly correlated with the magnitude of each
interval's perceived consonance.
RESULTS
Table 3 summarizes the analytic results for various pitch-class sets ac-
cording to both conceptions of optimum consonance described above.
Each section of the table pertains to pitch-class sets consisting of a differ-
ent number of tones. Table 3F, for example, tabulates the results for the
class of 7- tone sets. The sets are identified (column 3) by an ascending
(scalelike) recipe of semitone steps. In addition, the sets are identified
according to standard labels devised by Forte (1973). The corresponding
interval vectors are shown in column 4. The first column indicates the
aggregate dyadic consonance for the set, whereas the second column gives
the correlation between the interval vector and the index of perceived
tonal consonance. Only the "best" sets have been listed for each cardinal
class
- that is, only those pitch-class sets showing the highest aggregate
dyadic consonance values, plus all sets displaying consonance correlations
greater than +0.5 are tabulated. At the top of each section of the table, the
total number of unique interval vectors is noted. Excluding transpositions,
set inversions, and modal variants, a total of 189 unique sets of three or
more pitch classes can be constructed given the 12 chromatic pitches.
As an initial observation, we might note that the sets that display the
highest aggregate dyadic consonance are found in five-, six-, and seven-
tone sets. Both smaller and larger set sizes give generally lower aggregate
dyadic consonance values. In the case of the consonance correlation val-
ues, the results are somewhat different. As the number of tones in the set is
reduced, it is generally possible to form pitch-class sets that display a
higher correlation between the interval inventory and the interval conso-
nance data. For classes that have fewer than nine tones, for example, it is
always possible to find at least one pitch-class set that exhibits a conso-
nance correlation better than +0.62. However, this is most likely an arti-
fact of the reduced degrees of freedom.
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300 David Huron
In the case of seven-tone (heptatonic) sets, the common major and
harmonic minor scales are ranked first and third, respectively,
with respect
to the aggregate dyadic consonance (of 31 heptatonic sets having unique
interval vectors). With respect to the consonance correlations, the har-
monic minor and diatonic major scales are ranked first and second, respec-
tively. As our study enumerates all possible sets within each set class, the
ranking probability can be calculated directly. In the case of the aggregate
dyadic consonance values, the probability of the major and harmonic
minor scale being ranked among the top three sets is .0023. With respect
to the consonance correlations, the probability of the major and harmonic
minor scale being ranked among the top two sets is .0022.
Note that the pitch-class set for the major diatonic scale has close
parallels in several non- Western musics. Most notably, the major scale is
virtually identical to some of the most common thâts used in the music of
northern India: Bilâval, Khamâj, Kâfi, Äsävri, Bhairvi, and Kalyän
-
which in turn are comparable to the medieval modes: Ionian, Mixolydian,
Dorian, Aeolian, Phrygian, and Lydian, respectively.
Notice also that both
the ascending and descending forms of the melodic minor scale can be
found among the top five heptatonic sets given in Table 3F. Moreover, the
combined nine-tone form of the melodic minor scale (Table 3D) ranks
third of the 12 possible unique nine-tone pitch sets.
Because harmonic intervals are insensitive to direction, note that the
consonance correlation value for any pitch-class set will be the same as the
inverse for that set. In other words, the interval vector for any given
ascending pattern of tones will be the same if the ascending pattern is
treated as a descending pattern. In some cases, a set is symmetric
- with
identical ascending and descending patterns. The major diatonic scale is
one such symmetric scale. If the ascending scale pattern (2, 2, 1, 2, 2, 2, 1)
is reversed, the result is yet another major diatonic pitch-class set (Phryg-
ian mode). It has been noted by many music theorists that several common
pitch-class sets used in Western music (such as the major diatonic scale,
the whole-tone scale, major/minor triads) show such symmetries (Forte,
1973; Rahn, 1980). The harmonic minor scale is not symmetric, and so is
matched with its inverse scale in Table 3F.
In the case of the hexatonic (six-tone) pitch-class sets, the highest ranking
set in terms of aggregate dyadic consonance is the equally tempered equiva-
lent of the Japanese Ritsu mode (the tunings are not quite the same)
. Like the
major diatonic scale, the Ritsu mode is also inversionally symmetric. The
equally tempered equivalent of the common "blues" scale ranks fifth of the
35 unique hexatonic pitch-class sets in aggregate dyadic consonance and
ranks fourth in consonance correlation. In the case of pentatonic (five-tone)
pitch-class sets, the "common pentatonic" scale (anhemitonic pentatonic
Pitch-Class Sets and Tonal Consonance 301
scale) ranks the highest in terms of the aggregate dyadic consonance value.
With regard to the consonance correlation measure, the common penta-
tonic scale is ranked fourth of 35 pentatonic scales with unique interval
vectors. In addition, the common pentatonic scale is the only ranking five-
note scale that is inversionally symmetric.
Having examined the preeminent scales, we can now summarize the
results. According to both interpretations of optimum consonance out-
lined here, there is indeed a notable conformity between the perceived
consonance of harmonic intervals and the availability of such intervals in
commonly used musical scales. This conformity is significantly higher than
would be expected by a chance selection of tones from the equally tem-
pered set. Indeed, the scales we have considered rank foremost of all
possible pitch-class sets.
The results of Tables 31 (four-tone) and 3J (three-tone) are suggestive in
the context of Western harmonic practice. The pitch-class sets that most
conform to the index of tonal consonance turn out to be the most common
chords in Western music: the major and minor triads, and four common
four-note chords: (in order) the minor-minor seventh, the dominant sev-
enth, the half-diminished seventh, and the major-major seventh chords.
The augmented and diminished triads rank fifth and sixth (of 12) in Table
3J, but these sets show little correlation with the tonal consonance index.
These results replicate those of Hutchinson and Knopoff (1978, 1979;
Table 1) and Danner (1985; Tables 1-3) where pitch-class consonance
was calculated for three-note chords by using a model of tonal consonance
perception.
In the case of these three- and four-note sets, it is important to recognize
that the consonance measures do not reflect the consonance of the com-
plete set of concurrently sounding tones (such as the consonance of "a
major triad"). Rather, the consonance values reflect the collective conso-
nance of all possible pairs of tones drawn from the given pitch set.
Finally, we might consider which of the two conceptions of optimum
consonance best predicts the analytic results. That is, is musical practice
most consistent with the simple pursuit of consonant harmonic intervals to
the exclusion of dissonant intervals? Or is musical practice most consistent
with a rank ordering of the availability of intervals with their respective
degrees of perceived consonance? Table 4 compares the rank ordering of
the aggregate dyadic consonance values and the consonance correlation
values for common scales and vertical pitch sets. Evidence in favor of
either conception of optimum consonance would be reflected in a signifi-
cantly higher ranking for either the aggregate dyadic consonance values or
the consonance correlation values. However, Table 4 shows no systematic
or significant differences between the results of the two different analyses.
302 David Huron
TABLE 4
Comparison of Results for Two Conceptions of Optimum Consonance
Consonance
Ranking Correlation
Ranking
Melodic minor scale 3 3
Major diatonic scale 1 2
Harmonic minor scale 3 1
Ascending melodic minor 5 5
Descending melodic minor 1 1
Ritsu mode 1 4
"Blues" scale 5 4
Common pentatonic scale 1 4
Minor-minor seventh chord 1 1
Major-major seventh chord 3 3
Dominant seventh chord 5 2
Major triad 1 1
Minor triad 1 1
Augmented triad 4 4
Diminished triad 5 5
Hence we are unable to claim that one conception
provides
a better
ac-
count of musical
practice.
Conclusion and Discussion
Any set of pitch chromas
(such as defined
in a scale) delimits
a set of
possible harmonic
(and melodic)
intervals.
A useful analogy
is to liken a
musical
scale to a painter's palette in which a limited set of resources is
preselected.
Pitches can be paired together
in a manner
analogous
to the
way an artist mixes paints. Depending
on the available
paints on the
palette, certain colors or hues can or cannot be readily
produced.
Simi-
larly,
depending
on the pitch-class
set, various
interval classes
(and
hence
intervals)
occur
with greater
or lesser
frequency.
A frequency
distribution
of interval
classes
is provided
by a set's interval
vector. Pitch
intervals can
be rated
according
to their
perceived euphoniousness
or tonal
consonance.
If
one of the composer's
aesthetic
goals
is to generate predominantly
conso-
nant
music,
an appropriate
choice of palette
would maximize
the availabil-
ity of consonant harmonic intervals
while minimizing
the presence
of
dissonant harmonic intervals.
Alternatively,
a composer might
aim
to main-
tain the availability
of dissonant
intervals,
but only in inverse
propor-
Pitch-Class Sets and Tonal Consonance 303
tion to their degree of dissonance. That is, as intervals become more
dissonant, fewer of them can be generated from the pitches available on
the composer's palette.
Beginning with the 12 equally tempered pitch chromas, a large number
of unique sets (or scales) can be generated by selecting varying numbers of
tones from this initial set. When the interval vectors for these sets are
compared with measures of perceived consonance, certain sets display
elevated consonance values. Among the pitch-class sets whose interval-
class inventories conform most strongly with an index of perceived conso-
nance are the three preeminent scales in Western music: the major diatonic
scale, and the harmonic and melodic minor scales. The high consonance
values displayed by the harmonic and melodic minor scales are especially
notable, because the minor scales have resisted previous attempts to ac-
count for them using theories of consonance (see discussions in Parncutt,
1989 and Krumhansl, 1990, p. 53). Although several of the most popular
pitch-class sets are symmetric with respect to inversion, the harmonic
minor and complete melodic minor scales are notable exceptions. The fact
that these scales exhibit a high aggregate dyadic and correlational conso-
nance suggests that the pursuit of tonal consonance provides a more parsi-
monious account of the popularity of these sets than the formal property
of symmetry. Moreover, whereas the perceptual relevance of set inversion
is difficult to interpret, the perceptual relevance of tonal consonance is
clear.
In addition to the Western major and minor scales, the results are also
suggestive for scales that are less directly associated with equal tempera-
ment. Equally tempered equivalents of the common pentatonic scale, the
Japanese Ritsu mode, and the common "blues" scale all displayed high or
optimum consonance values. Other pitch-class sets
- such as those corre-
sponding to major and minor triads, and various seventh chords
-
similarly produce interval-class inventories that conform strongly with an
index of perceived consonance. In the case of aggregate dyadic conso-
nance, values peak for pitch-class sets containing between five and seven
tones
- a range that corresponds to the number of tones in most of the
world's scales.
It would be wrong to view the results of this study as endorsing Western
musical practice to the implied detriment of some non-
Western musics.
First, the initial set of 12 equally tempered pitches is obviously biased
toward Western musical practice. Second, in comparison to most of the
world's music, Western music tends to be highly harmonically oriented.
Where scales provide the basis for predominantly melodic music, examin-
ing the harmonic properties of these scales may be inappropriate. (Indeed,
non-Western musicians frequently cite the comparatively small number of
scales used by Western musicians as symptomatic of the relative melodic
304 David Huron
impoverishment of Western music.) Note, however, that the hypothesis
tested here and the method of investigation used can be applied to any
fixed-pitch system of tuning for any musical culture
- provided the appro-
priate tonal consonance data is available for all intervals that might arise.
In addition, it is important to emphasize that this investigation has dealt
only with pitch pairs. Little is known about the perception of three or more
concurrent pitches (although see Parncutt, 1989). Both the dependent and
independent variables examined in this study consider only simple pitch-
pairing. With these caveats in mind, it remains
a significant
observation that
the most commonly used pitch sets in Western
tonal music provide interval
inventories that conform well with ratings of perceived consonance.
Finally, in this study no inferences can be drawn regarding causality.
As noted earlier, it is possible that empirical measures of tonal conso-
nance are determined largely by listeners' experience with music com-
posed within the major and harmonic minor scales. Even if consonance is
entirely a learned phenomenon, the results of this investigation suggest a
significant coadaptation of scales and perceived consonance. However, it
is noteworthy that the most popular accounts of tonal consonance pro-
pose physical and psychophysical mechanisms and exclude experiential
factors (Greenwood, 1991; Kameoka & Kuriyagawa, 1969a, 1969b;
Plomp &c
Level
t, 1965; although see Terhardt, 1974). If these accounts of
tonal consonance are correct, then it suggests that scales have adapted to
the idiosyncratic properties of human hearing, rather than vice versa.
In conclusion, it is true that the inventory of possible intervals found in
Western musical scales roughly correspond to the ranking of consonant
and dissonant intervals. The more consonant intervals are more readily
available in these scales than in other possible pitch sets that can be drawn
from the 12 chromatic pitches. Whereas the origin of these scales can be
attributed to the simple pursuit of tonal consequence, it might be argued
that the ongoing use of these scales is a result of cultural inertia rather
than
the continued pursuit of tonal consonance as a shared goal in music-
making. However, in Huron (1991) is was shown that for common prac-
tice period music, composers avoid dissonant intervals in preference to
consonant intervals
- even when the interval inventory is controlled. Simi-
larly, Huron and Sellmer (1992) showed that the spacing of pitches in
Western sonorities is consistent with the goal of minimizing sensory disso-
nance. Together, these three studies imply that, in common practice, musi-
cians have been motivated
- at least in part
- by the remarkably simple
goal of creating euphonious vertical moments.2
2. The author extends his thanks to Charles Morrison for comments on an earlier draft
of this article. Software support provided by Mortice Kern Systems is gratefully acknowl-
edged. This research was supported in part through funds provided by the Social Sciences
and Humanities Research Council of Canada.
Pitch-Class Sets and Tonal Consonance 305
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