Interval-Class Content in Equally Tempered Pitch-Class Sets: Common Scales Exhibit OptimumTonal ConsonanceAuthor(s): David HuronSource: Music Perception: An Interdisciplinary Journal, Vol. 11, No. 3 (Spring, 1994), pp. 289-305Published by: University of California PressStable URL: http://www.jstor.org/stable/40285624Accessed: 06/10/2010 18:08

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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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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 thts used in the music of northern India: Bilval, Khamj, Kfi, svri, Bhairvi, and Kalyn - 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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Issue Table of ContentsMusic Perception: An Interdisciplinary Journal, Vol. 11, No. 3 (Spring, 1994), pp. 219-331Front MatterAn Exploration of the Use of Tempo in Jazz [pp. 219-242]Detection of Implied Harmony Changes in Triadic Melodies [pp. 243-264]The Role of the Perception of Rhythmic Grouping in Musical Performance: Evidence from Motor-Skill Development in Piano Playing [pp. 265-288]Interval-Class Content in Equally Tempered Pitch-Class Sets: Common Scales Exhibit Optimum Tonal Consonance [pp. 289-305]Music Perception and Musical Communities [pp. 307-320]The Pitch of Speech as a Function of Linguistic Community [pp. 321-331]Back Matter