Tonal Consonance versus Tonal Fusion in Polyphonic Sonorities

Abstract

An analysis of a sample of polyphonic keyboard works by J.S. Bach shows that the prevalence of different vertical intervals is directly correlated with their degree of tonal consonance. A major exception to this pattern arises with respect to those intervals that contribute to tonal fusion. The prevalence of the latter intervals is negatively correlated with the degree to which each interval promotes tonal fusion. Bach's avoidance of tonally fused intervals is consistent with the objective of maintaining the perceptual independence of the contrapuntal voices. In summary, two factors appear to account for much of Bach's choice of vertical intervals: the pursuit of tonal consonance and the avoidance of tonal fusion.

Introduction

Stumpf (1890) summarized centuries of observations regarding the tendency for some sound combinations to cohere into a single sound image through a process of Tonverschmelzung or tonal fusion. Tonal fusion arises when the auditory system interprets certain frequency combinations as comprising partials of a single complex tone (DeWitt & Crowder, 1987). Normally, such partials occur in a harmonic relationship where the frequencies are related by simple integer ratios. Tonal fusion occurs both in the case of pure tones, and also where concurrent complex tones contain coincident or complementary partials -- consistent with the possible existence of a single complex tone.

In the construction of musical works, one could imagine tone fusion to be either a welcome or unwelcome perceptual phenomenon -- depending on the musical objective. In some cases, the musical goal may be to create an integrated or consolidated sound image. In such cases a composer might purposely arrange vertical sonorities so as to enhance tonal fusion. In particular, a succession of such tonally fused sonorities would virtually ensure the perception of a single auditory stream (Bregman, 1990). A number of examples of (evidently intentional) tonal fusion can be found in Western orchestral literature, including the beginning of Aaron Copland's Billy the Kid ballet suite and a well-known passage in Maurice Ravel's Bolero. In other cases, tonal fusion may be contrary to the musical goal. Insofar as an objective of polyphonic music is to maintain the perceptual independence of the contrapuntal voices, one might expect polyphonic composers to avoid circumstances that could lead to the inadvertent perceptual integration of the parts. A variety of both "horizontal" and "vertical" factors have been identified as contributing to auditory streaming (Bregman, 1990). Horizontal factors that enhance the perception of independent streams include the maintenance of close pitch proximity within the voices (Dowling, 1967) and the avoidance of part-crossing (Huron, 1991). Vertical factors that encourage stream segregation include asynchronous tone onsets (Bregman, 1990) and the avoidance of tonal fusion (McAdams, 1982, 1984). Thus in polyphonic music, there may be good cause to avoid sonorities that promote tonal fusion.

The pitch interval that most encourages tonal fusion is the aptly named unison. The second most fused interval is the octave (1:2), and the third most fused interval is the perfect fifth (2:3) (DeWitt & Crowder, 1987; Stumpf, 1890). There is no general agreement in the literature concerning the rank ordering of subsequent intervals. Some commentators consider the perfect fourth (3:4) to be the next most fused intervals, whereas others have suggested the double octave (1:4). Experimental data collected by DeWitt and Crowder (pp. 77, 78) paradoxically suggests that major sevenths are more prone to tonal fusion than are perfect fourths.

Were it the case that fused intervals are avoided in polyphonic music in order to maintain voice independence, one might expect the degree of avoidance of a given interval to be proportional to the strength with which that interval promotes tonal fusion. Eliminating from consideration those intervals whose ranking is contentious, one might predict that the least common vertical interval to be found in polyphonic music would be the unison, followed by the octave, followed by the perfect fifth.

Juxtaposed against such a prediction is the recognition that the choice of harmonic intervals[1] in a musical work may be motivated by other factors apart from the desire to maintain the perceptual segrgation of the voices. Specifically, one might suppose that a composer also endeavors to choose intervals that conform to some harmonic plan and that preserve a certain degree of tonal consonance. Like theories of tonal fusion, traditional theories of consonance have also relied on the concept of simple integer frequency ratios -- the simpler ratios being the most euphonious. If traditional theories of consonance were true, and composers endeavored to maximize the degree of consonance in their works, then such an objective would appear to be in direct contradiction to the objective of reducing the amount of tonal fusion. One would not know, for example, whether a (hypothetical) absence of perfect intervals would indicate the composer's desire to avoid tonal fusion, or whether it might be symptomatic of the composer's desire to maintain a certain degree of dissonance. [1] The term "harmonic interval" is used here in contrast to "melodic interval." By harmonic interval, music theorists mean the pitch distance between two concurrent pitches. The term is not intended to connote a simple frequency ratio such as might arise from the harmonic series.

In a well-known paper by Plomp and Levelt (1965), it was shown that the concept of simple frequency ratios alone does not provide a satisfactory account of perceived consonance. Plomp and Levelt recalled work by Kaestner (1909), showing that the perceived consonance of intervals depends on the spectral content of the participating tones and on their pitch distance.[2] [2] It is well known in studies of perceived consonance that trained musicians respond differently from nonmusicians (see for example, Guernsey, 1928). Specifically, musicians may distinguish between "pleasantness" or "euphoniousness" on the one hand, and "consonance" on the other. Van de Geer, Levelt, and Plomp (1962) have shown that haive subjects make no such distinction and have argued that musicians' responses are mediated by an a prior theoretical understanding. Following Plomp and Levelt, in this paper we will avoid the ambiguous term "consonance" and use the more circumspect term "tonal consonance" -- which Plomp and Levelt define as the degree of perceived pleasantness or euphoniousness of a static pitch interval. The influence of timbre on tonal consonance had been predicted as early as 1863 by von Helmholtz. Plomp and Levelt invoked the concept of the critical band to explain various experimental results that contradict the "simple ratio" theory of consonance. They also showed that the degree of consonance for various intervals depends on the absolute pitch register of the interval. For example, the interval of a major third will sound much more dissonant in the bass register than it will in the treble register. This phenomenon appears to arise from changes of bandwidth fro critical bands with respect to log-frequency. Hence transposing a musical work to a different register will alter the perceived consonance of its vertical intervals.

Figure 1 plots three sets of published data showing the degree of tonal consonance (i.e., pleasingness or euphoniousness) as a function of semitone distance between two tones. The dotted and dashed lines reproduce data from Kaestner (1909) as given in Plomp and Levelt (1965). The dotted line represents data for intervals consisting of pure tones, whereas the dashed line represents Kaestner's data for intervals consisting of complex tones.

Fig. 1. Major and minor key profiles of Krumhansl & Kessler (1982). The vertical axis indicates how well, on average, each pitch-class (realized as a Shepard tone) is judged to follow or fit with a preceding diatonic chord progression in C major or C minor.

The solid line in Figure 1 reproduces data assembled by Kameoka and Kuriyagawa (1969b) for complex tones consisting of eight spectrally shaped harmonics -- where the second and third harmonics display the greatest amplitude. The effect of spectral content on tonal consonance has been explored in some detail by Kameoka and Kuriyagawa (1969a, 1969b) and by Vos (1986). Kameoka and Kuriyagawa confirmed that pitch register is important in the perception of tonal consonance. They also proposed that consonance is influenced by sound level and spectral masking.[3] [3] Vos (1986) 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's terminology) depends on interval size in addition to beats and roughness. In this paper, the intent is to compare composers' practices directly to empirical records of listeners' judgments -- and so sidestep the theoretical models and the associated controversy. Vos's own carefully collected data pertain only to the intervals of the perfect fifth and the major third, and so cannot be used in this study. The difference between Kaestner's data for complex tones and the Kameoka and Kuriyagawa data can be attributed to differences of spectral content in the participating tones and to the fact that the intervals lie in different pitch regions. In the case of Kaestner data, the lower tone for all intervals has a frequency of 320 Hz (E4), whereas the lower tone for the Kameoka and Kuriyagawa data has a frequency of 440 Hz (A4).

In interpreting Figure 1, note that few data have been assembled concerning the perception of tonal consonance for intervals larger than the octave. Som supraoctave data assembled by Kameoka and Kuriyagawa (1969a) cast considerable doubt on the traditional musical penchant to equate compound intervals that are octave-equivalent (as, for example, in the case of the major tenth and major third).

Let us suppose that a polyphonic composer's choice of harmonic interval is motivated primarily by two concerns: (1) the need to avoid inadvertent tonal fusion of the voices and (2) the need to maintain a certain degree of intervoice tonal consonance. Stumpf tells us which intervals must be avoided in order to pursue the first goal (i.e., avoid perfect consonances), whereas Plomp and Levelt tell us which intervals to avoid in order to pursue the second goal (i.e., avoid the dissonances: m2, M2, m7, M7, etc.).

If the first goal is pursued by composers, we ought to see evidence of attempts to minimize occurrences of unisons, octaves, fifths, and perhaps fourths. Moreover, we ought to see evidence of a rank ordering of these intervals such that perfect unisons should be the most avoided, followed by perfect octaves, followed by perfect fifths. If the second goal is pursued by composers, we ought to see evidence of attempts to minimize occurrences of dissonance intervals. Moreover, we ought to see evidence of a rank ordering of the various intervals according to their degree of tonal consonance. Expressed more formally, we might identify two specificy hypotheses. Hypothesis 1: The frequency of occurrence of an intervals is positively correlated with its degree of tonal consonance. Hypothesis 2: In polyphonic music, the frequency of occurrence of an interval is negatively correlated with the degree to which it promotes tonal fusion.

As noted above, the phenomenon of tonal fusion may be sought or avoided depending on the musical goal. The genre of music dubbed "polyphony" is appropriate for our study because polyphonic composers explicitly construct multiple concurrent musical lines/parts/voices/streams whose perceptual independence is deemed important. Thus one might assume that there exists in polyphonic music a compositional intent to preserve stream segregation between the voices -- an intention that may not be present in other types of music. In order to test our hypotheses, an initial sample of polyphonic keyboard works by Johann Sebastian Bach was selected -- specifically, Bach's 15 two-part keyboard Inventions. Subsequent experimental controls warranted switching the analytic sample to Bach's 15 three-part Sinfonias.

In examining harmonic intervals, a number of measurement issues arise. In brief, these issues include the manner in which interval instances are determined, the question of enharmonic equivalence, the influence of interval context, the issue of interval tuning, and the problem of timbre.

A distinction can be made between two approaches to the measurement of pitch intervals: the figural method and the time-base method. The figural method determines the type of interval for each novel vertical sonority -- that is, a new interval is deemed to occur each time a new note is articulated in either one of the voices forming the interval. The time-base method determines the pitch distance between two voices at regular metric divisions -- such as every sixteenth duration. The metric division used for the time-base method can be defined as equivalent to the shortest duration found in the work. The prevalence of a given interval can then be expressed as a percentage of the total number of intervals identified using the time-base mthod. To the extent that intervals with long durations are perceptually more salient than intervals with short durations (i.e., agogic accent), the time-base approach is arguably the preferred method for measuring intervals. The latter approach will be used throughout this article.

A second measurement issue arises with respect to enharmonic interval spellings. Musical notation permits the same (equally tempered) pitch distance to be spelled using several aliases -- as in the case of the diminished fifth and augmented fourth. It is likely that harmonic contexts dispose listeners to perceive one enharmonic "meaning" in preference to another -- even though the pitch distances may be identical (Cazden, 1980; Krumhansl, 1979). Suggestive results have come from Schackford (1961, 1962a, 1962b), who measured interval sizes as performed by three professional string quartets. Shackford (1961, p. 201) found, for example, that augmented fourths are typically performed 18 cents wider than diminished fifths. To the extent that performance practice is shaped by perceptual goals, this difference between two types of tritone suggests that harmonic context may be perceptually influential. It is possible that the tonal consonance for a given fixed pitch distance might be rated differently in different harmonic settings.

Unfortunately, systematic psychoacoustic data concerning enharmonic context have not bee collected. Published perceptual studies of tonal consonance -- such as Kaestner (1909), Guthrie and Morrill (1928), Plomp and Levelt (1965), and Vos (1986) -- have disregarded the possible effect of harmonic context on perceived euphoniousness. Without such data against which to compare musical practice, there is little reason to collect enharmonically differentiated data. Hence, in this study, pitch distances are measured in semitones without regard to interval spelling. For convenience, interval sizes will be referred to by standard diatonic terms; thus, we will use the label "minor seventh" in preference to an interval of 10 semitones. Nevertheless, it should be remembered that the actual notational spellings may differ: what we will call a minor seventh may be rendered as an augmented sixth or a doubly diminished octave, and so on.

A third measurement issue arises with respect to tuning: is it important to establish the precise tuning system under which the sampled works were composed? The debate here focuses predominantly on the contrast between just intonation and equal temperament. For perfect unisons and octaves, equally tempered intervals are identical to those for just intonation. In the case of successively more dissonant intervals, equal temperament tuning diverges more and more from just intonation so that for a minor second the difference is about 12 cents. Stumpf claimed that tuning differences do not especially affect judgments of consonance, whereas more recent data suggest that tuning differences are not insignificant.

With regard to tonal fusion, tuning differences appear to be more directly influential. DeWitt and Crowder (1987) showed that tonal fusion is slightly more pronounced in just intonation than in equal temperament tuning. However, the degree of tonal fusion for different interval types was found to correlate closely across the two tuning systems (p.77). The rank ordering of intervals in promoting tonal fusion remains the same in both systems. The differences between equal temperament and just intonation notwithstanding, given the small size of the effect of tuning on the rank ordering of intervals according to tonal consonance and tonal fusion, it is reasonable to proceed without attempting to control for tuning -- either in the sampled works or in the tonal fusion and tonal consonance data.

A fourth measurement issue concerns the timbre of the original sampled works. Because spectral content influences tonal consonance, it is difficult to relate the sampled works to any of the published data pertaining to tonal consonance. Ideally, we would like to be able to sample the instrumental timbres used (or imagined?) by Bach in the composition of the selected works. Using these timbres, we could then collect independent data measuring the perceived consonance for intervals of various sizes and in different pitch ranges or tessituras. It would then be possible to relate the intervallic practices in the sampled works to the instrumental timbres used. Although this approach has considerable merit, it was discounted as impractical. In pursuing this study, we will accept the confounding effect of timbre, and presume that either one or both of the Kaestner (1909) and Kameoka and Kuriyagawa (1969b) data wil provide an adequate template against which Bach's interval practices may be correlated. Because music normally consists of complex tones, the tonal consonance data for pure tones will not be used in the analysis.

Preliminary Analysis

Using the time-base method of measurement, harmonic interval data were collected for the 15 two-part Inventions. Figure 2 shows the prevalence of various vertical intervals in the Inventions. Figure 2 shows the prevalence of various vertical intervals in the Inventions. The data are displayed according to three categories of intervals distinguished in traditional music theory: perfect consonances (P1, P4, P5, P8, etc.), imperfect consonances (m3, M3, m6, M6, etc.), and dissonances (m2, M2, TT, m7, M7, etc.). The bell-shaped contour in these data at once revelas that the frequencies of occurrence for various intervals are confounded by between-voice pitch proximity and/or pitch range (tessitura) of the individual voices. Bach seems disposed to keep the two voics separated by an interval of about a tenth.

Contrary to our two hypothese, we might entertain the possibility that the predominance of various intervals might arise from the composer's preference for maintaining an optimum between-voice pitch distance -- or by a preference for certain tessituras for the two voices. In the case of the two-part Inventions, about three-quarters of the data lie beyond the interval of an octave. As noted earlier, detailed published data concerning both tonal consonance and tonal fusion pertain only to intervals up to an octave; hence only the within-octave interval data given in Figure 2 can be used in our analysis. From Figure 2, it is clear that the dozen smallest intervals are located on the rising slope of the distribution. This skews the data toward the larger intervals, and biases the sample. It is tempting to amalgamate octave-equivalent data -- however, as we have noted, octave equivalence of compound intervals is not a valid assumption.

Fig. 2. Harmonic interval prevalence: two-part Inventions.

Another problem with the data from the two-part Inventions is that the average lower pitch in each interval is somewhat low compared with the lower pitch in Kaestner's interval study and is quite low compared with the lower pitch in Kameoka and Kuriyagawa. The average pitch for the bass voice of the two-part Inventions lies a little more than 9 semitones below Kaestner's lower pitch (E4) and more than 14 semitones below the lower pitch of Kameoka and Kuriyagawa (A4). It is difficult to estimate the magnitude of the confounding effect of interval tessitura. Certainly, it would be reassuring to find interval data that more closely match the pitch region of at least one of our two sets of tonal consonance data.

In light of these problems, a more favorable sample of musical works was sought. A better sample would exhibit a higher average pitch in the lower voice and a closer proximity between the two voices -- such that the majority of harmonic intervals would lie within the interval of an octave. A somewhat improved sample is provided by the upper two voices of Bach's three-part Sinfonias -- an interval distribution for which is given in Figure 3. Figure 3 reveals that more than 85% of the intervals formed by the middle and upper voices are an octave or smaller in size. In addition, the mean pitch of the middle voice of the three-part Sinfonias is slightly higher. The average pitch of the middle voice lies less than 5 semitones away from Kaestner's lower pitch -- although it remains almost 10 semitones away from the lower pitch in Kameoka and Kuriyagawa. There remains some difficulties with these data, but let's proceed in any event with a preliminary comparison of Bach's interval practice and research on tonal fusion and tonal consonance.

Fig. 3. Harmonic interval prevalence: three-part Sinfonias (soprano-mid voices).

Interval Prevalence in the Upper Voices of Bach's Sinfonias

Figure 4 overlays the first octave interval distributions for the upper two voices of the three-part Sinfonias along with Kaestner's measurements of consonance for complex tones. Broadly speaking, there is a good correlation between the Bach data and Kaestner's data (r=.76); however there is no correlation to the data of Kameoka and Kuriyagawa (r=-.04). The discrepancies between the Bach data and Kaestner data are evident in Figure 4. Specifically, the unison and octave intervals are substantially less prevalent in the Bach data than would be suggested by their relative tonal consonance. In addition, the major third occurs less than expected, while occurrences of the minor third appear somewhat raised.

Fig. 4. Comparison of tonal consonance for complex tones (line) from Kaestner (1909) with interval prevalence (bars) in the upper two voices of Bach's three-part Sinfonias.

The suppression of unisons and octaves is predicted by Hypothesis 2 -- namely, avoiding intervals that promote tonal fusion. If we compare the prevalence of the perfect intervals (P1, P4, P5, & P8) with the data concerning the disposition of these intervals to promote tonal fusion, the correlations are -0.75 for the Stumpf data and -0.80 for the DeWitt and Crowder data. These results support Hypothesis 2 -- that is, that Bach is endeavoring to avoid tonal fusion. If we now eliminate the perfect intervals from consideration, we might recalculate the correlations for tonal consonance. Excluding the perfect intervals, the correlations between interval prevalence and the data on tonal consonance rise considerably: r=.92 in the case of Kaestner and r=.64 in the case of Kameoka and Kuriyagawa. The latter results are consistent with both Hypotheses 1 and 2. In short, it appears that Bach endeavors to promote tonal consonance while concurrently avoiding tonal fusion.

These results are highly suggestive. Nevertheless, it can be argued that the relative paucity of octaves and unisons can be attributed to the fact that these intervals are located at the lower and upper regions of the interval distribution -- where frequencies naturally decline. Amore robust demonstration of our hypotheses would remove the confounding artifacts of the interval distribution. Clearly, we need to address more directly the problem of how to eliminate the effect of intervoice pitch proximity.

Controlling for Voice Proximity: Prevalence versus Preference

In order to address this problem, Bach's interval data ought to be recast such that the effects of intervoice pitch proximity are strictly controlled. Specifically, a method is needed whereby the actual distribution of intervals can be contrasted with a distribution that might be expected to arise by chance.

In Huron (1991) an auto-phase method was described that is well suited to the tasks of controlling the effects of pitch proximity. An auto-phase can be likened to an autocorrelation. The method can be conceived of by using the following metaphor. A two-part work may be imagined to be notated on a single long strip of paper. The beginning and end of this strip are connected together to form a loop. The two parts are cut apart to from two independent but parallel loops in the manner of a circular slide rule. One of the voices can be shifted with respect to the other through a complete circle of 360 degrees, but only when the parts are aligned at zero degrees does their relationship correspond to the original musical score. The proportions of various harmonic intervals can be measured for each novel configuration as the parts are shifted with respect to each other by a fixed metric division (such as a sixteeth duration). The interval distributions for all of the non-zero-degree arrangements of a work can be amalgamated into a single controlled distribution against which the actual distribution of intervals in the work can be compared. This may be referred to as a pitch-proximity-controlled distribution.

The advantage of this method is that each rearrangement preserves the identical pitch distributions for the two voices, the same within-voice melodic contouring, durations, and within-voice rhythmic structure. Thus this method allows us to factor our the mean pitch proximity in the distributions of various harmonic intervals.[4] [4] A slight complication arises from the fact that, in imitative polyphonic, entries tend to occur at the intervals of the fourth, fifth, or octave. This means that at certain angles in the autophase, the parts will be shifted so that two previously asynchronous entries are now aligned. A this angle, the tally of various harmonic intervals will include a plethora of fourths (fifths, or octaves) due to the parallel melodic contour shared by the parts. In contrasting these ostensibly controlled results with the original interval distribution, the original scores will appear to be comparatively devoid of concurrent fourths (fifths, or ocatves) thus confounding our results. In order to determine the magnitude of this confound, an analysis of three works was done in which the autophase values for angles displaying synchronized entries were excluded. The corresponding Z-scores for the perfect consonants did rise slightly (by about 3%) when th synchronized entries were omitted. The reason that the change is so small is that, in the autophase method, the original distribution is compared with several hundred controlled distributions. Each Z-score is calculated with respect to a distribution containing several hundred phase-shifted values. In short, values arising when voice-entries are in-phase tend to be swamped by data for all other phase values. Thus, the confounding effect of synchronized entries was deemed to be insignificant.

In light of this new analysis method, our two initial hypotheses can be reformulated as follows. Hypothesis 3: Compared with a pitch-proximity-controlled distribution of intervals, the frequency of occurrence of an interval is positively correlated with its degree of tonal consonance. Hypothesis 4: Compared with a pitch-proximity-controlled distribution of intervals, the frequency of occurrence of an interval (in polyphonic works) is negatively correlated with the degree to which that interval promotes tonal fusion.

The autophase method does not eliminate the effect of the tessitura of the intervals: the data remain confounded by the absolute-pitch placement of the intervals. Nevertheless, in eliminating the effect of intervoice pitch proximity, one of the major confounds (which earlier led us to dismiss our initial repertoire sample) has been removed. For this reason, we need no longer restrict the analysis to the upper two voices of the three-part Sinfonias. As long as we bear in mind the continued confounding effect of interval tessitura, we can profitably expand our analysis to include the entire analytic sample of 30 polyphonic works.

Using this approach, we calculated auto-phase functions for the 15 two-part Inventions and for all three voice-pairings in the Sinfonias: soprano-mid, bass-mid, and soprano-bass. An aggregate (controlled) distribution was generated for each interval in each voice-pairing for each work. These distributions were subsequently standardized, and the Z-score was determined for the actual occurrence of various intervals in the given voice-pairing for individual works. Average Z-socres were then calculated for each interval for all voice-pairs in the sampled repertoire. A positive Z-score indicates that the interval is promoted or encouraged by the composer, whereas a negative Z-score indicates that the interval is avoided or suppressed. Z-score values near zero indicate that the interval is neither more common nor less common than would be expected in a chance juxtaposition of voices.

Plotting the mean Z-scores for the different intervals provides a good contrast to the interval histograms in Figures 2 and 3. It is tempting to assume that a frequency distribution of intervals (e.g., Figures 2 & 3) represents Bach's intervallic "preferences." However, it is a version of the Naturalist Fallacy to assume that the most common is the most preferred. (One's favorite food is not necessarily the food that one east most often.) By contrast, Z-scores indicate how a given interval fares with respect to a chance distribution. Intervals exhibiting high positive Z-scores may be deemed "preferred" in the sense that the are "sought-after" -- and this fact is independent of the actual prevalence of the interval.

MORE

Fig. 5a. Harmonic interval preference for Two--part Inventions.

Fig. 5b. Harmonic interval preference for three-part Sinfonias (soprano-mid voices).

Fig. 5c. Harmonic interval preference for three-part Sinfonias (bass-mid voices).

Fig. 5d. Harmonic interval preference for three-part Sinfonias (bass-soprano voices).

Interval Prevalence and Preference in Bach's Inventions and Sinfonias

A synopsis of the analytic results is presented in Tables 1 and 2. Table 1 tabulates the correlation coefficients relating Bach's intervallic practice to published data for tonal consonance; Table 2 tabulates the correlations pertaining to tonal fusion research. Insofar as Bach endeavors to pursue consonant intervals, the coefficient values in Table 1 ought to be predominantly positive. Conversely, to the extent that Bach avoids intervals that promote tonal fusion, the values in Table 2 ought to be predominantly negative. Both tables provide separate correlations for each voice-pairing in each of the repertoires studied. In the case of Table 1, upper-row values indicate the correlations with Kaestner's data (1909), whereas lower-row values indicate the correlations with data from Kameoka and Kuriyagawa (1969b). Values in parentheses give correlation coefficients calculated by using all the interval data. Table 1 values not in parentheses give correlation coefficients for all intervals except the perfect intervals: P1, P4, P5, and P8.

In Table 2, the upper-row values in each group indicate the correlations with data from Stumpf (1890), whereas the lower-row values indicate the correlations with data from DeWitt and Crowder (1987). Both Tables 1 and 2 provide separate columns for analyses of interval prevalence (i.e., interval frequency) and pitch proximity-controlled interval preference (i.e., interval Z-score).

In interpreting the results of these two tables, two points are appropriate. First, we should give greatest credence to the pitch proximity-controlled data (i.e., interval "preference"). Second, in the case of the tonal consonance correlations (Table 1), we ought to give greater credence to the data for the soprano/mid-voice pair in the three-part Sinfonias -- because the pitch region of the intervals more nearly approximates that of the independent data -- especially the data from Kaestner. The mean lower pitches in each of the other voice-pairings in the musical sample are significantly lower than the comparison data, and so other voice-pair comparisons are likely to be less reliable.

Hypothesis 1 would predict predominantly positive values for r in column 1 of Table 1, consistent with the pursuit of consonant intervals. Hypothesis 2 would predict predominantly negative values for r in column 1 of Table 2, consistent with the avoidance of tonal fusion. Hypothesis 3 would predict predominantly positive values for r in column 3 of Table 1, consistent with the pitch proximity-controlled pursuit of consonance. Hypothesis 4 would predict predominantly negative values for r in column 2 of Table 2, consistent with the pitch proximity-controlled avoidance of tonal fusion. Hypotheses 1 and 2 would concurrently predict large values for r in column 2 of Table 1, and that these values should be greater than the corresponding values for column 1; the data are consistent with the mutual pursuit of tonal consonance while avoiding tonal fusion. Hypotheses 3 and 4 would concurrently predict large values of r in column 4 of Table 1, and that these values should be greater than the corresponding values for column 3. Finally, we would expect that the tonal consonance correlations (Table 1) for the mid-soprano voice-pair would be the highest of all the table values, because this voice-pair corresponds best to the interval tessitura of the independent data.

TABLE 1

TABLE 2

As can be seen, the analysis results are consistent with all of the predictions arising from the hypotheses. Compared with a random distribution, the most avoided interval is the unison, followed by the octave, followed by the perfect fifth. Bach's evident suppression of these intervals suggests that he does indeed choose intervals in inverse proportion to the degree to which they promote tonal fusion. At the same time, Bach chooses intervals in a manner consistent with the pursuit of tonal consonance. Moreover, the data are most consistent, not with one or another of the hypotheses, but with the concurrent pursuit of tonal consonance and avoidance of tonal fusion.

Multiple Regression Analysis

A multiple regression analysis would seem to provide the most appropriate type of statistical test, given the nature of this study. Multiple regression enables us to identify the degree to which two or more independent measures are able to predict the behavior of a dependent measure. In order to pursue such an analysis, data must be available for all of the independent measures, plus the dependent measure. Unfortunately, between the tonal consonance and tonal fusion measures there are a considerable number of missing observations. This means that more than half of the interval classes studied must be omitted from a multiple regression analysis -- casting some doubt about the merit of such an approach. Nevertheless, there is some utility in such an analysis because it can help establish the relative importance of tonal fusion and tonal consconance as compositional goals.

Four sets of analyses were carried out using each of the four combinations of tonal fusion and tonal consonance data (Kaestner/Stumpf, Kaestner/DeWitt & Crowder, Kameoka & Kuriyagawa/Stumpf, Kameoka & Kuriyagawa/DeWitt & Crowder). The interval prevalence and preference data for all voice pairs in the Inventions and Sinfonias provided the dependent variables, resulting in 32 individual analyses. For most of these analyses, data pertaining to only six intervals could be used; hence the probability of achieving statistically significant results are reduced.

Although many analyses hovered near the 95% confidence level, only two analyses were found to give statistically significant results. Both analyses pertained to the soprano-mid-voice combination in the three-part Sinfonias. In using the Kaestner and Stumpf data to predict interval prevalence, R² was determined to be .8933 (F=16.74; p=.0014). In this case the Stumpf tonal fusion data were found to account for 56.4% of the observed variance, whereas the Kaestner consonance data were found to account for a further 33.0% of the variance. In using the Kaestner and DeWitt and Crowder data to predict interval prevalence, R² was determined to be .8720 (F=10.22; p=.0458). In this latter analysis, the DeWitt and Crowder tonal fusion data were found to account for 41.3% of the observed variance, whereas the Kaestner consonance data were found to account for a further 45.9% of the variance. In general, these results imply that in Bach's choice of harmonic intervals, the avoidance of tonal fusion is about equally important to pursuing tonal consonance.

Conclusion

An analysis of 30 polyphonic keyboard works by J.S. Bach suggests that the choice of harmonic intervals is governed by two predominant goals: (1) the pursuit of tonal consonance and (2) the avoidance of tonal fusion. Specifically, the prevalence of intervals (other than perfect consonances) is correlated with their degree of tonal consonance. In the case of the perfect consonances, the prevalence of an interval is inversely correlated with the interval's propensity to promote tonal fusion. Bach endeavors to minimize the occurrence of those intervals that most promote tonal fusion while concurrently pursuing tonal consonance.

These results reinforce the view that Stumpf was mistaken in regarding the phenomenon of tonal fusion as the source or cause of tonal consonance. As Bregman (1990, p. 508) has noted, Stumpf failed to distinguish properly between "heard as one" and "heard as smooth." It would appear that in his polyphonic compositions, Bach attempted to produce music that is "heard as smooth" without being "heard as one."

The avoidance of tonal fusion is in accord with other polyphonic practices used by Bach: Bach avoids inner voice entries (Huron & Fantini, 1989) and also avoids part-crossing (Huron, 1991). Together, these results suggest a significant agreement between compositional practices in polyphonic music and empirical research concerning the segregation of auditory streams.[5]

Footnotes

[1]

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