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PERCEPTION OF TONAL DISSONANCE IN PURE-TONE COMPOUND INTERVALS

Sang Yeop Kwak and Suk Won Yi

Department of Musicology, Seoul National University

  

... . Introduction

Most authors of today seem to acknowledge the statement, "no roughness is produced by pure-tone pairs exceeding critical band" which Terhardt(1974a) insisted. He indicated that V-shaped curve presented by Plomp and Levelt(1965) and Kameoka and Kuriyagawa(1969a, 1969b) exhibits no singular points corresponding to simple frequency ratios. In addition, he said that "hence it must be concluded that frequency distance rather than frequency ratio is the decisive parameter of the consonance of pure-tone intervals". However, there must be some corrections in Terhardt's statement for the following five reasons:

1 Past experimental studies on the tonal consonance and dissonance were mostly limited to simple intervals;

2 In spite of pure-tone pairs exceeding critical bandwidth, roughness can be produced by aural harmonics(harmonic distortion) in mistuned consonance(1:2, 1:3);

3 The direct interaction model presented by Plomp's study(1967) on the beats of mistuned consonances needs to be corrected because it lacks the comprehension of the different meaning between the threshold of pitch perception and of loudness perception;

4 As an appropriate traveling wave model on pure-tone pairs evoking roughness, composite model which is compromised between Plomp's direct interaction model and Clack's aural harmonics model must be considered;

5 Aural harmonics are significant parameters in the perception of tonal dissonance.

 

.... Tonal dissonance in pure-tone pairs

1. Review on the experimental results by Plomp and Levelt(1965)

The tonal dissonance of pure-tone pairs is closely related to the frequency difference between two pure tones. To clarify the assumption that the tonal dissonances of pure-tone pairs depend upon frequency difference, Plomp and Levelt performed the following experiments.

Fig. 1. Plomp and Levelt's experimental results of tonal consonance and dissonance perception for a number of pure-tone pairs. The abscissa indicates the frequency difference of two pure tones, and the ordinate indicates the degree of tonal consonance and dissonance based upon the 7-point scale. The musical notes and arrows above each graphs are by the author. (Adapted from Plomp & Levelt, 1965.) (Critical bandwidth(CB): 125-CB 100, 250-CB 100, 500-CB 115, 1000-CB 160, 2000-CB 300)

As predicted by Plomp and Levelt, the results in the graphs above might be shown as good findings reflecting their assumption that the tonal dissonances of pure-tone pairs depend upon the frequency difference. However, if the fine aspects of the above graphs are considered, several questions can rise relating to such a conclusion.

' Paradoxes of Plomp and Levelt's experiments

Their experiments adopted SPL(sound pressure level, unit: ) as intensity scale, one of several sound intensity scales. As a physical scale, SPL is different from loudness level(unit: phon) which is a sensational scale. In the case of frequency difference 240 in graph a, for example, the subjects come to hear the tone pair in which the higher tone is much louder than the lower tone. To put it concretely, if the frequency difference of a tone pair holding the geometric mean frequency 125 becomes 240, the frequency of lower tone is 53 and higher tone, 293. Because the intensity of tone pairs used in their experiments was 65(SPL), consequently, the lower tone 53 65(SPL) has about 35phons loudness level, and the higher tone 293 65, about 62phons(Fastl, Jaroszewski, Schorer, and Zwicker, 1990). In this situation, if the loudness level of the lower tone, 35phons is increased to 62phons, the resulting degree of tonal dissonance ends up increasing(Rakowski, Miskiewicz, and Rosciszewska, 1998).

The number of tone pairs used in each of their experiments was 121/414 pairs. Considering the position of dots on the graphs, in the context of musical intervals, it shows that simple intervals were adopted far more than compound intervals, i.e., the number of simple intervals were 101/412 pairs while the compound intervals, 21/44. Moreover, about a half of the total number of tone pairs used in their experiments were simple intervals within the critical bandwidth. The logarithmic coordinates, which have the geometric mean frequency as a constant and the frequency differences as a variable, are useful for comprehensively observing the aspects of consonance and dissonance between the tone pairs. However, as shown in Plomp and Levelt's graphs, the simple interval-based logarithmic coordinates exaggeratively reveal only the results of tone pairs which have relatively small frequency differences. In the case of the tone pairs within critical bandwidth, the aspect of results gets even more exaggerated. Based upon this reason, while the logarithmic coordinates by Plomp and Levelt are available for investigating the aspects of consonance and dissonance between tone pairs within the critical band, they are invalid for considering the fine aspects between tone pairs which have relatively large frequency differences, e.g., the compound intervals(sign of a quadrangle in the graphs).

 

' Graphs a(125) and e(2000)

The results of graphs a and e are well coincided with their assumption that the degree of dissonance is closely related to the frequency differences of pure-tone pairs. There are no singular points on the solid lines. As the reasons of such fact, the following cue could be taken: the frequency and loudness relation between aural harmonics of the lower tone, nf1 and the higher tone f2.

Depending on the increase of the frequency differences, the higher tone f2 can be interfered with the larger multiple aural harmonics of the lower tone f1. However, to evoke the dissonance sensation or a considerable roughness, it is required that the loudness levels of f1 and f2 have to be similar to some extent. In graph a, loudness level of the lower tone f1 gradually reduces depending on the frequency decrease. Thus, the loudness level of its aural harmonics nf1 is reduced too. On the contrary, the loudness level of the higher tone f2 gradually increases depending on the frequency increase. After all, in graph a, the difference between loudness level of nf1 and f2 gradually increases based upon the increase of frequency difference between f1 and f2, in which roughness as the cue of dissonance sensation is difficult to be produced.

In graph e which has rather high geometric mean frequency, likewise, no aural harmonics of the lower tone produce a considerable interference with the higher tone f2. For example, in 1500 frequency difference on graph e, each of f1 and f2 is 1386 and 2886. If it is possible to produce maximal roughness between 2886(f2) and 2772(2f1, i.e., second aural harmonic of f1), the loudness levels of both tones have to be same. However, in graph e, while the loudness level of f2(2886) is about 70phons, that of aural harmonic 2f1(2772) is no more than 20phons(Clack, 1972; Fastl et al., 1990)

' Graphs b(250), c(500), and d(1000)

It seems that the aspects of graphs b, c, and d are slightly different from that of graphs a and e. In common, also in graphs b, c, and d, the monotonous V-shaped curve aspect where the frequency differences of pure-tone pairs are within critical bandwidth, are broken after the point of frequency difference exceeding critical bandwidth.

In graph b, the first peak of consonance is about at 53 frequency difference. At this point, the higher tone is about 278 and the lower tone, 225(278/225 5/4). Now, look at the point of frequency difference 210. At this point, the higher tone is about 376 and the lower tone, 166(376/166 9/4). In terms of musical intervals, the former is major 3rd and the latter, major 9th(or major compound 2nd). Based upon the 7-point in graphs(ordinate), major 9th in the geometric mean frequency 250 is more dissonant than major 3rd, though it has a larger frequency difference. This aspect appears also in graph d.

In graph d, the first peak of consonance is about at 165 frequency difference, in which the higher tone is 1086 and the lower tone, 921(1086/921 20/17). This tone pair is a slightly narrower interval than minor 3rd(6/5), and its consonance value in the graph indicates about 6.5. Now, look at the point of frequency differences, 330 and 660. In the frequency difference 330, each of the higher and lower tone is 1179 and 849(1179/849 7/5), and in 660, each tone is 1383 and 723(1383/723 15/8). Approximately, the former is augmented 4th and the latter major 7th, in which each of the consonance values is about 5.55 and 5.75. In the geometric mean frequency 1000, the consonance values for the frequency differences, 165, 330, and 660 are in turn 6.5, 5.55, and 5.75., by which is shown that there is almost no relation between the aspects of tonal consonance and dissonance of the tone pairs exceeding critical band and the frequency difference.

Also in graph c, it is shown that there is no relationship between the aspect of consonance and dissonance and the frequency difference after the frequency difference 105. If more number of tone pairs exceeding critical bandwidth had been used in the experiments, such an aspect would have been shown more clearly. Anyhow, all of the graphs b, c, and d support that the V-shaped curve aspect by Plomp and Levelt is valid only in the tone pairs within the critical bandwidth.

 

2. The results of Kameoka and Kuriyagawa(1969a, 1969b)

An extensive experimental study of tonal consonance and dissonance for pure tone and complex tone was performed by Kameoka and Kuriyagawa(1969a, 1969b). In their notable two papers, there are several conclusions as follows:

-Dyads consisting of two partials with equal sound pressure level show simple V-curve characteristics.

-Over an octave separation, the interaction between two partials becomes negligibly small.

-For a given level difference, a dyad with a spectrum form (L1>L2) is more dissonant than that with its opposite form (L1<L2). This asymmetrical property is explained by the pure tone masking and nervous response patterns.










' V-curve

As shown in Fig. 1, the degree of dissonance for pure tone pairs decisively depends upon the frequency difference. Especially, the maximal dissonance of two pure tones was produced at the frequency difference corresponding to the position of about a quarter of critical bandwidth. After this position, the consonance value smoothly increases and almost recovers at the frequency difference corresponding to critical bandwidth. This aspect is also presented by Kameoka and Kuriyagawa(1969a, 1969b). However, there are some differences between Fig. 1 by Plomp and Levelt(1965) and graphs by Kameoka and Kuriyagawa(1969a). These differences are very important for considering discrepancy between both of the authors' experiments. At first, in comparison with the graphs by Plomp and Levelt(Fig. 1), the graphs by Kameoka and Kuriyagawa are more effective in investigating the aspect of consonance and dissonance between the pure-tone pairs within the critical band. They adopted the frequency deviation rate as a parameter to acquire the perfect V-shaped curve, and in addition, applied the simple interval-based logarithmic scale to their graphs. Thereby, it seems that they could systematically confute the 25% theory of the maximal dissonance by Plomp and Levelt(1965). According to Kameoka and Kuriyagawa, the most dissonant frequency difference varies with the frequency range. It also varies with the sound pressure level of two pure tones.

Above all, however, one of the most interesting aspects in graphs between both of authors is found at the range of compound intervals whose aspects of consonance and dissonance are apparently opposite each other. In graph by Kameoka and Kuriyagawa(1969a) the transition aspect of consonance and dissonance is indicated over three octaves. According to this graph, the maximal dissonance in f1=440 is produced at the interval consisting of f1 440 and f2 484, where the frequency deviation rate is 10%. This result agrees well with that of graph e in Fig. 1. However, compare the aspect of consonance and dissonance for compound intervals in graph by Kameoka and Kuriyagawa(1969a) with that of Fig. 1. In Kameoka and Kuriyagawa's graph, the aspect of compound intervals is a V-shaped curve, whereas in Fig. 1, it is a Λ-shaped curve!

' The change of dissonance degree depending on the level difference between two pure tones

Through their extensive experimental study, Kameoka and Kuriyagawa proved the fact that dissonance degree can be systematically changed by the level difference between two pure tones. When the level of lower tone(f1) is L1 and that of higher tone(f2), L2, the tone pair with a spectrum form L1>L2 is more dissonant than that with the opposite spectrum form L1<L2. With regard to the reason why the spectrum form L1>L2 is more dissonant than the opposite form L1<L2, Kameoka and Kuriyagawa suggested that such an asymmetrical property is explained by the pure-tone masking effect and nervous response patterns, but these cues seem to be somewhat obscure.

The magnitude of roughness produced between two pure tones, f1 and f2 attains to maximal value when the levels of f1 and f2 are same(Terhardt, 1974b). On the contrary, the conclusion by Kameoka and Kuriyagawa shows that L1>L2 is more dissonant than L1=L2. This fact is in conflict with the existing experimental results on the relation between amplitude modulation and the magnitude of roughness. However, if it is considered that two pure tones within critical band are not f1 and f2 but nf1(aural harmonics of f1) and f2, their conclusion is correct. The loudness difference between nf1 and f2 in L1>L2 is smaller than that in L1=L2. If nf1 and f2 are within the critical band, a pure-tone f2 causes roughness through the interference(or intermodulation) with not f1 but nf1. Therefore, with respect to the existing experimental results for the relationship between amplitude modulation and the magnitude of roughness, the magnitude of roughness between nf1 and f2 in L1>L2 is larger than that in L1=L2.

 

3. The assumption of Terhardt(1974a, 1974b, 1976)

According to Terhardt's definition, psychoacoustic consonance is the undisturbed simultaneous sounding of pure tones(1974a). In this context, a pure-tone interval, perfect 5th at the lower frequency range can be a psychoacoustic dissonance. On the contrary, a pure-tone interval, major 7th which is considered to be dissonant in music theory, can be considered as a consonance in the concept of psychoacoustic consonance. In a pure-tone interval, major 7th, no roughness as a cue of tonal dissonance is produced since two pure tones exceed the critical band.

"In music theory, an interval with a more complex frequency ratio is considered to be dissonant even in the cases where no roughness is produced, e.g., in the case of a pure-tone pair with a frequency distance exceeding critical bandwidth"(1974a).

According to the critical band theory, roughness as a cue of psychoacoustic dissonance should not be produced at the pure-tone intervals exceeding critical bandwidth. However, as shown in Fig. 1, there is a certain systematic aspect of incomplete dissonance in spite of the compound interval exceeding the critical band. This is an indirect evidence for the fact that even when two pure tones exceed the critical band, roughness can be produced by intermodulation with aural harmonics. The assumption of Terhardt(1974a) such as "in the cases of a pure-tone pair exceeding critical bandwidth, no roughness is produced" could be right or wrong according to the following premises. If his assumption was involving adequate recognitions for the systematic existence of aural harmonics which are physiological products, and for the assumption that an aural harmonic could be a pure tone, it is not a wrong assumption. However, if his assumption was overlooking the existence of aural harmonics because of focusing only the maximal dissonance phenomena, it ends up to be incorrect assumption in which the systematic aspect of 'incomplete dissonance' e.g., for pure-tone compound intervals is unfairly excluded.

 

 

.... Experiments

 

1. Purpose

This study investigates the aspects of incomplete dissonance of pure-tone pairs, especially, pure-tone compound intervals(CIs) exceeding critical bandwidth. Because the past researches are mostly limited to simple intervals(SIs), the experimental approach to pure-tone CIs is very suggestive. To verify the fact that aural harmonics affect the perception of incomplete tonal dissonance in pure-tone CIs, it is essential to make a thorough investigation of the experimental results in both of the same and different loudness conditions for several intervals such as augmented 4th, minor 6th, major 7th, minor 9th(compound 2nd), augmented 11th(compound 4th), and minor 13th(compound 6th).

 

2. Method

Through experiment 1(L1=55, L2=55) and 2(L1=55, L2=40), the total 3978 intervals were randomly presented to the 13 subjects(6 males, 7 females, average age=26.5). The subjects were chosen regardless of the degree of musical training. The frequency of invariable lower tone f1 was 262, and the variable higher tone f2 included 24 tones which are in an equal temperament scale within the distance of two octaves from the lower tone 262. Based upon 7-point scale, each subjects had to judge the degree of tonal consonance and dissonance of presented pure-tone intervals. The total number of intervals presented to an individual subject was 306(experiment 1=153, experiment 2=153). The intervals were presented in mono to both ears through the headphone and pure-tone stimuli were generated by a sine wave generator in E-synth(E-mu Systems). Each of the experiment 1 and 2 consists of three tasks varying in the interval types. In task A, 13 SIs were randomly presented three times(total 39 SIs). In task B, 13 CIs were presented as in task A(total 39 CIs). In task C, 25 intervals within two octaves(the fundamental 262, C4) were presented in the same method(total 75 SIs&CIs).

 

3. Results

' In the condition of L1=55, L2=40, augmented 11th(compound tritone) was perceived to be more dissonant than tritone(F(1,12)=14.74, P<.01).

' In L1=55, L2=40, minor 13th(minor compound 6th) was perceived to be more dissonant than minor 6th(F(1,12)=8.14, P<.025).

' Augmented 11th in L1=55, L2=40 was perceived to be more dissonant than that in L1=55, L2=55(F(1,12)=9.90, P<.01).

' Minor 13th in L1=55, L2=40 was perceived to be more dissonant than that in L1=55, L2=55(F(1,12)=12.28, P<.01).

The above results are interesting in the historical context with relation to whether or not aural harmonics exist. Incomplete dissonance in the CIs can be discussed in parallel with the problem of beat and roughness sensation in the mistuned consonances. Paradoxically, past studies for beats of mistuned consonances(Tonndorf, 1959; Plomp, 1967) suggest that beats can be produced by pure-tone pairs exceeding the critical band depending on the frequency ratio between two pure tones. Because, physically, beat is closely related with roughness, a lengthy discussion for the results ' , ', ', ' presented by this study would be inevitable when relating to the following three subjects: the origin of beats of mistuned consonances, Clack's experiments for aural harmonics, and the aspects of traveling wave in basilar membrane by pure-tone pairs exceeding critical band.

 

.... Discussion

 

1. The origin of beats of mistuned consonances

Traditionally, the origin of beats of mistuned consonances has been considered to be aural harmonics produced in auditory system(Stevens & Davis, 1938). In 1967, however, Plomp argued that aural harmonics are not the fundamental evidence for beats of mistuned consonances. In Plomp's paper(1967), three evidences are suggested as the reason for aural harmonics not being the origin of beats of mistuned consonances.

Evidence (a): Beats are produced even in the presence of masking noise for aural harmonics and combination tones. In the experiments for mistuned consonances 200:301 and 500:1001, he presented the broad-band noise, which lacks the frequency bands 200-301 and 500-1001 individually, in order to exclude the possibility for beat by aural harmonics and combination tones.

Evidence (b): The subjects who could not identify aural harmonics in the test of aural harmonics audibility, could hear the beats of mistuned consonances. For example, in the test of aural harmonics audibility for 125(SL 65), the subjects who could not identify aural harmonics, 250, 375, 500, etc., could hear the beat produced between 125(SPL 100) and 251(SPL 90).

Evidence (c): Beats are produced even in the mistuned consonances, 5:9 and 4:9. Traditionally, beats of mistuned consonance types of 1:2, 1:3, 1:4,......1:n have been made out by aural harmonics. On the other hand, beats in 5:9 and 4:9 are difficult to be explained by aural harmonics.

On the basis of these evidences, Plomp concluded that the origin of beats of mistuned consonances is not aural harmonics but a waveform variation(phase interference) by direct interaction between two primary tones(Plomp, 1967, 1976). In the view of the direct interaction theory, it is presumable that roughness could be produced by phase effect depending upon the frequency ratio even when two primary tones exceed the critical band. This fact contradicts with Plomp and Levelt's critical band theory(1965) and Terhardt's assumption(1974a) for the production of roughness, because the generation of phase effect and roughness in pure-tone pairs was depending on the frequency ratio of two primary tones. Also Kameoka and Kuriyagawa's assumption, "for a given level difference, a dyad with a spectrum form (L1>L2) is more dissonant than that with its opposite form (L1<L2)", lacks the consideration of roughness by phase effect depending on the frequency ratio. If they were considering that beats of mistuned consonances have an origin of two tones interaction depending on the frequency ratio, their assumption would be more complicated. Anyway, the studies of tonal dissonance sensation by all the authors above overlook the influences by phase effect or roughness by aural harmonics in pure-tone pairs exceeding the critical band.

Apart from all arguments above, however, one of the purposes of this paper is not to give support to the direct interaction theory that phase effect resulted directly from two primary tones is an origin of beats of mistuned consonances, but to reveal its limit and to indicate an error in Plomp's experiment(1967) for beats of mistuned consonances. Previously, three evidences for direct interaction theory were mentioned as follows: (a) Beats are produced even in the presence of masking noise for aural harmonics and combination tones; (b) The subjects who could not identify aural harmonics in the test of aural harmonics audibility, could hear the beats of mistuned consonances; (c) Beats are produced even in the mistuned consonances, 5:9 and 4:9.

Unlike the listening task of beat, the listening task of aural harmonics is an experiment in that subjects have to identify an individual pitch of aural harmonics separately. To identify an individual pitch of aural harmonics, two different abilities are required. One is an analysis ability of spectral pitch and the other, a perceptibility of loudness. Generally, in the measurement of the threshold of hearing for beats, only the perceptibility of loudness is required for subjects. On the contrary, in the measurement of the threshold of hearing for aural harmonics, both different perceptibilities mentioned above are required for subjects. The subjects have to identify an individual pitch of aural harmonics before perceiving the loudness of an aural harmonic. Such an analysis ability of spectral pitch is similar to a note-tracking ability by which it is possible to identify a particular tone among the 4-voiced tones accompanied by a piano. However, the note-tracking of an individual pitch of aural harmonics is far more difficult than that of 4-voiced tones by a piano because, in the former case, the fundamental note f1 is considerably loud compared with any other aural harmonics. Sensationally, loudness precedes pitch, i.e., the threshold of loudness perception is below that of pitch perception. If χ㏈LL(loudness level) is required for the perception of minimal loudness in a certain tone, the threshold of pitch perception will become at least (χ+α)LL(α>0) in the same tone.

First, Plomp's evidence (a) is nothing more than a clue suggesting the possibility of direct interaction. It can not be an evidence for the assumption that aural harmonics wouldn't be an origin of beats of mistuned consonances. In his experiment, the level of masking noise was adjusted up to the level of aural harmonics. In the condition where a masking noise is presented, subjects are not able to analyze spectral pitch. In spite of this situation, if the subjects could perceive the beats of mistuned consonances, it might be easily considered as that direct interaction is the only origin of beats of mistuned consonances. However, such inference is incorrect. Beat is by an amplitude variation. In the same phase, loudness level by phase effect is maximal and in the opposite phase, minimal. A loudness variation(beat) resulted from the phase relation between two tones can be produced by aural harmonics even if the aural harmonics are masked. Although the subjects couldn't perceive the spectral pitch of aural harmonics masked by a broad-band noise, it is possible enough to perceive the loudness variation by phase effect resulted from the interaction between aural harmonics nf1 and the higher tone f2, or aural harmonics nf1 and nf2. It is because the threshold of loudness perception is below that of pitch perception.

Second, Plomp's evidence (b) also can not be a proper evidence for direct interaction in the same reason. To perceive the spectral pitch of aural harmonics is very difficult in the condition when modulation is present(McAdams, 1982). In Plomp's experiment, a modulation factor interrupting the perception of aural harmonics was beats of mistuned consonances. Strictly speaking, perceiving beat is easier than perceiving an individual pitch of aural harmonics as if a dynamic image is more easily perceived than a static image.

Third, evidence (c) seems to be the most persuasive evidence among the three evidences because beats in 5:9 and 4:9 are difficult to be explained by aural harmonics. However, unfortunately, beats in 5:9 and 4:9 were generated only at the lower tone 125 in his experiment. None of the other cases generated beats in 5:9 and 4:9 mistuned consonances. The fact that beats in 5:9 and 4:9 mistuned consonances were produced only in 125 would be an evidence for aural harmonics rather than direct interaction. In the lower tone 125(SPL 100), 5th, 6th, and even 7th aural harmonic can be perceived easily. In addition, the frequencies of aural harmonics in question, evoking beats of mistuned consonances 5:9 and 4:9, lie in the frequency band(500-4000) at which we are sensitive. Virtually, it seems that beats of 5:9 and 4:9 mistuned consonances in the lower tone 125 have a combined origin of direct interaction, aural harmonics, and combination tones.

 

2. Clack's study of aural harmonics and phase effect

Aural harmonics have been called 'subjective tones' for a long time. Occasionally, they were even considered to be 'fictitious tones'. Many authors of today also regard aural harmonics as 'subjective tones'. On the contrary, authors such as Stevens and Newman(1938), Fletcher(1940), Bekesy(1972), and Clack(1967a, 1967b, 1968, 1971, 1972, 1975, 1977) are representative figures who considered aural harmonics as existent biophysical tones produced by harmonic distortion effect in the inner-ear.

A number of papers on aural harmonics are published by Clack, T. D.. He argued that Plomp's direct interaction theory(1967) supported by Schubert(1969) cannot provide a proper explanation for that the monaural phase effect in tone-on-tone masking was evident only in the case of 1:2(among the ratios, 1:1.5, 1:2, 1:2.5) when f1=1000(Clack, Erderich, and Knighton, 1972).

Clack's experimental results are very persuasive in respect of that phase effect by interference of two primary tones was evident only in the frequency ratio 1:n. In his advanced study in 1977, he observed that also in a lower tone 500(f1), the monaural phase effect in tone-on-tone masking is produced at two higher tones 1000 and 1500. His tone-on-tone masking paradigm was useful to prove the existence of aural harmonics nf1. However, his explanation by the tone-on-tone masking paradigm does not seem to be faultless for revealing that the direct interaction does not become the origin of the phase effect in the frequency ratio 1:n. It is because, in the Plomp's experiment(1967) on beats of mistuned consonances, beats were observed also in the case of 2:3 mistuned consonances(e.g., 125:189, 250:376, 500:751, 1000:1501) although, in Clack's experiment, phase effect in 1500(f2) was not evident when f1=1000. If Clack's assertion were to be right, the reason why beats in 2:3 mistuned consonances were produced in Plomp's experiment should have been fairly elucidated in Clack's study. What is, indeed, the origin of beats in 2:3 mistuned consonances? Is it either direct interaction or aural harmonics? It is explained by the following.

According to Plomp's results, in the presence of the lower tone 1000 at 90, the sound pressure level(SPL) of the higher tone 1501 which produced best beats was approximately 91, whereas SPL of the higher tone 2001 produced best beats at 84. In the 80 lower tone, SPL of the higher tone 1501 which produced best beats was 74, whereas in the higher tone 2001 no beats were produced. According to Clack's results, on the contrary, in the presence of the 90 lower tone 1000, the higher tone 1500 at 71(masked threshold) did not produce systematic phase effect, whereas the higher tone 2000 at 66(masked threshold) produced an evident phase effect. In the 80 lower tone, the higher tone 1500 at 62(masked threshold) did not produce systematic phase effect, whereas the higher tone 2000 at 58(masked threshold) produced an evident phase effect. In both author's cases, the narrow-band noise was presented to mask combination tones. Plomp's experimental results do not seem to be successful in rejecting aural harmonics theory in terms of the following reasons. First, he did not measure the peak level of best beats. If the beat level of 1:2 mistuned consonance was higher than that of 2:3 in the results above, this could rather be an evidence for aural harmonics. Second, masking noise for aural harmonics was not presented. In this condition, the best beat's level of mistuned consonances could be higher than that in the presence of masking noise for aural harmonics. Also Clack's experimental results, when strictly speaking, do not only support aural harmonics theory. It is because a little phase interference between two primary tones in his experiment was produced also in 2:3(1000:1500) although phase effects were not evident. After all, if both author's results are synthetically considered, the origin of beats of mistuned consonances can be regarded as both direct interaction and aural harmonics.

 

3. Incomplete dissonance in pure-tone compound intervals

To investigate the incomplete dissonance aspect in pure-tone pairs exceeding the critical band, it needs to observe the results of the particular intervals; augmented 4th, minor 6th, major 7th, minor 9th(compound 2nd), augmented 11th(compound 4th), and minor 13th(compound 6th).

Fig. 2. Average results for tonal dissonance values of simple intervals(SIs) and compound intervals(CIs) through experiment 1(L1=55, L2=55) and 2(L1=55, L2=40). The frequency of invariable lower tone f1 was 262. Tonal dissonance values of SIs and CIs in experiment 1(L1=L2) are shown in graph a and that of experiment 2(L1>L2) in graph b. The shift of CIs' dissonance values in experiment 2 are observed in graph c. In graph d, the differences between the values of SIs in experiment 1 and CIs in experiment 2 are shown. Especially, the differences of augmented 4th and 11th, and minor 6th and 13th are noticeable.

 

Four graphs in Fig. 2 show the dissonance aspects of SIs and CIs in experiment 1 and 2. First, observe the difference of dissonance aspects in L1=L2=55(graph a) and in L1=55, L2=40(graph b). The noticeable shifts are found at major 7th, minor 9th, augmented 11th, and minor 13th. In addition, it is found that augmented 11th and minor 13th are more dissonant than augmented 4th and minor 6th in both graphs a and b. Especially, the differences of augmented 4th and 11th, and minor 6th and 13th are significant in graphs b and d.

An evidence for that the differences of augmented 4th and 11th, and minor 6th and 13th can be explained by both theories of direct interaction and aural harmonics is shown in graph c. The findings that augmented 11th and minor 13th in experiment 2(L1>L2) were more dissonant than those of experiment 1(L1=L2), suggest that the more perceivable roughness could be generated by the direct interaction and aural harmonics in the case of L1>L2.

Plomp presented an illustration of interference of the vibration patterns along the basilar membrane produced by two pure tones in his paper(1967). His model was shown as an example a in Fig. 3. However, when it is considered in view of such classical scholars as Stevens and Newman(1938), Fletcher(1940), Bekesy(1972) including Clack(1967a, 1967b, 1968, 1971, 1972, 1975, 1977), a different model as an example b in Fig. 3 can be presented.

 

Fig. 3. Illustrations of the vibration patterns along the basilar membrane produced by a pure-tone interval, minor 9th. A direct interaction model presented by Plomp is shown in a, and an aural harmonics model in b. According to critical band model, Plomp's direct interaction model contradicts his own critical band model in the cases of a higher intense lower tone and the low frequency range below 1000. Model d is a compromised model between the models of direct interaction and aural harmonics.

 

 

 

 

Now, we need to consider a problem related to the critical band model presented by Plomp and Levelt(1965). The V-shaped aspect is evident only in the relatively lower intensities, and have to be limited only in the range of a single critical band. Paradoxically, an application of Plomp's critical band model in the higher intensities of a lower tone or for intervals exceeding critical band is invalidated by his own direct interaction model. His direct interaction model predicts essentially not that "consonance and dissonance of pure-tone pairs depend on frequency difference rather than frequency ratio(Terhardt, 1974a)" but that "consonance and dissonance of pure-tone pairs depend on both of the frequency difference and the frequency ratio".

An influence of aural harmonics on dissonance perception is not trivial. Particularly, it is more evident in CIs adjacent to 1:n harmonic intervals. If more CIs exceeding critical band were used in the experiments of Plomp and Levelt, and Kameoka and Kuriyagawa, the Terhardt's assumption, "no roughness is produced in a pure-tone pair exceeding the critical band" could not be accomplished. A direct interaction theory and an aural harmonics theory are not essentially in antagonistic relationship. Although aspects of incomplete tonal dissonance in CIs is variable depending on a given level of two pure tones(especially, the lower tone) and frequency range, the results from this study for the lower tone 262 at 55 are explained by a compromised model as presented in model d.

 

 

.... Conclusion

 

In this study, the aspects of incomplete dissonance of pure-tone pairs, especially, pure-tone compound intervals(CIs) exceeding critical bandwidth were investigated. The experimental results suggest several evidences on the fact that V-shaped curve aspect as a basal psychoacoustic concept explaining dissonance mechanisms in pure-tone pairs must be applied only within a single critical band. Furthermore, with regard to the fact that roughness can be produced even when both primary tones of pure-tone pairs exceed the critical band, this study argues that aural harmonics can be crucial cues for the formation of dissonance sensation.

 

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