PERCEPTlON AND iNTERNAL REPRESENTAHON 0F MUSWAL \NTERVALS Thesis for the Degree of M. A. MlCHlGAN STATE UNNERSHY MARY CATHERmE FYDA 1975 ’ u mum; lllzllgljflllll Ln! MI 111 Jill ”mum u f-W' or ‘ ABSTRACT PERCEPTION AND INTERNAL REPRESENTATION OF MUSICAL INTERVALS BY Mary Catherine Fyda Four untrained subjects and four musically trained sub— jects made similarity judgments of 14 musical intervals under five heard conditions, which included simple tone intervals that were pure sinusoids, complex tone intervals consisting of nine harmonics, and truncated tone intervals containing only harmonics between 900 and 1800 Hz. In the attention condition the stimuli contained nine harmonics, but the sub- jects were instructed to attend to only those components above 600 Hz. While all the intervals just described were constructed on a just scale, those in the equal tempered condition were constructed on an equal tempered scale. The four musically trained subjects also participated in the imagined condition, in which they were instructed to imagine the sounds of the intervals, and make similarity judgments between pairs of imagined intervals. Multidimensional scaling of the similarity data re- vealed several factors influencing similarity ratings of in- tervals, including pitch, ratio complexity and interval width. Inversion was also an important factor in the Mary Catherine Fyda truncated and imagined conditions. The relative importance of each of these dimensions varied across subjects, and also depended on the condition. Each subject tended to be con- sistent in his judgments across conditions, and this was particularly true for the musically trained subjects. PERCEPTION AND INTERNAL REPRESENTATION OF MUSICAL INTERVALS BY Mary Catherine Fyda A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS Department of Psychology 1975 To my parents, John and Lela Fyda ii ACKNOWLEDGMENTS I would like to express my appreciation to Dr. Lester Hyman and Dr. Gary Olson for their help in serving as mem- bers of my thesis committee. My special thanks go to Dr. David Wessel, my committee chairman, whose assistance in every phase of this research was invaluable. iii TABLE OF LIST OF TABLES 0 O O O O O O 0 LIST OF FIGURES. . . . . . . . INTRODUCTION . . . . . . . . . METHOD 0 O O O O O O O O O O 0 Subjects. . . . . . . . . Stimuli O O O O O O O O 0 Procedure . . . . . . . . RESULTS 0 O O O O O O O O O O 0 Phase I . . . . . . . . . Phase II. . . . . . . . . DISCUSSION . . . . . . . . . . REFERENCES . . . . . . . . . . CONTENTS iv vi Table LIST OF TABLES Frequencies and frequency ratios of the 14 intervals Harmonics included in each tone for the truncated condition Page LIST OF FIGURES Figure Page 1 Two dimensional MDSCAL solution with embedded l4 STRUCTR clustering for all subject-condition matrices 2 Two dimensional MDSCAL solution with embedded 16 STRUCTR clustering for musically trained sub- jects' subject-condition matrices 3 Two dimensional MDSCAL solution with embedded 18 STRUCTR clustering for untrained subjects' subject-condition matrices 4 Two dimensional INDSCAL stimulus space and 19 weight space for the simple condition 5 Three dimensional INDSCAL stimulus space for 21 the simple condition 6 Two dimensional INDSCAL stimulus space and 23 weight space for the complex condition 7 Three dimensional INDSCAL stimulus space for 24 the complex condition 8 Two dimensional INDSCAL stimulus space and 25 weight space for the equal tempered condition 9 Three dimensional INDSCAL stimulus space for 26 the equal tempered condition 10 Two dimensional INDSCAL stimulus space and 28 weight space for the imagined condition 11 Two dimensional INDSCAL stimulus space and 29 weight space for the truncated condition 12 Two dimensional INDSCAL stimulus space and 31 weight space for the attention condition 13 Two dimensional INDSCAL stimulus space and 32 weight space for all subjects vi Figure 14 15 16 17 18 Three dimensional INDSCAL stimulus space for all subjects Two dimensional INDSCAL stimulus space and weight space for musically trained subjects Three dimensional INDSCAL stimulus space for musically trained subjects Two dimensional INDSCAL stimulus space and weight space for untrained subjects Three dimensional INDSCAL stimulus space for untrained subjects vii Page 34 35 36 37 38 INTRODUCTION Both perception and imagery involve a correspondence between real world objects and internal representations of these objects. To the extent to which these internal repre- sentations resemble in some way the external events giving rise to them, the perception or imagery is veridical. Shepard (1968; Shepard & Chipman, 1970; Shepard, Kilpatric, 8 Cunningham, 1975) has proposed that a second order rela- tionship exists between external objects and correSponding internal representations. He is not concerned with first order structure which may exist in a single internal repre- sentation considered within itself or solely in relation to its external object. His principle of "second order isomor- phism" involves structure existing in the relations among a set of internal representations, and the parallelism between these relations and relations among the corresponding exter- nal events. To illustrate second order isomorphism, Shepard, Kilpatric, and Cunningham (1975) studied the relationships among similarity data for numbers presented and judged in various forms, including spoken and printed English names, Arabic and Roman numerals, several concrete counting repre— sentations, and the abstract concepts of the integers 2 themselves. They found that similarity ratings were unaf- fected by the form in which the stimuli were presented, but were correlated highly, moderately, or not at all depending upon whether the forms judged in different conditions were identical, structurally related, or unrelated. Multidimen- sional scaling solutions were very similar for conditions judged in the same form, and these solutions were readily interpretable in terms of features appropriate to the form judged. The present study used a similar methodOIOgy to assess the effects of form of presentation on similarity judgments of musical intervals. The purpose of this experiment was to determine the extent to which there is second order isomor- phism between the representations of perception and imagery of intervals, and the extent to which these relations paral- lel physical prOperties of the stimuli. Levelt, Van de Geer and Plomp (1966) had subjects judge the similarity of harmonic tone intervals by the method of triadic comparisons. Stimuli were of two kinds: intervals consisting of two simultaneous simple tones (fundamental only) and intervals consisting of two simultaneous complex tones (fundamental plus harmonics). Multidimensional scal- ing solutions for both conditions showed a common horseshoe- 1ike structure with narrow intervals at the left lower end and wide intervals at the right lower end. The authors sug- gested that this type of structure results from the presence of a reference point, with narrow and wide intervals sharing 3 the common characteristic of extremity from this norm. A third dimension found for the complex tone intervals seemed to order intervals on the basis of their ratio complexity. Unfortunately the explanations proposed by Levelt, Van de Geer and Plomp are not particularly convincing, especial- ly since the horseshoe shaped structure is probably an art- ifact of the scaling program they employed. Shepard (1974) discusses the fact that when one dimensional data are sub- mitted to the MDSCAL program, it attempts to use the extra degrees of freedom to fit any random noise in the data, and a C- or S- shaped curve results. The findings of this study do suggest, however, that interval width and ratio complex- ity may be relevant dimensions in judging interval similar- ity. Earlier work by the same authors (Van de Geer, Levelt & Plomp, 1962) showed that pitch is another dimension which determines a considerable proportion of the variance in judgments about intervals. Some researchers (Sorkin & Pohlmann, 1973; Sorkin, Pohlmann & Gilliom, 1973) have suggested that the basilar membrane is analogous to an array of filters, each of which is maximally sensitive to one particular frequency, and less sensitive to other similar frequencies. In their discussion of the attentional process, they imply that we are capable of "tuning in“ to the particular set of frequencies in which we are interested, and filtering out extraneous frequencies. To test this assumption, the present study included an at- tention condition in which subjects were asked to direct 4 their attention to a particular set of frequencies in the stimuli (those higher in pitch than the reference tone), and ignore other frequencies in making their similarity judg- ments. Another condition (truncated) included only those frequencies to which the subjects were asked to attend in the attention condition. If the subjects were indeed able to focus their attention, multidimensional scaling solutions should be very similar for these two conditions. Another factor which might play a role in similarity judgments of intervals is the tuning or temperament in which the intervals are constructed. Just intervals are those in which there is an integer ratio between the frequencies of the two tones comprising the intervals. Nearly all studies dealing with musical intervals have employed this type of interval. But the modern piano is not tuned to a just scale but rather an equal tempered scale. In equal temperament, the ratio between successive frequencies of the twelve tones of the scale is a constant: the twelfth root of two. Since this is an irrational number, none of the frequency ratios for intervals constructed on an equal tempered scale is an integer ratio. Each interval (except the octave) is slight- 1y 'mistuned', with the fourths and fifths least distorted. If ratio complexity is an important dimension, as Levelt, Van de Geer and Plomp (1966) suggested, the scaling solutions for equal tempered and just intervals might look quite dis- similar. To test this possibility, an equal tempered con- dition was included in the present study, as well as several 5 conditions using just intervals. Levelt, Van de Geer and Plomp chose frequencies for their stimuli so that for each interval the mean value was 500 Hz. This was done in order to eliminate pitch as a source of variation. The result of this manipulation was to create intervals similar to those heard in the piano tune "ChOpsticks". Since most music is constructed on a scale, these “chopsticks” intervals are artificial and unfamiliar. Therefore in the present experiment all intervals were con- structed within an octave, and on the same scale. This nec- essarily meant that pitch would still be present as a source of variation, but this was not considered a problem since other dimensions could still appear in the solutions as well. The seconds and sevenths, thirds and sixths, and fourths and fifths were constructed so that they were inversions of each other. This was done because inversion is another possible factor in rating the similarity of intervals. Intervals in the imagined condition were the same as those in the beard conditions, except that instead of being played on tape, they were written on paper, and the subjects were asked to imagine how they would sound. It was expected that the scaling solution for this condition would be most similar to that of the complex or equal tempered condition, since intervals in these two conditions are most similar to those frequently heard in music. In summary, subjects were asked to rate similarity of intervals under five heard conditions: simple, complex, 6 equal tempered, truncated, and attention, and also to rate the similarity of the intervals as they imagined them. It was expected that some of the factors influencing these judgments would be pitch, interval width, ratio complexity, tuning and temperament, and inversion. METHOD Subjects Eight undergraduate students attending Michigan State University volunteered to serve as subjects and were paid $2.50 per hour for their participation. The untrained group consisted of four students who had had no formal training in interval recognition. Four music majors who had completed at least one course in ear training constituted the musical- ly trained group. A fifth subject in this group did not complete the experiment because of inability to attend some of the experimental sessions; his data are not included in the analyses. Each group included two males and two females. Stimuli The stimuli consisted of 14 harmonic intervals, includ— ing the 12 musical intervals within the octave, with the in- tervals of perfect fourth and perfect fifth represented twice. For all conditions except the equal tempered condi- tion, the intervals were constructed on the just scale, i.e. there was a fixed integer ratio between the frequencies of the two tones comprising each interval. In the equal temp- ered condition, the intervals were constructed on a twelve tone equal tempered scale. All intervals lay in the octave 300—600 Hz., and were constructed so that the inversion of 8 each interval in the set was also in the set (except for the octave and the tritone). Table 1 lists the frequencies and frequency ratios of the stimuli used. The stimuli were constructed using a CDC 6500 computer and the MSU MUSIC program developed by Mark Dionne. An ex- tension of Max Mathews' MUSIC V, this program allows music- al sounds to be generated with Specified frequencies, in- tensities, and waveforms. Three types of waveform were used in this experiment. In the simple condition, both tones comprising each interval were pure sinusoids. Each tone in the complex, equal tempered, and attention conditions includ- ed nine harmonics, with the intensity of successive harmon- ics decreasing by 6 db per octave for the first four harmon- ics, and 12 db per octave for the last five harmonics. In the truncated condition, the intensity of successive harmonics decreased at this same rate, but only harmonics between 900 and 1800 Hz. were included. Thus the number of harmonics comprising each tone in the truncated condition depended on the frequency of the tone. Table 2 lists the harmonics in- cluded for each tone in this condition. The attention condition utilized the same stimuli as the complex condition, except that a constant 600 Hz. refer- ence tone was added to the attention condition tape on an- other channel. The intensity of this reference tone was ad- justed so that it was audible, but much softer than the in- tervals. Within each of the experimental conditions, there were Frequencies 9 Table l and frequency ratios of the 14 intervals Musical name Equal tempered condition Frequencies (Hz.) Other conditions Frequencies (Hz.) Ratios major second major third perfect fourth tritone perfect fifth major sixth major seventh octave minor seventh minor sixth perfect fifth perfect fourth minor third minor second 300.0000 300.0000 300.0000 300.0000 300.0000 300.0000 300.0000 300.0000 336.7386 377.9763 400.4520 449.4921 504.5378 566.3246 336.7386 377.9763 400.4520 424.2641 449.4921 504.5378 566.3246 600.0000 600.0000 600.0000 600.0000 600.0000 600.0000 600.0000 Table 2 300.0 300.0 300.0 300.0 300.0 300.0 300.0 300.0 337.5 375.0 400.0 450.0 500.0 562.5 337.5 375.0 400.0 420.0 450.0 500.0 562.5 600.0 600.0 600.0 600.0 600.0 600.0 600.0 H mmuwmsowoowwwwnoo Hmbme-‘NHUIWNubU'ikp U1 0‘ 0‘ Harmonics included in each tone for the truncated condition Frequency (32.) Harmonic numbers 300.0 337.5 375.0 400.0 420.0 450.0 500.0 562.5 600.0 NNNN WWUUWWUWU hbbhbb 10 182 possible pairs of intervals, where order within a pair was considered. All pairs were generated in pseudorandom order, with a different order for each condition, and were recorded on tape. A .25 second warning signal followed by a 1 second rest preceded each pair of intervals on the tape. Each interval had a duration of 1 second, and the two in- tervals of a pair were separated by a .5 second rest. Fol- lowing each pair was a 4 second rest. Thus the total time for each pair presentation was 7.75 seconds, and the total time for presenting all 182 pairs in each condition was ap— proximately 23.5 minutes. For ease of presentation, each tape was divided into four tapes, each lasting about 6 min- utes . Procedure Subjects were tested alone or in pairs, with untrained 'and musically trained subjects run separately. The experi- ment required four one-hour sessions for the untrained sub- jects, and five one-hour sessions for the musically trained subjects. The task of each subject was to rate the per- ceived similarity of each pair of intervals by marking a position along a scale which ranged from identical to very dissimilar. In the attention condition, subjects were instructed to pay attention to only those components of the intervals which were above the reference tone in pitch. They were to totally ignore the lower components in forming their similarity judgments for this condition. 11 In the imagined condition, the response sheets con- tained the letter names of the notes comprising each inter- val (e.g. D A perfect fifth). Subjects were instructed to imagine how each interval would sound if played in the oc- tave beginning with the D above middle C (the same octave used in the heard conditions). They were then to compare the two imagined intervals in each pair and rate their sim- ilarity as in the heard conditions. Only musically trained subjects participated in this condition, since it was con- sidered too confusing and difficult for the untrained sub- jects. There were 20 blocks of trials for the untrained sub- jects (four tapes for each of the five heard conditions), and 24 blocks for the musically trained subjects (20 heard tapes and 4 imagined blocks). Approximately five blocks were presented in each session. The order of presentation was random, with the constraint that the first block of each session be heard rather than imagined. Before each block a list of the intervals in that condition was played, to familiarize subjects with the set of intervals to be judged. RESULTS Since four musically trained subjects each participated in six conditions, and four untrained subjects each partici- pated in five conditions, there was a total of 44 sets of similarity data. Each set consisted of 182 similarity mea- sures, arranged in a 14x14 matrix with diagonal absent. As in other studies which have used this general type of meth- odology (Shepard & Chipman, 1970; Shepard, Kilpatric, & Cunningham, 1975), the analysis of these matrices was divid- ed into two phases. The purpose of the first phase was to examine relationships between the data sets obtained for different subjects under different conditions. Specifically, the question to be answered was whether different subjects tend to show similar data structures when judging intervals under the same condition, or whether subjects tend to be in— ternally consistent, judging intervals in the same way a- cross a variety of conditions. The second phase of the an- alysis examined the structure of the stimulus space within each condition, and attempted to relate this structure to identifiable prOperties of the stimuli. PHASE I Product-moment correlations were computed, for each pair of the 44 data sets, between the 182 similarity 12 13 measures from one member of the pair, and the correSponding 182 similarity measures from the other member of the pair. The result of this computation was the below-diagonal half of a symmetric 44x44 matrix, in which each entry was a de- rived measure of proximity between two of the subject- condition data sets. This matrix was submitted to a multi- dimensional scaling pregram MDSCAL (Kruskal, 1964), and to STRUCTR, a modified form of Johnson's hierarchical cluster- ing scheme (Johnson, 1967). The hierarchical clustering was then embedded within the scaling solution, as suggested by Shepard (1972). Figure 1 shows the obtained solution, in which each subject-condition matrix is represented by a letter-number combination. The first letter indicates the condition: S; simple, C; complex, E; equal tempered, T: truncated, A; attention, and I; imagined. The second letter and number indicate the subject, with M1 designating the first subject in the musically trained group, 02 designating the second subject in the untrained group, etc. This two dimensional solution is not a particularly good fit to the data (stress formula 2 = .474), as reflected in the fact that the embed— ded curves of the non-dimensional clustering solution are not always round and convex. The most striking thing about this solution is the fact that conditions do not, in general, tend to cluster together. What does seem to be true is that data sets for a particular subject cluster across conditions. This is especially clear 14 OAU3 .EUZ om FIGURE 1. Two dimensional MDSCAL solution with embedded STRUCTR clustering for all subject-condition matrices (MDSCAL stress formula 2 - .474) 15 for subject Ul, whose data sets for all conditions except the attention condition cluster rather closely. For the other subjects, data sets for most conditions tend to clus- ter with other data from the same subject. Where data sets do not cluster within a subject, they do tend to group with data from other subjects with the same amount of musical training. An exception to this pattern is subject M3, whose matrices consistently cluster with those of the untrained subjects. It is interesting to note that M3 happens to be the subject in the trained group with the least amount of ear training, and so might be said to be more similar to the untrained group than are the other members of the trained group. The truncated and attention conditions do not follow the generalization about subjects clustering with themselves across conditions. For some subjects the truncated and at- tention conditions stand apart from other conditions. Note that, while truncated conditions may cluster with each other, and attention conditions cluster with other attention condi- tions, the truncated and attention conditions do not cluster together. Figure 2 shows the results of MDSCAL and STRUCTR anal- yses for the musically trained group alone. Again the tend- ency is for matrices to cluster within subjects rather than within conditions. This tendency is particularly marked for subject M3, as would be eXpected from the fact that her data clustered with the untrained group rather than the 16 FIGURE 2. Two dimensional MDSCAL solution with embedded STRUCTR clustering for musically trained subjects' subject-condition matrices (MDSCAL stress formula 2 - .382) 17 trained group. The attention conditions stood apart, as in the previous analysis, but this was less true of the trun- cated conditions. Similar results were obtained for the untrained sub- jects (Figure 3): clusters formed within subjects, except for the truncated and attention conditions. A notable dif- ference between the trained and untrained groups is that the truncated condition clustered more closely with other con- ditions for the same subject in the trained group, and less closely in the untrained group. PHASE II This phase of the analysis attempted to find the struc- ture implicit in the data within each of the experimental conditions. To do this, the eight similarity matrices for each condition (four in the case of the imagined condition, where only trained subjects participated) were submitted to INDSCAL (Carroll & Chang, 1970). This multidimensional scaling prOgram generates a group stimulus space as well as a weight Space which shows the relative weights assigned to each of the dimensions in the solution by each subject. Simple_condition Figure 4 shows the two dimensional INDSCAL solution and weight Space obtained for the simple condition. Examination shows that all the intervals on the left side of the plot are those with a base frequency of 300 Hz., while those on the right side have an upper frequency of 600 Hz. FIGURE 3 . 18 Two dimensional MDSCAL solution with embedded STRUCTR clustering for untrained subjects' subject-condition matrices (MDSCAL stress formula 2 - .348) 19 Ace. I sowuoamuuou doauaufloo mamaao usu you macaw .uan03 one ounce osaaawuo A H sowosmsan .2” as. war «to mac as. 235 a. he . m2. 333 2 Sr Soc .3. Boo. cad . Sod ma. . Hay . w ma. €53 E . Aswanv mm. he. 0 Boo. 21b AHB. I soauoamuuoov .nouuuvaou mHmsHo gnu pom momma usaaauuo AdamnzH Homoeosusqv dunno o no saucepan Hmsowososav one .nm MMDUHM m scamsusan .o> N souososfln Na . .ij NS. he», a. 98.; oz. :83 E . . _.ma. cad mm 2. . 235. .E 0 O 235 2 . 22 Complex condition The two dimensional plot for this condition is shown in Figure 6. Dimension 1 is related to pitch, with the 300 Hz. base intervals on the left and the 600 Hz. t0p intervals on the right. The second dimension corresponds to ratio com- plexity, with dissonant intervals near the bottom and conso- nant intervals near the top. Again, musically trained sub- jects emphasize ratio complexity, while the untrained sub- jects weigh pitch more heavily. Figure 7 shows that the first two dimensions of the three dimensional solution for this condition are essential- ly the same as in the two dimensional plot. The third di- mension which emerges is interval width. It is interesting that the same three dimensions appear for both the simple and complex conditions: pitch, interval width, and ratio complexity. While pitch appears important in both conditions the salience of the other two dimensions varies. Interval width appears in the two dimensional plot for the simple con- dition, and so is more important than ratio complexity, which emerges only in the three dimensionalsolution. The reverse is true for the complex condition. Equal tempered condition Both the two and three dimensional solutions (Figures 8 and 9) are essentially similar to those in the complex con- dition, with pitch and ratio complexity being the most im- portant dimensions, and interval width emerging as a third dimension. The only differences arise from some local 23 30. I soauoHoHHOUV soauavsoo ”Bans—co 05 you women “Emacs one woman .3933 38% ~338an as. .o go: .mao as.- not m2. «:0 SA. E. use me... Asmanv em we. rams a. ADO ASA. Hg. as. oz. . 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While there is some tendency for narrow intervals to lie near the bottom of the plot, and wide intervals near the tOp, it is not entirely consistent. There is also a tendency for inversions to be close together which is incompatible with the interval width dimension. The minor second and major seventh pair, and the minor third and major sixth pair, are clear examples of the proximity of inversions. This trend is understandable in view of the fact that the letter names for inversions (9.9. D E major second, and E D minor seventh) contain the same letters in reverse order. Thus the similarity may be merely visual or verbal, rather than a true similarity between auditory images. Truncated condition The first dimension in the two dimensional solution (Figure 11) is related to ratio complexity. The second di— mension is more difficult to label; it is neither pitch nor interval width. 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DISCUSSION The fact that matrices in the MDSCAL solutions tended to cluster across conditions within subjects indicates that subjects are consistent in their similarity judgments over a variety of conditions. Each subject seemed to have a pre- ferred dimension which largely determined his ratings of the stimuli. Musically trained subjects M1, M2, and M4 empha- sized the ratio complexity dimension; untrained subjects UZ, 03, and U4 and trained subject M3 emphasized the pitch di- mension; and subject 01 weighed the interval width dimension most heavily. One effect of musical training seems to be the focus of attention on ratio complexity, which is not surprising in view of musicians' concern with consonance and harmony. The same three dimensions of pitch, interval width, and ratio complexity appeared in both the simple and complex condition solutions. Ratio complexity was least important in the simple condition. Since there were no higher harmon- ics to create beating in this condition, much of the rough- ness which characterizes dissonant intervals was absent. This tended to minimize the differences between intervals with simple and complex frequency ratios, and thus made the complexity dimension less salient. These results contradict 39 40 those of Stumpf (1890), who stated that consonance is inde- pendent of the number of harmonics present. A probable ex- planation for this contradiction is the fact that all Stumpf's subjects were musically trained, as was Stumpf him- self. They were undoubtedly able to identify the intervals which were used as stimuli, and give them their proper musical names. Since Stumpf usually asked them to respond verbally, or to rank the intervals in terms of consonance, they would probably act on the basis of their training and rank the major second as very dissonant, even though it might not sound very rough under conditions of few harmonics. The finding that the solutions for the complex and equal tempered conditions were nearly identical was rather surprising. Musicians have long argued about the best tun- ing, and have reluctantly accepted equal temperament as an imperfect compromise. These results suggest that under cer- tain circumstances it may not be such a bad compromise after all; at least in the conditions of this experiment there is little difference between reSponses to just and equal tem— pered intervals. The solution for the imagined condition shows that sub- jects are not entirely veridical in their imaging of inter- vals. Although the complexity dimension is clearly present in this condition, the other dimensions are difficult to in- terpret, and are not similar to those found in the heard conditions. There is a tendency for inversion pairs to group together, but this may be due to the fact that the 41 letter names for the notes are the same for these pairs. It would be possible to eliminate this name similarity by rep- resenting the intervals in standard musical notation, but this would introduce another factor of visual similarity. This problem is not unique to the present study, but is shared by most experiments dealing with imagery. It is dif- ficult to determine what a subject is doing when he is asked to imagine. Since the experimenter must communicate to the subject what he is to imagine, there is always the possibil- ity that he is merely responding on the basis of some clue given by the experimenter which is not related directly to imagery. As was expected, the solutions for the truncated condi- tion showed ratio complexity as an important dimension. Since the lower harmonics were absent, the pitch dimension assumed less importance. This explains the finding that the truncated condition tended to cluster with other conditions for the same subject in the musically trained group, while the truncated condition stood apart for the untrained sub- jects. Since musically trained subjects emphasized ratio complexity in all the other conditions, the truncated condi- tion was no different. When the pitch dimension which un- trained subjects weighed heavily in most conditions became unavailable, they were forced to change their criteria. This result suggests that using truncated intervals for ear training might facilitate the learning of interval recogni- tion by focusing attention on ratio complexity. 42 The only clear dimension which emerged in the attention condition was that of pitch. Instructions to the subjects to attend only to the higher harmonics of the intervals ap- parently did direct their attention to pitch, but they were unable to ignore the lower harmonics. 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