The role of symmetry in the good continuation ratings of two-part tonal melodies

Transcription

1 Perception & Psychophysics 1981,29 (I), The role of symmetry in the good continuation ratings of two-part tonal melodies WILLIAM R. BALCH Pennsylvania State University, Altoona, Pennsylvania Subjects heard a series of two-part melodies, in which each part was a random sequence of eight pitches (Experiment 1) or five pitches (Experiment 2) from the diatonic scale. The task was to rate each melody on how well the second part followed the first. It was predicted that the presence of symmetry between the two parts would increase the perception of good continuation. In Experiment 1, two symmetrical relations-inversion and retrograde-yielded melodies that were more highly rated than the control melodies, which consisted of nominally different parts. A third symmetry, the retrograde inversion, did not enhance good continuation ratings. In Experiment 2, inversions and retrograde inversions were compared with control melodies, using shorter sequences and pitches with equal durations. Again, inversions, but not retrograde inversions, were significantly preferred. The results suggest that the aesthetic judgment of good continuation depends at least partially on a cognitive analysis of the relation between the melody parts. The positive symmetry effects are further discussed in relation to other studies of symmetry transformations in the contexts of both musical sequences and visual arrays. Relatively little is known about the factors which predict aesthetic reaction to a given melody. Most of the work on this topic has related to amount of information in random tone sequences (e.g., Bragg & Crozier, 1974; Vitz, 1966). While these studies generally associate high preference with intermediate levels of uncertainty, they do not directly address the question of which melodic structures are preferred. Explicitly structural variables, such as contour and symmetry, are more commonly studied in experiments on the recognition of musical stimuli. For instance, a number of studies show that the simple up-down contour of the successive pitches is an abstraction which figures importantly in melody recognition. Specifically, test stimuli have been studied in which the exact pitches and intervals of an original melody are transformed, but contour is left intact (e.g., Dowling, 1971; Dowling & Fujitani, 1971). In these studies, it has generally been found that contourpreserving melodies tend to be mistaken for the originals on which they are based. The research reported in this article was supported by a Faculty Development Grant from The Pennsylvania State University. The author wishes to acknowledge assistance from Deborah W. Ault, David Carothers, David Lieber, and Thomas Russo. Joann Treese, Julie Corl, and Jean Emery assisted in the preparation of the manuscript; and helpful comments and advice were provided by George Balch, Samuel Merrill, Alan Mikula, and by Steven Currier, Sukhjit Singh, Valerie Stratton, and Annette Zalanowski. Correspondence concerning the article or requests for reprints should be sent to William R. Balch, Department of Psychology, The Pennsylvania State University, Altoona Campus, Altoona, Pennsylvania In structural terms, symmetry transformations are similar to those which preserve contour. Three such transformations are of interest here-the inversion (I), the retrograde (R), and the retrograde inversion (RI). The inversion turns the intervals of an original stimulus upside down, thus inverting contour but not otherwise changing it. For example, Figure IB is a diatonic inversion of the melody shown in Figure la. That is, each successive interval in 1B has the same number of diatonic steps as its counter- A B c o E Figure 1. Examples of melody plll1s used In the experiment. S is the invenion of A; C, the retropade; D, the retrograde invenion; and E is a different random melody. Each experimental melody consisted of two such plll1s. Copyright 1981 Psychonomic Society, Inc /81/ $01.15/0

2 48 BALCH part in la, but in each case the direction of the interval is opposite. The inversion can thus be considered a reflection operation with respect to pitch. Another type of reflection, which is defined with respect to time, is the retrograde. This transformation is simply a backwards rendition of an original sequence. In Figure I, plate 1C illustrates the retrograde of plate la. Note that the sequence of pitches is backwards, but not the rhythmic values of the notes. Finally, the retrograde inversion consists of both transformations taken together. This operation is illustrated by plate 10, which is the retrograde inversion of plate la. Again, rhythm is not included in the analysis. Using a short-term recognition procedure, Dowling (1972) has studied melodic inversions, retrogrades, and retrograde inversions. The nature of each transformation was explained to subjects. Then they heard a five-note atonal sequence, followed by a comparison sequence. The subjects had to indicate whether or not the comparison item was a particular transform of the original sequence. Inversions and retrogrades were recognized with an accuracy that was significantly above chance. Retrograde inversions were more difficult, and were recognized at above-chance level only in some of the experimental conditions. Dowling argues for the psychological reality of melodic transformations that are commonly used by composers. While little is known about the aesthetic consequences of musical symmetries, there are good reasons for expecting that such transformations would enhance melodic preference. One reason is that reflection symmetries are common and effective in visual art and design. For instance, Szilagyi and Baird (1977) have studied how subjects arrange elements in one-, two- and three-dimensional visual arrays when instructed to produce aesthetically pleasing designs. Perfect symmetry of pattern parts was the most common condition for all dimensions. More theoretically, Garner (1970) has pointed out that patterns with axes of symmetry are redundant, because the whole is predictable from any part. Garner identifies pattern goodness with this concept of redundancy. Of particular relevance here is melodic symmetry, which can be created by following one sequence of pitches by its inversion, retrograde, or retrograde inversion. Symmetrical melodies should be perceived as better patterns than melodies in which the two sequences are unrelated. In each of two experiments reported here, subjects rated two-part melodies on good continuation. Each part was a random sequence of tones containing eight notes (Experiment 1) or five notes (Experiment 2). The subjects were asked to rate how well the second part followed from, and went along with, the first part. Moreover, they were instructed not to base their ratings on either part taken by itself, and to wait until the end of the second part before rating. Good continuation was chosen as a rating criterion over one of general preference, because the former stresses a comparison between the parts of the melody. In addition, good continuation is an intuitively easy concept for subjects to understand. It is one ofthe Gestalt principles ofperceptual organization, as illustrated in Figure 2. Plate A of the figure is an adaptation of a popular example of visual good continuation. In this example, the best continuation for the curve MO seems to be OQ rather than OP. Interestingly, the two curves are visual inversions of one another. An example of musical symmetry is given in plate lb. Here, two fivenote parts of a melody are inversions of each other. If symmetry is associated with aesthetic preference, this melody should be judged relatively good in continuation. The concept of good continuation is potentially ambiguous. Sometimes it is defined as an independent manipulation which is assumed to produce the perception of good continuation. For instance, in a visual study by Prinzmetal and Banks (1977), good continuation was defined as a straight-line relation between the elements in an array. However, in the present study it was treated strictly as a judgment of the subjects. Therefore, no visual analogues or references to structure were offered to the subjects. Since the effect of symmetry on the perceived goodness of continuation was the issue in question, it seemed important to make no mention or suggestion of symmetry as a criterion of good continuation. Another methodological choice concerned the tonal A.P M. 0.Q N Fiaure 1. Plate A is a typical visual figure represedddg the Gestalt pridciple of good coddduadod. Plate B illustrates bow a similar pridciple of symmetry can be applied to perceived good coddduadod ida two-part melody.

3 SYMMETRY 49 system to be used in the experimental melodies. This factor of scale has been shown to be important in musical perception (Dowling, 1978; Krumhansl, 1979). Since the present study required subjects to make an aesthetic judgment, it seemed advisable to use a tonal system that was familiar to them. Diatonic scaling was used here, since it is predominant in music of Western cultures (as opposed to atonal scaling). The diatonic scale is exemplified by major or minor "keys." For instance, the white keys on a piano form the scale of C major. In diatonic scales, there is a tone center called the tonic. Every other tone in the scale has a definite tonal function in relation to the tonic (see Krumhansl, 1979). Since aesthetic effects were sought in this study, tonal factors were controlled as fully as possible, particularly in Experiment I. This experiment employed the eight notes spanning one octave of the 0 major scale, from 0 4 to Os. Each melody part was a random order in which every note of the scale appeared once. Both the highest and the lowest tones were tonic, thus reinforcing the sense ofscale. While all melody parts were tonally uniform, a dilemma concerning rhythm developed during pilot research. Originally it had been intended that every note in a sequence have the same duration. However, to the author, these original sequences sounded rather unmelodius and similar to one another. Several other listeners agreed in this judgment. To alleviate the problem, a constant 3/4 rhythm was used for all sequences. As shown in Figure 1, the fourth and the eighth notes lasted three beats, and the others, one beat. This modification seemed to create a modest degree of melodic character and musical naturalism in the sequences, while maintaining a controlled rhythm for all sequences. On the other hand, the rhythmic pattern used in Experiment I complicated the interpretation of the symmetry relations being manipulated, as discussed at the end of Experiment 1. The issue was addressed in Experiment 2, in which the melody parts had five notes of equal duration. The major independent variable in this study was the type of symmetry relation between the two parts of each experimental melody: inversion, retrograde (Experiment 1 only), retrograde inversion, and different. The last condition served as a control, in which the two melody parts were different, randomly generated sequences. In Figure I, plates la and IE are "different" sequences. Since the intention of this study was to examine differences in aesthetic rating due to symmetry type, every subject heard melodies having every relation type. Thus, it was necessary to eliminate the effect of the sequences themselves on the ratings. To this end, counterbalancing was done in each experiment. For different groups of subjects, different melodies were associated with each symmetry relation type. This procedure created a second independent variable: counterbalanced lists. These lists are described in detail in the Method section of each experiment. A final independent variable was musical experience (Experiment 1 only). This variable was introduced because musical training has been shown to influence memory for musical material, using measures of both recall (Jellison, 1976; Sloboda, 1976, 1978) and recognition (Bever & Chiarello, 1974; Dowling, 1978). Aesthetic judgments of the present melodies may be affected by the subject's ability to retain both parts long enough to detect the presence of a particular relationship between these parts. Since musical experience affects melodic memory, it may also influence the good continuation ratings used here. The central hypothesis in this study is that symmetry relation type between the two parts of a melody will affect the judgment of good continuation. If symmetry is a condition that promotes perception of good continuation in melodies, then the inversion (I), retrograde (R), and retrograde inversion (RI) should all be rated higher than the control condition, which involved melodies with different (D) parts. An interesting question arises in the case of the retrograde inversion (RI). Although it is formally a symmetrical relation, it is only marginally recognizable by subjects (Dowling, 1972). The assumption is that subjects cannot prefer a relation they are unable to detect. Therefore, it seems reasonable to expect that the preference for the retrograde inversion continuation should not necessarily be any higher than that for the control continuation, involving two different melody parts. EXPERIMENT 1 Method Subjects. The participants in this experiment were 30 first- and second-year undergraduates enrolled at the Altoona Campus of The Pennsylvania State University. Of these subjects, 15 (6 female and 9 male) were classified as having low musical experience. They had taken no lessons on any musical instrument, did not play an instrument, and did not read music. Another 15 subjects (II female and 4 male) were designated as high in experience. All had taken at least 4 years of lessons on at least one instrument, played an instrument currently, could read music, and had performed either formally or informally on their instruments within the past year. All students participated on a voluntary basis and received a small amount of extra academic credit for serving as subjects. Materials and Design. Every subject rated a series of two-part melodies on good continuation. Each melody part in the experiment was a randomly ordered sequence of eight pitches drawn from the G major scale: G. (392 Hz), A. (440 Hz), B. (494 Hz), C, (523 Hz), D, (587 Hz), E, (695 Hz), F~ (740 Hz), and G, (784 Hz). In every sequence, each of the eight pitches occurred exactly once. The fourth and eighth pitches were 3 beats in duration; the other pitches were I (see Figure I). The melody parts

4 50 BALCH used in Experiment 1 are given in Table 1. Thirty original random sequences are shown in the column marked "Base." The inversions, retrogrades, and retrograde inversions of these originals appear inthe next three columns to the right. Thus. there are 30 different sets of four related sequences. which are numbered in the column labeled "SequenceGroup." The main independent variable in the experiment was the symmetry relation between the two parts of each experimental melody. There were four relations employed: (1) Inversions (I)-The first and second parts were inverted with respect to each other. Each note of one sequence was reflected to form the other. The point of reflection was the midpoint of the pitch scale. As a result. G. became G A. became F1. B. became E and so forth. This procedure produced a diatonic inversion. The successive intervals of each part were opposite in direction to each other. but the intervals had the same number of diatonic scale steps. In terms of Table 1. the first part of every inversion melody was one of the sequences in the column labeled "Inversion." The second part was one of the sequences in the column labeled "Base." (2) Retrograde (R)-The two melody parts were the backwards versions of each other with respect to pitch. Since rhythm was a control factor, the rhythmic pattern was not reversed. The first part of retrograde melodies was drawn from the "Retrograde" column of Table I, while the second part was a "Base" melody. (3) Retrograde Inversion (Rl)-The two sequences were simultaneously backwards and inverted versions of each other. One of the "Retrograde" sequences in Table 1 appeared as the first part of the melody, while a "Base" sequence was the second part. Note that in this and the preceding two symmetry cases. the order in which the parts occur does not alter the relation. As a convention, "Base" sequences were always used as second parts. (4) Different (D)-The two parts were independent random sequences, bearing no nominal relation to each other. Each of the parts was a sequence drawn from the "Base" column of Table 1. Different melodies served as a control against which the melodies having each of the other symmetry relations were compared. Every subject rated melodies of all symmetry types. Therefore, a counterbalancing variable was necessary to separate out the effect of the melody materials from that of the symmetry relations between the sequences. The 30 sequence groups in Table 1 were divided into five sets of six each. Although there were only Table 1 Specifications for the 120 Sequences in Experiment I Sequence Type Sequence Group Base (B) Inversion (I) Retrograde (R) Retrograde Inversion (RI) 1 G.FfA.D.G4C. E, B. G.A~Ffc,G.D.B.E. B.E.C.G.D.A.FfG, E.B.D,G.C.FfA.G. 2 B.A.C.G.G. F;t'E,D. E, F!D,G,G. A. B. C. D.E,FfG.G.C.A.B. C.B.A.G.G.D.F~, 3 G.E,G.C.A.Ff'B.D. G.B.G,D.FfA.E.C. D.B.FfA.C.G.E,G. C.E,A.F!D.G,B.G. 4 A.E.B.C.G.G.D,Ff Frn4E,D.G.G,C.A. FfD.G.G.C.B.E.A. A.C.G,G.D,E,B.Ff 5 FfG.C. D. G. E, A. B. A.G,D.C.G.B.FfE. B.A.E.G.D.C.G.Ff E,Ff'B.G.C,D,G,A. 6 B.C.D.G, A. F;tG.E, E.D.C.G.FfA.G.B. E.G4FfA4G.D.C.B. B.G. A. Fre.C. D.E. 7 A.FfD.B.C.G4E,G, FfA4C.E.D,G.B.G4 G.E,G.C, B4D, FfA. G.B.G,D.E,C,A.Ff 8 G..E. A. C. Fro. B. G. G.B..Fro.A4C, E, G4 G,B4D.Ffc.A.E.G. G.E.C.A.D.FfB.G. 9 D.B.G..E.A.G.C,Ff C,E.G.B.FfG.D.A. Ffc.G.A.E.G.B.D. A.D.G.FfB.G.E.C, 10 E.G.D,C.FfG.A.B. B.G.C.D.A.G.FfE. B.A.G.Ffc.D,G.E. E.F;tG,A4D.C.G4B. 11 G.B. F:'c. A. E. D. C. G.E.A.G4FfB.c~D. C.D.E.A.G.Ff'B.G4 D.C.B4FfG.A4E.G. 12 B.A.FfD,C.E.G.G. E.FfA4C.D.B.G.G. G.G.E.C,D.FfA4B4 G.G.B.D.C.A.FfE. 13 E. Ffo. A4C, G. G. B. B.A.C. Ffo,G.G.E. B4G. G4C. A. D. FfE. E.G.G. D, Ffc. A.B. 14 G,A4C.B.FfE.D.G. G.Ffo.E.A4B.C.G. G.D.E,Ff'B.C.A4G. G.C.B.A.E.D,FfG. 15 D.A.G.E.FfG,B.C. C. FfG, B.A. G4E.D, C.B.G.Ff'E.G.A4D. D.E.G4A.B4G.FfC. 16 C.B4E.G.G.D.A.Ff D,E,B4G.G,c.FfA4 FfA.D.G.G.E.B.C. A.Ffc,G.G.B.E.D. 17 A4G.Ff'B.D,G.E,C. FfG.A4E.C.G.B.D. C.E.G. D.B. F;tG.A. D.B.G.C.E.A.G.Ff 18 D.Ffa.G,G 4C.A.E. C.A.E.G.G,D,Frn. E.A.C.G.G.B.Ffo, B. F!D.G.G.E. A.C. 19 G.A.E.B.D.C.G.Ff G.FfB.E.C.D.G.A. FfG.C.D.B.E.A.G. A.G.D.C. E.B.FfG. 20 B.C.G.Ff'E.A.D.G. E.D.G.A.B.Ffc.G. G.D.A.E.FfG.C.B. G.c.FfB.A.G.D.E. 21 Ffc.D.B.A.G.E.G, A.D.C,E.FfGsB.G. G.E.G.A.B.D.C.Ff G.B.G,FfEsC.D.A. 22 Ffo.G.A.E.G.C,B. A.C.G. Ff'B.G.D.E. B.C.G.E.A.G4D.Ff E,D,G4B.F;*G.C.A. 23 C. B. G. FfA. D. G. E, D.E.G.A.Ffc.G.B. E.G. D,A.FfG. B.C. B.G.C,FyA.G,E.Ds 24 E.C.G.FfB.G.A.D, B.D,G4A..E,G. Ffc. D.A.G.B. FfG.C.Es C,FfG,E.A.G.D.B. 2S C.A.E,D.FfB.G.G, D,FfB4C A Es G,G. G.G.B.FfD.E.A.C. G.G,E.A.C,B.Ff'D, 26 G.G.D.C.FYA.B4E, G.G,C.D.A.FfE.B. E.B.A.Frc.D.G.G. B4E.F;A.D,C,G.G. 27 D,G.FfB.E,G.C,A. C,G.A.E,B4G.D,Fy A.C.G.E.B4F fg.d. Ffo,G, B4E.A.G.C. 28 E.D.C,G.A.B.G.Fy B.C.D.G.FfE.G.A. FfG,B..A..G.C,E,D, A.G.E. FfGsD.C,B. 29 G,C.B.FfE,G.DsA. G.D.E.A.B.G.CsFf A.D.G..E.FfB.C.G. Ffc,Gs B. A. EsD.G. 30 G.E.B.D,A.Frc.G. G4B.EsC.FfA.D.Gs G.C. FfA.D, B.E.G, G,D,A.FyCsE,B4G. Note-The fourth and eighth notes are three beats in duration, and the other notes last one beat. Beat duration =.5 sec. The sequences were played on an alto recorder. Approximate frequencies ofthe pitches were: G. = 392 Hz, A. 0= 440 Hz, B. 0= 494 Hz, C. 0= 523 Hz, D, =587 Hz, E. = 659 Hz, Ff= 740 Hz, G. = 784 Hz.

5 SYMMETRY 51 four symmetry relations, five sets of sequences were necessary for the following reason. I, R, and RI melodies each involve two sequences from the same group. However, D melodies use two different base sequences, requiring two groups. Therefore, a total of five groups were needed to define every four symmetry melodies (i.e., I, R, RI, and D). Using five sets of six groups each, five counterbalanced lists were generated which contained 24 two-part melodies. The counterbalancing scheme is detailed in Table 2. It can be seen that each list uses different sequence groups in different types of symmetry melodies. To sort out the effects of the melodic materials themselves, different groups of subjects were run on each of the five counterbalanced lists, making lists a second independent variable in the experiment. 1 The final independent variable was musical experience. A low and a high group each included l5 subjects, as detailed in the Subjects section. All subjects came from the same population of iritroductory psychology students. A three-way mixed factorial design was employed. The randomized factors were lists and musical experience, consisting of five and two levels, respectively. These factors combined to generate 10 groups in which there were three subjects each. All groups received all four levels of the repeated factor, symmetry. Although the group size was small, all the sources of variance due to treatment involve cell sizes of at least six subjects, except for the three-way interaction. Procedure. Sequences used in the experiment were played by a musician on a Yamaha alto recorder, at a rate of.5 sec per beat. The tempo was controlled by a Franz LM-FB-4 metronome, which established the beat by means of a flashing light. Sequences were recorded on a Sony TC-270 stereo recorder, with the same input received on each channel. From this master tape, an experimental tape was made which contained the five counterbalanced lists. A Sony TC-377 stereo recorder was used as the copy and editing machine for the latter tape. In each list, the two parts of each melody were separated by a pause of 2 sec. The melodies were presented at an intensity of approximately 70 db (SPL). The subjects listened through Sony DR-7A headphones. The subjects participated in individual experimental sessions. Each was instructed that he or she would be listening to a series of two-part melodies. There would be a sequence of 8 notes, a brief pause, and then a second sequence of 8 notes. The subjects were informed that their task was to judge how well the second part followed from the first part. They were to make the rating after the last note of the second part. For examples, they were asked to think of good continuation-between the first several notes and the next several notes-in songs or melodies. Regarding the experimental melodies, it was explained to subjects that the characteristics of either part taken individually were not to be considered in their judgments. They were advised that there were no right or wrong answers in the experiment, but rather that its purpose was to find out how subjects themselves rated the melodies. Not.hing at all was mentioned about the various transformations employed. A rating was given for each two-part melody. The subjects responded on a series of continuous 15-cm-long line scales. They were labeled "very poor continuation" at one end, and "very good continuation" at the other. The subjects made their ratings by making a vertical mark somewhere on each line. They were asked to try to avoid responding in the same place each time, and to use as much of the full range of the scales as possible. Before rating the experimental set of melodies, the subjects first rated two warm-up examples. Then they were asked if they had any questions. Most subjects indicated that they understood the task well. Three subjects asked if they were supposed to rate good continuation on the basis of how similar the two parts were. They were told that the criterion was not necessarily one of similarity. Rather, they were to rate the goodness of continuation between the two parts, whatever that judgment meant for them. In rating the 24 experimental melodies, the subjects were given approximately 8 sec to rate each one. After the 12th melody, they were given a break of about 2 min. Results and Discussion The outcome of this experiment is graphed in Figure 3. The average good-continuation ratings of the four symmetry conditions are shown for subjects of both high and low musical experience. The rating scores were collapsed across lists. This variable was not significant as a main effect [F(4,20)=.98]. Moreover. there were no significant or near-significant interactions between lists and the other experimental variables. In particular, there was no significant interaction between lists and symmetry [F(12,60) =.28]. One positive finding was that subjects with high musical experience rated all the melodies better in continuation than did subjects with low experience. This tendency is reflected in a significant main effect of musical experience [F(1,20) = p <.05]. The effect appears uniform across all symmetry types, since there was no significant interaction between experience and symmetry [F(3,60) = 1.56, p >.10]. Counterbalance List Table 2 Design of the Counterbalanced Conditions in Experiment 1 Melody Type Inversion Retrograde Retrograde Inversion Different IrIB II R-IIB IIIRrIII B IVB-VB IIrIIB IIIR-III B IVRrIVB VB-IB IIIrIIIB IVR-IVB VRrVB IB-IIB IVrIVB VR-VB IRrIB IIB-IIIB VrVB IR-IB IIRrIIB IIIB-IVB Note-The entries under each melody type are two-part melodies. Each part is drawn from one of five sets of six sequence groups (see Table 1 for specifications). The sets are labeled I through V. Set I = Sequence Groups 1-6, Set II = 7-12, III = 13-18,IV= 19-24, V = Each row on this table represents a counterbalanced list of 24 different two-part melodies. Total number ofmelodies = 120. Subscripts refer to the sequence types in Table 1. B = base, I =inversion, R =retrograde, RI=retrograde inversion. The two sequences for each melody produce the melody types on which the columns ofthis table are based.

6 52 BALCH <!) z ~ a:: z o ~Z I Z 8 8 o<!) LOW HIGH MUSICAL EXPERIENCE Figure 3. The results of the ratings. Melodies in which the two parts were related by a retrograde (R) symmetry were rated highest in good continuation, followed by tbe inversion (I), retrograde inversion (RI), and different (D) relations. The IlISt is a control condition, in which tbe parts consisted of two different random sequences. The result of primary interest concerns symmetry. As Figure 3 shows, the R relation was rated highest in continuation, followed by the I and the RI. Melodies with 0 parts were rated lowest. Collapsed across the other independent variables, the mean ratings were: R = 8.80, I = 8.20, RI = 7.58, and D = 6.99, for which the maximum rating was 15. The main effect of symmetry is significant [F(3,60) = 5.52, p<.005]. The hypothesis was that symmetry was a condition facilitating good continuation. To examine the effect of each of the three experimental symmetries (R, I, and RI), a comparison was made between each of these types and the control condition (0). The three effects were R - 0 for the retrograde, 1-0 for the inversion, and RI - 0 for the retrograde inversion. Three Bonferroni confidence intervals (116 df) were obtained, one for each of the differences. The R and I effects were each significant, the intervals being 1.81 ± 1.28 and 1.21±.98, respectively. However, the RI was not a significant effect, the interval being.59 ± Therefore, the experimental hypothesis was confirmed for the R and the I symmetries, but not for the RI symmetry. Presumably, the failure to find a significant preference for the latter relation stems from the low detectability of retrograde inversions (Dowling, 1972). A complication in the interpretation of the results arises from the rhythmic pattern imposed on the melody parts. The durations of the eight notes were not equal, the fourth and eighth pitches being 3 beats and the others, 1. For inversions, the analogous tones of each part have identical rhythmic emphases. However, for Rs and RIs, the durations of analogous notes would sometimes differ between the first and second parts. As a result, Rand RI relations might have been less detectable in this experiment than they would have been without the complication in rhythm. In particular, the RI symmetry might have had a nonsignificant effect merely because of the rhythmic pattern used in Experiment 1. This problem was addressed in Experiment 2. Specifically, the melody parts were changed to sequences of 5 notes each, with every pitch having the same duration. The number of pitches per part was reduced to alleviate an anticipated difficulty arising from having all notes be equal in duration: all sequences sound much the same to subjects. As mentioned earlier, this difficulty had been encountered in pilot research. Another modification was introduced in Experiment 2. Only inversion, retrograde inversion, and different melodies were included. The retrograde condition was dropped. In Experiment 1, it was the only type of melody in which the last note of the first part was always the same as the first note of the second part. The high ratings obtained for retrogrades could have been due, at least partially, to the identity of these two notes. If subjects had used such a criterion in evaluating retrogrades, they would not have been responding to each two-part R melody in its entirety. Therefore, they might not have been judging retrogrades entirely on the basis of symmetry per se. A final change was that, in Experiment 2, subjects were not grouped according to level of musical experience. In Experiment 1, experience was significant as a main effect but did not interact with symmetry. The specific effects of interest in this study involve comparisons between the symmetry types. Since experience affected only the absolute values of the good continuation ratings-but not the differences between them-it was not used as an independent variable. EXPERIMENT 2 The specific hypothesis for Experiment 2 was that Is would be rated significantly better than Ds in continuation, but that Rls would not. The nonsignificant preference obtained for the RI melodies in Experiment 1 was assumed to result from the low detectability of the RI relation, not from the rhythmic pattern imposed on the melodies. The equal pitch durations used in Experiment 2 served to test this assumption.

7 SYMMETRY 53 Method Subjects. The 28 subjects in this experiment were drawn from the same population used in Experiment 1. However, the participants were selected without regard to amount of musical experience. Materials and Design. The two-part melodies were constructed in the same manner as in Experiment I. This time the parts were composed of sequences containing only five notes. each having a duration of I beat (.5 sec). The 72 sequences used in Experiment 2 are given in Table 3. The symmetry relations between melody parts were the same as in Experiment I, except for retrogrades, which were omitted. In addition, a similar counterbalancing scheme was employed, as shown in Table 4. A two-way, mixed factorial design was used. Four different groups of seven subjects each heard one of the counterbalanced lists. All subjects in Experiment 2 received all three symmetry conditions. Procedure. The method of recording the experimental tapes was identical to that used in Experiment I. Likewise. subjects were instructed and tested in this same manner. An average good continuation rating for each symmetry type was thus obtainable for each subject, as in Experiment I. The only difference in procedure was that the subjects rated all experimental stimuli without a midway break, since there were only 18instead of 24 melodies. Results and Discussion A two-way mixed-design analysis of variance was performed to see if the rating scores could be col- lapsed across counterbalanced lists. The main effect of lists was nonsignificant [F(3,24) = 1.02]. Moreover, there was no significant interaction between lists and symmetry relation [F(6,48) = 1.46, p >.10]. When the good continuation ratings were pooled across lists, the individual means for the symmetry conditions were: I = 8.93, RI = 8.09, and D = The hypothesis-that inversions are preferred continuations, but retrograde inversions are not-was directly tested by examining two differences: 1- D and RI - D. For each of these effects, a two-tailed t test was performed on the 28 individual difference scores. The inversion effect (I - D) was 1.08 em. The difference was significant [t(27) = 2.16, p <.05]. On the other hand, the retrograde inversion effect (RI - D) was only.24 em. This difference was not significant [t(27) =.48, p >.10]. The results of Experiment 2 are comparable to those of Experiment 1. In each case, the inversion was significantly preferred as a good continuation, but not the retrograde inversion. Moreover, the difference scores for these effects were quite similar across the two experiments. The inversion (I - D) effects were 1.21 and 1.08 em in Experiments 1 and Sequence Group Table 3 Specifications for the 72 Sequences in Experiment 2 Base (B) Sequence Type Inversion (I) Retrograde Inversion (RI) I IS B.GsDsEsA. B.A.CsG.Ds A.G.CsB.Ds GsEsDsCsFf' GsDsEsCsA. Ff'CsEsA.Ds G.Ff'osB.A. Ff'EsG.A.Gs CsEsB.G.Ff' EsGsB.A.Ff' EsB.GsG.A. G.CsA.GsD. A.G.EsFf'B. DsB.G.EsFf' DsGsEsCsA. A.CsGsG.B. EsG.A.CsFf' G. t, A. Ff'B. CsEsB.Ff'Gs CsFf'EsG.Gs CsDsB.EsGs B.GsCsDsA. B.DsA.GsCs CsEsGsA.G. EsG.CsB.Ff' EsFf'osGsCs Ff'GsDsEsCs G.B.CsDsA. G.CsB.DsFf' A.DsB.Ff'Cs GsA.CsEsFf' A.B.GsFf'G. DsB.EsGsA. B.G.EsFf'A. B.EsG.GsFf' GsDsFf'G.Cs Ff'GsB.A.Es CsEsGsB.A. CsG.B.DsFf' Ff'osG.GsEs B.GsFf'osA. GsB.Ff'A.Es DsB.EsA.G. DsA.B.GsG. DsCsEsB.G. EsG.DsCsFf' EsCsFf'G.Ds DsB.G.Ff'Gs Ff'B.CsG.Es CsGsDsFf'Es c, s, n,c, Ff' A.DsCsB.G. Ff'osB.CsG. CsFf'B.DsA. Ff'EsCsA.Gs G. Ff'GsB.A. A.GsEsB.Ds A.Ff'EsG.B. Ff'GsG.EsB. CsG.Ff'DsGs EsA.B.GsFf' A.B.GsEsCs Ff'osB.G.Cs EsGsG.DsFf' A.DsFf'GsB. EsA.Ff'B.Gs G.A.EsB.Ds G.GsB.A.Ds G.B.EsCsDs Ff'CsDsG.Es DsG.Ff"csEs GsFf'G.B.Ds Note-All notes in each sequence last one beat. Beat duration =.5 sec. Approximate frequencies of notes are the same as in Experiment 1 (Table 1).

8 54 BALCH Table 4 Designof the Counterbalanced Conditions in Experiment 2 Counter- Melody Type balance Retrograde List Inversion Inversion Different I II-IB IIRrlIB liib-ivb 2 III-lIB IlIRrIlIB IVB-IB 3 IlII-lIIB IVRrIVB IB-IIB 4 NrIVB IRrIB IIB-IIIB Note-The entries under each melody type are two-part melodies. Each part is drawn from one of the four sets of six sequence groups (see Table 3 for specifications). The sets are labeled I through IV. Set I =Sequence Groups1 6, Set II = 7-12, Set III =13-18, Set IV= Each counterbalance list contains 18 different two-part melodies. Total number ofmelodies = 72. Subscripts refer to the sequence types in Table 3. B = base, I =inversion, RI =retrograde inversion. The two sequences for each melody share the relation on which the columns of this table are based. 2, respectively. The retrograde inversion (RI - D) effects were.59 cm in Experiment 1 and.24 em in Experiment 2. GENERAL DISCUSSION One implication of these results is that melodic symmetry must be recognized to be preferred. Inversions (in both experiments) and retrogrades (in the first experiment) were the preferred continuations, while retrograde inversions were not rated significantly better than were the control melodies. This pattern of results was comparable to Dowling's (1972) findings using a recognition procedure. In the present study, the symmetries were not explained or demonstrated to subjects, as in Dowling's study. Rather, the judgment involved was an aesthetic one, in which the specific criteria depended on the subjects' own preferences. Since the rating results obtained here paralleled the recognition results found by Dowling, it seems plausible that an aesthetic preference for a symmetry depends on the recognition of the relation involved in that symmetry. In turn, the recognition of a symmetry suggests a cognitive process involving both memory and cognition. Dowling (1972, 1978) suggests that musical contours are held in short-term memory, and mentally rotated and compared with one another. Presumably, subjects in the present experiment were using a similar process in evaluating the continuation between the parts of a melody. Whether or not the comparison was between each part taken in its entirety is a question that requires further research. For example, it has already been mentioned that retrogrades might have been preferred at least partially because the last pitch of the first part and the first pitch of the second part were the same. The inversion relation used in this study avoided that problem, However, subjects still may have been basing their judgments about inversions on selected pitches in each part. For instance, several subjects reported after the experiment that they preferred melodies in which the last two or three notes of each part "went in opposite directions." While the details of the processing mode used by the subjects cannot be precisely determined, the judgments must have been based to some extent on a cognitive analysis on the part of the subjects. This conclusion seems particularly appropriate for inversions, which appeared in both experiments and were significantly preferred in each. The present fmdings thus appear to be a case in which an aesthetic reaction depends upon cognitive processing. This phenomenon is interesting in the light of Zajonc's (1980) recent argument that affective judgments in general may be independent of cognitive processing. The evaluation of good continuation in melodies may be a counterexample of his assertion that "preferences need no inferences." Of course, the good continuation rating was chosen purposely for the present study because of its emphasis on the relationship between two melody parts. The results would not necessarily have been the same with a still more subjective judgment, such as like/ dislike ratings, a more common index of affect. A second aspect of the results which is of theoretical interest is the direction of the obtained symmetry effects. Clearly, detectability of a relation does not require that it be a preferred continuation. However, the positive effect found here is consistent with the results of a study in the aesthetics of visual symmetry (Szilagyi & Baird, 1977), as well as with Garner's (1970) theory of pattern goodness. Thus, it seems reasonable that the redundancy of the two melody parts due to symmetry should facilitate an aesthetic response to that melody. Beyond the consideration of formal redundancy, symmetrically related sequences may also be similar perceptually. This interpretation is suggested by the finding that retrogrades and inversions have been found to have nearly equivalent characteristics of short-term recall. In one study, Royer and Garner (1970) studied subjects' recalled descriptions of dichotomous tone sequences. These auditory patterns could be organized in a number of ways, since their starting and ending points were arbitrary. Subjects' descriptions showed that when a particular organization was preferred, so was its retrograde. Regarding the inversion, Restle (1970) obtained serial position curves for his subjects' recall of temporal visual patterns. The inversion of a given sequence was found to have an error profile resembling that of the original sequence. In both studies, then, the response to a transformed sequence resembled the response to the original one. These findings suggest

9 SYMMETRY 55 that symmetrically related sequences are similarly perceived. Thus, the subjects in the present study may have heard the two melody parts as resembling, or being consistent with, one another in the case of inversion and retrograde relations. In contrast, the control melodies had two unrelated parts. These "different" sequences may have been perceived as less consistent with each other, and therefore poorer in continuation from the first to the second part. Regarding the measurement of the symmetry effects in this study, certain methodological choices were made in an effort to achieve some degree of musical naturalism. First, rhythms were chosen which avoided melody parts of more than five pitches with equal rhythm. Secondly, diatonic scaling was used because it is prevalent in Western music. An important decision to control for the tonal function of the notes was made, particularly in Experiment 1, in which all eight diatonic pitches within an octave appeared in each sequence. However, this procedure resulted in average interval sizes that were larger than those typically found in Western music (see Dowling, 1978; Jeffries, 1974). This artifact serves to limit the degree to which the experimental sequences represent conditions used in normal composition. In any study of melody perception, the considerations of naturalism and experimental control must be traded offagainst each other. A useful attribute of the symmetry manipulations studied here is that an internal axis of symmetry, across which the two melody parts reflect each other, is created. Although internal symmetries have been studied here in an aesthetic context, they should also be examined in regard to their role in melodic memory. That is, work should be done on the effect of symmetry within a melody on the memorability of that melody. REFERENCES BEVER, T. G., & CHIARELLO, R. J. Cerebral dominance in musicians and non-musicians. Science, 1974,185, BRAGG, B. W. E., & CROZIER, J. B. The development of age of verbal and exploratory responses to sound sequences varying in uncertainty level. In D. E. Berlyne (Ed.), Studies in the new experimental aesthetics. Washington, D.C: Hemisphere, DOWLING, W. J. Recognition of inversions of melodies and melodic contours. Perception & Psychophysics, 1971,9, DOWLING, W. J. Recognition of melodic transformations. Perception & Psychophysics, 1972, 12, DOWLING, W. J. Scale and contour: Two components of a theory of memory for melodies. Psychological Review, 1978, 85, DOWLING, W. J., & FUJITANI, D. S. Contour, interval and pitch recognition in memory for melodies. Journal of the Acoustical Society ofamerica, 1971,49, GARNER, W. R. Good patterns have few alternatives. American Scientist, 1970, 58, JEFFRIES, T. B. Relationship of interval frequency count to ratings ofmelodic intervals. Journal ofexperimental Psychology, 1974,102, JELLISON, J. A. Accuracy of temporal order recall for verbal and song digit-spans presented to right and left ears. Journal of Music Therapy, 1976,13, KRUMHANSL, C. L. The psychological representation of musical pitch in a tonal context. Cognitive Psychology, 1979, 11, PRINZMETAL. W., & BANKS, W. P. Good continuation affects visual detection. Perception & Psychophysics, 1977, 21, RESTLE, F. Theory of serial pattern learning. Psychological Review, 1970,77, RoYER. F. L., & GARNER, W. R. Perceptual organization of nine-element auditory temporal patterns. Perception & Psychophysics, 1970,7, 1l SLOBODA, J. A. Visual perceptionof musical notation: Registering pitch symbols in memory. Quarterly Journal ofexperimental Psychology, 1976,28,1-16. SLOBODA. J. A. Perception of contour in reading music. Perception, 1978, SZILAGYI. P. G., & BAIRD, J. C. A quantitative approach to the study of visual symmetry. Perception & Psychophysics, 1977, 22, VITZ, P. C. Affect as a function of stimulus variation. Journal ofexperimentalpsychology, 1966,71, ZAJONC, R. B. Feeling and thinking: Preferences need no inferences. American Psychologist, 1980,35, NOTE I. The melodies of each counterbalanced list were presented in a random order. The actual list sequences can be obtained from the author. (Received for publication July25, 1980; revision accepted November 5,1980.)