examples of these transformations Recognition of melodic transformations: such as were used in the present experiment: (C) is an inversion of (A), (E) is a retrograde of (A), and (G) is a Inversion, retrograde, and retrograde inversion of (A). One issue raised by the present retrograde inversion* study is whether transformations of melodic material can function as an actually perceived aspect of musical W. J. DOWLING structure or whether they must be University of California, Los Angeles, California 90024 relegated to the category of merely intellectual conceits of the composer. The melodic transformations of inversion, retrograde, and retrograde inversion The case is strongest for inversions. occur in pieces of music. An important question is whether such manipulations Dowling (1971) has shown that of melodic material are perceptually accessible to the listener. This study used a listeners do better than chance in short-term recognition-memory paradigm and found that in the easier conditions recognizing inversions in a task similar all these transformations were recognized with better than chance accuracy. The to that of the present study. And the ascending order of difficulty was: inversion, retrograde, retrograde inversion. inversion transformation is very similar There was no evidence that listeners distinguish between transforms that to the "mirror" transformation Restle preserve the exact interval relationships of the standard stimulus and those that (1970) includes in his structural merely preserve its contour (pattern of ups and downs). In view of the order of theory of serial pattern learning. (The difficulty of the transforms, two theoretical explanations of performance are difference lies in the fact that Restle's possible (1) Listeners may perform the mental transformation required by the theory allows the mirror recognition task on a representation of the vector of pitches in the standard-an transformation of a single element, operation that is very like transforming a mental image of the written notation. whereas melodic inversion is only (2) Listeners may handle inversions differently from the other transformations, possible on a pattern of at least two comparing the standard and the comparison contour element by contour elements.) There is more question element, in temporal order. In this view, the temporal dimension would appear concerning the perceptual accessibility to have precedence over the pitch dimension in the musical structure, in of retrogrades and retrograde consideration of the consequences of disturbing it. inversions. In various periods of western music, these transformations How the perceiver recognizes the common and is found in nonwestern have figured in the construction of stimuli in his environment under their (for example, Indonesian) as well as in riddle canons and the like, where the various transformations of shape and western music. Retrograde and composer wrote down only a cryptic size is a basic problem in the study of retrograde inversion, though more set of notes with enough hints so that perception. One version of this rare, can be found throughout the the whole piece could be problem in audition is how the listener history of western music. Tovey reconstructed, as for example with recognizes melodic transformations (1956) gives several examples from Machaut's hint "Ma fin est mon commonly used in music. In music, various periods. Figure 1 shows commencement [Apel, 1944]." For melodies already known to the listener may be speeded up or slowed STANDARD DIFFERENT down-diminution and augmentation. The pitch-interval sizes between the notes of the melody may be altered. • b. • • • Melody recognition with altered pitch O. • .,.... -.. relationships has been studied by ~ Frances (1958), White (1960), and Dowling & Fujitani (1971). These transformations may operate on a TRANSFORMS whole melody or on only isolated elements-as in the temporal spacing EXACT CONTOUR of the intact phrases of a hymn in a baroque chorale prelude, or as in the alteration of only certain pitch relationships in the "tonal answer" to INVERSION c. d. a fugue subject (Tovey, 1956). ~ • -;- *. Another set of transformations operates on the melodic pattern as a whole, turning it upside down, backwards, or upside down backwards RETROGRADE •. ~ J;JEf: f. • in the pitch-duration domain: the transformations of inversion, retrograde, and retrograde inversion. Of these, inversion is the most RETROGRADE ~ h. *This research was supported in part bY a INVERSION g. . grant from the UCLA Academic Senate. Computing facilities were provided by the Department of Psychology, UCLA. I thank Michael Lisinsky, Edward Carterette, Fig. 1. Examples of stimuli used in the experiment. (A) Standard stimulus; Caroline Dowling, Dan Rourke, James (B) different comparison stimulus; (C), (E), and (G) exact-interval size-preserving Thomas, Roberta Greer, Kelyn Roberts, and Dane Harwood for valuable assistance and transforms of the standard; (D), (F), and (H) contour-preserving transforms of comments. the standard.

Perception &; Psychophysics, 1972, Vol. 12 (5) Copyright 1972, Psychonomic Society, Inc., Austin, Texas 417 examples of riddle canons by Bach, see sizes in semitones. (See Dowling & memory for inversions, the contour David and Mendel (1966). Tovey Fujitani, 1971, for more details.) For seems to be all that is remembered. (1956) doubts that retrogrades and example, the intervals of Fig. 1A and That result is checked here with retrograde inversions are effective Expression 1 are: retrogrades and retrograde inversions perceptually, except in a very few as well as inversions. cases, and in those cases he points out +2 +1 -2 +3 . (5) A question raised by the present that rhythm seems to be the decisive study is: Which of the above factor, not melody. And Frances The inversion may be derived by characterizations of the process of (1958) expresses reservations about multiplying Expression 5 by -1: transforming a melody is more like the effectiveness of these melodic what listeners do when they recognize transformations as a cohesive force in -2 -1 +2 -3 . (6) transformed melodies? Plausible 12-tone compositions. The present arguments can be marshaled on both study will not attempt to demonstrate The retrograde inversion may be sides of the question. In favor of the effectiveness of these obtained by reversing the order of the operations on vectors of interval sizes, transformations in actual music, but intervals in (5). Observe that the there is (A) the fact that melodies are only the existence of one of the retrograde inversion of the intervals is not changed by transposition, at least necessary conditions of that obtained by taking the retrograde of in long-term memory. The fact that effectiveness: the fact that listeners the pitches. Musicians use the pitch melodies become less recognizable can recognize such transformations terminology rather than the interval with transposition in short-term when they hear them under certain terminology, and I will do the same memory has been explained in two limited circumstances. here to avoid confusion. Experssion 7 ways: first, Dowling and Fujitani A second issue raised here is: given represents the retrograde inversion of (1971) argue that nontransposed that listeners can succeed in (5 ): melodies are easily recognized (in recognizing these three types of distinction to their particular transformation, what psychological +3 -2 +1 +2 . (7) confusion set) simply on the basis of processes might be involved? There are their (unordered) set of pitches and two convenient ways of characterizing The retrograde may be obtained by not the pitch relationships which make melodic transformations in formal reversing the order of (5) and then up the "melody." Second, Frances terms: as operations on intervals and multiplying by -1: (1958) and Dowling! argue that, at as operations on pitches. The former is least for tonal melodies, a scale and a more convenient for purposes of -3 +2 -1 -2 . (8) tonality provide a framework of pitch formal description, but there is reason functions for remembering the to believe that people tend to think Note that the retrograde of a melody melodic pitch relationships which and perceive in terms of the latter. The takes two operations on the vector of carries over from standard to characterization of melodic interval sizes, while the inversion and comparison stimuli. (B) The impor­ transformations in terms of pitches retrograde inversion take only one. In tance of contour for short-term can be derived by first assigning pitch actual musical practice and in the memory noted above makes the numbers in semitones above some present experiment, these interval size characterization, which arbitrary reference tone, for example, transformations are almost always codes contour directly, more middle C. Figure 1A would then be accompanied by transposition, which attractive. represented as: in the pitch characterization amounts In favor of operations on vectors of to adding a constant to the pitches and pitches is (A) musicians' tendencies to 7 9 10 8 11 . (1) in the interval characterization think in terms of pitches (or images of amounts to selecting a different pitches represented in notation) when The inversion of (1), shown in starting point. describing the transformations-their Fig. 1C, is obtained by subtracting the Some of the transformed stimuli use of descriptions like "upside down" pitches of (1) from a constant, for used in the present study are for inversion and "upside down example, 12: transforms of the contour (the pattern backwards" for retrograde of ups and downs) of the melody, inversion-which recall the operations 53241. (2) without preserving the exact interval on the pitch vectors outlined above. sizes of the original. In Fig. 1, Sometimes this tendency to operate The retrograde of (1) (Fig. IE) is Examples D, F, and H are such on the notes themselves has led obtained by reversing the order of contour transformations. They are composers to write so that turning the pitches: most conveniently described as page upside down would lead to a preserving the vectors of signs of viable retrograde inversion, or reading 11 8 10 9 7 . (3) Expressions 6; 7, and 8, but not the each line of music backwards a viable magnitudes. Thus the contour retrograde. (B) Inversion and The retrograde inversion of (1) inversion of (5) is: retrograde, the simpler (Fig. 1G) is obtained by subtracting transformations of pitches, are the the pitches from a constant and -- +- (9) more prevalent transforms in Western reversing their order-that is, the music, while retrograde inversion, a reversal of (2): simpler transform than retrograde in the interval characterization, is 14235. (4) Contour transformations are included comparatively rare. along with exact transforms in this The results of the present study Note that the inversion and study because in actual musical bear directly on the question of which retrograde transformations take one practice these transforms, at least in kind of process listeners use when they operation on the vector of pitches, the two kinds of inversion, tend to be recognize these transforms. If they use while retrograde inversion takes two contour-preserving, since exact interval transformations on intervals, then operations. size preservation would destroy the retrograde should be the hardest to The description in terms of intervals tonal scale relationships of the original recognize; if they use pitch relies on a characterization of a (Apel, 1944). Second, Dowling (1971) transformations, then retrograde melody as a vector of signed interval showed that in short-term recognition inversions should be hardest. It is

418 Perception & Psychophysics, 1972, Vol. 12 (5) beyond the scope of this study to effectively to the listeners the undergraduates from in troductory distinguish between processes which definitions of the various transforms psychology courses assigned to the 15 operate on pitches as a vector of pitch to be recognized. (For example, until groups. Groups varied in size due to numbers and processes which operate they actually worked it out in the scheduling problems, and the smallest on images of pitches (as of notes on a ineluctable visual modality, many Ss group contained 9 Ss. For this reason, page). I will not try to maintain that found it hard to believe that the an unweighted-means analysis of the latter is the case, but only that in retrograde inversion of the contour of variance model was used. Ss came view of listeners' and my own a monotonically ascending melody was from a population shown in previous introspections it seems highly itself a monotonically ascending studies (Dowling, 1971; Dowling & plausible. Ss ' reports of mental melody.) Fujitani, 1971) to have a median manipulation of the shapes sound very amount of musical experience of 1.0 similar to the mental rotations of METHOD year and a mean of 2.25 years. Groups visual shapes described by Shepard and The experiment was organized into were assigned randomly to conditions. Metzler (1971). a 5 conditions by 3 transforms by 2 The present experiment utilizes a stimulus types factorial design. Stimuli short-term recognition-memory Orthogonal comparisons were planned A Hewlett-Packard 211GB computer paradigm in which the comparison to test differences among the generated the stimuli, which consisted stimulus is one of the three transforms conditions. The first four conditions of sawtooth waves. Stimuli were defined above of the standard. constituted a 2 by 2 factorial design of recorded on tape and played to Ss over Separate groups of listeners performed instructions [exact recognition (E) and high-quality sound-reproduction the task with different contour recognition (C)] vs rate [fast equipment. On each trial, the transformations. I expected that (F) and slow (S)]. Planned computer generated a new five-note (1) performance would be better than comparisons were designed to test the standard stimulus by starting on chance, thus demonstrating the main effects of instructions and rate. middle C (262 Hz) and selecting possibility of the transforms' being The first four conditions were all succeeding notes such that intervals perceived in music. (2) Inversions caried out without the analogous between successive notes occurred should be more easily recognized than visual task preceding the recognition with the following probabilities: P(±1 the other transforms, since these are task. The fifth condition (VEF) was a semitone) = .5, P(± 2 semitones) = P(± 3 most common in music and both of replication of the exact-recognition semitones) = .25. The comparison the theoretical formulations above fast condition (EF) but included the melody started on a different note place inversions among the more easily visual task. The effect of the visual selected at random from the 14 notes handled perceptually. Whether task was tested by comparing of a chromatic scale 1-7 semitones retrogrades or retrograde inversions are Condition VEF with the other four higher or lower than middle C. Stimuli easier should help decide whether the conditions in a planned orthogonal were presented at rates of 5 tones/sec pitch vector or the interval vector, comparison and also informally (Conditions EF, CF, and VEF) or respectively, represents the more checking against Condition EF. The 2 tones/sec (Conditions ES and CS). nearly correct description of the three transforms were inversion (I), Comparison stimuli followed standard listener's behavior. (3) The issue of retrograde (R), and retrograde stimuli by 2 sec. Ss had 5 sec to whether the exact interval sizes or just inversion (RI). The two stimulus types respond. A warning tone (4,250 Hz) the contours are preserved in these were exact-interval size preserving and preceded the onset of each trial by transformations is tested by presenting contour preserving. This last variable 2 sec. both exact and contour transforms to proved ineffectual, and the results Exact-interval size-preserving all groups of listeners. Half of the discussed below are presented as comparison stimuli were exact groups had instructions to respond collapsed across it. inversions, retrogrades, and retrograde positively only to exact transforms, Fifteen groups of Ss served in the inversions of the standard stimuli, each the other half to all contour 15 cells of the 5 conditions by 3 beginning on a new note. transforms including the exact ones. transforms design. Each group Contour-preserving comparisons Any tendency of Ss to differentiate performed the recognition-memory contained newly selected intervals, between these two types of transforms task with the two types of comparison different from the corresponding should show up either as a main effect stimuli: transforms preserving the intervals of the standard but with the of the comparison-stimulus types exact interval sizes of the standard and same contour transformed in the (exact vs contour) or as an interaction transforms of the contour of the appropriate way, beginning on a new of instructions (exact vs contour) with standard. Positive recognition (hit) note. Different comparisons were the stimulus types. rates for each of these comparison selected just as the standards, but with (4) In pilot studies, I noticed that Ss types were plotted against the constraint that the contour be found the task extremely difficult, false-positive response rates to different in at least one interval especially in the retrograde-inversion random, different comparison stimuli direction from the appropriate condition. Therefore, I devised a more for purposes of calculating areas under transform of the standard. extensive set of instructions involving the memory operating characteristic New selections of standards were an analogous visual task. Three groups (MOC) as an estimate of percentage made randomly by the computer for. of Ss did the visual task before correct in a two-alternative each of the 15 experimental groups. performing the rest of the experiment. forced-choice task (see Norman & Each new standard was constructed The visual task consisted of a Wickelgren, 1965, for details). That is, independently of all previous recognition test of transforms of visual all groups were scored the same standards. forms similar to notational regardless of instructions. Each group representations of the melodies in the had 15 trials of each stimulus Procedure auditory experiment. I expected that type-exact transform, contour On each of the 45 trials of the doing this visual task would lead to transform, and different-making 45 experiment, S heard a standard improved performance on the auditory trials in the session. melody followed by a comparison task. The main function I saw in the melody. S's task in Conditions EFI visual task was as a set of improved Subjects and ESI was to say whether the instructions communicating more Th ere were 355 UCLA comparison melody was an exact

Perception & Psychophysics, 1972, Vol. 12 (5) 419 Table 1 conditions. Retrogrades were Areas Under the MOe Averaged Across Types of Stimuli recognized better than chance in all Condition except Condition EF. Retrograde inversions were recognized better than Transforms EF ES CF CS VEF Mean chance only in the CS and VEF Inversion .67 .70 .70 .80 .70 .70 conditions. Retrograde .54 .79 .59 .67 .65 .64 Retrograde Inversion .53 .53 .50 .68 .60 .55 DISCUSSION These results clearly demonstrate Mean .59 .64 .61 .73 .65 t hatinversions, retrogrades, and retrograde inversions of brief melodies inversion of the standard or not. Ss were exact. Ss were given feedback as can be recognized with better than responded on a four-category to their correctness after the 10 chance accuracy. The most confidence level scale with categories problems had been completed. Ss then troublesome of these, the retrograde labeled "sure same" "same" proceeded to the main part of the inversion, was recognizable in one "different," and "sure different." The experiment. condition at the slow rate and was response "same" in this case meant recognizable at the fast rate when the that the comparison stimulus was an RESULTS session was preceded by the visual task exact inversion of the same melody as In the 5 by 3 by 2 analysis of clarifying the notion of retrograde the standard. No feedback was given variance, the main effects of inversion. Retrogrades were recognized Ss. The tasks in Conditions EFR, ESR, conditions [F(4,680) = 3.73, P < .01] with better than chance accuracy in all EFRI, and ESRI were similar except and transforms [F(2,680) = 11.10, but Condition EF. Although the that Ss in those conditions had to P < .001] were significant. The present experiment presented brief recognize exact retrogrades and descending order of difficulty was melodies in isolation from any retrograde inversions. These conditions inversion, retrograde, and retrograde confusing background and in that way were preceded by instructions inversion. The main effect of stimulus made them easier to recognize than explaining the nature of the transform types was not significant, and, for they would have been in an actual to be recognized and the importance purposes of further description, the musical context, there are several of distinguishing between exact data are collapsed across stimulus reasons why recognition in the transforms and merely types. The only significant effect experiment should be more difficult contour-preserving transforms. Three involving stimulus types was a than in music. (1) Atonal melodies samples of each of the three trial Stimulus Types by Transforms such as those used here are typically types-exact, contour, and interaction [F(2,680) = 4.48, P < .01] difficult to deal with in recognition different-accompanied the due almost entirely to the easiness of experiments because of their departure instructions. the contour stimuli in the R from the well-learned scale functions Ss in Conditions CFI, CSI, CFR, conditions, especially the RS the listener hears in the rest of his CSR, CFRI, and CSRI did a similar condition-overall a difference of 71 % musical experience (Frances, 1958). task to Ss in the other conditions, correct vs 57% correct. Whether this is (2) The intervals used in the present except that instead of recognizing to be attributed to the ease of stimuli are much smaller than those exact transforms they had only to recognizing retrograde contours at encountered in normal melodies. The recognize transforms of contour. slow speeds or to the nature of the median interval size of one semitone is Instructions to these Ss explained the particular stimuli generated in this much smaller than the median of nature of the transform to be experiment is not settled. The planned about three semitones found in most recognized and the concept of the comparison of fast vs slow rates was melodies (Dowling & Fujitani, 1971). countour-preserving transform as well significant [F(1,680) = 4.25, P < .05], (3) In actual music, there is a rhythmic as of the exact transform, but with the with the slow rate easier (60% vs 68% dimension present that was avoided in added stipulation that for the purposes correct). The planned comparisons of this experiment. White (1960) found of this experiment the two were' to be instructions and visual task were that familiar melodies could be considered equivalent. Again, sample nonsignificant. In order to check recognized with better than chance trials accompanied the instructions. further on the effect of the visual task, accuracy on the basis of their rhythm Ss in the three VEF conditions did I made a post hoc test on the al on e. Presumably rhythmic the same task as those in the EF difference between the VEF and EF differences in actual music would serve conditions, but with the whole session conditions. The difference, although in to differentiate among alternatives, initiated with a visual recognition task the predicted direction (65% vs 59% not only for inversions in which analogous to the auditory task. The correct), was not significant. rhythmic patterns remain unaltered, task consisted of 10 problems. For Table 1 shows the data collapsed but also in the retrogrades in which each problem, there appeared at the across stimulus types so that, in effect, characteristic clusters of longer or left-hand side of the page a visual form the "same" responses to both exact shorter notes would still be (standard stimulus) consisting of five and contour stimuli are counted distinguishable. (4) In the construction dots equally spaced from left to right correct. It is evident that the ascending of a piece of music, composers usually of the standard, and, separated from it order of difficulty of transforms is I, choose melodic material so that by a vertical rule on the page, were R, RI; and that this is roughly true of separate melodies will be clearly four comparison stimuli constructed in all but the ES conditions. All of the distinguishable from each other. In a manner similar to that of the slow presentation rate groups are recognition experiments like the standard. One of these comparisons superior to their corresponding fast present one, however, materials are was the appropriate transform of the groups, except ESRI, which is equal to selected to provide maximal standard, and S's task was to select the EF RI. There is little difference homogeneity. Thus, the set of right one and put its number in the between groups EFI and VEFI but alternative stimuli in music is divided right-hand margin of the page. The greater difference due to the visual into a few clearly defined sets with problems included all the various task in the VEFR and VEFRI groups. corresponding responses. This contour shapes made with five points Inversions were recognized with situation is made explicit in Frances's and their inversions. All transforms better than chance accuracy under all (1958) Experiment X, in which Ss

420 Perception & Psychophysics, 1972, Vol. 12 (5) listened to a Beethoven scherzo, the pitch-vector characterization of performance over straight melody tapping once each time they heard the the process by which Ss handle the recognition (Dowling, 1971). In such a first theme and tapping twice each task seems the more plausible view, the temporal dimension would time they heard the second theme. In psychological model than does the take precedence over pitch in contrast to Frances's situation with interval-vector characterization. Ss importance, in the sense that its stable stimulus classes distinguished often described their method of doing disturbance is more disruptive of along several dimensions at once, the transformations in terms of recognition behavior. When Dowling's stimuli in the present experiment were reversing an image of the notes, or of (1971) data are used as a baseline, distinguished on only a few turning it upside down-descriptions performance is seen to drop from 87% dimensions and the positive stimulus compatible with the pitch-vector correct to 70% or 73% with pitch set changed on each trial. model. Thus, Ss' introspections agreed inversion. However, performance Dowling's (1971) result that exact with those of most musicians. drops from 87% to 64% with temporal interval size information becomes lost However, some reservations about this reversal. The tendency of some in recognition of inversions was conclusion should be noted. The contemporary music theorists to view replicated and extended to retrogrades retrograde and inversion transforms the pitch and duration domains as and retrograde inversions. The only were not equally well recognized, as analogues to the two dimensions of exception to the failure of either the pitch-vector model would predict. the painter's canvas seems to run into contour stimuli or contour As good a case for the easiness of difficulties with the nature of musical instructions to make a difference in Ss' inversions as opposed to the other two perception. behavior compared with exact stimuli transformations could be made as for and instructions was the comparative the difficulty of retrograde inversions. REFERENCES 2 APEL. W. The Harvard dictionary of music. easiness of the retrograde stimuli in Shepard has suggested that the Cambridge, Mass: Harvard University the slow conditions. This could be inversions might be easier because Press, 1944. because the particular stimuli among the three they allow for the DAVID, H. T., & MENDEL, D. A Bach generated by the computer in those reader. New York: Norton, 1966. element-by-element comparison of the DOWLING. W. J. Recognition of inversions conditions were easier to recognize in comparison stimulus with memory of of melodies and melodic contours. their transformations. (In my opinion, the standard in the same temporal Perception & Psychophysics, 1971, 9. linear ascending and descending forms 348-349. order. Suppose the standard is coded DOWLING, W. J.. & FUJITANI, D. S. were easily recognized, but such an as in Expression 9: "Down, Down, Up, Contour, interval. and pitch recognition opinion would require checking by a Down." Then, if S hears a stimulus in memory for melodies. Journal of the separate experiment.) It is a defect of going "Up, Up, Down, Up," he need Acoustical Society of America,.1971, 49, the present design that the computer 524·531. only compare the two, using the rule FRANCES. R. La perception de la musique. was left to generate each new standard that for each "Up" there should be a Paris: J. Vrin. 1958. independently rather than select it corresponding "Down" in memory, NORMAL, D. A., & WICKELGREN, W. A. from a controlled population. I and vice versa. This kind of procedure Short term recognition memory for digits and pairs of digits. Journal of attribute the unexplained variance will not work for retrogrades and Experimental Psychology. 1965, 70, here to lack of control over the retrograde inversions. In those cases, 479-489. standard stimulus forms and suggest the order of one of the stimuli must be RESTLE, F. Theory of serial pattern reversed at some point and the learning: Structural trees. Psychological that future experiments exert more Review, 1970, 77, 481-495. control. comparison made with one reversed SHEPARD, R., & METZLER, J. Mental Slow presentation conditions were and one nonreversed stimulus. rotation of three-dimensional objects. easier than fast. Further work is Reversal can only occur after storage Science, 1971, 171, 701-703. TOVEY, D. F. The forms of music. needed to decide whether the of the whole stimulus has occurred. Cleveland: Meridian Books, 1956. advantage of slow presentation lies Thus, in this view, performance with WHITE. B. Recognition of distorted chiefly in Ss' being better able to store the retrogrades and retrograde melodies. American Journal of the stimulus when he hears it or in the inversions would be considerably Psychology, 1960. 73, 100-107. added time he has to manipulate the complicated over that for inversions. NOTES material. Future experiments should In this view, the increased difficulty of 1. Dowling, W. J. Musical scales as explore the relative effects of changing retrograde inversions over retrogrades cognitive structures. In preparation. 2. R. Shepard, personal communication, presentation rates within stimulus would be explained simply by the 1972. patterns and of changing interstimulus added complication of having to intervals. perform the inversion Inasmuch as retrograde inversions transformation-an operation that (Accepted for publication April 3, 1972; were the most difficult to recognize, causes a significant decrement in revision received July 2,1972.)

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