questioning a melodic archetype: do listeners use gap-fill to

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Questioning a Melodic Archetype: Do Listeners Use Gap-Fill to Classify Melodies? Author(s): Paul von Hippel Source: Music Perception: An Interdisciplinary Journal, Vol. 18, No. 2 (Winter, 2000), pp. 139-

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Music Perception © 2000 by the regents of the university of California Winter 2000, Vol. 18, No. 2, 139-153 all rights reserved.

Questioning a Melodic Archetype: Do Listeners Use Gap- Fill to Classify Melodies?

PAUL VON HIPPEL

Ohio State University and Stanford University

Leonard B. Meyer (1973) argued that listeners' experience of melodies is shaped by certain melodic "archetypes." Among these archetypes is "gap- fill," a name for melodies in which an early skip is followed by some of the pitches that have been skipped over. In experiments conducted with Rosner, Meyer tested gap-fill's effect on the ways in which listeners compare and classify melodies (B. S. Rosner & L. B. Meyer, 1982, 1986). The present reanalyses of Rosner and Meyer's experimental results, however, suggest that gap-fill played little or no role. Together with an earlier study suggesting that gap-fill has no influence on melodic shape (P. von Hippel & D. Huron, 2000), these reanalyses tend to weaken the claim that gap-fill is an important concept for classifying melodies.

Received June 14, 1999; accepted July 8, 2000.

is one of the oldest tasks in music analysis. Throughout history, scholars have classified music according to mode, meter, char-

acter, and social function, as well as other qualities. Ordinary listeners, too, are remarkably adept at classifying music, as demonstrated by their

split-second ability to recognize the format of a radio station (Perrott & Gjerdingen, 1999).

Psychologists studying musical classification have often invoked the con- cepts of prototype or schema. A schema is a mental representation for the way that features fit together in a familiar setting - for example, the objects and events to expect in a restaurant (Graessner & Nakamura, 1982; Schank & Abelson, 1977). In music, schematic knowledge enables listeners to pre- dict the pitches and chord progressions that are most likely to occur in a given key (Krumhansl & Castellano, 1983). Schémas are often related to prototypes, which are exemplary or idealized representatives of a class; a robin, for example, is a highly prototypical bird. Prototypes are used as

Address correspondence to Paul von Hippel, School of Music, Ohio State University, 1866 College Ave., Columbus, OH 43210. (e-mail: [email protected]).

ISSN: 0730-7829. Send requests for permission to reprint to Rights and Permissions, University of California Press, 2000 Center St., Ste. 303, Berkeley, CA 94704-1223.

139



140 Paul von Hippel

mental references to which other members of a class are compared (Posner & Keele, 1970; Rosch, 1978). In music, for example, listeners tend to hear chromatically inflected melodies as departures from diatonic prototypes (Bartlett & Dowling, 1988; Cohen, Thorpe, & Trehub, 1987).

Among psychological theories of melodic classification, the ideas of Leonard Meyer (1973) are strikingly ambitious. Instead of classifying melodies by such basic qualities as chromatic or diatonic content, Meyer has proposed a more refined classification based on a handful of melodic "archetypes." Archetypes, according to Meyer's colleague Robert Gjerdingen, are "[Meyer's] term for innate or universally valid schemata" (Gjerdingen, 1988, p. 7). Meyer sometimes refer to archetypes as "archetypal schemata" - a usage that corroborates Gjerdingen's definition (Meyer, 1973). Meyer's archetypes have also been discussed under the heading of prototypes (Gjerdingen, 1991, p. 131). The connection between archetypes and prototypes is not hard to see: when Meyer (1973) discusses archetypes in terms of exemplary cases, or when he rates musical examples according to how well they exemplify a given archetype (Rosner & Meyer, 1986, Tables 1-5), one could certainly imagine that it is prototypes that are under dis- cussion.

Despite these correspondences, it is not clear that Meyer would claim for archetypes all of the measurable effects that are associated with prototypes and schémas. For example, schémas often induce memory errors: features that violate the prevailing schema tend to be forgotten, and features that fit the prevailing schema tend to be remembered even if they did not occur (Brewer &c Treyens, 1981). A distinctive feature of prototypes, on the other hand, is asymmetries in perceived similarity: when a prototype is at work, the perceived similarity of two items depends on the order or the grammatical relationship in which they are presented (Tversky, 1977; Bartlett & Dowling, 1988). These characteristic effects of prototypes and schémas have never been claimed for Meyer's archetypes. Instead, with the exception of two experiments to be discussed in this article (Rosner &c Meyer, 1982, 1986), Meyer's archetypes remain a theoretical concept rather than an operational one.

The archetypes identified by Meyer include "linear," "triadic," "comple- mentary," "changing-note," and "Adeste Fidèles" melodies (Meyer, 1973; Rosner &c Meyer, 1986). Perhaps the most important of Meyer's archetypes, however, is "gap-fill." A gap-fill melody typically begins with a large skip (or gap), then continues by filling in scale tones that have been skipped over. Figure 1 displays Meyer's most straightforward examples of gap-fill: the chorus from the show tune "Over the Rainbow" and the fugue subject from Geminiani's Concerto Grosso in E Minor, op. 3, no. 3 (Rosner & Meyer, 1982).



Do Listeners Use Gap-Fill to Classify Melodies* 141

Fig. 1. Two gap-fill melodies. Top: the chorus from "Over the Rainbow," by Harold Arlen and E. Y. Harburg. Bottom: the first-movement fugue subject from Geminiani's Concerto Grosso in £ Minor, op. 3, no. 3. Both melodies begin with large skips or gaps that are gradually filled in.

Meyer has written more about gap-fill than about any other archetype (Meyer, 1956, 1973; Rosner & Meyer, 1982, 1986). Perhaps this is because the gap-fill concept fits so well with Meyer's ideas about melodic shape and expectation. Meyer (1956) has claimed that listeners, after hearing a gap, expect it to be filled in - a claim that fits the results of several cognitive experiments (e.g., Schellenberg, 1997; Schmuckler, 1989). Meyer (1956) further claims that many melodies are constructed to satisfy an expectation for gap-fill. Indeed, he argues, this may be the reason why centuries of pedagogues have taught that a skip should be followed by a con-

trary step (e.g., Nanino & Nanino, ca. 1600; Fux, 1725/1943; Prout, 1890; Kostka & Payne, 1995). The prevalence of gap-fill melodies, in turn, might explain why listeners would develop an archetypal schema for representing them.

This web of ideas related to gap-fill has recently frayed. Statistical analyses have shown that, contrary to centuries of teaching, melodies are not

generally constructed to fill gaps. Instead, the melodic shape that Meyer calls gap-fill seems to be an artifact of constraints on range or tessitura (von Hippel & Huron, 2000). Skips tend to land near the extremes of a

melody's tessitura, and from those extremes, a melody has little choice but to retreat by changing direction - little choice, that is, but to regress to- ward the mean.

If melodies are not constructed to fill gaps, it seems reasonable to reopen the question of whether gap-fill is really a psychological archetype. To address this question, I reanalyzed two sets of experiments designed to test the archetypal status of the gap-fill pattern (Rosner & Meyer, 1982, 1986). To the degree that the experimental results can be interpreted, the reanalyses suggest that gap-fill had little if any effect on listeners. These findings encourage further skepticism regarding the psychological importance of

gap-fill.



142 Paul von Hippel

An Experiment on Learning

The earliest experiment on the gap-fill archetype was carried out by Rosner and Meyer in 1982. This experiment was designed to test the authors' claim that "archetypes are easily learned" (p. 319) - a claim that could also apply to prototypes or schémas. Specifically, Rosner and Meyer sought to test whether listeners could learn by example - that is, without explicit instruction - to recognize the presence or absence of the gap-fill pattern.

At the beginning of the experiment, listeners heard two gap-fill melodies. Listeners were not told that these melodies were called gap-fill, nor were they told the melodies' defining attributes. Instead, listeners were simply told that the melodies were "Type A." After hearing these two examples, listeners began a training session in which they learned to categorize 16 melodies by using the Type A label. Eight of these melodies were meant to illustrate gap-fill (though none was as clear a specimen as the examples in Figure 1); the remaining 8 melodies were intended to be foils. After hearing each melody, listeners guessed whether the melody was Type A, and the experimenter told them whether their answer was correct. This training session could go on until the entire set of 16 melodies had been played 12 times.

For the introductory stage of a cognitive experiment, this is quite a lot of training. Because each playing of the training set took about 8 minutes (Rosner & Meyer, 1982, p. 329), the training stage could last up to 96 (12 x 8) minutes - more than an hour and a half. To put the point another way, listeners could receive feedback on up to 192 (12 x 16) practice classifica- tions. Not all of the listeners received so much feedback, however. If a listener classified 14 of the 16 melodies correctly, and did so two times in a row, the training session ended, and the listener was considered to have passed. Of the 17 listeners who completed the study, 14 passed this training stage.

The chances are extremely remote that 14 of 17 listeners could have passed the training stage by indiscriminate guessing (p < 1058).1 It does not

1. It is not clear from Rosner and Meyer's (1982) article whether listeners were told how many "Type A" and "non-Type A" melodies they would hear. If listeners were privy to this information, then the independence of the data would be reduced, because the likelihood of a Type A answer would depend partly on the number of Type A answers that were given earlier. If the independence of the data is compromised in this way, then statistical tests in the first half of this article - and in Rosner and Meyer's (1982) original analyses - exagger- ate the significance of the results.

Assuming that the data are independent, however, the probability given in the text (p < 10"58) was calculated in five steps:

1. Suppose that a listener is guessing. With two classes available to her, her chance of correctly classifying any one melody is .5.



Do Listeners Use Gap-Fill to Classify Melodies? 143

necessarily follow, however, that listeners were attending to the gap-fill archetype. Instead, they could have passed the training stage simply by memorizing which melodies were given the Type A label. To test this possi- bility, Rosner and Meyer administered a test of generalization, in which listeners classified 12 melodies that they had not heard in the training stage. Six of these melodies were meant to be instances of gap-fill; the remaining 6 melodies were meant to be foils. Without benefit of feedback, listeners tried to determine which melodies should take the Type A label. Of the 12 melodies in the generalization test, listeners classified a mean of 7.5 correctly.

This level of accuracy, Rosner and Meyer found, was only marginally inconsistent with the idea that listeners were guessing. When a two-tailed Kolmogorov-Smirnov test (Conover, 1980) was applied to the 14 listeners who had passed training, the results only verged on statistical significance (Dmax(14) = .3271, two-tailed p < .10). The conventional threshold for significance was crossed only when the analysis included the 3 listeners who had failed the training stage (Dmax(17) = .3356, two-tailed p < .05). The statistical tests used by Rosner and Meyer, however, were two-tailed. Be- cause this experiment was concerned only with the claim that listeners' accuracy was better than chance (not worse), Rosner and Meyer could have improved the apparent significance of their results by conducting one- tailed tests - in which case the p values would be half as large.

Regardless of the level of significance, these results do not necessarily support the authors' claim that listeners could learn to classify gap-fill melodies "easily." The mean score of 7.5 correct answers is lower than it may seem. Listeners had only two class names available, so that, given 12 melodies, on average they could provide 6 correct answers simply by guessing. A mean of 7.5 correct answers, therefore, is just one quarter of the way from a chance score to a perfect one. To put the point another way, the results suggest that listeners recognized the class of just one quarter of the melodies (3 of 12), because, by guessing the class of half the remainder (4.5

2. In a single playing of the training set, a listener's binomial probability of correctly identifying at least 14 of the 16 melodies is .002. (See Howell, 1997, pp. 121-129, for the pertinent formula.)

3. Given two consecutive play ings of the training set, the listener's probability of both times reaching the specified level of accuracy is .0022.

4. Because a listener could hear the training set up to 12 times, she had 11 opportunities to reach the specified level of accuracy in two consecutive play ings. Given 1 1 opportunities, therefore, her chance of two consecutive successes is close to 11 x .0022, or 4.4 x 10"5. (The situation is analogous to a Bonferroni correction for 11 nonindependent tests - see Darlington, 1990, p. 252. The exact probability, which is closer to 4.39 x 10~5, can be obtained by a recursive procedure [Mario Peruggia, personal communication, February 24, 2000]).

5. If each listener's chance of two consecutive successes is 4.4 x 10"5, the binomial probability of 14 out of 17 listeners having two consecutive successes is less than 10"58, as claimed above.



144 Paul von Hippel

of 9), listeners could reach a total score of 7.5. A skeptic might argue that, after up to an hour and a half of training, one-quarter recognition does not suggest that a system of classification can be learned easily.

To assess ease of learning more formally, we would need to know the results of a control condition - one in which listeners learned a melodic category that was not based on a nominal archetype. Unfortunately it would be hard to design a suitable control, because Meyer offers no theoretical criteria for deciding whether a melodic pattern is or is not archetypal. With- out a control condition, however, we cannot compare the gap-fill results with a non-archetypal standard.

In sum, Rosner and Meyer's classification experiment provided no evidence that gap-fill could be learned easily - and only weak evidence that it could be learned at all. On the basis of these experimental results, it is not clear that gap-fill should be granted the status of an archetype.

Two Experiments on Melodic Similarity

In two later experiments, Rosner and Meyer (1986, Experiments 3 and 5) tested the effect of gap-fill on listeners' judgments of melodic similarity. In these experiments, listeners were given no instruction or training as to how they should classify melodies. Instead, listeners simply rated the similarity of pairs of musical passages. Twelve passages were used in these experiments. Six were meant to be gap-fill melodies; the other 6 represented an archetype that Rosner and Meyer called "Adeste Fidèles."

An Adeste Fidèles melody, according to Rosner and Meyer (1986, p. 19),

always involves two characteristic skips. The first spans a fourth and the next one a fifth. Both usually occur in the first half of the melody; the second skip leads to upward motion to the third or fourth of the scale, followed by downward resolution often to the tonic.

Two of Rosner and Meyer's examples of this archetype are displayed in Figure 2. The first example is the eponymous Christmas carol, "Adeste Fidèles (O Come, All Ye Faithful)." The second example is the opening melody from the minuet of Handel's Flute Sonata in G, op. 1, no. 5.

With respect to gap-fill, a basic question that could be asked of these experiments is whether listeners found the gap-fill melodies more similar to one another than to the Adeste Fidèles melodies. Unfortunately, Rosner and Meyer did not address this question, and their listeners' similarity judgments are no longer available for analysis (B. Rosner, personal communication, 1997). Rosner and Meyer did, however, represent the aggregate




Fig. 2. Two Adeste Fidèles melodies. Top: the Christmas carol "Adeste Fidèles (O Come, All Ye Faithful)." Bottom: the opening of the minuet from Handel's Flute Sonata in G Major, op. 1, no. 5. Both melodies have early downward skips of a fourth or fifth, followed by movement up to the third of fourth degree of the scale, followed by at least partial resolution.

data graphically in the form of multidimensional scaling plots (Kruskal & Wish, 1978) and hierarchical clustering trees (Aldenderfer & Blashfield, 1984).

On the multidimensional scaling plots, redrawn here as Figure 3, melodies that listeners rated as similar are plotted close together, whereas melodies that listeners rated as dissimilar are plotted far apart. Gap-fill melodies are represented by filled circles; Adeste Fidèles melodies, by open circles. The plot for Experiment 3 represents 90.5% of the variance in listeners'

similarity judgments, whereas the plot for Experiment 5 represents 88.8% of the variance (Rosner & Meyer, 1986, pp. 21, 31). Similarity judgments were summed across listeners before entering the scaling algorithm; the

plotted distances therefore represent an aggregate tendency, rather than the

judgments of individual listeners. If listeners generally found gap-fill melodies similar to one another, those

melodies would be closer to one another than to Adeste Fidèles melodies. On neither plot, however, is such a pattern evident. To the contrary, on both plots the median distance between gap-fill melodies is actually longer than the median distance between gap-fill and Adeste Fidèles melodies. On Rosner and Meyer's plot for Experiment 3, the median distance between

gap-fill melodies is 31 mm, half a millimeter longer than the 30.5 mm median distance between gap-fill and Adeste Fidèles mélodies.2 On the plot for Experiment 5, the median distance between gap-fill melodies is 32 mm, one-third longer than the 24 mm median distance between gap-fill and Adeste Fidèles melodies.

2. All distances were measured from Rosner and Meyer's (1986) original plots (Figures 8a and 13a) and are tabulated in the Appendix to this article. Distances measured on the redrawn plots in the present article will have the same proportions but may differ by a

scaling factor.



146 Paul von Hippel

Fig. 3. These multidimensional scaling plots summarize the results for two of Rosner and Meyer's (1986) experiments on melodic similarity. The filled circles represent gap-fill melodies, and the open circles represent Adeste Fidèles melodies. Melodies close together on the plot were judged similar by listeners; melodies far apart were judged dissimilar. This figure is redrawn from Rosner and Meyer's (1986) Figures 8a and 13a.

Neither plot, then, suggests that gap-fill melodies sound exceptionally similar to one another. The plots do, however, suggest some kinship among the Adeste Fidèles melodies. To begin with the plot for Experiment 3, although the gap-fill melodies are "widely scattered," "the Adeste Fidèles passages [occupy] a remarkably compact region" near the left edge of the plot (Rosner & Meyer, 1986, p. 22). To describe this difference quantita- tively, the median distance between Adeste Fidèles melodies is just 12 mm, less than half the 30.5-mm median distance separating Adeste Fidèles from gap-fill melodies; a Wilcoxon rank-sum test (Howell, 1997) indicates that the Adeste Fidèles melodies are significantly closer to one another than they are to the gap-fill melodies ( W (15,36) = 208.5, one-tailed p = .0001). A similar, though weaker, pattern is evident in the plot for Experiment 5; here the gap-fill points are "scattered peripherally" (Rosner & Meyer, 1986, p. 31), but five of the six Adeste Fidèles melodies clump together in the lower right. More formally, the Adeste Fidèles melodies are significantly




closer to one another than to the gap-fill melodies, though the level of significance is mild (Ws( 15,36) = 307, one-tailed p = .04). The median distance between Adeste Fidèles melodies is 15 mm, about one-third shorter than the 24-mm median distance between gap-fill and Adeste Fidèles melodies.

Analysis of the hierarchical clustering trees, redrawn here as Figure 4, tells a similar story. Here the perceived similarity of two melodies is represented by the vertical length of the branches that connect them. Again, gap- fill melodies are represented by filled circles, and Adeste Fidèles melodies are represented by open circles. Like the multidimensional scaling plots, these trees represent judgments that have been summed across listeners; they do not necessarily reflect the judgment of individuals. The tree for Experiment 3 represents 79% of the aggregate variance in listeners' simi-

Fig. 4. These hierarchical clustering trees offer another representation for Rosner and Meyer's (1986) experimental results. Again, the filled circles represent gap-fill melodies, while the open circles represent Adeste Fidèles melodies. Melodies connected by short branches were judged similar by listeners; melodies connected by long branches were judged dissimilar. This figure is redrawn from Rosner and Meyer's (1986) Figures 8b and 13b.



148 Paul von Hippel

larity judgments; the tree for Experiment 5 represents 81% of the variance (Rosner & Meyer, 1986, pp. 21, 31).3

If listeners found gap-fill melodies notably similar to one another, then the branches connecting gap-fill melodies would be shorter - and at a lower level of the hierarchy - than the branches connecting gap-fill to Adeste Fidèles melodies. In neither tree is such a pattern evident. To the contrary, in both trees the branches connecting gap-fill melodies are actually longer than the branches connecting gap-fill to Adeste Fidèles melodies. Specifi- cally, in Experiment 3 the branches connecting gap-fill melodies have a median vertical length of 12. 6,4 slightly longer than the median length of 12.15 for branches joining gap-fill and Adeste Fidèles melodies. Likewise, in Experiment 5, the branches connecting gap-fill melodies have a median length of 13.1, slightly longer than the median length of 12.4 for branches connecting gap-fill to Adeste Fidèles melodies.

The hierarchical clustering trees, then, corroborate the multidimensional scaling plots in suggesting that gap-fill melodies did not sound notably similar to one another. However, the trees also corroborate the plots in suggesting that there was notable similarity among the Adeste Fidèles melodies. To begin with Experiment 3, the Adeste Fidèles melodies are significantly closer to one another than they are to the gap-fill melodies (Ws( 15,36) = 172, one-tailed p < .0001); the branches connecting Adeste Fidèles melodies have a median length of 9.8, nearly one-fifth shorter than the 12.15 median length for branches joining Adeste Fidèles to gap-fill melodies. In

Experiment 5, the differences are more modest; branches connecting Adeste Fidèles melodies have a median length of 10.9, just over one-tenth shorter than the 12.4 median length for branches connecting Adeste Fidèles to gap- fill melodies. The difference between branch lengths only borders on significance (Ws(15,36) = 321, one-tailed p = .07), but it does reinforce the

pattern observed in the corresponding multidimensional scaling plot. Taken at face value, both the trees and the plots suggest that, although

similarity judgments were not affected by the gap-fill archetype, they may have been affected by the Adeste Fidèles archetype. This convergence of results is not surprising, because both the trees and the plots were derived from the same similarity data.

Interpretation of these results, however, is complicated by a number of factors. The musical excerpts used in these experiments were not isolated

3. For the hierarchical clustering trees, Rosner and Meyer did not directly report ex-

plained variance directly. Instead, they reported the "cophenetic" correlations (Aldenderfer & Blashfield, 1984) between ratings of perceived similarity, summed across listeners, and the lengths of branches in the hierarchical clustering trees. These correlations- r = .89 for Experiment 3, r = .90 for Experiment 5 (Rosner & Meyer, 1986, pp. 22, 32)- must be squared to compute explained variance (r2).

4. The units of measurement are taken from the unlabeled vertical axes of Rosner and Meyer's (1986) Figure 8b and Figure 13b. All branch lengths are tabulated in the Appendix.




melodies played with a uniform timbre; instead, they were commercial re- cordings of solo, chamber, and orchestral works written over a period of one hundred and fifty years. Although Rosner and Meyer ensured that every excerpt had a similar AA'B form, they did not control other variables such as mode, rhythm, tempo, harmony, or instrumentation. It is possible that some of these uncontrolled variables confounded the experimental results.

Figure 5 suggests a plausible alternative interpretation for the data from Experiment 3. Here the points from that experiment's multidimensional scaling plot have been labeled with the name of each melody's composer. These labels show that the nine melodies on the left were written before 1800 by composers from the late baroque period (Hàndel) to the late classical (early Beethoven). In contrast, the three melodies on the right were written in or after 1830 by the romantic composers Chopin, Brahms, and Verdi. Although melodic archetype might explain why the six Adeste Fidèles melodies are all near the left side of the plot, it cannot explain why three of the gap-fill points are there as well. Stylistic period, however, can explain the horizontal positions of all 12 melodies.

In Figure 6, the hierarchical clustering tree for this data has been similarly labeled with the names of the melodies' composers. The visual effect here is not quite as striking, but after a little inspection one can see that the tree separates into two major limbs. Consistent with a stylistic interpretation of the results, the left limb branches out to connect the three romantic melodies, whereas the right limb branches out to join the nine baroque and classical melodies.

It seems at least plausible, then, that similarity judgments in this experiment were not influenced by archetypes at all, but instead were shaped by

Fig. 5. The points plotted here are the same as in the top half of Figure 3. In this version, however, each melody is labeled with its composer, as given in Rosner and Meyer's (1986) Table 3. Relabeling the points in this way shows that the nine melodies on the left were written before 1800 by baroque and classical composers, whereas the three melodies on the right were written in or after 1830 by romantic composers. This distribution of melodies suggests that stylistic features, rather than melodic archetypes, may explain the pattern of listeners' similarity judgments.



150 Paul von Hippel

Fig. 6. This hierarchical clustering tree offers another view of the pattern shown in Figure 5. Again, each melody is labeled with the name of its composer. The tree is separated into two large limbs; the left limb branches out to join the three romantic melodies, while the right limb branches out to join the nine classical and baroque melodies. Like the multidimensional scaling plot in the previous figure, this tree suggests the influence of stylistic features.

differences in stylistic period. These stylistic differences depend on a complex of factors, and the use of the Adeste Fidèles archetype could be a part of that complex. There are so many stylistic differences present, however, that pinning an interpretation on any one of them seems rash.

Conclusion

In summary, a lack of controls made Rosner and Meyer's (1982, 1986) experiments difficult to interpret. Even if these difficulties are ignored, however, the results provide little if any evidence for the gap-fill archetype. When Rosner and Meyer (1982) trained their listeners, those listeners re- mained near chance levels in their ability to classify melodies in terms of gap-fill. When Rosner and Meyer (1986) did not train their listeners, judgments of melodic similarity showed no effect of gap-fill at all. Together with a recent study suggesting that gap-fill has no influence on melodic shape (von Hippel & Huron, 2000), these results tend to weaken the claim that gap-fill is an important concept for classifying melodies.5

5. 1 thank Jonathan Berger, Chris Chafe, David Huron, Eleanor Selfridge-Field, and espe- cially David Temperley for comments on this article before its submission. During the review process, I benefited from the feedback of Burton Rosner and two anonymous reviewers.




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152 Paul von Hippel

Appendix

Table Al Distances Between Melodies on Rosner and Meyer's (1986)

Multidimensional Scaling Plots (in Millimeters)

Experiment 3

Gap-Fill Melodies Adeste Fidèles Melodies

3GFt 3GF2 3GF3 3GF4 3GF5 3GF6 3AFt 3AF2 3AF3 3AF4 3AF5

3AF6 13 31 45 29 7 29 16 13 21 22 17

3AF5 30 40 36 13 10 34 12 4 4 7 3AF4 34 39 29 7 15 31 19 10 5 3AF3 34 42 34 10 14 35 15 8

3AF2 26 37 38 17 6 32 10 3AFj 28 45 48 25 12 41

3GF6 29 9 31 36 29

3GF5 20 33 40 22

3GF4 41 43 26

3GF3 53 39

3GF2 27

Experiment 5


3GF6 2GF7 2GF9 2GF12 3GF14 3GF15 2AF7 2AF8 2AF9 3AF10 3AFn

3AF12 32 36 20 12 5 11 14 33 15 9 9 3AFn 30 27 23 20 13 19 16 25 15 14 3AF10 41 40 29 13 5 16 6 39 9 2AF9 45 37 35 22 13 24 4 38 2AF8 24 8 32 43 38 40 40 2AF7 45 41 34 18 11 22 3GF15 32 44 16 6 11 3GF14 37 40 24 9 2GF12 38 47 22 2GF9 17 39 2GF7 32

Note - The acronyms in the table margins (3GF , etc.) were used by Rosner and Meyer to indicate the position of each melody on the multidimensional scaling plots (Rosner &c Meyer, 1986, Figures 8a and 13a). Acronyms including the letters "GF" represent Gap-Fill melodies; acronyms containing "AF" represent Adeste Fidèles melodies. In order to calculate the distances reported here, the multidimensional scaling plots were scanned and viewed using image-processing software. The x and y coordinates of each melodic acronym were measured against an on-screen ruler, then converted from the ruler units (sixty-fourths of an inch) to millimeters. Finally, the distance between acronyms was computed by the Euclid- ean distance metric, Vx2 + y2. For the purpose of these measurements, the coordinates of each acronym were taken at the upper-left corner of its letter F. As a check for gross errors of measurement or transcription, the ordering of the transcribed x and y coordinates was visually compared with the ordering of horizontal and vertical positions on the plots. This error check revealed no discrepan- cies.




Table Bl Vertical Lengths of Branches Joining Melodies in Rosner and Meyer's

(1986) Hierarchical Clustering Trees

Experiment 3


^GFt 3GF2 3GF3 3GF4 3GFS 3GF6 3AFt 3AF2 3AF3 3AF4 3AFS

3AF 11.7 12.6 12.6 10.7 9.2 12.6 9.2 9.8 7.3 9.2 9.8

3AF5 11.7 12.6 12.6 10.7 9.8 12.6 9.8 7.3 9.8 9.8

3AF4 11.7 12.6 12.6 10.7 8.7 12.6 7.1 9.8 9.2

3AF3 11.7 12.6 12.6 10.7 9.2 12.6 9.2 9.8

3AF2 11.7 12.6 12.6 10.7 9.8 12.6 9.8

3AFT 11.7 12.6 12.6 10.7 8.7 12.6

3GF6 12.6 7.6 11.1 12.6 12.6

3GF. 11.7 12.6 12.6 10.7

3GF4 11.7 12.6 12.6

3GF3 12.6 11.1

3GF2 12.6

Experiment 5


7gf6 2gf7 2gf9 2gf12 3gf14 3gf15 2af7 2af8 2af9 3af10 3AFn 7S Î3Â Ï3A 12^4 12^4 1O9 9^9 \03 13^1 1Q3 1O9 5^9 ,AFn 13.1 13.1 12.4 12.4 10.9 9.9 10.9 13.1 10.9 10.9

3AF10 13.1 13.1 12.4 12.4 8.9 10.9 8.9 13.1 8.9

2AF9 13.1 13.1 12.4 12.4 6.9 10.9 4.4 13.1

2AF8 11.3 10.6 13.1 13.1 13.1 13.1 13.1

2AF7 13.1 13.1 12.4 12.4 6.9 10.9

3GF15 13.1 13.1 12.4 12.4 10.9

3GF14 13.1 13.1 12.4 12.4

2GF12 13.1 13.1 10.5

2GF9 13.1 13.1

2GF7 11.3

Note - The acronyms in the table margins (3GF15 etc.) were used by Rosner and Meyer to

specify the melody that terminated each branch of the hierarchical clustering trees (Rosner & Meyer, 1986, Figures 8b and 13b). The branch lengths were measured by hand against the unlabeled vertical axes of Rosner and Meyer's figures. As a check for errors, the tabulated values were measured and transcribed twice. Four errors were corrected in this way.



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