Highpoints: A Study of Melodic Peaks by Zohar Eitan

Reviewed by David Huron

Music Perception, Vol. 16, No. 2 (1999) pp. 257-264.
Zohar Eitan, Highpoints: A Study of Melodic Peaks. Philadelphia: University of Pennsylvania Press, 1997. 187pp., with 61 mus. ex., $59.95, ISBN 0-8122-3405-7 (hard cover).

This handsome book by Zohar Eitan focuses on a single aspect of musical organization -- the melodic peak. Peaks are defined simply as the highest pitch in some musical passage, such as the highest pitch in a phrase, in a formal section, or in a whole movement. Eitan sets out to determine whether there are any regularities to peaks. Where do peaks occur? How are they approached and followed? What other musical features tend to be linked with peaks? What musical purposes (if any) are served by peaks?

The structure of the book is a model of clarity. The first chapter reviews the analytic and perceptual literatures pertaining to melodic peaks. The second chapter describes in detail the method of analysis used for the remainder of the book. Three ensuing chapters report on analyses of musical samples from Haydn, Chopin, and Berg, respectively. The final chapter summarizes the results and draws general conclusions.

Highpoints is primarily a contribution to music analysis. Roughly 100 works/movements were analyzed in preparing this book. Some 60 musical examples are reproduced -- some examples spanning several pages each. Most music theorists will find Eitan's analytic approach to be highly novel: Eitan embraces a rigorous methodology that starts with the assumption that the analyst is likely to see things in music that are not there.

Properties of Melodic Peaks

Eitan begins by proposing a series of hypotheses about melodic peaks: peaks will tend to be emphasized by durational (agogic) accent, tend to coincide with strong metric positions, tend to be located closer to the end of a piece or segment, tend to be approached by melodic leaps, tend not to involve immediate repeated pitches, tend to appear uniquely (only once) in a segment, tend to correspond with points of harmonic tension, and tend to coincide with the culmination of crescendi or otherwise receive a dynamic accent.

In order to test these hypotheses, Eitan selected a control group of pitches. These pitches were randomly selected from the upper-most line in the sample repertoires. The purpose of control pitches will need no explanation to psychologists, but for many music theorists this practice is likely to be unfamiliar -- especially in the context of music analysis. For each hypothesis, Eitan compares the peak pitches with the control pitches. This allows Eitan, for example, to contrast the durations of the peak pitches with the durations of the control pitches and so test the hypothesis that peaks tend to receive an agogic accent. In total, 302 peaks were examined from roughly 100 works/movements, and contrasted with a comparable number of randomly selected control pitches. Common statistical tests were then used to determine whether the control pitches and peak pitches differ significantly.

The results not only highlight general features of melodic peaks, but also reveal differences between the three composers studied. In the case of Haydn, peaks do not exhibit the intensifying or emphatic features one might expect. In fact, with only a few exceptions, Haydn's peak pitches don't differ at all from the control pitches. Compared with the control pitches, Eitan notes that the peak pitches are more likely to be approached and followed by large intervals. In Haydn, melodic peaks appear to be simply outlying pitches with little special preparation or intensifying treatment. In other words, Haydn's peak pitches aren't built-up the way one might expect of a musical climax. Eitan suggests that the prosaic matter-of-factness of Haydn's melodic peaks is symptomatic of the demure Galant style with its aristocratic reserve and understatement.

Haydn's Melodic Peaks

In the Haydn repertoire, Eitan was able to find two major differences between peak pitches and control pitches. First, there is a tendency for the peak pitch not to recur within a segment -- peak pitches are likely to be unique. Second, the distributions of scale degrees differ significantly. Specifically, peak pitches are more likely to be the sixth scale degree and less likely to be the seventh scale degree. The latter tendency to avoid the leading-tone is reasonable, since one would expect leading-tones to rise to the tonic (and so the tonic would be the peak pitch). Eitan suggests that the sixth scale degree might commonly descend to the root of a dominant chord (as in a half cadence or when tonicizing the dominant).

Chopin's Melodic Peaks

In the sample of Chopin Waltzes and Mazurkas, Eitan found that peak pitches are (1) longer than the control pitches, (2) more likely to occur in metrically stressed positions, (3) more likely to be approached and followed by large intervals, (4) more likely to occur at the culmination of a crescendo, (5) more likely to occur uniquely within a segment, and (6) more likely to belong to the tonic triad. Unlike Haydn, Chopin prepares and emphasizes peak pitches in a manner consistent with the notion of an expressive climax.

Berg's Melodic Peaks

The sample of music from Alban Berg includes the six movements from Berg's Lyric Suite, plus three songs from Der Wein, the four pieces for clarinet and piano, as well as dozens of `numbers' from the opera Lulu. In total the sample includes roughly 14 works, analyzed as 86 segments. In Berg's music, Eitan found that peak pitches are more likely to be approached and followed by large intervals and that peak pitches are more likely to occur uniquely within a segment. The tendency for peaks to exhibit an agogic accent is stronger than in Chopin. In Berg, a tendency to place peaks late in a segment is exhibited at all levels of segmentation, up to entire pieces. By contrast, late peak placement is entirely absent in the Haydn sample, and in Chopin it applies only to one level of segmentation. Perhaps the most interesting finding in Highpoints is the similarity between Chopin and Berg in their crafting of melodic peaks. In both of these tonal and posttonal repertoires we see peaks used in an effusive romantic manner that contrasts with the nonchalant, incidental peaks in Haydn.

Analytic Confounds

In summarizing the book's findings, Eitan suggests that there are two general properties of peaks that are evident in all three repertoires: "First, in each style peaks tend to be approached (and to a certain degree left) by relatively large intervals ... Second, all three styles have a strong inclination to present the peak pitch only once in a segment." There are statistical reasons to be suspicious of these (and other) conclusions. Recall that peak pitches are contrasted with control pitches that are randomly selected from the same repertoire. Since peak pitches are by definition located at the extreme end of the pitch distribution for a melody, there are some statistical artifacts that can be expected.

Consider first the hypothesis that peak pitches tend to occur uniquely (only once) in a musical segment. Since peak pitches lie at the end of the pitch distribution, one would expect them to be rare for the same reason that a person with the biggest feet in a room full of people will tend to be the only person wearing that particular shoe size. Comparing the frequency of occurrence for peak pitches with the frequency of occurrence for a randomly selected pitch is inappropriate. A better control group would be to compare highest pitches in a melody with, say, the lowest pitches in a melody. Since music theorists have not posited a role for melodic low points comparable to the presumed role of high points, this choice of control pitches ought to highlight those differences that are due to musical goals rather than artifacts of the pitch distribution.

Some Tests

In order to test for the possible existence of pitch-distribution confounds, I carried out some measurements using 1,000 randomly selected melodies from the Essen Folksong Collection (Schaffrath, 1995). This computer database consists of traditional folksong melodies drawn from European sources. It would be preferable to use the same Haydn, Chopin and Berg samples used by Eitan, but these works aren't available on-line. Nevertheless, the issues can be illustrated using the Essen database. First, I simply counted the total number of occurrences of the highest pitch in each melody. In 1,000 works there were 2,755 highest pitches -- meaning that the average European folk melody contains 2.755 instances of the highest pitch. By contrast there were 3,164 lowest pitches -- meaning that the average folk melody in this sample contains 3.164 instances of the lowest pitch. It is indeed the case that peak pitches are relatively rare -- at least when compared with the lowest pitches X²=28.2; df=1; p<0.001).

Next consider the claim that peak pitches tend not to be immediately repeated. Once again, comparing peak pitches with randomly selected pitches is suspect since outliers are rare. Repetitions might be more common for randomly selected notes simply because pitches near the middle of the tessitura occur more frequently. Once again I tested this by contrasting the lowest and highest pitches in 1,000 melodies from the Essen Folksong Collection. In this sample I found 562 instances where the highest pitch was followed by an immediate pitch repetition (i.e. a melodic interval of a unison). By contrast, I found 713 instances where the lowest pitch was followed by an immediate pitch repetition. Since there are 2,755 highest notes and 3,164 lowest notes in the sample, the frequencies of repeated pitches are 20.4 percent for peak pitches and 22.5 percent for lowest pitches; this difference is not statistically significant X²=3.11; df=1; p>0.05).

Consider now the hypothesis that the highest pitches tend to be approached and followed by larger than average melodic intervals. For extremes of pitch, large antecedent and consequent intervals would be expected by chance. (Imagine a line of people of different heights; if we encounter a person who is especially tall, then we would expect the difference in height between this person and his/her neighbors to be larger than the average height difference.) Using randomly selected pitches in this case is an inappropriate control. A better way of testing whether melodic peak points tend to be approached by larger or smaller intervals is by randomly rearranging the order of the notes in a melody and determining whether the antecedent and consequent intervals surrounding peak pitches are now larger or smaller. Changing the order of the pitches preserves the identical pitch distribution while destroying the composed interval relations. Contrasting the actual melodies with the re-ordered melodies allows us to isolate the interval activity while eliminating the influence of the pitch distribution on the frequency of various intervals.

Once again I used the Essen Folksong Collection to test the claim that peak pitches are more likely to be framed by relatively large intervals. For each melody, I determined the peak pitch (or pitches) and measured the antecedent and consequent intervals in semitones: the average antecedent interval to a peak point is 3.12 semitones, and the average consequent interval is 2.29 semitones. (This difference in average interval size is consistent with Eitan's finding that antecedent intervals tend to be larger than consequent intervals.) For each of the 1,000 folk melodies, I separately scrambled the order of the notes (treating tied-notes as a single note) and then determined the interval approaching and following the peak pitch points. The average antecedent interval to a peak point in the scrambled melodies is 5.41 semitones; the average consequent interval is 5.36 semitones. In other words, although peak points are approached and followed by larger intervals than is the case for other notes, peak points are not approached and followed by intervals that are larger than would be expected in a random ordering of the notes of a melody. The larger melodic intervals surrounding peak pitches seems to be little more than an artifact of the fact that peak pitches are simply further away from the majority of other pitches in the melody.

A more telling story arises by comparing the antecedent and consequent intervals for the lowest pitches in the scrambled melodies. The lowest pitches in the scrambled melodies were found to have an antecedent interval of 6.02 semitones and a consequent interval of 5.94 semitones. These intervals are larger than those surrounding peaks, suggesting that peak pitches are closer to the mean pitch for a melody, even though lowest pitches are more common.

Of course the above results may apply only to European folksongs, and may not apply to the Haydn, Chopin and Berg repertoires. However, the presence of these confounds in even a single repertoire casts doubts on the validity of the conclusion for Haydn, Chopin and Berg. In order for Eitan to sustain many of his claims about melodic peaks, he needs to redo the analyses using different control pitches.

Identifying Appropriate Controls

The issue of identifying appropriate controls highlights some interesting issues concerning the interpretation of statistical results. Consider the question of the size of antecedent intervals prior to melodic peaks. On the one hand, it is perfectly true that melodic peaks in European folksongs are approached by intervals that are larger than the average melodic interval. Hence, one can truthfully conclude that, on average, "peak pitches are approached by large intervals." On the other hand, these intervals are smaller than would be expected in a random arrangement of notes. Hence, one can truthfully conclude that, on average, "peaks are approached by smaller intervals than would be expected given their extreme registral position." If we conceive of the act of composition as one of arranging a collection of notes within a predefined tessitura, then the composer's actions have resulted in reducing the size of the intervals framing a melodic peak. Alternatively, if we conceive of the act of composition as one of creating a succession of intervals, then the composer's actions have resulted in an increase in the size of the intervals framing a melodic peak.

The temptation is to throw up one's arms and declare that statistical approaches to music analysis will get us nowhere. On the contrary. Such inferential statistical approaches will help us generate and test much more refined hypotheses about the precise nature of compositional processes. What indeed are composers doing when they arrange notes on a page? Are they arranging pitches, or intervals, or scale degrees, or contours, ... or some combination? A systematic statistical approach allows us to answer such questions.

Type I and Type II Skepticism

The problem of antecedent intervals in melodic peaks raises a further broader issue concerning the use of statistics -- an issue that is often ignored or overlooked by those of us interested in empirical approaches in music scholarship. In statistical inference, a distinction is made between so-called Type I Errors and Type II Errors. A Type I error occurs when we claim something to be true/useful or knowable that is, in fact, false/useless/unknowable. A Type II error occurs when we claim something to be false/useless/unknowable that is, in fact, true/useful/knowable.

Throughout the history of the physical and social sciences, researchers have endeavored to minimize or reduce the likelihood of making Type I errors. That is, traditional scientists are loath to make the mistake of claiming something to be true that is, in reality, false. This practice has arisen from the awareness that we are commonly wrong in our intuitions and all too eager to embrace suspect evidence in support of our pet theories.

In the past decade or so, research in medicine has raised serious challenges to this orthodox scientific position. The U.S. Food and Drug Administration formerly approved only those drugs that had been proved to be effective according to criteria minimizing Type I errors. (That is, drugs that might be useful were never approved.) However, for those patients facing imminent death, it is the enlightened physician who will recommend that the patient seek out the most promising of recent quacks. [Footnote 1] In other words, the medical community has drawn attention to the possible detrimental effects of committing Type II errors (i.e. claiming something to be useless that is, in fact, useful).

In their attitudes towards hypotheses, the dispositions of traditional scientists and traditional arts/humanities scholars are starkly contrasting. Traditional humanities scholars (including most music scholars) have tended to be more fearful of committing Type II errors. For many arts and humanities scholars, a common fear is dismissing prematurely an interpretation or theory that might have merit -- however tentative or tenuous the supporting observations. In all arts disciplines, a premium is placed on sensitive observation and intuition: no detail is too small or too insignificant when describing or discussing a work of art. Moreover, for many traditional humanities scholars, dismissing an observation as a "mere coincidence" is problematic; for these scholars, apparent coincidences are more likely to be viewed as "smoking guns."

It is this difference in attitude, I believe, that partly accounts for the widespread suspicion of statistical arguments in arts disciplines. Both traditional scientists and traditional humanities scholars are motivated by skepticism, but they are motivated by two different forms of skepticism. One might characterize these two communities as Type I-phobic and Type II-phobic respectively. For one group, a premium is placed on avoiding false positive claims; for the other group, a premium is placed on avoiding false negative claims.

To the knowledgeable statistician there is nothing new in this discussion. Statisticians have always understood the reciprocal relationship between Type I and Type II errors, and have long recognized that whether a researcher endeavors to reduce one or the other depends entirely on the moral (or esthetic) repercussions of making either error. In most traditional arts and humanities scholarship, committing a Type I Error (i.e. making a false positive claim) rarely has onerous moral or esthetic repercussions. Simultaneously, Type II errors have often been seen as reckless. [Footnote 2]

Historically, statistical tests have been used almost exclusively to minimize Type I errors. Yet there is nothing in statistical inference per se that is contrary to the traditional arts/humanities scholar's penchant for false negative skepticism.

All of this is changing. In the same way that contemporary medicine has become more cognizant of the importance of Type II errors, many of us in the arts and humanities disciplines are becoming more aware of the problems of Type I errors. Several hundred years of musical speculation has lead to the promulgation of many ideas that lack substance. Until recently, there was little one could do about this. The scarcity of pertinent data in many humanities fields simply made it impossible to satisfy statistical criteria for minimizing Type I Errors. The opportunities to address these problems have been immensely expanded due to the growing availability of musical databases.

Conclusion

In this context, Eitan's work on melodic analysis is seminal. Highpoints is a paradigm for future music analysis. It establishes a new and welcome rigor by beginning with the assumption that the analyst will occasionally see things in music that are not there. Music theorists must rely foremost on their intuitions about what may be going on; but wherever possible, these intuitions should be subjected to careful Type I-phobic testing. Eitan has illustrated how this can be done in a traditional music analysis. Moreover, this "systematic analysis" approach does not simply affirm intuitions that are already obvious -- a common criticism of empirical approaches voiced by humanities scholars. For example, in identifying how Berg and Chopin utilize peaks in similar ways (while showing that Haydn's practice differs significantly) Eitan has shown how systematic analysis can reveal compositional patterns which were not otherwise evident.

The flaws in Eitan's work are clearly evident for the simple reason that his methods are clearly described and meticulously followed. Good research does not guarantee coming up with the right answers. Rather, good research exposes its assumptions and so lays itself open to criticism and improvement. Eitan's approach to music analysis deserves the flattery of imitation.

David Huron
Ohio State University


References

Schaffrath, H. (1995). The Essen Folksong Collection. D. Huron, (ed.). Stanford CA: Center for Computer Assisted Research in the Humanities.


Return to Publication List

Return to Huron's Home Page

Return to "Music Cognition at Ohio State University"