Cognitive Foundations of Musical Pitch. by Carol L. Krumhansl

Reviewed by David Huron

Psychology of Music, Vol. 20, No. 1 (1992) pp. 180-185.
Carol L. Krumhansl, Cognitive Foundations of Musical Pitch. Oxford: Oxford University Press. 1990. 307pp. ISBN 0-19-505475-X.

The title of this book has been carefully chosen. It is not a book about the perceptual foundations of musical pitch. (The reader will not find discussions about pitch shifting, residue tones, or other topics traditionally associated with the perception of pitch.) Nor is this a book about tonality in a sense to which all music theorists would readily accede. Although the phrase "tonal hierarchy" appears throughout the text, Krumhansl's use of this phrase is quite circumspect. There is no discussion of large-scale key-areas, or other formal topics that music analysts might associate with the concept of tonal hierarchy. It is a book primarily about the perception of key. But entitling this book the Cognitive Foundations of Musical Key would very much understate the scope and significance of Krumhansl's work. This book is something of a sleeping giant. As the chapters unfold, what begins as a trickle of modest empirical observations is transformed into a cascade of musical insights.

To many readers, Krumhansl is known primarily for her pioneering work using the so-called "probe-tone" experimental method -- a method developed in collaboration with Roger Shepard. The idea is that a musical passage or auditory stimulus is repeated several (normally twelve) times. Each successive trial ends with the playing of one of the twelve pitch chromas. Listeners are then asked to rate the "stability" or "appropriateness" of the given chroma in light of the preceding musical context. In essence, the method allows the experimenter to probe the cognitive state of the listener at a given moment.

Musicians will hardly be surprised to learn that in tonal contexts, the most stable pitch is the tonic, followed by other pitches of the tonic triad, followed by the remaining scale tones, followed by the non-scale tones. This four-level response pattern is what Krumhansl means by the term "tonal hierarchy." The stability profiles for major and minor keys turn out to be highly consistent. Krumhansl gets a lot of mileage out of this fact. First, the response patterns alone provide a reliable index of key perceptions. Second, the tonal hierarchy correlates well with the distribution of pitch chromas for musical passages; play a pitch often enough, and the tonic will tend to drift towards that pitch. The correlation between the tonal hierarchy and probabilities of various pitches within tonal music are consistently high (average r=0.88). Krumhansl exploits this fact to develop a remarkably successful yet simple key-finding algorithm. Third, by cross-correlating the distributions for different keys it is possible to generate a spatial representation of interkey distances. In one of the more magical moments of the book, Krumhansl applies multi-dimensional scaling to her response data: the "circle of fifths" pops right out -- showing that this theoretical construct is not simply a fanciful abstraction, but bears real cognitive import. Later in the book a repeat performance is given for music from Northern India, where an even more complex theoretical representation is replicated through multi-dimensional scaling of the response data.

In presenting her experimental results Krumhansl adopts an overtly Gestalt approach. Three Gestalt principles -- contextual identity, contextual distance, and contextual asymmetry -- provide the framework for discussions of both monophonic pitch sequences and chord progressions. When two instances of the same pitch are separated by intervening pitches, the two pitches are more apt to be recognized by listeners as the same if they are rated more stable in the given key context. Thus successive tonics are more similar to each other than are successive dominants -- which in turn are more similar to each other than leading-tones, etc. In short, the "psychological distance" between two instances of the same pitch (or chord) depends upon their relationship to the prevailing key. This phenomenon illustrates precisely what the Gestalt psychologists dubbed the principle of contextual identity.

In the case of pairs of different pitches, we are more likely to perceive them as being closely related if both pitches are rated highly in the tonal hierarchy. Hence, the dominant and mediant tones are perceived as more closely related than the subdominant and supertonic pitches (contextual distance). In addition, there is an order effect (contextual asymmetry); when one pitch follows another, the psychological distance between the two pitches is judged closer when the second pitch has a greater stability in the tonal hierarchy than the first pitch. Thus the "distance" between the subdominant followed by the dominant pitch is smaller than the "distance" between the dominant pitch followed by the subdominant. Similar asymmetries also exist for successions of chords. These results affirm the view that tonality provides a system of reference points within which pitch events can be efficiently deciphered or coded. In short, tonality provides a paradigmatic example of a "cognitive schema."

Krumhansl rightly draws attention to the implications of these logical asymmetries to the broader subject of psychological scaling and measurement. Theoretical discussions concerning psychological scaling widely regard the properties of transitivity, identity, and symmetry to be axiomatic. In light of the experimental evidence however, these measurement assumptions must be viewed much more skeptically.

Krumhansl provides an experimentally-determined table of values showing the pitch-relatedness for all two-pitch sequences in major and minor key contexts (p.125). The table shows, for example, that moving from the supertonic to the dominant has a higher pitch-relatedness (6.08) than the reverse movement from dominant to supertonic (4.25). Although Krumhansl doesn't address this issue, an interesting musical question is whether those pitch successions having the greatest pitch-relatedness are more commonly used in typical melodies. Since pitch perceptions are contextually asymmetrical, a simple test of this hypothesis is to compare the mean pitch-relatedness value for a given melody with the mean relatedness for the same melody played backwards (retrograde). Using Krumhansl's experimental values, I carried out a brief study of five melodies by Stephen Foster (a sample of about 500 intervals). Alas, the results showed no significant differences between the pitch-relatedness for prime and retrograde versions. Should these results hold true in general, it would imply that the pitch-relatedness values cannot be attributed to the frequency of exposure to various melodic dyads.

By chapter 9, the reader's diligence begins to pay big dividends. Krumhansl uses the probe-tone technique to address questions pertaining to modulation, bitonality, and atonality. Truly musical questions arise:

When a modulation occurs, how quickly does a sense of the new key develop and is a sense of the initial key maintained after the modulation? How do these effects depend on the distance between the initial and final keys? ... Can listeners attend to ... two tonalities separately and, if not, what attributes of the passage make selective attention difficult?

Krumhansl discusses a set of wonderfully tedious experiments used to illuminate the psychological dynamics of various harmonic progressions. She describes the process as follows:

The first chord of each sequence was followed by each of the 12 probe tones from the chromatic scale. That is, the first chord was sounded, then followed by a single tone; then the first chord was sounded again, followed by another tone. This process continued until all 12 probe tones had been sampled. The listeners' task was to rate how well the probe tone fit with the preceding chord. This generated a rating profile for the sequences after the first chord of the [harmonic progression]. Then, the first two chords were sounded, followed by all possible probe tones, each of which was rated in terms of how well it fit with the two-chord sequences. This generated a rating profile for the sequences after the first two chords. Probe tone ratings were then given following each of the remaining chords in an analogous manner, generating ratings profiles for sequences of lengths three through nine [chords].
Using this method, Krumhansl was able to trace dynamically how a listener's sense of key is established and evolves over the course of a harmonic progression. Cogent answers emerge for each of the questions posed above.

A thrilling chapter 10 applies the probe-tone method to the study of non-Western and atonal musics. The chapter begins by recounting Krumhansl, Sandell, and Sergeant (1987) in which listeners were exposed to sequences of 3, 6, 9 or 12 tones from two tone rows used by Arnold Schoenberg. In rating how well a given probe tone fits with the preceding context, listeners divided into two diametrically opposed response patterns. One group of listeners had internalized atonal conventions and judged as ill-fitting those pitches that had appeared recently (recency effect). Another group of listeners continued to hear the sequences according to tonal expectations. The two groups differed also with respect to musical background -- the former group being more highly trained. Krumhansl found no evidence for a truly unique "atonal" way of listening, however. Rather, her results suggest that diatonic tonal hierarchies continued to be used by all listeners, but that some listeners systematically responded in a manner contrary to the tonal schema -- a sort of musical "reverse psychology." For these listeners, the most appropriate subsequent pitch was the one that had the lowest tonal stability given the current context. Furthermore, Krumhansl found that while listeners were sensitive to recency effects, they showed little awareness of the specific ordering of pitches -- even for tone rows to which they had received considerable exposure (see also Millar, 1984). In short, the perceptual experience of 12-tone rows is more atonal than serial. More specifically, the term "atonal" (without tonality) is something of a misnomer since tonal schema continue to inform listener expectations -- albeit in reverse. From a perceptual point of view, a more accurate term might be "contra-tonal" (against tonality).

It is interesting to note that Arnold Schoenberg's method of composing with 12 tones assumes a theory of tonality that is in close accord with Krumhansl's experimental presentation. Schoenberg himself argued that the frequent occurrence of a given pitch-class will raise it to the rank of tonic (Schoenberg, 1948/1975; p.246). The goal of a tone-row is to circumvent the creation of a tonal hierarchy and thereby avoid tonal music. Like Krumhansl, Schoenberg assumed octave-equivalence. Like Krumhansl, Schoenberg regarded simple pitch occurrences as more important in defining tonality than the avoidance of intervallic implications. Schoenberg might have devised a system in which certain intervals or sequences having tonal implications would be forbidden, but he was content to define a simpler system of pitch-class constraints rather than intervallic constraints. In light of Krumhansl's experimental work, Schoenberg's 12-tone prescription must be recognized as a remarkably succinct heuristic for composing atonal music. It is difficult to imagine a more efficient rule for ensuring atonal music-making than to require successive presentations of all twelve pitch-classes and to avoid "the emphasis given to a tone by a premature repetition" (Ibid. p.246). There is a certain poetic vindication in this efficient rule, since Schoenberg's espoused aesthetic goal was "economy of expression."

The chapter continues with discussions of two seminal cross-cultural studies -- by Castellano, Bharucha, and Krumhansl (1984), and Kessler, Hansen, and Shepard (1984). In experiments involving North Indian, Balinese, and Western listeners, these studies showed that listeners are remarkably adaptable to different tonal schemas.

Rather than demonstrating a psychological system that imposes a fixed set of expectations based on the style with which the listeners are most familiar, the results showed that listeners can set aside these expectations and hear the pitch events in style-appropriate terms quite independently of their prior musical experience. For example, both Western and Indian listeners produced probe tone ratings that were consistent with the hierarchical differentiation of tones in North Indian music. Western listeners produced probe tone ratings for pélog contexts of Balinese music that were similar to those for Balinese listeners, and some of the Balinese listeners produced probe tone ratings for Western diatonic music that were similar to those for Western listeners. In no case was there evidence of residual influences of the style more familiar to the listeners on ratings of how well the probe tones fit with the musical contexts. [p.268]
It would appear that people are reasonably good musical tourists (at least with respect to the perception of tonality). Listeners tend not to import their culture-specific tonal schemas to the experience of listening to music from other cultures. On the other hand, listeners do make full use of their existing culture-specific tonal schemas in listening to music from their own culture. For example, Castellano, Bharucha, and Krumhansl (1984) found that Western listeners relied solely on the relative durations of the various tones in forming a suitable tonal schema when listening to selections of North Indian music. For Indian listeners, by comparison, thãt membership had an effect on their probe-tone ratings above and beyond the relative durations of the various tones. When an experienced listener recognizes the appropriate tonal genre, then the genre-specific tonal scheme is automatically invoked.

The main conclusion of Krumhansl's book emerges directly out of the experimental results:

Listeners appear to be very sensitive to the frequency with which the various elements [pitch chromas] and their successive combinations are employed in music. It seems probable, then, that abstract tonal and harmonic relations are learned through internalizing distributional properties characteristic of the style. [p.286]
Krumhansl's language here is quite temperate ("seems probable") despite the fact that nearly all of the evidence reported in this volume is directly consistent with the above view. But Krumhansl recognizes that there is some need for caution in advancing this claim. There are ways in which this view can be tested more directly, although such tests are not addressed in this book. For example, a critical experiment would need to show that the appropriate tonic response is evoked with stimuli consisting of random tones whose frequencies of occurrence conform to the major or minor tonal hierarchy. Such an experiment would show that the distribution of pitch chromas alone is sufficient to evoke the appropriate tonal experience.

Of course showing that tonal hierarchies are linked with the predominance of various chromas in musical compositions still begs the question of why musical chromas would be distributed this way. There may well be some special property shared by the world's existing scales and modes that make them especially suited to the establishment of tonal schemas. An appropriate empirical question is whether we can establish an arbitrary tonal hierarchy for any chroma set? A suitable experiment would involve the selection of random chroma sets (e.g. F, F#, G, C#) with a randomly selected tonic (say F). By exposing listeners to randomly generated tone sequences conforming to some concocted tonal hierarchy, can we evoke appropriate probe tone responses in all cases? Are some chroma sets better suited to the production of stable tonal hierarchies, and if so, why? Moreover, within a given set, are some chromas better suited to the role of tonic?

Krumhansl does consider one possible explanation concerning the origin of the tonal hierarchy. Krumhansl compares the stability values for tones in the major and minor tonal hierarchies with measures of tonal consonance between successive chromas and the tonic pitch. By my own calculations, the correlation between the tonal hierarchy for the major scale and measures of tonal consonance (with the tonic pitch) is about +0.77. Despite this moderately strong correlation, Krumhansl draws attention to some systematic deviations between the the tonal hierarchy for major keys and tonal consonance measures. For example, the leading tone is more important in the tonal hierarchy than the lowered leading tone (the so-called "subtonic" pitch); but the minor seventh interval is more consonant than the interval of the major seventh. In the case of the minor-key tonal profile, the deviations are even more marked.

In truth, it doesn't make much sense to correlate key profiles with measures of tonal consonance for various harmonic intervals generated from the tonic pitch. Very little Western music consists of single scale tones played against a tonic drone. Rather, scale tones are paired together in a variety of ways. Why, therefore, would we characterize the "consonance" of a scale tone merely by how it sounds in conjunction with the tonic pitch? A better test of the relationship between key profiles and tonal consonance would be to compare the pitch stability values with the aggregate consonance for all pitch-pairings which can be made for each tone in the scale.

Whether it is necessary to maintain a certain distribution of chromas in order to evoke appropriate tonal responses is a muddier question. It is a question that has already attracted some contradicting evidence. For example, a ii-V7 chord progression produces a strong tonic key (I) sense, although the tonic pitch itself is entirely absent from the progression. Helen Brown (1988) has shown that a simple rearrangement of pitches is sufficient to evoke quite different tonic perceptions. Clearly, the story of tonality remains incomplete.

Like any new book, this work has its share of seemingly inescapable glitches. There are some unfortunate notational errors in the musical examples (misplaced sharps in Figs. 6.1, 6.2, and wrong clefs in Fig. 9.12). At least two of my former theory instructors would disapprove of Krumhansl's habit of referring to serial music as a style. More disconcerting are several graphs that plot correlation values greater than +1.0 (e.g. pp.224-225).

These minor problems fail to tarnish the polish on this outstanding volume. This book represents another landmark contribution to the field of music perception. Krumhansl provides a wealth of information carefully assembled and lucidly presented. Moreover, it is not a work merely for the specialist. It is a book that both musicians and music theorists will find readable, and unfailingly stimulating. As a work of research, Krumhansl laudably avoids premature closure on the issues she raises. The Cognitive Foundations of Musical Pitch will surely shape and inspire all subsequent work in the area of tonality.

David Huron
Ohio State University


References

Brown, H. (1988). The interplay of set content and temporal context in a functional theory of tonality perception. Music Perception, 5 (3), 219-250.

Castellano, M.A., Bharucha, J. J., and Krumhansl, C. L. (1984). Tonal hierarchies in the music of North India. Journal of Experimental Psychology: General, 113, 394-412.

Kessler, E. J., Hansen, C., and Shepard, R. N. (1984). Tonal schemata in the perception of music in Bali and the West. Music Perception, 2, 131-65.

Krumhansl, C. L., Sandell, G. J., and Sergeant, D. C. (1987). The perception of tone hierarchies and mirror forms in twelve-tone serial music. Music Perception, 5, 31-78.

Millar, J. K. (1984). The Aural Perception of Pitch-Class Set Relations: A Computer-Assisted Investigation. PhD dissertation, North Texas State University.

Schoenberg, A. (1948/1975). Composition with twelve tones (2). In L. Stein (ed.), Style and Idea. London: Faber & Faber, 1975; first published in 1948.
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