Carol L. Krumhansl, Cognitive Foundations of Musical Pitch
Kris Shaffer, Sound and Mind, May 2007

Carol L. Krumhansl: Cognitive Foundations of Musical Pitch
Oxford University Press (1990).

The purpose of Cognitive Foundations of Musical Pitch (CFMP), according to author Carol L. Krumhansl, is ‘to describe the knowledge listeners have about musical structure and characterize how this knowledge affects the way listeners perceive and remember music’ (p. 281). She focuses specifically on the pitch structure of ‘tonal-harmonic music’ (p. 12) of the common-practice period of the Western classical tradition. Of course, the structure of tonal music and the effect that knowledge of it has on a listener’s perception have long been hot topics in the study of music, and they are particularly salient subjects of inquiry in cognitive musicology. Cognitive musicologists have approached these topics from various angles: some lend toward the traditionally music theoretical or analytical, some are scientific but speculative (e.g. Lerdahl 2001), some are primarily computational (e.g. Temperley 2001, 2007), some are experimentally and/or empirically based (e.g. the work of Aniruddh D. Patel), and some function as a broader survey of a number of kinds of studies (e.g. Huron 2006). Though CFMP has an air of survey, it most decidedly takes an experimental approach, incorporating the ‘methods and theoretical approach’ of ‘cognitive psychology’ (p. 5). This study also ‘employ[s] statistical and other analytical methods’ (p. 8 ) in interpreting the data gained from the experiments cited, and in generalizing from those data and interpretations.

Before diving into the specifics of the research in this book, I would like to highlight a point of tension in this book which is contained within Krumhansl’s stated purpose and appears continually throughout her discussion of the research. This point of tension is one between the idea of a ‘musical structure’ and a phenomenology of musical perception. Musical structure is often held to be a static structure, with pitches (or pitch classes), chords, or keys occupying fixed points in a conceptual musical space, with the cognitive distance between two musical elements being a fixed value. Such a musical structure can be found in any of the numerous geometrical representations of musical space, such as those of Euler, Riemann, Cohn, Shepard, Lerdahl, Tymoczko, and many others. Krumhansl cites a number of these Euclidean musical spaces in CFMP and explores how such a structure affects the perception and memory of music in real-time. However, a number of other theorists—e.g. Meyer (1956), Narmour (1990), Huron (2006), and others—have problematized this notion, arguing instead that since music happens in real-time and musical activities happen in a time-dependent context, a conceptualization of musical structure must be context dependent, as well. Musical elements cannot be fixed points in a Euclidean space, as their identity and relationships to other elements change with each context. The notion of a dynamic, contextual representation of musical structure can be found at least as early as the writings of J. P. Kirnberger, who writes in chapter 7 (‘Modulation’) of Die Kunst des reinen Satzes (1771-79) that a modulation to a key with a tonic a fifth lower than the original key is a more distant modulation than if one were to modulate up a fifth (for reasons which I will not address here). Kirnberger thus implies that two keys whose tonics lie a fifth apart cannot be represented by a static measurement of distance. The cognitive distance traveled depends not only on the location of two points in a fixed space, but also on the direction in which one moves. In fact, for theorists like Kirnberger, the notion of fixed points in Euclidean space is a useless—even deceptive—representation of musical structure.

This perspective on musical structure is also active in CFMP. In fact, Krumhansl devotes an entire chapter (ch. 6 ‘Perceptual organization and pitch memory’) to the discussion of the ‘dynamic’ nature of cognitive distances between musical elements, drawing on principles from gestalt psychology, as well as aspects of her data which are consonant with such principles. Nevertheless, Krumhansl does not presume to have settled the issue between static and dynamic representations of musical structure. Nor does she avoid generating her own geometrical representations of musical space (though she is quick to point out their limitations). Thus, the tension between these two ideas of musical structure plays an active role in her study, and throughout my review I will draw attention to the way in which this tension is played out throughout the book, particularly in the ways in which these two competing views of structure affect Krumhansl’s interpretation of her experimental data.

A second key point to consider throughout this study is Krumhansl’s reliance on her ‘probe tone’ methodology. In general, this methodology involves a musical stimulus comprised of a key-defining context (typically a diatonic scale or cadential chord progression ending on tonic), followed by a single tone or chord. The subjects of the experiment, upon hearing the stimulus, are then asked to provide a value (typically from 1 to 7) of the probe tone’s or probe chord’s goodness of fit within the preceding context. There are a number of variants of this methodology, which will be discussed throughout this review. This methodology provides interesting—and useful—data, which confirms a number of intuitions regarding musical structure and perception. Krumhansl also gets a lot of mileage out the the data, subjecting it to numerous carefully considered statistical analyses. However, the probe-tone method is only one amongst many methods of testing listeners’ conceptualizations of musical structure and the expectations based on those conceptualizations (c.f. Huron 2006, p. 41ff. for a list of various methods which have been employed to the same ends). Further, the probe-tone method has a number of significant limitations (some of which will be explored in this review), only a few of which Krumhansl noted in CFMP. Thus the probe-tone methodology has a significant effect on the data produced by these experiments, and on Krumhansl’s interpretation of those data, which one must keep in mind when considering her results and their potential impact on the larger project of cognitive musicology.

Krumhansl groups her results into three general categories. First is ‘the hierarchical differentiation of musical elements’ (p. 271). That is, musical elements (pitch-classes, chords, keys) are perceived as having varying degrees of stability. The function with which these elements can be employed, the degree of relationship between these elements, and the frequency with which those elements are employed in the literature all correlate to or depend upon the relative stability of the element. Second is ‘the dependency of perceived relations on context’ (p. 271). That is, the identity, functions, degrees of relationship, and frequency of use of musical elements are dependent on the context in which they are employed (i.e. the notes, chords, keys immediately preceding and following). And third are the results relating to ‘the mappings between objective and subjective aspects of music’ (p. 271). These results include the correlation between the placement of a particular element in the hierarchical structure and that elements frequency of use within the common-practice repertoire, as well as the ability of humans to learn new stylistic norms and abstract hierarchies through statistical learning (p. 286; c.f. Meyer 1956).

These general results are indicative of the aforementioned tension between static and dynamic representations of musical structure as they relate to the purpose of the book. The tonal hierarchy and the statistical distribution of musical elements within the repertoire are static structures. However, the dependency on context illustrates the significance of the gestalt-psychological principles and the dynamic nature of musical structure. And the tension between those two types of structure is central to the interaction between the ‘objective’ properties of the music and the ‘subjective’ properties of the cognitive representations of its structure. Krumhansl’s data engages with both static and dynamic conceptualizations of musical structure, and this tension will frame the bulk of my discussion as I now proceed through the content of CFMP in greater detail.

I will begin in chapter 2 ‘Quantifying tonal hierarchies and key distances,’ as I have already unpacked the bulk of the contents of chapter 1, ‘Objectives and methods’ in the above discussion. In chapter 2, Krumhansl examines the relations of tones to tonal centers, quantifying the ‘tonal hierarchy’ for major and minor keys and thus the ‘relative stability or structural significance of tones’ within major and minor keys (p. 16). This idea of representing tonality as a frame of reference (tonic) and the degree of relationship of all other elements to that frame of reference is a standard gestalt and cognitive psychological approach which Krumhansl and a number of others have found useful in the study of cognitive musicology.

Figure 1. Results of Krumhansl and Shepard 1979. Group 1 - most musical training; group 2 - intermediate musical training; group 3 - least musical training.

This chapter lays out the methodology of the two simplest probe-tone experiments. In Krumhansl and Shepard (1979), the experimenters played an incomplete ascending or descending major scale (e.g., C-D-E-F-G-A-B), followed by a probe tone (one of the twelve chromatic pitch classes in the octave of the scale, or one of the twelve quarter-tones between those chromatic pitches). The subjects (Westerners of various levels of musical training) were asked to judge how well the probe tone completes the scale, on a numbered scale from 1 to 7. As shown in Figure 1, those with little musical training (group 3) primarily judged goodness of completion by pitch height, with added weight given to the actual tonic on the opposite end of the scale. Those with substantial musical training (groups 1 and 2) rated the tonic highest, followed by notes which belong to the diatonic scale, followed by the remaining chromatic pitches (quarter-tones were left out of the data reported, for lack of significance). This demonstrates that, for those with substantial musical training, there is a 3-level hierarchy for scale completion: tonic, diatonic pitches, chromatic pitches.

Figure 2. Results of Krumhansl and Kessler 1982.

Krumhansl and Kessler (1982) follow up on this study, this time requiring that subjects have substantial musical training, but minimal exposure to formal music theory. They also modified the experiment, this time providing a complete scale or a cadential chord progression, asking the listeners to judge goodness of fit (not completion). They also used Shepard tones (tones in multiple octaves, rather than single pitches), in order to eliminate registral bias. As Figure 2 shows, a more differentiated 5-level hierarchy emerged from this data, consisting of the tonic pitch, the fifth scale-degree, the third, other scale tones, and non-scale chromatic tones (for minor keys, the fifth and third are switched). One potential problem with using the probe-tone method to judge goodness of fit like this is the potential for the subject to weigh finality strongly in their consideration of goodness of fit, as the probe tone is always last in the stimulus. However, to the degree that Krumhansl interprets this as a hierarchy of stability, that is probably not significant. Further, these results do confirm the intuitions of speculative theorists like Meyer (1956), and this hierarchy does significantly correlate with the statistical frequency of pitches in the repertoire, as will be seen later in CFMP and in Temperley (2001 and 2006). Thus in this case, Krumhansl’s assumptions seem reasonable.

Where Krumhansl perhaps goes too far is in extending this data into a measurement of key distance. She transposes the major and minor key profiles to all twelve tonics, generating profiles for all twenty-four keys. She then measures the statistical correlation for each pair of profiles (fixed pitch, no transpositions), and generates a geometrical model of key distance from those correlation values. This, of course, is problematic merely by considering the disparity between the data (a single tone following a scale or cadential pattern) and the breadth of the model (a spatial arrangement of keys from which one can measure cognitive distance). It is also problematic given the lack of a study of intervals, chords, and sequences of pitches, intervals, and chords in the context of key definition. Further, she also assumes that the distance between two keys in this geometric space is a representative of a listener’s experience of a tonal modulation in real-time. The latter is particularly problematic given the theories of Kirnberger, Meyer, Narmour, and the information theorists (many of whom Krumhansl cites throughout CFMP) claim that key distances are asymmetrical (C is closer to F than F is to C). In fact, some of Krumhansl’s own data concerning intervallic distance (chapter 5) supports such theories. However, Krumhansl proceeds with the creation of a Euclidean representation of tonal space and the cognitive distance between keys based on the statistical correlation of their probe-tone key profiles. While the static nature of the probe-tone based key profiles is reasonable (since a key-defining context is built into the probe-tone study, and since a given pitch class’s place in the hierarchy changes from key to key, demonstrating its dynamic character as well), using those static profiles as the basis for a representation of key distance—rather than concocting a new experiment that derives key distance from contextual modulations—is far less reasonable and leaves Krumhansl’s geometrical model of key distance with little solid empirical or theoretical ground on which to stand. However, the key profiles and the hierarchies which the studies of this chapter suggest do provide useful starting points for other studies, as will be seen throughout the book.

Chapter 3, ‘Musical correlates of perceived tonal hierarchies,’ explores two potential ways in which the hierarchies of chapter 2 may be internalized by humans: objective properties of sound—namely consonance and dissonance—or pitch-class distribution in the common-practice repertoire. To discern if either of these factors contribute to the perceived tonal hierarchies (as quantified by the probe-tone experiments of chapter 2), Krumhansl takes quantifications of intervallic consonance from a number of studies ranging from Helmholtz (1885) to Hutchinson and Knopoff (1978) as well as statistical values of pitch class distribution from a few information-theoretical studies of common-practice tonal music and nursery rhymes. She then measures the statistical correlation of these various models of consonance or pitch-class distribution to the probe-tone key hierarchies. She finds that both the consonance models and the pitch-class distribution profiles significantly correlate with the probe-tone hierarchies. However, the pitch-class distribution has a stronger correlation, and a multiple-regression analysis found that the consonance models added nothing beyond the correlation of the pitch-class distribution with the probe-tone profiles. Krumhansl thus concludes that while consonance could play some part in the creation of a tonal system, statistical learning based on pitch-class distribution in the repertoire is the most likely candidate for the source of our internalization of the tonal hierarchy (p. 76).

This would be very interesting data—and it would offer substantial ammunition to those who would critique a prominent view in the history of music theory that the tonal system is a result of the natural properties of consonance and dissonance as found in rational intervals or the overtone series—were it not for some fundamental problems with the project of this chapter. First, the consonance profiles she compares to the probe-tone profiles are flawed. They are a measure only of the consonance value (derived according to several different methods in the various studies cited) of a particular scale degree with the tonic. Thus the lowered seventh degree is more consonant than the leading tone, the fourth degree is more consonant than the third, and there is no difference between minor and major keys. This is no surprise to tonal theorists, who would never hold that the leading tone is part of the scale because of its consonance with the tonic! In fact, dissonance is an integral part of the tonal system, and it is of the utmost significance in determining the role of the leading tone and the fourth degree for most tonal theorists. Further, if consonance is to be the basis of a tonal hierarchy, at the very least one must explore the consonance of each pitch class in the diatonic collection with each other pitch class in the collection. For instance, it is more likely that the leading tone would receive its consonance value in relation to the fifth scale degree (particularly in a just-tuned or mean-tone system) or the third degree (particularly in a Pythagorean system) than with the tonic. Thus the consonance-based key profiles Krumhansl utilizes in this chapter are little more than useless.

The pitch-class distribution profiles suffer from a less severe problem. Namely, the data Krumhansl cites count pitch classes in a piece relative to the tonic of the home key. Modulation is not taken into account. On the other hand, the probe-tone key profiles are heavily context dependent; the probe-tone is evaluated relative to the immediately preceding key-defining stimulus. By taking modulation into account, it is possible that Krumhansl would find an even stronger correlation between the probe-tone profiles and pitch-class distribution. She also may find better explanations for anomalies in the comparison. For example, Krumhansl writes, ‘there is also a consistent discrepancy for the second scale degree, which is sounded more frequently than would be expected based on its position in the tonal hierarchy’ (p. 69). She explains this by the melodic significance of scale degree 2 as it approaches the tonic. However, her probe-tone profile actually predicts that if a piece spent significant time in the dominant key (the most common modulation in major), scale degree 2 of the original key would get a boost, as it is the fifth scale degree (the second highest pitch in the hierarchy) of the dominant key. By not considering modulation, then, she receives less accurate data, and provides less compelling explanations of the anomalies found in that less accurate data. Further, the repertoire upon which the distribution data is based is common-practice classical music. However, the probe-tone subjects—though probably familiar with this repertoire, given their musical training—have likely been exposed to a large amount of Western popular music, which may have a different pitch-class distribution profile. It certainly possesses different norms of harmonic sequence. Again, accounting for this repertoire may actually increase the correlation of her probe-tone profiles with pitch-class distribution in the repertoire with which her listeners are familiar. However, more work is needed to establish that connection. In all, this chapter suggests the possibility of statistical learning by pitch-class distribution in the repertoire, but it establishes nothing. Her flawed methodology renders her conclusions regarding both consonance and pitch-class distribution too problematic to be useful as they stand in this chapter.

Chapter 4, ‘a key finding algorithm based on tonal hierarchies,’ provides a much more convincing case for the correlation between the probe-tone key profiles and pitch-class distribution in the repertoire. This chapter outlines the Krumhansl/Schmuckler key-finding algorithm, which evaluates the correlation of the pitch-class distribution of a musical passage with the twenty-four key profiles from the probe-tone studies. The algorithm was tested on the opening bar(s) of preludes of J. S. Bach, Shostakovich, and Chopin, as well as fugues of J. S. Bach and Shostakovich. The algorithm was highly accurate—significantly more accurate than the Longuet-Higgins and Steedman (1971) model—and only produced incorrect results on passages which are clearly ambiguous (i.e., the tonic or third is missing, and the algorithm picks the relative or parallel key, respectively). Given the accuracy of this algorithm in determining key based on pitch-class distribution, it seems reasonable to consider the possibility of our internalization of a tonal hierarchy based on the distribution of pitch-classes in the repertoire (though there is still a need to account for other Western tonal repertoires than the classical keyboard tradition).

Chapters 5 and 6, ‘Perceived relations between musical tones’ and ‘Perceptual organization and pitch memory,’ move beyond the consideration of a single tone in a context, and thus begin to exhibit more directly the tension between static and dynamic conceptualizations of musical space. Krumhansl presents a number of geometrical models of musical space which represent pitch classes, chord roots, or key tonics as fixed points in a two-, three-, or four-dimensional Euclidean space, such as Euler’s Tonnetz (p. 116) or Shepard’s torus (pp. 114-115). In her discussion of these geometrical models of musical space, Krumhansl points out the limitations of such models, chiefly that they are order-agnostic; that is, the cognitive distance between two points is fixed and does not vary depending on the context or the order in which those two points are perceived. Thus it seems that through her critique Krumhansl may be positioning herself as a phenomenologist, arguing against such geometrical models of human music perception.

She even offers experimental data which supports the phenomenological position. Her 1979 probe-tone study is much like the aforementioned experiments, except that the single probe tone is replaced by two successive probe tones. The subjects are asked to judge how well the second tone follows the first. She found that not all intervals of a class are perceived equally (figure 3). Namely, the stability of the two notes in the tonal hierarchy of the key whose scale or cadence precedes the probe tones significantly affects the judgments of how well the second tone follows the first. The stability of the second tone was weighted significantly higher than the stability of the first tone.

Figure 3. ‘Relatedness ratings of ordered pairs of tones (with C as a reference tonic),’ p. 125.

Thus both the placement of the interval within the key and the order of the two pitch classes affected the judgments. In other words, the cognitive distance between two pitches, at least according to the data of this experiment, is not a fixed value.

Figure 4. ‘Idealized configuration showing multidimensional scaling solution of judgments of pairs of tones in a C major key context (left) and a C minor key context (right) from the experiment. The tonic is located at the vertex of the cone. For other tones, the distance from the vertex is an inverse function of its position in the quantified tonal hierarchy (Krumhansl & Kessler, 1982). Around the cone, tones are arranged as on the chroma circle’ (p. 128).

However, Krumhansl proceeds into averaging the values for the two different orders of each interval (averaging the values for C-F and F-C, for example), and using the resulting values to generate a three-dimensional geometric representation of pitch-class space in a given key, her well-known cone model, shown in figure 4. This model suggests that the cognitive distances between pitch classes within a key context are indeed fixed. However, such a model does not follow logically from the data. Rather, Krumhansl made an ideological choice to reduce out the order dependencies in her data in order to generate such a model. Perhaps that act is the strongest critique of a static geometric model of musical space as it relates to the perception of tonal music. The data does not support the model; it derives directly from the ideology of the researcher. (I do not mean to imply that any scientific conclusion is free from ideological bias, of course—only that here it is quite blatant and difficult to reconcile with the data.)

If Krumhansl avoids the issue of phenomenology in chapter 5, then in chapter 6 she confronts it head-on. Here she engages gestalt theory in the interpretation of her probe-tone data and other related studies. Gestalt theory has contributed much to the field of cognitive psychology, and it maintains that no part can be considered apart from the whole to which it belongs. In the previous chapters, discussion always took place within the context of a particular key, but once that key was established, everything within it is fixed in its relationship to the tonic. However, in applying gestalt theory to her data—particularly the interval probe-tone study of chapter 5 and a memory study of 1979—Krumhansl begins to look more at the note-to-note or chord-to-chord context and the way that the probe-tone-derived tonal hierarchy affects note-to-note and chord-to-chord relationships. In this analysis and within the general framework of gestalt theory, Krumhansl derives three principles of cognitive distance: contextual identity, contextual distance, and contextual asymmetry (Figure 5).

Figure 5. Three gestalt-based ‘principles of tonal relations’ (p. 141).

These principles boil down as follows:

• Stable tones in the tonal hierarchy are better remembered than unstable tones. E.g., in C major, listeners tend to remember hearing more stable tones like C, E, and G more than less stable tones like B, B-flat, or F-sharp.
• Stable tones are heard as more related than unstable tones. E.g., in C major, listeners tend to judge the tones C and G more related than the tones F-sharp and C-sharp, despite the intervallic equivalence of the two pairs of tones.
• Unstable tones are heard as more related to stable tones than vice versa. E.g., in C major, listeners tend to hear B as more related to C than C to B.
• Stable tones are preferred in all positions, especially in final positions. E.g., in C major, the minor second motion from A-sharp to B covers more cognitive distance than the minor second motion from B to C; further, the minor second motion from C to B (more stable to less stable) covers more cognitive distance than the minor second motion from B to C (less stable to more stable).

In my opinion, this is the most unique and the most significant discovery of the research in CFMP. Not only does it severely problematize the concept of a static geometrical representation of musical space and cognitive distance; it also lays out a clear, empirically supported conceptualization of the interaction between a static cognitive model of tonality (the probe-tone based tonal hierarchy) and a dynamic, phenomenologically dependent perception of cognitive distance. This is at the heart of Krumhansl’s project and the broader project of cognitive musicology (‘to describe the knowledge listeners have about musical structure and characterize how this knowledge affects the way listeners perceive and remember music’), and it makes a claim that any theorist embarking on the creation of a geometric representation of cognitive distance between tonal elements must address.

Chapter 7, ‘Quantifying harmonic hierarchies and key distances,’ essentially repeats earlier probe-tone experiments, this time incorporating target chords instead of target pitch classes. Because of the similarity between these and previous studies, I will not go into detail about the research of this chapter, except to point out that Krumhansl found ‘a strong interdependency between tonal [probe-tone] and harmonic [probe-chord] hierarchies’ (p. 177).

Chapter 8, ‘Perceived harmonic relations,’ likewise applied the principles of the chapter 5 interval study to chords; that is, each key-defining context was followed by a pair of chords, where the subjects were asked to judge the relatedness of the two chords. Though these results also mimic the results from the tone level on the chord level, it is worth exploring this chapter in some detail. First, Krumhansl finds the same gestalt principles at work, where stable chords are heard as more related than unstable chords, and the stability of the final chord outweighed the stability of the first chord. More specifically, though, Krumhansl attempts to manipulate the judgments of the relatedness of the target chords by changing the key context in which they are heard, and thus their stability in the harmonic hierarchy. To control this, Krumhansl selects five key contexts—C major, G major, A major, B major, and F-sharp major—and uses only the diatonic triads from C major and F-sharp major as target chords. She then separates the subjects’ judgments by key context, in order to observe any differences in relatedness based on the key. Krumhansl analyzes the data in two ways, one which explores the phenomenological aspects of perception (examining specifically the effects of key context and phenomenological order on relatedness ratings), and one which examines the general relatedness of the harmonies of each key (represented as a static geometric graph) within the various key contexts. For the purposes of this study (which include the examination of general, order-agnostic chord relationships in various contexts), I did find the geometric model insightful, especially when considered alongside the phenomenological results. The results of these analyses are shown as figures 6 and 7.

Figure 6. ‘Average ratings for pairs of diatonic triads in the same key (top) or in different keys (bottom) presented in the context of C major, G major, A major, B major, or F# major (from Bharucha & Krumhansl, 1983, and Krumhansl, Bharucha, & Castellano, 1982). The top effect is an example of the contextual-distance principle; the bottom effect is an example of the contextual-asymmetry principle’ (p. 198).

Figure 7. ‘Multidimensional scaling solution of judgments of pairs of diatonic traids [sic] in C and F# major presented in the context of C major, G major, A major, B major, or F# major (from Bharucha & Krumhansl, 1983, and Krumhansl, Bharucha, & Castellano, 1982). The key context enhances the perceived relatedness of chords that function in that key and in closely related keys, an example of the contextual-distance principle. Adapted by permission of the publishers’ (p. 197).

What is perhaps most striking in these results is the dependency of judgments of cognitive distance on the tonal context of the target chords. Specifically, Krumhansl’s gestalt principles of contextual distance and contextual asymmetry demonstrate a strong effect on these results. (Contextual identity demonstrates a strong effect on a different memory-testing task in this chapter, which I do not have time to discuss here.) As shown in figures 6 and 7, no matter what the pair of target chords, their relatedness to the contextual key (measured as number of steps apart on the circle of fifths) determines in large part their degree of relatedness. This leads Krumhansl to the following conclusion: ‘The hierarchical differential of tones and chords seems to reflect a general tendency in perception and cognition to encode, name, and remember elements in terms of their relations to a few stable, psychologically central reference points’ (p. 210). This ties back to a statement made by Krumhansl in chapter 2: ‘the results obtained here parallel those found in other areas of human cognition and perception, suggesting that a general psychological principle is operating in the particular musical case considered’ (p. 17). That general psychological principle, she states, is the theory of reference points or schemata as articulated by the gestalt psychologists, and as predicted to be musically relevant by Leonard Meyer (CFMP, pp. 19-21). Her gestalt-based principles, then, are not only supported by the data given in CFMP, they also correlate to theories in other domains of cognitive psychology. So while I have reservations about some aspects of the probe-tone methodology, and while I find some of her interpretations and generalizations of the earlier chapters problematic (particularly her geometric model of musical space), I find her gestalt principles of musical perception quite compelling.

Chapters 9 and 10, ‘Perceiving multiple keys: modulation and polytonality’ and ‘Tonal hierarchies in atonal and non-Western tonal music,’ are somewhat removed from the general purpose of the book and seem to be more of an attempt to get as much mileage out of the probe-tone methodology as possible. Most of the studies contained in these chapters are also rather preliminary and require additional research before any significant conclusions may be drawn. Thus I will only briefly mention a few points from these chapters which are relevant to the topic at hand. First, Krumhansl—in her probe-tone-based study of modulation—presented data which suggests that listeners do not make a once-for-all decision regarding the identity or function of a particular harmony when processing music in real time. Rather, listeners tend to examine context and constantly reinterpret chords throughout the hearing of a passage of music. Thus the context upon which a particular chord depends for its functional interpretation is not only the music immediately preceding it, but also the music immediately following it. This is, of course, congruent with a large corpus of linguistic research regarding the real-time processing of spoken language, suggesting again that there may be general psychological principles at work in the perception and memory of music. Second, in comparing subjects’ behavior when listening to various other musical styles—such as atonal Western music, Balinese gamelan music, North Indian ‘classical’ music, etc.—to the data gathered relative to the Western classical common-practice repertoire, Krumhansl found that ‘listeners’ perceptions adapt quite readily to [stylistic] differences’ and that ‘listeners can set aside these expectations [of the style of music with which they are must familiar] and hear the pitch events in style-appropriate terms quite independent of their prior musical experience’ (p. 268). So while the tonal hierarchy may be specific to Western tonal-harmonic music, it seems that—according to Krumhansl’s data—the cognitive strategies are not.

As chapter 11, ‘Music cognition: theoretical and empirical generalizations,’ primarily summarizes the data, interpretations, and generalizations of CFMP, I will do so now as well. In CFMP Krumhansl quantifies the ‘tonal hierarchy’ (built of pitch-classes in the context of a key) and the ‘harmonic hierarchy’ (built of diatonic triads within the context of a key). While some of her extensions of these key profiles—for example, into quantification of the cognitive distances between keys or into the exploration of their correlation with ‘tonal consonance’—are problematic and could easily be discarded, Krumhansl accomplishes much good with them as well. Primarily, she demonstrates their correlation with the repertoire via a successful key-finding algorithm based on the probe-tone key profiles (suggesting strongly the possibility that the hierarchies are internalized through statistical learning), and she demonstrates quite convincingly the dependency of musical perception and memory on tonal context, as expressed in her gestalt-based principles of contextual identity, contextual distance, and contextual asymmetry. These principles and the data that supports them confirm the intuitions of theorists from Kirnberger to Meyer and pose a strong challenge to those who represent an internalized conceptualization of tonal space with a Euclidian geometrical model. While cognitive musicologists following Krumhansl have focused primarily on her probe-tone-based key profiles or the Krumhansl/Schmuckler key-finding algorithm (c.f. Temperley 2001 and 2007, Huron 2006), I find her gestalt-based principles to be the most interesting, useful, and revolutionary aspect of CFMP, both in the fields of cognitive musicology and traditional music theory. I hope that more research in music cognition will pursue this line of thought in the future.

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