Melodic Anchoring - mp.ucpress.edu · Melodic A nchoring 385 Overview of the Model Stable tones in a tonal context serve as potential anchors. The stability of a tone is the activation

Music Perception © 1996 by the regents of the Spring 1996, Vol. 13, No. 3, 383-400 university of California

Melodic Anchoring

JAMSHED J. BHARUCHA Dartmouth College

Unstable tones in a tonal context demand resolution to stable pitch neigh- bors. The stable tones serve to anchor unstable tones in their pitch neighborhood. Two constraints characterize this process of melodic anchoring: the anchor and the anchored tones are close in pitch (proximity), and the anchor always follows the anchored tone (asymmetry). Given these two constraints, anchors function like cognitive reference points, and melodic anchoring can be thought of as a special case of a more general cognitive phenomenon. This paper represents an attempt to understand the cognitive mechanisms underlying this phenomenon. The pervasiveness of these constraints is demonstrated by analyzing the use of nonchord tones in the exposition of Mozart's Piano Sonata in El» Ma- jor, K. 282. Nonchord tones were almost always followed by a chord tone at an interval of two semitones or less, and no such stricture held for how these tones were approached or for how chord tones were followed or approached. The proposed model postulates that an unstable tone attracts auditory attention to its pitch neighborhood. The focus of frequency-selective auditory attention accounts for the pitch proximity constraint. The most active representational units in that neighborhood -

typically the most stable tones - then drive the expectation. This process is inherently asymmetric. The expectation underlying melodic anchoring is modeled as a vector, called the tonal force vector. The strength of the tonal force vector in each direction is proportional to the activation of the nearest anchor in that direction and inversely proportional to the distance to that anchor. The overall directional expectation, modeled as the sum of all possible expectation vectors, including the tonal force vector, is called the yearning vector.

enduring characteristic of tonal music is that unstable tones behave in highly predictable ways. They seek resolution to stable tones that

are proximal in pitch. In Western tonal music, this is typified in two ways: (1) by the resolution of nonchord tones to neighboring chord tones as in the case of appoggiaturas, suspensions, or passing tones, and (2) by voice leading from one chord to the next, as in the movement of the leading tone to the tonic or the subdominant to the mediant in a V7-I chord progression,

Requests for reprints may be sent to Jamshed J. Bharucha, Department of Psychology, Dartmouth College, 6207 Gerry Hall, Hanover, NH 03755. (e-mail: [email protected])

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384 Jamshed J. Bharucha

or in the voice leading involved in the resolution of an augmented sixth chord. I shall focus only on the first of these two manifestations, although the model presented generalizes to the second. The demand or yearning for resolution seems greatest when the tones to be resolved occur on a strong metrical beat and when they are nondiatonic or otherwise dissonant. In Indian music, it is typified by the performer dwelling on the semitone on either side of the Sa (tonic) or Pa (dominant) and finally resolving to the Sa or Pa with dramatic effect.

Although these procedures for voice leading are elementary tenets of music theory, little is understood about the underlying cognitive processes. From Meyer's (1956, 1967, 1973) and Narmour's (1990, 1992) work, the central role of expectations and implications in music perception has be- come almost axiomatic, and increasing attention is being paid to expectations in the work of psychologists (Bharucha & Stoeckig, 1986; Carlsen, 1981; Jones, 1981a, 1981b, 1986; Schmukler, 1989). In this paper, I focus on one particular case of expectation and attempt to cast some light on it psychologically. Melodic expectation is complex and involves a number of factors (Narmour, 1990, 1992). This paper is limited to the tonal factors that lead to the expectation of a chord tone following a nonchord tone.

This process of resolution by step may be considered a form of anchoring - in this case, melodic anchoring (Bharucha, 1984). An account of melodic anchoring must deal with two constraints. First, a nonchord tone seems to want to be anchored by an interval of no more than two semitones. This is the pitch proximity constraint. Pitch proximity has been known to play a special role in the processing of tones (Deutsch, 1978). Second, pitch proximity is most starkly manifested in the interval following rather than preceding the nonchord tone. This is the asymmetry constraint. Implicit in both constraints is the additional constraint that the anchor is a chord tone. The perceived asymmetry of pairs of tones as a function of their relative stability is well known (Krumhansl, 1979), and several mathematical models exist to account for them (Krumhansl, 1978; Tversky & Hutchinson, 1986). One recent and intriguing model (Katz, 1995) has addressed aspects of the resolution of dissonance in the context of affect, using a neural net approach. The model presented here is an attempt to account for the constraints on anchoring, using two processing mechanisms that have cognitive generality across domains: activation and attention.

The next section of the paper provides an overview of the model. The following section is an analysis of the use of nonchord tones in Mozart's piano sonata, K. 282, as a way of demonstrating the almost rigid applica- tion of these constraints. This is followed by a formal development of the model.

Melodic A nchoring 385

Overview of the Model

Stable tones in a tonal context serve as potential anchors. The stability of a tone is the activation of its representational unit. Here I assume a neural net model in which there are representational units for tones, which, through a process of neural self-organization, connect to representational units for chords and keys (Bharucha, 1987a, 1987b, 1992; Bharucha & Mencl, in press). In the neural net model, activation of a tone unit has both bottom-up and top-down components. The bottom-up component is a result of the occurrence of that tone; the more often and the more recent the occurrences of the tone, the greater the activation. The top-down component is a result of priming from more abstract representational units, such as those for chords and keys, which have themselves been activated by bottom-up processes. At any given time during a piece of music, there ex- ists a particular pattern of activation across the tone units, and this pattern represents the relative stability of pitches at that point in time. It is assumed, for example, that when a subject in an experiment reports how well a probe tone fits into the preceding context (as in the experiments of Krumhansl, 1990), it is the activation level that the subject reports.

The anchors could be considered cognitive reference points in Rosch's (1975) sense. Krumhansl (1979) proposed that stable tones in a tonal context are cognitive reference points, based on the asymmetry of the perceived relationship between two tones differing in stability. Given two tones $ and w, such that s is more stable than u, subjects in Krumhansl's study judged the two tones to be more closely related when s followed u than in the reverse order. Activation peaks in a layer of neural representational units can function as cognitive reference points when coupled with attention. This occurs when there is sufficient differentiation between the highest peaks and the lowest troughs, as is the case in a tonal context.

The nearest-neighbor model of Tversky and Hutchinson (1986) gives a

good account of relationships of this sort. Tversky and Hutchinson used Krumhansl's (1979) rating data to compute each tone's nearest psychological neighbor. A tone is a focus if it is the nearest neighbor of more than one tone. Several levels of foci emerge, along with the nearest neighbor rela-

tionships to foci, which match the anchoring relationships discussed here (see Krumhansl, 1990).

The present model draws on the finding that attention operates over the

frequency domain as a limited capacity system whose resources can be focused to specific regions of the continuum (Lukas, 1980; Scharf, Quigley, Aoki, Peachey, & Reeves, 1987; Swets, 1963, 1984). The width of the re-

gion of attention selectivity is roughly a critical bandwidth (Greenberg 6c


Larkin, 1968). The role of attention over the frequency domain in audition is analogous in some respects to the role of spatial attention in vision: a signal facilitates the detection of subsequent signals proximal to it (Posner, 1980; Scharf et al., 1987). The processing advantages that accrue to tones proximal in pitch (Deutsch, 1978) could perhaps be accounted for by frequency-selective auditory attention. The role of attention has been under- estimated in research on music cognition, with the notable exception of work by Jones (e.g., Jones, Boltz, & Kidd, 1982). In the present paper, attention is proposed as a cognitive mechanism that drives the sense that a nondiatonic tone induces a need for resolution.

According to the present model, a tone attracts attention to a pitch neighborhood around it. The more dissonant, unexpected, or otherwise salient the tone, the greater the resulting attention selectivity favoring the pitch neighborhood centered at that tone. Nonchord or nonscale tones, being dissonant or unexpected, are particularly effective at drawing attention. Once attention is focused on a pitch neighborhood, the most active representational units within that neighborhood drive the melodic expectation. An unstable tone thus serves to draw attention to neighboring stable tones.

An event in the world attracts attention for one of two reasons: (1) it is salient (for example, intense) compared with its surrounding context or (2) it is unexpected (it may be possible for the second reason to subsume the first). It follows from the second reason that an event attracts attention to the extent that its mental representation is inactive before the event occurs. The representational units for nondiatonic tones have low levels of activation in a tonal context (Bharucha, 1987a, 1987b); hence nondiatonic nonchord tones will be particularly effective at drawing attention to themselves.

Attention is attracted to internal objects - mental representations - to the extent that they are activated. When several objects are salient, either in the perceptual environment or as activated representations, attention can be oriented toward a select subset of these objects by means of an attention-getting device such as pointing. When a nonchord tone attracts attention to its pitch neighborhood, the most active representational units within that neighborhood then rivet attention to themselves. The resulting attentional focus on the representation of the highly activated chord tone neighbor while the nonchord tone is still sounding gives rise to the subjec- tive demand or yearning for resolution to the chord tone.

All things being equal, the more stable a pitch class, the more highly expected it is. However, the extent to which a tone is expected depends also on the locus of frequency-selective attention and the relative proximity of other stable tones within that neighborhood. Without an attentional focus, the expectation is ambiguous because alternative anchors exist. Expecta- tion becomes specific or unambiguous when attention is directed to the

Melodic Anchoring 387

immediate neighborhood of a potential anchor and when alternative anchors within that neighborhood are further away. This results in a strong, conscious expectation, a yearning. The yearning resulting from this expectational specificity is most clearly manifest in an appoggiatura that is a semitone away from a chord tone, because the alternative anchor is two or three semitones away. The appoggiatura attracts attention because it is dissonant and on a strong metrical beat. The greater proximity of the appoggiatura to one potential anchor over the other creates a directional expectation. Thus, according to the model, it is not just the proximity to a potential anchor that drives the expectation, but the relative proximities and activations of the nearest candidate anchors on either side. The strength and specificity of the expectation depends on the strengths of the activation of the neighboring stable tones and their distances from the unstable one. Two equally activated peaks equidistant on either side of the nonchord tone will result in no directional expectation - the expectation will be ambiguous and will not be felt as a yearning or demand for resolution. The model predicts that the stronger the directional expectancy, the greater the likelihood of that anchor being selected in music.

Because the expectations in melodic anchoring involve both direction and magnitude, they can be modeled as vectors. It should be remembered throughout this discussion that other factors, such as those discussed by Narmour (1990, 1992), may serve to disambiguate an expectation. Be- cause a number of factors may determine the overall expectation, the model involves the addition of vectors.

Analogies to melodic anchoring as attention orientation abound and serve to illustrate it. If someone points in a general direction, you orient toward that direction, and then the most salient object in that general direction stands out. In the case of melodic anchoring, the nonchord tone is the pointer or attention-getting cue and the salient object is a mental object - a strongly activated mental representation.

Some analogies are inspired by Rosch's (1975) work on cognitive reference points. Rosch claimed that straight lines oriented horizontally and vertically function as cognitive reference points. A line that is 5° to the horizontal is judged to be almost horizontal, whereas a horizontal line is not judged to be almost 5° in most contexts. In terms of our attentional model, a painting hanging slightly askew leads us to want it straightened. The skewed frame in the context of a room with horizontal floors and ceilings and vertical walls draws attention to particular neighborhoods along the mentally represented dimension of orientation. The most salient orien- tations - the cognitive reference points - within these neighborhoods then drive one's expectations of how the painting should hang. The almost horizontal edges of the frame seem to want to be exactly horizontal, and the almost vertical edges seem to want to be exactly vertical.

388 Jamshed J. Bhar ucha

Melodic Anchoring in Mozart's K. 282

Figure 1 shows the nonchord tones (marked by plus signs) in the exposition (ending on the Bl> major chord on the third beat of measure 15). As is

Fig. 1. Mozart's Piano Sonata in El> Major, K. 282. Nonchord tones in the exposition are marked with "+".

Melodic A nchoring 389

Fig. 1. (Continued).

typically the case, some segments of the piece may lend themselves to alternative harmonic analyses. As the obvious alternatives do not change the conclusions drawn below on the basis of the analysis in Figure 1, I will stick to this one analysis. It should be noted that a harmonic analysis is far from an exhaustive analysis of a piece of music and that such an analysis is depicted here only to focus on the chord structure of the piece as it drives the anchoring of tones. In this respect, it is striking that the bottom stave consists for the most part of broken chords and contains no more than 10 tones that cannot be analyzed as chord tones.Given the paucity of nonchord tones and the difficulty of determining the melodic intervals in the bass, the following comparison of chord tones and nonchord tones is based on the top stave. The 10 nonchord tones in the bass will nevertheless be examined to see if they conform to the constraints being investigated.

Using Narmour's (1990, 1992) terminology, I will refer to the interval preceding a note as the implicative interval and the interval following a note as the realized interval. The criteria for counting implicative and realized intervals were as follows. The first note in the piece obviously has no implicative interval. When the voice divides in measure 8, the implicative interval of the first note of the lower voice (Bl>) is omitted because it is counted for the top voice. The realized interval of the last note in the exposition was counted. Grace notes were treated literally as shown in the score. Tied notes were counted as one. Since the number of notes in a trill can vary, a minimal trill was assumed; for example, a trill on Bl> was treated as B-F-Ek


The distribution of implicative and realized intervals (in semitones) is shown in Figure 2 for chord tones and in Figure 3 for nonchord tones. Intervals of 0 semitones (repetition of a note) are omitted. Figure 4 shows the mean intervals of each of these four distributions. A two-way analysis of variance was performed with chord membership (chord tone vs. nonchord tone) and temporal direction of the interval (implicative vs. realized). This analysis was performed with the caveat that the very same intervals (with one exception) were counted as implicative in one condition and realized in another. Although this may raise questions regarding the use of analysis

Fig. 2. Distribution of implicative and realized intervals (in semitones) for chord tones in the top stave of the exposition.

Fig. 3. Distribution of implicative and realized intervals (in semitones) for nonchord tones in the top stave of the exposition.


Fig. 4. Means of the distributions shown in Figure 2.

of variance, the analysis is offered solely to supplement the descriptive sta- tistics.

In general, intervals associated with chord tones (mean = 2.53 semitones) are significantly larger than those associated with nonchord tones (mean = 1.75 semitones) [F(l,396) = 16.54, p < .001]. To digress for a moment, this means that a leap strongly predicts a chord tone rather than a nonchord tone. In an instrument such as the violin, in which the performer can com- municate that a leap is in progress before reaching the next note, this infor- mation would lead to the expectation of a chord tone; a nonchord tone following a leap would be unexpected and therefore salient.

There is no overall difference between the sizes of implicative and realized intervals [F( 1,396) = .37]. This should not be surprising given that, with one exception, these two conditions represent the same intervals. What is most interesting is the significant interaction between chord membership and temporal direction [F(l,396) = 5.08, p = .025]: the realized intervals are smaller than the implicative intervals for nonchord tones but not for chord tones. For chord tones, implicative intervals range between 0 and 7 semitones, and realized intervals range between 0 and 8 semitones. For nonchord tones, implicative intervals range between 0 and 8 semitones, but realized intervals range only between 1 and 2 semitones.

Only once is a nonchord tone not followed immediately by a chord tone. In this case (m. 14), it is followed by a semitone to another nonchord tone, which is in turn followed by a semitone to a chord tone. The eventual chord tone, while delayed, is still only a step removed from the first nonchord tone. And only twice does a chord change occur between the nonchord tone and the anchor: in measures 13-14, Ftt-G, and in measure 14, Dt-D.


Figure 5 shows the distance (in semitones) of each possible nonchord tone from the closest upward and downward chord tone. Diminished seventh chords were not scored. The anchor points occur at the root, the mi- nor third (for minor chords), the major third (for major chords), the perfect fifth, and the minor seventh (for dominant seventh chords). The number of instances of each case of anchoring is shown beside each arrow. Bold- face numbers represent cases in which the chord was minor, and under- lined numbers represent cases in which the chord was a dominant seventh. Where the major and minor chords have the same anchor points on either side (i.e., between the perfect fifth and the tonic above), or where the major and dominant seventh chords have the same anchor points on either side (i.e., between the tonic and the perfect fifth above), the instances are col- lapsed into the numbers in plain font. The minor second and minor seventh above the root were never used as nonchord tones.

Fig. 5. A grid representing distance (in semitones) above the root of the anchoring chord. Each arrow represents the trajectory of a nonchord tone to its anchor. Solid lines represent tones that serve as anchors in this musical example. The number beside an arrow represents the number of cases with that trajectory.


A Model of Melodic Anchoring

YEARNING VECTOR

Each factor, i, that exerts a pull on the current event, E, can be represented as a psychological force vector, fEi, in unidimensional pitch-height space. It depicts the direction (up or down) and strength of the pull. The vector sum of all such forces yields the net psychological force pulling the event up or down. This net force is an expectation. If strong enough, as in the case of an appoggiatura or a suspension, it may demand or evoke a conscious sensation of yearning for a particular resolution. I will use the term yearning vector to represent the net psychological force pulling a musical event up or down. The yearning vector, yE, is the resultant of the various force vectors acting on the event E, and is given by:

TONAL FORCE VECTOR

Among the components of the yearning vector are the bottom-up and top-down factors detailed by Narmour (1990, 1992). I will focus on one component, which I shall call the tonal force vector. The tonal force vector is generated by the current event's relationship to the tonal context and is the expectation that underlies melodic anchoring. It is the expectation of the anchor before it occurs - the psychological force that underlies an unstable tone's demand for resolution to a stable neighbor.

The tonal force vector in a particular direction is proportional to the activation of the nearest activation peak in that direction and is inversely proportional to the pitch distance from that peak.

The upward tonal force vector, tE+, is thus

where <* means "proportional to," a+ is the activation of the closest activation peak above the current event, and d+ is the distance (in semitones) of that peak to the current event (Figure 6). Similarly, the downward tonal force vector, tE_, is

tr £,- oc-a/d - £,- - -

where a_ is the activation of the closest activation peak below, and d_ is the distance (in semitones) to that peak. The net tonal force vector, tE, is thus proportional to the difference between the upward and downward vectors:

tFoc(a/d)-(a_/dJ.


Fig. 6. A schematic diagram showing the current event E (indicated by the downward arrow) and the nearest major activation peaks above and below.

In a Western tonal context, the strongest peaks at any given point in time are associated with the chord tones. Figure 7 shows the activation of the tone units of the MUS ACT neural net model following the first event (an Et major chord) and following the second event (a nonchord tone, C). Before the nonchord tone occurs, there is little activation at C. C is thus unexpected and serves to draw attention to its neighborhood, within which the nearest peaks are associated with Et and Bk Bt is closer and does indeed anchor the C in this passage.

Note that the pattern of activation closely resembles the profiles resulting from Krumhansl's (1990) experiments. The chord tones are the most highly activated (in this example, they were actually sounded). Even though they were presented to the network with the same bottom-up activation, the tonic is slightly higher than the remaining chord tones because of top- down activation from the chord and key units. Among the nonchord tones, the remaining diatonic tones in the key of Et major are more highly activated than the nondiatonic tones, again as a result of top-down activation. When the chord changes, the new chord tones emerge as the most highly

Fig. 7. Pattern of activation across the tone units of the MUS ACT model (Bharucha, 1987a) following the first event (a Bt major chord) and the second event (a nonchord tone, C) of the piece. The activation is scaled logarithmically.


activated, while the remaining diatonic tones show amongst themselves the typical tonal hierarchy.

Factors likely to contribute to the proportionality constant include (1) the attention, A, drawn to the pitch region of the current event and (2) the prior activation, a0, of the unit representing E. The tonal force vector is here hypothesized to be directly proportional to A and inversely proportional to a0. Hence:

tE=(A/a0)x[(a+/d+)-(a./d.)] The term A represents factors that may draw attention to E. Among these is metrical position. A strong metrical position can be thought of as a tem- porally specific focus of attention (Jones, 1986; Jones &c Boltz, 1989; see also Desain, 1992; Gjerdingen, 1989). An appoggiatura triggers a stronger tonal force vector than does the same pitch in the same tonal context on a weaker beat. In addition, the term A might be influenced by the number of events placed on a particular beat (Palmer 6c Krumhansl, 1990).

Another factor likely to draw attention to the current event is its dissonance. A more dissonant event triggers a stronger tonal force vector than a less dissonant event in the same context. According to this model, dissonance alone cannot trigger a tonal force vector, suggesting that the lay intu- ition that dissonance demands resolution is mistaken. Dissonance simply strengthens a tonal force vector generated by the tonal context. The current event's relationship to the tonal context determines the direction of the tonal force vector, and dissonance amplifies the magnitude of the vector.

The degree to which the current event is itself unexpected following the prior context is a third factor influencing attention. The more unexpected E is, the greater the attention drawn to it and the stronger the tonal force vector demanding its resolution. This is one mechanism by which violating an expectation can heighten the demand for resolution: the attention drawn by E's violation of expectancy strengthens the expectation it generates for its successor. Among the determinants of this factor are those enumerated by Narmour.

Other factors that might draw attention to the current event, thereby increasing A, include those discussed in the literature on performance (see Palmer, 1989), for example, duration, accent, novel timbres, and other performance cues. It is not within the scope of this paper to advance a full- blown model of factors that drawn attention. The attention parameter in this model is therefore a place holder for any and all factors that might contribute to attention and in doing so magnify the tonal force vector.

The model's measure of the stability of E just before its occurrence is

designated a0. As in the neural net model (Bharucha, 1987a, 1987b), it is determined by the number of its prior occurrences weighted by their recency, as well as by top-down activation received indirectly from activated chords


and keys. The greater a0 is, the smaller is the magnitude of the tonal force vector. In other words, the more stable the current tone, the less the expectation to move to another tone.

Consider the case in which the distances to the lower and higher peaks are equal (d_ = d+). If the activations of the lower and higher peaks are roughly equal (a_ = a+), then there is no tonal force from either above or below, so the net tonal force is zero (tE = 0) and no tonally based expectation occurs. This equidistant condition characterizes the major second above the root in major triads and the perfect fourth in minor triads. In the Mozart sample, the probability of anchoring upward given that the two anchors are equidistant from the nonchord tone is 0.33, which is not significantly less than chance [t(8) = 0.96, p > .05]. Therefore, no evidence of a directional preference is apparent in these cases, in accord with the model. If there is indeed a preference for a downward anchoring here, that is not measurably significant because of the small number of cases, it is in the expected direction, because most of these instances (seven of nine) involve the anchoring of the major second, and the activation is likely to be greater at the root than at the major third.

Consider next the case in which the distance to the lower peak is less than the distance to the higher peak (d_ < d+). If the activations of the lower and higher peaks are roughly equal (a_ = a+), the net tonal force will be less than zero (tE < 0), implying a downward expectation. This condition characterizes the minor sixth, the major sixth, and the perfect fourth in the major chord. In the Mozart sample, the probability of anchoring upward in this case is 0.17, which is significantly less than chance [£(17) = 2.73, p < .01], supporting the model.

Now consider the case in which the distance to the lower peak is greater than the distance to the higher peak (d_ > d+). If the activations of the lower and higher peaks are roughly equal (a_ = a+), the net tonal force will be greater than zero (tE > 0), implying an upward expectation. This condition characterizes the major seventh, the minor third and augmented fourth in the major chord, the major second in the minor chord, and the major sixth in the dominant seventh chord. In the Mozart sample, the probability of anchoring upward in this case is 0.86, which is significantly greater than chance [£(20) = 3.21, p < .005], supporting the model.

Consider two cases, p and q (see Figure 8), in which the distance to the higher peak is the same in the two cases, but the distance to the lower peak is greater in case q. Consequently, tE > tE > 0. For the same distance to the nearest peak, expectations are stronger the greater the distance from the competing peak.

Table 1 shows the probability of resolving by x semitones given that the alternative anchor is y semitones away, in the Mozart sample. It is clear that when the alternative anchor is three or more semitones away, it is never chosen. It is also clear, particularly from the bottom panel, that the


Fig. 8. Two examples that differ only in the distance from the current event to the nearest major peak below.

TABLE 1

Anchoring Transition Probabilities Given Alternative Anchors

x y p

1 2 0.76 1 3 1.00 1 4 1.00 2 1 0.36 2 2 0.50 2 3 1.00

note. - Probability of resolving by x semitones if the alternative is y semitones.

probability of resolving by a fixed distance (in this case, two semitones)^ increases as the distance to the alternative anchor increases.

The model predicts that stronger tonal contexts yield stronger directional expectations. Although this cannot be tested in the Mozart sample, it is worth noting. Figure 9 shows a comparison of cases, p and q, in which the distances to the nearest peaks are the same in the two cases, but the activations of the peaks are greater in p than in q. Consequently, tEq < tEp.

Table 2 shows the tonal force vector computed by the model for each nonchord tone above the root of a chord, assuming equal activation for all

Fig. 9. Two examples that differ only in the strength of the activation of the neighboring peaks, that is, that differ in the strength of the tonal context.


TABLE 2

Yearning Vector for Nonchord Tones in Major Chords

Interval p (Anchoring to Above Root Yearning Vector Nearest Neighbor) K(Up)-K(Down)

Min2nd -0.67 - -1.91 Maj2nd 0.00 0.78 -0.92 Min3rd 0.67 1.00 -0.09 Perf4th -0.50 0.64 0.42 Aug4th 0.50 1.00 1.50 Min6th -0.75 1.00 -0.35 Maj6th -0.17 1.00 -0.50 Maj7th 0.75 1.00 2.09

note. - Assuming constant activation for all chord tones.

chord tones. The tones with the highest expectancies are the minor sixth and major seventh. The major second is zero, because the anchors on either side are equidistant and hence exert equal pulls. If instead one assumes that the root has a higher activation than the other chord tones, the major seventh emerges as having the highest expectancy.

The middle column shows the probability that this expectation was satisfied by Mozart (i.e., that the anchoring was to the nearest neighbor in the Mozart sample). Two points are noteworthy. First, in the vast majority of cases, Mozart satisfied the expectation as given by the vector. Second, the only cases in which Mozart did not satisfy the vector (i.e., where the probability was less than 1.00) were when the magnitude of the vector was small (10.51 or less).

The last column in the table shows the difference between Krumhansl's (1990) observed ratings for the two possible anchorings. K(up) and K(down) represent the rating from the nonchord tone in question to the nearest up- per and lower anchors, respectively. The difference, K(up) - K(down), should be a measure of the preference for one anchoring over the other, where a positive difference means a preference for an upward anchoring and a nega- tive difference means a preference for a downward anchoring. The yearning vectors and the difference of the rating scores are highly correlated [r = 0.67, *(7) = 2.24, /?<. 05].

Discussion

In the exposition of the Mozart sonata, K. 282, not a single nonchord tone was followed by an interval of greater than a step, and no such stricture held for how these tones were approached, or for how chord tones were followed or approached. Although there is little by way of discovery


in this analysis (these constraints on the use of nonchord tones are elementary tenets of harmony), this paper is an attempt to provide an explanation of underlying cognitive mechanisms.

It is important for psychologists to adopt the stance of a music theorist on occasion by attempting to understand aspects of specific pieces of music, as I have done here. Similarly, it is exciting to see music theorists delve into psychological process, as the theorists contributing to this issue have done. For example, Lerdahl (this issue) describes how his broader theory accounts for anchoring, and in doing so, he provides further grist for the psychologist's mill.

The statistical analysis of nonchord tones presented earlier serves only to underscore to psychologists just how strong - even rigid - the constraints are on the use of nonchord or nonscale tones in tonal music.

An informal analysis of the use of nonchord or nonscale tones in samples of popular music and in Indian music suggests that these constraints are pervasive there too. The expectations generated by nonchord tones - particularly diatonic ones - are subjectively strong and specific. Although lis- teners without formal musical training have considerable difficulty learn- ing to map statements about the structure of music onto their experience of it, the experience of tones demanding resolution seems to be relatively eas- ily recognized.1

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Melodic Anchoring - mp.ucpress.edu · Melodic A nchoring 385 Overview of the Model Stable tones in a tonal context serve as potential anchors. The stability of a tone is the activation