Probing the Minor Tonal Hierarchy

Home , Major and minor

Dominique T. Vuvan, Jon B. Prince, and Mark A. Schmuckler

Volume 28, Issue 5, pp 461-472, Music Perception

DOI: https://doi.org/10.1525/mp.2011.28.5.461 Probing the Minor 1

Running head: MINOR TONAL HIERARCHIES

Probing the Minor Tonal Hierarchy

Dominique T. Vuvan

University of Toronto Scarborough

Jon B. Prince

Murdoch University

Mark A. Schmuckler

University of Toronto Scarborough Probing the Minor 2

Abstract

One facet of tonality perception that has been fairly understudied in the years since Krumhansl and colleagues’ groundbreaking work on tonality (Krumhansl & Kessler, 1982; Krumhansl &

Shepard, 1979) is the music theoretical notion that the minor scale can have one of three distinct forms: natural, harmonic, or melodic. The experiment reported here fills this gap by testing if listeners form distinct mental representations of the minor tonal hierarchy based on the three forms of the minor scale. Listeners heard a musical context (a scale or a sequence of chords) consisting of one of the three minor types (natural, harmonic, or melodic) and rated a probe tone according to how well it belonged with the preceding context. Listeners’ probe tone ratings corresponded well to the minor type that had been heard in the preceding context, regardless of whether the context was scalar or chordal. These data expand psychological research on the perception of tonality, and provide a convenient reference point for researchers investigating the mental representation of Western musical structure.

Keywords: tonality, music perception, minor key, pitch, probe tone Probing the Minor 3

Probing the Minor Tonal Hierarchy

Krumhansl and colleagues’ pioneering studies of Western tonal hierarchies (Krumhansl

& Kessler, 1982; Krumhansl & Shepard, 1979) have become of central importance to work in music cognition. Numerous researchers have used these seminal findings to investigate the cognitive representation of musical pitch in particular, and the brain’s processing of complex hierarchical stimuli in general (for reviews, see Krumhansl, 1990, 2000). This research revealed that after hearing a key-defining musical context, listeners’ goodness-of-fit rating for each pitch class was analogous to the pitch class’ position in the theoretical tonal hierarchy defined by that context. Therefore, it was concluded that listeners possess stored internal representations of both major and minor tonal hierarchies that mirror music theoretical descriptions of tonality (see

Lerdahl, 2001 for a detailed description of diatonic pitch space).

In Western tonal music, the tonal hierarchy organizes the 12 possible chromatic pitch classes around a central reference pitch, resulting in four levels of psychological stability (see

Table 1). The tonal hierarchy has two fundamentally important properties. First, it is transposable, meaning that a hierarchy can be built with any of the chromatic tones as a tonal centre. Second, and most relevant here, is that there are actually two distinct categories of tonality in Western music. The first is called a major tonality, and along with being employed most commonly in musical composition, is also used in the majority of research employing tonal stimuli. The second is called a minor tonality, and embodies a similar hierarchical structure to the major, but with notable changes that are detailed subsequently.

The four levels of the major tonal hierarchy are defined as follows. The top level of the hierarchy contains the central reference pitch, or tonic (scale degree 1̂), and is the most stable pitch class within the key. The second level contains pitches that are 4 and 7 semitones above the Probing the Minor 4 tonic (scale degrees 3̂ and 5̂), with the semitone being the smallest unit of pitch distance routinely employed in Western music. Taken together, the first and second levels of the major or minor tonal hierarchy form the tonic triad. The third hierarchic level contains pitches that are 2,

5, 9, and 11 semitones above the tonic (scale degrees 2, 4̂, 6̂, and 7̂), and when combined with the first two levels comprises what is called the diatonic set. Finally, the lowest level of the hierarchy contains the remaining five tones of the chromatic set. These tones are considered to be outside the key, and are the least important and least stable tones of the chromatic set; these tones are called the non-diatonic tones (Schmuckler, 2004, 2009).

In contrast, minor tonal hierarchies exhibit not one, but three distinct (and related) forms depending on the musical context. As compared to the major tonal hierarchy, the basic form of the minor tonal hierarchy, called the natural minor, includes pitches that are 3 rather than 4 semitones above the tonic in the tonic triad, and pitches that are 8 and 10, rather than 9 and 11, pitches above the tonic in the diatonic set. The two additional forms of the minor tonal hierarchy, named for the musical functions they fulfill, are the harmonic minor and melodic minor, and constitute modifications of the natural minor hierarchy described above. The harmonic minor takes on the pitch that is 11 rather than 10 semitones above the tonic in the diatonic set. The structure of the melodic minor, on the other hand, depends on whether the musical context is ascending or descending in pitch. The descending form of the melodic minor is identical to the natural minor, whereas the ascending form includes pitches that are 9 and 11 rather than 8 and 10 semitones above the tonic in the diatonic set. The major and minor tonal hierarchical structures are summarized in Table 1.i

Krumhansl and Kessler (1982) convincingly demonstrated that the psychological representation of tonality obeyed the theoretical relations described here. However, one element Probing the Minor 5 of tonality that was not tackled by their landmark study, nor has it been considered with much emphasis in empirical research that has followed, is how the minor scale is represented psychologically by listeners in its three forms. Given the important theoretical differences between the three possible structures of the minor tonal hierarchy, it is somewhat surprising that music cognition researchers have not previously questioned whether listeners perceive these differences. For instance, Krumhansl and Kessler’s (1982) study tested only harmonic minor contexts, and did not explicitly distinguish listeners’ representation of harmonic minor from the other possible minor types.

To get a more accurate sense of the degree to which the various forms of the minor have been used in research, a review of psychological studies investigating tonality was conducted using PsycInfo. A search of this database, employing a variety of key words,ii revealed 67 studies on Western tonality and tonal hierarchies in music. Of these studies, all have employed major keys, whereas only 26 have employed minor keys. Of the 26 studies employing minor keys, 22 did not specify the minor type, three used natural minor contexts, one used harmonic minor contexts, and none used melodic minor contexts. These 26 minor key studies focused on an array of themes, including key finding (Brown, 1988; Cohen, 1991; Corso, 1957; Frankland & Cohen,

1996; Schmuckler & Tomovski, 2005b; Toiviainen & Krumhansl, 2003; Yoshino & Abe, 2004), the representation of tonality across the life span (Delzell, Rohwer, & Ballard, 1999; Feierabend,

Saunders, Holahan, & Getnick, 1998; Guilbault, 2004), the neural instantiation of tonality

(Janata, et al., 2002; Mizuno & Sugishita, 2007; Otsuka, Kuriki, Murata, & Hasegawa, 2008), the relationship between emotion and tonality (Kellaris & Kent, 1993; Mizuno & Sugishita, 2007), the effect of tonality on memory (Feierabend, et al., 1998; Mead & Ball, 2007), and tonal priming (Hutchins & Palmer, 2008; Otsuka, et al., 2008). Finally, of the four studies that did use Probing the Minor 6 a particular minor type context, none explicitly distinguished between representations of the minor key in general and the representation of the particular minor type that was employed.

Given that the distinctions between the various forms of minor keys are quite subtle, why might one believe that these distinctions would be important enough to result in variations in listeners’ representations of tonal information? One source of evidence in support of this possibility are the numerous studies examining listeners’ mental representations for musical structures that lie outside of those used in Western tonal music, studies that have observed variation in the nature of listeners’ representations. Such work includes studies examining polytonality (Krumhansl & Schmuckler, 1986; Thompson & Mor, 1992), atonality (Krumhansl,

Sandell, & Sergeant, 1987), diatonic modes (Creel & Marvin, 2000, November; Hershman,

1995), as well as novel tonalities, both artificial (Oram & Cuddy, 1995) and from unfamiliar cultures (Castellano, Bharucha, & Krumhansl, 1984; Kessler, Hansen, & Shepard, 1984;

Krumhansl, et al., 2000). Overwhelmingly such work demonstrates that listeners can learn to distinguish a variety of organizations other than the Western major tonal hierarchy. Accordingly, it is reasonable to question whether listeners would also be able to differentiate among the three forms of the minor tonal hierarchy. Thus, the goal of this study was to explore listeners’ mental representations of natural, harmonic, and melodic minor tonal structure.

Method

Participants

Sixteen participants (mean age = 18.9 years, SD = 1.4 years) with a minimum of five years of musical training (mean = 8.2 years, SD = 2.3 years) were recruited from the University of Toronto Scarborough community using the introductory psychology participant pool. With respect to other musical activity, participants had an average of 3.5 years of musical theory Probing the Minor 7 training (SD = 3.8 years), listened to music for 12.8 hours per week (SD = 7.7 hours per week), and played music for 2.2 hours per week (SD = 3.5 hours per week). None of the participants had ever participated in a music psychology experiment before, nor did any participants report a familiarity with the research literature surrounding the tonal hierarchy. Finally, none of the participants reported having absolute pitch.

Stimuli

All stimuli were produced using a grand piano sound on Finale (2009). Two different context types (scale vs. chord) establishing the three minor types (natural vs. harmonic vs. melodic) were produced for a total of six musical contexts (Figure 1). Scalar and chordal contexts were used in order to assess the possibility of a congruency interaction between minor type and context type. Given that the harmonic minor is associated with harmonic (chordal) contexts and the melodic minor is associated with melodic (scalar) contexts, it is possible that the cognitive representations of the harmonic and melodic minors would be better elicited by chordal and scalar contexts, respectively.

All contexts were in A minor, lasted four bars, and contained a falling contour in the first two bars and a rising contour in the last two bars. Contexts were presented only in one key due to practical concerns about the length of the experiment, and because previous research has shown that tonal hierarchical representations generalize across tonal centres (Krumhansl & Kessler,

1982). All contexts were played at 120 beats per minute; accordingly, each half note chord in the chordal contexts was 1 s long, each quarter note in the scalar contexts was 0.5 s long, and each context was 8 s long. Twelve 1 s long probe tones were produced, representing each semitone in the range from A3 (220.00 Hz) to G#4 (415.30 Hz).iii Probing the Minor 8

These stimuli were presented to participants using an Intel Pentium 4 personal computer, with code written and run in MATLAB 7.0 (Moler, 2004), using the Cogent toolbox (Romaya,

2002). The experiment interface was viewed on an LG Flatron L1710S monitor, while the auditory components of the experiment were heard through a pair of Sennheiser HD 280 pro headphones connected to a Creative Sound Blaster Audigy 2 ZS soundcard.

Procedure

At the beginning of each trial, participants were told to listen carefully to a musical context followed immediately by a probe tone. Following presentation of the context and a probe tone, listeners were asked to indicate on a seven-point Likert scale how well the probe belonged

(in a musical sense) with the context they had just heard, with “1” corresponding to “belongs very poorly” and “7” corresponding to “belongs very well”. Trials were blocked by minor type, and the order of the three minor type blocks was counterbalanced across participants to avoid carry over effects. Each block included every possible combination of the two context types and

12 probe tones repeated three times. Therefore, each participant heard 216 trials consisting of three minor types (natural, harmonic, melodic), two context types (chord, scale), 12 probe tones

(from A3 to G#4), and three repetitions.

Prior to beginning the experimental trials, listeners were given up to five practice trials

(randomly chosen from the experimental trials), until they reported being sufficiently comfortable with the task to continue. Following the experimental trials, listeners completed a survey regarding their musical experience. The entire experimental session lasted approximately one hour.

Results Probing the Minor 9

The first step in analyzing these data examined the degree of consistency between listeners. Probe tone ratings across the three repetitions were averaged (preserving each unique combination of minor type, context type, and probe tone), and were used to calculate intersubject correlations. Ratings were similar across participants, with an average intersubject correlation of r (70) = .27 (SD = .20), p < .05. This correlation is relatively low compared to previous studies

(see Krumhansl & Kessler, 1982) but is not altogether unexpected due to the difficulty of processing minor tonalities (Vuvan & Schmuckler, in press). Nevertheless, given the significant degree of consistency among listeners, probe ratings were averaged across listeners for some of the subsequent analyses.

First, listeners’ probe tone ratings were analyzed in a four-way repeated measures

Analysis of Variance (ANOVA), with four within-subjects factors: Minor Type (natural vs. harmonic vs. melodic), Context Type (scale vs. chord), Repetition (1 vs. 2 vs. 3), and Probe Tone

(A3 – G#4). There were significant main effects for Context Type, F(2,30) = 6.89, MSE =

2 2 257.85, p < .05, η p = .32 and Probe Tone, F(11,165) = 13.52, MSE = 112.64, p < .001, η p = .47.

These effects reveal that participants provided higher probe tone ratings when the context was a scale that when it was a series of chords (this effect will be discussed in more detail later), and, as expected, in accordance with the identity of the probe tone itself. Neither Minor Type nor

Repetition was significant as a main effect (all F scores < 1, n.s.). In addition, the interaction

2 between Minor Type and Repetition was significant, F(4,60) = 2.55, MSE = 4.57, p < .05, η p =

.15, indicating a difference in overall ratings of the different minors as a function of repetition.

Because this effect collapses across the ratings for the various probe tones, however, it has no bearing on the nature of the perceived tonal hierarchy as a function of these two variables.

Accordingly, this effect is theoretically unimportant, and as such will not be explored further. Probing the Minor 10

Most critically, the interaction between Minor Type and Probe Tone was significant,

2 F(22,330) = 4.67, MSE = 14.24, p < .001, η p = .32, indicating that the pattern of listeners’ probe tone ratings depended upon which of the three minor types they had heard in the preceding context. Accordingly, this result indicates that listeners did indeed differentiate between the three forms of the minor key. Figure 2 presents the average probe tone ratings for the three minor types, with the probe tone ratings themselves listed in Table 2. This experimentally central effect will be explored further in subsequent analyses.

The interaction between Context Type and Probe Tone was also significant, F(11,165) =

2 6.11, MSE = 23.46, p < .001, η p = .29, indicating that the pattern of listeners’ probe tone ratings also depended upon whether the preceding context was chordal or scalar. This variation in probe tone rating as a function of context type is presented in Figure 3. Further scrutiny reveals that this effect (and the main effect of Context) can be explained by the influence of pitch height for scalar contexts but not for chordal ones. This effect was likely driven by the structure of the scalar stimuli, which shared greater textural similarity with the probe tone than the chordal contexts, and always ended with a rising contour. These properties might result in the perceptual salience of notes higher in the register at the time that listeners made their probe tone judgments.

Thus, a simple perceptual comparison between probe tone and the last portion of the stimulus would lead to higher ratings for higher pitched probe tones, which would lead to the significant main effect of Context Type reported above. Interestingly, this result is akin to the concept of representational momentum (Finke & Freyd, 1985; Freyd & Finke, 1984), a phenomenon that has been demonstrated in a variety of auditory contexts (Freyd, Kelly, & Dekay, 1990; Henry &

McAuley, 2009; Johnston & Jones, 2006). Indeed, removing the pitch height effect from the scale context trials greatly reduced the difference between scale and chord contexts (see Figure Probing the Minor 11

3), with the effect size for the Context Type by Probe Tone interaction falling from η2 = .29

2 before detrending to η p = .15 after detrending. Thus, although this detrending does not completely remove the effect of the different contexts on probe tone ratings, it does account for a large proportion of this interaction. Most importantly, and despite the differences between the ratings for scale and chord contexts, the averaged original ratings for the scale and chord contexts were significantly correlated, r(10) = .72, p < .01, suggesting comparable patterns of probe tone ratings for each of the context types. None of the remaining two- or three-way interactions were significant. Most importantly, the three-way interaction between Minor Type,

Context Type, and Probe Tone was not significant, F(22,330)= 1.24, n.s., indicating that the ratings for the probe tones relative to the different minor forms were not differentially influenced by scalar versus chordal presentation.

Although the previous analyses demonstrate that the different forms of the minor key did give rise to variation in listeners’ probe tones ratings, it does not address the principal question of experimental interest. Specifically, did the different minor forms produce probe tone ratings that are systematically related to the theoretical tonal hierarchies shown in Table 1? Although the significant Minor Type by Probe Tone interaction raises the possibility of this relation, it does not specifically address this concern. Accordingly, to examine this question, a series of correlational analyses were conducted. For this analysis 12-element probe tone profiles were created for each of the three minor types, based on the theoretical position that each probe tone should occupy in each minor type’s tonal hierarchy. Thus, the tonic was assigned a value of 4, the remaining tonic chord members were assigned a value of 3, diatonic tones were assigned a value of 2, and non-diatonic tones were assigned a value of 1. For instance, scale degree 11 is a diatonic tone in the natural minor, a non-diatonic tone in the harmonic minor, and both a diatonic Probing the Minor 12

(descending) and non-diatonic (ascending) tone in the melodic minor. Therefore, it received a value of 2 in the natural minor theoretical profile, 1 in the harmonic minor, and 1.5 (an average of 1 and 2) in the melodic minor (Table 3).iv

Next, the probe tone ratings were collapsed across listener, context type, and repetition to produce a single 12-element rating profile for each of the three minor types. Bivariate correlations were then calculated between these listener-produced rating profiles and (1) the theoretical tonal hierarchy profiles just described, and (2) the minor tonal hierarchy profile of

Krumhansl and Kessler (1982). In terms of the correlations with the theoretical tonal hierarchies, as expected, the correlation between the natural minor rating profile and its theoretical profile was significant, r(10) = .73, p < .01, as was the correlation between the melodic minor rating profile and its theoretical profile, r(10) = .64, p < .05. In addition, the correlation between the harmonic minor rating profile and its theoretical profile was marginally significant, r(10) = .57, p

= .051.

Next, it was of interest to discern whether the correlation between the rating profile for each minor type context and the corresponding theoretical profile (i.e., natural context ratings correlated with natural theoretical profile from Table 3) was significantly higher than the correlation between that rating profile and the other two theoretical profiles (i.e., natural context ratings correlated with harmonic theoretical profile or melodic theoretical profile from Table 3).

Toward this goal, each listener’s ratings were averaged across repetition to form a rating profile for the natural, harmonic, and melodic minor. Next, each rating profile was correlated with the three possible theoretical minor type profiles (Table 3), creating nine correlations for each listener (3 minor types x 3 theoretical profiles). Finally, these correlations were standardized using Fisher’s z’ and entered into a repeated measures ANOVA with Minor Type (natural, Probing the Minor 13 harmonic, melodic) and Theoretical Profile (natural, harmonic, melodic) as factors. There were no significant main effects, but the interaction between Minor Type and Theoretical Profile was

2 significant, F(4,60) = 12.81, MSE = 0.01, p < .001, η p = .46.

Planned contrasts illustrated that the source of the interaction was due to the predicted pattern of differences arising among the correlations for each of the minor types and theoretical profiles. Specifically, for the natural minor type, the correlation between ratings and the theoretical natural profile was significantly higher than the correlation with the theoretical harmonic, F(1,15) = 10.69, MSE = 0.05, p < .01, or melodic, F(1,15) = 9.58, MSE = 0.02, p <

.01, profiles. For the harmonic minor type, the correlation between ratings and the theoretical harmonic profile was significantly higher than the correlation with the theoretical natural,

F(1,15) = 15.48, MSE = 0.02, p = .001, or melodic, F(1,15) = 16.00, MSE = 0.01, p = .001, profiles. Finally, for the melodic minor type, the correlation between ratings and the theoretical melodic profile was not significantly higher than the correlation with the theoretical natural or harmonic profiles, both F(1,15) < 2.06, n.s., perhaps reflecting the fact that the melodic is in fact a hybrid of the natural and harmonic forms.

Importantly, the correlation between each minor type’s rating profile and the corresponding theoretical profile was the highest of the three possible correlations with theoretical profiles; these correlations are shown in Figure 4. This finding indicates not only that listeners differentiated between the three minor types in their probe tone ratings, but also that these ratings reflected the minor type presented in the preceding context.

As for the correlations with Krumhansl and Kessler (1982) minor tonal hierarchy, the strongest correlation was with the natural minor form, r(10) = .67, p < .05, followed by the melodic minor form, r(10) = .59, p < .05, and finally the harmonic minor form, r(10) = .47, n.s. Probing the Minor 14

Two points are interesting to note in this regard. First, for none of the minor forms did correlations between probe ratings and the Krumhansl and Kessler (1982) profiles exceed the correlation with the corresponding theoretical hierarchy. In other words, tonal hierarchies which included the subtle distinctions between the three theoretical minor types were at least as good, if not better, at predicting listeners’ probe ratings for contexts based on each of those types than was Krumhansl and Kessler’s (1982) “general” minor tonal hierarchy. This result could be expected considering that these authors had employed harmonic minor contexts as a proxy for the minor in general, although it is interesting that the harmonic tonal hierarchy was still better at predicting harmonic context probe tones than Krumhansl and Kessler’s (1982) minor tonal hierarchy. Second, these correlations are illuminating in regards to the Krumhansl and Kessler

(1982) ratings themselves. Although these ratings were generated based on musical contexts employing the harmonic minor type, it is clear that these ratings most closely correspond to what might be expected based on the natural minor form.

Discussion

The experiment reported here addresses a facet of the psychological representation of the

Western tonal hierarchy that has been long neglected by researchers. This work has demonstrated that listeners possess sophisticated knowledge of the minor key that not only is distinguishable from the major key, but also includes detailed representations of the three forms embodied by the minor key in different contexts. More specifically, following natural, harmonic, or melodic minor key contexts, listeners gave probe tone ratings that corresponded well to theoretical descriptions of the structure of the natural, harmonic, and melodic minor tonal hierarchies. Most fundamentally, these data can serve as a reference point for future perceptual investigations of Probing the Minor 15 tonality, and specifically of the minor key, in addition to the well-known data provided by

Krumhansl and Kessler (1982).

Moreover, these data show not only that listeners can distinguish between tonal representations with different modes (major or minor) and different tonics (as seen in Krumhansl

& Kessler, 1982), but that they can distinguish between representations with the same tonic and the same mode. The fact that listeners were sensitive to the pitch information in musical contexts in which both the mode and tonic were controlled illustrates a level of fine-grained distinction between tonal representations that has not been seen in previous research.

One implication of the idea that listeners can represent the three versions of the minor tonal hierarchy is that there might actually be some competition among these representations in music listening. In this case, when a participant listens to a musical piece in a minor key, it is conceivable that all three minor type representations, natural, harmonic, and melodic, are initially activated in parallel. Upon further listening, the musical context might then lead to the strengthening of one minor type representation over the other two. Such a process would be dynamically competitive, with changes in the context leading to the changes in the differential strengths of each of the minor type representations over the course of the piece. Thus, if a listener is presented with an ambiguous minor context that does not strongly characterize any one of the three minor types, none can dominate, and the three representations remain in competition.

Even in cases where the minor context does specifically embody one of the three minor types, it is likely that weakened representations of the two other minor types remain mentally accessible.

One consequence of such competing representations is that minor contexts might lead to somewhat more ambiguous tonal percepts in cognitive-perceptual tasks as compared to major contexts. Consequently, performance in these tasks would be poorer with minor tonalities than Probing the Minor 16 major tonalities. This result might help explain why some researchers have found that the minor key may be less well-represented than the major key by the cognitive system (Delzell, et al.,

1999; Harris, 1985; Krumhansl, Bharucha, & Kessler, 1982; Vuvan & Schmuckler, in press). For instance, Vuvan and Schmuckler (in press) investigated imagined tonal hierarchy representations using a probe tone task. This study found that probe ratings based on an imagined minor context were much less reliable between participants and showed much less delineation of tonal hierarchical levels than those ratings that were based on an imagined major context.

One particularly fascinating finding from the current study was that listeners were able to distinguish successfully the three different versions of the minor key regardless of whether the context took on a melodic (scale) or harmonic (chord) form. This finding is especially intriguing given that one might expect that a chordal context would elicit a stronger representation of the harmonic minor, whereas a scalar context would be expected to elicit a stronger representation of the melodic minor. Despite this straightforward prediction, context type had little effect on the perception of the form of the minor. Such a finding suggests that listeners could accurately extract pitch distribution information from the contexts heard in the experiment, and did not seem affected by any prior knowledge of correlations between minor type and context type in real musical excerpts.

This finding also, however, suggests that there is more to perceptually distinguishing between the different forms of the minor than simply picking up on the pitch distribution information contained in the contexts. One important difference between the scalar and chordal contexts involves their relative pitch class distributions. The two contexts are similar in that both contained pitches drawn solely from the respective diatonic sets for their form. However, the two contexts are different in terms of the relative frequency of occurrence of the various pitch Probing the Minor 17 classes. For the scalar context, the pitch distribution was essentially flat, with each of the pitch classes occurring twice, except for the tonic, which occurred three times. In contrast, the chordal contexts contained a graded pitch distribution, with pitch classes higher in the hierarchy occurring more often than pitch classes lower in the hierarchy. One might expect that the scalar context would produce a weaker representation of that tonal hierarchy than the chordal context, given that it contained only a single source of information regarding its component structure

(distinguishing between diatonic and non-diatonic tones), and one that does not differentiate the various levels of the theoretical hierarchy. This intuition, however, was not borne out in this study, as there was no reliable difference between scalar and chordal contexts in their respective abilities to induce tonal percepts (as measured by correlations between probe ratings and theoretical natural, harmonic, and melodic tonal hierarchies as well as to Krumhansl and

Kessler’s (1982) classic ratings).

To restate this finding in a more general way, this study provides some evidence that participants were relying on internal representations of the three minor types to perform the probe tone task, particularly in trials employing scalar contexts. Given a scale, participants gave probe ratings that demonstrated an understanding of the appropriate four-level tonal hierarchy despite the fact that, as stated previously, scalar contexts do not convey hierarchical information via tone distributions. These findings are in line with past work by Smith and Schmuckler

(2004), who showed that fairly limited surface pitch statistical information can elicit a robust tonal hierarchy in listeners, provided that this information corresponds with stable memory representations built on previous exposure. However, when this information conflicts with such internalized representations, variation in distributional properties is relatively ineffective in inducing the percept of hierarchical structure in listeners. In contrast, Oram and Cuddy (1995) Probing the Minor 18 provided surface information about novel tonal hierarchies in the form of frequency of occurrence. Because these hierarchies were novel and did not conflict with existing long-term memory representations, these authors were able to elicit tonal hierarchy-like ratings in listeners through exposure to their stimuli. Nevertheless, Smith and Schmuckler’s (2004) study seems more relevant to these results, since listeners would have plenty of previous exposure to music in the minor key.

The most obvious application of these findings is in the domain of key-finding, which seeks to model how the cognitive system is able to determine the key of a musical excerpt with very short exposure. One such prominent model, the Krumhansl-Schmuckler key-finding algorithm (Krumhansl, 1990; Krumhansl & Schmuckler, 1986; Schmuckler & Tomovski,

2005a), uses values derived from Krumhansl and Kessler (1982) to match the pattern of the pitch class duration distributions in the musical excerpt with its best-fit tonal hierarchy in order to resolve the key of the excerpt. Temperley (2001) has proposed a modified version of the model, in which musical excerpts are segmented and then pitch class present/absent distributions are calculated (in place of the duration distributions used by Krumhansl and colleagues). In later work, Temperley (2002) has also suggested that the match between the empirically derived pitch class distribution and tonal hierarchy profiles should be calculated using Bayesian methods, rather than simple correlations or vector products. Although these algorithms have been relatively successful at key determination for a wide variety of musical passages, they could be made more accurate by the replacement of the single minor tonal hierarchy vector with the three minor tonal hierarchies measured here.

Specifically, this addition to previous algorithms would involve the implementation of the natural, harmonic, and melodic minor tonal hierarchies derived in this study (in addition to Probing the Minor 19

Krumhansl and Kessler’s (1982) major tonal hierarchy) as bases of comparison to the pitch class distribution derived from the musical excerpt of interest. The improvement offered by this refined algorithm would be most obvious where a decision must be made between a major key and its relative minor. Major keys share a pitch set with their relative natural minor, but not with their relative harmonic or melodic minors. Theorists have noted that pieces written in the minor key readily shift between the pitch sets for the natural, harmonic, and melodic minors (e.g.,

Laitz, 2008). Therefore, confusions between a major key and its parallel minor would be prevented if the piece’s pitch class distribution correlated better with any one of the three minor type tonal hierarchies than with the major tonal hierarchy.

Finally, some consideration should be given to the issue of the ability to generalize these results to non-musicians. Would non-musicians be able to differentiate among the three forms of the minor key as musicians were able to do in this study? Previous research has demonstrated that, given a probe tone task, the ratings profiles produced by non-musicians are similar to those produced by trained musicians, but that non-musicians do not distinguish the different levels of the tonal hierarchy as clearly as musicians (Cuddy & Badertscher, 1987; Halpern, Kwak,

Bartlett, & Dowling, 1996; Krumhansl & Shepard, 1979). Thus, some researchers have turned to using tasks that implicitly test for knowledge of the statistical properties of music, demonstrating that non-musicians are in fact able to demonstrate a sophisticated understanding of the statistical properties of Western music. For instance, in tonal priming studies, preceding a chord upon which a judgment must be made by another chord that is highly related increases judgment speed and accuracy, whereas the opposite is seen when the preceding chord is highly unrelated (see

Tillmann, 2005 for a review). In fact, simply changing the instructions in a probe tone task by asking listeners to rate melody completion rather than belongingness can ameliorate non- Probing the Minor 20 musician performance to musician levels (Bigand & Poulin-Charronnat, 2006). Additionally,

Loui, Wessel, and Kam (2010) demonstrated that musicians and non-musicians alike were able to implicitly learn the statistical distributions of novel tonal systems. Speculatively, then, it is highly likely that non-musicians possess refined knowledge of the three minor forms just as musicians do, and would empirically distinguish them given the right instructions and task.

Overall, the work presented here has improved our understanding of listeners’ mental representation of minor tonal hierarchies. The finding that listeners are able to form a mental representation of the subtle discrepancies among the three versions of the minor tonal hierarchy fills an important gap in the music cognition literature, improving the resolution with which researchers can study Western tonal hierarchies. This work will hopefully provide a basis for more complete investigations of tonal perception in the future. Probing the Minor 21

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Probing the Minor 27

Author Note

This research was supported by grants from the Natural Sciences and Engineering

Research Council of Canada to the first author and third author. Portions of this work were presented at the Society for Music Perception and Cognition Conference (August 3-7, 2009,

Indianapolis, IN). Correspondence concerning this article should be addressed to the first author at [email protected].

Probing the Minor 28

Footnotes

1 From a music theoretical point of view, the three forms of minor are generally only considered to be distinct in a pedagogical sense. Music also tends to drift from one form to another through time (see Aldwell & Schachter, 2003; Clendinning & Marvin, 2005; Laitz,

2008). However, the question of whether listeners perceive the subtle distinctions between the three related minor forms remains important, even in this practical context.

2 The specific keywords we employed for this search were “tonality or tonal hierarchy or probe tone” and “music”, with the restrictions that the results be peer-reviewed primary research articles, Western music, experimental, published in English.

3 We realize that the stimuli are somewhat artificial (for example, the soprano voice in the harmonic chordal progression moving through F - G# - A). The decision to deviate from common practice was made so that all stimuli would be uniform in musical contour.

4 These types of dummy profiles assume a linear additive relationship across the levels of the theoretical hierarchy and have been commonly used in past research to represent schematically hierarchical relationships. For examples, see Palmer and Krumhansl (1990) similar quantification of Lerdahl and Jackendoff’s (1983) metrical framework, Jones’ (1987) joint accent structure, which adds linearly across levels, and Krumhansl and Schmuckler’s (1986) dummy coded bitonal representation of bitonality. Probing the Minor 29

Table 1

Tonal hierarchies for major key and natural, harmonic, and melodic (ascending) minor keys Tonal Hierarchy Level (Scale Degree) Semitone Interval from Tonic

Major

Tonic (1̂) 0

Tonic triad members (3̂, 5̂) 4 7

Diatonic tones (2̂, 4̂, 6̂, 7̂) 2 5 9 11

Non-diatonic tones 1 3 6 8 10

Natural Minor

Tonic (1̂) 0

Tonic triad members (3̂, 5̂) 3 7

Diatonic tones (2̂, 4̂, 6̂, 7̂) 2 5 8 10

Non-diatonic tones 1 4 6 9 11

Harmonic Minor

Tonic (1̂) 0

Tonic triad members (3̂, 5̂) 3 7

Diatonic tones (2̂, 4̂, 6̂, 7̂) 2 5 8 11

Non-diatonic tones 1 4 6 9 10

Melodic (Ascending) Minor

Tonic (1̂) 0

Tonic triad members (3̂, 5̂) 3 7

Diatonic tones (2̂, 4̂, 6̂, 7̂) 2 5 9 11

Non-diatonic tones 1 4 6 8 10 Probing the Minor 30

Table 2

Averaged probe tone ratings for natural, harmonic, and melodic minor types

Pitch Class Semitones from Natural Minor Harmonic Minor Melodic Minor Tonic Rating Rating Rating

A 0 5.08 4.62 4.75

A# 1 3.03 2.63 3.26

B 2 3.73 3.74 3.76

C 3 4.23 4.23 4.46

C# 4 3.64 3.63 3.49

D 5 3.85 3.81 4.09

D# 6 3.13 4.15 3.67

E 7 5.29 5.21 5.08

F 8 4.43 4.77 4.14

F# 9 3.95 3.95 4.43

G 10 5.26 3.79 4.51

G# 11 3.99 5.30 4.91 Probing the Minor 31

Table 3

Theoretical probe tone profiles for natural, harmonic, and melodic tonal hierarchies in A minor

Pitch Class Semitones from Natural Profile Harmonic Profile Melodic Profile Tonic Value Value Value

A 0 4 4 4

A# 1 1 1 1

B 2 2 2 2

C 3 3 3 3

C# 4 1 1 1

D 5 2 2 2

D# 6 1 1 1

E 7 3 3 3

F 8 2 2 1.5

F# 9 1 1 1.5

G 10 2 1 1.5

G# 11 1 2 1.5 Probing the Minor 32

Figure Captions

Figure 1. Musical contexts produced by crossing three minor types with two context types.

Figure 2. Averaged probe tone ratings and theoretical dummy profiles for natural, harmonic, and melodic minor types. Error bars illustrate standard error of the mean for each data point.

Figure 3. Averaged probe tone ratings for scale contexts (original and detrended) and chord contexts. Error bars illustrate standard error of the mean for each data point.

Figure 4. Bivariate correlations between rating profiles (averaged across subjects) and theoretical profiles. * - Correlation significant at the .05 level. ** - Correlation significant at the .01 level. †

– Correlation marginally significant. Probing the Minor 33

Figure 1 Probing the Minor 34

Figure 2

Averaged probe tone ratings Theoretical dummy profile

6 Natural

0 A A# B C C# D D# E F F# G G#

n i

t 6 Harmonic a

R 5

s 4

e n

g 3

n i

g 2

n o

l 1 e

B 0 A A# B C C# D D# E F F# G G# 7

6 Melodic

0 A A# B C C# D D# E F F# G G#

Probe Tone Pitch Class Probing the Minor 35

Figure 3

7 Scale (original) 6

2 A A# B C C# D D# E F F# G G#

g n

i Scale (detrended) t

a 6

s 5

e n

g 4

i g

n 3

l e

B 2 A A# B C C# D D# E F F# G G# 7 Chord 6

2 A A# B C C# D D# E F F# G G#

Probe Tone Pitch Class Probing the Minor 36

Figure 4

** p < .01 * p < .05 .8 † p = .051 ** * * *

.6 * †

)

(

i t

a .4

o C .2

0 Natural Harmonic Melodic Natural Harmonic Melodic Natural Harmonic Melodic Natural Ratings Harmonic Ratings Melodic Ratings