tone profiles following short chord progressions: top-down ... · chord progression may then be...

Music Perception © 2000 by the regents of the university of California Fall 2000, Vol. 18, No. 1, 25-57 all rights reserved.

Tone Profiles Following Short Chord Progressions: Top-Down or Bottom-Up?

RICHARD PARNCUTT

University of Graz

ALBERT S. BREGMAN McGill University

Three experiments explored the relationship between chroma-salience profiles of individual chords and tone profiles obtained after short chord progressions. Musicians' tone profiles for diatonic progressions of one, two, and three chords were compared with predictions of three models: a bottom-up stimulus model (number of times each chroma occurs in the progression), a top-down or schema-driven key model (best-fitting key profile of C. L. Krumhansl & E. J. Kessler, 1982), and an intermediate pitch model that includes both top-down and bottom-up components (cumulative pitch salience; R. Parncutt, 1989, 1993). For single chords, all predictors significantly matched tone profiles, except the key model applied to the diminished triad. For pairs of chords, the pitch and key models consistently outperformed the stimulus model, consistent with the assumption that a (top-down) key had been established; in the pitch model, the second chord influenced the tone profile more than the first (recency effect). Progressions of three chords comprised forward (e.g., F- G-C) and backward (C-G-F) cadences in major and minor keys. The pitch and key models were successful for all progressions, but the key model predicted the tonic of backward cadences in C major and minor to be F. Predictions of the stimulus model were clearly worse than those of the other models, especially for backward cadences. Both primacy and recency effects were observed. In summary, the pitch model was the most consistently successful model over all experiments. To successfully predict tone profiles following chord progressions, it was necessary to account not only for recency (and primacy) but also for variations in pitch salience within chords. Results are consistent with a model of tonality induction in which bottom-up processes interact in real time with top-down processes of two kinds: recognition of harmonic pitch patterns and recognition of key profiles.

Received September 3, 1998; accepted for publication March 7, 2000.

Address correspondence to Richard Parncutt, Department of Musicology, University of Graz, Mozartgasse 3, 8010 Graz, Austria, (e-mail: [email protected])

ISSN: 0730-7829. Send requests for permission to reprint to Rights and Permissions, University of California Press, 2000 Center St., Ste. 303, Berkeley, CA 94704-1223.

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26 Richard Parncutt & Albert S. Bregman

Top-down or schema-based processes involve memory, knowledge, at-

tention, and so on, whereas bottom-up or primitive processes depend on information drawn directly from the stimulus (e.g., Cuddy & Lunney, 1995; Eysenk & Keane, 1990, pp. 84-95; Krumhansl, 1995; Narmour, 1992). The key profiles of Krumhansl and Kessler (1982), reproduced in Figure 1, have been interpreted and explained in both these ways. In a top- down interpretation (Jordan & Shepard, 1987; Krumhansl & Kessler, 1982; Leman, 1995), the profiles have been regarded as schemata, applied by listeners to passages of tonal music in order to determine the prevailing key. A schema may be regarded as a "control system in the human brain that is sensitive to some frequently referring pattern, either in the environment, in ourselves, or in how the two interact" (Bregman, 1990, p. 401). In a bottom-up interpretation, the key profiles reflect the distribution of pitches occurring in the immediately preceding stimulus and held in short-term memory: for example, most of the information contained in the key profiles is already contained in the notation of typical cadential progressions, in particular in the chord progressions used in Krumhansl and Kessler's experiments (Butler, 1989).

A purely bottom-up explanation of the key profiles can immediately be rejected. A bottom-up model would predict flat profiles in the case when probe tones follow scale passages in which all seven diatonic tones occur equally often. But the profiles obtained in that case are far from flat (Brown, Butler, & Jones, 1994; Krumhansl & Shepard, 1979; Lamont, 1998; Lamont &C Cross, 1994). Such a model would also predict baseline results for events not present in the stimulus; but probe-tone ratings for such events regu-

Fig. 1. Key profiles of Krumhansl and Kessler (1982). Vertical axis: mean goodness-of-fit ratings of probe tones following chords and cadences in major and minor keys. Horizontal axis: pitch of probe tone relative to the tonic.

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Tone Profiles of Chord Progressions 27

larly exceed the minimum value of the response scale and vary relative to each other in a way that depends on musical structure and context.

The predictions of a bottom-up model can nonetheless be useful. In most harmonic-contrapuntal contexts, the tone profile at any given point of observation is well predicted by the frequency of occurrence of chroma in the passage preceding that point (Cuddy, 1997; Knopoff & Hutchinson, 1983; Krumhansl, 1990a; Oram &c Cuddy, 1995). In this article, predictions based on frequency of occurrence ("stimulus model") are used as a baseline against which the predictions of more sophisticated models are compared.

A purely top-down interpretation to tonality perception is similarly problematic. The results of Krumhansl and Shepard (1979) were clear evidence for the existence of top-down processes in tonality perception, and the high correlations observed between profiles for scales, chords, and chord progressions observed by Krumhansl and Kessler (1982) are evidence that the key profiles are robust in the face of considerable variation in the acoustic makeup of stimulus. However, Krumhansl and Kessler (1982, Experiment 2) also showed that the underlying process is not purely top-down: in modulating chord progressions, the most recently heard chord typically influenced the profile most strongly. Tone profiles following chord progressions may thus be determined by an interaction between a top-down process in which the profile of the strongest prevailing key is matched to the stimulus, and a bottom-up process in which the specific details of the stimulus influence the tone profile directly. This is essentially what Oram and Cuddy (1995) found in the case of melodies with atypical chroma distributions: profiles were influenced both by the top-down process of key-profile matching and by the bottom-up process of sensitivity to frequency of occurrence.

A third approach to the perception of key integrates both top-down and bottom-up processes into one model - but with a different kind of top- down process. According to Parncutt (1989), the tone profile measured after a chord progression depends on the relative perceptual salience of pitches within each sonority in the progression. Pitch salience within each sonority may be determined by a top-down, pattern-matching procedure in which the schema represents the ear's familiarity with the harmonic structure of typical complex tones in the human environment - that is, the harmonic series (Terhardt, 1972). The specific shape of the profile following a chord progression may then be determined by a bottom-up procedure in which more recent sonorities play a stronger role than earlier ones. Devel- oping this approach, Huron and Parncutt (1993) assumed that the last chord of a sequence contributes more to the composite profile than previous chords (recency effect) and that sensory memory decays exponentially with a half-life of roughly 1 s. On this basis, they were able to accurately model the gradual changes in tone profiles measured by Krumhansl and Kessler (1982) during modulating chord progressions.

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Yet another kind of top-down process is assumed by Browne (1981). Listeners may determine the tonic, and thereafter maintain their tonal bear- ings, by comparing groups of tones as they occur in real time with the 12 different diatonic subsets of the chromatic scale. When a set of tones be- longs to just one of these scales, the scale is uniquely defined by that set. According to the theory, this process of elimination then allows the tonic of that scale to be perceived. Browne showed that the smallest set of tones that can uniquely determine a major scale is a set of three of which two span the interval of a tritone. This theory is problematic in several respects (cf. Krumhansl, 1990b). First, it can only apply to music in which chromaticisms are relatively rare; this, for example, would count out much of the minor-mode music of J. S. Bach. In its simplest form, the tritone- plus-one principle clearly breaks down for common chromaticisms such as the augmented sixth chord. Second, even in purely diatonic music, the tritone-plus-one principle can apply only to major scales, because the harmonic minor includes two tritones. Third, a single melodic perfect fourth interval appears able to induce a clear tonality at least as strongly as a tritone-plus-one configuration (Vos, 1999). Fourth, it has not yet been possible to develop an algorithm based on the tritone-plus-one theory that would allow experimental data to be quantitatively and generally predicted and the model to be falsified. Notwithstanding these caveats, the underlying principle of position finding by matching diatonic fragments to diatonic scales remains credible, and some experimental data exist (Brown, 1988; Brown & Butler, 1981) that support it. In this article, we will apply and test the principle in a few special cases.

The general aim of the present study is to further explore the interplay between top-down and bottom-up processes in progressions of two and three chords that, according to everyday tonal music theory, either weakly or strongly instantiate major and minor keys. Our primary hypotheses are as follows:

First, we hypothesize that top-down processes of key determination of the kind described by Krumhansl and by Browne will play a more important role in the perception of longer than of shorter chord progressions. Rationale: In music theory, at least two chords (typically dominant-tonic), and preferably three chords (e.g. subdominant-dominant-tonic), are required to instantiate a key. Individual chords are not generally thought to suggest key centers - or if they do, the implication is not clear (e.g., a repeated major triad may be either a tonic or a dominant: Tovey, 1944). l In Browne's

1. Cohen's (1991) listeners successfully identified the tonic of Preludes from Bach's Well- Tempered Clavier from the first chord. Our prediction does not contradict her finding, for in that repertoire the first triad is always the tonic. The task of her listeners may be interpreted as one of root (chord) rather than key identification.

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(1981) approach, this is because they do not uniquely specify a diatonic scale.

Second, inspired by the results and modeling of Krumhansl and Kessler (1982, Experiment 2), we expect that the relative importance of top-down and bottom-up processes will depend on the clarity of the key of a passage. In progressions that more clearly instantiate a key, key perception should be determined more by top-down processes, whereas in the case of tonally ambiguous progressions, bottom-up processes should play a stronger role. Rationale: Krumhansl, in agreement with basic music theory, assumes that an ambiguous passage may be in two or more keys simultaneously. Leman (1995) speculated that listeners require a finite period of time to lock onto a key in a modulating passage, and thereafter resist modulation once a key has been established. He described this dragging effect as a kind of hysteresis, and modeled it by "attr actor dynamics with an elastic snail-like meta- phor" (p. 127).

Third, we hypothesize that the relative importance of top-down and bottom-up processes will depend on musical training.2 We predict that musicians will be more sensitive to bottom-up influences than nonmusicians. Rationale: Both musicians and nonmusicians are sensitive to tonality, so (top-down) key profiles are presumably relevant for tonality cognition in both cases. But only musicians can know what specific tones in a melody or chord typically lead to the establishment of a specific key. Moreover, the musicians in our study (students at a Western postsecondary music institu- tion) tended to be more familiar with music in ambiguous tonalities than were the nonmusicians to whom we had access.

To quantify and investigate the role of harmonic pitch pattern recognition in tonality perception, we used Parncutt's (1993) adaptation of Terhardt's pitch algorithm (Terhardt, Stoll, & Seewann, 1982) and chord- root model (Terhardt, 1982). In this approach, tone profiles are predicted for chords presented in isolation, based purely on their acoustic structure (spectrum of frequencies and amplitudes). A pitch-salience profile is a graph of (relative) perceptual saliences of pitches in a sonority, operationally defined by an experiment in which listeners are asked whether, or how strongly, a probe tone is present in a preceding stimulus. In this article, pitches are always octave-generalized, that is, the octave register is not specified; so the term chroma-salience profile may be substituted for pitch-salience profile. There are 12 chroma (or pitch classes) in an octave: C, Ci, D, etc. A chroma-salience profile shows the relative perceptual importance of these 12 tones, regardless of which octave register they fall in. This may be inter-

2. Recall that Krumhansl and Kessler's (1982) listeners were all musicians, so those authors were unable to investigate effects of musical training.

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preted as a measure of the probability of noticing a tone whose pitch corresponds to a particular tone of the musical scale, when a sonority is sounded. In a chord of C major (CEG), for example, the tone C typically has higher salience than the tones E and G (cf. below). Other tones are implied but not played, and also vary in salience relative to each other; for example, in the chord CEG, F is more salient than Ft.

Parncutt's (1993) algorithm takes a chord of octave-complex (Shepard) tones as input and outputs a chroma-salience profile. Any chord playable in the chromatic scale (i.e., any Tw-set, Rahn, 1980) can be processed. The algorithm first accounts for mutual masking among the tones of a chord. Then a pattern-matching procedure is carried out, involving intervals from an (octave-generalized) harmonic series: octave (frequency ratio approx. 2:1), perfect fifth (approx. 2:3), major third (approx. 4:5), and so on. The tuning of these so-called root-support intervals is not specified, but assumed to be variable within a musically typical range. The assumed extent of each interval's influence on the root is expressed in the model by its root-support weight. In this article, root-support weights are set as follows: P1/P8 = 10, P5 = 5, M3 = 3, m7 = 2, M2/M9 = 1, and all other intervals including m3 = 0. The same series of weights was used in several of Parncutt's studies (1994, p. 160; 1997, p. 187). The algorithm is described in more detail later.

The chroma-salience approach may shed light on Krumhansl's top-down concept of tonality. Key profiles may be modeled on the basis of the acoustic structure of typical cadential chord progressions in tonal music (subdominant-dominant-tonic), by calculating chroma-salience profiles for individual, isolated chords, and adding them across time to obtain the tone profile for the cadence (Huron & Parncutt, 1993; Parncutt, 1989, 1994). However, the application of pitch-perceptual theory to the modeling of key profiles is beyond the present scope. Instead, we accept the key profiles provided by Krumhansl and Kessler as given, and invoke chroma-salience profiles only to model the perception of specific chord progressions.

The present study is organized as follows. In Experiment 1, profiles are obtained for individual chords. In Experiment 2, these profiles are regressed onto profiles for progressions of two chords, using the same group of sub- jects. This procedure allows the two main parts of the model of Huron and Parncutt (1993) - the prediction of chroma salience within individual chords, and the combination of chroma-salience profiles across time - to be investigated independently of one another. Finally, in Experiment 3, profiles are measured following selected progressions of three chords (both cadential and noncadential).

As in Krumhansl and Kessler (1982), effects of pitch height (register) are avoided in all experiments reported here. All chords are constructed from octave-complex tones, and all results are averaged over transpositions of stimulus materials around the chroma circle.

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Experiment 1

Experiment 1 had three main aims. The first and primary aim was to replicate previous tone-profile data for harmonic dyads and triads presented in isolation and to establish profiles that were valid for our group of sub- jects and for the precise spectral and temporal characteristics of our stimuli. A second aim was to provide new data for selected harmonic dyads and to test the model of Parncutt (1993) for these sonorities. Third, we wished to familiarize listeners with the experimental paradigm before moving onto the more difficult task of Experiment 2.

To satisfy the first aim, profiles were obtained for major, minor, and diminished triads of octave-complex tones (corresponding to the triad types presented by Krumhansl &c Kessler, 1982). Regarding the second aim, profiles were determined for m3, M3, P4, and TT dyads (harmonic intervals of 3, 4, 5, and 6 semitones).3 Because all sounds were composed of octave- complex tones, dyads were equivalent to their inversion within the octave (e.g., P4 and P5 intervals are physically identical) and the inversion in which triads were played was not defined (e.g., the root position of a chord is physically identical with its first inversion).

Various authors had previously established tone profiles for some of these sonorities. Krumhansl and Kessler (1982) obtained tone profiles for major, minor, and diminished triads. Parncutt (1993) obtained profiles for major and minor triads. Thompson and Parncutt (1997) obtained profiles for the major triad and the M3 and P4 harmonic dyads (4 and 5 semitones). These data are presented in Table 1. Their main qualitative features are as follows: First, ratings tended to be higher if the probe tone was actually in the chord. Second, ratings were higher for music-theoretic chord-roots than for other tones. Third, ratings were higher for nonchord tones that are "implied by" or "go with" the chord in music, than for other nonchord tones. Fourth, ratings were sometimes higher for tones that represent satis- factory resolutions of, or progressions from, the chord in question (i.e., implied leading tone to tonic progressions; compare, e.g., results for the probe tone F following the C-major triad). The first three effects may be described as harmonic (depending only on the isolated sonority) and the fourth as melodic (depending on resolution, voice leading, or streaming between target and probe). Parncutt (1993) showed that the harmonic effects may be accounted for by models of pitch perception in isolated sonorities (especially Terhardt, 1972, 1982). The melodic effect depends on

3. The intervals m2 and M2 were avoided because (1) they are less common (as harmonic intervals) than thirds and fourths; (2) results for such (harmonic) intervals would

depend to a considerable extent on masking, which was not a main focus of the present study; and (3) we wished to avoid confounding effects of roughness or dissonance, because these are not accounted for by the models tested here.

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Table 1 Experimental and Calculated Tone Profiles for Sonorities Tested in

Experiment 1

(a) C-major triad

C d E f G a b

K&K 6.20 2.55 3.45 2.85 4.22 4.57 2.67 5.25 2.45 3.35 2.70 2.70 Pex93 2.37 0.78 1.00 1.07 1.85 1.52 1.22 2.30 0.81 1.19 0.70 1.04 T&P 6.3 2.5 3.5 2.9 4.7 4.7 2.9 5.8 3.5 4.1 4.3 3.3

Pmo93 12.59 0.00 2.11 1.97 6.19 4.38 1.24 6.57 2.23 4.41 0.74 0.00 Pearson correlation coefficients (r)

K&K Pex93 T&P

Pe^93 j ^39 T&P I .917 .851 Pmo93 I .932 .894 .883

(b) A-minor triad

C d E f g A b

K&K 5.28 2.58 3.55 2.87 4.80 4.25 2.67 3.50 3.42 6.03 3.35 3.67 Pex93 1.96 1.07 1.19 0.96 2.15 1.26 1.11 1.44 0.93 2.15 1.11 1.52

Pmo93 8.42 0.00 5.27 0.00 7.45 5.07 1.49 0.66 1.8610.29 0.62 1.31 Pearson correlation coefficients (r)

K&K Pex93

Pe^93 I ^887 Pmo93 I .931 .826

(c) B-diminished triad

c D e F g a B

K&K 4.65 3.13 4.63 3.95 3.97 4.05 2.95 4.12 2.90 3.05 3.30 4.75 Pmo93 0.55 3.39 5.50 0.68 4.49 6.77 0.00 6.14 0.00 0.68 5.04 6.77 r = .527

(d) M3 dyad (CE)

C d E f g a b

T&P 63 17 3^6 Js ~53 3^8 3jTTÔ 33 4.9 7Ï 3~6 Pmo93 10.38 0.00 2.39 0.00 7.98 3.99 1.60 0.00 2.39 3.99 0.80 0.00 r = .779

(e) P4 dyad (GC)

C d e f G a b

T&P 6.3 2.8 4.7 3.9 4.1 3.9 2.3 6.1 ~33 4.1 3.9 Js Pmo93 12.97 0.00 1.73 2.59 0.00 5.19 0.00 8.65 2.59 1.73 0.86 0.00 r = .851

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Table 1 (continued) (f) m3 dyad (EG)

c d E f G a b

Pmo93 5.59 0.00 0.70 2.10 6.99 0.70 1.40 6.99 0.00 4.89 0.00 0.00

(g) TT dyad (FB)

c d e F g a B

Pmo93 0.00 4.55 0.00 0.91 4.55 9.09 0.00 4.55 0.00 0.91 4.55 9.09

Note - The experimental data are from three sources: Krumhansl and Kessler (1982), indicated "K&K," based on a rating scale from 1 to 7, kindly provided by Professor Krumhansl; Parncutt (1993), labeled "Pex93," 0 to 3; and Thompson and Parncutt (1997), labeled "T&P," 1 to 7. These are compared with predictions (audibility of each chroma) of the model of Parncutt (1993) (labeled "Pmo93") with root-support weights P1/P8 = 10, P5 = 5, M3 = 3, m7 = 2, M2/M9 = 1, m3 = 0 and free parameters kM = 6 dB/oct and kw = 1. Note names are written above the figures for reference; those in capitals are physically present in the stimulus.

familiarity with Western tonal-harmonic syntax and cannot be accounted for by a simple psychoacoustic model.

Also included in the table are predicted chroma-salience profiles for the various sonorities according to the algorithm of Parncutt (1993). I will return later to details of these data and predictions.

In the present study, we wished to collect data that could be easily interpreted and against which psychoacoustic models could reasonably be tested. One concern was to eliminate, as far as possible, the melodic effect from the data. To achieve this, we asked listeners to rate the similarity of the chord and the tone in each trial. Parncutt's (1989) question "Is the tone in the chord?" was avoided because it implies that the experiment might be an ear-training task and that correct answers may exist; this may encourage listeners to recognize chords and intervals and allow music-theoretic knowledge to guide responses. Other possible questions that we considered were "How expected or surprising is the tone?" and "How consonant is the tone in the context of the previous chord?" Both these were rejected because, like goodness-of-fit ratings, they are broad enough to include melodic effects and seem to re- fer to musical experience and style.

In pilot runs of the experiment, sonorities were compared with all 12 octave-complex tones. Listeners' similarity ratings were found to be skewed toward the dissimilar end of the response scale. To encourage more normally distributed data, the mean perceptual similarity of the sound pairs in the experiment was increased by comparing each sonority with only seven tones taken from one of the diatonic scales to which the sonority belonged.

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The scale chosen was the one in which the more salient pitches of the sonority (according to the data presented in Table 1) were best represented.4

METHOD

Listeners

Fifteen musicians and fifteen nonmusicians participated. Each musician had played (practiced or performed) a (Western) musical instrument for at least 6 years, whereas each nonmusician had played for less than 2 years (mostly, not at all). The musicians were 28 ± 6 years old (mean ± standard deviation [SD]) and had practiced or performed a musical instrument for 15 ± 6 years. The nonmusicians were 22 ± 4 years old and had played for 0.5 ± 0.5 years.

Equipment

The experiment was run from a Macintosh Classic II computer. Sampled waveforms were produced by calculating digitized waveforms for pure tones and adding them up to produce complex tones. Sounds were presented at a sampling rate of 22 kHz over head- phones in a quiet room. Listeners responded by pressing keys on the computer keyboard.

Sounds

Twelve octave-complex tones were created. Each comprised pure-tone components at octave intervals across the range C3 (130 Hz) to Cg(4200 Hz). All pure-tone components were tuned to the standard (unstretched) equally tempered scale with A4 = 440 Hz. The amplitude envelope increased linearly from C3 to C4, was flat from C to C7, and decreased linearly from C7 to Cg. All sounds in the experiment (probe tones, narmonic dyads, and chords) were created by combining the original 12 octave-complex tones in different combinations and without attenuation. As a result, chords had greater physical intensity than dyads, and dyads had greater physical intensity than single tones. The duration of all sounds (tones and chords) was 300 ms. Pauses between sounds in each trial also lasted for 300 ms.

Design

In each block, 7 test sounds were each compared with 7 probe tones, making 49 trials. Test sounds were CEG, ACE, BDF, EG, CE, GC, and BE Probe tones were C, D, E, F, G, A, and B. To promote holistic listening, probes followed chords (rather than preceding them). Each listener did two blocks of trials. The order of trials within each block was random and was newly calculated for each block and each listener.

4. The elimination of almost half of the trials allowed more data to be collected for each point, but reduced the number of degrees of freedom available for correlations between profiles from 12 - 2 = 10 to 7 - 2 = 5. This reduction in statistical power did not, however, prevent consistent patterns from emerging in the data (see correlations between profiles below). The reason is presumably that the five elements eliminated from each profile typically did not differ significantly from each other and therefore would have contributed little extra information to the profile as a whole. Consider, for example, the variations in mean ratings for the nondiatonic tones Ct, Df, Fit, Gtt and Alt in KrumhansPs C-major key profile (the valleys in Figure 1). These variations are small in comparison with variations in the mean ratings for the seven diatonic tones.

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Procedure

Each trial (both sound and probe) was transposed randomly through an interval of 0 to 11 semitones (around the chroma cycle) before presentation. The pattern of transpositions was different for each listener and each block. To further inhibit serial effects, the probe- tone pitch was constrained to differ from trial to trial. Listeners heard the test-probe pair once and then rated the similarity of the two sounds on a 9-point scale marked "DIS ... SIM." A relatively broad rating scale was chosen, to ensure that the data would be approxi- mately normally distributed; listeners were instructed to confine most of their responses to the range 3 to 7, reserving the endpoints of the scale for extreme cases. Listeners could take as long as they wished to respond but were encouraged to do so spontaneously. It was stressed that this was not a test of musical ability - it was the opinions of the listeners that were being sought, and there were no right or wrong answers. The experiment was preceded by a short practice session. The two blocks of the experiment were performed in one sitting. Listeners were encouraged to take a break between the blocks.

RESULTS

The tone profiles for individual sounds are shown in Figure 2. These are

presented separately because results varied with combinations of test sound and probe tone, for both the musicians, F(36, 504) = 16, p < .0001, and nonmusicians, F(36, 504) = 2.7, p < .0001.

Consider first the results for the major triad. For musicians, results varied significantly as a function of probe tone, F(6, 84) = 22, p < .0001; the mean result for chord tones (C, E, G) exceeded the mean result for other tones (D, F, A, B), F(l, 14) = 64, p < .0001; the result for C, the music- theoretic root of the chord, exceeded the mean result for E and G, F(l, 14) = 18, p < .002; and no significant differences were observed among the other probes. Results for nonmusicians did not vary significantly as a function of probe.

Results for the minor triad again varied significantly as a function of

probe tone, but only for musicians, F(6, 84) = 11, p < .0001; the mean result for chord tones (C, E, A) exceeded the mean result for other tones

(D, F, G, B), F(l, 14) = 29, p < .0005. Results for the diminished triad varied significantly as a function of probe

tone, but again, only for musicians, F(6, 84) = 9.8, p < .0001: the mean result for chord tones (D, F, B) exceeded the mean result for other tones (C, E, G, A), F(l, 14) = 19, p < .001; the mean result for D and B exceeded the result for F, F(l, 14) = 7.4, p < .02; and the mean result for E and A exceeded the result for C, F(l, 14) = 6.2, p < .05.

For musicians, results for the m3 dyad EG varied significantly as a function of probe, F(6, 84) = 18, p < .0001: the mean result for tones in the

sonority (E, G) exceeded the mean result for other tones, F(l, 14) = 140, p < .0001, and the result for C, the music-theoretic root of the dyad, exceeded the mean result for D, F(l, 14) = 10, p < .01. This time, results for nonmusicians also varied significantly as a function of probe, F(6, 84) =

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Fig 2. Results of Experiment 1. Tone profiles for all seven test sounds (three harmonic triads and four harmonic dyads).

6.3, p < .002; again, the mean result for E and G exceeded the mean result for other tones, F(l, 14) = 20, p < .001, and the result for F exceeded the

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mean result for C, D, A, and B, F(l, 14) = 6.2, p < .05 (presumably, an effect of pitch proximity).

Results for the M3 dyad CE varied significantly as a function of probe, but only for musicians, F(6, 84) = 21, p < .0001. The mean result for tones in the sonority (C, E) exceeded the mean result for other tones, JF(1, 14) = 332, p < .0001. The result for G (the missing fifth of a C-major triad) exceeded the mean result for the other nonplayed tones (D, F, A, and B), F(l, 14) = 22,p<.001.

Results for the P4 dyad GC varied significantly as a function of probe, but only for musicians, F(6, 84) = 24, p < .0001. The mean result for tones in the sonority (G, C) exceeded the mean result for other tones, F(l, 14) = 110, p < .0001. The result for E, the missing third of a major triad, exceeded the mean result for the other nonplayed tones (D, F, A, and B), F(l, 14) = 10, p < .01 - but compare Thompson and Parncutt (1997), who observed no difference between E and F in this case. The result for B, the leading tone, was lower than the mean result for D, F, and A, F(l, 14) = 6.1, p< .05.

Results for the tritone (TT) dyad BF varied significantly as a function of probe both for musicians, F(6, 84) = 17, p < .0001, and for nonmusicians, F(6, 84) = 3.8, p < .005. For musicians, the mean result for tones in the sonority (B, F) exceeded the mean result for other tones, F(l, 14) = 39, p < .0001, and the mean result for C, D, and G exceeded the mean result for E and A, F(l, 14) = 4.7, p < .05. For nonmusicians, the mean result for tones in the sonority (B, F) exceeded the mean result for other tones, F(l, 14) =

13,p<.005.

DISCUSSION

In most cases, only musicians' results varied as a function of probe, sug- gesting that the task was simply too difficult for the nonmusicians or that the task was not well enough defined. In this regard, our findings are consistent with probe-tone ratings following melodies as investigated by Oram and Cuddy (1995). They observed that "The ratings of untrained listeners were influenced less by frequency of occurrence than were trained listeners, and more influenced by pitch proximity" (p. 110). In our experiment, nonmusicians seemed unable to respond to the exact pitch of a tone in a chord, but Figure 2 suggests that they tended to give higher responses for probe tones nearer in pitch to chord tones. Owing to the design of our experiment, it is only possible to see this trend in the data for the dyads in the lower part of the figure.

The experimental task was similar to the task of hearing out the partials of a complex tone, in that it requires separation of parts of a global sound. Fine and Moore (1993) found that the superior ability of musicians to hear out partials was not due to smaller critical bandwidths, as critical band-

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width is not significantly affected by musical training, but by differences in the efficiency of the detection process after filtering. On this basis, the superior performance of the musicians in the present task was probably due to training rather than any innate disposition.

Results for musicians were broadly consistent with Parncutt's (1988) model of chord roots. The root of the major triad was more similar to the triad than its other tones, consistent with the music-theoretic idea that the root in some sense represents the chord. The same was true for the minor triad, but the result in that case was not significant (although it apparently was significant in the corresponding Krumhansl-Kessler experiment, see Table lb), consistent with the prediction that the root of a minor triad is weaker or more ambiguous than that of the major. In the diminished triad BDF (b°), the prediction that the "missing root" G would be more salient than the other nonchord tones (see Table lc) was again not confirmed, but the results were not inconsistent with the prediction, which had previously been confirmed by Krumhansl and Kessler (see again Table lc).

The root model was less successful in the case of the dyads. For the dyad EG (see predictions in Table If), the predicted pitch C was perceived, but not the predicted pitch A. For the CE dyad (see predictions in Table Id), G was perceived but not predicted, and A was predicted but not perceived (whereas Thompson and Parncutt found no significant difference between G and A in this dyad, see Table Id). For the GC dyad (Table le), F was predicted but not perceived, and E was perceived but not predicted. As suggested by Thompson and Parncutt (1997), these disagreements may be explained as follows: Listeners may have heard the dyads CE, EG, and CG as incomplete major triads and may have rated highly those tones that would have completed the triads.

COMPARISON WITH PREDICTIONS

Figure 3 graphs correlations between results of Experiment 1 (musicians only) and predictions according to three models. In the stimulus model, a constant value is assigned to tones that appear in each test sound and a value of zero is assigned to other tones in the diatonic scale: for example, the prediction for the major triad, beginning with the root, is the vector {1 0 1 0 1 0 0}. In the key model, the profile obtained experimentally for each test sound is compared with all 24 major and minor key profiles of Krumhansl and Kessler in turn; the profile producing the highest correlation coefficient (vertical axis) is interpreted as the prevailing key (tonality) and labeled on the graph as C, c, D, and so on. For the purpose of modeling, the key profile is assumed to be rigid; we did not account for the possibility that, like most schemata, key profiles include some inbuilt flexibil- ity, acquired by exposure to different tonal styles (Krumhansl, 1990a).

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Fig. 3. Comparison of predictions of three models (stimulus, key, pitch) with results of Experiment 1 for musicians. Each model predicted a tone profile for each of the seven test sounds presented in the experiment. Vertical axis: Pearson correlation coefficient over seven values of each tone profile. Correlations above the horizontal line are significant (p < .05). Keys found by the key model are marked on the graph (C = C major, c = C minor, etc.). For the F-B tritone, the correlation between the key model's prediction and the experimental profile (r = .34) falls outside the graph area.

The pitch model is that of Parncutt (1993)- not Parncutt (1988) or Parncutt (1989) - because, of these three versions, only Parncutt (1993) is tailored directly to the perception of chords of octave-complex tones (see experimental data reported in that article). The model of Parncutt (1988) is

essentially the same as that of Parncutt (1993) except that it does not account for masking, so its predictions would probably have been inferior to those of Parncutt (1993) in this case (as demonstrated in the 1993 article). The models of Parncutt (1989) and Parncutt (1993) are functionally equivalent to each other in the case of chords of octave-complex tones, so their

predictions should be identical; of these two, the model of Parncutt (1993) is preferred here for its relative mathematical transparency.

The specific details of Parncutt's (1993) model are as follows. The input consists of pure-tone components within a single octave register, assumed for the purpose of argument to lie in the dominance region of pitch perception (say, 500 to 1000 Hz). Their SPL is assumed to be constant at 50 dB. First, masking between the components is evaluated. The masking effect of one component on another (masking level) is set equal to the SPL of the masker minus 6 dB for each semitone between the components (up to a maximum of 6 semitones, because the tones are octave-complex). This effect is assumed to be independent of pitch direction (i.e., masking patterns are assumed to be symmetrical). For example, the masking effect of B on Ct (and vice versa) is calculated to be 50 - (6 x 2) = 38 dB; in the absence of any other simultaneous components, the effective audible level,

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or level above masked threshold, of these two components is then 50-38 = 12 dB. Because the input sounds are octave-complex tones, the distance between tone components is calculated by modulo 12 arithmetic, where 12 is the number of semitones in an octave; for example, the distance between components at 11 and 1 semitones above a reference pitch is not 10, but 2 semitones. The effect of masking by more than one component (e.g., in a triad, the masking of each tone by the other two) is calculated by adding masking amplitudes (i.e., by converting SPL to amplitude, adding, and converting back to SPL). This procedure outputs an audible level for each component in decibels such that tone components that are relatively close to their neighbors (e.g., in the triad BDF, the tone D) have lower audible levels than do tone components that are relatively distant from their neighbors (e.g., in the triad BDF, the tones B and F). The perceptibility of a pure-tone component saturates as its audible level increases, so perceptibility is almost independent of audible level at high audible levels; to account for this, audible levels are converted to pure-tone audibility values by an exponential function that approaches an asymptote for audible levels exceeding about 20 dB. Harmonic pitch pattern recognition (the central idea in Terhardt's approach) is modeled as described in Parncutt (1988), except that the input components are weighted according to their audibility. An octave-cyclic harmonic template (consisting of weighted intervals PI, P5, M3, m7, M2 above a reference pitch) is matched to the pattern of pure-tone audibility values. Each time there is a match between these values and the template, a complex-tone audibility value is calculated and applied to the template's reference pitch. It is given by a linear sum of contributions from the template-component matches, where each contribution is a simple product of the corresponding pure-tone audibility and template-component weight. These complex-tone audibility values are the predictors of the model for the present purpose.

Figure 3 shows that the most successful model for the triads is generally the pitch model, whereas the most consistently successful for the dyads is the stimulus model. As noted above, the reason for the pitch model's failure to account for these dyads may be the overriding effect of familiarity with the sound of major triads, acquired by exposure to tonal music in which major triads typically outnumber all other sonorities including mi- nor triads (Eberlein, 1994). A clear example of this effect may be seen in Table 1 (d, e). For the M3 and P4 harmonic dyads, the correlations (r = .78, .85) between the Thompson-Parncutt data and the predictions of Parncutt (1993) are considerably less impressive than those for the major and minor triads. The discrepancy at G for the CE dyad and at E for the GC dyad in the present experiment suggest that listeners perceived these dyads as incomplete major triads, perceptually filling in the missing tones.

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The relative failure of the key model to predict profiles for single chords is unsurprising, given that several chords are generally needed to establish a musical key. Note that the p = .05 line in Figure 3 applies only to correlations involving the stimulus and pitch models; the key model involves a comparison of several correlation coefficients, so the p = .05 line for the final result is effectively higher.

Experiment 2

Experiment 2 involved progressions of two chords. The following questions were posed. First, do the profiles for the individual chords simply add? In other words, can the data be well predicted by a bottom-up model? This question was addressed by comparing experimental profiles following pairs of chords with profiles for individual chords obtained in Experiment 1 from the same set of listeners (multiple regression). Oram and Cuddy's (1995) finding that "greatest responsiveness was found when musically trained listeners were presented with sequences generated from the diatonic toneset" (p. 114) suggested that the bottom-up effect would be stronger if chords were diatonic in the same key; this constraint was also adopted in the present design. Second, what is the relative importance of the chords for the final profile? Does the second chord (recency) or the first (primacy) have the greater effect? Third, does the progression V-I determine a major or minor key more clearly than any other diatonic progression, including the reverse order, I-V? This question had been addressed by Huron and Parncutt (1994) by correlating calculated profiles of pairs of chords with the key profiles of Krumhansl and Kessler (1982); the V-I progression was found to produce the highest correlation coefficient. Here, a similar technique was used, except that the profiles for the pairs of chords were determined experimentally. Finally, do progressions that uniquely define a major scale produce a clearer sense of key than those that do not, as Browne (1981) suggests? This question was tested by comparing the tone profile for the progression b°-C, which is diatonic in only one key (C major), with the tone profile for the progression G-C, which could be in either of two keys (C or G).

METHOD

Listeners, equipment, and sounds were the same as in Experiment 1.

Design

In each block, 5 pairs of chords were each presented in 2 different orders and compared with 7 different probe tones, making 70 trials. Each listener did two blocks of trials. The

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42 Richard Parncutt Ôc Albert S. Bregman

order of presentation of trials in each block was random and was different for every combination of listener and block.

The two orders of each chord pair will here be referred to as "forward" and "backward." The "forward" orders consisted of a diatonic triad in the key of C major (d, e, G, a, or b°, where the superscript circle denotes the diminished triad) followed by a C-major triad; to avoid chord repetitions and duplications of chord progressions, the first triad was never C major or F major (the progression F major to C major is equivalent to the progression G major to C major in transposition). The five "forward" progressions were thus d-C, e-C, G-C, a-C, and b°-C.

Procedure

In each trial, the listener heard two chords followed by a probe tone and rated the similarity of the tone to the two chords taken together on the same 9-point scale as before. Similarity judgments were used to enable direct comparison with data from Experiment 1 and for the other reasons set out in the introduction to that experiment. Again, each trial was transposed randomly through an interval of 0 to 1 1 semitones around the chroma cycle before presentation. The chords and tones lasted 300 ms each, the pause between the two chords was 100 ms, and the pause between the chords and the probe tone was 700 ms. (The pauses were adjusted to facilitate the grouping of the two chords into one perceptual unit before comparison with the probe tone.) The experiment was immediately preceded by Experiment 1, and followed by Experiment 3.

RESULTS

A three-way interaction was found between probe tone, chord pair, and order both for musicians, F(24, 336) = 2.2, p < .005, and for nonmusicians, P(24, 336) = 2.1, p < .005. The shape of the profile obtained as a function of probe for each individual chord pair also depended on the order of the chords, but only for musicians, for example, d-C: F(6, 84) = 5.7, p < .0002. Results for each chord pair in each order are thus graphed separately in Figure 4.

Inspection of the figure reveals first a general tendency for the results to be higher for tones present in the chords of each progression than for other tones. For example, the tone B is not present in the progressions d-C and C-d, so the corresponding rating is low. Exceptions to this rule may be accounted for by pitch-perceptual theory. In progression G-C, for example, A is rated higher than B, even though, of these two tones, only B is present in one of the preceding chords. According to the theory, A is strongly implied by the C chord, but the B in the G chord has relatively low perceptual salience. The difference is strengthened by recency, as the data for the reverse-order progression (C-G) confirm. A second general tendency is that ratings for chord roots are higher than for other tones (e.g., G and C in the progressions G-C and C-G). Moreover, ratings for the roots of major triads tend to be higher than for the roots of minor triads (e.g., compare ratings for C and D in the progressions C-d and d-C), consistent with the prediction that the root of the major triad is less ambiguous than that of the minor (Parncutt, 1988). Third, ratings tend to be higher for the tones (and root) of the second of the two chords in each case; recency effects of this

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Fig. 4. Results of Experiment 2. Tone profiles for all 10 chord progressions. Left side: original order. Right side: reverse order.

kind were discussed by Krumhansl and Kessler (1982), Parncutt (1989) and Huron and Parncutt (1993). The nonmusicians' responses for indi-

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vidual chord pairs generally did not vary significantly with probe pitch and so are not discussed further.

Turning to the orthogonal contrasts, consider first the profiles obtained for the dominant-tonic progressions G-C and b°-C, and their reverse orders. For the progressions G-C and C-G taken together; musicians' responses to C, the root of the C-major triad, exceeded responses to other probes, F(l, 14) = 39, p < .0002; and to G, the root of the G-major triad, exceeded the rest except C, F(l, 14) = 45, p < .0002. Similarly, in the progressions b°- C and C-b° taken together, musicians' responses to C were higher than to the other probe tones;5 G was higher than D, E, F, and A; F was higher than D, E, and A; and E was higher than D and A. The results are consistent with the music-theoretic expectation that the chords G (or b°) and C together imply C as a tonic, regardless of the order of the chords, but that tonicization is stronger in the order dominant-tonic than in the reverse order (as already discussed, e.g., by Krumhansl & Kessler, 1982).

We turn now to the other diatonic progressions. Consider first the musicians' data for the progressions d-C and C-d, taken together. The mean rating for C exceeded the mean rating for the other probes, F(l, 14) = 32, p < .0005, and the mean rating for F (the second most salient root candidate of the D-minor triad according to Parncutt, 1988, and the only other major key to which the chords C and d both belong) exceeded that for D, E, G, A, and B, F(l, 14) = 31, p < .0005. Regarding the progressions e-C and C-e, musicians' ratings for C exceeded ratings for other probes, F(l, 14) = 29, p < .0005; moreover, G (the second most salient root candidate in both chords, and the only other major key to which both chords belong) exceeded D, E, F, A, and B; E exceeded D, E, A, and B; and B exceeded D, F, and A. In the progressions a-C and C-a, all tested (orthogonal) differences in the musicians' data between probe tones (C versus the rest, G versus the rest but G, and so on) were significant.


Figure 5 compares the predictions of three different models with the results of Experiment 2. The stimulus model simply involves counting the number of times each chroma occurs physically in each progression. Here, the stimulus model correlates significantly (p < .05) only for 2 out of 10 progressions (C-e, C-G), and one of these correlations might well have oc- curred by chance. It is therefore fair to say that the stimulus model com- pletely failed to predict the data. There is, however, an interesting pattern in the predictions of the stimulus model: it tended to be least successful for progressions that clearly define a key, such as the dominant-tonic progres-

5. For clarity, statistical details are omitted for this and following orthogonal contrasts.

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Fig. 5. Upper panel: Comparison of predictions of three models (stimulus, key, pitch) with results of Experiment 2 for musicians. Each model predicted a tone profile for each of the 10 chord progressions presented in the experiment. Vertical axis: Pearson correlation coefficient over seven values of each tone profile. Correlations above the horizontal line are significant (p < .05). Lower panel: Results of multiple regression of each profile onto the profiles of the individual chords as determined in Experiment 1. Vertical axis: coefficients in regression equation, for each chord in each progression.

sions, G-C and b°-C (and reverse orders). The strength of the top-down effect in these cases could explain the failure of the bottom-up model. (Note that the progression C-G, for which the stimulus model performed quite well, does not clearly define a key - it may be interpreted either as I-V in C or as IV-I in G.)

The key model is represented by the Krumhansl-Kessler key profile that most closely corresponds to the experimental profile. As before, the p = .05 line on the graph does not strictly apply to this model, because several correlations are required for each point. The model predicts keys that make sense from a music-theoretic viewpoint: the progression C-d can indeed be heard as V-vi in the key of F, and C-G can of course be heard as IV-I in the key of G. The figure also confirms that the key model is most successful

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where the key is most clearly defined. Here, that is the case for the dominant-tonic progressions G-C and b°-C. The model's failure to correlate significantly with one of the progressions (C-e) is also consistent with expectations based on music theory: the key of this progression seems quite ambiguous (C major or E minor). Recall that Krumhansl's key profiles were obtained for clear key-defining contexts; there is no reason why they should correlate strongly with profiles of tonally ambiguous progressions.

Note that neither Figure 4 nor Figure 5 provides any support for the prediction, on the basis of Browne's 1981 theory, that the progression b°-C more clearly instantiates the key of C than does the progression G-C. If anything, the reverse may be true: The data suggest that the effect of root progression (especially through falling fifths: see Eberlein, 1994; Parncutt, 1996; Vos, 1999) was more significant than the effect of position finding relative to diatonic scale sets, at least in this case.

The pitch model is a weighted6 sum of the experimental tone profiles of the two chords in each progression, as obtained in Experiment 1. The idea of accumulating pitch information over time was discussed by Krumhansl and Kessler (1982), Parncutt (1989), and Huron and Parncutt (1993). Here, the weighting was optimized by multiple regression to give the best fit to the data. The degrees of freedom in the optimization process again mean that the p = .05 line in the upper panel of Figure 5 does not strictly apply.

The coefficients in the regression equation of the pitch model are graphed in the lower panel of Figure 5. There is a clear trend for the second chord in the progression to have a greater effect on the experimental profile than the first (recency effect). In four cases, the first chord seems to have no significant effect on the profile. Three of these cases (G-C, b°-C, e-C) include the leading-tone progression B-C. In the corresponding panels of Figure 4, the salience of B tends to be lower than the salience of other tones in the first chord. A possible explanation combines pitch salience with streaming. Re- garding streaming, the memory for a tone can be erased by a second tone presented immediately afterward at an interval of 1 semitone higher or lower (Deutsch, 1973). Larson (1997, p. 105) explained the effect as follows: "In a melodic step, the second note tends to displace the trace of the first, leaving one trace in musical memory; in a melodic leap, the second note tends to support the trace of the first, leaving two traces in musical memory." Thus, the tone C can erase the memory of an immediately preceding tone B, or vice versa. In the present harmonic contexts, the two tones additionally differed a priori in perceptual salience. According to Terhardt's pitch theory (e.g., Parncutt, 1988), the tone B in a G-major triad

6. A weighted sum and multiple regression could also have been used for the stimulus model. Presumably, similar effects of primacy and (in Experiment 3) recency would have been found.

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(the third) has relatively low salience, whereas the tone C in a C-major triad (the root) has relatively high salience. This can explain the considerable difference in stability between the leading tone and the tonic in Krumhansl's key profiles. In the aforementioned progressions, the difference in salience is consistent with the observation that C more often erases memory for B than vice versa.

Experiment 3

The experiments of Krumhansl and Kessler (1982) included presentations of three chords followed by a probe tone. Huron and Parncutt (1993) and the present Experiment 2 provided evidence that profiles following more than one chord may be determined by linear combination of the profiles produced by each chord alone and that in such a procedure the most recent chord typically carries the most weight.

The present experiment involved progressions of three chords and again addressed the issue of the relative importance of the chords. Standard cadential chord progressions (selected from those used by Krumhansl and Kessler) were presented both in their original order and in reverse order.

Experimental profiles following the progressions were compared with profiles for individual chords by multiple regression, and coefficients in the

regression equation were compared. Three possible outcomes of such an

experiment may be envisaged a priori: First, the key of the progression may be largely determined by the last

chord, as the findings of Experiment 2, Krumhansl and Kessler (1982), Huron and Parncutt (1993), and Parncutt (1995) suggest. In other words, if the last chord is C, the key is also C. This idea is, of course, problematic in cases in which the last chord is not the conventional tonic; it would

predict, for example, that the key of the progression C-G-F is F. A second possibility is that tonality depends on the relationship between

chords presented, without regard to their order. Implicit in tonal music

theory is the idea that any isolated chord progression that is diatonic in

only one key (minimum requirement: two tones in a tritone relationship plus one other tone of the scale; Browne, 1981) will imply that key regardless of the order of the chords. According to this argument, the chord progressions F-G-C and C-G-F are equally, and unequivocally, in C major. But

why are the reverse-order cadences (subdominant-dominant-tonic) relatively uncommon in the Western tonal literature, and so less familiar to Western

ears, than their forward-order counterparts (Eberlein, 1994; McHose, 1947)? Key implications might be somehow responsible for, or influenced

by, this asymmetry.

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A third possibility combines the previous two. It is possible that the key of a chord progression depends on the order of chords but does not necessarily correspond to the root of the final chord. This would be consistent with the observation that the harmonic-contrapuntal structure of cadential chord progressions developed historically under the influence of a range of constraints, including perceptual "uni versais," pedagogical functions of voice-leading rules, performance practice, and sociological considerations (Eberlein, 1994). According to this approach, tonality is primarily determined by familiarity with somewhat arbitrary pitch-time patterns. But again there is a problem: it is not generally possible on this theoretical basis to predict which tone or chord of a progression will be the tonic.

METHOD

Listeners, equipment, and sounds were the same as in Experiment 1.

Design

In each block, 2 chord progressions (F-G-C, f-G-c) were each presented in 2 different orders (forward, backward) and compared with 12 different probe tones, making 48 trials. The number of probe pitches per octave was increased from 7 to 12, for two reasons: first, to allow a meaningful comparison of major and minor key profiles, both with each other and with those of Krumhansl and Kessler, and second, because in pilot runs of this experiment the data were no longer markedly skewed (as they had been in Experiment 1). As before, each listener did two blocks of trials. The order of presentation of trials in each block was random and was different for every combination of listener and block.

Procedure

The procedure was identical to that of Experiment 2, except that there were three chords (rather than two) before each probe tone. The chords and tones lasted 300 ms each, the pauses between the two chords lasted 100 ms, and the pause between the chords and the probe tone was 700 ms. The experiment was immediately preceded by Experiment 2.

RESULTS

The shape of the profiles depended on the order of presentation of the chords (forward versus reverse),7 for both musicians, F(ll, 154) = 9.9, p < .0001, and nonmusicians, F(ll, 154) = 3.1, p < .002, meaning that the second prediction set out in the introduction (that tonality depends only on the chords presented, regardless of their order) was not supported. The shape of the profiles also depended on mode (major versus minor), but -

surprisingly - only for musicians, F(ll, 154) = 9.7, p < .0001; for nonmusicians the interaction between probe tone and mode was nonsig- nificant, F(ll, 154) = 0.9, p = 0.6. Thus, the profiles of the nonmusicians

7. That is, there was a significant interaction between probe (12 levels) and order (2 levels).

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distinguished between forward and backward progressions, but not between major and minor keys.

For musicians, there was also a significant interaction between probe tone, mode (major/minor), and order (forward/backward), F(ll, 154) =

2.9, p < .005. Results are therefore presented as a function of probe tone for all four combinations of mode and order in Figure 6.

In Experiments 1 and 2, nonmusicians' results for individual trials had usually not varied significantly with probe tone. That was not the case here. When results for each of the four progressions were analyzed separately, there was a significant effect of probe tone in each case, for both musicians and nonmusicians. In each case, the effect was less significant for the nonmusicians than for the musicians. Statistical results for the nonmusicians were as follows: for F-G-C, F(ll, 154) = 4.1, p < .0002; for C-G-F, F(ll, 154) = 2.3, p < .02; for f-G-C, F(ll, 154) = 4.7, p < .0001; and for c-G-f, F(ll, 154) = 2.4, p < .01. In these experiments, then, nonmusicians needed to hear three chords before they could clearly per- ceive differentiations in salience or stability among the tones of the corre-

Fig. 6. Results of Experiment 3. Tone profiles for chord progressions in major (F-G-C) and minor (f-G-c) keys and the same progressions in reverse order.

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sponding scale. Musicians, by contrast, could make such differentiations after hearing only one chord. This suggests that, for nonmusicians, the top- down process of key recognition becomes stronger or more important as the length of a diatonic chord progression increases from one to three chords.8

As observed in Experiment 2, the peaks in the profiles of Figure 6 tend to correspond to the tones of the last chord in the progression, and the highest peak tends to correspond to the root of the last chord. The peaks in the "forward" progressions correspond to the conventional tonic C, whereas the peaks in the "backward" progressions correspond to the root of the subdominant chord F. Again as before, the predominance of the final root is clearer when the last chord is major than when it is minor.


As before, the results were compared with predictions of three different models. The stimulus model involved counting the number of times each chroma occurs physically in each progression, and this model produced the same prediction for both orders of each progression. The key model worked exactly as before, finding the Krumhansl-Kessler key profile that corresponds most closely to the experimental profiles (but over 12 values, rather than 7 as in the previous experiments). The pitch model took as its input predictions of Parncutt (1993) with parameter values as for Experiment 1; note that profiles from Experiment 1 could not be used, as they covered only 7 chroma. As in Experiment 2, the pitch model involved multiple regression between the profiles of the individual chords and the experimental profile of the progression.

The upper panel of Figure 7 compares the predictions of the three different models with the results of Experiment 3 for musicians. On the whole, the key and pitch models are successful and clearly superior to the stimulus model. However, details of the comparisons cast doubts on the validity of both the key and pitch models. The predictions of the key model disagree with music theory for the reverse-order progressions (no music theorist would assign a tonal center of F to these progressions) but are consistent with the widely held assumption (Auhagen, 1994; Brown et al., 1994; But-

8. The difference in significance between the profiles obtained in Experiments 1 and 2 and in Experiment 3 was at least partly due to the number of probe tones, of which 7 (diatonic scale) were used in Experiments 1 and 2, and 12 (chromatic) in Experiment 3. (Recall that the rationale for this difference was to encourage normally distributed data in each experiment.) Our conclusion is supported by visual inspection of Figure 6, in which nonmusicians9 profiles are consistent with stimulus structure for all four chord progressions: peaks correspond to tones of the final chord in each case (all three tones of major triads, and the root and fifth of minor triads). In Experiments 1 and 2 (Figures 2 and 4), however, this was mostly not the case.

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Tone Profiles of Chord Progressions 5 1

Fig. 7. Upper panel: Comparison of predictions of three models (stimulus, key, pitch) with musicians' results in Experiment 3. Each model predicted a tone profile for each of the four chord progressions presented in the experiment. Vertical axis: Pearson correlation coefficient over 12 values of each tone profile. Correlations above the horizontal line are significant (p < .05). Lower panel: Results of multiple regression of each profile onto the profiles of the individual chords as determined in Experiment 1. Vertical axis: coefficients in regression equation for each chord in each progression.

1er, 1989; Krumhansl & Kessler, 1982) that tonal centers depend on the order in which musical elements are presented. On the other hand, the surprisingly poor prediction by the pitch model for the reverse-order minor progression c-G-f (by comparison to the key model) suggests that Krumhansl's top-down process is more important than the corresponding bottom-up process in this case. This casts doubt on the first prediction above, that the tonic corresponds to the root of the last chord. The third prediction - that an arbitrary, historically determined relationship exists between chord progressions and tonics - could not be tested against these data.

These results highlight the absence of a widely accepted operational definition of perceptual key. They suggest that the best-fitting key profile does

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not necessarily correspond to the music-theoretic key (as assumed by Krumhansl, 1990a); nor does the peak of the tone profile necessarily correspond to the music-theoretic tonic. A possible solution to the problem may be to operationally define the tonic as a triad rather than a tone, consistent with Schenker's (1906/1954) idea that a tonal work is a temporal unfold- ing, prolongation, or "composing out" of its tonic triad, and to adopt a method of tonic determination that uses a probe triad rather than a probe tone. For example, the reverse-order cadence C-G-F would presumably sound more complete if followed by a C chord than an F chord.

The lower panel of the figure shows the coefficients in the regression equations for the pitch model. As in Experiment 2, there is a clear trend for the last chord to have the greatest effect on the profile for the whole progression. This time, however, the chord that has the least effect on the final profile is not the first but the second (except in the progression C-F-G, where the coefficients for the second and third chords are almost the same). The data thus point to a strong effect of recency for the third chord and a somewhat weaker effect of primacy for the first chord.

Only four progressions were investigated in this initial, exploratory study, limiting the conclusions that can be drawn. In future work, it would be desirable to investigate a larger number of progressions or the same progressions presented in a larger number of different orders.

General Discussion

The experiments explored the relationship between chroma-salience profiles of individual chords and tone profiles obtained after short chord progressions. Results were compared with predictions of three different models: a baseline stimulus model, a top-down or schema-driven key model, and a pitch model that combined both bottom-up and top-down elements.

The stimulus model (number of times each chroma occurs in the progression) was relatively successful for individual sonorities, although it was consistently less successful than the pitch model in the case of major, minor, and diminished triads. The stimulus model broke down seriously in the case of progressions of two or three chords, in comparison to the other two models. This may be taken as evidence for the importance of primacy, recency, and top-down processes in the perception of tonality in chord progressions.

The key model (best-fitting key profile of Krumhansl & Kessler, 1982) was most successful in the case of clear tonal "forward" progressions such as G-C, b°-C, F-G-C and f-G-c. It was less successful than the other models in predicting tone profiles for individual sonorities, and for tonally ambiguous elements such as the diminished triad and chord progressions that

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could equally well be in two different keys (e.g., C-e). These findings are consistent with the procedures by which the profiles were initially determined and with the existence of a limited number of key templates that are matched "top down" to familiar tonal progressions. The predictions of the key model contradicted music theory in the case of the reverse-order progressions C-G-F and c-G-f. Two possible explanations may be advanced: either music theorists are wrong and the tone center in these cases is F, or Krumhansl's method of key determination breaks down in this case. The difference would appear to rely on the exact definition of perceived tonality. Because no single, widely accepted perceptual method of determining tonality exists (Auhagen, 1994; Eberlein, 1994), it is difficult to shed further light on this distinction at this time.

The pitch model (cumulative chroma salience) was generally successful in predicting tone profiles in all three experiments. This may be taken as evidence for the existence of both the bottom-up process of pitch (salience) accumulation over time (Parncutt, 1989; Huron & Parncutt, 1993) and the top-down process of harmonic pitch pattern recognition (Terhardt, 1972) in progressions of chords of any length. In the case of isolated harmonic

dyads, the pitch model could not, however, account for a tendency for listeners to hear dyads as incomplete triads. Predictions of the pitch model were also less successful for the progressions d-C and a-C (in both orders) and for the reverse minor-key cadence f-G-c; the reasons for these fluctua- tions in performance are not clear but presumably involve key implications and associated top-down processes.

In the introduction, we made three hypotheses. First, we hypothesized that top-down processes of key determination would play a more important role in the perception of longer chord progressions than in the perception of shorter chord progressions. Our findings were consistent with this

expectation: the bottom-up stimulus model was consistently successful only for isolated chords, whereas the top-down key model was generally more successful for progressions of two or three chords. This is to be expected on the basis of Browne's (1981) theory. But our results are also consistent with the idea that both top-down and bottom-up processes play a role in chord progressions of any length. It is not easy to compare the relative

importance of top-down and bottom-up processes in specific cases, owing to the quantitative similarity of their outcomes: predictions made on the basis of top-down and bottom-up models often correlate significantly with each other.

Second, we hypothesized that the relative importance of top-down (key recognition) and bottom-up processes would depend on the clarity of the

key of a passage: bottom-up processes should play a stronger role in ton-

ally ambiguous passages. As described earlier, our findings were again largely consistent with these expectations.

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Third, we hypothesized that the relative importance of top-down and bottom-up processes would depend on musical training, such that musicians would be more sensitive to bottom-up effects. This prediction could not be tested because of the lack of significant data for nonmusicians in Experiments 1 and 2, the quantitative similarity of the three predictors, and the limited number of tested chord progressions in Experiment 3. Vi- sual inspection of Figure 6 suggests that peaks in nonmusicians' profiles correspond to tones of the final chord in each case (all three tones of major triads, and the root and fifth of minor triads). This is weak evidence for the role of recency and of bottom-up processes and neither confirms nor re- futes our hypothesis.

On the basis of these findings and arguments, the cognition of short chord progressions may be conceptualized as follows. First, chroma-salience profiles are determined for individual sounds (top-down pitch-pattern recognition). These profiles are then combined (bottom up) such that more recent sounds have a greater effect than earlier ones. Finally, the re- sultant profile is compared with key profiles stored in memory (top down). The latter process may involve a kind of categorical perception for musical keys (cf. Leman's 1995 hysteresis model) but we are aware of no direct evidence for this idea in the empirical literature.

In summary, the main theoretical issues, and our conclusions based on available evidence, are as follows:

1. Pitch salience. When calculating tone profiles on the basis of the frequency of occurrence of tones (Cuddy, 1997; Knopoff & Hutchinson, 1983; Krumhansl, 1990a; Oram & Cuddy, 1995), one should ideally consider not only recency and accentuation of tones and sonorities (including, e.g., durational accent due to variations in interonset interval, and other accent types, including dynamic, metrical, and structural; see Lerdahl &c Jackendoff , 1983, for an overview), but also variations in pitch salience within sonorities. Caveat: Streaming between chords may reduce the salience of pitches that are implied (according to Terhardt's theory) but not played (cf. Eberlein, 1994); this effect is not yet incorpo- rated into the relevant models.

2. Harmonic pitch pattern recognition. Both present and previous results are consistent with the assumption that tone profiles are influenced not only by the bottom-up process reflected by preva- lence distributions of tones and by the top-down process reflected by KrumhansPs key profiles, but also by the top-down process of harmonic pitch-pattern recognition in tonal simultaneities (vir- tual pitch perception). The main evidence for this was a general trend in all experiments for ratings for roots of chords to exceed ratings for other tones - an effect that was consistently greater

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for major than for minor triads, and consistent with the prediction (e.g., Parncutt, 1988) that the root of the minor triad is considerably more ambiguous than that of the major.

3. Primacy and recency. The tone profile was generally influenced most by the last chord. There was also a subsidiary effect of primacy: the first chord in a three-chord sequence was more important than the second.

4. Familiarity with tonal musical syntax. Like some results of Thompson and Parncutt (1997), some results of Experiment 1 here could be explained in terms of familiarity with major triads, such that when two of the three tones were heard simultaneously, the other was imagined. Cognitively, this suggests the existence of a relatively robust schema for major triads that can be fit to incomplete presentations, just as Krumhansl's relatively robust key profiles may be fit to different tonal progressions, and - in Browne's theory - diatonic scales can be fit to diatonic fragments.

5. Musical training. Three different processes (one bottom-up, two top-down) influenced the results of musicians. Nonmusicians produced significantly varying tone profiles only for progressions of three chords. Experiments 1 and 2 seemed to have been simply too difficult for them, possibly due to problems with pitch discrimination. Although nonmusicians produced significant profiles in Experiment 3, it is not clear which of the three above- mentioned processes might have played a role in their perception.

6. Definition of "key." The best-fitting key profile does not necessarily correspond to the music-theoretic key; nor does the peak of the tone profile necessarily correspond to the music-theoretic tonic. A possible solution may be to operationally define the tonic as a triad rather than a tone.

The various recently developed approaches to tonality perception sometimes contradict each other and so do not build a unified or complete picture. In this article, we have attempted to bring two of these approaches closer together: Shepard and Krumhansl's cognitive structural approach, and Terhardt and Parncutt's psychoacoustic approach. Considering both old and new data, we have shown that both these approaches can account for observable behav- iors, but in different and complementary ways. This article may be regarded as a step towards a comprehensive perceptual-cognitive theory of harmony and tonality perception that can explain both the (cognitive) nature and the (perceptual) origins of everyday music-theoretic phenomena.

As in much music-psychological research, in this article, we have walked a tightrope between experimental thoroughness and ecological validity. To

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56 Richard Parncutt &C Albert S. Bregman

allow definite conclusions to be drawn from the quantitative data, it was necessary to limit the spectral and temporal characteristics of our stimuli. Appropriate temperance is required before generalizing our results to real, tonal music made up of different kinds of (harmonic complex) tone with variations in timbre, expressive timing, tempo, and so on.

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9. This research was carried out at the Faculty of Music, McGill University, Montreal, during R. P.'s tenure of an International Fellowship granted by the Natural Sciences and

Engineering Council of Canada (NSERC). Results were first reported at the 4th Interna- tional Conference on Music Perception and Cognition (ICMPC) held at the Faculty of Music, McGill University, in August 1996. We thank Bruce Pennycook for provision of facilities

during the data collection phase, and Adrian Houtsma, Carol Krumhansl, Ernst Terhardt, and Bill Thompson for helpful comments on the manuscript.

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