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Alma Mater Studiorum University of Bologna, August 22-26 2006

The perception of non-adjacent harmonic relations

Matthew Woolhouse Centre for Music & Science, Faculty of Music, Cambridge

University, UK

mhw31@cam.ac.uk

Ian Cross Centre for Music & Science, Faculty of Music, Cambridge

University, UK

ic108@cam.ac.uk

Timothy Horton Centre for Music & Science, Faculty of Music, Cambridge

University, UK

tjh20@cam.ac.uk

ABSTRACT

Most research on cognition of harmonic relations has fo-cused on the experience of relations between successive events. Yet a major feature of theories of harmony is the elucidation of relationships between non-adjacent events. The present study seeks to address this issue empirically. The experiment demonstrates that listeners can be sensitive to relations between non-adjacent harmonic events and, in the process, provides new evidence concerning the cogni-tive structure of fifth-relations.

Keywords

Harmony; hierarchy; non-adjacent dependencies.

INTRODUCTION

A significant feature of language is that relationships be-tween elements that are temporally non-adjacent have structural significance in perception and production; the functional identities of elements of language can depend on other, temporally non-contiguous, elements as well as on immediate temporal context. This attribute, which appears to be operational at several levels (although for some con-

straints on these see Newport and Aslin, 2004), is most evident in the use of relative clauses in language, where significant syntactic and semantic relationships can exist between elements of linguistic structure that are widely separated in time, yet appear to be easily – indeed, effort-lessly – accessible to language perceivers and producers. Relationships between such temporally non-adjacent ele-ments can be thought of as exemplifying principles of re-cursion or hierarchicality, which have been proposed as foundational in respect of the human language faculty (Hauser, Chomsky and Fitch, 2002). Principles of hierar-chical organisation have also been proposed as underlying aspects of music perception and production (Lerdahl and Jackendoff, 1983). However, to date little clear empirical evidence exists (see Deutsch, 1981; Cook, 1987) to indi-cate the operation of hierarchical principles in music per-ception that are either of the same order of complexity, or are temporally extended over similar durations, as are those evident in language.

Theories of music suggest that it is structured hierarchically in respect of different parameters, including pitch and rhythm; while most such theories bear on western com-mon-practice period tonal music (e.g., that of Lerdahl & Jackendoff, 1983), it has been suggested that similarly hi-erarchical principles apply to a range of other music (Hughes, 1991). Sequences of events such as pitches or chords are conceived of as being grouped at multiple levels on the basis of both their temporal contiguity and their ab-stract structural relationships (which may pertain to rela-tions of both constituency – which elements should be grouped with which – and dependency – which elements depend on others for their structural and functional identity, and in what ways: see Horton 2004), with events of greater structural significance within groups determining the func-tional identity of the group at higher levels of abstraction.

In: M. Baroni, A. R. Addessi, R. Caterina, M. Costa (2006) Proceedings of the 9th International Conference on Music Perception & Cognition (ICMPC9), Bologna/Italy, August 22-26 2006.©2006 The Society for Music Perception & Cognition (SMPC) and European Society for the Cognitive Sciences of Music (ESCOM). Copyright of the content of an individual paper is held by the primary (first-named) author of that pa-per. All rights reserved. No paper from this proceedings may be repro-duced or transmitted in any form or by any means, electronic or me-chanical, including photocopying, recording, or by any information retrieval systems, without permission in writing from the paper's primary author. No other part of this proceedings may be reproduced or transmit-ted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information retrieval system, without permission in writing from SMPC and ESCOM.

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Few experimental studies have addressed the issue of hier-archicality in the experience of music. Deutsch (1981) found that melodic sequences which could be described recursively were more stably represented in listeners' memories than were less structured sequences. Serafine, Glasmann & Overbeeke (1989) conducted a series of stud-ies which required listeners to judge the similarity between complex melodic stimuli and 'reduced' versions of these that could be regarded as hierarchical abstractions from the musical surface. Their results were unclear, though Dibben (1994), in a study using similar methods but employing more complex musical excerpts, found that analytic reduc-tions were consistently judged as more similar to the origi-nal music than were foils, though this finding held only for tonal musical excerpts.

In experiments exploring the extent to which listeners were capable of using overt or covert knowledge of tonal musi-cal principles in generating pieces (using a 'puzzle' para-digm, in which subjects were presented with randomly-ordered visual representations of segments of an existing brief tonal work and asked to construct a piece from these), Deliège Mélen, Stammers & Cross (1996) found some de-gree of sensitivity to structure at a global level, but only for highly trained musicians. Non-musician subjects did appear sensitive to structural relations between segments but only at a local level. An extensive series of studies by Bigand and his collaborators using priming, probe and puzzle paradigms (for an overview, see Tillmann & Bigand, 2004) extended these findings in respect of both perception and music construction. Tillmann & Bigand conclude (ibid., p219) that "…numerous experiments highlight the strength of local structure for music perception. In short temporal windows, listeners understand the local function of ca-dences, perceive changes in tonality, and go somewhat beyond the musical surface. However, their perception fails to integrate these structural markers into more global struc-tures." In other words, while abstract structural relation-ships of dependency and constituency appear to be opera-tional in listeners' perceptions and preferences, these ap-pear limited to local, immediate levels.

Hence there is little if any evidence to support the notion that musical perceptions engage with hierarchical structure in music. It may of course be that hierarchical structures are instantiated in principles of music composition rather than perception as Cook (1990) has suggested. However, most experimental approaches to the exploration of the experience of hierarchy in music perception have tended to focus on the moment-to-moment unfolding of musical events and inter-relationships rather than on listeners' ca-pacities to abstract connections between events in the mu-sical stream that are temporally non-contiguous. Hierarchi-cal structure as manifested in the linguistic domain in the form of the relative clause relies on precisely this latter type of relationship, and it seems reasonable to expect that the issue of hierarchicality in musical perception should be

addressed by exploring the issue of sensitivity to relations between temporally non-adjacent events in music. The pre-sent study adopts this approach, exploring the relations that are perceived by listeners as holding between both adjacent and non-adjacent harmonic events.

EXPERIMENT

The method used in the present study is analogous to that employed by Krumhansl & Kessler (1982), who used a probe-tone approach to study the evolving sense of key that listeners experienced as a sequence of chords progressed and modulated. However, here the focus was not on mo-ment-to-moment changes in key, but on the extent to which the initial key of a sequence might be represented in sub-jects' responses to a probe-cadence presented at the end of a complete modulating sequence. Hence a method was de-veloped which enabled the effect of the initial key (tempo-rally distant from the probe, hence non-adjacent) to be dis-tinguished from the effect of the final key (temporally con-tiguous with the probe, hence adjacent).

Stimuli

The experiment used two basic sequence types: non-modulating plus probe cadence (Figure 1), and modulating plus probe cadence (Figure 2).

Non-modulating sequence

Adjacent key

2 beat rest

Probe cadence

Figure 1. Non-modulating sequence

Modulating sequence

Non-adjacent key

Adjacent key

2 beat rest

Probe cadence

Figure 2. Modulating sequence

Each non-modulating sequence consisted of nine chords: I-IV-V-I-vii o-I-IV-V-I in a single key, 2 beats silence, and a V-I cadence in any key. Each modulating sequence also consisted of nine chords: I-IV-V-I in the non-adjacent key, viio-I-IV-V-I in the adjacent key, 2 beats silence, and a V-I probe cadence in any key. ‘Adjacent’ refers to the position of each key with respect to the probe cadence. All possible modulations were used in the generation of experimental sequences, with each modulation being probed by all pos-sible probes, yielding 132 modulating sequences and 12 modulating sequences (144 in all). All stimuli, including the probe cadences, were in major keys – minor key rela-tionships were not explored in the present study. Tempo was 90 beats per minute, one chord being presented per beat; 6 seconds thus elapsed between the last chord of the non-adjacent key and the last chord of the probe cadence.

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Stimuli were produced using an Apple Macintosh G5 com-puter running SuperCollider3 software. ‘Shepard’ tones (Shepard, 1964), consisting of superposed octave-related sinusoids with overall amplitude controlled by a Gaussian function, were used for the presentation of the stimuli. The technique produces tones with approximately equal overall pitch height and with no clear pitch maxima or minima. Subjects listened via headphones and adjusted the volume to a comfortable level before testing commenced.

Task

For both sequence types, subjects were required to rate on a 7-point scale the degree to which the probe cadence com-pleted the preceding sequence (i.e., the induced sense of tonal closure): 7 for a very strong sense of completion, 1 for a very weak sense of completion. To control for order effects, each subject was presented with the stimuli in a different random order and transposition.

Subjects

Subjects were 57 adults (30 females, 27 males) with two distinct levels of musical experience – henceforth referred to as “Musicians” and “Non-musicians”. There were 27 Musicians and 30 Non-musicians. The Musicians included 16 females and 11 males aged from 19 to 27 years old, with a mean age of 21 years (SD = 2.1). The Non-musicians included 14 females and 16 males aged from 20 to 41 years old, with a mean age of 25 years (SD = 4.2).

The Musicians in the study were overwhelmingly involved with western classical music, having received on average over 10 years formal musical training, and professing to practice/play/perform on average 10 hours per week. The Musicians’ practical involvement with classical music was reflected in their listening habits which shows a 60% - 40% split in favour of classical music over popu-lar/jazz/rock/folk music, and which was on average 12.7 hours per week.

The Non-musicians in the study had received on average less than 1.5 years formal musical training and were not currently active players and/or singers. On average at least 9 years had elapsed since 43% of the Non-musicians had played an instrument, while the remaining 57% had never played a musical instrument. A greater degree of plurality in non-musicians’ listening habits was shown by a 78% - 22% split in favour of popular/jazz/rock/folk music over classical, and which is on average 10.5 hours per week.

The Musicians were undergraduate and graduate members of the Faculty of Music, Cambridge University, and re-ceived no financial remuneration for their participation. The Non-musicians were largely drawn from other depart-ments of the University and received payment for participa-tion. Each experiment session lasted approximately 45 minutes.

Data

Single variables with 3 or more conditions were investi-gated using one-way within-subjects ANOVA (Tukey HSD is used for multiple comparisons). Interactions were inves-tigated using a 2-factor (musical experience ✕ variable) mixed factorial ANOVA. The 2 levels of musical experi-ence formed the between-subject factor; the within-subject factor was probe cadence. The relative contributions of a number of different variables to the data were assessed using multiple regression analysis.

Prior to analysing the data the stimuli presentation order was unscrambled and re-transposed; if the non-adjacent key was under investigation, the adjacent key = C; if the adjacent key was under investigation, the non-adjacent key = C. Subjects’ ratings were normalised (Z-scores) and re-expressed in terms of the original 7-point scale.

Non-modulating probe cadence profiles

Intersubject differences

Intersubject correlations of the probe cadence profiles for the non-modulating sequence were taken as a measure of experiment task competence. From the original subject groups, two musicians and three non-musicians were found to have produced probe cadence profiles in response to the non-modulating sequence that failed to correlated with over 75% of the profiles produced by the other members of their respective groups. The data of these subjects were therefore eliminated from the analysis as outliers. For the musician subjects the average inter-subject correlation for the non-modulating probe cadence profile was 0.26, while for the non-musicians the average correlation was 0.15. Individual differences were not explored further and the analysis was based either on individual subject ratings, or on the mean rating for each probe cadence.

Results 1

Musicians’ mean probe cadence profile

The mean rating for each probe cadence was subject to initial analysis, the symmetry around C of both groups' mean probe cadence profiles (particularly that of the musi-cians) warranting particular attention (see Figure 3). Treat-ing the subjects’ data as the dependent variable, multiple regression analysis was used to assess the relative predic-tive capacities of two independent variables: one linear, one non-linear. The linear variable was the proximity of the probe cadence to the context key around the cycle of 5ths (5ths distance); the non-linear variable involved a simple square-root function (Sqrt+1).

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Figure 3. Mean probe cadence ratings for musicians and non-musicians following non-modulating sequences in C.

Giving C the nominal value 6, 5ths distance was calculated by subtracting 1 from the nominal value for each step around the cycle of 5ths travelled. Therefore, G (1 step from C) = 6-1 = 5, D (2 steps from C) = 6-2 = 4, and so on. Again giving C the nominal value 6, Sqrt+1 was calculated by taking the square root of the nominal value and adding a constant (1), and then passing this value to the next step around the cycle of 5ths and re-performing the calculation. Therefore, G = (√6)+1 = 3.45, D = (√3.45)+1 = 2.86, A = (√2.86)+1 = 2.69, and so on, producing a Gaussian-like distribution (see Figure 4). The correlation and standard-ised beta coefficients of the multiple regression analysis are independent of the chosen initial nominal value (in this case, 6).

Figure 4. Musicians’ mean ratings for the non-modulating sequence in C, and variables 5ths distance and Sqrt+1.

Table 1. Multiple regression summary of the Musicians’ non-modulating probe cadence profile

Factor Pearson correlation

Beta coefficient

t-statistic df = 11

Probe cadence data

R square = .97**

5ths distance .79** .20 2.46*

Sqrt+1 .98** .84 10.37**

*Significant at p < .05; **Significant at p < .005

The Pearson correlation values for 5ths distance and Sqrt+1 are significant at .79 and .98 respectively (n=12), and the R square value indicates that the independent vari-ables account for 97% of the dependent variable. However, the beta coefficients from the regression equation indicate that the non-linear variable, Sqrt+1, has over 4 times the predictive capacity compared to the linearly derived vari-able, 5ths distance (.84 versus .20).

Non-musicians’ mean probe cadence profile

The same regression procedure above was used to analyse the Non-musicians’ non-modulating probe cadence profile.

Table2. Multiple regression summary of the Non-musicians’ non-modulating probe cadence profile

Factor Pearson correlation

Beta coefficient

t-statistic

df = 11

Probe cadence data

R square = .73**

5ths distance .60** -.02 -.07

Sqrt+1 .85** .87 3.5*

*Significant at p < .05; **Significant at p < .005

The Pearson correlation values for 5ths distance and Sqrt+1 were significant at .60 and .85 respectively (n=12), and the R square value (.73) indicates that the independent variables account for 73% of the dependent variable. How-ever, the beta coefficients from the regression equation show that only the non-linear variable, Sqrt+1, has a sig-nificant predictive capacity in relation to the dependent variable.

Musicians’ and Non-musicians’ probe cadence profiles

Individual subject responses were then explored in a 2-factor (musical experience ✕ probe cadence) mixed ANOVA, which examined the possible discrepancies shown between the Musicians’ and Non-musicians’ probe cadence profiles in the above regression analysis.

In the test of within-subject effects, the factor cadence was significant: F(11,605) = 14.469; p < 0.005. Given the data spikes on C for both groups, that there should be a signifi-cant main effect for cadence is perhaps not surprising. However, the tests of between-subject effects showed no interaction between the two groups: F(1,55) = 0.000; n.s.

In sum, the ANOVA indicates that there are significant differences between probe cadence ratings, but that these differences are similar across the two levels of musical experience. As a result, the remainder of the analysis is based on the pooled data of the two groups. This yields a single mixed musical experience subject group with 57 members.

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Modulating probe cadence profiles

The experiment explored all modulations: rising and falling minor and major 2nds; rising and falling minor and major 3rds; rising and falling perfect 4ths; and the diminished 5th modulation. Each sequence was probed using a V-I ca-dence presented at all 12 possible transpositional levels, a process that gave rise to 132 (11 modulations ✕ 12 probe cadences) modulating sequences in total.

Rationale

The main objective of the study was to determine non-adjacent key effects on the perception of tonal closure, i.e., the sense of musical completion created by the probe ca-dence. Using a key matching process, the effect on probe cadence ratings of the non-adjacent or adjacent key of the modulating sequence can be investigated. Referred here to as ‘residual’ analysis, the following is a brief outline of the procedure. If the adjacent key of the modulating sequence is the same as the non-modulating sequence, by subtracting the probe rating of the non-modulating sequence from the probe rating of the modulating sequence the effect of the non-adjacent key on the perception of tonal comple-tion/closure can be isolated (i.e., the residual from the sub-traction). For an example, see Figure 5:

Non-modulating sequence

Key 2: C

2 beat rest

Probe

cadence: B

Rating of the probe cadence above subtracted from the rating of the probe cadence below

Modulating sequence

Key 1: B Key 2: C

2 beat rest

Probe

cadence: B

Figure 5. Isolation of the effect of the non-adjacent key of B on the perception of tonal closure for the modulating sequence

B-C: B (probe cadence)

Conversely, if the non-adjacent key of the modulating se-quence is the same as key of the non-modulating sequence, by subtracting the probe rating of the non-modulating se-quence from the probe rating of the modulating sequence the effect of the adjacent key on the perception of tonal completion can be isolated. For an example, see Figure 6.

Non-modulating sequence

Key 1: C

2 beat rest

Probe

cadence: F

Rating of the probe cadence above subtracted form the rating of the probe cadence below

Modulating sequence

Key 1: C Key 2: Db

2 beat rest

Probe

cadence: F

Figure 6. Isolation of the effect of the adjacent key of Db on the perception of tonal closure for the modulating sequence C-

Db: F (probe cadence)

A significant difference between the probe cadence ratings of the non-modulating and modulating sequence pairs indi-cated that the key under investigation (either the adjacent or non-adjacent key) was having a noticeable effect on the perception of tonal closure of the sequence. Subtracting the modulating from non-modulating probe cadence ratings yielded a total of 132 residual values. These values were used as a dependent variable – referred to as Residual – and modelled using a number of variables relating to music theory and key distance on the cycle of 5ths (Table 3).

Music theory variables

The music theory related variables were the following key relationships: tonic (T), dominant (D), flattened supertonic (b2), and flattened submediant) (b6). The rationale for choosing these key relationships related to their strong theoretical association with the tonic key, and, ipso facto, with the probe cadence. It was hypothesised that these key relationships, when established in relation to the probe ca-dence, would have a positive effect upon the perception of tonal closure. That is, if the isolated key created any of these key relationships with the probe cadence, ratings would increase; if the isolated key destroyed any of these relationships with the probe cadence, ratings would de-crease. Central to the study is whether or not this hypothe-sised phenomenon can be actuated by non-adjacent key relationships.

The method for calculating each independent variable is briefly as follows. First, if the isolated key – compared to the non-modulating sequence – created the specified key relationship with the probe cadence, then the appropriate table entry was given a score of 1 (Table 3, light grey squares). For example, the second row in column T in Ta-ble 3 has the value 1 due to the isolated key (Db) having (in comparison with the non-modulating sequence) a tonic relationship with the probe cadence.

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Table 3. Table of independent variables. The table shows the first 18 entries of each variable. In total there

are 132 entries (rows) per variable.

Second, if the isolated key – compared to the non-mod se-quence – destroyed the specified key relationship with the probe cadence, then the appropriate table entry was given a score of –1 (Table 3, dark grey squares). For example, the first row in column T in Table 3 has a value of -1 because of the isolated key (Db) not having (in comparison with the non-modulating sequence) a tonic relationship with the probe cadence. Third, if the isolated key – compared to the non-mod sequence – neither created nor destroyed the specified key relationship with the probe cadence, then the appropriate table entry was given a score of 0. For exam-ple, the first row in column D in Table 3 has a value of 0 due to the isolated key (Db) neither creating nor destroying (in comparison with the non-modulating sequence) a domi-nant relationship with the probe cadence.

Key distance variable

One other variable is used to model the dependent variable Residual. Key distance was obtained by calculating the distance – around the cycle of 5ths – between the isolated key and the probe cadence, and between the non-modulating key and the probe cadence, and then subtract-ing one from the other. For example, if the isolated key is five step on the cycle of 5ths from the probe cadence and the non-modulating key is zero, the residual key distance is 0 – 5 = -5 (see Figure 7 and Table 3, first row). If the iso-lated key is zero steps on the cycle of 5ths from the probe cadence and the non-modulating key is five steps, the re-sidual key distance is 5 – 0 = 5 (see Table 3, second row), and so on.

The relative contribution of each independent variable (four Music theory and Key distance) to the dependent variable, Residual, was assessed using multiple regression. This analysis was carried out for both non-adjacent and adjacent key effects.

Non-modulating sequence

Key: C (0 steps from probe cadence on the cycle of 5ths)

2 beat rest

Probe

cadence: C

Modulating sequence

Key: Db (5 steps from the probe cad. on the cycle of 5ths

Key: C

2 beat rest

Probe

cadence: C

Figure 7. Calculation of Key distance. C is zero steps on the cycle of 5ths from C (probe cadence, upper box); Db (the isolated key) is five step on the cycle of 5ths from C (probe cadence, lower box). The net effect of the isolated key is to be five steps further away on the cycle of 5ths

from the probe cadence.

Results 2

Non-adjacent key effect

Data used: mean rating for each probe cadence.

Table 4. Multiple regression summary for dependent vari-able Residual and independent Music theory and Key dis-tance variables. Non-adjacent key analysis.

Factor Beta coefficient

t-statistic df = 131

Residual data

R square = .20**

T (tonic) .34 3.36**

D (dominant) .22 2.41*

b2 (flat supertonic) .00 -.26

b6 (flat submediant) -.02 -.36

Key distance .08 .73

*Significant at p < .05; **Significant at p < .005

Only two of the four music theory variables had a consis-tent relationship to the data: T and D; b2 and b6 were not significant predictors of Residual. The R square value (.20) indicates that the independent variables accounted for 20% of the dependent variable, which although low is, neverthe-less, significant. The beta coefficients show that variable T was a stronger predictor than D. Key distance was not a significant predictor of the dependent variable in the non-adjacent position.

Adjacent key effect

Again, of the music theory variables only T and D had a consistent relationship to the data; b2 and b6 were not sig-nificant predictors of Residual. The R square value (.90) indicates that the independent variables account for 90% of the dependent variable, which is highly significant. The beta coefficients show that variable T was a stronger pre-

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dictor than D. In the adjacent position, Key distance was a significant predictor of the dependent variable.

Table 5. Multiple regression summary for dependent vari-able Residual probe cadence and independent Music theory

variables. Adjacent key effect.

Factor Beta coefficient

t-statistic df = 131

Residual data

R square = .90**

T (tonic) .752 21.69**

D (dominant) .10 3.14**

b2 (flat supertonic) .02 .66

b6 (flat submediant) -.02 .04

Key distance .08 7.92**

*Significant at p < .05; **Significant at p < .005

The correlation between T and Residual can be seen in the scatterplot in Figure 8. Although showing considerable variability, the distribution clearly demonstrates the signifi-cant, positive non-adjacent effect of the tonic key on the ratings of the 132 modulating sequences: r = .353; p < 0.005.

Figure 8. Scatterplot of variable T (tonic) against Residual,

non-adjacent effect.

Assessment of variables

In instances where an independent variable relates signifi-cantly and positively to the dependent variable, a one-way ANOVA was carried out on the mean rating for each probe cadence to determine whether the differences between the variable’s levels were significant.

Tonic

In the multiple regression, T significantly related to the dependent variable in both non-adjacent and adjacent posi-tions. The result of the ANOVA for T in the non-adjacent position was significant (F(2, 129) = 9.217; p < 0.001) and a Tukey HSD test of pairwise comparisons between the 3 levels revealed significant differences also (see Table 6).

The result of the ANOVA for T in the adjacent position was highly significant, F(2, 129) = 261.206; p < 0.005, and a Tukey HSD test of pairwise comparisons between the 3 levels in the AVOVA showed highly significant differ-ences between all levels.

Table 6. Multiple comparisons between T levels –1, 0, 1

Dominant

In the multiple regression, D significantly related to the dependent variable in both non-adjacent and adjacent posi-tions. The result of the ANOVA for D in the non-adjacent position was significant (F(2, 129) = 3.607; p < 0.05) and a Tukey HSD test of pairwise comparisons between the 3 levels revealed some significant differences also (see Table 7).

Table 7. Multiple comparisons between D levels –1, 0, 1

The result of the one-way ANOVA for D in the adjacent position was not significant.

Key distance

In the multiple regression, Key distance significantly re-lated to the dependent variable only when in the adjacent position. The result of the ANOVA for Key distance was significant, F(2,118) = 10.801; p < 0.001, and a Tukey HSD test of pairwise comparisons between the 11 levels in the AVOVA showed a number of significant differences. This can be seen in the homogeneous subsets in Table 8.

Table 8. Homogeneous subsets table for Key distance

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Figure 9. Mean Residual values for each Key distance level

The positive correlation for Key distance against Residual in the adjacent position (r = .71) is reflected in the bar chart of mean Residual value of each Key distance level (Figure 9).

Discussion

In summary, the experiment found an effect of non-adjacent key on the perception of tonal closure; in addition to this, a non-linear variable, Sqrt+1, was found to be a highly significant predictor of Musicians’ mean probe ca-dence ratings in non-modulating sequences.

The latter finding, explored by analysing mean probe ca-dence ratings following non-modulating sequences by means of multiple regression, implies that key relatedness is not a linear function of distance on the cycle of 5ths, rather a function in which discrimination between distantly related keys is less acute than between closely related keys. That is, our abilities to differentiate between closely related keys would appear to be greater than between distantly related keys. Indeed, it could be suggested that, at least for trained musicians, a more accurate topology for key relat-edness in cognition than a 'circle of fifths' might be an 'oval of 5ths'.

The former finding, of a sensitivity to relationships be-tween non-adjacent harmonic events amongst both highly trained musicians and non-musicians, was investigated by using a process of probe cadence rating subtraction to iso-late the effect on the probe cadence of either the adjacent or non-adjacent key. The Residuals of the subtraction were

used as the dependent variable in a multiple regression analysis involving a number of substantive independent variables. Of the four Music theory variables (i.e., those involving music-theoretic key relations) only the tonic and dominant were found to have a non-adjacent and adjacent effect on the perception of tonal closure at the probe ca-dence. This finding clearly demonstrates that the subjects were able to hold in memory – at some level – the non-adjacent key, and that this was responsible for significant and predictable variations in probe cadence ratings. The keys identified as those having a non-adjacent effect formed tonic and dominant relationships to the probe ca-dence, and it would appear that when presented with these particular stimuli listeners hierarchically nested the 'adja-cent' key within the overarching structure of the outer keys (Figure 10). To the authors' knowledge, sensitivities to relationships between such non-adjacencies in music have hitherto not been empirically observed.

Figure 10. Hypothesised nesting of the adjacent key within the outer keys

The many studies that have previously explored the effects of harmonic structure at larger timescales than the immedi-ate have generally been concerned with articulating the perceptual identity of key as a sequence unfolds (as in Krumhansl & Kessler, 1982), have explored hierarchical perceptions via changes in adjudged tension (Bigand & Parncutt, 1999), or have investigated interactions between perceptions of melody and harmony and effects of key dis-tance (Cuddy & Thompson, 1992). The present study finds a previously unreported effect of key distance that is some-what at variance with the findings of Cuddy & Thompson (1992) as well as demonstrating that listeners can be sensi-tive to harmonic relationships between non-adjacent events, a finding which calls into question Tillmann & Bi-gand's (2004) conclusion that listeners' "perception fails to integrate these structural markers [cadences and changes in tonality] into more global structures". Nevertheless, the present findings must be regarded as indicative rather than conclusive. They derive from a single experiment (albeit with a large subject sample) in which stimuli were highly 'reduced' in consisting of Shepard tones; neither voice-leading effects or those of melodic or motivic structure were addressed, nor were any explicit effects of rhythmical

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or metrical organisation. Moreover, the temporal scope of the effective relationship shown here between non-adjacent harmonic events was fixed for all stimuli, and it might be expected that variable sequence length would lead to somewhat different patterns of results. Further experiments will be required in order conclusively to claim that music in perception exhibits types of hierarchical structuring that are of the same order of those of language.

Conclusion

Notwithstanding these caveats, the present study has shown that listeners are sensitive to relationships between non-adjacent key events in music in ways that may be analo-gous to typical linguistic capacities. Nevertheless, many questions remain. Are the effects identified here truly analogous to those manifested in linguistic syntactical rela-tions, or might they be better conceived of as analogous to the morphological level? Are the effects limited to a single hierarchical level immediately superordinate to the musical surface, or might more complexly nested relationships be found? What are the 'basic units' that may be experienced as hierarchically related in music (it is important to note that events between which non-adjacent relationships are shown to hold in the present study are not single chords but key defining units). But if the effects shown here are gen-eral, then it calls into question the claim that recursion is a defining and domain-specific property of human language (Hauser, Chomsky & Fitch, 2002), lending support to the suggestions of those such as Arbib (2005) who argue that a degree of recursion is a feature of many human behavioural and cognitive domains – which can now be said to include in their number music.

ACKNOWLEDGMENTS

The authors wish to thank Nick Collins for programming the ex-periments in Supercollider3. The research reported here was sup-ported by a grant from the UK Arts & Humanities Research Council to the second author (AHRC APN18519).

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