Analysis

Primary analysis was performed in R (3.3.2) on mean complexity ratings, averaged across the four versions and across participants for each of the 24 basic stimuli. Mean complexity ratings are plotted in Figure 7.5. This primary analysis consisted of three multiple linear regression models answering three questions about the relationship between the stimuli and the collected complexity ratings. These are specified in Section 7.3.4 below. These models included fixed effects only and were constructed according to the question being posed. Variance explained by each model is given by R-squared, calculated by a correlation test between the model’s predictions and the data. The overall F-statistic of the model is also given. Statistical significance of each predictor is given by the result of a likelihood-ratio test between a null model (intercept only) and a model containing the single evaluated predictor. Statistical significance of each individual factor level for a given predictor is evaluated by the t statistic

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given by the lme4 package for a multiple linear regression model, and by the 95% confidence intervals, where an interval not including zero indicates a significant predictor.

The influence of musical training on complexity ratings was also evaluated for each analysis question posed. This was done by adding a fixed effect for training, reflected by Gold- MSI scores for each participant, where each trial’s response was treated individually (long format data). In this format, and to model with maximal effects in accordance with the experimental design (Barr et al., 2013), random intercepts on participant and stimulus number were added to all models including musical training to create multiple linear mixed effects models. To evaluate the significance of musical training, mixed effects models with and without the predictor were compared using a log likelihood ratio test.

Finally, the multiple linear mixed effects model was evaluated using a correlation test between the model’s predictions and the data.

7.3.4 Results

As described in Section 7.3.3 above, multiple linear regression analyses were carried out to answer three questions about the relationship between the stimuli and the complexity ratings. The influence of musical training (fixed effect) was then evaluated for each question.

First, does the objective manipulation of complexity predict subjective ratings? For this model, objective complexity (1-8), parameter (melody, harmony or rhythm) and version (four types of target melody) were included as predictors of mean ratings. For objective complexity, all factor levels were compared to Level 1, for parameter factor levels were compared to rhythm and for version, comparisons were made to Version A. All but the second and fourth levels of objective complexity had a significant impact (t (83) = -0.19, p = .27, t (83) = 0.43, p = .01, t (83) = 0.27, p = .12, t (83) = 0.54, p = .002, t (83) = 0.82, p < .0001, t (83) = 0.76, p < .0001, t (83), 1.10, p < .0001 for levels 2 through 8 respectively) along with one factor

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Table 7.1. Summary of the fixed effects manipulation model, including coefficients, 95% confidence intervals and R2 _{of each predictor. Each factor of objective complexity is in relation}

to Level 1; each factor of parameter is in relation to rhythm; and each factor of version is in relation to version A. Predictor Coefficient 2.5% 97.5% R2 (Intercept) 3.25 2.93 3.57 - Level 2 -0.19 -0.54 0.15 .68 Level 3 0.43 0.07 0.78 Level 4 0.27 -0.07 0.62 Level 5 0.54 0.19 0.89 Level 6 0.82 0.46 1.17 Level 7 0.76 0.41 1.12 Level 8 1.10 0.75 1.45 Melody 0.19 -0.01 0.41 .03 Harmony 0.27 0.06 0.49 Version B -0.13 -0.38 0.11 .01 Version C -0.02 -0.27 0.22 Version D 0.07 -0.17 0.32

of parameter, harmony (t (83) = 2.55, p = .01 for harmony and t (83) = 1.86, p = .07 for melody); no level of version was significant (all p > .05). Overall, objective complexity was a significant predictor, F (7, 88) = 11.36, p < .0001, parameter was not, F (3, 93) = 1.89, p = .15 and version was not, F (3, 92) = 0.47, p = .69. This model has an R2 _{of .72 and F (12, 83) = 7.78, p < .0001.}

The addition of musical training to predict ratings in long format marginally improved a model without it, χ2 _{(1) = 3.58, p = .05, but the model’s fit to the data was low, r}2 _{= .27, t (2686) =}

14.94, p < .0001. A summary of the fixed effects model predicting mean ratings and of the maximally fitted mixed effects model predicting individual ratings can be found in Tables 7.1 and 7.2 respectively.

The second question is whether complexity is accurately simulated by information content? This model included mean melodic information content, mean harmonic information content and mean rhythmic information content, based on the mean IC of the outer voices

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Table 7.2. Summary of the maximally fitted manipulation model, including coefficients, 95% confidence intervals and R2 _{of each predictor. Each factor of objective complexity is in relation}

to Level 1; each factor of parameter is in relation to rhythm; and each factor of version is in relation to version A. Predictor Coefficient 2.5% 97.5% R2 (Intercept) 2.72 2.22 2.96 - Level 2 -0.10 -0.48 0.27 -.01 Level 3 0.22 -0.15 0.60 Level 4 0.11 -0.25 0.49 Level 5 0.22 -0.15 0.60 Level 6 0.50 0.02 0.78 Level 7 0.34 -0.04 0.72 Level 8 0.54 0.16 0.92 Melody 0.11 -0.11 0.34 .00 Harmony 0.12 -0.11 0.35 Version B 0.01 -0.25 0.27 -.00 Version C -0.05 -0.33 0.21 Version D 1.16 -0.10 0.43 Gold-MSI -0.15 -0.32 0.00 _.00 Predictor Variance Random

intercepts Participant _Stimulus 0.15 _0.06 .20 _.08

Residual

variance 4.53

calculated from a selection of pitch and timing viewpoints (see Section 7.2.1 for details). As the range of mean information content for these predictors varies (mean melodic IC range = 2.35 – 7.62; mean harmonic IC range = 4.20 – 9.55; mean rhythmic IC range = 1.27 – 3.04), each predictor was transformed into z-scores (mean = 1, SD = 1) so that the mean melodic IC range became -1.42 – 3.48, mean harmonic IC range became -2.03 – 1.32 and mean rhythmic IC range became -0.55 – 3.20. To confirm its influence, or lack thereof, melodic IC of the target voice was also included as a predictor (rhythmic IC was not included because it does not vary). Melody and harmony IC predictors were significant (F (1, 94) = 16.18, p = .0001 and

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Table 7.3. Summary of the fixed effects information content model, including coefficients, 95% confidence interval and R2 _{for each predictor.}

Predictor Coefficient 2.5% 97.5% R2 (Intercept) 3.86 3.75 3.96 - Melody IC 0.17 0.06 0.28 .38 Harmony IC 0.22 0.11 0.33 .15 Rhythm IC -0.02 -0.12 0.08 .00 Target Pitch IC -0.02 -0.12 0.08 .00

Table 7.4. Summary of the maximally fitted information content model, including coefficients, 95% confidence interval and R2 _{for each predictor.}

Predictor Coefficient 2.5% 97.5% R2 (Intercept) 2.93 2.75 3.10 - Melody IC 0.09 -0.00 0.19 .00 Harmony IC 0.11 0.01 0.21 -.01 Rhythm IC -0.01 -0.11 0.08 .00 Target Pitch IC 0.03 -0.06 0.12 .00 GoldMSI -0.15 -0.32 0.00 .00 Predictor Variance Random intercepts Participant 0.15 .20 Stimulus 0.06 .08 Residual variance 4.54

F (1, 94) = 23.05, p < .0001 respectively) but not rhythm IC nor melodic IC for the target voice (F (1, 94) = 0.09, p = .75 and F (1, 94) = 0.11, p = .73 respectively), with an R2 _{of .53 and F (3,}

92) = 12.03, p < .0001. The addition of musical training, predicting ratings in long format, marginally improved the model, χ2 _{(1) = 3.57, p = .05, but correlation to the data was low, r2 =}

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and of the maximally fitted mixed effects model, predicting individual ratings, can be found in Tables 7.3 and 7.4 respectively.

The third question is: what musical properties might correspond to the variations in complexity ratings? In other words, how can the differences in complexity perception be characterised in musical terms? For this model, the mean interval size (in semitones; range 0.41 – 8.41) of the two outer voices of each stimuli, the mean pitch (MIDI; range 72.14 – 78.14 for the upper voice and 43.85 – 51 for the lower voice) of each of the two outer voices separately, the mean note duration (in milliseconds, where 1 beat = 24ms; range 16 – 27.42) of the two outer voices, the proportion of out-of-key notes in relation to the total number of pitches in the two outer voices, key proportion (range 0 – 0.38) and syncopation score, where a lower score equates to more syncopation (Lerdahl & Jackendoff, 1983) were calculated. Mean duration and key proportion were significant predictors in this model, though only key proportion was significant when tested against a null model, F (1, 94) = 50.63, p < .0001. Mean duration has significance only in the context of the rest of the model, t (89) = -3.21, p = .001. The model overall has a total R2 _{of .68 and F (5, 90) = 12.53, p < .0001. The addition of musical}

training marginally improved the model, χ2 _{(1) = 3.588, p = .05, but correlation to the data}

remains low, r2 _{= .27, t (2686) = 14.75, p < .0001. A summary of the fixed effects model and}

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Table 7.5. Summary of the fixed effects musical properties model, including coefficients, 95% confidence intervals and R2 _{for each predictor.}

Predictor Coefficient 2.5% 97.5% R2 (Intercept) 8.83 1.46 16.20 - Key proportion 3.59 2.65 4.53 .59 Mean Duration -0.11 -0.18 -0.04 .06 Syncopation Score -0.06 -0.14 0.006 .01 Mean Interval Size -0.02 -0.09 0.04 .00 Mean Pitch – Upper voice -0.02 -0.09 0.05 .00 Mean Pitch – Lower voice 0.06 -0.006 0.13 .02

Table 7.6. Summary of the maximally fitted musical properties model, including coefficients, 95% confidence intervals and R2 _{for each predictor.}

Predictor Coefficient 2.5% 97.5% R2 (Intercept) 4.84 -2.75 12.43 - Key proportion 1.77 0.80 2.74 -.01 Mean Duration -0.05 -0.12 0.01 .00 Syncopation Score -0.03 -0.10 0.04 .00 Mean Interval Size -0.01 -0.08 0.05 .00 Mean Pitch – Upper voice -0.00 -0.08 0.07 .00 Mean Pitch – Lower voice 0.03 -0.03 0.11 .00 GoldMSI -0.15 -0.32 0.00 .00 Predictor Variance Random intercepts Participant 0.15 .20 Stimulus 0.05 .08 Residual variance 4.53

194 7.3.5 Discussion

In order to validate the operational definition stating that information content is proportional to complexity, complexity ratings were collected for a series of stimuli manipulated in terms of melodic, harmonic and rhythmic information content as calculated by IDyOM. The primary analysis results support the definition, as both the experimental manipulations, based on information content, and raw information content successfully predicted mean ratings, explaining 72% and 53% of the variance in the data respectively. When random effects are added to account for individuals and stimulus, the fixed effects lose almost all their predictive power, but do not consistently explain more of the data. Musical training did not significantly improve any models when added as a fixed effect. These additions will be addressed in more detail below. While the information content model demonstrates a correlational relationship between information content and complexity, the explicit manipulation of information content in the experimental design crucially also provides causal evidence. The coefficients and the relative R2 _{of these models can be interpreted to draw some}

interesting conclusions.

First, the intercepts are both close to the centre of the rating scale, indicating a slightly lower than mid-scale baseline rating. Continuing with the manipulations model, there is an imperfect, yet steady increase in the coefficients for every factor level of the objective complexity predictor, and positive coefficients for both factor levels of the parameter predictor. In the case of objective complexity, this equates to stimuli with higher objective complexity resulting in higher subjective complexity ratings as compared to Level 1, as expected. In the case of parameter, these coefficients indicate that melodic and harmonic complexity stimuli receive higher complexity ratings overall than rhythmic complexity stimuli, with harmony being the highest and significantly different from those stimuli manipulated in terms of rhythm.

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This suggests that listeners consider unpredictable harmonic progressions to be more complex than unpredictable melodies (e.g., large leaps or out-of-key notes), and these more complex than rhythmic variety. However, note that the explanatory power of parameter is negligible compared to objective complexity. Finally, version was not a significant predictor, supporting the assumption that the four versions of each stimulus would be rated similarly. The information content model provides somewhat conflicting evidence for the interpretation of the above predictor parameter, where melodic IC carries the most explanatory power, followed by harmonic IC and finally rhythmic IC, with the same pattern of magnitude seen in their coefficients. However, as mean information content is a more fine-grained measure than objective complexity level and parameter, melody IC captures some of the features of unpredictable harmonic progressions, such as out-of-key pitches, resolving the discrepancy between the results of the two models. Thus far, rhythmic complexity is rated as simpler, and onset information content negligible when explaining ratings.

Several raw musical properties of the stimuli were also tested for potential predictive power to attempt to locate the musical features the models and the listeners are responding to when assessing complexity. Mean interval size for both outer voices, mean pitch of each outer voice, mean note duration for both outer voices, the proportion of out-of-key notes among both outer voices and the degree of syncopation of both outer voices was calculated. Together, these represent melodic, harmonic and rhythmic dimensions of music. Analysis on mean ratings reveals an effect of note duration and proportion of out-of-key notes. Both are in the expected direction, where higher proportions of out-of-key notes lead to higher complexity ratings, and longer mean durations, corresponding to stimuli in the lower levels, result in slightly lower complexity ratings. The proportion of out-of-key notes not only provides the most explanatory power, its coefficient combined with the intercept add up to a complexity rating of

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approximately 6.5, near the top end of the rating scale. Knowing that the range of the key proportion predictor is small (ranging from 0 – 0.38), this interestingly puts all ratings quite high on the scale when ratings are predicted by a model based on musical features. The importance of the out-of-key versus in-key proportion is in line with the two previous models where experimental manipulations and information content are predictors: the composed unexpected harmonic progressions contain more out-of-key pitches, which is accounted for in melodic information content since these will have low probability in context.

There was, perhaps surprisingly, no effect of musical training on ratings, where it might be expected that increased exposure to music would yield better models and therefore lower information content, and lower perceived complexity. Additionally, when random effects for participants were included in the mixed effects models, this explained the majority of the variance. Thus, it can be concluded that individual differences supersede any effects of experimental manipulation, information content, musical features or musical training on ratings, and in the cases of information content, increase the predictive power of the model. However, when results are collapsed across participants and more general patterns are considered, these same experimental manipulations, measures of information content and musical features explain at least half the variance in the data in each case, up to as much as 72% in the case of experimental manipulations.

In summary, we find that the results demonstrate a strong link between information content and perceived complexity and furthermore, that harmonically manipulated stimuli, melodic information content and out-of-key notes – all closely related – have the largest influence on complexity ratings, followed by melodic manipulations and harmonic information content and finally rhythmic manipulations and information content, where other melodic and rhythmic musical features have negligible influence.

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