## 6 Evaluation
### Evaluating the Objective Function  
Firstly, we address the validation of the objective function \(M\) as a proxy for the perceptual evaluation of tonal tension profile in chord progressions. We expect the objective function to reflect the perceptual quality of the chord progressions independently of the tree structure associated. We designed a listening test that presents several chord progressions generated automatically. We created a total of 12 chord progressions with a total between 5 and 8 chords each one. Each chord progression had a different tree structure. All of them were in major or minor keys. We used a total of four different keys. The rhythm in which the chords are played was always regular to put the focus on the harmonic properties instead of in rhythmic features. The chord progressions were generated by our system and all of them follow some of the curves presented as options in Figure 4. Each chord progression was presented with a reproduction bar, and they were synthesized with a standard piano. The presentation of each chord progression in the listening test is shown Figure 4.  
Figure 4: Question of the listening test. The listeners were presented the chord progressions in a random order. The progressions could be played as many times as they need. The listeners are given the options in Figure 4, and they have to select which curve adapts best to the tonal tension profile of the chord progression. In total, 38 people took the test, among which 11 declared no musical training, 10 considered themselves amateurs, and 17 professional musicians. We grouped the results according to the curves that the chord progressions represented. To analyze the results we calculated the hit rate of the subjective ratings versus the fitness values for the chord progressions included in the listening test. Note that the fitness value is the correlation between the curve expected by the system and the curve obtained evaluating \(M\) for all the chords. In general, chord progressions with higher fitness values have a higher hit rate for all curves tested. We also aim to demonstrate that \(M\) captures the tonal tension profile of any given chord progressions. Thus, we compared the fitness values of the chord progressions with all the curves given in the options. We calculated the correlation between the fitness of each curve and the hit rate that the users gave. As they were two datasets, we can apply the Pearson correlation. We present the Pearson correlation value \(\rho\) and the \(p\)-value in Table 1. The \(p\)-value has been calculated by transforming the correlation to create a t-statistic measure to proof the null hypothesis. The null hypothesis states that there is no relation between the objective measure and the subjective values obtained from the participants of the listening test. Therefore, if one of the \(p\)-values is lower than 0.05 the Pearson correlation is considered significant. The first column gives the statistical measures corresponding to the tonal tension profile calculated by the measure \(M\). The results suggest that the objective function captures the perceptual quality of the chords. The \(p\)-values are all much lower than 1% for the null hypothesis and the Pearson correlation coefficients \(\rho\) are above 0.8. This indicates that the subjective ratings of the users correlate strongly with the tonal tension profile that our model predicts. To validate \(M\) against other proposals, we use two works proposed previously: the Tonal Tension Model proposed by [PERSON], which is one of the most prominent models in the community, was described in Section 2 and captures hierarchical and linear properties following a different topological space, and ChordAIS, a previous work proposed in [24], which captures only linear properties of chord progressions. ChordAIS captures the following properties: consonance, perceptual relatedness to the previous chord, perceptual relatedness to the key and the harmonic function it represents.