## 6 Discussion  
The developed model of hardness uses the features: log weighted average maximum bandwidth (\(b\)); mean attack centroid (\(c\)); log weighed-average midband level (\(m\)); log percussive-to-harmonic ratio (\(r\)); mean onset strength (\(s\)); and log attack time (\(a\)). The overall hardness can then be predicted with the linear regression:  
\[ \begin{split}\text{hardness}=13.53b+18.15c+13.17m+\\ 5.031r+5.226s-3.710a-89.89.\end{split}  \tag{9}\]  
Coefficients in Equation (9) are expressed to four significant figures and are non-standardised (each feature during the regression modelling was not standardised). This means that each feature's coefficient can be directly implemented into a model of hardness, but does not indicate its relative contribution to hardness. Figure 9: Performance of the hardness model with the between-source-type validation data. The majority of the variance in the data was explained by the log-weighted-average maximum bandwidth of the attack portion, the feature identified first in the iterative modelling process. This is a novel feature created by the authors. Similar previously explored bandwidth-related features, such as taking the difference between the spectral roll-on and roll-off, or the spectral spread [31], did not predict the perceived hardness well (nor did they relate to the authors impressions of bandwidth). The method described in this paper for calculating the bandwidth relies on a threshold method, and it is possible that some signals will never cross this threshold, resulting in an undetectable bandwidth. In this case the reported bandwidth defaults to the resolution of the frequency analysis (10.8 Hz) and might not reflect the actual signal bandwidth. The log attack time of a sound has previously been proposed as a feature that relates to the softness of sounds [24]. This proposal was supported by the findings in this research: the log attack time improved the performance of the model; however, it should be noted that this feature was the last added and, therefore, does not account for the majority of the variance in the data. Additionally, the model of timbral hardness developed in this study incorporates two features similar to features used in the musical hardness model developed by [PERSON] et al. [6; 7]. The percussive-to-harmonic ratio is calculated in a similar manner to the level of percussive energy feature in the musical hardness model. The midband level feature is similar to the intensity of signal components between 2 kHz and 4 kHz feature used in the musical hardness model, but focuses on a lower range of frequencies. Using [PERSON]'s four features (see Section 3.6) to create a linear regression model on the current dataset resulted in a performance of \(R^{2}=0.49\) and rho = 0.72. The hardness model developed in this paper performs significantly better, \(R^{2}=0.76\) and rho = 0.84. It should be noted that the stimuli used by [PERSON] were controlled recordings of single impacts, where the variables were the type of mallet and object struck. This limited variation in the type of data that [PERSON]'s model was trained on may explain its poorer performance with the current stimuli: sounds of multiple sources and unknown recording methods. In the between-source-type validation, the hardness model achieved a performance of \(R^{2}=0.57\) and rho = 0.79, lower than that obtained with the training data. As discussed in Sections 4 and 5.2, technical deficiencies in the stimuli will impact the features extracted, yet may be ignored by subjects rating perceived hardness. It was shown that the performance of the model with the between-source-type validation data improves when removing signals with these degradations, increasing \(R^{2}\) to 0.69, and rho to 0.83. Future development work might usefully target methods for handling low-quality recordings.