# Populating the Mix Space: Parametric Methods for Generating Multitrack Audio Mixtures
## 3 Applications
### Estimation of Spectral Centroid in Sets of Mixes  
It is common to use the spectral centroid as a feature to describe the timbre of an audio signal, specifically as an approximation to perceptual brightness [22; 23; 24]. However, where the spectral centroid of a mixed recording is evaluated, it is not clear that the value obtained is typical of the recording as a whole, or if it simply relates to that specific mix of the recording. This is especially problematic in an object-based audio broadcast, where no reference mix exists. This applies to any signal feature, not just the spectral centroid. As studies of features across multiple alternate mixes are still rare in the literature [8; 25; 26], this issue has not been adequately investigated. A previous work by the authors [8] reports on the spectral centroid of 1501 user-generated mixes of 10 songs. The number of mixes per song ranges from 97 to 373. The estimated probability distributions of spectral centroid are shown (among other signal features relating to amplitude, timbre and spatial properties), indicating that the median spectral centroid can vary by song, although it is still possible for significant overlap in distributions to exist. The work in this section investigates the distributions of the spectral centroid that occur for artificial mixes drawn from different mix-space distributions. Equation (7) describes a simple model for mixes while Equation (8) shows the result of a perceptual level-balancing experiment. What is it about the mix that changes when these levels are adjusted? In this section, an estimation of the median spectral centroid produced by these two sets of mixes is made using Monte Carlo methods. The experiment was conducted as follows. Using \(\mu=\) Equation (7) and \(\kappa=200\), a set of 500 gain vectors was generated. For each of these vectors, a mix was created and the spectral centroid was measured. This resulted in 500 measurements of spectral centroid, the density of which was estimated using Kernel Density Estimation (KDE). This procedure was repeated for a second set of 500 mixes, generated using \(\mu=\) Equation (8) and \(\kappa=200\). The estimated density distribution of both is plotted in  
Figure 14: Estimated tempo for three songs, 500 mixes each using Equation (8). In each histogram, the data is split into 100 bins. Overall, performance is better for the metre-based method, as it demonstrates greater accuracy and improved robustness to changes in the mix; (**a**) “Burning Bridges”—The correct tempo is \(\approx\)100 bpm; (**b**) “I’m Alright”—The correct tempo is \(\approx\)96 bpm; (**c**) “What I Want”—The correct tempo is \(\approx\)99 bpm. Figure 15. These distributions were compared using a Wilcoxon rank sum test, which tests the null hypothesis that the distributions of both samples are equal. This null hypothesis was rejected in each case, as shown by the \(p\)-values in each subplot of Figure 15 (\(p<0.05\) in each case). The significant difference between the medians of the two groups illustrates that there is a coarse perceptual difference in timbre, generally, between mixes drawn from the two distributions. This is true for all three songs considered. Of course, whether or not there is a significant difference between the medians of the two groups depends on the chosen parameters: the \(\mu\) vectors must be perceptually different but if \(\kappa\) is low enough, then the distributions will overlap, regardless of the choice of \(\mu\) (recall that as \(\kappa\to 0\), the distribution approaches uniformity). The choice of \(\kappa\) depends on the application. Figure 15. Probability distribution of spectral centroid as a function of mix-space parameters; (a) “Burning Bridges”; (b) “I’m Alright”; (c) “What I Want”. The higher spectral centroid in the simple equal-gain-with-vocal-boost approach (Equation (7)) is caused by an overestimation in the level of the drum overheads and vocal, and an underestimation of the level of bass and kick drum, when compared to the results of the perceptual test (Equation (8)).