# Populating the Mix Space: Parametric Methods for Generating Multitrack Audio Mixtures
## 2 Theoretical Framework
### Track Panning  
Thus far, only mono mixes have been considered, where all audio tracks are summed to one channel. In creative music production, it is rare that mono mixes are encountered. The same mathematical formulations of the mix-space can be used to represent panning. Consider Figure 4, which shows track gains in the range \([-1,1]\). Should these be replaced with track pan positions \(p_{n}\) (with \(-1\) and \(1\) corresponding to extreme left and right pan positions, for example) then the  
Figure 6: Boxplots of track gains for two generated datasets of mixes, drawn from separate distributions. (a) \(\mu=\text{Equation}\) (7), \(\kappa=200\); (b) \(\mu=\text{Equation}\) (8), \(\kappa=200\)  
mix-space (or \"pan-space\") can be used to generate a position for each track in the stereo field. To avoid confusion with the earlier use of \(\phi\), the pan-space is denoted by \(\theta\), although the formalism is identical.  
\[\Big{(}\underbrace{p_{1},p_{2},p_{3},\ldots,p_{n}}_{\text{absolute }\text{panning}}\Big{)}=\Big{(}\underbrace{r_{\text{pan}}}_{\text{width- scaling}}\underbrace{\theta_{1},\theta_{2},\ldots,\theta_{n-1}}_{\text{pan-space}}\Big{)} \tag{9}\]  
However, the mix-space for gains (\(\phi\)) takes advantage of the fact that a mix (in terms of track gains only) is comprised of a series of inter-channel gain ratios, meaning that the radius \(r\) is arbitrary and represents a master volume. In terms of track panning, one obtains a series of inter-channel panning ratios, the precise meaning of which is not intuitive. Additionally, the radius \(r_{\text{pan}}\) would still be required to determine the exact pan position of the individual tracks. Therefore, the pan-space describes the relative pan positions of audio tracks to one another.  
For a simple example with only two tracks, the meaning of \(r_{\text{pan}}\) and \(\theta\) is relatively simple to understand. Consider the unit circle in a plane where the Cartesian coordinates \((x,y)\) represent the pan positions of two tracks, as shown in Figure 7. Mix \(A\) is at the point \((\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})\): both tracks are panned at the same position. As this is a circle with arbitrary radius, \(r_{\text{pan}}\), then the radius controls how far positive (right) the two tracks are panned, from 0 (centre) to \(+1\) (far right). Mix \(B\) does the same but towards the left channel. One may ask whether \(A\) and \(B\) are identical \"panning-mixes\", as \(p\) and \(p^{\prime}\) in Figure 1 were identical \"level-mixes\"?  
Now consider mix \(C\), where one track is panned left and the other right. Mix \(D\) is simply the mirror image of this. Are _these_ to be considered as the same mix, or as different mixes? Here, \(r_{\text{pan}}\) adjusts the distance between the two tracks, from both centre when \(r_{\text{pan}}=0\), to \((-1,1)\) when \(r_{\text{pan}}=\sqrt{2}\) (as indicated by mix \(C^{\prime}\)). Does a change in \(r_{\text{pan}}\) change the mix, or is it simply the same mix only wider/narrower? Overall, the angle \(\theta\) adjusts the panning mix and \(r_{\text{pan}}\) is used to obtain absolute positions in the stereo field, at a particular width-scale (i.e., to zoom in or zoom out).