# Populating the Mix Space: Parametric Methods for Generating Multitrack Audio Mixtures
## 2 Theoretical Framework
### Track Gains  
Consider the trivial case where two audio signals are to be mixed, where only the absolute levels of each signal can be adjusted. In Figure 1, the gains of two signals are represented by \(x\) and \(y\), where both are positive-bound. Consider the point \(p\) as a configuration of the signal gains, i.e., \((p_{x},p_{y})\). From this point, the values of \(x\) and \(y\) are both increased in equal proportion, arriving at the point \(p^{\prime}\). The magnitude of \(p\) is less than that of \(p^{\prime}\) (\(\|p\|<\|p^{\prime}\|\)) yet since the ratio of \(x\) to \(y\) is identical, the angles subtended by the vectors from the \(y\)-axis are equal (\(\angle p=\angle p^{\prime}\)). In the context of a mix of two tracks, what this means is that the volume of \(p^{\prime}\) is greater than \(p\), yet the blend of input tracks is the same. As an alternate to Equation (1), a mix can be thought of as the relative balance of audio signals. From this definition, the points \(p\) and \(p^{\prime}\) are the same mix, only \(p^{\prime}\) is being presented at a greater volume. If the listener has control over the master volume of the system, then any difference between \(p\) and \(p^{\prime}\) becomes ambiguous. **Definition 1**: _Mix: an audio stream constructed by the superposition of others in accordance with a specific blend, balance or ratio._  
From \(p\), the level of fader \(y\) can be increased by \(\Delta_{y}\), arriving at \(q\). In this particular example, the value of \(\Delta_{y}\) was chosen such that \(\|q\|=\|p^{\prime}\|\). However, for any \(|\Delta_{y}|>0\), \(\angle q\
eq\angle p^{\prime}\). Therefore, \(q\) clearly represents a different mix to either \(p\) or \(p^{\prime}\). Consequently, the definition of a mix is clarified by what it is not: when two audio streams contain the same blend of input tracks but the result is at different overall amplitude levels, these two outputs can be considered the same mix. For this mixing example, where there are \(n=2\) signals, represented by \(n\) gain values, the mix is dependant on \(n-1\) variables; in this case, the angle to the vector. The \(\ell_{2}\) norm of the vector is simply proportional to the overall loudness of the mix. Figure 2a shows a similar structure, with \(n=3\). Here, the point \(p^{\prime}\) is also an extension of \(p\). As in Figure 1, \(q\) is located by increasing the value of \(y\) from the point \(p\) and \(\|q\|=\|p^{\prime}\|\). Here, the values of each angle are explicitly determined and displayed. All three vectors share the equatorial angle of \(60^{\circ}\). The polar angle of \(p\) and \(p^{\prime}\) is \(50^{\circ}\), while the polar angle of \(q\) is less than this, at \(\approx\)\(37^{\circ}\). As in the two-dimensional case, it is the angles which determine the parameters of the mix and the norm of the vector is related to the overall loudness. Figure 1: Points \(p\), \(p^{\prime}\) and \(r\), in 2-track gain space. Note that the audio output at points \(p\) and \(p^{\prime}\) is the same ‘mix’. While Figures 1 and 2a show a space of track gains, there is clearly a redundancy of mixes in this space. What is ultimately desired is a space of mixes. **Definition 2**. : _Mix-space: a parameter space containing all the possible audio mixes that can be achieved using a defined set of processes._  
It becomes apparent that a Euclidean space with track gains as basis vectors is not an efficient way to represent a space of mixes, according to Definition 2. This explains why Equation (1) would not be appropriate when searching for mixes. If, in Figure 2a, a set of \(m\) points randomly selected on \(\mathbb{R}^{3}\) were chosen, the number of mixes could be less than \(m\), as the same mix could be chosen multiple times at different overall volumes. A set of \(m\) randomly selected points on a sphere of any radius (\(\mathbb{S}^{2}\)) would result in a number of mixes equal to \(m\).